Laterally confined THz Sources and
Graphene based THz Optics
Shruti Badhwar
St John’s College
Department of Physics, University of Cambridge
A thesis submitted for the degree of
Doctorate in Philosophy
31st May 2014
2
To the hands who nourished the seed,
To the winds that made it stronger,
To the sun that gave it warmth,
To the earth which held it on.
Declaration
This dissertation is the result of work carried out in the Semiconductor Physics
group at the Cavendish Laboratory from October 2009 to May 2014. This
dissertation is the result of my own work and includes nothing which is the
outcome of work done in collaboration except where specifically stated in the
text. This work is not the same as any I have already submitted, or I am in
the process of submitting, for any degree at this, or any other university. This
dissertation contains 95215 words including appendices, bibliography, tables
and equations and has 200 figures.
Shruti Badhwar,
Cambridge, May 2014
"Behind the wall of objective science, a
thousand emotions collide"
Acknowledgements
I am grateful to St. John’s College, Cambridge for the generous PhD stu-
dentship which made it possible for me to carry out this research at the Cavendish
Labs. I would also like thank the Semiconductor Physics group for providing
funding for the final year.
I would like thank Prof. David Ritchie for his kind supervision and support
through out my PhD. His encouragement, understanding and patience at every
step allowed me to grow as a researcher.
I have learnt a lot from Dr. Harvey Beere through out my PhD. His systematic
approach to scientific problems, attention to detail, critical analysis and clarity
in presentation has helped shape many of the ideas in this work. I can only
hope to imbibe some of his many excellent qualities. He has also patiently
grown the wafers presented in this thesis.
I would like to thank Dr. Joshua Freeman and Anthony Brewer, from whom
I learnt valuable lessons in clean room processing and electro-optic measure-
ments of quantum cascade lasers. Thank you Ant for coffee and cake in times
that were good and bad.
Some of the clean room processing was carried out the National Center for
III-V materials at Sheffield. I would like to thank Ken Kennedy for making it
possible to use dry etching systems at Sheffield.
I would also like to thank Dr. Owen Marshall and Dr. Subhasish Chakraborty
for the use of FTIR and Si-Ge bolometer that made it possible to test the ultra-
thin quantum cascade laser.
Jon Griffiths provided incredible help in electron-beam writing of the nanopil-
lars, many a times, on tight schedules.
I would also like to thank my collaborators, Dr. Axel Zeitler, Juraj Sibik and
Nicholas Tan for help with the THz time domain spectroscopy.
Without the generous support of graphene samples from Piran Kidambi and Dr.
Stephan Hofmann, much of this work would not have been possible. I would
iii
also like to thank Reuben Puddy whose experience with graphene helped some
of the ideas to kick-start.
Without Yash Shah, a lot of Helium transfers would not have been possible.
He made the THz corner a lot more memorable by bringing chocolates, and a
great sense of humour. I am also grateful to David Jessop for proofreading and
Varun Kamboj for help with last-minute experiments.
Ayesha Jamil and Deepyanti Taneja provided their shoulders to cry on when
many a devices failed. I will always remember the good dinners that helped
ride the tide on many occasions. Thank you Ayesha and Deepyanti for being
such wonderful friends and colleagues. Your friendship is very dear to me.
I would like to acknowledge all the members of the Semiconductor Physics
group for many a useful discussions. Adam Klimont, David Herbshleb, Cas-
sandra, Chong and so many more people have contributed to my graduate
experience in ways that can not be expressed in words.
I would also like to thank my family members - my parents and my sister, for
their love, patience and support that made it possible to reach where I am. I
am eternally grateful for their encouragement throughout my studies. Finally,
I am grateful to the love of my life who inspite of all the difficulties made the
world look beautiful to me and kept me sane. Thank you Anurag.
Abstract
The region between the infrared and microwave region in the electromagnetic
spectrum, the Terahertz (THz) gap, provides an exciting opportunity for fu-
ture wireless communications as this band has been underutilized. This doc-
toral work takes a two-pronged approach into closing the THz gap with low-
dimensional materials. The first attempt addresses the need for a compact THz
source that can operate at room temperature. The second approach addresses
the need to build optical elements such as filters and modulators in the THz
spectrum.
Terahertz quantum cascade lasers (THz QCLs) are one of the most compact,
powerful sources of coherent radiation that bridge the terahertz gap. How-
ever, their cryogenic requirements for operation limit the scope of the ap-
plications. This is because of the electron-electron scattering and heating of
the 2-dimensional free electron gas which leads to significant optical phonon
scattering of the hot electrons. Theoretical studies in laterally confined QCL
structures have predicted enhanced lifetime of the upper state through sup-
pression of the non-radiative intersubband relaxation, which leads to lower
threshold, and higher temperature performance. Lithographically defined ver-
tical nanopillar arrays with electrostatic radius less than tens of nm offer a
possible route to achieve lateral confinement, which can be integrated into
QCL structures. A typical gain medium in a QCL consists of at least 100 repeat
periods, with a thickness of 6-14 µm. For practical implementation of the top-
down approach, restrictions are imposed by aspect ratios that can be achieved
in present dry-etching systems. Typically, for sub-200 nm radius pillars, the
thickness ranges from 1-3.5 µm. It is therefore necessary to work with THz
QCLs based on 3-4 quantum well active regions, so as to maximize the number
of repeat periods (hence gain) within a ≤ 3.5 µm thin active region.
After an introductory chapter, Chapter 2 presents a theoretical treatise on the
realistic electrostatic potential in a lithographically defined nanopillar by scal-
ing from a single quantum well (resonant tunneling diode) to a THz QCL.
Chapter 2 also discusses, the effect of lateral confinement on the intersub-
band states and the plasmonic mode in a THz QCL. One of the key experi-
mental challenges in scaling down from QCLs to quantum-dot cascade lasers
is the electrical injection into the nanopillars. This involves insulation and pla-
narization of the high aspect-ratio nanopillar arrays. Furthermore, the choice
of the planarizing layer is critical since it determines the loss of any optical
mode. This experimental challenge is solved in Chapter 3. Chapter 4 presents
the electro-optic performance of low-repeat period QCLs with an active region
thickness ≤ 3.5 µm.
Another topic of recent interest in the THz optics community is plasmonics in
graphene. This is because the bound electromagnetic modes (plasmons) are
tightly confined to the surface and can also be tuned with carrier concentration.
Plasmonic resonance at terahertz (THz) frequencies can be achieved by gating
graphene grown via chemical vapor deposition (CVD) to a high carrier con-
centration. THz time domain spectroscopy of such gated monolayer graphene
shows resonance features around 1.6 THz superimposed on the Drude-like
frequency response of graphene which may be related to the inherent poly-
crystallinity of CVD graphene. Chapter 5 discusses these results, as an un-
derstanding of these features is necessary for the development of future THz
optical elements based on CVD graphene. Chapter 5 finally describes how the
gate tunability of THz transmission through graphene can be exploited to in-
directly modulate a THz QCL.
Chapter 6 presents ideas from this doctoral work, which can be developed in
future to address the issues of enhanced temperature performance of Thz QCLs
and to realize realistic THz devices based on graphene.
Publications
Some ideas and concepts have appeared or will appear in the following publi-
cations and conferences.
PUBLISHED JOURNAL ARTICLES
1. Intrinsic terahertz plasmonic signatures in chemical vapour deposited graphene,
Shruti Badhwar, Juraj Sibik, Piran R. Kidambi, J. Axel Zeitler, Stephan
Hofmann, Harvey E. Beere and David. A. Ritchie, Applied Physics Letters,
11, 074337 (2013)
2. Indirect modulation of a terahertz quantum cascade laser using gate tun-
able graphene, Shruti Badhwar, Reuben Puddy, Piran Kidambi, Juraj
Sibik, Anthony Brewer, Joshua Freeman, Harvey E. Beere, Stephan Hoff-
man, Axel Zeitler, David A. Ritchie, IEEE Photonics Journal, 4, 1776-1782
(2012)
TO BE SUBMITTED
1. Lateral confinement in Terahertz quantum cascade lasers, Shruti Badhwar,
Harvey E. Beere, Jon Griffiths, Geb Jones, David A. Ritchie
2. Fabrication of suspended contact to high aspect ratio nanopillars, Shruti
Badhwar, Harvey E. Beere, Jon Griffiths, Geb Jones, David A. Ritchie
3. Parallel conduction through 500 x 500 laterally confined RTDs, Shruti Bad-
hwar, Joshua R. Freeman, Anthony Brewer, Harvey E. Beere, Jon Grif-
fiths, Geb Jones, David A. Ritchie
CONFERENCES
1. Lateral confinement in THz quantum cascade lasers, Shruti Badhwar, Nan-
oTech Fest, Institute of Manufacturing, Cambridge, UK (December 2013)
2. Bridging the THz gap with low dimensionality - the route to faster wireless
communications, Shruti Badhwar, Graduate Student Conference, Cam-
bridge, UK (December 2013)
3. Graphene based THz optics, Shruti Badhwar, Juraj Sibik, Piran Kidambi,
Juraj Sibik, Stephan Hofmann, J. Axel Zeitler, Harvey E. Beere and David
A. Ritchie, UK Terahertz Day, Cambridge, UK (July 2013)
4. Graphene as a THz filter, Shruti Badhwar, Riccardo DeglâA˘Z´Innocenti,
Piran Kidambi, Juraj Sibik, Nicholas Tan, Stephan Hofmann, Axel Zeitler,
Harvey Beere and David A. Ritchie, International Workshop on Optical
Science and Technology, Kyoto, Japan (April 2013)
5. Indirect modulation of a terahertz quantum cascade laser using gate tunable
graphene, Shruti Badhwar, Reuben Puddy, Piran Kidambi, Juraj Sibik,
Anthony Brewer, Joshua Freeman, Harvey E. Beere, David A. Ritchie,
Stephan Hoffman, Axel Zeitler , IEEE Photonics, San Francisco, US (Septem-
ber 2012)
6. Electrical contacts to nanopillar superlattice arrays, Shruti Badhwar, Joshua
R. Freeman, Anthony M. Brewer, Harvey E. Beere, David A. Ritchie, UK
Semiconductor Conference, Sheffield, England (July 2011)
7. Inter-sub-stack layers in heterogeneous terahertz quantum cascade lasers,
Shruti Badhwar, Joshua Freeman, Anthony Brewer, Harvey E. Beere,
David A. Ritchie, International Quantum Cascade Laser Summer School
and Workshop, Florence, Italy (September 2010)
8. Terahertz quantum cascade lasers with focused ion beam milled photonic
structures, Anthony Brewer, Joshua Freeman, Owen Marshall, Shruti Bad-
hwar, Harvey E. Beere, David A. Ritchie, UK Semiconductor Conference,
Sheffield, England (July 2010)
INVITED TALKS
1. THz optics - the route to faster wireless communications, Shruti Badhwar,
Indian Institute of Technology, Gandhinagar, India (December 2013)
2. Terahertz optics with graphene and nanostructured materials Shruti Bad-
hwar, Department of Physics, Ecole Normale Superieure, Paris, France
(June 2013)
Abbreviation
THz............... Terahertz
Mid IR............... Mid infrared
RTD............... Resonant tunneling diode
QCL............... Quantum cascade laser
BtC............... Bound to continuum
LO phonon............... Longitudinal optical phonon
TDS............. Time domain spectroscopy
RIE............. Reactive ion etching
ICP RIE............. Inductively coupled plasma reactive ion etching
SOG............. Spin on glass
BHF............. Buffered hydroflouric acid
CVD............. Chemical vapour deposition
PECVD............. Plasma enhanced chemical vapour deposition
NDR............. Negative differential resistance
LIV............. Light current voltage
QW............. Quantum well
AR............. Active region
FOM............. Figure of merit
ix
0. ABBREVIATION
x
Contents
Abbreviation ix
1 Introduction 1
1.1 Terahertz (THz) gap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 THz sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2.1 Broadband sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2.2 Narrowband sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 THz quantum cascade lasers (THz QCLs) . . . . . . . . . . . . . . . . . . . . . . 3
1.3.1 The paradigm change: intersubband transitions . . . . . . . . . . . . . . 3
1.3.2 The journey from mid-infrared QCLs to THz QCLs . . . . . . . . . . . . 4
1.3.3 THz QCL active regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3.4 THz QCLs: quest for room temperature operation . . . . . . . . . . . . 10
1.4 Zero-dimensional confinement in QCLs . . . . . . . . . . . . . . . . . . . . . . . . 11
1.4.1 Bottom up approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.4.2 Top down Cavendish approach . . . . . . . . . . . . . . . . . . . . . . . . 15
1.5 THz waveguides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.5.1 Surface plasmons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.5.2 Plasmonic waveguides in THz QCL . . . . . . . . . . . . . . . . . . . . . . 19
1.5.3 Graphene: a material for THz plasmonics . . . . . . . . . . . . . . . . . . 20
1.6 THz detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.7 Thesis outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2 Nanopillars in RTDs and QCLs: Theory 25
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.2 General description of RTDs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.3 Confinement in a single nanopillar RTD . . . . . . . . . . . . . . . . . . . . . . . 30
2.4 Reed’s simplistic model for a zero-dimensional RTD . . . . . . . . . . . . . . . . 31
xi
CONTENTS
2.5 Effect of surface states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.5.1 Interface charge density . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.5.2 Charge neutrality level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.5.3 Chia et. al’s model for homogeneous GaAs nanopillar . . . . . . . . . . 36
2.6 Results: extension of Chia’s model to double barrier RTD . . . . . . . . . . . . 38
2.6.1 Undoped GaAs Well . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.6.2 Doped GaAs Spacer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.6.3 Doped GaAs contact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.6.4 Depletion as a function of doping density . . . . . . . . . . . . . . . . . . 45
2.6.5 Implication of surface states on NDR peaks . . . . . . . . . . . . . . . . . 45
2.7 Three dimensional potential in laterally confined RTDs . . . . . . . . . . . . . . 48
2.8 Three dimensional potential in laterally confined QCLs . . . . . . . . . . . . . . 50
2.9 Effect of lateral confinement on THz QCL bandstructure . . . . . . . . . . . . . 52
2.9.1 Comparison between RTD and THz QCL bandstructure . . . . . . . . . 53
2.9.2 Magnetic field vs. lithographic confinement of carriers . . . . . . . . . . 53
2.10 Results: radiative states in laterally confined THz QCLs . . . . . . . . . . . . . . 57
2.10.1 Radiative states in laterally confined 3 THz 3 QW Paris design . . . . . 58
2.10.2 Radiative states in laterally confined 3 THz 4 QW ETH design . . . . . 61
2.10.3 Accurate estimation of radiative states . . . . . . . . . . . . . . . . . . . . 64
2.11 Effect of lateral confinement on plasmonic mode in THz QCLs . . . . . . . . . 66
2.11.1 Plasmon mode simulations in THz QCLs . . . . . . . . . . . . . . . . . . 67
2.11.2 Effective medium approximation in laterally confined QCLs . . . . . . 69
2.11.3 Simulation results: effect of active region thickness . . . . . . . . . . . 72
2.11.4 Simulation results: effect of pitch/diameter ratio . . . . . . . . . . . . . 74
2.11.5 Simulation results: effect of planarising material . . . . . . . . . . . . . 77
2.12 Summary and conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
3 Nanopillar arrays in RTDs and QCLs: Experimental realization 81
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
3.2 RTD active region used: relevance to QCLs . . . . . . . . . . . . . . . . . . . . . 82
3.3 Molecular beam epitaxy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
3.4 General methods: fabrication of nanopillar arrays . . . . . . . . . . . . . . . . . 84
3.4.1 Electron beam lithography . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
3.4.2 Optical lithography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
3.4.3 Dry etching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
3.4.3.1 Reaction ion etching system (RIE) . . . . . . . . . . . . . . . . 91
xii
CONTENTS
3.4.3.2 Inductively coupled plasma (ICP) RIE system . . . . . . . . . 93
3.5 Fabrication of electrically connected nanopillar arrays . . . . . . . . . . . . . . 95
3.5.1 Processing guidelines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
3.5.2 Etch mask definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
3.5.2.1 Etch-mask assessment parameters . . . . . . . . . . . . . . . . 98
3.5.2.2 Dielectric mask: results . . . . . . . . . . . . . . . . . . . . . . . 99
3.5.2.3 Metal mask: results . . . . . . . . . . . . . . . . . . . . . . . . . 102
3.5.2.4 Etch mask summary . . . . . . . . . . . . . . . . . . . . . . . . . 103
3.5.3 Etch of high aspect ratio pillars . . . . . . . . . . . . . . . . . . . . . . . . 104
3.5.3.1 Factors affecting the aspect ratio . . . . . . . . . . . . . . . . . 104
3.5.3.2 Etch assessment parameters . . . . . . . . . . . . . . . . . . . . 105
3.5.3.3 Etch in ICP using a dielectric mask: results . . . . . . . . . . . 105
3.5.3.4 Etch in ICP using a metal mask: results . . . . . . . . . . . . . 111
3.5.3.5 Etch in RIE using a metal mask: results . . . . . . . . . . . . . 113
3.5.3.6 Nanopillar etching summary . . . . . . . . . . . . . . . . . . . . 115
3.5.4 Insulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
3.5.4.1 Insulation assessment parameters . . . . . . . . . . . . . . . . . 117
3.5.4.2 Trilayer spin-on-glass process . . . . . . . . . . . . . . . . . . . 118
3.5.4.3 Results from trilayer spin-on-glass process . . . . . . . . . . . 122
3.5.4.4 Device recovery after ineffective planarization . . . . . . . . . 124
3.5.4.5 Nanopillar insulation summary . . . . . . . . . . . . . . . . . . 128
3.5.5 Fabrication of top-contact and bond pads . . . . . . . . . . . . . . . . . . 128
3.5.6 Suspended top-contact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
3.5.6.1 Possible routes to suspended contact . . . . . . . . . . . . . . . 130
3.5.6.2 Spin on glass route to suspended contact: results . . . . . . . 134
3.6 Fabrication of RTDs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
3.6.1 Large-area RTDs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
3.6.2 Nanopillar arrays in RTDs: bottom-contact dilemma . . . . . . . . . . . 137
3.7 Experimental set-up for characterizing RTDs . . . . . . . . . . . . . . . . . . . . 138
3.8 Results from RTDs: A1700 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
3.8.1 Large-area RTDs: ohmic contact . . . . . . . . . . . . . . . . . . . . . . . 139
3.8.2 Large-area RTDs: Schottky contact . . . . . . . . . . . . . . . . . . . . . . 141
3.8.3 Nanopillar arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
3.8.3.1 Results from 500 x 500 array, Rp = 100 nm . . . . . . . . . . 145
3.8.3.2 Challenges associated with NDR mapping . . . . . . . . . . . . 147
3.8.3.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
xiii
CONTENTS
3.9 Results from RTDs: W0021 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
3.9.1 Large-area RTDs: ohmic contact . . . . . . . . . . . . . . . . . . . . . . . 150
3.9.2 Nanopillar arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
3.9.2.1 500 x 500 array, Rp = 150 nm . . . . . . . . . . . . . . . . . . . 153
3.9.2.2 500 x 500 array, Rp = 100 nm . . . . . . . . . . . . . . . . . . . 157
3.9.2.3 500 x 500 array, Rp = 50 nm . . . . . . . . . . . . . . . . . . . 162
3.10 Fabrication of nanopillar arrays in metal-metal THz QCLs . . . . . . . . . . . . 167
3.11 Summary and conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
4 Ultrathin QCLs 169
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
4.1.1 Effect of active region thickness on gain . . . . . . . . . . . . . . . . . . . 169
4.1.2 Ultrathin QCLs: finding the right active region . . . . . . . . . . . . . . 171
4.2 Ultrathin QCLs in literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
4.3 Fabrication of ultrathin QCLs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
4.3.1 Wafer bonding and host-substrate removal . . . . . . . . . . . . . . . . . 175
4.3.2 Mesa etch and metallization . . . . . . . . . . . . . . . . . . . . . . . . . . 176
4.3.2.1 Dry etching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
4.3.2.2 Wet etching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
4.4 Characterization of ultrathin QCLs: methods . . . . . . . . . . . . . . . . . . . . 179
4.4.1 Mounting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
4.4.2 Electro-optic measurements . . . . . . . . . . . . . . . . . . . . . . . . . . 179
4.4.3 Spectral measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
4.4.3.1 Above lasing threshold . . . . . . . . . . . . . . . . . . . . . . . 180
4.4.3.2 Below lasing threshold . . . . . . . . . . . . . . . . . . . . . . . 181
4.5 Ultrathin active regions used . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
4.6 2.7 THz 3 QW Paris design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
4.6.1 Ultrathin QCL (V671) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
4.6.2 Ultrathin QCL (W0798) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
4.6.3 Ultrathin QCL (V692) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
4.7 3 THz 3 QW Paris design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
4.7.1 Full-stack (V703) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
4.7.2 Comparison of V703 with Literature . . . . . . . . . . . . . . . . . . . . . 193
4.7.3 Ultrathin QCL (V702) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
4.8 3 THz 4 QW MIT design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
4.8.1 Full-stack V705 and comparison to V569 . . . . . . . . . . . . . . . . . . 196
xiv
CONTENTS
4.8.2 Ultrathin QCL (V704) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
4.9 3 THz 4 QW ETH design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
4.9.1 Full-stack (V674) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
4.9.2 Comparison of full-stack V674 with literature . . . . . . . . . . . . . . . 202
4.9.3 Comparison of full-stack V674 and full-stack V696 . . . . . . . . . . . . 203
4.9.4 Chemical etching of ultrathin QCLs . . . . . . . . . . . . . . . . . . . . . 205
4.9.5 Ultrathin QCL (V674T ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
4.10 Waveguide losses in ultrathin THz QCLs . . . . . . . . . . . . . . . . . . . . . . . 209
4.10.1 Effect of contact thickness and doping on waveguide losses . . . . . . . 209
4.10.2 Effect of removal of top contact on waveguide losses . . . . . . . . . . . 211
4.11 Measurement of spontaneous emission . . . . . . . . . . . . . . . . . . . . . . . . 213
4.12 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
5 Graphene based THz optics 217
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
5.2 Electronic structure of graphene . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
5.3 Optical conductivity of graphene . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
5.4 Plasmons in graphene . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
5.5 Growth and processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
5.5.1 Chemical vapour deposition (CVD) of graphene . . . . . . . . . . . . . . 223
5.5.2 Transfer of CVD graphene to arbitrary substrate . . . . . . . . . . . . . . 225
5.5.3 Characterization of growth quality . . . . . . . . . . . . . . . . . . . . . . 226
5.5.4 Device fabrication post transfer . . . . . . . . . . . . . . . . . . . . . . . . 227
5.6 Transport measurements of CVD graphene . . . . . . . . . . . . . . . . . . . . . 228
5.6.1 Results from back-gated graphene . . . . . . . . . . . . . . . . . . . . . . 229
5.6.2 Results from top-gated graphene . . . . . . . . . . . . . . . . . . . . . . . 232
5.7 THz time domain spectroscopy (THz TDS) of CVD graphene . . . . . . . . . . 237
5.7.1 Experimental set-up in transmission mode . . . . . . . . . . . . . . . . . 237
5.7.2 Experimental set-up in reflection mode . . . . . . . . . . . . . . . . . . . 240
5.7.3 Transmission mode vs. reflection mode set-up . . . . . . . . . . . . . . . 241
5.7.4 Theory behind THz transmission through graphene . . . . . . . . . . . 242
5.7.5 Results from back-gated graphene . . . . . . . . . . . . . . . . . . . . . . 243
5.7.6 Results from top-gated graphene . . . . . . . . . . . . . . . . . . . . . . . 247
5.7.7 Role of domains in spectroscopic response . . . . . . . . . . . . . . . . . 251
5.7.8 THz imaging of top-gated graphene . . . . . . . . . . . . . . . . . . . . . 254
5.7.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
xv
CONTENTS
5.8 Graphene ribbons as THz filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
5.9 THz modulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257
5.9.1 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258
5.9.2 Results from indirect modulation of a THz QCL by graphene . . . . . . 259
5.9.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262
5.10 Summary and conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263
6 Conclusions and future work 265
6.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265
6.2 Opportunities for future . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
6.2.1 Optical spectroscopy of nanopillars processed from THz QCLs . . . . . 267
6.2.2 Waveguides for quantum-dot cascade structures . . . . . . . . . . . . . 267
6.2.3 Lateral confinement in THz QCLs - Nanofences . . . . . . . . . . . . . . 269
6.2.4 Characterization of graphene domain boundaries by THz TDS . . . . . 273
6.2.5 THz filters from graphene anti-dots . . . . . . . . . . . . . . . . . . . . . 274
6.2.6 THz amplitude modulators . . . . . . . . . . . . . . . . . . . . . . . . . . . 274
7 Appendix A 275
7.1 Epitaxial structure of RTDs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275
8 Appendix B 277
8.1 X-ray diffraction technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277
8.2 Rocking curves of THz QCLs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278
References 301
xvi
List of Figures
1.1 Electromagnetic spectrum showing the THz gap that lies between the radio
and microwave energies and the infrared energy. Reproduced from [1]. . . . 1
1.2 Interband and Intersubband transitions . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Schematic conduction band diagram of the first quantum cascade laser . . . . 5
1.4 Schematic representation of the dispersion relationship, where E32  ELO . . 6
1.5 Schematic representation of the dispersion relationship, where E32 < ELO . . 7
1.6 Schematic conduction band diagram of the first THz quantum cascade laser . 7
1.7 Schematic conduction band diagram of common THz QCL active regions . . . 9
1.8 Operating temperature of THz QCLs as a function of frequency, . . . . . . . . . 11
1.9 Schematic representation of the dispersion relationship, where E32 < ELO
and the momentum is discretized . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.10 Schematic energy band diagram of first quantum-dot cascade proposals . . . 13
1.11 Energy band diagram of first experimentally realized quantum-dot cascade
structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.12 Energy band diagram of quantum dot cascade structure showing polarised
mid-infrared emission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.13 Terahertz absorption from self assembled quantum dot cascade structures . . 16
1.14 Surface plasmons at the interface between a metal and a dielectric . . . . . . 17
1.15 Optical profile of THz QCL waveguides . . . . . . . . . . . . . . . . . . . . . . . . 19
1.16 Typical absorption spectrum of graphene, with associated intra and inter-
band processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.1 Band diagram and transmission characteristics of an RTD . . . . . . . . . . . . 28
2.2 Electrical characteristics of an RTD, showing alignment of energy states at
the NDR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.3 Current voltage characteristics of the first laterally confined RTD . . . . . . . 31
xvii
LIST OF FIGURES
2.4 Schematic diagram showing lateral and longitudinal potential in a zero-
dimensional RTD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.5 Band bending due to interface states . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.6 Electrostatic radius as a function of interface states . . . . . . . . . . . . . . . . 40
2.7 Schematic diagram of surface depletion in undoped GaAs nanopillar . . . . . 41
2.8 Schematic diagram of surface depletion in doped GaAs nanopillar . . . . . . . 43
2.9 Electrostatic radius as a function of doping density . . . . . . . . . . . . . . . . 46
2.10 Implication of surface states on NDR peaks . . . . . . . . . . . . . . . . . . . . . 47
2.11 Schematic diagram showing hour-glass potential in a nanopillar RTD . . . . . 48
2.12 Longitudinal potential in a nanopillar defined in an RTD . . . . . . . . . . . . . 50
2.13 Schematic diagram showing external gates controlling the lateral potential
in an RTD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
2.14 Longitudinal potential in a nanopillar defined in an QCL . . . . . . . . . . . . . 52
2.15 Schematic diagram of RTD and QCL bandstructure . . . . . . . . . . . . . . . . 53
2.16 Schematic diagram of RTD bandstructure under the influence of (a) an ex-
ternal magnetic field (b) lithographic confinement of carriers . . . . . . . . . . 54
2.17 Schematic diagram of THz QCL bandstructure under the influence of (a) an
external magnetic field (b) lithographic confinement of carriers . . . . . . . . 56
2.18 Band diagram of 3 THz 3 QW Paris design showing energy states at alignment 57
2.19 Band diagram of 3 THz 4 ETH design showing energy states at alignment . . 58
2.20 Ladder states as a function of nanopillar radius Rpin 3 THz 3 QW Paris
design at an electric field, E = 12.1 kV/cm . . . . . . . . . . . . . . . . . . . . . 60
2.21 Ladder states as a function of nanopillar radius Rp in 3 THz 3 QW Paris
design at an electric field, E = 12.1 kV/cm, showing optically permitted
radiative transitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
2.22 Ladder states as a function of nanopillar radius Rpin 3 THz 4 QW ETH design
at an electric field, E = 7.6 kV/cm . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
2.23 Ladder states as a function of nanopillar radius Rp in 3 THz 4 QW ETH
design at an electric field, E = 7.6 kV/cm, showing optically permitted ra-
diative transitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
2.24 Schematic diagram showing a THz QCL in a double metal waveguide . . . . . 66
2.25 Schematic diagram showing the cross-section of the Fabry Pèrot ridge in a
double metal waveguide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
2.26 Schematic diagram showing optical constants associated with a THz QCL . . 69
2.27 Schematic diagram of laterally confined THz QCL . . . . . . . . . . . . . . . . . 70
2.28 Schematic diagram showing optical constants associated with a THz QCL . . 71
xviii
LIST OF FIGURES
2.29 Plasmonic mode calculated in a THz QCL and a nanopillar array, with active
region thickness equal to 3 µm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
2.30 Comparison of waveguide properties of a THz QCL with a laterally confined
THz QCL, as a function of active region thickness . . . . . . . . . . . . . . . . . 74
2.31 Effect of p/2Rp on the optical properties of a laterally confined THz QCL . . . 76
2.32 Schematic diagram illustrating the optimum spacing of nanopillars when
the array is insulated by air . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
2.33 Effect of planarizing material on the optical mode inside a QCL nanopillar
array . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
2.34 Schematic diagram illustrating the optimum spacing of nanopillars when
the array is insulated by a dielectric . . . . . . . . . . . . . . . . . . . . . . . . . . 78
3.1 Schematic diagram of a typical molecular beam epitaxy system . . . . . . . . . 84
3.2 General flowchart for fabrication of electrically connected nanopillar arrays . 85
3.3 Schematic diagram describing the photolithographic process flow . . . . . . . 88
3.4 Schematic diagram showing plasma chamber. The particle motion is shown
for the positive half cycle of the RF bias. Adapted from [2]. . . . . . . . . . . . 89
3.5 Schematic diagrams showing four basic methods of plasma etching . . . . . . 90
3.6 Schematic diagram of the reactive ion etching system, used for etching sili-
con oxide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
3.7 Schematic diagram of the reactive ion etching system, used for etching GaAs
/ AlxGa1−xAs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
3.8 Schematic diagram of the in-house ICP RIE . . . . . . . . . . . . . . . . . . . . . 94
3.9 Schematic diagram showing (a) the nanopillar pattern written using e-beam
lithography (b) the top and bottom-contact pads relative to the the nanopil-
lar pattern. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
3.10 Process schematic: dielectric etch mask . . . . . . . . . . . . . . . . . . . . . . . 99
3.11 Scanning electron micrograph of Ti/Al disks after e-beam lithography and
lift-off . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
3.12 Scanning electron micrograph of SiO2 disks after etching . . . . . . . . . . . . 101
3.13 Isometric view of the dielectric etch mask . . . . . . . . . . . . . . . . . . . . . . 101
3.14 Process schematic: metal etch mask . . . . . . . . . . . . . . . . . . . . . . . . . . 102
3.15 Scanning electron micrograph: metal etch mask . . . . . . . . . . . . . . . . . . 103
3.16 Schematic diagram of the nanopillar showing etch profile . . . . . . . . . . . . 106
3.17 Nanopillars imaged after being etched at 55 W table power and 100 W
chamber power in ICP-RIE using 100 nm thick SiO2 as an etch mask . . . . . 107
xix
LIST OF FIGURES
3.18 Nanopillars imaged after being etched at 55 W table power and 70 W cham-
ber power in ICP-RIE using 100 nm thick SiO2 as an etch mask . . . . . . . . . 108
3.19 Schematic diagram, showing etching of nanopillars in ICP-RIE using a di-
electric etch mask . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
3.20 Nanopillars imaged after being etched at 55 W table power and 100 W
chamber power in ICP-RIE using 100 nm thick SiO2 as an etch mask. The
samples are mounted to enhance thermal dissipation. . . . . . . . . . . . . . . . 109
3.21 Nanopillars imaged after being etched in three stages, interspersed with
two pump cycles at 55 W table power and 100 W chamber power in ICP-RIE
using 100 nm thick SiO2 as an etch mask. . . . . . . . . . . . . . . . . . . . . . . 110
3.22 Nanopillars imaged after being etched at 55 W table power and 100 W
chamber power in ICP-RIE using 100 nm thick Ti/Au as an etch mask. The
samples are mounted to enhance thermal dissipation. . . . . . . . . . . . . . . . 112
3.23 Nanopillars imaged after being etched in three stages, interspersed with
two pump cycles at 55 W table power and 100 W chamber power in ICP-RIE
using 100 nm thick Ti/Au as an etch mask. . . . . . . . . . . . . . . . . . . . . . 113
3.24 Etch rate vs. process parameters to etch nanopillars in an RIE, with a
NiCr/Au etch mask . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
3.25 Planarization schematic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
3.26 Trilayer planarization process schematic . . . . . . . . . . . . . . . . . . . . . . . 119
3.27 Schematic diagram, showing post bake and curing procedure . . . . . . . . . . 120
3.28 Leakage current through insulation . . . . . . . . . . . . . . . . . . . . . . . . . . 122
3.29 Scanning electron micrograph of nanopillars imaged after planarization by
SOG - optimization of parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
3.30 Schematic diagram showing a poor etchback . . . . . . . . . . . . . . . . . . . . 124
3.31 Nanopillars imaged after successful planarisation and etchback . . . . . . . . . 125
3.32 Device recovery after ineffective planarization . . . . . . . . . . . . . . . . . . . 127
3.33 Schematic diagram showing fabrication of top-contact and bond pads . . . . . 129
3.34 Suspended top-contact process schematic . . . . . . . . . . . . . . . . . . . . . . 131
3.35 Lateral etch of insulation in relation to all nanopillar arrays . . . . . . . . . . . 131
3.36 Schematic diagram showing fabrication of suspended top-contact using Of-
fermans approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
3.37 Scanning electron micrographs of nanopillars formed in planarized and etched
back nanopillars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
3.38 Scanning electron micrographs of suspended top-contact formed in pla-
narized nanopillars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
xx
LIST OF FIGURES
3.39 Processing steps of large-area RTDs . . . . . . . . . . . . . . . . . . . . . . . . . . 136
3.40 Flowchart for fabrication of electrically connected nanopillar arrays in RTDs 138
3.41 Experimental set up for characterizing RTDs . . . . . . . . . . . . . . . . . . . . 139
3.42 Electrical characterization of large-area RTD processed from A1700, with an
ohmic top-contact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
3.43 Electrical characterization of large-area RTDs with different mesa sizes, pro-
cessed from A1700, with an ohmic top-contact . . . . . . . . . . . . . . . . . . . 141
3.44 Negative differential peak corresponding to the ground state of large-area
RTDs processed from A1700 with a Schottky top-contact . . . . . . . . . . . . . 142
3.45 Extended characterization of large-area RTDs processed from A1700 with a
Schottky top-contact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
3.46 Characterization of large-area RTDs processed from A1700 with a Schottky
top-contact, with different mesa dimensions . . . . . . . . . . . . . . . . . . . . 145
3.47 Electrical characteristics of 500 x 500 nanopillar array processed from A1700,
with Rp = 100 nm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
3.48 Negative differential peak corresponding to the ground state of nanopillars
processed from A1700, at negative bias. . . . . . . . . . . . . . . . . . . . . . . . 146
3.49 Electrical characterization of large-area RTD processed from W0021, with
an ohmic top-contact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
3.50 Electrical characterization of large-area RTD processed from W0021, with
an ohmic top-contact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
3.51 Electrical characterization of large-area RTDs with different mesa sizes, pro-
cessed from W0021, with an ohmic top-contact . . . . . . . . . . . . . . . . . . 152
3.52 Electrical characteristics of 500 x 500 nanopillar array processed from W0021,
with Rp = 150 nm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
3.53 Electrical characteristics of 500 x 500 nanopillar array processed from W0021,
with Rp = 150 nm, during forward and reverse voltage sweeps . . . . . . . . . 154
3.54 Electrical characteristics of 500 x 500 nanopillar array processed from W0021,
with Rp = 150 nm, at higher resolution . . . . . . . . . . . . . . . . . . . . . . . 156
3.55 Electrical characteristics of 500 x 500 nanopillar array processed from W0021,
with Rp = 100 nm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
3.56 Electrical characteristics of 500 x 500 nanopillar array processed from W0021,
with Rp = 100 nm, during forward and reverse voltage sweeps . . . . . . . . . 158
3.57 A comparison of electrical characteristics of a 500 x 500, 100 nm array with
a 500 x 500, 150 nm array . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
xxi
LIST OF FIGURES
3.58 A comparison of differential conductance of a 500 x 500, 100 nm array with
a 500 x 500, 150 nm array . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
3.59 Energy distribution of peaks in a 500 x 500, 100 nm nanopillar array . . . . . 161
3.60 Electrical characteristics of 500 x 500 nanopillar array processed from W0021,
with Rp = 150 nm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
3.61 Electrical characteristics of 500 x 500 nanopillar array processed from W0021,
with Rp = 50 nm, during forward and reverse voltage sweeps . . . . . . . . . 163
3.62 Differential conductance of a 500 x 500, 50 nm array . . . . . . . . . . . . . . . 165
3.63 Energy distribution of peaks in a 500 x 500, 50 nm nanopillar array . . . . . . 166
4.1 Schematic diagram showing radiative and non-radiative lifetimes across the
lasing levels in a THz QCL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
4.2 IV and spectral characteristics of ultra-thin 3 quantum-well LO phonon de-
sign [3] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
4.3 Temperature performance of 3 quantum-well LO phonon design [3] . . . . . . 173
4.4 Characteristics of ultrathin 4 QW LO phonon design [4] . . . . . . . . . . . . . 174
4.5 Process steps in the wafer-bonding of a double metal QCL . . . . . . . . . . . . 176
4.6 Processing steps in dry etching of QCL . . . . . . . . . . . . . . . . . . . . . . . . 177
4.7 Processing steps in wet etching of QCL . . . . . . . . . . . . . . . . . . . . . . . . 178
4.8 Electro-optic setup for measuring LIV characteristics when QCL is operated
in the pulsed mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
4.9 Experimental set-up used to collect spectra above lasing threshold. Diagram
from [5]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
4.10 Experimental set-up used to collect spectra below lasing threshold. Diagram
from [5]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
4.11 Energy band diagram of 2.7 THz 3 QW Paris design . . . . . . . . . . . . . . . . 183
4.12 LIV characteristics of ultrathin 2.7 THz 3 QW Paris design (V671) double-
metal ridge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
4.13 LIV characteristics of ultrathin 2.7 THz 3 QW Paris design (W0798) double-
metal ridge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
4.14 LIV characteristics of ultrathin 2.7 THz 3 QW Paris design (V692) double-
metal ridge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
4.15 Energy band diagram of 3 THz 3 QW Paris design . . . . . . . . . . . . . . . . . 189
4.16 LIV characteristics of full-stack 3 THz 3 QW Paris design (V703) . . . . . . . . 191
4.17 Differential resistance of full-stack 3 THz 3 QW Paris design (V703) . . . . . . 192
4.18 Spectral characteristics of full-stack 3 THz 3 QW Paris design (V703) . . . . . 192
xxii
LIST OF FIGURES
4.19 LIV characteristics of V702 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
4.20 Energy band diagram of 3 THz 4 QW MIT design . . . . . . . . . . . . . . . . . 195
4.21 Comparison of LIV characteristics of 3 THz 4 QW MIT design (V705) with 3
THz 4 QW MIT design (V569) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
4.22 Comparison of differential resistance characteristics between 3 THz 4 QW
MIT design (V705) and 3 THz 4 QW MIT design (V569) . . . . . . . . . . . . . 198
4.23 LIV characteristics of 3 THz 4 QW MIT design (V704) . . . . . . . . . . . . . . . 199
4.24 Energy band diagram of 3 THz 4 QW ETH design . . . . . . . . . . . . . . . . . 200
4.25 LIV characteristics of 3 THz 4 QW ETH design (V674) . . . . . . . . . . . . . . 202
4.26 Differential resistance of 3 THz 4 QW ETH design (V674) measured in
pulsed mode, at 4 K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
4.27 LIV characteristics of 3 THz 4 QW ETH design as reported in the literature [6]204
4.28 Comparison of LIV characteristics of full-stack 3 THz 4 QW ETH design
(V674) with full-stack 3 THz 4 QW ETH design (V696) . . . . . . . . . . . . . . 205
4.29 Optical image of ultrathin QCLs after etching . . . . . . . . . . . . . . . . . . . . 206
4.30 LIV characteristics of ultrathin 3 THz 4 QW ETH design (V674T ) . . . . . . . . 207
4.31 Differential resistance of 3 THz 4 QW ETH design (V674T ) . . . . . . . . . . . 208
4.32 Frequency spectra of 3 THz 4 QW ETH design (V674T ) . . . . . . . . . . . . . . 209
4.33 Plasmonic mode as a function of active region thickness . . . . . . . . . . . . . 212
4.34 Frequency spectra of full-stack 3 THz 4 QW MIT design (V705) . . . . . . . . 214
4.35 Below threshold measurements of full-stack 3 THz 4 QW MIT design (V705) 215
4.36 Survey of reported peak powers in THz QCLs . . . . . . . . . . . . . . . . . . . . 216
5.1 Structure of graphene, and it’s comparison to graphite . . . . . . . . . . . . . . 218
5.2 Honeycomb lattice and its Brillouin zone . . . . . . . . . . . . . . . . . . . . . . . 218
5.3 Electronic dispersion of graphene . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
5.4 Drude weight as a function of gate bias . . . . . . . . . . . . . . . . . . . . . . . 222
5.5 Growth of chemical vapor deposited (CVD) graphene . . . . . . . . . . . . . . . 224
5.6 Scanning electron micrograph of CVD graphene [7] . . . . . . . . . . . . . . . . 224
5.7 Process schematic: Transfer of as grown CVD graphene onto the substrate. . 225
5.8 Raman spectroscopy of monolayer CVD graphene . . . . . . . . . . . . . . . . . 226
5.9 Microscopic characterization of monolayer CVD graphene . . . . . . . . . . . . 227
5.10 Process schematic for fabricating back-gated graphene field effect transistor . 228
5.11 Process schematic for fabricating top-gated graphene field effect transistor . . 229
5.12 Schematic diagram of the band structure at different doping levels . . . . . . 230
5.13 Circuit diagram for transport measurements . . . . . . . . . . . . . . . . . . . . 230
xxiii
LIST OF FIGURES
5.14 Drain-source current as a function of back-gate bias . . . . . . . . . . . . . . . . 231
5.15 Schematic diagram illustrating gating using an ion gel . . . . . . . . . . . . . . 233
5.16 Drain-source current as a function of top-gate bias . . . . . . . . . . . . . . . . . 234
5.17 Drain-source current as a function of top-gate bias, at different ramp rates . . 234
5.18 Leakage current in a top-gated graphene field effect transistor . . . . . . . . . 235
5.19 Transport parameters in a top-gated graphene field effect transistor . . . . . . 236
5.20 THz time domain set-up in transmission mode . . . . . . . . . . . . . . . . . . . 238
5.21 THz waveform through dry air in time domain and frequency domain, mea-
sured using reflection mode THz TDS . . . . . . . . . . . . . . . . . . . . . . . . . 239
5.22 THz time domain waveform transmitted through the SiO2/Si substrate . . . . 240
5.23 THz waveform through dry air in time domain and frequency domain, mea-
sured using reflection mode THz TDS . . . . . . . . . . . . . . . . . . . . . . . . . 241
5.24 THz time domain waveform reflected from a SiO2/Si substrate . . . . . . . . . 241
5.25 THz time domain waveform transmitted through monolayer graphene as a
function of back-gate voltage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244
5.26 THz absorption spectra of large-area back-gated graphene . . . . . . . . . . . . 245
5.27 AC conductivity of graphene as a function of back-gate voltage, . . . . . . . . 246
5.28 THz time domain waveform transmitted through monolayer graphene as a
function of top-gate voltage Vbg < VCN P . . . . . . . . . . . . . . . . . . . . . . . 247
5.29 THz time domain waveform transmitted through monolayer graphene as a
function of top-gate voltage Vbg > VCN P . . . . . . . . . . . . . . . . . . . . . . . 248
5.30 THz absorption spectra of large-area top-gated graphene derived from the
primary transmitted pulse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
5.31 THz absorption spectra of ionic gate (LiClO4), derived from the primary
transmitted pulse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
5.32 THz absorption spectra of top-gated large area graphene evaluated from the
secondary transmitted pulse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
5.33 THz absorption spectra of a second graphene device, as a function of top-
gate bias . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252
5.34 THz reflection image of CVD graphene gated at charge neutral condition . . . 255
5.35 THz absorption spectra of graphene microribbons, as a function of top-gate
bias . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256
5.36 Experimental set-up for THz modulation . . . . . . . . . . . . . . . . . . . . . . . 258
5.37 Characteristics of THz QCL used in Modulation . . . . . . . . . . . . . . . . . . . 259
5.38 Average THz power measured as a function of modulation frequency . . . . . 260
5.39 Amplitude modulated THz signal recorded with time . . . . . . . . . . . . . . . 261
xxiv
LIST OF FIGURES
5.40 Amplitude modulated THz signal recorded with time, as a function of fluence263
6.1 Process schematic of suspended top contact in surface emitting quantum-dot
cascade laser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268
6.2 First demonstration of resonant tunneling of one-dimensional electrons . . . 270
6.3 Scanning electron micrograph of nanofences etch mask . . . . . . . . . . . . . 271
6.4 Processing route for nanofences . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272
6.5 Optical alignment using Moire patterns . . . . . . . . . . . . . . . . . . . . . . . 273
8.1 XRD rocking curve from ultrathin 2.7 THz 3 QW Paris design (V671) . . . . . 278
8.2 XRD rocking curve from ultrathin 2.7 THz 3 QW Paris design (V692) . . . . . 279
8.3 XRD rocking curve from ultrathin 2.7 THz 3 QW Paris design (W0798) . . . . 279
8.4 XRD rocking curve from full-stack 3 THz 3 QW Paris design (V702) . . . . . . 280
8.5 XRD rocking curve from ultrathin 3 THz 3 QW Paris design (V703) . . . . . . 280
8.6 XRD rocking curve from full-stack 3 THz 4 QW MIT design (V704) . . . . . . 281
8.7 XRD rocking curve from ultrathin 3 THz 4 QW MIT design (V705) . . . . . . . 281
8.8 XRD rocking curve from full-stack 3 THz 4 QW ETH design (V674) . . . . . . 282
8.9 XRD rocking curve from full-stack 3 THz 4 QW ETH design (V696) . . . . . . 282
xxv
LIST OF FIGURES
xxvi
"The story I am going to tell has more
than one beginning and without an
end."
T. Afsin Ilgar
Chapter 1
Introduction
1.1 Terahertz (THz) gap
The Terahertz (THz) gap lies in a very unique position in the electromagnetic spectrum. On
the lower end of the gap are the radio and microwave frequencies, commonly associated
with communications owing to the distance these long wavelengths can propagate. On the
higher end lie the infrared frequencies (shorter wavelengths), which are exploited by the
optical fiber communication techniques owing to higher bandwidth and data rates.
Figure 1.1: Electromagnetic spectrum showing the THz gap that lies between the radio
and microwave energies and the infrared energy. Reproduced from [1].
Applications of the THz band are not limited to communication. The THz gap is so
called as it is sandwiched between the realms of electronics and photonics. The gap exists
as there is no easy solution to generating and detecting such low energies (1-10 THz, 4-
40 meV). Historically, much of the interest in developing optical elements in this region
1
1. INTRODUCTION
- sources and detectors - came from the fields of astrophysics and atmospheric sciences,
where the small energy associated with THz frequencies corresponds to strong rotational
and vibration modes of molecules [8, 9]. Development in the field of THz sources over the
past few years have expanded the potential to biomedical imaging, ultrafast spectroscopy,
security screening, and non-destructive evaluation [8, 9, 10].
1.2 THz sources
THz sources in the literature can be classified as (i) broadband sources (ii) narrow band
sources.
1.2.1 Broadband sources
Broadband sources use a variety of mechanisms to generate THz radiation. These include
(i) photocarrier acceleration in photoconducting antennas [11], (ii) second-order non-
linear effects in electro-optic crystals such as ZnTe, ionic salts [12, 13], (iii) non-linear
generation by optical parametric conversion in materials such as LiNbO2 [14] and (iv)
plasma oscillations [15]. However, these sources are indirect in nature and the conversion
efficiencies in all of these sources are very low. Consequently, average THz beam powers
tend to be in the nW to µW range, whereas the average power of the femtosecond opti-
cal source is in the region of 1 W. A comprehensive review of broadband THz sources is
provided elsewhere [9, 16, 17].
1.2.2 Narrowband sources
Narrowband sources are relevant for high-resolution spectroscopy applications. In the past,
the only fundamental terahertz oscillators available were optically pumped molecular gas
lasers using continuous wave CO2 lasers. However, not only were these lasers bulky, the
type of gas (for example, methanol, chloroform and ethanol) determined the operating fre-
quency, denying tunability of operation [18]. Tunable emission from semiconductor lasers
was obtained from exotic gain mechanisms such as the p-Ge hot-hole lasers. These lasers
achieved population inversion between the heavy-hole and the light-hole valence band in
crossed electric and magnetic fields [19]. High power THz radiation (1W-103W range)
can be obtained from electron beams, such as free electron lasers [20], relativistic acceler-
ators [21] and synchrotrons [22]. More recently there have been claims on achieving peak
powers from laser-driven particle accelerators, which are greater than synchrotrons [23].
These high power sources require dedicated facilities and involve prohibitive costs.
2
1.3 THz quantum cascade lasers (THz QCLs)
THz sources motivated from oscillators in electronics are limited by transit time and
resistance-capacitance effects at high frequencies, operating slightly above 1 THz (λ ≈
300 µm) at below mW level at best [24]. These approaches include the Gunn diode,
IMPact ionization Avalanche Transit Time (IMPATT) diode and resonant tunneling diodes
(RTD) all of which employ negative differential resistance [9]. Some of the electronic
approaches to generate THz radiation employ frequency up conversion using nonlinear
reactive multiplication of lower-frequency microwave oscillators.
The photonic approaches on the other hand, are restricted by the lack of appropriate
materials with sufficiently small band gaps to allow optical transitions below 15 THz. A
few semiconductor lasers, operating at ≈ 6.5 THz based on narrow gap lead-salt materials
were demonstrated in the 1980s [25]. However, these devices once fabricated suffer from
rapid aging and are particularly sensitive to Auger processes in which the energy and
momentum of the recombining electron-hole pair are transferred to a third carrier, rather
than to a photon. Semiconductor lasers based on intersubband transitions in quantum
wells dramatically altered the focus from finding a material with a band gap in the THz
range to engineering band gaps with THz frequencies. These semiconductor lasers were
given the name THz quantum cascade laser, and are described below.
1.3 THz quantum cascade lasers (THz QCLs)
Among the THz sources discussed above, THz quantum cascade lasers (THz QCLs) are
unrivalled in performance in terms of their size and power. Here, we review the principles
behind THz QCLs and how they evolved from their mid IR counterparts. Note that this
review here is by no means complete and the interested reader is directed to comprehensive
books written on QCLs [10, 26].
1.3.1 The paradigm change: intersubband transitions
Conventional semiconductor lasers, including quantum well lasers are bipolar in the sense
that they rely on transitions between conduction and valence bands in a forward-biased p-
n junction device where the electrons and holes radiatively recombine across the material
bandgap. It is this band gap, which determines the emission wavelength. Furthermore,
the gain spectrum is broad and reflects the distribution of charge carriers in the conduc-
tion and valence bands (having opposite curvature) according to the Fermi’s statistics, see
figure 1.2(a).
In the early 1970s, Kazarinov and Suris proposed a paradigm shift from interband
3
1. INTRODUCTION
transitions to intersubband transitions in quantum wells to achieve generation and light
amplification by electrical pumping [27], [28]. This proposal followed on the seminal
work by Esaki and Tsu [29] who introduced the concept of negative differential resistance
(NDR) in a superlattice. The proposal by Kazarinov and Suris was based on unipolar pho-
ton assisted tunneling. Intersubband transitions unlike interband transitions, are unipolar
in nature, which rely on optical transitions within the quasi-two-dimensional electronic
states in semiconductors (called subbands) formed due to confinement of the electron
wavefunction in the growth dimension. In these unipolar devices, the radiative frequency
can be engineered by changing the well width. Also, the emission frequency is limited only
by the heterostructure offset, ∆Ec (∆Ev) for conduction (valence) band. Since the initial
and final states have the same curvature if non-parabolicity is ignored, the joint density of
states is very sharp which gives rise to a narrow gain linewidth, see figure 1.2(b).
Figure 1.2: (a) Interband laser transition (b) Intersubband laser transition. Here, ρ(E) and
g(E) refer to the joint density of states and gain as a function of optical transition [10].
1.3.2 The journey from mid-infrared QCLs to THz QCLs
Since the energy separation between subbands is independent of the fundamental band
gaps of the material system, the bands can be engineered such that operation at any long
4
1.3 THz quantum cascade lasers (THz QCLs)
wavelength is possible. However, the biggest stumbling block in designing any arbitrary
frequency is the reststrahlen band, or the longitudinal phonon energy of the constituent
material. The initial attempts at achieving emission from intersubband transitions focussed
on energies below the restrahlen band so as to avoid the extremely fast non-radiative in-
tersubband relaxation of electrons via the emission of optical phonons[30, 31]. The major
breakthrough came in 1994 when Faist et al. [32] demonstrated the first intersubband
laser in the mid infrared region. This laser operated at energy E32  ELO (λ≈ 4.6µm, f =
70 THz, E = 290 meV), see figure 1.3. Here, E32 is the energy difference between the diag-
onal levels 3 and 2. Aided by LO phonon scattering, the electrons in the lower lasing level
2 are depopulated to level 1. In addition to the three quantum wells, a digitally graded
alloy is implemented within the superlattice period. This region forms a miniband of levels
that helps in the extraction of electrons from the lower lasing levels 2 and 1, to the upper
lasing level 3 of the next period. This doped region termed as the injector serves as a Bragg
reflector for hot electrons in the upper lasing level 3, preventing them from escaping into
the continuum over the barriers.
Figure 1.3: Schematic conduction band diagram of the first quantum cascade laser show-
ing the radiative levels 3 and 2. Reproduced from [32].
Note that the electron phonon scattering element is proportional to 1/∆k, where ∆k
is the momentum exchanged in the scattering process. In order to achieve E32  ELO, the
momentum exchange involved for LO phonon scattering from level 3 to level 2 is an order
of magnitude higher than the momentum exchange involved for LO phonon scattering
from level 2 to level 1 [32, 33]. This is illustrated in figure 1.4.
Since the inception of the first mid infrared quantum cascade laser, the field developed
rapidly. Within a span of two years, the mid infrared quantum cascade laser started to
5
1. INTRODUCTION
Figure 1.4: Schematic representation of the dispersion relationship E(k//), where E32 
ELO, and k// is the corresponding wavevector parallel to the layers. Adapted from [33].
operate at room temperature [34]. A comprehensive review on mid infrared quantum
cascade laser is provided by Gmachl et al. [35].
Inspite of the developments in the mid infrared QCLs, the move to THz QCLs (E = 4
to 20 meV) was met with several roadblocks. The first promising example of THz elec-
troluminescence was reported in 1998, at a frequency of 8 THz (λ = 37.4 µm, E = 33
meV) [36]. Figure 1.5 shows the dispersion relation for a three level quantum cascade
laser where E32 < ELO. In GaAs, ELO = 36 meV. Due to the small energy spacing between
the upper lasing level 3 and lower lasing level 2, various intersubband mechanisms such
as electron-electron, electron-impurity and interface roughness come into play, which re-
duces the amount of population inversion [33]. A small transition energy E32 makes it
difficult to selectively inject into the upper lasing level. At the same time, it becomes diffi-
cult to selectively extract from the lower lasing level. Note that the momentum exchange
involved for electron LO phonon scattering from level 3 to level 1 (∆k31) is very close to
the momentum exchange involved for electron LO phonon scattering from level 2 to level
1 (∆k21).
Besides these challenges, one of the major hurdles in development of the first THz QCL
was the waveguiding of the THz radiation. Conventional dielectric based waveguides used
in the mid IR could not be used without hitting against practical limitations of growing
waveguides to support long wavelengths. Owing to these difficulties, it took eight years
to go from mid IR QCL to a THz QCL, with the breakthrough coming in the year 2002,
when Koehler et al. demonstrated a THz laser operating at 4.4 THz (λ = 67µm, E = 18
meV) [37]. This laser required a waveguiding approach that was radically different to it’s
mid infrared counterpart. Making use of surface plasmons that exist at the interface of two
6
1.3 THz quantum cascade lasers (THz QCLs)
Figure 1.5: Schematic representation of the dispersion relationship E(k//), where E32 <
ELO, and k// is the corresponding wavevector parallel to the layers. Adapted from [33].
materials with dielectric constants of opposite signs, the authors were able to confine the
mode to the active region.
Figure 1.6 shows the conduction band diagram of the first THz QCL demonstrated in
2002. Adopting a chirped superlattice approach, the design favours population inversion
between the second and first miniband states, even when precise tailoring of the lifetimes
is too complex. Carriers are injected into state 2 via resonant tunneling from the injector
state labelled g, and extracted from the lower lasing level 1 through a broad phase space
of states available for electron scattering. Owing to the relatively small widths of the
minibands, the LO phonons are not directly involved in the relaxation of carriers from the
lower lasing level [37].
Figure 1.6: Schematic conduction band diagram of the first THz quantum cascade laser
showing the 18 meV minigap between states 2 and 1. Reproduced from [37, 38].
Since 2002, a lot of progress has been made in achieving better lasing performance
7
1. INTRODUCTION
through active region design [39]. In the next section, we describe some of the most
common active regions used for THz QCLs.
1.3.3 THz QCL active regions
The merit of different active regions is evaluated in terms of material gain GP and the
normalized oscillator stength ful . Here, ful is the ratio of the quantum optical strength of
the optical transition between upper (u) and lower (l) lasing levels to that of a classical
electron oscillator. The oscillator strength characterizes the overlap and symmetry of the
initial and final wavefunctions. The material gain GP , discussed in section 4.1.1, is propor-
tional to the product of oscillator strength and population inversion. GP is also inversely
proportional to linewidth of the optical transition.
Figure 1.7(a-d) shows the schematic conduction band diagram of common THz QCL
active regions. Apart from the chirped superlattice (CSL) active region, QCL designs come
in two main variants - the bound to continuum (BtC) design and resonant phonon (LO
phonon) design. More recently, the scattering assisted (SA) design has gained prominence
in the THz community.
The BtC design employs the same miniband based extraction scheme as the CSL de-
sign, see figure 1.7(b). The difference in the two designs comes in the upper lasing level,
which is made to be a bound ’defect’ state in the minigap. Designed to be more diagonal in
real space, the upper state lifetime increases as the non-radiative scattering reduces. How-
ever, the diagonality of the transition also impairs the oscillator strength of the radaiative
transition. Compared to the CSL design, the oscillator strength reduces from f21 = 2.5 - 3
to f21 = 1.5 - 2.
The LO phonon design deviates from BtC and CSL design in that it uses resonant tun-
neling to inject into the upper state, but employs the LO phonon scattering to depopulate
the lower lasing level, see figure 1.7(c). In section 1.3.2, we brought out the difficulty of
achieving population inversion posed by LO phonon scattering when the energy of radia-
tive transition is lower than the phonon energy. This challenge was addressed by Williams
et al. [41] where the authors identified that the lower lasing state must be brought into a
broad tunneling with the adjacent quantum wells so that it’s wavefunction is spread over
several quantum wells. The lack of a miniband state implies that the LO phonon designs
have a smaller oscillator strength when compared to BtC and CSL designs, f21 is approx-
imately 0.5 - 1. However, one of the biggest advantages of LO phonon design lies in the
reduction of the period length from 105-110 nm to nearly half.
Like the LO phonon design, the scattering assisted (SA) design, also has a small period
8
1.3 THz quantum cascade lasers (THz QCLs)
Figure 1.7: Schematic conduction band diagram of common THz QCL active regions: (a)
chirped superlattice (CSL) (b) bound to continuum (BtC) (c) resonant phonon (LO phonon
/ RP) (d) scattering assisted (SA). Adapted from [39, 40]
9
1. INTRODUCTION
length. However, the SA design does not employ resonant tunneling to inject electrons into
the upper lasing level, instead it uses resonant phonon relaxation to selectively inject into
the upper lasing level as well as extract the electrons out from the lower lasing level [40].
The oscillator strength (ful ≈ 0.6) is similar to that of the LO phonon. Note that currently,
the SA design holds the record for the highest operating temperature within the literature.
This is important since the operating temperature holds the key to commercialization of
THz QCLs for various applications.
1.3.4 THz QCLs: quest for room temperature operation
Over the last decade, rapid advancements in the design of gain medium as well as improve-
ment in the fabrication and wave guiding techniques have led to significant improvements
in the operating performance of the THz QCLs. THz QCLs now cover a wide spectral range
(0.45 - 5.0 THz) [39]. They also exhibit high bandwidth (up to 1 THz)[42] and operate
at a power exceeding 1 W [43]. However, room temperature operation still continues to
elude THz QCLs, making this subject a holy grail in this field. What is encouraging is that
THz QCLs have come a long way in terms of their operating temperature since their first
demonstration in 2002 at a maximum operating temperature of 50 K. Figure 1.8 shows
the maximum operating temperatures achieved with improvements made in active region
design, heat sinking and confinement of optical mode. As one can see, the relationship
between the operating temperature T and the operating frequency ω is linear. This is
because at low frequencies, the energy carried by the photon (Eul ∼ 8 meV at 2 THz) is
similar to the typical energy broadening of the subbands (a few meV). Hence, it becomes
difficult to selectively inject from the injector into the upper lasing level.
There are however three design principles that deviate from this trend. The first ap-
proach makes use of a unique scattering assisted injection scheme, first shown to work
at 1.8 THz by Kumar et al. Based on this work, the SA design was made more diagonal
to achieve a record operating temperature of 200 K at a frequency of 3.6 THz [44]. The
second approach is more recent and employs intracavity difference frequency generation
from mid IR QCLs, to achieve room temperature emission between 3.3 to 4.6 THz [45].
The third approach makes use of very high magnetic field to suppress the non-radiative
scattering by LO phonon emission. By imposing zero-dimensional confinement on the
carriers, this technique has shown to improve the operating temperature of a standard
resonant phonon design from ≈ 178 K to 225 K [46]. Below, we focus our attention to
zero-dimensional confinement as a route to higher temperature performance.
10
1.4 Zero-dimensional confinement in QCLs
Figure 1.8: Operating temperature of THz QCLs as a function of frequency, reported in the
literature. Adapted from [39].
1.4 Zero-dimensional confinement in QCLs
Figure 1.9 shows the schematic dispersion relationship for a laterally confined three-level
THz QCL. Note that the non-radiative LO phonon scattering from the upper lasing level 3
to the lower level 1, which is particularly dominant at high electron density in the upper
subband [47, 48], is suppressed due to quantization of the in-plane momentum. The in-
plane quantization of the momentum wavevector can be imposed in several ways. First,
we survey the different approaches explored in the literature to attain zero-dimensional
confinement, aside from a very high magnetic field. The term bottom-up refers broadly to
quantum-dots, quantum-dashes or nanowire structures grown by epitaxial methods.
1.4.1 Bottom up approach
Theoretical proposals for quantum dot cascade structures
The first proposal by Wingreen et. al (figure 1.10(a)) invisioning a quantum-dot cas-
cade structure [49], predates the demonstration of THz electroluminescence from quantum-
well cascade structures [37, 51]. Suggested for operation at mid-infrared frequencies, this
proposal made use of a vertical cascade of a pair of quantum dots to achieve radiative tran-
sition between energy levels with an energy difference E ≥ ħhωLO. Following this proposal,
11
1. INTRODUCTION
Figure 1.9: Schematic representation of the dispersion relationship E(k//), where E32
< ELO. The wavevector parallel to the layers k// is discretized due to zero-dimensional
confinement.
another design which made use of radiative transition in a single quantum-dot with energy
splitting E ≥ 2ħhωLO was suggested by Chia-Fu et. al (figure 1.10(b)). Infact, Chia-Fu was
one of the very few authors who actually proposed a lasing architecture for a quantum-dot
cascade structure, with a dielectric cladding surrounding the quantum-dot gain medium
to act as a waveguide. Both, Wingreen et. al and Chia-Fu et. al estimated the lateral
dimension of the quantum dot to be ∼ 20 nm. With increasing electron-population, the
Coulomb interaction between electrons may start to effect radiative transitions described in
figure 1.10. Compared to a noninteracting system, this interaction may result in a blueshift
of the luminescence spectra[52, 53].
Mid infrared emission from quantum dot cascade structures
The first experimental observation of electroluminescence from a quantum-dot cas-
cade structure, at mid-infrared frequencies came a few years after the theoretical proposals
were made. Instead of replacing the quantum wells with quantum dots, as suggested by
Wingreen et. al, this design employed a cascade with each period consisting of self assem-
bled InAs dots grown on top of GaAs wells, see figure 1.11. While pure quantum-well
systems (without dots) showed emission that was transverse-magnetic (TM) polarized, the
radiation emitted from these ’quantum-dot in quantum-well’ system was observed to be
only slightly polarized. The lack of polarization was attributed to an elliptical cross section
of the quantum-dots [54]. Similar cascade structures, with InAs dots grown on top of AlIn-
GaAs wells were shown to emit p-polarized electroluminescence due to hybrid radiative
transitions. These transitions occurred from the excited and ground state of the quantum-
12
1.4 Zero-dimensional confinement in QCLs
Figure 1.10: (a) Schematic energy band diagram showing diagonal radiative transition in a
pair of coupled dots, i.e. between the first-excited state of one dot and the ground state of
a second dot. The density of the coupled dots in an array determines the gain. To achieve
lasing, these coupled dots are stacked vertically as well as repeated in a planar array [49].
(b) Schematic energy band diagram, showing radiative transition in a single quantum dot.
The Bragg mirror in the direction of growth achieves the same purpose as the coupled dots
in figure (a) i.e it reduces the leakage of electrons in the upper lasing level. The dots are
arranged in a planar array to achieve the required gain, and make use of only one repaeat
period in the vertical direction [50].
dot to the ground state of the following quantum well, at mid-infrared frequencies [55]
and did not depend upon the size of the quantum-dot.
Anisotropically polarized mid-infrared emission from quantum-dot cascade structures
that relied on transitions within the InAs quantum dot was observed in an active region
very similar to the vertical transition design proposed by Chia-Fu [56]. The schematic band
diagram showing vertical transition from the excited to the ground state of the quantum
dot is shown in figure 1.12. Transitions within a single InAs quantum dot on AlInAs, sup-
ported with superlattices on either side for efficient injection and extraction were exploited
recently, to achieve room temperature anisotropic, transverse-electric (TE) polarized emis-
sion at mid-infrared frequencies, which depended on the amount of lateral confinement
[57]. A quantum dot in InAs modeled as a semi-cylinder with width = 15 nm, length = 65
nm and height = 2.5 nm, corresponds to an energy difference of |2,1, 1〉 - |1,1, 1〉 = 120
meV, which agrees with an electroluminescence peak at 110 meV [57] when the minibands
are aligned to the dot energy states.
THz absorption from quantum dot cascade structures
The transition energies in a vertical design, quantum-dot cascade structure depend
upon the quantum dot dimensions and the growth material. Absorption at THz frequen-
13
1. INTRODUCTION
Figure 1.11: Energy band diagram indicating the radiative transition levels in the quantum-
dot cascade structure (3 and 2) used in [54]. The excited energy levels of the quantum dot
(3 and 2) are coupled to the quantum-well states. The schematic shows the energy levels
of the ’quantum-dot embedded in quantum-well’ structure for a single period, under the
conditions of (a) no electrical bias and (b) with electrical bias [54].
cies was reported in 2009 from THz pump-probe experiments on quantum-dot cascade
structures based on self-assembled InGaAs quantum dots [58]. Transition energy between
the first excited state and the ground state, in these InGaAs dots lie in the range of 40-60
meV (f = 9.6 - 14.5THz ). By annealing the quantum-dots post-growth, the transition
energy can be brought down below the longitudinal phonon energy, ħhωLO. The InGaAs
quantum-dots modeled as truncated cones of base (top) radius 12 nm (6 nm) and height
4.9 nm, correspond to a transition energy of 48 meV (f = 12 THz), which reduces to 10
meV (f = 2.4 THz) when the radius of the dot is increased to 25 nm by post-growth an-
nealing. Unlike previous designs in the mid-infrared, this quantum-dot cascade structure
did not employ any Bragg mirror to improve electron injection and extraction. However,
each dot layer was separated from the other by a thick GaAs barrier to prevent structural
and electronic coupling between the layers. Figure 1.13 shows the absorption spectra be-
low Reststrahlen band due to intersublevel transitions between the excited p-state and the
s-like ground state in a stack of self-assembled InGaAs quantum dots. While the quantum-
dot structure holds promise at higher energies, the spectra around 10 meV (f = 2.4 THz) is
quite weak and broad possibly due to intermixing, suggesting the need for better dot size.
14
1.4 Zero-dimensional confinement in QCLs
Figure 1.12: Energy band diagram, under an electrical bias, showing the radiative transi-
tion levels in the quantum-dot cascade structure used in [56]. The superlattice minibands,
grown on top of the quantum-dot, are tailored to block leakage of electrons from the
upper lasing levels (excited states of the dot) at the injector energy. Furthermore, these
minibands (shaded in blue) promote extraction of electrons from the ground state of the
quantum dot. This superlattice structure is similar to the Bragg mirror shown in figure
1.10(b). However, unlike [50], in spite of the vertical transition in a single quantum dot,
this structure makes use of stacking of the active periods in the growth direction to improve
gain [56].
Growth of nanowires
In the year 2010, Williams et. al, proposed that intersubband transitions within semi-
conductor nanowires with axial and core-shell geometeries might have potential for THz
lasers operating at high temperature 1. However, while such nanowire geometeries have
shown promise as photodetectors through intersubband absorption [59, 60], their appli-
cability to THz emitters still remains as an area of active research.
1.4.2 Top down Cavendish approach
We can now introduce the subject of this thesis - zero dimensional confinement in THz
QCLs through a top down nanofabrication approach. So far, the most optimistic result
of zero-dimensional confinement leading to intersubband transitions at THz frequencies
comes from Zibik et. al [58]. However, the need to realize a 25 nm wide quantum dot
or wider through epitaxial growth, for emission at ≤ 2.4 THz has been inhibiting the
development of THz quantum dot cascade lasers through a bottom up approach. The
challenge of the top down approach to realize such structures is exactly the opposite.
While it is relatively straightforward to fabricate structures that are a few µm wide over
1Url: https://www.collectiveip.com/grants/NSF:1002387
15
1. INTRODUCTION
Figure 1.13: (a) Calculated wavefunction of the s and p states in the quantum dot (b)
Schematic energy diagram showing vertical transitions between s and p states. The two
transitions, px -s and py -s are separated by an energy difference equal to ∆pp. (c) Absorp-
tion spectra of quantum dots as a function of annealing temperature [58].
the thickness of a THz QCL, it is not trivial to realise structures which are sub-200 nm
in size that can be supported throughout the active region. The applicability of a top
down approach to realize zero dimensional confinement in THz QCLs was first suggested
by Tredicuccu et. al in the year 2009 [61]. Since the confinement in growth direction is
implicit, we use the term lateral confinement interchangeably with a top down approach
to zero dimensional confinement. In this work, we look to study the feasibility of this
approach as a means of suppressing non-radiative relaxation mechanisms within a THz
QCL.
1.5 THz waveguides
In section 1.3.2, we mentioned that the development of THz QCLs struggled initially
because of the absence of a low loss THz waveguide. Dielectric waveguides cannot be used
at THz frequencies, as the thickness of the cladding layer scales linearly with the free space
wavelength, which makes it prohibitive to implement. On the other hand, the very high
free-carrrier absorption at THz frequencies prevents the use of doped layers. The optimal
solution for THz frequencies uses an alternate approach to waveguiding as opposed to
techniques borrowed from the microwave and the optical frequencies [62, 63, 64, 65, 66].
This solution relies on using surface plasmon waves at the interface of two materials with
dielectric constants of opposite signs, typically a metal and a semiconductor. Below, we
16
1.5 THz waveguides
describe the theory behind surface plasmons.
1.5.1 Surface plasmons
Surface plasmons are charge density oscillations that exist at the interface of a dielectric
and a conductor, and propagate along the surface of the conductor, see figure 1.14. Tech-
nically, surface plasmons should be addressed as surface plasmon polaritons owing to the
hybrid nature of the interaction between a photon (of electromagnetic radiation) and free
electrons (on the surface of the conductor) which respond collectively by oscillating with
the light wave. The normal component of the electric field decays exponentially with dis-
tance from the surface. The decay length in the metal δm depends upon the skin depth
of the conductor and the decay length δd in the dielectric is approximately of half of the
incident wavelength.
Figure 1.14: Surface plasmons at the interface between a metal and a dielectric (a) Elec-
tromagnetic wave coupled to surface charge density fluctuation. (b) Exponential decay
of the normal component of the electric field |Ez|. (c) Dispersion characteristics of the
surface plasmon wave. Reproduced from [66]
If the surface plasmon propagates in the x-direction, the electric and magnetic fields in
the dielectric (Ed and Hd , z > 0) and in the metal (Ed and Hd , z < 0) can be expressed as:
Ed(r , t) = (ex Ed,x + ezEd,z)e
−κdzei(kx x−ωt) (1.1)
Hd(r , t) = (eyHd)e
−κdzei(kx x−ωt) (1.2)
Em(r , t) = (ex Em,x + ezEd,z)e
κmzei(kx x−ωt) (1.3)
Hm(r , t) = (eyHm)e
−κmzei(kx x−ωt) (1.4)
17
1. INTRODUCTION
The surface wave corresponding to a frequency of ω is characterized by the wavenum-
bers kx , ky and kz . Solving the generic wave equation
∇2E = εµ∂
2E
∂ t2
(1.5)
using for electric and magnetic fields given by eqns. 1.2 - 1.4 yields the dispersion rela-
tions:
k2x − k2d = εd
ω2
c2
(1.6)
k2x − k2m = εm
ω2
c2
(1.7)
Here, εd and εm are the relative permittivity values of the dielectric and the metal. The
permeability values for both the dielectric and the metal are assumed to be equal to 1, for
non-magnetic materials. The ratio of κd to κm can be expressed in terms of the propgation
wavevector kx and frequency ω as:
κ2d
κ2m
=
k2x − εdω2/c2
k2x − εmω2/c2 (1.8)
Applying the Maxwell equation, curl(E) =−µ∂H/∂ t, the respective magnetic fields can be
written in terms of the electric field as:
Hd,y =−iεo εdωκd Ed,x (1.9)
Hm,y = iεo
εmω
κm
Em,x (1.10)
Since, Hd,y = Hm,y and Ed,x = Em,x at the surface, the boundary conditions reduce
eqn. 1.9 and eqn. 1.10 to
εd
εm
=− κd
κm
(1.11)
On substituting eqn. 1.11 into eqn. 1.8 one obtains the dispersion relation of surface
plasmons, which can be written as:
kx =
ω
c

εdεm
εd + εm
1/2
(1.12)
Thus, the wavevector kx of the surface plasmon wave is larger than the wavevector
18
1.5 THz waveguides
of the incident electromagnetic radiation, given as ko = ω/c. This dispersion relation-
ship is plotted in figure 1.14(c). This implies that the electromagnetic energy is confined
within a dimension that is much smaller than the wavelength of light. This is the reason
why an incident THz wave, which has a long wavelength, can be supported by a surface
plasmon which can propagate in subwavelength dimensions. Below, we describe the two
main categories of waveguide configurations based on surface plasmons, which enable the
functioning of THz QCLs.
1.5.2 Plasmonic waveguides in THz QCL
The waveguide developed for the first THz QCL relied on semi-insulating GaAs substrates,
which were relatively free from absorption related losses. This approach also made use of
the highly doped bottom contact layer, grown directly beneath the active region [37]. A
surface plasmon mode then exists between the top metal contact and the bottom doped
n+ GaAs layer, with a substantial extension into the GaAs substrate, see figure 1.15(a).
This kind of a waveguide is commonly known as a single plasmon waveguide. Although
the mode extends into the substrate, the overlap with the doped semiconductor layer is
small, so the free carrier absorption loss is curtailed. In the single-plasmon design, the
overlap factor, Γ typically ranges from 0.1 to 0.5. Here, the overlap factor is defined as the
percentage of the optical mode interacting with the active region. The exact formalism is
discussed in section 2.11.1.
Figure 1.15: THz QCL waveguides: schematic diagram and 2-dimensional mode profile
(a) single-plasmon (b) double-metal [39]
19
1. INTRODUCTION
Recent developments have seen the emergence of the double metal waveguides, which
provide an overlap factor Γ ≈ 1 [67]. Double metal waveguides have metal layers on the
top and bottom of the active region, causing two surface plasmon modes to be formed, see
figure 1.15(b). As compared to single plasmon waveguides, the double-metal waveguides
supports smaller ridge widths and have better heat sinking properties, which allows for a
greater temperature performance. Furthermore, as the mode is confined between the two
metal surfaces, the ridge width can be considerably shrunk, leading to smaller devices with
lower drive currents. However, because of the strong confinement, there exists no cutoff
frequency for the TM mode in this waveguide structure. This implies greater impedance
mismatch with vacuum at the ridge facets, which considerably increases the reflectance
at the facet. Although, the increased reflectance minimizes the mirror loses, it also acts
detrimental to the beam profile and the output powers extracted from the device.
1.5.3 Graphene: a material for THz plasmonics
In a THz QCL, the gain medium and the waveguide are two disparate entities. The optical
gain arises from intersubband transitions in GaAs/AlGaAs materials, and the optical wave
is supported by the plasmonic mode at the metal/active region interface. In the year 2008,
another material came to the attention of the THz community. This exotic material was
called graphene, a two dimensional monolayer of carbon atoms arranged in a hexagonal
lattice, with zero bandgap. Pitched as a conducting material that could support THz plas-
mons as well as provide plasmon gain, monolayer graphene quickly gained wide interest
from the optical community [68].
Compared to traditional metals, graphene is only one atom thin. Therefore, the bound
electromagnetic modes (plasmons) are tightly confined to the surface. Furthermore, the
conductivity of graphene can be tuned externally with bias by changing the carrier concen-
tration. This implies that the plasmon frequency ωp, which is function of carrier concen-
tration n, can be controlled externally. The plasmon frequency ωp of a conventional two
dimensional electron gas (2DEG) can also be varied externally with carrier concentration.
However, to obtain plasmon frequencies in the THz range, one generally requires low tem-
peratures (4.2 K) [69, 70]. In contrast, the plasmon frequencies in graphene can approach
THz frequencies at room temperature by external gating [71]. The differences between
graphene and a 2DEG are summarized in table 1.1.
Figure 1.16(a) illustrates the typical absorption spectrum from graphene. Note that the
20
1.5 THz waveguides
Property Monolayer graphene 2DEG
Energy dispersion Linear, no gap Quadratic, energy gap > 1 eV
Mass of electron Massless 0.067 me
Carrier concentration 109 to 1013 cm −2 109 to 1012 cm −2
Gating efficiency Dielectric/Chemical Dielectric
Plasmon frequency ωp depen-
dence on carrier concentration n
Varies as n1/4 Varies as n1/2
Table 1.1: Difference between monolayer graphene and 2DEG. Data sourced from [71, 72].
Figure 1.16: (a) Schematic diagram showing typical absorption spectrum of graphene,
with Drude peak at THz frequencies, minimal absorption at mid-infrared frequencies due
to Pauli blocking and transition to universal 2.3 % absorption in near infrared to visible
frequencies. (b) Illustration of the various optical processes corresponding to THz, mid-
infrared and visible regions in the absorption spectrum. Reproduced from [73]
21
1. INTRODUCTION
Drude peak lies in the THz spectrum, indicated by 1. This is followed by minimal absorp-
tion at mid-infrared frquencies due to Pauli blocking, indicated by 2 and finally, a transi-
tion to universal 2.3 % absorption in the near infrared to visible frequencies, indicated by
3 [73]. The optical processes corresponding to absorption in the THz, mid-infrared and
near-infrared to visible regions is shown in figure 1.16(b). At low energies, where ħhω is
less than the thermal energy, transitions occurs via the intraband processes, indicated by
1. At a finite ħhω < 2 EF , where EF marks the position of the fermi level, disorder can play
a role in imaparting the momentum for the optical transition, indicated by 2. A transi-
tion occurs around ħhω = 2 EF , where direct interband processes lead to a universal 2.3 %
absorption, indicated by 3 [73].
The potential for graphene based THz optics ranging from modulators, filters, polariz-
ers and photodetectors is a rapidly evolving field and is discussed in references [73, 74].
Room temperature optical elements that can be a part of an optical system involving com-
pact THz sources such as THz QCLs, would bring the THz technology one step closer to
applications in the real world. In this thesis, we explore whether graphene can fulfill some
of these needs at the THz frequency.
So far, our definition of a THz gap was restricted to THz sources and waveguides. At
times, this absence of a fast, reliable THz detector cripples the development of other THz
optical elements. In the next section, we discuss some common detectors that are used in
the measurement of a THz response.
1.6 THz detectors
Most detectors at THz frequencies (4-40 meV, 1-10 THz) face a problem from high thermal
background noise (kBT at 300 K ∼ 25.8 meV). In general all THz detection schemes can
be classified as (i) incoherent detection systems, which involve direct detection of signal
amplitude and (ii) coherent detection system, which involve detection of phase and ampli-
tude of the THz signal. For an excellent review of current THz detection techniques, the
reader is directed to reference [75, 76].
Direct detection scheme
Under the direct detection scheme, there are several techniques that can be used to
measure the signal amplitude. Some of these schemes are summarized in table 1.2 and are
primarily based upon thermal absorption. These detection schemes include thermopiles,
golay cells, pyroelectrics and bolometers. Thermopiles are essentially arrays of thermocou-
22
1.6 THz detectors
ples, which convert temperature to an electrical output. Golay cells rely upon changes in
gas pressure with heat absorption, to detect the THz signal amplitude. Pyroelectrics detec-
tors are based on crystals such as Lithium Niobate and Lithium Tantalate, which measure
change in polarization as a function of temperature. Bolometers typically operate at cryo-
genice temperatures and are based upon the temperature dependent electrical resistance
of semiconducting materials such as Si, Ge or InSb. They can also be based upon super-
conducting materials such as Niobium, where the change in resistance with temperature
measures the amplitude of the THz signal. The merit of a detector is assessed in terms of
noise equivalent power (NEP), optical responsitivity and rise time, see table 1.2.
Detector Operating
temperature
Noise equiva-
lent power
Optical responsi-
tivity
Rise time Source
K 10−10 W/pHz V/W ms
Thermopile 300 4.8 102 16 Roithner-laser 1
Golay Cells 300 100 4.8 x 103 25 Microtech in-
struments 2
Pyroelectric
detector
300 2 4.2 x10 3 10-100 Gentec-EO 3
Composite
Bolometer
4.2 0.02 104 0.4 QMC Instru-
ments 4
Table 1.2: Comparision between different commercially available THz detectors. Data was
sourced from manufacturer specifications and reference[76].
Although, the specifications listed in table 1.2 reflect only a small subset of detec-
tors, in general, the uncooled bolometers display the highest senstivity (low noise equiv-
alent power) among the commercially available detection schemes. At liquid He temper-
atures, the bolometers display a noise equivalent power approaching pW/
p
Hz. Cryo-
genic bolometers also measure the THz signal with the lowest response times (< 1 ms).
More recently, the field of THz detection has caught renewed interest with development of
plasma wave detection in field effect transistors based on low-dimensional materials such
as graphene and semiconducting nanowires [77]. Some THz detectors based on graphene
use Landau level quantization to detect energies of 10 meV, at a very small magnetic field
(0.5 T) [78]. Reports on the bolometric response of graphene and carbon nanotube con-
1http://www.roithner-laser.com
2http://www.mtinstruments.com
3https://www.gentec-eo.com/
4http://www.terahertz.co.uk/
23
1. INTRODUCTION
tinues to gain traction [79, 80, 81], but so far these techniques are still in their infancy.
Coherent detection scheme
The methods discussed above only measure the intensity of THz radiation and not the
temporal profile of the radiation. Electro-optic sampling and photoconductive antennas
enable the measurement of the THz waveform over time. Both these methods work in
the same manner as their emitter counterparts, except in reverse. In the first case, an
optical beam enters along with the THz pulse into an electro-optic crystal. The THz field
modulates the polarization ellipticity of the optical probe passing through the crystal. The
x and y components of the polarization give information about the amplitude and phase
of the THz pulse. The photoconductive antenna measures an induced current across the
antenna as a result of the incident THz radiation, which is probed by the femtosecond
pulse. Other methods of detection THz radiation include heterodyne techniques, which
requires a local oscillator emitting at the THz frequency of interest. The THz signal is
mixed with local oscillator, for example a Schottky diode and is down converted signal.
This technique offers a high spectral resolution and a high sensitivity.
1.7 Thesis outlook
This thesis is organized as follows. Chapters 2 to Chapter 4 are driven by the motivation
to achieve higher operating temperature in THz QCLs through lateral confinement. In
Chapter 2, we develop our understanding of lateral confinement from a simple system - a
resonant tunneling diode to a more complicated structure - a THz quantum cascade laser.
Building on from theoretical predictions, in Chapter 3 we discuss the various processing
routes explored to meet the requirements of lateral confinement in THz QCLs. Chapter 4
discusses THz quantum cascade lasers with an active region of thickness≤ 3.5 µm. Termed
as ultrathin THz QCLs, these lasers form the foundations of a gain medium that is suitable
for studying effects of lateral confinement. In Chapter 5, we move away from THz QCLs
and investigate plasmonic properties of a two-dimensional material - monolayer graphene,
which promises interesting applications at THz frequencies owing to voltage tunability of
carrier concentration. Finally, conclusions and future work discussed in Chapter 6 reflect
on some of ideas built upon in this doctoral work.
24
"Hope doesn’t come from calculating
whether the good news is winning out
over the bad. It’s simply a choice to
take action."
Anna Lappe
Chapter 2
Nanopillars in RTDs and QCLs:
Theory
2.1 Introduction
Considering the compact and powerful nature of QCLs as a source bridging the THz gap,
achieving room temperature operation would open up doors to a plethora of oppurtunities
in both academic and commercial areas of sensing, spectroscopy and imaging.
In Chapter 1, we discussed that the maximum operating temperature of THz QCLs is
limited by optical phonon scattering of hot electrons in the upper lasing level. These elec-
trons gain sufficient in-plane kinetic energy to relax non-radiatively into the lower lasing
level. This decay which occurs at a very fast time scale (2-3 ps), lowers the population
inversion (and hence gain), in THz QCLs. Discretization of inplane momentum may sup-
press this non-radiative inter-sub-band relaxation mechanism, and enhance the upper state
lifetime [61]. By imposing a lateral potential, i.e. confining the electrons in-plane, the mo-
mentum can be discretized. These predictions have been verified by application of high
magnetic fields to THz QCLs [82], where electrons are confined by the cyclotron radius,
and also in a cascade of self-assembled quantum dots [58], where electrons are confined
spatially by the dot radius.
In this chapter, we explore a top down approach to obtain lateral confinement in THz
QCLs. This lateral confinement is imposed in the radial direction by nanofabrication meth-
ods. This radial potential is superimposed on top of the epitaxially imposed confinement in
the longitudinal growth direction, which gives rise to 3-D confinement in these nanopillar
arrays.
Before we delve into the energy spectrum and optical properties of a laterally confined
25
2. NANOPILLARS IN RTDS AND QCLS: THEORY
THz QCL, it is useful to study the effect of lateral confinement in a simpler III-V heterostruc-
ture - a resonant tunneling diode. Consisting of a single well, sandwiched between two
barriers, a resonant tunneling diode (RTD) exhibits negative differential resistance (NDR)
as the applied voltage aligns with energy states associated with the longitudnal confine-
ment of the two-dimensional electron gas [83]. Unlike a THz QCL, which is a cascade of
several repeat periods, an RTD is a single stage oscillator. RTDs by themselves have been
proven to oscillate at frequencies approaching 1 THz, at room temperature [84]. However,
their gain is derived from the negative differential resistance, unlike QCLs which rely on
intersubband transitions. Here, we do not use the RTD as a THz oscillator, but exploit it’s
simplicity to understand the effects of lateral confinement on intersubband transitions in
THz QCLs.
The enormous advantage of RTDs is that they exhibit a very clear NDR peak upon
alignment in their transport characteristics. This transport behavior will be influenced by
discretization of energy states due to an additional potential - imposed by nanopillars.
Therefore, RTDs act as a very useful testbed to probe effects of a top-down approach on
vertical transport of electrons. We use the RTDs to answer the following questions. The first
question, is whether enough optical gain can be achieved by the array of nanopillars. In
the first approximation, this translates to whether the nanopillar array can carry a current
density that approaches the threshold current densities of THz QCLs, which is typically in
the order of hundreds of A/cm2. The second question relates to the cascade nature of the
QCL. Given a fixed nanopillar radius, what is the true lateral potential across the height
of the nanopillar, as this may depend upon the surface states and doping of each layer ?
The third question relates to electron transport through more than a thousand nanopillars,
connected in parallel to achieve sufficient gain required for lasing. The question therefore
is how statistical deviations arising from fabrication, influences the vertical transport in an
ensemble of quantum dots?
In this chapter, we seek to learn from existing models of lateral confinement in an RTD,
and translate the ideas to the more complicated QCL structure. We also probe the effect of
such lateral confinement on the band structure of the THz QCLs used in this work. Vertical
transport through a nanopillar array necessitates the use of a planarizing material and
formation of a top electrical contact. This process can influence the optical mode inside a
nanopillar array. Therefore, in the final section, we discuss the effect of lateral confinement
on the plasmonic mode in a THz QCL nanopillar array and identify optimum geometries.
26
2.2 General description of RTDs
2.2 General description of RTDs
In this section we provide a general discussion on the band diagram and electrical charac-
teristics of large-area RTDs. The electrons in these large-area RTDs have in-plane momen-
tum, and are confined only in the longitudinal growth direction.
First, let us look at the material composition of an RTD. Figure 2.1(a) shows the band
gaps and conduction (valence) band∆Ec (∆Ev) discontinuties for GaAs, Al0.33Ga0.67As and
AlAs. As is seen from the figure, Al0.33Ga0.67As and AlAs are higher band gap materials
as compared to GaAs. Electrons in GaAs surrounded by either Al0.33Ga0.67As layers on
both sides, or AlAs layers on both sides, will be confined in a potential well, leading to
quantization of energy states in the growth direction. It is this quantization, which leads
to negative differential resistance (NDR) in RTDs.
Figure 2.1(b) shows the schematic diagram of an epitaxially grown double barrier RTD.
This system is described by the time independent Schrodinger equation in the growth (z)
direction, given as:
− ħh
2
2m∗
∂ 2
∂ z2
ψ(z) + V (z)ψ(z) = Eψ(z) (2.1)
Here, m∗ is the effective mass of electrons inside the quantum well and V(z) is the po-
tential in the growth direction. At the well boundaries, V(z) = Vo =∆Ec , which is typically
0.26 eV in Al0.33Ga0.77As/GaAs/Al0.33Ga0.77As structures and 1.1 eV in AlAs/GaAs/AlAs
structures. The wavefunction of the electron is described by the eigen function ψ(z).
By solving eqn. 2.1 under appropriate boundary conditions, the energy states in the
quantum well of finite height Vo, can be calculated by solving the transcedental equation:
En =
ħh2
2mo L2w

pi(n+ 1)− arcsinpEn/Vo2 , n= 0,1, 2... (2.2)
It is easy to see from eqn. 2.2, that the energy levels can be engineered by changing
the width, Lw of the quantum well [87]. The tunneling rate depends upon the width of the
barrier Lb. Figure 2.1(c) shows the ground and excited energy states in an Al0.33Ga0.67As /
GaAs / Al0.33Ga0.67As double barrier RTD, with Lw = 10 nm and Lb = 5 nm. The transmis-
sion probabilities of electrons tunneling through the Al0.33Ga0.67As / GaAs / Al0.33Ga0.67As
double barrier RTD is described by the following equation, and plotted in figure 2.1(d).
T(E) = Tn
(1/2Γ)2
(1/2Γ)2+ (E − En)2 . (2.3)
27
2. NANOPILLARS IN RTDS AND QCLS: THEORY
Figure 2.1: (a) Band gaps, conduction (valence) band ∆c (∆v) discontinuties for
GaAs, Al0.33Ga0.67As and AlAs(b) Schematic of the epitaxially grown barrier/well/barrier
structure. (c) Conduction band diagram of unbiased double barrier resonant tunnel-
ing diode showing the energy levels in the well (d) Transmission function as a func-
tion of energy. The bound states and the transmission coefficients have been calculated
for an Al0.33Ga0.67As/GaAs/Al0.33Ga0.67As double barrier structure using RTD NEGF tool
[85, 86]. The thickness of the GaAs well is 10 nm, and the thickness of the Al0.33Ga0.67As
barrier is 5 nm.
Here, Γ is the full width half maximum (FWHM) of the resonance peak and Tn is the
transmission when energy E = En.
The tunneling current density J , for each contact, can be evaluated by integrating the
density of states g(E), with the Fermi function f(E), the electron charge e, the electron
velocity ν and the transmission coefficient T(E). This is given as
J =
∫ ∞
0
g(E) f (E)eνT (E)dE (2.4)
In the case of a large area RTD, the contacts can be treated as bulk and the electrons
confined in the well can be treated as a two-dimensional electron gas.
Figure 2.2 plots the theoretical current density of an Al0.33Ga0.67As / GaAs / Al0.33Ga0.67As
28
2.2 General description of RTDs
Cu
rr
en
t	
  D
en
sit
y	
  
	
  	
  	
  
	
  	
  	
  
(A
/c
m
2 )
	
  
Voltage	
  (V)	
  
0	
   0.5	
  0.3	
   0.4	
  0.2	
  0.1	
  
15000	
  
10000	
  
5000	
  
0	
  
(ii)	
  
(i)	
  
(iii)	
  
(iv)	
  	
  	
  	
  	
  	
  On	
  	
  Resonance	
  
	
  
Off	
  
Resonance	
  
E0	
   E0	
   E0	
   E0	
  Ε
l
F	
  
Εc	
   Εc	
  eV	
  
	
  
ΕrF	
  
Vp	
  
Jp	
  
(i)	
  
(ii)	
  
(iii)	
   (iv)	
  
Vth	
   VV	
  
JV	
  
ΔVNDC	
  
	
  	
  	
  	
  	
  At	
  	
  
threshold	
  
Figure 2.2: Theoretically evaluated current-voltage characteristics of an RTD, showing (i)
current before alignment due to non-resonant processes (ii) current at alignment of the
Fermi level with the ground state (iii) maximum current at alignment of the conduction
band with the ground state followed by negative differential resistance due to misalign-
ment (iv) increase in current through barriers due to thermionic emission and scattering
assisted tunneling. The current was calculated for a double barrier RTD with 5 nm thick
Al0.33Ga0.67As barriers and a 10 nm GaAs well, using RTD NEGF tool [85, 86].
29
2. NANOPILLARS IN RTDS AND QCLS: THEORY
double barrier RTD, with the energy states indicated in figure 2.1. Also shown in figure 2.2
is the band structure of the RTD under different bias conditions. At a low bias, case (i), a
small current flows due to non-resonant processes such as thermionic emission over barri-
ers and scattering assisted tunneling. As the bias is increased to a point (ii) on figure 2.2,
the Fermi level of the left contact (El F ) aligns with the ground state of the well (Eo). Here,
the current starts to flow due to resonant tunneling of electrons. This point is indicated by
a threshold bias Vth. On further increase in bias, the conduction band of the left contact
(Ec) aligns with the ground state of the well (Eo). When this occurs, maximum current
tunnels into the resonant level. This is the bias at which peak current is measured, and is
indicated by (iii) in figure 2.2. The voltage and current density at the point of resonance
is indicated by Vp and Jp, respectively. Since Vp is the voltage dropped across the two
contacts, the resonance condition for the ground state is met when Eo = Vp/2. In an RTD,
a clear NDR signature is obtained when the conduction band aligns with the ground state,
and the current reduces as the bias is increased, creating a so called valley. The voltage
and current density at the valley is indicated by Vv and Jv , respectively. The difference
between the peak voltage Vp and the voltage Vv at the valley is called ∆VN DC , and the
ratio between the peak to valley currents (Jp/Jv) is called PVCR. The current rises again
due to non-resonant processes, case (iv) in figure 2.2, until the Fermi level meets the next
energy state given by eqn. 2.2.
The exact mechanism of resonant tunneling depends upon the width of the barrier Lb.
For example, resonant tunneling may occur by coherent (Fabry-Perot) mechanism, which
preserves the phase of the electron wavefunction or by incoherent (sequential) mechanism,
which destroys the phase coherence of electrons [88]. The latter mechanism dominates
RTD transport in the case where Lb is greater than 7 nm. Since the barrier width for
RTDs used in this work is Lb ≤ 7 nm, we will assume coherent tunneling of electrons.
Theoretical description of RTD transport is detailed in [83] and the interested reader is
directed towards the references listed within [83].
2.3 Confinement in a single nanopillar RTD
In the previous section, we discussed quantization of energy states in a double barrier RTD
due to confinement of electrons in the longitudinal (growth) direction. When a lateral
potential is imposed upon the two-dimensional electrons, the energy levels in the well split
into multiple energy levels. This is shown in figure 2.3(a), in a schematic diagram, where
the energy states denoted as E0, E0, E1 in figure 2.1 split into multiple energy states, with
a difference ∆E. The splitting of the energy levels, ∆E depends upon the radial potential
30
2.4 Reed’s simplistic model for a zero-dimensional RTD
V(r) that is imposed upon the electrons.
Figure 2.3: (a) Schematic diagram, showing splitting of energy levels due to lateral con-
finement (b) Current voltage characteristics of the first zero-dimensional RTD, as a func-
tion of temperature. The arrows indicate the voltage positions corresponding to lateral
quantization. Reproduced from [89].
In the year 1988, Reed et. al were the first to demonstrate lateral confinement in RTDs
[90]. Using a top-down nanofabrication approach to impose a lithograhically imposed
potential, Reed et. al showed that discretization of energy states in laterally confined
RTD can be observed by measuring the transport characteristics of such structures, see
figure 2.3. They were the first to correlate lateral quantization with the fine structure
observed at low temperatures, appearing as multiple NDR peaks in the current-voltage
spectrum.
We will now discuss the simplistic model that Reed et. al employed to compute the
energy splitting ∆E, in laterally confined RTDs.
2.4 Reed’s simplistic model for a zero-dimensional RTD
Figure 2.4 shows the schematic diagram of the longitudinal and lateral potential in a zero-
dimensional RTD. Lateral confinement is imposed by etching a nanopillar of physical radius
Rp. The epitaxial composition of the double barrier RTD is given in figure 2.4(a).
31
2. NANOPILLARS IN RTDS AND QCLS: THEORY
Figure 2.4: (a) Epitaxial structure of the double barrier RTD used by Reed et. al (b)
Schematic description of the conduction band diagram plotted in the longitudinal direc-
tion. The schematic shows energy splitting of energy states due to lateral confinement. (c)
Schematic description of the parabolic potential, in the radial direction, which arises due
to surface depletion. Adapted from [90].
32
2.4 Reed’s simplistic model for a zero-dimensional RTD
The density of interface states at the exposed surface of the nanopillar causes the con-
duction band to bend upwards with respect to the Fermi level, causing a depletion of
carriers over a length Rd , in order to maintain charge neutrality in the complete structure.
The electrostatic radius is then described as Re= Rp-Rd . The bending of conduction band
leads to the formation of a parabolic potential in the radial direction.
Reed et al. [89] described this parabolic potential as:
Φ(r) = ΦT (1− (Rp − r)/Rd)2 (2.5)
where ΦT is the height of the potential determined by Fermi pinning, at r = Rp. The
above model is equivalent to a harmonic potential of the form V(r) = m∗eω2r2/2, with the
boundary condition that V(Rp) = m∗eω2R2p/2 = ΦT . Solving the Schro¨dinger equation in
the radial direction, the energy level splitting is evaluated as
∆E = ħhω=
ħh
Rp
È
2ΦT
m∗e
(2.6)
where m∗e is the effective mass of electron and ΦT is assumed to be 0.7 eV for GaAs.
The model proposed by Reed et al. [89] suffers from significant simplifications. Firstly,
Reed et. al’s model does not include intersubband mixing effects, that occurs if the lateral
potential is not harmonic. Therefore, any slight deviation from cylindrical geometry is
likely to have an influence on the energy splitting ∆E. Secondly, Reed et. al did not
calculate the three-dimensional potential across the RTD, which arises due to differences
in the doping densities within each epitaxial layer. This problem has been addressed in
later work, for example see [83, 91, 92]. Differences in doping densities can also lead to
mode mixing. For a bulk material, the depletion width Rd is inversely proportional to
p
Nd
[93]. This implies that the undoped well will have a greater Rd , as compared to the doped
contact [94]. In three-dimensions, this leads to the formation of an hour-glass potential
with a neck in the undoped well. Calculation of energy states for this potential requires
a scattering matrix approach, shown by Mizuta et. al in [83]. The third simplification
lies in the estimation of the potential ΦT at r = Rp. The potential φT depends upon the
doping of the material and the density of interface states, which exist at the sidewalls of
the nanopillar. However, this has been kept constant at 0.7 eV in the above calculation.
In 2012, Chia et. al calculated the lateral potential for a homogeneous GaAs nanopillar,
taking this dependence into account [95]. However, so far, these calculations have not
been extended to understand electron transport in laterally confined RTDs. Therefore,
in the next subsection, using Chia et. al’s model for homogeneous GaAs nanopillars, we
33
2. NANOPILLARS IN RTDS AND QCLS: THEORY
attempt to estimate the lateral potential φ(r) taking into account (i) the doping of each
layer, and (ii) the density of interface states, in a nanopillar RTD.
2.5 Effect of surface states
Figure 2.5 shows the schematic band diagram of an n-doped homogeneous GaAs nanopillar
in the radial direction proposed by Chia et. al, which suggests that parabolic potential φT
= 0.7 eV, is not a constant but dependent upon many interdependent parameters.
2.5.1 Interface charge density
From figure 2.5, we note that the bands bend upwards to maintain charge neutrality, as
an opposite and an equal space charge Qsc builds up to compensate for the interface states
Qi t (per unit area), which exists at the surface. It is this space charge Qsc that gives rise to
depletion of carriers inside the nanopillar. The interface charge Qi t can be written in terms
of the charge neutrality level ECN L , the Fermi enegy EF and the density of interface states
(traps per unit area per eV), Di t as:
Q i t =−e
∫ EF
ECN L
Di tdE (2.7)
Before we delve into the origin of the surface states, let us examine eqn. 2.7 qualita-
tively. Q i t depends upon charge neutrality level ECN L which depends upon the doping of
the material and the processing conditions of the nanopillar. The Fermi level EF remains
constant as a function of nanopillar radius and depends only on the doping of the mate-
rial. The density of interface states Di t may depend upon the doping. Out of these three
parameters, only EF can be estimated accurately.
2.5.2 Charge neutrality level
Although Chia et. al evaluate the parabloic potential in terms of ECN L and Di t , they assume
ECN L to be 0.53 eV, referred from the top of the valence band [95]. From our survey of
literature, we feel that this common assumption may not be accurate. We find that ECN L
depends upon material doping, processing conditions and measurement method. In some
cases, the error associated with the measurement of ECN L is nearly 60 %.
The origin of the interface states can be traced to the Ga-type or As-type dangling
bonds in a UHV cleaved GaAs (110) surface. These dangling bonds have energies lying in
34
2.5 Effect of surface states
Ec	
  
EF	
  
Ei	
  
Ev	
  
ECNL	
  
Φo	
  
Φs	
  
Rp	
  
Rd	
  
Re	
  
Neutral,	
  empty	
  
acceptors	
  
Nega9vely-­‐charged	
  
Filled	
  acceptors	
  
Neutral,	
  full	
  
donors	
  
+	
   +	
  
+	
  
+	
  
r	
  
Qsc	
  
Qit	
  Δ	

ECNL	
  	
  Charge	
  neutrality	
  level	
  
	
  
Ei	
  	
  	
  	
  	
  	
  Intrinsic	
  level	
  
	
  
EF	
  	
  	
  	
  	
  Fermi	
  level	
  
	
  
Φo	
  	
  	
  	
  	
  Difference	
  between	
  Fermi	
  level	
  	
   	
  
	
  and	
  intrinsic	
  level	
  at	
  at	
  r	
  =	
  0	
  
	
  
Δ  	
  	
  	
  Poten9al	
  difference	
  charge	
  	
  	
   	
  
	
  neutrality	
  level	
  ECNL	
  and	
  the	
  	
  
	
  intrinsic	
  level	
  Ei	
  ,	
  at	
  r	
  =	
  Rp	
  
Φs	
  	
  	
  	
  Surface	
  poten9al,	
  also	
  the	
   	
  
	
  difference	
  between	
  Fermi	
  level	
  	
  
	
  and	
  intrinsic	
  level	
  at	
  r	
  =	
  Rp	
  
Figure 2.5: Schematic diagram of conduction and valence band of an n-doped GaAs
nanopillar, in the radial direction. The diagram shows band bending due to interface
states. Adapted from [95].
between the band gap of the material [96]. The charge neutralitly level ECN L separates
the donor As states from the acceptor Ga states. In an n-doped GaAs, the Fermi level EF is
pinned close to ECN L such that the bands bend upwards to compensate for the negatively
charged occupied acceptor states with ECN L close to midgap. In p-doped GaAs the bands
bend downwards to compensate for the positively charged empty donor states with ECN L
in the lower half of the band gap. The position of the charge neutrality level, ECN L (and
consequently Qi t) is also effected by surface processing such as etching, ion-bombardment
and exposure to gases like oxygen and atomic hydrogen, varying from 0.65 to 0.8 eV for
n-doped GaAs and 0.45 to 0.55 eV for p-doped GaAs [96].
Reported values of ECN L also vary depending upon the measurement method. For
example, measurements based on metal-GaAs studies report ECN L to be equal to 0.53 eV,
with an error of ± 0.33 eV [93, 97]. However, measurements based on photoelectric
35
2. NANOPILLARS IN RTDS AND QCLS: THEORY
studies suggest an ECN L of 0.76 eV for n-doped GaAs. According to the authors, this value
may further depend upon the crystallographic plane [97].
Since we are interested in the depletion width of the undoped GaAs well to understand
lateral confinement in nanopillar RTDs, we would like to point out that it is difficult to
rely completely on ECN L evaluated from studies on n-doped or p-doped GaAs. However,
independent of this assumption, Chia’s model provides an analytical understanding of sur-
face depletion in nanopillars as a function of ECN L and Di t . Therefore, in the following
discusssion, we use the same value of ECN L as the authors, i.e. ECN L = 0.53 eV, referred
from the top of the valence band.
2.5.3 Chia et. al’s model for homogeneous GaAs nanopillar
The potential distribution in a GaAs nanopillar can be described by writing Poisson’s equa-
tion in cylindrical coordinates. For a long homogeneous nanopillar, i.e. neglecting any
confinement in the growth direction, Poisson’s equation can be simplified to an ordinary
differential equation in r, as follows:
∂ 2φ
∂ r2
+
1
r
∂ φ
∂ r
=−ρ
ε
(2.8)
Here r is the radial distance, ρ is the density of charge carriers in bulk GaAs and ε is
the permittivity of GaAs. The solution to the differential eqn. 2.8 is given by
Φ(r) =− ρ
4ε
r2+ C1 ln(r) + C2 (2.9)
where C1 and C2 are constants. The constants can be determined by imposing appro-
priate Neumann and Dirichlet boundary conditions. From figure 2.5, we see that in the
case of a partially depleted nanopillar, Rd < Rp, the following boundary conditions hold.
Φ(Re) = Φo (Rd < Rp) (2.10)
dΦ(r)
dr r=Re
= 0 (2.11)
where eqn. 2.11 merely states that the radial component of the electric field is zero in
the steady state. We note that the above boundary conditions are also applicable to the
case of undoped GaAs, where EF approaches midgap. In the case of undoped GaAs, φo in
eqn. 2.10 approaches 0.
On substituting the boundary conditions (eqns. 2.10- 2.11) in to eqn. 2.9, the following
36
2.5 Effect of surface states
solution is obtained for the radial potential :
Φ(r) =
ρR2e
2ε
‚
− r
2
2R2e
+ ln

r
Re

+
1
2
Œ
+Φo (2.12)
Eqn. 2.12 dictates that Φ(r)r=Rp = Φs, where φs is the surface potential and is equal to
ρR2e
2ε

− R2p
2R2e
+ ln

Rp
Re

+ 1
2

+Φo.
Under the assumption of complete ionization in the space charge region, the charge
density ρ = eNd where, Nd is the number of dopants per unit volume. Therefore, the
radial potential given by eqn. 2.12 can be rewritten in terms of doping density as:
Φ(r) = Φo , 0< r < Re (2.13)
Φ(r) =
eNdR
2
e
2ε
‚
− r
2
2R2e
+ ln

r
Re

+
1
2
Œ
+Φo , Re < r < Rp (2.14)
The position of the Fermi level EF referred to the intrinsic level Ei at the center of the
nanopillar is determined from the Boltzmann approximation as
Nc exp
−(Ec − EF )
kT

= Nc exp
‚−(Eg/2−Φo)
kT
Œ
= Nd (2.15)
Here, Ec is the conduction band and Nc = 2
€
m∗ekT/2piħh2
Š3/2
is the effective density of
states in the conduction band. Nc = 4.7 x 1017 cm−3 in GaAs 1. Given the doping density
Nd , we can estimate the potential φo at the center of the nanopillar.
Finally, the depletion width Rd and hence, the electrostatic radius Re are determined
by invoking the charge neutrality condition given below, where the sum of all bulk and
surface charges is zero.
pi(R2p − R2e)eNd + 2piRpQ i t = 0 (2.16)
where Q i t is determined by eqn. 2.7. From figure 2.5, we note that Q i t can be expressed
in terms of the surface potential φs and ∆, which is the difference between Ei and ECN L at
the surface of the nanopillar. Therefore Q i t can be written as:
Q i t =−e
∫ EF
ECN L
Di tdE =−e
∫ Ei+φs
Ei−∆
Di t dE =−eDi t(∆+φs) (2.17)
The surface potential φs can be determined by substituting eqn. 2.17 into eqn. 2.16.
1From the online semiconductor database, url: http://www.ioffe.ru/SVA/NSM/Semicond/GaAs/bandstr.html
37
2. NANOPILLARS IN RTDS AND QCLS: THEORY
This is given below.
φs =
(R2p − R2e)eNd
2RpeDi t
−∆ (2.18)
The surface potential φs can be equated to eqn 2.12 at r = Rp. Therefore, the
eqns. 2.15, 2.14 and 2.16 can be written as:
(R2p − R2e)eNd
2RpeDi t
−∆= eNdR
2
e
2ε
 
− R
2
p
2R2e
+ ln
Rp
Re

+
1
2
!
+Φo (2.19)
Eqn. 2.19 is a transcendental equation. Given Nd , Di t and ECN L , eqn. 2.19 can be
solved to estimate Re. Before proceeding on to the results, let us reiterate the key features
of Chia et. al’s model.
The first point of significance is that the electrostatic radius Re and the surface potential
φs are dependent upon each other. Knowledge of φs determines Re according to eqn. 2.17.
However, φs depends upon Re as
ρR2e
2ε

− R2p
2R2e
+ ln

Rp
Re

+ 1
2

+Φo. Solution of eqn. 2.12,
and therefore φs and Re requires us to invoke the charge neutrality condition (eqn. 2.16).
Note, that the charge neutrailty condition in bulk is different from charge neutrality
condition in the nanopillar. Text book calculations of depletion width are typically based
on p-n junction models. In the case of a one-sided abrupt semiconductor junction (i.e
p+n or n+p ), the depletion width is equal to
q
2εφT
eNd
, where φT is the amount of band
bending at the surface of the junction [93]. Previous studies on lateral confinement in
RTDs inaccurately translate this depletion width, derived for bulk Schottky junctions, to
surface depletion in nanopillars, with Rd = Rp−Re =
q
2ε(φo−φs)
eNd
, where φo−φs = 0.7 eV
[91, 94]. In a finite sized homogeneous nanopillar, the correct charge neutrality condition
is calculated by balancing the interface charge around the curved surface of the nanopillar,
with the space charge inside the volume of the nanopillar. This is the second point of
importance, and is described by eqn. 2.16.
The third salient feature of Chia et. al’s model is that the electrostatic radius Re depends
upon the doping density Nd , the density of interface states Di t and ECN L . We know from
section 2.5.2 that ECN L may depend upon Nd as well as other material and measurement
parameters. Furthermore, Di t , which is typically quoted as 1.25 ± 1 x 1014 cm−2eV−1 in
GaAs may not be a constant or uniformly distributed in the forbidden gap [93, 97, 98].
In the calculations below, we neglect any functional or phenomenological dependence of
ECN L and Di t on other parameters.
38
2.6 Results: extension of Chia’s model to double barrier RTD
2.6 Results: extension of Chia’s model to double barrier RTD
Figure 2.6(a) shows the schematic diagram of a double barrrier RTD. The n+ doped GaAs
contact, GaAs spacer, AlAs (or, AlxGa1−xAs) barrier and undoped GaAs well are indicated
in the diagram. For surface depletion calculations, we ignore the thin barrier layers, typ-
ically ≤ 7 nm. The structure and doping concentrations listed in figure 2.6(a) are repre-
sentative of real RTD structures used in this thesis, see Chapter 3. In the results discussed
below, we neglect the depletion of carriers at the interface of each epitaxial layer, i.e.
n+ doped GaAs contact, GaAs spacer and undoped GaAs well are assumed to be non-
interacting.
In figure 2.6(b-e), we plot the electrostatic radius Re, estimated from eqn. 2.19, as a
function of density of interface states Di t for a homogeneous GaAs nanopillar of a constant
Rp. The doping Nd is varied according to the epitaxial layer considered in the calculation.
For example, the doping Nd of the homogeneous nanopillar in figure 2.6(b), is equal to the
intrinsic carrier concentration in an undoped quantum well. Therefore, we use Nd = Ni =
2 x 106 cm−2, which is the intrinsic carrier concentration in GaAs1. Figure 2.6(c) shows
the variation of Re as a function of Di t , in a homogeneous GaAs nanopillar with a doping
concentration Nd equal to 1 x 10
16 cm−2, which is the typical doping concentration of a
GaAs spacer. Figure 2.6(d-e) shows the electrostatic radius Re as a function of Di t , in a
homogeneous GaAs nanopillar, with doping concentration Nd equal to 5 x 10
16 cm−2 and
8 x 1016 cm−2, respectively, which is an order of magnitude less than the heavily n+ doped
GaAs contact. The reason for these values of Nd is explained in section 2.6.3.
2.6.1 Undoped GaAs Well
Let us first consider the behavior of Re vs Di t , in the undoped well. We note from fig-
ure 2.6(a) that there exists a critical density of states D∗i t , for each pillar size, beyond
which the nanopillar is fully depleted, or pinched off. This density depends upon the phys-
ical radius of the nanopillar, i.e. D∗200i t for a 200 nm radius nanopillar is higher than the
D∗50i t for a 50 nm nanopillar. This is a consequence of the charge neutrality condition.
Therefore, while a high Q i t may cause complete depletion of charge carriers in a 50 nm
pillar, it will only cause partial depletion in a 200 nm pillar. From the calculations at Nd =
Ni , we infer that the electrostatic radius Re falls rapidly with Di t for all sizes of nanopillars,
i.e. Rp = 50 nm to Rp = 200 nm as a function of Di t , as it approaches the critical density,
which separates the partially depleted regime from the fully depleted regime.
1From the online semiconductor database, url: http://www.ioffe.ru/SVA/NSM/Semicond/GaAs/bandstr.html
39
2. NANOPILLARS IN RTDS AND QCLS: THEORY
Figure 2.6: (a) Schematic diagram of a double barrier RTD, showing the n+ doped GaAs
contacts, GaAs spacer, AlAs (or, AlxGa1−xAs) barrier and undoped GaAs well. (b) Elec-
trostatic radius Re as a function of interface states Di t , in the undoped GaAs well. The
doping density is assumed to be equal to intrinsic carrier concentration, which is Nd = 2
x 106 cm−2 (c) Electrostatic radius Re as a function of interface states Di t , in the GaAs
spacer doped with a carrier concentration Nd = 1 x 1016 cm−2 (d) Electrostatic radius Re
as a function of interface states Di t , in n
+ GaAs doped with a carrier concentration Nd =
5 x 1016 cm−2 (e) Electrostatic radius Re as a function of interface states Di t , in n+ GaAs
doped with a carrier concentration Nd = 8 x 1016 cm−2.
40
2.6 Results: extension of Chia’s model to double barrier RTD
Figure 2.7: Schematic diagram of surface depletion in undoped GaAs nanopillars. Four
different cases of Rp are discussed, i.e Rp 50 nm, 100 nm, 150 nm and 200 nm. Top panel
shows partial depletion in undoped GaAs nanopillars, in all four cases of Rp. The density of
interface states is constant (2 x 1013 cm−2eV−1) and less than the critical density of states
for all the nanopillars. Bottom panel shows partial depletion in undoped GaAs nanopillars
with Rp ≤ 100 nm, and complete depletion in the smallest nanopillar with Rp = 50 nm. The
density of interface states is constant in all four cases and is equal to 1 x 1014 cm−2eV−1.
This density is greater than D∗50i t but less than D∗100i t , D∗150i t and D∗200i t .
41
2. NANOPILLARS IN RTDS AND QCLS: THEORY
Figure 2.7 illustrates this idea in a schematic diagram. The top panel in figure 2.7
shows partial depletion in undoped GaAs nanopillars with size Rp varying from 50 nm to
200 nm. Note, that the density of interface states is constant (2 x 1013 cm−2eV−1) for all
four nanopillars. Theoretically, Q i t depends upon Di t and the surface potential φs, which
is ∝ R2e , see eqn. 2.17. For simplicity in understanding the result, we have used Q i t ∝ Di t
and R2p in figure 2.7. At low doping densities, dopants within the nanopillar volume, ionize
to balance the interface charge. At a density of states equal to 2 x 1013 cm−2eV−1, the 200
nm pillar has an electrostatic radius Re = 192 nm, the 150 nm pillar has an electrostatic
radius Re = 141 nm, the 100 nm pillar has an electrostatic radius Re = 91 nm and the 50
nm pillar has an electrostatic radius Re = 42 nm.
The bottom panel in figure 2.7 shows the smallest nanopillar with Rp = 50 nm pinching
off when Di t is increased to 1 x 10
14 cm−2eV−1, which is greater than D∗50i t . At this den-
sity of states, the larger nanopillars are still partially depleted, but their depletion width
increases to accommodate the higher amount of interface charge. The 200 nm pillar has
an electrostatic radius Re = 159 nm, the 150 nm pillar has an electrostatic radius Re = 109
nm, the 100 nm pillar has an electrostatic radius Re = 54 nm and the 50 nm pillar is fully
depeleted (Re = 0).
2.6.2 Doped GaAs Spacer
Compared to the undoped GaAs well, the doping density in a GaAs spacer is much higher,
typically Nd = 1 x 1016 cm−2. At this density, the results plotted in figure 2.6(c) show
counter-intutive behavior. Similar to the results evaluated for the undoped well, the elec-
trostatic radius Rp in a GaAs spacer falls rapidly with Di t , as it approaches the critical
density. However, at this doping density, the critical density D∗i t for a doped 50 nm pillar
is lower than the D∗i t for an undoped 50 nm pillar, see figure 2.6(b). This is true for all
sizes of nanopillars, between Rp = 50 nm to Rp = 200 nm. We believe that this behavior
occurs because Re does not depend upon Nd in a straightforward way for a finite-sized,
cylinder. This is unlike the case of a planar slab (bulk surface) where the depletion width
is ∝ 1/pNd . The dependence of Re on Nd , at different values of Di t is shown in figure 2.9
and will be discussed in section 2.6.4.
2.6.3 Doped GaAs contact
At Nd = 5 x 1016 cm−2 and 8 x 1016 cm−2, see figure 2.6(d-e), the behavior of Re as a
function of Di t is different from what we observe in figure 2.6(b-c). At Nd = 5 x 1016
cm−2, see figure 2.6(d), Re is nearly equal to Rp, for a 200 nm pillar, decreasing to ≈
42
2.6 Results: extension of Chia’s model to double barrier RTD
180 nm as Di t approaches 1 x 10
15 cm−2eV−1. As Nd is increased further, upto 8 x 1016
cm−2, see figure 2.6(e), the depletion width for a 200 nm pillar is negligible. The interface
charge Q i t due to dangling bonds outside the nanopillar sidewall is balanced by an equal
and opposite space charge Qsc due to a high density of ionized donors inside the sidewall.
Therefore, at these high doping densities, the 200 nm pillar never pinches off for the entire
range of Di t plotted in figure 2.6. In the case of a 50 nm and a 100 nm pillar, the surface
area is small when compared to the 200 nm nanopillar. Hence, to maintain the charge
neutrality condition, carriers from inside the nanopillar volume provide the charge needed
to compensate the Q i t . This can cause a 50 nm (100 nm) pillar to deplete completely at
the critical density of interface states given by D∗50i t (D∗100i t ). Note, the critical density of
states D∗i t required to pinch off a 50 nm or a 100 nm pillar is higher when the doping
density Nd is 8 x 10
16 cm−2, as compared to 5 x 1016 cm−2. This agrees with our intuitive
understanding of depletion, i.e. it requires a higher amount of interface charge Q i t at the
surface to completely deplete a volume that contains a higher density of ionized donors
forming the total space charge. Qsc . The electrostatic distribution of space charge as a
function of Di t in a 150 nm pillar, lies in between the two responses, i.e. (i) flat Re in a
200 nm pillar, and (ii) rapidly varying Re in a 50 nm (100) nm pillar. Even though the
electrostatic radius Re decreases with increase in Di t , the roll-off does not cause the 150
nm pillar to pinch off completely.
Figure 2.8 shows the schematic diagram illustrating surface depletion in GaAs nanopil-
lars, doped at Nd = 5 x 1016 cm−2 at a Di t equal to 2 x 1013 cm−2eV−1 (greater than D∗50i t ).
At this density of states, the 200 nm pillar shows negligible depletion (Re = 198 nm), the
150 nm and 100 nm pillars show partial depletion (Re = 137 nm and 77 nm, respectively)
and the 50 nm pillar is depleted completely (Re = 0).
At Nd = 2 x 1018 cm−2, we find that the doping density at the surface of the nanopillar,
in all four cases, RP = 50 nm, 100 nm, 150 nm and 200 nm, is sufficient to balance the
interface charge. Therefore, GaAs contacts doped to a density of Nd = 2 x 1018 cm−2 have
Re = Rp, independent of the density of interface states. In the next section, we summarize
the dependence of Re on the doping density Nd .
43
2. NANOPILLARS IN RTDS AND QCLS: THEORY
Figure 2.8: Schematic diagram of surface depletion in doped GaAs nanopillar at a constant
density of interface states, labeled as DCit , showing neglegible depletion in pillars with Rp =
200 nm, partial depletion in 150 nm and 100 nm pillars and pinch off in 50 nm pillar. The
schematic is shown for the case where Nd = 5 x 1016 cm−2 and Di t = 2 x 1013 cm−2eV−1,
which is greater than D∗50i t
44
2.6 Results: extension of Chia’s model to double barrier RTD
2.6.4 Depletion as a function of doping density
In figure 2.9, we plot the electrostatic radius Re as a function of doping density Nd for all
four cases of Rp discussed in the previous section. In GaAs Di t is typically quoted as 1.25
± 1 x 1014 cm−2eV−1 [93, 97, 98]. To remain consistent with realistic estimates of Di t , we
discuss the behavior of Re vs. Nd as it varies from 1 x 10
12 cm−2eV−1 to 1 x 1014 cm−2eV−1
on a log scale.
We note three distinctive regimes in all four sets of calculations. The first regime,
occurs at low doping densities Nd , where for certain values of Di t , the extent of depletion
increases with the number of ionized donors available in the volume. This occurs for all Nd
to the left of the dashed red line in figures 2.9(a-d). Decrease in Re with increase in Nd is
counter intuitive to the textbook understanding of surface depletion due to interface states.
As discussed previously, this is because the finite size of the nanopillar and the density of
interface states was not accounted in the textbook explanation surface depletion[93]. The
solution to Re as a function of Nd is determined by the transcendental equation, given by
eqn. 2.19. The second regime, occurs at values of Nd in between the dashed red line and
the continuous red line in figure 2.9. In this regime, the electrostatic radius Re increases
with increase in Nd . In the third regime, indicated by the right side of the continuous red
line, no surface depletion occurs. In this regime, the dopant density is sufficiently high at
the surface to balance any interface charge on the sidewall.
The transition between the first regime and the second regime occurs at a point de-
termined by the physical radius Rp and the density of interface states Di t . Similarly, the
transition between the second regime and the third regime depends upon Rp and Di t . For
certain values of Di t and Nd , the solution to Re approaches zero, i.e. the nanopillar is com-
pletely pinched off. For example, in the case of a 200 nm nanopillar, no solutions exist for
13.3 cm−3 ≤ log10(Nd)≤ 15.5 cm−3 at Di t = 1 x 10 14 cm−2eV, see figure 2.9(d). We know
from eqn. 2.12 that in the limit of Re → 0, the surface potential φ(r) reaches a singularity.
Therefore, it is difficult to identify a clear transition between the three regimes discussed
above at these values of Di t and Nd .
In the next section, we discuss the implication of these results on the observation of
fine NDR features in a laterally confined RTD.
2.6.5 Implication of surface states on NDR peaks
In 1988, the same year as Reed et. al’s experimental observation of NDR peaks in laterally
confined RTDs (see section 2.3), Chou et. al argued that if the lateral confinement (Re)
is the same throughout the structure (i.e. contacts, barriers, spacers and well), then one
45
2. NANOPILLARS IN RTDS AND QCLS: THEORY
6 8 10 12 14 16 1810
20
30
40
50
6 8 10 12 14 16 180
25
50
75
100
6 8 10 12 14 16 180
50
100
150
6 8 10 12 14 16 180
50
100
150
200
log10(Nd)	
  in	
  cm-­‐2	
  
El
ec
tr
os
ta
7
c	
  
ra
di
us
,	
  R
e	
  
(n
m
)	
  
10	
  
2 	
  
30	
  
40	
  
5 	
  
	
   	
   0	
   12	
   14	
   	
   18	
  
El
ec
tr
os
ta
7
c	
  
ra
di
us
,	
  R
e	
  
(n
m
)	
  
El
ec
tr
os
ta
7
c	
  
ra
di
us
,	
  R
e	
  
(n
m
)	
  
log10(Nd)	
  in	
  cm-­‐2	
  
6	
   8	
   10	
   	
   14	
   16	
   18	
  
0	
  
	
  
	
  
100	
  
	
  
0	
  
50	
  
	
  
150	
  
	
   	
   	
   12	
   	
   	
   18	
  
log10(Nd)	
  in	
  cm-­‐2	
  
0	
  
50	
  
100	
  
	
  
	
  
El
ec
tr
os
ta
7
c	
  
ra
di
us
,	
  R
e	
  
(n
m
)	
  
log10(Nd)	
  in	
  cm-­‐2	
  
6	
   	
   	
   2	
   4	
   	
   	
  
(a)	
   (b)	
  
(c)	
   (d)	
  
Dit	
  =	
  1x	
  1012	
  
D
it 	
  =	
  1x	
  1013	
  
D
it
	
  =
	
  2
	
  x
	
  1
0
13
	
  
D
it
	
  =
	
  3
	
  x
	
  1
0
13
	
  
D
it 	
  =	
  4	
  x	
  10 13	
  
D
it 	
  =	
  5	
  x	
  10
13	
  
Dit	
  =	
  1x	
  1012	
  
Dit	
  =	
  1x	
  1013	
  
Dit	
  =	
  2	
  x	
  1013	
  
D
it
	
  =
	
  5
	
  x
	
  1
01
3	
  
D i
t	
  =
	
  4
x	
  
10
13
	
  
D
it 	
  =	
  6	
  x	
  10
13	
  
D
it 	
  =	
  7	
  x	
  10 13	
  
D
it 	
  =	
  8	
  x	
  10
13	
  
D
it 	
  =	
  9	
  x	
  10
13	
  
D
it 	
  =	
  1	
  x	
  10
14	
  
Dit	
  =	
  1	
  x	
  1012	
  
Dit	
  =	
  1	
  x	
  1013	
  
Dit	
  =	
  2	
  x	
  1013	
  
D
it 	
  =	
  3	
  x	
  1013	
  Dit 	
  =	
  4	
  x	
  1013	
  
D i
t	
  =
	
  6
	
  x
	
  1
0
13
	
  
D
it 	
  =	
  7	
  x	
  10 13	
  
D
it 	
  =	
  8	
  x	
  10
13	
   D
it
	
  =
	
  7
	
  x
	
  1
0
13
	
  
Dit	
  =	
  1	
  x	
  1012	
  
D i
t	
  =
	
  9
	
  	
  x
	
  1
0
13
	
  
D
it 	
  =	
  6	
  x	
  10 13	
  
D
it 	
  =	
  5	
  x	
  10 13	
  
Dit	
  =	
  4	
  	
  x	
  1013	
  
Dit	
  =	
  3	
  	
  x	
  1013	
  
Dit	
  =	
  2	
  	
  x	
  1013	
  
Dit	
  =	
  1	
  	
  x	
  1013	
  
Physical	
  radius,	
  Rp	
  =	
  50	
  nm	
   Physical	
  radius,	
  Rp	
  =	
  100	
  nm	
  
Physical	
  radius,	
  Rp	
  =	
  150	
  nm	
   Physical	
  radius,	
  Rp	
  =	
  200	
  nm	
  
N
o	
  
Pi
nc
h	
  
off
	
  
N
o	
  
Pi
nc
h	
  
off
	
  
N
o	
  
Pi
nc
h	
  
off
	
  
D
it
	
  =
	
  1
	
  x
	
  1
01
4	
  
	
  Pinch	
  off	
  	
  Pinch	
  off	
  
	
  Pinch	
  off	
  	
  Pinch	
  off	
  
Figure 2.9: Electrostatic radius Re as a function of doping density, Nd plotted for four
different values of physical radius Rp (a) 50 nm (b) 100 nm (c) 150 nm (d) 200 nm. The
density of interface states varies from 1 x 1012 cm−2eV to 1 x 1014 cm−2eV on a logarithmic
scale.
46
2.6 Results: extension of Chia’s model to double barrier RTD
applied bias will align all of the subband states in the well with the corresponding subbands
in the contact [99]. This would then destroy any fine structure (NDR peaks) associated
with the splitting of energy levels in the current-voltage spectra. Following this work,
Bryant et. al suggested that fine structure may be observed if the lateral confinement is
different in each region of the RTD [91].
Extending this analysis to include the effect of surface states, we find that depending
upon the physical radius of the pillar Rp, the density of interface states in each region of
the nanopillar Di t and the doping density Nd , the lateral confinement (Re) may be same for
the undoped well and the doped GaAs contact. For example, in a 50 nm nanopillar RTD,
if the density of interface states = 6 x 1013 cm−2eV on the sidewalls of an undoped well,
then the electrostatic radius Re = 23 nm. See point A in figure 2.10. This undoped well
may be contacted in the growth direction with a doped GaAs layer, which has a density
of interface states = 2 x 1013 cm−2eV on the sidewalls. Under this scenario, the lateral
confinement in the contact equals 23 nm, which is the same as the undoped well, see point
B in figure 2.10. If these conditions were to exist, no fine feature will be observed even
when quantization due lateral confinement occurs [91, 99].
Figure 2.10: Electrostatic radius Re as a function of doping density, Nd plotted for Rp =
50 nm. The density of interface states varies from 1 x 1012 cm−2eV to 1 x 1014 cm−2eV
on a logarithmic scale. For certain values of Di t , the electrostatic radius may be same for
undoped GaAs well, and doped GaAs contacts. The points of intersection A and B highlight
these combinations.
In the next section we discuss the potential along the growth direction in laterally
confined RTD.
47
2. NANOPILLARS IN RTDS AND QCLS: THEORY
2.7 Three dimensional potential in laterally confined RTDs
Figure 2.11 shows the schematic diagram of the longitudinal potential in a nanopillar RTD
[100]. Numerical calculations of the longitudinal potential in a laterally confined RTD,
predict a non-uniform hour-glass potential along the growth direction. The neck of this
hour-glass shaped potential exists in the undoped well, where depletion is maximum.
Figure 2.11: Schematic diagram showing hour-glass potential in a nanopillar RTD [100]
We now extend our understanding of surface depletion in homogeneous nanopillars to
heterogeneous nanopillars, where all regions i.e. doped GaAs contacts, doped spacer and
undoped GaAs well are integrated into a single structure. Using the modified expression
for lateral confinement estimated from eqn. 2.19, we calculate the longitudinal potential
of the lithographically defined nanopillar RTD.
Given a constant Di t and Nd , the surface potential φ(r = Rp) can be calculated for
each epitaxial layer using eqns. 2.19 and 2.18. We assume azimuthal symmetry of the
potential, which agrees with realistic devices.
Neglecting surface depletion at the interface of the epitaxial layers, the complete Pois-
son equation, given below:
1
r
∂ ζ(r, z)
∂ r
+
∂ 2ζ(r, z)
∂ r2
+
∂ 2ζ(r, z)
∂ z2
=
eN id
ε
(2.20)
can be solved numerically by applying the Dirichlet condition ζ(r, z)r=Rp for all z ∈
[Li−1, Li] = φo-φi(r)r=Rp , where Li - Li−1 is the thickness of i-th layer and φi(r)r=Rp is
the surface potential calculated using eqn 2.19 at that layer. At the interface between two
48
2.7 Three dimensional potential in laterally confined RTDs
layers, the tangential component of the electric field is zero and the normal component of
the displacement field can be written as:
εi
∂ ζ(r, z)
∂ z
|z→L−i = εi+1
∂ ζ(r, z)
∂ z
|z→L+i (2.21)
We use COMSOL 4.3 to numerically solve eqn. 2.20 for the RTD structure shown in
figure 2.12. We ignore the barrier layers in the simulation, and only consider the GaAs
layers. The three dimensional potential calculated using eqn. 2.20, agrees with the hour-
glass potential shown in figure 2.11 under certain conditions. These conditions exist when
the density of interface states Di t is different for each epitaxial layer, and are such that
Re in the undoped well is less than the Re in the injector. Therefore, if for example, Di t
in the injector is equal to 1 x 1013 cm−2eV and Di t in the undoped well is equal to 5 x
1013 cm−2eV, then the three-dimensional potential ζ(r, z) shown in figure 2.12 is shaped
as an hour-glass. Under this scenario Re = 27 nm in the undoped well, and is equal to
35 nm in the GaAs spacer (Nd = 1 x 1016cm−3). Note that the depletition width of each
epitaxial layer evaluated by solving the complete three-dimensional potential in an RTD
(see figure 2.12) is slightly different from the width evaluated by solving eqn. 2.19. This is
because eqn. 2.19 assumes homogeneity throughout the nanopillar and does not account
for boundary conditions at the interface of each epitaxial layer.
There are other combinations of Di t and Nd , which satisfy the above mentioned con-
dition for an hour-glass potential. These values can be evaluated from figure 2.9(a). The
solution to energy states En, in a laterally confined RTD, may then determined by three-
dimensional scattering matrix theory that takes the non-uniform potential into account.
This approach is beyond the scope of this work, and has been discussed in detail by Mizuta
et. al [83].
While the first device structure to obtain laterally confined RTDs involved lithographic
definition of nanopillars [89, 101], most of the latter techniques used an external gate bias
to control the depletion of charge carriers [102, 103]. This allowed an external control
over the electrostatic radius such that the lateral potential imposed on the quantum well
remained uniform along the z-direction. Figure 2.13 shows the schematic diagram of an
RTD with external gates that control the lateral potential in the quantum dot, with it’s
associated electrical characteristics.
This approach can not be scaled to achieve lateral confinemenet in QCLs, which re-
quires an array of hundred thousand laterally confined pillars connected in parallel to
achieve the gain required for lasing. An immediate consequence of restricting to the top-
down nanopillar approach implies an hour-glass (non-uniform) potential ζ(r, z) because
49
2. NANOPILLARS IN RTDS AND QCLS: THEORY
Figure 2.12: Three-dimensional potential ζ(r, z) in a nanopillar RTD, with Rp = 50 nm.
of surface depletion repeating with each period along the z-direction.
In the next section, we discuss lateral confinement in QCLs and calculate the three-
dimensional potential ζ(r, z) for a laterally confined QCL structure.
2.8 Three dimensional potential in laterally confined QCLs
As discussed above, although, the use of an external gate bias to control the lateral con-
finement is an appealing concept, it is difficult to scale this approach from a single dot in
an RTD to an array of dots in QCLs.
Figure 2.14 shows the epitaxial structure of a typical QCL used in this thesis. Each
period in the quantum cascade comprises of (i) a doped GaAs injector and, (ii) undoped
quantum GaAs wells that form the upper and lower lasing levels. If we assume, Di t in
the injector is equal to 1 x 1013 cm−2eV and Di t in the undoped well is equal to 5 x 1013
cm−2eV, then from figure 2.9(a) we obtain an Re = 27 nm in the undoped well and Re =
35 nm in the doped GaAs injector (Nd = 2 x 1016cm−3).
As before, we use COMSOL 4.3 to numerically solve eqn. 2.20 for the three dimensional
50
2.8 Three dimensional potential in laterally confined QCLs
Figure 2.13: (a) Schematic diagram showing an RTD, with variable lateral potential that
is achieved by external gating (b) Current-voltage characteristics at different gate bias.
Higher gate bias implies lower electrostatic radius. Reproduced from [100, 103].
potential ζ(r, z) in a laterally confined QCL. We ignore the barrier layers in the simulation,
and only consider the GaAs layers. The three-dimensional potential ζ(r, z) is shown in
figure 2.14.
We observe that the curvature of the parabolic potential in the undoped active region
in a laterally confined QCL is weaker than the laterally confined RTD. This is because the
thickness per period of the undoped GaAs is 31.9 nm, which is more than the thickness
of the undoped GaAs well in an RTD (9 nm). As opposed to a single undoped quantum
well RTD, there are multiple undoped quantum wells in a QCL, which form part of a
period. These periods repeat in the growth direction. Therefore, the parabolic potential
also repeats each time with the period.
The complete three dimensional potential ζ(r, z) can now be introduced into the three
dimensional time-independent Schro¨dinger equation, given below:
‚
− ħh
2m∗
‚
1
r
∂
∂ r

r
∂
∂ r

+
1
r2
∂ 2
∂ θ2
+
∂ 2
∂ z2
Œ
+ ζ(r, z)
Œ
Ψ(r,θ , z) = EΨ(r,θ , z) (2.22)
The solution to eqn. 2.22 yields the spectrum of energy states En in a laterally confined
QCL. Electron transport through such zero-dimensional cascade structure can be calculated
using a three dimensional multi-mode scattering matrix theory [104]. This is beyond the
scope of this work.
For simplicity in calculations, we use Reed’s simplistic model for harmonic potential in
51
2. NANOPILLARS IN RTDS AND QCLS: THEORY
Figure 2.14: Three-dimensional potential ζ(r, z) in a nanopillar QCL, with Rp = 50 nm.
The potential is calculated for the epitaxial structure of a QCL, labeled as the 3 THz 4 QW
ETH design in Chapter 4. The thickness of Al0.15Ga0.85As barriers is expressed in bold and
the thickness of the undoped GaAs wells is expressed in regular text.
nanopillars and use it to study the effect of lateral confinement on the bandstructure of a
THz QCL. This is discussed below.
2.9 Effect of lateral confinement on THz QCL bandstructure
In this section, we extend Reed’s simplistic model for lateral confinement in RTDs to under-
stand splitting of energy levels in a THz QCL. We then compare the quantized energy states
of carriers that are lithographically constricted in a THz QCL to a scenario where they are
confined by the application of an external magnetic field. Before we discuss the effect of
lateral confinement on THz QCLs, it is useful to understand the difference between the
bandstructure of a typical THz QCL with a single quantum well RTD.
52
2.9 Effect of lateral confinement on THz QCL bandstructure
2.9.1 Comparison between RTD and THz QCL bandstructure
We know from our understanding of an RTD that the thickness of the wells determine the
energy states in a simple double barrier structure. A THz QCL is far more complex than
an RTD - it comprises of multiple wells within each period. This period is repeated in
the growth direction to obtain gain. The thickness of the wells governs the energy states
(or subbands) within each period of a THz QCL. The energy states involved in radiation,
i.e. the injector, upper lasing level, lower lasing level and extractor are determined by
optimizing various design parameters to achieve maximum population inversion.
Figure 2.15: Schematic bandstructure diagram of (a) RTD and (b) THz QCL.
Figure 2.15 shows the schematic bandstructure diagram of an RTD and a THz QCL.
The ground state Eo and the first excited state E1 in a single quantum well RTD structure is
highlighted in red, see figure 2.15(a). On the right, we show only those energy states that
take part in four-stage lasing mechanism in a typical THz QCL, see figure 2.15(b). These
states are labeled from the bottom of the conduction band, in the order of energy. We call
these states as E1, E2, E3, E4 and E5. In this particular design of THz QCL, the extraction
of the electrons from the lower lasing level occurs through resonant LO phonon scattering.
The radiative transition occurs between the upper lasing E5 and the lower lasing level E4.
We next discuss splitting of energy states in RTDs and THz QCLs due to lateral confine-
ment.
2.9.2 Magnetic field vs. lithographic confinement of carriers
On application of an external magnetic field in the growth direction, the two-dimensional
electron gas gets constricted in the ’xy’ plane to a cyclotron radius. This cyclotron radius
is given by
Æ
ħhc
eB
. Here, c is the speed of light and e is the charge of the electron. Fig-
ure 2.16(a) shows the schematic diagram of an RTD under the influence of an external
53
2. NANOPILLARS IN RTDS AND QCLS: THEORY
magnetic field, B. Each subband |i〉, in the quantum well, splits into a ladder of Landau
states |i, m〉 with energies given by:
Ei,m = Ei + (m+ 1/2)ħhωc , m= 1,2, 3... (2.23)
whereωc = eB/cm∗e is the cyclotron frequency, and m∗e is the effective mass of electron.
Figure 2.16: Schematic diagram of RTD bandstructure under the influence of (a) an exter-
nal magnetic field, B (b) lithographic confinement of carriers to a physical radius Rp.
Application of a magnetic field is analogous to creating an artificial quantum dot. A
real quantum dot is created upon lithographic confinement of electrons in the ’xy’ plane,
which also quantizes the energy states. Figure 2.16(b) shows the schematic diagram of an
RTD when it is laterally constricted to a nanopillar of physical radius Rp. In this case, each
subband |i〉, in the quantum well, splits into a ladder of states |i, m〉 with energies given by
Ei,m = Ei + (m+ 1/2)ħhω, m= 1,2, 3... (2.24)
where ħhω = ħh
Rp
q
2ΦT
m∗e
is the in-plane quantization energy obtained using Reed’s sim-
plistic harmonic potential discussed in section 2.3.
An important difference in the energy states Ei,m of artificial quantum-dots (assisted
54
2.9 Effect of lateral confinement on THz QCL bandstructure
by magnetic field) and lithographically defined quantum-dots comes from the in-plane
quantization energy δE = ħhω which varies proportionally with B and inversely with Rp.
Note that as Rp → 0, Ei,m →∞.
In the case of a simple structure such as an RTD, the number of energy states Ei are
limited. Therefore, the energy spectrum of laterally confined RTDs Ei,m, is relatively easier
to interpret from transport or optical measurements. In the case of a THz QCL, there are
multiple energy states Ei which split upon lateral confinement, see figure 2.17. This lateral
quantization creates additional subbands that influences electron scattering and transport
in a THz QCL. While magnetotransport behavior of THz QCLs is generally understood [82],
it offers us useful insights into transport and spectroscopic behavior of lithographically
confined THz QCLs.
We extend Reed’s harmonic potential to evaluate ladder states in a THz QCL using
eqn. 2.17. Note that this is a very simple picture of the confining potential. In reality,
ħhω would be dictated by the true depletion width in each quantum well described by
figure 2.14.
The same resonance conditions that determine optical selection rules of THz QCLs un-
der a magnetic field, apply to lithographically confined THz QCLs (nanopillars). As is the
case with the application of a B field, we propose that the physical radius Rp would mod-
ulate the lifetime of the laser transition states. Similar to the magneto-photon resonance
conditions, the resonance condition in the case of zero-dimensionality can be written as:
Ei,m− E j,m′ − pħhωLO = 0, (2.25)
Here, i is the upper or lower lasing level, j is the lower subband. In the above equation,
p = 0, m = 0, m′ = 0, 1, 2 etc for elastic scattering and p = 1, m′ = 0, m = 0, 1, 2 etc for
inelastic scattering. The term ħhωLO = 36 meV in eqn. 2.25 is the energy associated with
resonant LO phonon scattering. When the upper lasing level is at resonance, the elastic
and inelastic scattering will cause quenching of any radiation. However, when the lower
lasing level is at resonance, the enhanced relaxation of charge carriers will lead to higher
power and operating temperature.
For the design shown in figure 2.17, the upper lasing level is labeled as E5, and the
lower lasing level is labeled as E4. Therefore, upon lateral confinement, for those radial
dimensions Rp that cause the upper lasing level E5 to be at resonance, no emission will be
observed. This happens when the following condition is met.
55
2. NANOPILLARS IN RTDS AND QCLS: THEORY
Figure 2.17: Schematic diagram of THz QCL bandstructure under the influence of (a) an
external magnetic field (b) lithographic confinement of carriers.
E5,0 = E4,m Elastic scattering (2.26)
E5,m− E4,0 = ħhωLO LO phonon scattering (2.27)
Here, m = 0, 1, 2 and so on. On the other hand, for those radial dimensions Rp that
cause the lower lasing level to be resonance, emission will be observed. In this particular
example there are two lower lasing levels E4 and E3 that are close to each other in energy
(coupled to each other) and the condition for enhanced relaxation is applied to the lower
most level E3.
E3,0 = E2,m Elastic scattering (2.28)
E3,m− E2,0 = ħhωLO LO phonon scattering (2.29)
56
2.10 Results: radiative states in laterally confined THz QCLs
Here, m = 0, 1, 2 and so on. Therefore, in a laterally confined THz QCL, the reso-
nance conditions determine which radial dimensions would lead to enhanced emission.
At these radial dimensions, the radiative states that take part in the energy transition are
determined by the optical selection rules which follow from conservation of energy and
momentum. For simplicity, we do not assume any broadening of states.
2.10 Results: radiative states in laterally confined THz QCLs
Here, we look at two important THz QCLs designs that are used in this thesis, with the
intention of improved temperature performance due to quantization of energy states in
the in-plane direction. The two designs (i) 3 THz 3 QW Paris design, and (ii) 3 THz 4
QW ETH design have been discussed in detail in Chapter 4. In this section, we are only
concerned with the position of the injector, upper lasing level, lower lasing level and the
extractor with respect to each other at the alignment field.
Figure 2.18: Band diagram of 3 THz 3 QW Paris design showing energy states at alignment
field E = 12.1 kV/cm. For epitaxial structure and simulation details, see section 4.7.
Figure 2.18 shows the band diagram of the 3 THz 3 QW Paris design at the alignment
field. Also shown are the energy levels E4, E3, E2 and E1, labeled as quantum well (QW)
states because they arise due to confinement in the growth direction. E4 and E3 form the
upper and lower lasing level, respectively. The lower lasing level E3 is coupled to E2. The
electrons resonantly relax from E2 to the extractor through LO phonon scattering.
Figure 2.19 shows the band diagram of the 3 THz 4 QW ETH design, at the alignment
field. In this design, the QW states E5 and E4 form the upper and lower lasing levels,
respectively. Compared to the 3 THz 3 QW Paris design, there are more number of states
57
2. NANOPILLARS IN RTDS AND QCLS: THEORY
Figure 2.19: Band diagram of 3 THz 4 QW ETH design showing energy states at alignment
field E = 7.6 kV/cm. For epitaxial structure and simulation details, see section 4.9.
that take part in the extraction of carriers from the lower lasing level. This is because this
designs employs a miniband of states E4, E3 and E2, that are resonantly coupled to the
extractor E1 through LO phonon scattering.
2.10.1 Radiative states in laterally confined 3 THz 3 QW Paris design
Figure 2.20 shows ladder states as a function of nanopillar radius Rp for the 3 THz 3 QW
Paris design. The QW states are calculated by solving Poisson-Schro¨dinger equation self-
consistently in the growth direction, as shown above. For clarity, we separate the ladder
states into four groups, i.e. those involved in (a) elastic scattering of electrons in the lower
lasing level E2 with the extractor E1 (b) inelastic scattering of electrons in the lower lasing
level E2 with the extractor E1 (c) elastic scattering of electrons in the upper lasing level E4
with the lower lasing level E3, and finally (d) inelastic scattering of electrons in the upper
lasing level E4 with the lower lasing level E3.
Scattering of electrons in the lower lasing level E2 leads to enhanced relaxation of car-
riers in E3, leading to enhanced population inversion. This happens when the resonance
condition for scattering are met, see figure 2.20 (a-b). At these resonance positions, the
two states E2 and E1 cross each other. The dashed arrows coloured in orange, indicate
the resonance positions that maximise carrier relaxation from E2 to the extractor state E1.
Scattering of electrons can also quench radiation of photons. This occurs when carriers in
the upper lasing level E4 scatter non-radiatively into the lower lasing level E3. These reso-
nance positions are highlighted with dashed black arrows in figure 2.20 (c-d). Table 2.1
58
2.10 Results: radiative states in laterally confined THz QCLs
summaries the resonance positions in the order of increasing Rp.
Nanopillars in 3 THz 3 QW Paris design
Scattering process Energy levels Physical radius, Rp
(nm)
Elastic scattering E2 E2,0 = E1,1 33.71
E2,0 = E1,2 67.21
E2,0 = E1,3 100.3
E2,0 = E1,4 133.6
Inelastic scattering E2 E2,2=E1,0 +3ħhωLO 35.5
E2,1=E1,0 +2ħhωLO 36.29
E2,3=E1,0 +3ħhωLO 53.33
E2,2=E1,0 +2ħhωLO 74.14
E2,3=E1,0 +2ħhωLO 110.8
E2,4=E1,0 +2ħhωLO 147.5
Elastic scattering E4 E4,0 = E3,1 97.14
E4,0 = E3,2 193.5
Inelastic scattering E4 E4,2=E3,0 +2ħhωLO 43.03
E4,1=E3,0 +ħhωLO 54.72
E4,3=E3,0 +2ħhωLO 64.04
E4,4=E3,0 +2ħhωLO 85.05
E4,2=E3,0 +ħhωLO 108.2
E4,3=E3,0 +ħhωLO 165.5
Table 2.1: Summary of resonance positions in laterally confined 3 THz 3 QW Paris design
at a constant electric field E = 12.1 kV/cm. The resonance positions are listed in the order
of increasing nanopillar radius.
Looking at figure 2.20, we propose that only those nanopillars whose radial dimensions
meet the criterion of enhanced relaxation of carriers in the lower lasing level will show
improved temperature performance. For radial dimensions that encourage scattering of
carriers in the upper lasing level, lateral confinement would be detrimental. Therefore,
we identify three windows of Rp, labeled as P, Q and R for nanopillars that meet the
requirements of emission discussed above. The position of the windows in figure 2.20 is
determined by the resonance positions shown in figure 2.20(a) and figure 2.20(b). The
width of the windows is governed by the resonance positions shown in figure 2.20(c) and
figure 2.20(d) that should be avoided as they quench emission.
The first window P is identified by including the resonances E2,0 = E1,1 at Rp = 33.71
nm, E2,2 = E1,0 + 3ħhωLO at 35.5 nm and E2,1 = E1,0 + 2ħhωLO at 36.29 nm. The range
of the first window is limited by the resonance E4,2 = E3,0 +2ħhωLO at 43.03 nm. The
resonance E2,3 = E1,0 + 3ħhωLO at 53.33 nm, which enhances relaxation lies very close to
the resonance E4,1 = E3,0 + ħhωLO at 54.72 nm that quenches emission. Therefore, the
59
2. NANOPILLARS IN RTDS AND QCLS: THEORY
Figure 2.20: Ladder states as a function of Rp in 3 THz 3 QW Paris design at an electric
field, E = 12.1 kV/cm, showing (a) elastic scattering of the lower lasing level (b) inelas-
tic scattering of the lower lasing level (c) elastic scattering of the upper lasing level (d)
inelastic scattering of the upper lasing level.
60
2.10 Results: radiative states in laterally confined THz QCLs
second window Q is identified in between the resonances E4,3 = E3,0 + 2ħhωLO at 64.04
nm and E4,4 = E3,0 + 2ħhωLO at 85.05 nm. This window includes the resonance E2,2 = E1,0
+ 2ħhωLO at 74.14 nm that enhances relaxation of carriers. At higher radial dimensions,
the resonance E2,3 = E1,0 + 2ħhωLO at 110.8 nm that enhances relaxation lies very close to
the resonance E4,2 = E3,0 + ħhωLO, which quences radiation. Hence, the third window R
is identified such that it lies between the resonance E4,2 = E3,0 + ħhωLO at 108.2 nm and
the resonance E4,3 = E3,0 + ħhωLO at 165.5 nm. This window includes the resonance E2,4
= E1,0 + 2ħhωLO that enhances electron relaxation from the lower lasing level.
Now that we know which radial dimensions permit THz emission, it is useful to esti-
mate the energy states that are involved in THz emission. These radiative states are de-
termined by the optical selection rule that dictates conservation of energy and momentum
during an intersubband transition. We therefore propose that in the case of nanopillars
fabricated from 3 THz 3 QW Paris design, only those transitions will radiate, that meet the
optical selection rule given below, i.e.
E4,m = E3,m′ for all m = m’ (2.30)
where, m and m’ = 0, 1, 2, 3 and so on.
For radial dimensions in the window P ≤ 42 nm, the transitions E4,0 → E3,0 at 3.16
THz and E4,1 → E3,1 at 3.15 THz are likely to be radiative levels. In the second window
Q, 65 ≤ Rp ≤ 84 nm, electron transition between the states E4,2→ E3,2 is likely to yield a
photon of energy at 3.14 THz. For larger pillars 110 ≤ Rp ≤ 164 nm, transition between
E4,0 → E3,0 and E4,4 → E3,4 are likely to form part of the radiative routes with an energy
difference of 3.52 THz and 3.13 THz, respectively.
2.10.2 Radiative states in laterally confined 3 THz 4 QW ETH design
Let us now look at the ladder states in a second design, that uses a miniband of energy
states as well as a resonant LO phonon stage for extraction of carriers from the lower
lasing level. Figure 2.22 shows a plot of the ladder states in nanopillars etched in 3 THz
4 QW ETH design as a function of Rp. As before, the ladder states are separated into
four panels, i.e. those involved in (a) elastic scattering of electrons in the lower miniband
state E2 with the extractor E1 (b) inelastic scattering of electrons in the lower miniband
state E2 with the extractor E1 (c) elastic scattering of electrons in the upper lasing level E5
with the lower lasing level E4, and finally (d) inelastic scattering of electrons in the upper
lasing level E5 with the lower lasing level E4. Table 2.2 summarises the energy states with
the corresponding radial dimensions that are involved in scattering processes that lead
61
2. NANOPILLARS IN RTDS AND QCLS: THEORY
Figure 2.21: Ladder states as a function of nanopillar radius Rp in 3 THz 3 QW ETH design
at an electric field, E = 12.1 kV/cm, showing optically permitted radiative transitions
within the windows P, Q and R.
enhanced relaxation or suppressed emission.
From figure 2.22, we identify four windows, labeled as P, Q, R and S that meet the
requirements of emission in laterally confined 3 THz 4 QW ETH design. The first window
P is identified by excluding the resonance E2,0 = E1,1 at Rp = 40.65 nm that leads to
suppressed emission. Therefore, the first window P lies in the range of Rp ≤ 39 nm. The
second window Q is identified by avoiding the resonances at E5,3 = E4,0 + 2ħhωLO at 64.04
nm and E5,4 = E4,0 + 2ħhωLO at 85.44 nm, that quenches emission. The third window R,
includes the resonance at E2,3 = E1,0 +2ħhωLO at 92.58 nm that enhances relaxation of
carriers from the lower miniband states. This window avoids the resonance at E5,4 = E4,0
+2ħhωLO at 85.44 nm and at E5,2 = E4,0 + ħhωLO at 107 nm, which leads to suppressed
emission. The fourth window S includes the resonances E2,0 = E1,3 at 122.3 nm and E2,4
= E1,0 +2ħhωLO at 123 nm. The width of this window is determined by the position of
the resonances E5,2 = E4,0 + ħhωLO at 107 nm and E5,3 = E4,0 + ħhωLO at 162.1 nm which
quenches emission.
In the case of nanopillars fabricated from 3 THz 4 QW ETH design, only those transi-
tions will radiate, that meet the optical selection rule given below, i.e.
E5,m = E4,m′ for all m = m’ (2.31)
62
2.10 Results: radiative states in laterally confined THz QCLs
Figure 2.22: Ladder states as a function of Rp in 3 THz 4 QW ETH design at an electric field,
E = 7.6 kV/cm, showing (a) elastic scattering of the lowest miniband state (b) inelastic
scattering of the lowest miniband state (c) elastic scattering of the upper lasing level (d)
inelastic scattering of the upper lasing level.
63
2. NANOPILLARS IN RTDS AND QCLS: THEORY
Nanopillars in 3 THz 4 QW ETH design
Scattering process Energy levels Physical radius, Rp
(nm)
Elastic scattering E2 E2,0 = E1,1 40.65
E2,0 = E1,2 80.88
E2,0 = E1,3 122.3
E2,0 = E1,4 163.3
Inelastic scattering E2 E2,1=E1,0 +2ħhωLO 30.54
E2,2=E1,0 +3ħhωLO 32.54
E2,2=E1,0 +2ħhωLO 61.86
E2,4=E1,0 +3ħhωLO 65.82
E2,3=E1,0 +2ħhωLO 92.58
E2,4=E1,0 +2ħhωLO 123.7
Elastic scattering E5 E5,0 = E4,1 100.5
E5,0 = E4,3 200
Inelastic scattering E5 E5,2=E4,0 +2ħhωLO 42.43
E5,1=E4,0 +2ħhωLO 52.14
E5,3=E4,0 +2ħhωLO 64.04
E5,4=E4,0 +2ħhωLO 85.44
E5,2=E4,0 +ħhωLO 107
E5,3=E4,0 +ħhωLO 162.1
Table 2.2: Summary of resonance positions in laterally confined 3 THz 4 QW ETH design
at a constant electric field E = 7.5 kV/cm. The resonance positions are listed in the order
of increasing nanopillar radius.
where, m and m’ = 0, 1, 2, 3 and so on. Figure 2.23 plots ladder states as a function
of nanopillar radius Rp in 3 THz 4 QW ETH design at an electric field, E = 7.6 kV/cm,
showing optically permitted radiative transitions. In the first window P, the intersubband
transition, E5,1 → E4,1 will dominate emission at 2.9 THz. In the second window, 66 ≤
Rp ≤ 84 nm, the transition would occur between the ground states E5,0 → E4,0, with an
energy difference equal to 3.07 THz. The transitions E5,3 → E4,3 at Rp = 92.58 nm and
at Rp = 123.7 nm would form part of the radiative routes in windows R and S, with an
energy difference equal to 3.82 THz and 3.00 THz respectively.
2.10.3 Accurate estimation of radiative states
In the calculations above, we have not evaluated the dipole moment 〈zul〉 and oscillator
strength f between the two radiative states. In a QD cascade structure, these design pa-
rameters need to be determined from the complete wavefunction Ψ(θ ,r,z) = Θ(θ)ψ(r,z).
Here,Θ(θ) is the azimuthal part of the solution to the 3D Schro¨dinger equation (eqn. 2.22).
The potential ζ(r,z) is determined from eqn. 2.20. Assuming azimuthal symmetry Θ(θ) =
64
2.10 Results: radiative states in laterally confined THz QCLs
Figure 2.23: Ladder states as a function of nanopillar radius Rp in 3 THz 4 QW ETH design
at an electric field, E = 7.6 kV/cm, showing optically permitted radiative transitions within
the windows P, Q, R and S.
eimθp
2pi
, which implies that Ψ(θ ,r,z) = e
imθp
2pi
ψ(r,z).
Furthermore, the intersubband transitions (QD states) identified as the radiative paths
in laterally confined 3 THz 4 QW ETH design and laterally confined 3 THz 3 QW Paris
design have only been calculated at the alignment field of the QW states. For a more
accurate estimate of the radiative states in laterally confined THz QCLs, the simulation has
to be swept over a range of applied bias values, taking into account alignment between the
extraction stage of one period with the injector stage of the neighbouring period.
What is important to note is that this model predicts the tolerance in Rp that allows THz
emission in laterally confined THz QCLs. This tolerance depends upon the QCL design and
width of the window. The minimum tolerance is for pillars with radial dimensions in the
window R, fabricated in 3 THz 4 QW ETH design, where ± 3.5 nm accuracy in fabrication
is required for observation of THz emission, see figure 2.21. For pillars fabricated from
3 THz 3 QW Paris design, the minimum tolerance is ± 10 nm, which is slightly higher.
This tolerance is estimated for pillars with radial dimensions in the window Q shown in
figure 2.23.
Inspite of the simplifications, the model provides a theoretical framework to predict
radiative states in laterally confined THz QCLs. This theoretical framework is necessary to
65
2. NANOPILLARS IN RTDS AND QCLS: THEORY
understand the THz energy spectra of laterally confined QCLs. So far, we have discussed
the effect of lateral confinement on the QCL bandstructure. We now focus on the plasmonic
mode of laterally confined THz QCLs.
2.11 Effect of lateral confinement on plasmonic mode in THz
QCLs
In the previous section, we focussed on the effect of nanopillar radius on intersubband
transitions in laterally confined THz QCLs. While it is essential to be within the range of
radial dimensions that permit emission of photons, it is also equally crucial to (i) achieve
enough gain to overcome losses, and (ii) support a plasmon with freespace wavelength
equal to the emitted photon (100 µm = 3 THz). The free space wavelength is 102-103
times larger than the nanopillar radius (50-200 nm). The two constrains necessiate the
fabrication of a nanopillar array that is packed into an area equal to the area of a THz QCL.
Typically, a standard THz QCL processed in a double metal waveguide geometry comprises
of a 2 mm long Fabry Pèrot ridge, that is 100 µm wide, see figure 2.24. Therefore, the
surface area of the ridge translates to ≈ 107 nanopillars of radius Rp = 50 nm, connected
in parallel.
Figure 2.24: Schematic diagram showing the three-dimensional view of a THz QCL in a
double metal waveguide.
66
2.11 Effect of lateral confinement on plasmonic mode in THz QCLs
2.11.1 Plasmon mode simulations in THz QCLs
Before we discuss the effect of various device parameters on the plasmonic mode of a
laterally confined THz QCL, it is useful to revisit the plasmonic mode in a double metal
THz QCL (discussed in section 1.5.2). Figure 2.25 shows the cross-sectional view of the
THz QCL ridge illustrating the different material layers.
Figure 2.25: Schematic diagram showing the cross-section of the Fabry Pèrot ridge in a
double metal waveguide.
The problem of evaluating the eigen mode solution to plasmons propagating inside a
THz QCL is analogous to the multilayer slab waveguide problem. The complex propagation
constant, ñ determined by solving the multilayer problem comprises of a real component
n that characterizes the effective refractive index of the medium and an imaginary compo-
nent κ, also known as the extinction coefficient that characterizes the losses in the medium.
This is given below:
n˜= n+ iκ (2.32)
The extinction coefficient κ characterizes the absorption losses in the waveguide. For
a freespace wavelength λ, the absorption losses are written as αw =
4piκ
λ
.
The overlap Γ, is another parameter that is crucial to the waveguide and is defined as
the percentage of the optical mode interacting with the active region. This is written as:
Γ =
∫
AR
|E(z)|2dz∫ +∞
−∞ |E(z)|2dz
(2.33)
Here E(z) is the z-component of the electric field and AR is the QCL active region
excluding the top and bottom n+ GaAs doped contacts.
The figure of merit (FOM) defines the quality of a waveguide in terms of both the
absorption losses αw and the overlap factor Γ, and has units of length. Defined as the ratio
67
2. NANOPILLARS IN RTDS AND QCLS: THEORY
between Γ and αw , FOM can be written as:
FOM =
Γ
αw
(2.34)
These parameters, i.e Re(ñ), αw , Γ and FOM depend upon (a) the geometry of each
layer, and (b) the complex dielectric constant of each layer. In the case of the GaAs active
region, the n+ doped GaAs contacts and the gold layers, the complex dielectric constants
can be estimated from the Drude model. The accuracy of this model to estimate the optical
behavior of semiconductors and metals at THz frequencies has been described in great
detail before [5, 41]. Here we only present the key formulae that we use to estimate the
complex dielectric constant, that is dependent on frequency (ω), effective mass m∗e and
doping Nd .
The complex dielectric constant can be written as:
ε(ω) = ε′+ iε”=
 
1− ω
2
pτ
2
1+ω2τ2
+ i
τω2p
ω(1+ω2τ2)
!
. (2.35)
Here,ωp =
q
Nd e2
m∗eεo
is the volume plasma frequency and εo is the permittivity of vacuum.
The carrier lifetime τ is equal to 0.1 ps for undoped GaAs and 1 ps for n+ doped GaAs.
We use COMSOL v4.3 to evaluate the mode profile supported by the waveguide. In the
COMSOL environment, for every material layer included in the COMSOL simulation, i.e
the active region, doped contacts and metal contacts, we input the real part of the dielectric
constant ε′ and the dc conductivity σdc . The dc conductivity σdc = Nd e
2τ
m∗e
is related to the
imaginary part of the dielectric constant as:
ε”=
σdc
1+ω2τ2
(2.36)
We notate ε′ as εr for every material. For simplicity we also drop the subscript in dc
conductivity and notate it as σ.
Figure 2.26(a) shows the optical constants associated with each material layer in a dou-
ble metal THz QCL. The active region is characterized by optical constants εGaAsr and σ
GaAs.
The top (bottom) doped n+ GaAs contact layers are characterized by the optical constants
ε
topGaAs
r (εbotGaAsr ) and σ
topGaAs (σbotGaAs). The n+ GaAs substrate is characterized by op-
tical constants εsubst rater and σ
subst rate, and finally, the metal contacts are characterized by
optical constants εAur and σ
Au. Figure 2.26(b-c) show that in a double metal THz QCL, the
THz wave is confined as a surface plasmon between the top and bottom metal layers. The
surface plasmon propagates perpendicular to the plane and the mode profile varies in the
68
2.11 Effect of lateral confinement on plasmonic mode in THz QCLs
growth direction. The dipole selection rule in intersubband lasers only permit radiation
that has it’s electric field polarized perpendicular to the growth direction, and hence only
transverse magnetic (TM) modes are considered in plasmon simulations [5].
Figure 2.26: (a) Schematic diagram showing the optical constants corresponding to the
material layers in a THz QCL in double metal waveguide (b) Plasmon mode calculated
for a full-stack THz QCL in a double metal waveguide, where the optical constants of the
different material layers correspond to the 3 THz 4 QW ETH design (c) Cross-sectional
view of the mode confinement in the QCL active region.
2.11.2 Effective medium approximation in laterally confined QCLs
As discussed before, the freespace wavelength of intersubband transitions in a THz QCL
(100 µm = 3 THz) is much larger than the radial dimensions (Rp ≤ 200 nm) of the
nanopillars. Therefore, an array of nanopillars is required to support the plasmonic mode
in a laterally confined THz QCL. The nanopillars, fabricated from a THz QCL active region
are planarized with an insulator and contacted on the top and bottom with metal layers.
Figure 2.27 shows the schematic diagram of a laterally confined THz QCL, detailing the
cross-sectional and top view of the nanopillars. The packing density of the nanopillar array
is determined by the pitch p, defined as the distance between the center of two closest
nanopillars. Here we use a square lattice to pack the nanopillars. In theory a different
lattice arrangement can also be used to pack the nanopillars. For the same p, a hexagonal
lattice allows for a higher packing density than a square lattice.
69
2. NANOPILLARS IN RTDS AND QCLS: THEORY
Figure 2.27: Schematic diagram showing (a) the cross-sectional view of laterally confined
THz QCL (nanopillars) in a double metal waveguide and (b) the top-view of nanopillars
fabricated in THz QCL and planarized with an insulator.
Assuming that the radiation emitted from a laterally confined THz QCLs is TM polarized
and is perpendicular to the plane, we suggest that the losses due to free carrier absorption
in laterally confined THz QCLs can be calculated in the same way as standard double metal
ridges in THz QCLs. From figure 2.27, we note that the optical constants εAur and σ
Au of
the two top and bottom metal layers which electrically connect the nanopillars remain
unaffected by lateral confinement. However, the optical constants of the laterally confined
active region layer as well as the top and bottom n+ doped GaAs layers must now include
the surrounding medium, be it air or a planarizing dielectric medium. The planarizing
material will have it’s own optical constant at THz frequencies. In this work, we label the
electrical permittivity as εinsr and conductivity as σ
ins. In the case of air, εinsr = ε
air
r = 1
and σins = σair = 0 S/m. For most planarizing materials εinsr ranges from 2.5 to 4 and
σins ranges from 10−10 - 10−15 S/m 1.
Figure 2.28 shows the schematic diagram of the laterally confined THz QCL illustrat-
ing the effective medium approximation approach to evaluate the optical constants of the
active region (2) and the top and bottom n+ doped layers (1 and 3). Note that the optical
constant of each epitaxial layer is governed by the surrounding medium, the radius Rp of
1Material database at http://www.mit.edu/ 6.777/matprops and references [105, 106, 107]
70
2.11 Effect of lateral confinement on plasmonic mode in THz QCLs
Figure 2.28: Schematic diagram showing (a) the cross-section view of laterally confined
THz QCL (nanopillars) in double metal waveguide and, (b) the top-view of nanopillars fab-
ricated in THz QCL and planarized with an insulator of permittivity εinsr and conductivity
σins.
the nanopillar array and, the pitch p of the nanopillar array. The optical constants of the
laterally confined active region εN Pr and σ
N P can be calculated as:
εN Pr = ε
GaAs
r piR
2
p + ε
ins
r (p
2−piR2p) (2.37)
σN P = σGaAspiR2p +σ
ins(p2−piR2p) (2.38)
Here, εGaAsr is the real part of the complex dielectric constant of the QCL active region
and σGaAs is the conductivity of the QCL active region. The optical constants of the lat-
erally confined doped layers can be evaluated using eqns. 2.37 and 2.38, by substituting
the permittivity (εtopGaAsr , εbotGaAsr ) and conductivity (σ
topGaAs, σbotGaAs) of the top and
bottom doped layers.
We find that the effective medium approximation is a very useful tool to analyse the
losses in a laterally confined THz QCL and optimize for the best device geometry. In the
next sub-sections, we discuss the effect of active region height, packing density of the
nanopillar array and planarizing material on the optical properties of laterally confined
THz QCLs. The active region of choice in all the calculations discussed below is the 3 THz
71
2. NANOPILLARS IN RTDS AND QCLS: THEORY
4 QW ETH design. This design has an average doping per period equal to 5.609 x 1015
cm−2. In the simulations, the thickness of the top and bottom n+ GaAs contact layer is 100
nm, which is doped with a carrier concentration Nd equal to 5 x 10
18 cm−2.
2.11.3 Simulation results: effect of active region thickness
In general the thickness of a full-stack THz QCL ranges from 6 - 14 µm depending upon
the design. In the case of the 3 THz 4 QW ETH design, a full-stack THz QCL with 150
repeats has a thickness of 10 µm. Ideally, to achieve the same gain as a standard THz
QCL, one should implement the top-down approach on full-stack structures. Experimental
considerations, however, limit the thickness of the active region to approximately 3 to 3.5
µm (ultrathin THz QCLs). This is discussed in detail in section 3.5. An important concern
while implemeting the top-down approach would be whether a plasmonic mode can be
supported in a THz QCL with active region thickness less than 3 µm in a standard double
metal geometry. Extending the argument to laterally confined THz QCLs, it would be im-
portant to know whether a common plasmonic mode can exist within the effective medium
comprising of nanopillars and insulation between the top and bottom metal contacts.
Figure 2.29: Plasmonic mode calculated for (a) a THz QCL in a double metal waveguide,
with active region thickness equal to 3 µm (b) a suspended nanopillar array (εinsr =ε
air
r ,
σins = σair), with active region thickness equal to 3 µm and pitch/diameter (p/2Rp) ratio
= 4. Also shown is the cross-sectional view of the mode profile confined between the top
and bottom metal contacts.
Figure 2.29(a) shows the plasmonic mode calculated for a 3 µm thick, Fabry Pèrot
ridge, with optical constants corresponding to a 3 THz 4 QW ETH design, in a double
metal waveguide configuration. The mode is confined within the top and bottom metal
72
2.11 Effect of lateral confinement on plasmonic mode in THz QCLs
contacts, similar to a standard full-stack laser. A comparison between the mode properties
of ultrathin active regions with full stacks is discussed in section 4.10. Here, we limit the
discussion to low-repeat period THz QCLs and their comparison with laterally confined
THz QCLs. Figure 2.29(b) shows the plasmonic mode calculated for the same design, with
optical constants corresponding to a nanopillar array of radius 50 nm, p/2Rp equal to 4
and active region height of 3 µm. For simplicity, we use air as the surrounding medium
around the nanopillars. The use of the effective medium approach eliminates the need
to define the number of nanopillars considered in the simulation and the same mesh can
be used for standard THz QCLs and the laterally confined THz QCLs. It is interesting to
note that a plasmonic mode exists in this effective medium. Furthermore, the plasmonic
mode shown in figure 2.29 is confined within the top and bottom metal contacts, similar
to THz QCLs with a double metal waveguide. This allows us to compare the waveguiding
properties of THz QCLs with laterally confined THz QCLs.
Figure 2.30 compares the THz QCL with laterally confined THz QCL as a function of
active region thickness. In a double metal ridge, the waveguide loss αw increases from 9.6
cm−1 to 10.5 cm−1 as the thickness of the active region is reduced from 5 µm to 2 µm.
This increase is explained in terms of greater field intensity at the metal/semiconductor in-
terface as the active region thickness is reduced [3]. The overlap factor Γ ∼ 0.97 increases
very slightly with increase in thickness. The figure of merit (FOM) also varies slightly (0.09
cm to 0.1 cm), when the thickness is increased from 2 µm to 5 µm. Finally, we note that
the effective refractive index or the propagation constant decreases with increase in active
region thickness. Between a thickness of 2-5 µm the effective refractive index of a THz
QCL in a double metal waveguide is estimated to be ≈ 3.65. Similar to the 100 µm wide
double metal ridge, the optical losses in a laterally confined THz QCL increases with de-
crease in thickness. This is shown in figure 2.30 for nanopillars surrounded by air as an
insulating medium (also called suspended nanopillars). What is striking is the reduction
in waveguide losses upon introducing lateral confinement. Let us compare a 3 µm thin
THz QCL, with a 3 µm thin laterally confined THz QCL. As opposed to an optical loss of
9.45 cm−1 in a 3 µm thin double metal ridge, the optical loss in 3 µm tall nanopillar array
yields a significantly lower value (2.47 cm−1). The low loss is attributed to the low con-
ductivity of the surrounding medium (σins), which is equal to 0 S/m in the case of air. The
disadvantage of suspended nanopillars is that the optical mode leaks into the surrounding
air. Hence, the overlap factor reduces from 0.97 in a 3 µm thin QCL to 0.70 in a 3 µm
tall suspended nanopillar structure. The figure of merit of suspended nanopillars, still re-
mains higher than the Fabry-Perot ridge. For example the figure of merit of a 3 µm tall
suspended nanopillar array is three times more than the double metal ridge of the same
73
2. NANOPILLARS IN RTDS AND QCLS: THEORY
Figure 2.30: Comparison of THz QCL with laterally confined THz QCL, as a function of
active region thickness in terms of (a) waveguide losses, αw (b) overlap, Γ (c) figure of
merit, FOM (d) propagation constant Re(n˜) = n. Simulations have been performed using
COMSOL v4.3. In the case of laterally confined THz QCL, the nanopillars have a suspended
top contact (εinsr = ε
air
r , σ
ins = σair) and the ratio p/Rp is equal to 4.
thickness. Note that the propagation constant of the optical mode supported by 1- 5 µm
tall suspended pillars is equal to 1.25. Therefore, the effective refractive index of a laterally
confined suspended THz QCL is closer to air, than it is to GaAs.
2.11.4 Simulation results: effect of pitch/diameter ratio
Here, we discuss the effect of pitch/diameter (p/2Rp) ratio on the waveguide properties
of the suspended nanopillar arrays. The p/2Rp characterises the packing density of the
nanopillar array. As p/2Rp → 0, the optical properties of laterally confined THz QCLs
(εN P , σN P) approaches the optical properties of a standard THz QCL (εGaAs, σGaAs), see
eqns. 2.37- 2.38. In reality, experimental considerations constrain the packing density
that can be achieved. For the smallest pillars, a p/2Rp < 2 suffers from two challenges.
74
2.11 Effect of lateral confinement on plasmonic mode in THz QCLs
The first challenge arises from the proximity effect while directly writing a pattern using
e-beam. The second issue pertains to incomplete etching of trench passage between the
nanopillars, also called as reactive ion etching (RIE) lag. These problems are discussed in
detail in Chapter 3. For these reasons we restrict our discussion to cases where the p/2Rp
ratio ≥ 2.
While it might seem that a small p/2Rp is ideal as it maximizes the number of pillars
in an area, figure 2.31(a) shows that the optical losses of a suspended nanopillar array are
highest when p/2Rp = 2. This loss is due to absorption by free carriers in the active region
and the doped layers. As the p/2Rp ratio is increased, the losses decrease from 4 cm−1 to
a lowest of 2.5 cm−1 at p/2Rp = 4. This decreases is due to the zero conductivity of air
which forms part of the effective medium that supports the optical mode. As the p/2Rp is
increased further, the optical mode weakens as it leaks out of the cavity. The overlap factor
shown in figure 2.31(b) decreases with the p/2Rp ratio. From figure 2.31, we note that
the highest figure of merit that is experimentally possible for laterally confined THz QCLs
is achieved when the top contact is suspended and the pillars are packed with a p/2Rp =
4.
Let us now discuss what these results mean for nanopillar dimensions highlighted in fig-
ure 2.23. The gain of a nanopillar array is decided by the strength of the optical transition.
The losses are dictated by the p/2Rp ratio. Therefore, if we were to pattern a nanopillar
array, with Rp = 35 nm that permits intersubband transition (window P in figure 2.23), at
a pitch equal to 4 times 2Rp (280 nm) , the figure of merit would be maximum. To achieve
the same figure of merit in a nanopillar array with Rp = 120 nm (window S in figure 2.23),
the pillars have to be separated by a larger distance. The pitch equals 4 times 2Rp = 960
nm. Figure 2.32 shows the schematic diagram illustrating the optimum spacing for Rp =
35 nm and Rp = 120 nm nanopillars when the array is insulated by air, i.e the nanopillars
have a suspended top contact.
75
2. NANOPILLARS IN RTDS AND QCLS: THEORY
Figure 2.31: Effect of p/2Rp on the optical properties of suspended nanopillar array fabri-
cated from 3 THz 4 QW ETH design. Shown in the figure is a comparison of (a) waveguide
losses (b) overlap (c) figure of merit and, (d) propagation constant Re(n˜) = n. Simulations
have been performed using COMSOL v4.3, with active region thickness kept constant at 3
µm.
76
2.11 Effect of lateral confinement on plasmonic mode in THz QCLs
Figure 2.32: Schematic diagram illustrating the optimum spacing of nanopillars when the
array is insulated by air (εinsr = ε
air
r , σ
ins = σair).
2.11.5 Simulation results: effect of planarising material
The planarizing material plays a critical role in the fabrication of potential quantum dot
cascade lasers based on a top-down approach. Commonly used planarizing materials such
as spin-on-glass, polyimide and cyclotene have a finite electrical conductivity, which can
range from 10−10 - 10−15 S/m [105, 106, 107]. Considering the variability in εinsr and σins
arising out of differences in curing temperature, humidity and other processing parameters,
we do not differentiate between available planarizing materials. Instead, we compare the
optical mode parameters of a nanopillar array planarized with an insulator characterized
with an εinsr =3.9 and σ
ins = 10−10 S/m, with a nanopillar array that has a suspended
top contact and is surrounded by air, which is characterized with an εinsr = 1 and σ
ins = 0
S/m.
Figure 2.33(a) shows the waveguide losses of a planarized array as a function of p/2Rp
ratio. Compared to a suspended array, the waveguide losses in an insulated array increase
nearly exponentially as p/2Rp increases from 2 to 6. The minimum loss in an insulated ar-
ray of pillars is achieved when p/2Rp ratio = 2, which is commensurate with the suspended
array. To achieve the minimum loss, nanopillars with radial dimensions in the range of 35
nm to 150 nm (figure 2.23) must be separated by a pitch p equal to 2(2Rp). Figure 2.34
shows the schematic diagram of the pillar arrangement, illustrating the optimum spacing
for Rp = 35 nm and Rp = 120 nm nanopillars when the array is insulated by a dielectric
medium.
In general, the minimum distance that an insulator can fill is determined by the fill
77
2. NANOPILLARS IN RTDS AND QCLS: THEORY
Figure 2.33: Comparison of (a) waveguide losses (b) overlap (c) figure of merit, FOM (d)
propagation constant, Re(n˜) = n as a function of p/2Rp ratio of the nanopillar array with a
suspended top contact (red) and an unsuspended/insulated top contact (black), calculated
using COMSOL v4.3. The nanopillar thickness is constant at 3 µm.
Figure 2.34: Schematic diagram illustrating the optimum spacing of nanopillars when the
array is insulated by a dielectric (εinsr = 3.9, σ
ins = 10−10 S/m).
78
2.12 Summary and conclusions
properties of the planarizing materials. Typically, this minimum distance is equal to 300
nm. For a pitch density p/2Rp = 2 that yields the lowest optical loss, the fill distance p -
2Rp equals 2Rp. In the case of pillars with radial dimensions in the range with Rp = 35
(to 150 nm), the optimum spacing that ensures the lowest optical loss is equal to 70 nm
(to 300 nm). This distance is less than the fill length of typical planarizing materials (>
300 nm). For these range of pillar dimensions, it is not experimentally viable to achieve
the lowest optical loss (5 cm−1) in insulated nanopillars. If we increase the p/2Rp ratio
to 4, the waveguide loss in an insulated array is ≈ 10 cm−1. At the same p/2Rp ratio,
the waveguide losses of a suspended array is 2.5 cm−1, which is four times lower than the
insulated array. An advantage of retaining insulation is that the optical mode is confined
more strongly in an insulated array as compared to a suspended array and decreases only
slightly with increasing p/2Rp. Infact, the overlap factor in an insulated array approaches
0.9, which is close to what is observed in standard double metal THz QCLs. However, the
suspended nanopillar array shows a higher figure of merit compared to the insulated array.
These simulations provide useful design considerations for optimizing the performance
of a laterally confined THz QCL. Finally, we summarize the key results of this chapter below.
2.12 Summary and conclusions
In the first part of the chapter, we explored the effect of lateral confinement on the energy
states of a simple structure, the resonant tunneling diode (RTD). Using Chia’s model for
homogeneous nanopillars, we calculated the effect of surface states on the electrostatic
potential of a laterally confined RTD. We found that the estimation of electrostatic radius
is non-trivial as it depends upon many factors, such as density of interface states, doping
of the epitaxial layer and the physical radius of the nanopillar. Using these results, we
simulated the three-dimensional potential of an RTD nanopillar. We also extended the
model to simulate the three-dimensional potential of a laterally confined THz QCL.
In the second part of the chapter, we used resonance conditions and optical selection
rules to identify intersubband states that would permit radiation in laterally confined THz
QCLs. We also identified windows of radial dimensions that permit emission between those
that suppress emission.
Finally, we developed an effective medium approach to simulate plasmonic modes in
laterally confined THz QCLs. From our simulations, we found that nanopillar arrays with a
suspended top contact and a pitch/diameter ratio = 4, yielded a 2 to 4 times higher figure
of merit compared to a double metal Fabry Pèrot ridge from the same active region.
79
2. NANOPILLARS IN RTDS AND QCLS: THEORY
80
"It’s always helpful to learn from your
mistakes because then your mistakes
seem worthwhile."
Garry Marshall
Chapter 3
Nanopillar arrays in RTDs and QCLs:
Experimental realization
3.1 Introduction
In Chapter 2, we identified geometrical constraints under which the top-down nanofab-
rication approach would possibly yield improved temperature performance of THz QCLs.
These constraints require the development of new processing routes that can achieve a
suspended electrical contact simultaneously to ∼ 107 nanopillars (section 2.11). Addi-
tionally, the constraints on the nanopillars only allow for a vertical sidewall profile with a
tolerance such that the radius lies within the dimension windows, which permit radiative
intersubband transitions.
Fabrication of such electrically connected nanopillar arrays is non-trivial as it can di-
rectly infuence the electro-optic performance of laterally confined THz QCLs. It requires
optimization of several experimental processes and their parameters, some of which are
unique to THz QCLs. Furthermore, one processing stage can impact subsequent process-
ing stages, which necessitates identification of processing steps that do not conflict with
each other. In this chapter, we discuss the experimental challenges that were encountered
during the fabrication of electrically connected nanopillar arrays in THz QCLs.
In chapter 2, we discussed that RTDs serve as a useful test bed to examine the electrical
robustness of this top-down fabrication approach as the effect of device geometry can be
clearly seen in the electrical behavior. However, fabrication of nanopillars in RTDs requires
either the addition or omission of certain processing stages, which pertain only to the RTDs.
In this chapter we demonstrate fabrication routes that are compatible with both RTDs and
QCLs. Therefore, we first test the validity of our nanofabrication approach on RTDs. We
81
3. NANOPILLAR ARRAYS IN RTDS AND QCLS: EXPERIMENTAL REALIZATION
also present electrical characteristics of laterally-confined RTDs that allow us to comment
on the electrical stability of these nanopillar arrays. In addition to electrical contact, we
show evidence that suggests quantization effects in these nanopillar arrays, which is in
agreement with quantization effects observed in a single nanopillar [89, 94, 102, 108].
In the next section, we discuss the RTD active regions that are compatible with the THz
QCLs in terms of similar current density and material system. These RTD active regions
are used to demonstrate the validity of our fabrication process.
3.2 RTD active region used: relevance to QCLs
The most pertinent material system to the growth of high-quality THz QCLs is the GaAs /
AlxGa1−xAs system. What is in favour of this particular system is that both GaAs and AlAs
have almost identical latttice constants [109]. Therefore, independent of the Aluminium
concentration x, the entire layered structure, that is typical of a QCL, is lattice matched.
Furthermore, this system allows flexibility in engineering the barrier height for the elec-
trons, based upon the concentration of Aluminium. Owing to these reasons, THz QCLs
discussed in this thesis, were grown in the GaAs / AlxGa1−xAs system, exclusively.
Naturally, the simple single quantum well structures, i.e. the RTDs, were also grown in
the same material system. The RTD active regions that we chose as test structures had to
meet the following two criteria. The first and foremost requirement is that of peak current
density Jp, which is defined as the maximum current density due to resonant tunneling
of electrons from the contacts into the quantum well (see section 2.2). In order to test
electrical robustness of nanopillar arrays, this peak current density must be comparable to
the threshold current density Jth of THz QCLs, which is defined as the current density at
which the QCL overcomes losses and starts to exhibit lasing (see section 4.2). The second
condition demanded that the RTD of choice must exhibit a negative differential peak at
room temperature, i.e. the tunneling current must be larger than the background current
due to crystal defects. This condition allowed us to test the electrical contacts of laterally-
confined RTDs at 4 K as well as at room temperature.
Within the existing repositry of wafers grown at Cavendish, we found that two RTD
structures met with our requirements. These two wafers were labelled W0021 and A1700.
While W0021 used AlAs barriers, A1700 used AlxGa1−xAs barrier where x = 0.33. The
barrier height of both these RTDs is higher than the barrier height of THz QCLs discussed
in Chapter 3 which use x= 0.15. However, this slight difference is not important to demon-
strate the proof of principle. The epitaxial structure of W0021 is given as 2/9/2 where the
thickness is expressed in nm and the AlAs barriers are highlighted in bold. The epitaxial
82
3.3 Molecular beam epitaxy
structure of A1700 is given as 5/10/5 where the thickness is expressed in nm and the
Al0.33Ga0.67As barriers are highlighted in bold. The complete layer structure that includes
the thickness of spacers and contacts is detailed in appendix A. The electrical characteristics
of these RTDs are discussed in sections 3.8 and 3.9.
Before proceeding to discuss the fabrication of nanopillar arrays, it is imperative to
describe the growth of high-quality GaAs / AlxGa1−xAs layered structures, fashioned with
atomic level precision. This precise control is enabled by molecular beam epitaxy, which is
described below.
3.3 Molecular beam epitaxy
Molecular beam epitaxy (MBE) relies on deposition of atoms or cluster of atoms, with
monolayer control, under an ultra high vacuum (typically 10−10 mTorr base pressure).
These atoms are produced by heating high-purity solid elemental sources to their sublima-
tion temperatures. Figure 3.1 shows the schematic diagram of a typical molecular beam
epitaxy system. After loading the sample, a few preparatory steps are taken. Moisture is
removed and a standard thermal degas is performed to remove the native oxides, following
which the layer-by-layer growth of the III-V structure is performed. This occurs under the
ultra high vacuum, where the vapourized elemental flux impinges upon a rotating, heated
substrate.
Growth rate is determined by the group III flux rate under standard growth condi-
tions, which includes optimal substrate temperature and group V excess. Individual layer
composition and thickness are governed by a combination of deposited materials by use
of shutters. Since the operation of the shutters is faster than typical growth rates, abrubt
’monolayer’ interface can be achieved.
Growth rates can be determined by several techniques, such as reflection high-energy
electron diffraction (RHEED), ex-situ X-Ray measurements, flux gauge and optical band
pass pyrometer [110, 111]. Wafers discussed in this thesis were monitored prior to growth
using an optical band pass pyrometer [110]. This technique detects oscillations in the
surface temperature occuring during the growth of heteroepitaxial structures due to Fabry-
Pèrot cavity interference. The oscillation period can be related to the growth rate of the
material that is being deposited [111]. Accurate growth rate are a key parameter in the
QCL performance [112].
83
3. NANOPILLAR ARRAYS IN RTDS AND QCLS: EXPERIMENTAL REALIZATION
Figure 3.1: Schematic diagram of a typical molecular beam epitaxy system
3.4 General methods: fabrication of nanopillar arrays
In the previous section, we described monolayer level control over the growth of the epi-
taxial layers to obtain confinement in the growth direction. In this section, we describe the
techniques that are employed to achieve the designed lateral potential in nanopillars that
are processed from layered structures grown using MBE.
Our philosophy behind fabrication of nanopillar arrays is given below.
The geometrical constraints described in Chapter 2 require precise control over the pro-
cessing of nanopillars. It is therefore, essential that the final device geometry after multiple
processing stages is as close as possible to the designed geometry, within the tolerance limits
described in section 2.9. For control over the energy states due to lateral quantization, any
deviation between the final geometry and the expected geometry must be accounted for at each
processing stage. This is of use to subsequent iterations, where the disparity between the final
and expected geometry is incoporated at the design stage itself.
The device geometry, i.e. the radial dimensions of the nanopillar and the pitch of
the nanopillar array is defined using electron beam lithography. This lithography helps
in defining the etch mask for the nanopillars. Although, there are other routes such as
deposition of a colloidal solution of nanoparticles, which can act as an etch mask [113],
such lithographic techniques do not allow control over the device geometry in terms of size
and pattern.
84
3.4 General methods: fabrication of nanopillar arrays
Following the definition of an etch mask, the nanopillars are etched using plasma pro-
cessing. Plasma processing (or dry etching) is far superior to wet chemical etching tech-
niques in terms of obtaining a vertical sidewall profile [114]. Moreover, for radial dimen-
sions in the range of sub-200 nm, dry etching is the only way to achieve pillars with a
high-aspect ratio (diameter/height ratio). Unlike wet chemical etching, which is isotropic
in nature and causes a rounded profile due to underetching of the mask, dry etching can be
used to used to obtain any desired profile. This can be achieved by adding reactive species
in the plasma. Therefore, the etch profile is controlled by a balance of physical sputtering,
which is anisotropic in nature and chemical etching, which reacts with the sample to be
etched.
Following etching, the nanopillar arrays are planarized with an insulator. This insula-
tion is etched back to expose the top n+ doped GaAs contact, without compromising on
the electrical integrity of the planarizing material. Finally, the nanopillars are connected in
parallel by lithography and deposition of a top metal contact. Figure 3.2 shows the general
process flowchart to fabricate electrically connected nanopillar arrays. These processing
stages are common to both RTDs and QCLs.
Figure 3.2: General process flowchart to fabricate electrically connected nanopillar arrays.
These processing stages are common to both RTDs and THz QCLs.
Below, we describe the general experimental techniques that we used to fabricate elec-
trically connected nanopillar arrays.
85
3. NANOPILLAR ARRAYS IN RTDS AND QCLS: EXPERIMENTAL REALIZATION
3.4.1 Electron beam lithography
Electron beam lithography (or e-beam lithography) uses a focused beam of electrons to
write patterns directly on to a resist, which is spin-coated on the sample [114]. The e-
beam pattern is generated in a computer aided design software (AutoCAD), and transferred
directly to a computer system which controls the exposure of the e-beam as it scans the
sample.
E-beam lithography has the benefit of ultra high resolution (1-5 nm), that is not limited
by the diffraction of light. Instead, the minimum pattern dimensions are determined by the
beam spot size that is achievable with the electron optics and the interactions of electrons
with the resist and substrate during the writing process.
In this work, e-beam patterns were written using Vistec VB6, ultra high resolution
(UHR), extra wide field (EWF) deflection system. This system employs a Gaussian beam
step and exposure strategy, with the thermal field emission gun operating at a maximum
accelerating voltage of 100 KV. The maximum writing speed of the system is 50 MHz. The
deflection coils allow the electron beam to be steered electromagnetically, around a fixed
region, allowing selective exposure within that area without any movement of the sample
relative to the column. Under the technical specifications listed for the above-mentioned
system, the maximum possible deflection is 1.3 mm at 100 KV.
The e-beam pattern is written on a Poly (methyl) methacrylate (PMMA), a positive
tone resist that has the adavantage of long film life. PMMA also offers extremely high-
resolution, has excellent surface adhesion characteristics and is not sensitive to white light
1. We use an undiluted solution of PMMA (A5) with a molecular weight of 950,000, which
is spun at 5000 rpm for 1 minute. After coating the sample, we follow 10 min post-bake
at 150oC to achieve a final resist thickness of ∼ 160-180 nm. The thickness of the e-beam
resist is chosen keeping further processing stages in mind. If the thickness of the PMMA
film is higher than 180 nm, then the dosage requirement would constrain the resolution of
the e-beam pattern. On the other hand, if the PMMA film is lower than 160 nm, it would
limit the thickness of the evaporated metal that serves as an etch mask (see section 3.5.2).
At this film thickness, the minimum radius (of a circular pattern) that can be written on
a ∼ 500 µm GaAs substrate under optimized dosage conditions is restricted to 50 nm. This
feature size is larger than the minimum resolution than can be theoretically achieved by
the Vistec e-beam tool. This is because any pattern that is written by the e-beam system is
broken down to smaller shapes, which are realiziable by the pattern generator. The fracture
process splits larger shapes into trapezoidal sub-shapes. In the case of writing an array of
1Microchem PMMA data sheet, URL: http://microchem.com
86
3.4 General methods: fabrication of nanopillar arrays
circles, each circle is broken into trapezoidal sub-shapes. Therefore, the e-beam tool only
writes a feature that best approximates the circle, with a grid size of 5 nm. Together with
these approximations, and the interaction of electrons with the PMMA and the substrate,
the minimum radius that can be conveniently achieved by the Vistec tool, is restricted to 50
nm. Therefore, the e-beam lithography step imposes a minimum radius of the nanopillar,
which can be written on the sample.
3.4.2 Optical lithography
For patterns larger than 1 µm, e-beam lithography can be prohibitively expensive and
time consuming. For such feature sizes, optical photolithography is a convenient alter-
native. Photoresists are patterned by selective exposure to a UV light source, when the
resist-coated sample is brought in contact with the optical mask. The selective exposure
is performed using this mask, which has pre-defined regions that are either transparent
or opaque to UV light. Patterns related to different processing stages can be included
on a single optical mask. Using carefully designed alignment marks, the sample can be
brought into alignment and subsequently exposed to the desired pattern. For all litho-
graphic stages, apart from the definition of nanopillars, we use UV light to expose the
desired feature. Depending upon the processing stage, these features may either be used
to form metal contacts or to etch material.
After a standard clean with actone and IPA, followed by a prebake step, we spin-coat
the sample with a positive photoresist. In this work, we use Shipley series: 1805 and 1813,
where the last two digits in the product number indicate the thickness of the photoresist
film after spin coating at 5000 rpm for 1 min. Therefore, a sample coated with Shipley
1805 (1813) yields a film thickness of 500 nm (1800 nm). The minimum resolution that
can be obtained by optical lithography often depends upon the thickness of the resist, the
exposure time and the development time.
A sample coated with Shipley 1805, exposed between 3 to 3.5 secs and developed in a
KOH based solution (MF319) for 15 to 30 secs, yields a minimum feature size of 1-2 µm.
This occurs when the optical mask is in complete contact with the resist. Shipley 1805
is used while performing lithography for metal contacts, where high alignment or high
resolution is critical. In the case where the photoresist serves to protect the underlying
features during a wet chemical etch, it is important to use a thicker photoresist to avoid
underetching of the mask. Under this scenario, we use Shipley 1813 exposed between 6.5
to 7 secs and developed for 1 min.
There is a slight difference in the photoresist profile when it is used as a mask for wet
87
3. NANOPILLAR ARRAYS IN RTDS AND QCLS: EXPERIMENTAL REALIZATION
chemical etching versus when the resist is used for metal evaporation. In the former case,
the resist is heated at 115 oC post exposure to hardbake it for chemical etching. Whereas,
in the latter case, the resist is softbaked at 90 oC post exposure and dipped in chorobenzene
before development to obtain a sloping undercut resist profile. This is necessary to pro-
duce a shadow profile to facilitate a discontinuity in the metal film deposition. Figure 3.3
summarizes the process flow for optical photolithography.
Figure 3.3: Schematic diagram describing the photolithographic process flow. The process
flow shows (a) photoresist spin-coated over the sample (b) exposure of the photoresist
under UV light (c) photoresist after UV exposure (d) photoresist after development. Also
shown in (d) is the resist profile when it is used as a mask during wet chemical etching
and when it is used for metal evaporation.
3.4.3 Dry etching
In this section we discuss the principle behind etching highly vertical nanopillars using
plasma processing (dry etching). Plasma processing involves a mechanism to generate
ions, and impart these ions with the momentum that is directed towards the sample. This
plasma consists of (i) neutral and ionized ions that etches material by physical sputtering,
and (ii) electrons and free radicals that etches material by reacting with it chemically.
In order to obtain highly vertical sidewall profile using plasma etching, it is necessary to
obtain the right balance between these two processes. Unlike sputtering which removes
material by physical bombardment of ions, chemical etching depends upon the formation
of volatile products with the exposed surface and offers etch selectivity over the mask.
88
3.4 General methods: fabrication of nanopillar arrays
Generation of this plasma requires a chamber that is maintained at a low pressure
(typically 0.01 to 1 Torr). When a high frequency voltage is applied between the electrodes,
current flows forming a plasma which emits a characteristic glow [115]. Figure 3.4 shows
the schematic diagram of the etching chamber after the plasma is struck upon introduction
of the feed gas. This plasma is an ionized gas with equal number of positive and negative
charges. In etching systems, the extent of ionization is quite small, typically only one
charged particle per 106 - 107 neutral atoms or molecules [115]. The difference in the
mobility of the positive and negative charges causes the formation of a plasma sheath at
the boundaries of the discharge.
Figure 3.4: Schematic diagram showing plasma chamber. The particle motion is shown for
the positive half cycle of the RF bias. Adapted from [2].
Figure 3.5 shows the multiple processes that are involved in etching. Under the in-
fuence of the electric field, the ions are directed perpendicular to the sample causing
anisotropic etching (figure 3.5(a)). If the pressure is high, these ions may cause (i) mask
erosion and (ii) mechanical damage of the sample. Therefore, one of the necessary condi-
tions to control when sputtering is dominant is to maintain a chamber pressure that is less
than 60 mT [115]. In the case where chemical etching is dominant, the etch is selective
but isotropic (figure 3.5(b)). Since chemical etching requires a suitable density of reactive
species to etch the surface, it is dependent upon the RF power that is applied to gener-
ate the plasma. Reliable chemical etching also depends upon the volatility of the reaction
products. Higher volatility of the reaction products would ensure a uniform etch rate of the
89
3. NANOPILLAR ARRAYS IN RTDS AND QCLS: EXPERIMENTAL REALIZATION
Figure 3.5: Schematic diagrams showing four basic methods of plasma etching: (a) Sput-
tering of ions physically removes substrate material, causing directional etching. (b) Ther-
malized neutral radicals chemically react with the substrate material to form volatile
products, causing isotropic etching. (c) In the case where neutral radicals can not form
a volatile product, the impinging ion flux can modify substrate reactivity, leading to en-
ergetic ion-enhanced anisotropic etching. (d) There may be a case where the substrate
and the etchant would react spontaneously and etch isotropically, if it were note for the in-
hibitor species - leading to inhibitor ion-enhanced etching. The inhibitors form a thin film
on vertical surfaces, which acts as a barrier to the etchant and prevents attack of sidewalls,
leading to anisotropic etching. Adapted from [115].
90
3.4 General methods: fabrication of nanopillar arrays
sample. Depending upon the sample and the reactant species in the plasma, there may be
other mechanisms that dominate etching. For example, neutral radicals alone may not be
able to react with the substrate. However, if the surface is modified by an impinging flux of
ions, either by formation of dangling bonds, or by introduction of defects/dislocations in
the lattice, the neutral radicals may chemically react with the substrate (figure 3.5(c)). This
ion-enhanced chemical etching occurs in a direction that is perpendicular to the growth,
and is therefore anisotropic. The etch rate is dependent upon many empirical parameters
such as ion-density, chamber pressure, RF power and substrate to reactant chemistry. In
the fourth case, neutral radicals may spontaneously react with the surface, causing rapid
isotropic etching. In the presence of an inhibitor species, a passivating layer protects the
sidewalls of the feature. Since the direction of the ion flux is perpendicular to the surface,
the ions remove the protective layer on the surface, thereby allowing the neutral radicals
to chemically react with the surface. This mechanism of etching falls under the category of
inhibitor ion-enhanced etching and leads to vertical sidewalls (figure 3.5(d)).
There is a wide choice of dry etching systems and gases that can etch III-V materials
[116, 117, 118]. However, not all chemistries and etch systems, can achieve high-aspect
ratio nanopillars with vertical sidewalls. These reasons may vary from corrosivenes of gas,
control over ion energies, control over ion trajectories, selectivity over the etch mask and so
on. Below, we discuss the systems and gas chemistries that we used to achieve nanopillars
in GaAs / AlxGa1−xAs materials.
3.4.3.1 Reaction ion etching system (RIE)
The schematic diagram of a typical reactive ion etching (RIE) system is shown in figure 3.4.
This system uses a single RF coil operating at 13.56 MHz, and an anode that is larger than
the cathode. Often, the entire chamber is used as an anode. Increasing the anode size
with respect to the cathode raises the voltage drop across the plasma sheath, increasing
the ion energy and therefore etch anisotropy [2, 115]. In this work, an RIE system is em-
ployed under two scenarios. In the first scenario, an RIE system is used to etch SiO2 (see
section 3.5.2.2 and section 3.5.4.3). In the second scenario, an RIE system is used directly
to etch nanopillars, see section 3.5.3.5.
I. OXIDE ETCH
Figure 3.6 shows the schematic diagram of the RIE system, a JLS designs Ltd bench-
top plasma pod, that is used to etch SiO2. The SiO2 layer is etched in a CHF3 and O2
91
3. NANOPILLAR ARRAYS IN RTDS AND QCLS: EXPERIMENTAL REALIZATION
plasma. The plasma etching system has an aluminium chamber (280 mm diameter, 150
mm high) with a substrate electrode (125 mm diameter) that is water-cooled. The top
electrode is grounded and is accompanied with an internal gas distribution head. The
plasma chemistry of CHF3 etching of SiO2 is quite complex and is mediated by several gas
phase reactions [119]. However, the overall reaction can be summarized as 2CHF3 + O2
→ 2CO + H2 + SiF4 [120].
Figure 3.6: Schematic diagram of the reactive ion etching system, used for etching silicon
oxide
II. GaAs / AlxGa1−xAs ETCH
Figure 3.7 shows the schematic diagram of a Plasmatherm load-locked RIE system (SLR
720), that is used to etch GaAs /AlxGa1−xAs. The samples are mounted onto a Si wafer
which rests on an alumina platen in a manually operated load-locked chamber. Prior to
etching, this alumina platen is transferred to the plasma chamber using a mechanical arm.
A load-lock system minimizes exposure of the plasma chamber to ambient moisture and
oxygen which can cause the aluminium in AlxGa1−xAs to oxidize, reducing it’s reactivity
to chemical etching.
The plasma etching system is equipped with a 500 W RF generator and has a substrate
92
3.4 General methods: fabrication of nanopillar arrays
electrode that is 280 mm in diameter. The feed gas that forms the reactive plasma uses
SiCl4 and Ar. Under ideal conditions, this chemistry, selective over a wide range of dielec-
tric, photoresist and metal mask, etches GaAs and AlxGa1−xAs at an equal rate. In the first
step SiCl4 dissociates into Si and 2Cl2. These chlorine molecules collide with electrons to
form atomic chlorine, which get’s adsorbed on the surface as GaAs-Cl. This leads to the
following reaction GaAs-Cl → GaClx + AsCly , where x and y may range from 1 to 3. In
the case of Al, atomic chlorine reacts with the surface to form AlCl3. Other reactive species
in the plasma such as energetic ions (Cl+2 , Cl
+) and electrons may assist the reaction on
the surface [121]. The benefit of introducing Si in the feed gas is that it acts as the in-
hibitor species in the plasma, reacting with the oxygen present in the chamber to form
SiOx protective layer over the sidewalls (see Figure 3.5(d)).
Figure 3.7: Schematic diagram of the reactive ion etching system, used for etching GaAs /
AlxGa1−xAs
3.4.3.2 Inductively coupled plasma (ICP) RIE system
As described in figure 3.5(a), one of the important criterion to maintain directionality of
the etch is low chamber pressure. Reducing the pressure increases the mean free path of
the ions which limits scattering within the reactive plasma and ensures a trajectory per-
93
3. NANOPILLAR ARRAYS IN RTDS AND QCLS: EXPERIMENTAL REALIZATION
pendicular to the sample. While this ensures vertical sidewalls of the feature (nanopillars),
reduction in pressure limits the etch rate and hence the etch depth of the feature. Furthe-
more, reduction in pressure adversely influences the convective cooling of the substrate,
which raises the temperature of the sample for plasma reactions. To compensate for the
reduced etch rate, one can increase the RF power in an RIE, which increases the density of
the reactive species. However, this may influence several other reaction parameters at the
same time.
To circumvent this problem, high aspect ratio structures are generally etched in in-
ductively coupled (ICP) RIE systems [122]. In an ICP RIE, two independent RF sources
operating at 13.56 MHz are used, where one RF coil (chamber) is used to create a high-
density plasma and the other RF coil (table) is used to direct and accelerate the reactive
species from the plasma to the sample. The two RF generators allow independent control
of ion density and the kinetic energy of the reactive species [116, 117, 118].
Figure 3.8: Schematic diagram of the in-house ICP RIE
Figure 3.8 shows the schematic diagram of the in-house inductively coupled reactive
ion etcher (ICP RIE). By adding an additional 300 W RF coil to an existing RIE80 system
(JLS Instruments Ltd), this system was modified to generate the ion density using this
94
3.5 Fabrication of electrically connected nanopillar arrays
additional coil, called the chamber power and control the energy of the ions using the
existing coil, which is called the table power. Commercial ICP RIEs are often accompanied
by a load-lock similar to a system shown in figure 3.7. As mentioned before, a load-lock
system minimizes exposure of the plasma chamber to atmospheric water. This is particu-
larly important for equirate etching of GaAs and AlxGa1−xAs layers, with high Al content,
i.e. x > 0.5. Fortunately, lack of a load-lock system does not significantly effect the dry
etching of THz QCL structures which use AlxGa1−xAs barriers with x = 0.15. However, the
absence of a load-lock system on the in-house ICP RIE system does influence the pumping
and preperation time, limiting the number of etches that can be performed in 24 hours to
at most two.
We overcame this limitation by careful experimental design, i.e. we included multiple
processing conditions and etch masks in the same etch, so as to maximize the process pa-
rameter sweep within a single etch. Upon identification of the optimum processing condi-
tions within one etch, we swept the etch variables to arrive at the most suitable parameters,
required to achieve high-aspect ratio nanopillars in GaAs / AlxGa1−xAs structures. Etch of
high-aspect ratio nanopillars using the in-house ICP RIE is described in section 3.5.3.3 and
section 3.5.3.4.
3.5 Fabrication of electrically connected nanopillar arrays
In this section, we present the sequential progress of fabricating electrically connected
nanopillar arrays, starting from definition of etch-mask to removal of insulation post top-
contact deposition. For each sequence, we specify the parameters that we use to gauge the
quality of processing and then discuss the experimental results.
3.5.1 Processing guidelines
Before listing our processing guidelines, it is useful to collate the ideal characteristics of a
top-down laterally confined THz QCL derived from our theoretical analysis in Chapter 2.
(a) The device should comprise of nanopillars with sub-200 nm radius. Pillars of only
those radial dimensions will lead to emission, which are allowed by the optical selection rules.
The sidewall profile should be vertical, with a tolerance that is dictated by the resonance con-
ditions.
(b) The height of the nanopillar must approach the height of a standard THz QCL. This is
95
3. NANOPILLAR ARRAYS IN RTDS AND QCLS: EXPERIMENTAL REALIZATION
required to achieve sufficient gain to overcome optical losses.
(c) The device architecture should comprise an array of nanopillars, electrically connected
in parallel. This is required to ensure that the material gain of a top-down laterally confined
THz QCL is comparable to the gain of a standard Fabry-Pèrot ridge. Under first approxima-
tion, this condition requires that the array must be electrically and structurally robust to carry
a current density that is comparable to the threshold current density of a THz QCL (100-900
A/cm2 [6, 37, 123]). As discussed in section 2.11, a Fabry-Pèrot ridge of dimensions that is 2
mm long and 100 µm wide is equivalent to 107 nanopillars x (pi R2p), where Rp = 50 nm.
(d) To minimize optical losses in the device architecture the nanopillar array must have
the flexibility of removal of planarization at a later stage to obtain a suspended contact if so
required.
(e) Finally, the nanopillar array should be packed such that the pitch/diameter ratio = 4.
For a square lattice, this ratio ensures maximum figure of merit of the device architecture.
Based on the simulation results described in section 2.11, table 3.1 summarizes the
ideal device parameters of a sub-200 nm pillar array in a QCL that is optimized for THz
emission.
Example radius,
Rp in nm
Ideal pitch,
p in nm
Ideal
Pitch/diameter
(p/2Rp) Ratio
Ideal number of
pillars in an array
Ideal equivalent
physical area in
µm2
35 280 4 5.1 x 107 200000
50 400 4 2.5 x 107 200000
120 960 4 4.4 x 106 200000
150 1200 4 2.8 x 106 200000
200 1600 4 1.6 x 106 200000
Table 3.1: Ideal device parameters of a sub-200 nm pillar array optimized for emission in
a THz QCL.
A direct implementation of this ideal device architecture on THz QCLs would be ar-
dous. Furthermore, it would impede troubleshooting of processing-related problems, as
well as hinder the understanding of electro-optic characteristics which are correlated with
geometry. Therefore, in this work, we have attempted to adhere to some of these ideal
characteristics, which at the same time allow us to demonstrate the proof of principle.
96
3.5 Fabrication of electrically connected nanopillar arrays
These processing guidelines have been designed such that when the nanopillar array is
fabricated in THz QCLs, it meets the stringent structural requirements of THz QCLs and
when fabricated in test RTDs, it meets the electrical requirements imposed by the test RTD.
These processing guidelines are given below.
(a) Multiple arrays of nanopillars are defined using e-beam lithography, where nanopil-
lars in each array have a constant radius. This radius varies between arrays and ranges
from Rp = 50 nm to 200 nm. While the minimum radius is limited by e-beam lithography
(section 3.4.1), we have not attempted to define the radius such that it necessarily permits
intersubband emission.
(b) The etching is optimized to achieve the maximum height that is possible for 50 - 200
nm radius pillars using available dry etching capabilities. This is to experimentally identify
the optimum device dimensions that are structurally stable in top-down laterally confined THz
QCLs.
(c) The device architecture is designed to electrically connect an array of nanopillars in
parallel. The optical mask is flexible to simultaneously connect to any number of pillars de-
fined by e-beam within a footprint of 500 µm x 500 µm. However, to be consistent in our
measurements, we keep the number of nanopillars in an array as constant and equal to 500
x 500. The electrical stability of the array is tested on those RTD active regions, which can
support current densities that are comparable to the threshold current densities of THz QCLs.
(e) The processing route is designed for flexibility in removal of insulation post top-contact
metallization. Therefore, the designed fabrication process allows for both suspended and un-
suspended top-contact.
(f) Independent of the diameter, the pitch of the nanopillar array is kept constant at 1 µm.
Therefore, the pitch/diameter ratio varies for each array. This distance was chosen to meet
gap-fill specifications of conventional planarizing materials in the first instance.
Table 3.2 summarizes the key parameters that were used in the definition of nanopillar
arrays while processing. Note that the parameters of the array shown below do not follow
the optimum array pattern predicted for optimizing the emission from a THz QCL, see
table 3.1. However, they bridge the gap in our understanding of lateral confinement in a
single nanopillar to an array of nanopillars. The physical transport area of the nanopillars
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3. NANOPILLAR ARRAYS IN RTDS AND QCLS: EXPERIMENTAL REALIZATION
is equivalent to the physical transport area of typical large-area RTDs (100 to 40000 µm2).
Radius, Rp
in nm
Pitch, p in
µm
Pitch/diameter
(p/2Rp) Ratio
Number of pillars
in an array
Equivalent physical
area in µm2
50 1 10 500 x 500 490
100 1 5 500 x 500 1963
150 1 3.3 500 x 500 4417
200 1 2.5 500 x 500 7854
Table 3.2: Device paramaters that define the nanopillar pattern while processing
Figure 3.9 shows the schematic diagram of the nanopillar pattern written using e-beam
lithography, keeping the above-mentioned processing guidelines in mind. Note that the
footprint of the array on the chip is constant. Also shown in figure 3.9 is the position of
the top and bottom-contact pads relative to the nanopillar pattern.
Figure 3.9: Schematic diagram showing (a) the nanopillar pattern written using e-beam
lithography (b) the top and bottom-contact pads relative to the the nanopillar pattern.
3.5.2 Etch mask definition
Upon e-beam lithography of the array pattern, the first stage of processing is to transfer the
pattern written on the resist to an etch mask. This etch mask may be a dielectric or a metal.
The choice of etch mask material determines the etch quality and the type of top-contact
to the nanopillar array. Below we discuss the parameters used to assess the merit of an
etch mask.
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3.5 Fabrication of electrically connected nanopillar arrays
3.5.2.1 Etch-mask assessment parameters
So far we have used Rp to indicate the radius of the nanopillar that is written on the
resist. After transferring the pattern from the e-beam resist to the etch mask, the radial
dimensions may vary depending upon material and process conditions. We call the radius
of the etch mask as Rmask. The difference between Rmask and Rp, quantifies the consistency
of transfer, which is our first assessment parameter. The second parameter that we use to
assess the integrity of an etch mask is it’s durability, i.e. the resistance to withstand physical
erosion and chemical reaction during plasma processing of nanopillars. The endurance of
the etch mask may depend upon the thickness, shape and material of the etch mask, but
can effect the aspect ratio and the sidewall profile of the etched nanopillar. While these
conditions may be relaxed for nanopillars in RTDs, they are of significant importance to
nanopillars in THz QCLs where etch depth and sidewall uniformity is a big concern (see
section 2.8).
3.5.2.2 Dielectric mask: results
Here, we describe the process of transferring the e-beam pattern on to a dielectric mask
which is generally used in inductively coupled plasma RIE systems to etch high aspect
ratio features. We review the experimental results for nanopillars in light of the above-
mentioned assessment parameters. The process of transferring the pattern from an e-beam
resist to a dielectric mask is a two stage process. In the first stage, the pattern on the resist
is transferred on to a sacrificial mask. In the second stage, the pattern from the sacrificial
mask is transferrred on to the dielectric layer. This process is summarized in Figure 3.10.
As described in section 3.4.1, the array pattern is written on an e-beam resist (160-
180 nm thick PMMA). This resist is spin-coated over a 100 nm thick dielectric that is
deposited using plasma enhanced chemical vapour deposition (PECVD) on a pre-cleaned
chip cleaved from the wafer of choice. This dielectric layer is essentially SiO2 deposited
at a temperature of 300 oC at a rate of 7 nm/sec. The dielectric thickness was chosen
to be equal to the minimum nanopillar diameter defined using e-beam (2 x 50 nm), to
maintain a mechanically stable mask profile, i.e. an aspect ratio of 1:1. Following e-
beam exposure the resist is developed for 3-4 sec in a solution of MIBK:MEK:IPA (5:1:15)
followed by a rinse in IPA for 30 sec. A thin layer of Ti/Al (5/50 nm) is evaporated, which
acts as a sacrificial mask for the underlying SiO2. Figure 3.11 shows the scanning electron
microscope (SEM) images of Ti/Al disks with a radius Rp = 50 nm, 100 nm and 150 nm
after metal liftoff. Note that the pattern defined by e-beam is transferred from the resist to
the sacrificial mask Ti/Al with a high degree of consistency, i.e. Rp = RTi/Al .
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3. NANOPILLAR ARRAYS IN RTDS AND QCLS: EXPERIMENTAL REALIZATION
Figure 3.10: Process Schematic (a) 100 nm SiO2 deposited on a clean 6 mm x 7 mm chip
(b) e-beam lithography and development of resist (c) evaporation of Ti/Al and lift-off (d)
dry etch of SiO2
Figure 3.11: Scanning electron micrograph of Ti/Al disks after e-beam lithography and
liftoff, shown for three different arrays (a) Rp = 50 nm (b) Rp = 100 nm (c) Rp = 150 nm.
100
3.5 Fabrication of electrically connected nanopillar arrays
In the second stage of pattern transfer, the Ti/Al serves as an etch mask for the under-
lying SiO2 layer. The SiO2 layer is etched in the plasma-pod in a CHF3 and O2 plasma,
see figure 3.6. At an RF power of 75 W, CHF3 (O2) flowrate at 30 (4) sscm and base pres-
sure of 60 mTorr, the plasma etches 100 nm of SiO2 in 6 mins. The sacrificial Ti/Al mask
is removed by rinsing in Microposit developer - MF319 (a N(CH3)
+
4 OH
− based alkaline
solution) for 3 mins, followed by a rinse in DI water.
Figure 3.12: Scanning electron micrograph of SiO2 disks etched in CHF3 and O2 plasma
using Ti/Al mask with (a) RTi/Al = Rp = 50 nm (b) RTi/Al = Rp = 100 nm (c) RTi/Al = Rp
= 150 nm.
Figure 3.13: (a) Schematic diagram of SiO2 disks after etching (b) schematic diagram of
SiO2 disks after removal of Ti/Al layer (c) Scanning electron micrograph of SiO2 disks after
etching, isometric view.
Figure 3.12 shows the scanning electron microscope (SEM) images of SiO2 disks after
etching, using the Ti/Al pattern shown in figure 3.11. We note that the etched SiO2 disks
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3. NANOPILLAR ARRAYS IN RTDS AND QCLS: EXPERIMENTAL REALIZATION
are larger than the Ti/Al masks, i.e. RSiO2 - RTi/Al = RSiO2 - Rp ≈ 50 nm. Figure 3.13(a-b)
shows the schematic diagram of the SiO2 disks immediately after etching, and after re-
moval of the Ti/Al sacrificial mask. This schematic is corroborated with the isometric view
of the SiO2 disks recorded immediately after etching, using scanning electron microscopy,
see figure 3.13(c). The increase in the difference between RSiO2 and RTi/Al ≈ 140 nm in-
stead of 50 nm, is due to reduced CHF3 content in the plasma. In the case of figure 3.13(c),
we used 30:10 sscm, instead of the optimized process involving CHF3 to O2 ratio of 30:4
sscm.
The positive slope associated with the sidewall profile of the etch mask indicates insuf-
ficient ion energy and density during reaction ion etching, leading to reduced sputtering
and chemical removal. Increasing the RF power and pressure of the reactive species can
correct the sidewall profile, reducing the difference between RSiO2 and Rp. However, at-
tempts in the direction of high pressure prompted plasma instabilities within the chamber.
Improvements in power from 75 W to 100 W, and changes in the ratio of CHF3 to O2 from
30:4 to 30:3, 50:2 returned similar enlargement of feature sizes. Under these constraints,
we found that a dielectric etch mask compromised on the consistency of pattern transfer,
even before etching the nanopillars. To correct this inconsistency, we removed the second
stage of pattern transfer and instead focused on using a metal mask directly. Below, we
discuss the fabrication procedure and reproducibility of pattern transfer from e-beam resist
to a metal mask.
3.5.2.3 Metal mask: results
In the case of a metal mask, the pattern is transferred from the e-beam resist to 100-140
nm, thick gold. The schematic diagram of the process is shown in figure 3.14.
Figure 3.14: Process Schematic (a) E-beam lithography and development of resist (c) evap-
oration of NiCr/Au
The thickness of the gold mask is limited by the thickness of the e-beam resist. Evapo-
ration of gold over GaAs requires the use of a wetting layer, which promotes the adhesion
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3.5 Fabrication of electrically connected nanopillar arrays
of gold to the surface. This wetting layer may be 5-10 nm of Ti or Nichrome (NiCr), which
is a non-magnetic alloy of Ni and Cr. While NiCr/Au forms a permanent self-aligned Schot-
tky contact with the top n+ doped GaAs layer, Ti/Au etch-mask can be easily removed in
buffered hydroflouric (BHF) acid if an additional non-aligned ohmic contact is desired.
However, the merits of NiCr/Au contact supersede that of Ti/Au. Unlike NiCr/Au, Ti/Au is
unsuitable for processes such as etch-residue and insulation removal (section 3.5.6). These
processes involve buffered hydroflouric acid, which rapidly reacts with Ti with an etch rate
of 1000 nm/min [124]. Figure 3.23 shows the Ti/Au mask partially removed after clean-
ing the samples in BHF. On these grounds, we prefer to use a NiCr/Au etch mask over a
Ti/Au etch mask.
Figure 3.15 shows the scanning electron micrograph of the NiCr/Au mask after lift off.
Note that the pattern is transferred from the e-beam resist to gold, with high degree of
fidelity, i.e. RNiC r/Au ≈ Rp.
Figure 3.15: Scanning electron micrograph of NiCr/Au etch mask after lift off, shown for
three different arrays (a) Rp = 50 nm (b) Rp = 100 nm (c) Rp = 200 nm.
3.5.2.4 Etch mask summary
In terms of pattern transfer, we note that a metal mask offers a higher degree of fidelity, as
compared to a dielectric mask. This metal mask may either comprise of NiCr/Au deposition
or a Ti/Au deposition. While NiCr/Au forms a permanent self-aligned Schottky contact to
the nanopillars, both Ti/Au and SiO2 can be removed in BHF to deposit a non-aligned,
shallow ohmic contact. This contact may be formed from (a) an AuGeNi slug annealed at
400oC for 40 secs or (b) the evaporation of Pd/Ge (25/75 nm) subsequently scintered at
350oC for 300 secs. However, a secondary contact that involves rapid thermal annealing
jeopardises the integrity of the planarising material [125].
A self-aligned ohmic contact can be formed using a two-stage metal etch mask. This
involves a primary evaporation step of AuGeNi or Pd/Ge, followed by a secondary evap-
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3. NANOPILLAR ARRAYS IN RTDS AND QCLS: EXPERIMENTAL REALIZATION
oration step involving NiCr/Au. However, the combined thickness of the mask increases
with two-stage evaporations, which poses a structural problem for masks with Rp ∼ 50 nm
[94].
Table 3.3 compares the different masks, in terms of consistency of pattern transfer, type
of contact and suitability for further processing. So far, we have not commented on the
endurance of the etch mask to plasma processing. In the following section, we describe
the etch of high aspect ratio pillars in GaAs / AlxGa1−xAs materials using both SiO2 and
NiCr/Au etch masks.
Etch mask assessment
parameters
Dielectric Metal Metal (reports)
SiO2 NiCr/Au Ti/Au Two stage evap-
oration: AuGeNi
(PdGe) + Ti/Au
[94]
Fidelity of transfer RSiO2 - Rp ≥ 50 nm,
Rp = 50 - 200 nm
RNiC r/Au ≈ Rp , Rp
= 50 - 200 nm
RTi/Au ≈ Rp , Rp =
50 - 200 nm
Combined metal
thickness ≥ 200 nm,
not suitable for Rp
= 50 nm
Type of contact Not aligned: Schot-
tky or ohmic
Aligned: Schottky Not aligned: Schot-
tky or ohmic
Aligned: ohmic
Suitability for planariza-
tion and insulation re-
moval
Yes, with Schottky
contact
Yes Yes, with Schottky
contact
Untested in this
work
Table 3.3: Comparison of etch masks in terms of fidelity of pattern transfer, type of contact
and suitability for further processing.
3.5.3 Etch of high aspect ratio pillars
The central step in realizing top-down laterally confined THz QCLs is the high quality
etching of high aspect ratio nanopillars. In this work, we have attempted to identify ex-
perimental processes and systems that maximize the aspect ratio and maintain a vertical
sidewall profile for pillars with sub-200 nm radial dimensions. Below, we describe the
factors that limit the aspect ratio of nanopillars in GaAs / AlxGa1−xAs materials.
3.5.3.1 Factors affecting the aspect ratio
There are two factors that limit the aspect ratio of the nanopillar during plasma process-
ing. The first factor that comes into play is the mask erosion. Ideally, the resistance of
an etch mask to plasma processing should be infinite. In reality, the energy of ions erodes
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3.5 Fabrication of electrically connected nanopillar arrays
the material of the mask, albeit at a much lower rate as compared to GaAs (AlxGa1−xAs).
Therefore, the height of the nanopillar that can be successfully etched is equal to the thick-
ness of the etch mask times the ratio of the etch rates between the mask and the GaAs
(AlxGa1−xAs). The second factor that influences the etch depth is the aspect ratio depen-
dent etching (ARDE) [126]. This second parameter encompasses multiple processes that
cause the etch rate to reduce with time. These processes include aspect ratio dependent
transport of neutrals, and shadowing of ions and neutrals [126]. In the first case, transport
of neutrals is attenuated due to reflections from the sidewalls as they travel to the surface.
This occurs when the neutrals have have mean free path length larger than the channel
length, which in the case of an array is p - 2Rp. In the second case, as the etch depth
increases, the angular spread function of the ions and neutrals decreases. This reduces the
number of ions and neutrals reaching the etch surface. Under this scenario, the radicals
dominate the etching and attack the sidewalls of the etched feature, which leads to under-
cutting and structural instability of the nanopillar array. This effect worsens if the channel
length, or the p - 2Rp reduces, and is commonly described as RIE lag [116, 126].
3.5.3.2 Etch assessment parameters
In this section we define the parameters employed to assess the quality of the etch. In
an ideal scenario, the radial dimensions of the nanopillar should be transferred with high
degree of fidelity, from the e-beam stage to the etch-mask stage and finally to the etched
pillar stage. This implies that Rp = Rmask = Retch, where Retch is the radial dimension of the
pillar after etching. Furthermore, the nanopillar of height H and aspect ratio Retch |z=0:H
should have a vertical profile, i.e. Retch |z=H = Retch |z=0, see figure 3.16(a). However,
the dimensions and the sidewall profile may alter significantly from the ideal geomtery
after processing the nanopillars in a reactive ion etching system. In order to quantify this
deviation, we characterize the quality of plasma processing with the following parameters.
The first parameter quantifies the consistency of pattern transfer from the e-beam stage
to the etch stage. Accounting for dimension variations along the growth direction (z), we
define an average radius of the nanopillar as Retch |z=H/2 = Retch|z=H+Retch|z=02 . The difference
between this average radius and the radius defined on the e-beam resist Rp, is defined
as ∆Retch and it quantifies the consistency or fidelity of pattern transfer. The second pa-
rameter quantifies the profile of the sidewall. The inclination of the sloping sidewall with
the vertical axis, defined as θ = tan−1 Retch|z=H−Retch|z=0
H
characterizes the closeness of the
sidewall profile to the ideal geometery. While a positive θ indicates insufficient etching,
see figure 3.16(b), a negative θ implies an undercut of the nanostructure due to enhanced
105
3. NANOPILLAR ARRAYS IN RTDS AND QCLS: EXPERIMENTAL REALIZATION
chemical reactivity, figure 3.16(c).
Figure 3.16: Sidewall profile of nanopillars after reactive ion etching, shown in the case
of (a) ideal conditions (b) insufficient physical and chemical etching, and (c) undercutting
due to greater chemical reactivity.
3.5.3.3 Etch in ICP using a dielectric mask: results
In this section we discuss the results obtained from etching nanopillars in the in-house ICP
RIE, see section 3.4.3.2, using a dielectric etch mask. Before etching, the chamber is pre-
cleaned in CHF3/O2 plasma at low and high pressures for 30 min each. This pre-treatment
cleans the ICP tube of any depositions, identified as containing Si-O-F-Cl [127] that can
start to detach from the tube walls after multiple etches. The chamber is cleaned further
in an O2 plasma to remove any hydrocarbon residue on the chamber walls. To ensure
reproducibility of the etch, the chamber is preconditioned with the SiCl4/Ar etch process
without loading the samples. This involves tuning a network of reactive elements to match
the impedance of the RF coil with the reactive load (the plasma). The temperature of the
substrate electrode is maintained at 18oC
I. ETCHING UNDER STANDARD RECIPE
The samples were post-baked at 80oC for 10 min to expel all solvents. Following the
bake, the samples were mounted on the substrate using fomblin oil onto a 6" Si wafer
clamped to the substrate electrode. After pumping the plasma chamber for 12 hours, post
106
3.5 Fabrication of electrically connected nanopillar arrays
mounting, the GaAs/AlxGa1−xAs nanopillars were etched in a SiCl4/Ar chemistry with a
flow of 5/2 sscm and a pressure of 4 mTorr.
Figure 3.17 shows the scanning electron microscope images of nanopillars etched at a
table power of 55 W and a chamber power of 100 W. It is encouraging to note that pillars
of Rp ≥ 200 nm show a high degree of fidelity (∆Retch ≈ -10 nm) and maintain a vertical
profile, with θ = -1.14o. However, pillars with Rp ≤ 100 nm do not survive the plasma
processing and collapse due to undercutting.
Figure 3.17: Scanning electron micrograph of nanopillars after being etched at 55 W table
power and 100 W chamber power, in ICP-RIE, using 100 nm thick SiO2 etch mask. The
profile is shown for radial dimensions (a) Rp = 50 nm, p/2Rp = 10 (b) Rp = 100 nm,
p/2Rp = 5 (c) Rp = 200 nm, p/2Rp = 2.5.
Below, we discuss the effect of chamber power on the fidelity and sidewall profile of
the nanopillars.
II. EFFECT OF CHAMBER POWER
Upon reducing the chamber power from 100 W to 70 W, the density of ions is reduced,
which improves the etch profile of nanopillars with Rp= 100 nm, see figure 3.18. We
measure an undercut angle θ = -1.96o and the fidelity of pattern transfer to be ∆Retch ≈
-20.5 nm. The undercut angle of pillars with Rp = 200 nm improves from θ = -1.14o,
figure 3.17(c) to θ = -0.16o, figure 3.18(c). However, the improvement in the sidewall
profile of pillars with Rp= 200 nm comes at the cost of a lower fidelity of pattern transfer,
with ∆Retch ≈ 16 nm. In both the scenarios, figures 3.17 and 3.18, Retch is less than Rmask.
This indicates mask erosion in the initial stages of the etch, followed by anisotropic etching
107
3. NANOPILLAR ARRAYS IN RTDS AND QCLS: EXPERIMENTAL REALIZATION
of nanopillars resulting in an undercut. The schematic diagram of the process is shown in
figure 3.19.
Figure 3.18: Scanning electron micrograph of nanopillars after being etched at 55 W table
power and 75 W chamber power, in ICP-RIE, using 100 nm thick SiO2 etch mask. The
profile is shown for radial dimensions (a) Rp = 50 nm, p/2Rp = 10 (b) Rp = 100 nm,
p/2Rp = 5 (c) Rp = 200 nm, p/2Rp = 2.5.
Figure 3.19: Schematic diagram, showing etching of nanopillars in ICP-RIE using a dielec-
tric etch mask.
III. EFFECT OF MOUNTING
We also note that the mounting procedure can often introduce variability in the etch
profile as the temperature influences the plasma reactions at the etch surface. If the sam-
108
3.5 Fabrication of electrically connected nanopillar arrays
ples are glued to a 2" Si wafer with a silver epoxy compound for enhanced thermal dissi-
pation instead of mounting them using the procedure followed in figure 3.17, the results
can vary. We note that if the epoxy is applied uniformly over the backside of the chip, the
undercutting of the nanopillars is reduced, see figure 3.20. At the same process conditions
as figure 3.17, we observe a positive inclination of the sidewall profile instead of undercut-
ting. We further note that although pillars of Rp = 50 nm survive the plasma processing,
the fidelity of the pattern transfer is severely compromised. For instance, pillars with Rp =
100 nm (200 nm) in figure 3.20, show ∆Retch = 52 nm (75 nm). This is nearly 2 (3) times
higher than ∆Retch = 20.5 nm (16 nm), in figure 3.18.
Figure 3.20: Scanning electron micrograph of nanopillars after being etched at 55 W table
power and 100 W chamber power, in ICP-RIE, using 100 nm thick SiO2 etch mask. The
samples are mounted to enhance thermal dissipation. The profile is shown for radial di-
mensions (a) Rp = 50 nm, p/2Rp = 10 (b) Rp = 100 nm, p/2Rp = 5 (c) Rp = 200 nm,
p/2Rp = 2.5.
IV. EFFECT OF INCLUDING PUMPING STAGE DURING ETCH
From figure 3.17 and figure 3.18, we infer that two processes limit the aspect ratio of
the nanopillars. In the initial stages of the etch, the sputtering of ions modifies the shape of
the etch mask. In the later stages of the etch, plasma reactions cause anisotropic etching of
the nanopillar. A common strategy to inhibit lateral etching and improve the aspect ratio, is
to employ alternate cycles of etching and passivation achieved by introducing appropriate
feed gas at each cycle [128]. This strategy was difficult to implement on the in-house ICP
109
3. NANOPILLAR ARRAYS IN RTDS AND QCLS: EXPERIMENTAL REALIZATION
RIE, with dedicated gas lines. We therefore replaced the cycle of passivation with pumping
to minimize the amount of non-volatile reactive species present on the surface. Therefore,
between etch cycle, we introduced a pumping cycle to bring the chamber back to the base
pressure.
Figure 3.21: Scanning electron micrograph of nanopillars after being etched in three
stages, interspersed with two pump cycles at 55 W table power and 100 W chamber power,
in ICP-RIE, using 100 nm thick SiO2 etch mask. The profile is shown for radial dimensions
(a) Rp = 50 nm (b) Rp = 100 nm (c) Rp = 200 nm.
Figure 3.21 shows the scanning electron micrograph of nanopillars etched in three
steps, interspersed with two pump cycles. The effective etch rate is 306 nm/min. The
aspect ratio of pillars with alternate etch-pump cycles (figure 3.21) is far superior to the
aspect ratio shown in figure 3.17. Compared to the collapse of Rp = 100 nm nanopillars
in figure 3.17(b), we observe pillars of the same dimension demonstrating an aspect ratio
of 1:12. It is interesting to note that the stability of the Rp = 100 nm pillars is not af-
fected by the undercut at the base of the etched surface as shown in figure 3.17(b). The
attenuated transport of the reactant species in the vertical direction, and scattering with
the sidewalls of the nanopillars leads to lateral etching of the nanopillars at various places,
figure 3.21(b). From figure 3.21, we further note that even though all the nanopillar ar-
rays are etched at the same time, the etch depth of Rp= 200 nm pillars is less than the etch
depth of the Rp= 100 nm pillars. This is a consequence of the RIE lag that arises due to
the packing density of the array which contains nanopillars of Rp= 200 nm. The distance
between the nanopillars of radial dimensions Rp= 200 nm (p - 2Rp = 1000 - 400 = 600
110
3.5 Fabrication of electrically connected nanopillar arrays
nm), is less than the distance between the nanopillars of radial dimensions Rp= 100 nm (
p - 2Rp = 1000 - 200 = 800 nm).
3.5.3.4 Etch in ICP using a metal mask: results
In the previous section, we discussed that high fidelity of pattern transfer from the e-beam
resist to the mask can be obtained using a dielectric mask. This requires etch conditions
that promote mask erosion and anisotropic etching. High consistency of pattern transfer
through the dielectric route comes at the cost of destroying nanopillars with radial dimen-
sions that are defined to be close to 50 nm. Etch conditions can be engineered to obtain
structurally stable nanopillars of Rp = 50 nm. However, such conditions are non-ideal
and lead to enlargement of nanopillar dimensions after the etch. In this section we assess
the consistency of pattern transfer from the e-beam resist to the nanopillars using a metal
mask.
I. ETCHING UNDER OPTIMIZED RECIPE
Figure 3.22 shows the scanning electron micrograph of nanopillars etched at 55 W ta-
ble power and 100 W chamber power in the in-house ICP RIE using a 100 nm thick Ti/Au
etch mask. The samples are mounted on a 2" Si wafer with a silver epoxy compound for
enhanced thermal dissipation and etched in a SiCl4/Ar chemistry with a flow 5/2 sscm and
a pressure of 4 mTorr. . Compared to figure 3.20, we observe an improvement in the side-
wall profile and consistency of pattern transfer for nanopillars with all radial dimensions,
i.e. between 50 nm and 200 nm, see figure 3.22. For example, nanopillars defined with
an Rp = 200 nm, but etched with a dielectric mask in the in-house ICP RIE show a fidelity
∆Retch = 75 nm and a sidewall inclination θ = 2.64o. On the other hand, nanopillars of
the same dimensions, but etched with a metal mask in the in-house ICP RIE show a fidelity
∆Retch = 9 nm and a sidewall inclination θ = 2.45o.
The scanning electron micrograph shown in figure 3.22 has been recorded after rinsing
the nanopillars in buffered hydroflouric acid to remove the polymer residue deposited on
the surface during the etch. However, the etchant reacts with Ti, removing the self-aligned
contact from the pillars. To correct this, we use NiCr/Au as the etch mask instead of Ti/Au,
which yields similar etch profile as the latter.
II. EFFECT OF INCLUDING PUMPING STAGE DURING ETCH
111
3. NANOPILLAR ARRAYS IN RTDS AND QCLS: EXPERIMENTAL REALIZATION
Figure 3.22: Scanning electron micrograph of nanopillars after being etched at 55 W table
power and 100 W chamber power, in ICP RIE, using 100 nm thick Ti/Au etch mask. The
samples are mounted to enhance thermal dissipation. The profile is shown for radial di-
mensions (a) Rp = 50 nm, p/2Rp = 10 (b) Rp = 100 nm, p/2Rp = 5 (c) Rp = 200 nm,
p/2Rp = 2.5.
Figure 3.23 shows the scanning electron micrograph of nanopillars, with a Ti/Au mask
etched in three steps, interspersed with two pump cycles. The effective etch rate is 293
nm/min. The pillars were etched in the same active region, and at the same time as
the pillars shown in figure 3.21. The results indicate one of the best aspect ratios that
have been demonstrated at Cavendish Labs. Although, nanopillars with Rp = 50 nm are
destroyed completely, nanopillars with Rp = 100 nm remain structurally stable with an
aspect ratio of ≈ 1:15. The maximum height of the nanopillars shown in figure 3.23(b) is
limited by underetching at the base of the nanopillar, unlike lateral etching near the mask
and sidewalls as shown in figure 3.21(b). This implies that perhaps the endurance of the
metal mask is better than the endurance of the dielectric mask. We also note once again,
that when the pillars are etched with a metal mask, the fidelity of pattern transfer (∆Retch
= 9 nm, Rp = 200 nm) is superior to the fidelity of pattern transfer, when the pillars are
etched under the same conditions with a dielectric mask (∆Retch = 40 nm, Retch = 200
nm).
Although, a metal etch mask yields better sidewall profile and consistency of pattern
transfer, it’s use as an etch material is limited in most ICP RIE systems due to possibilities
of micromasking. Micromasking is a phenomenon that occurs under high powers and feed
gas flow rates, which promotes sputtering of the gold mask on to the GaAs surface [129].
112
3.5 Fabrication of electrically connected nanopillar arrays
It is for this reason, we abstained from processing the nanopillars in the in-house ICP RIE
and shifted the process to a Plasmatherm load-locked RIE system, which permits metal-
based processing. The results obtained from etching nanopillars in an RIE system, with a
metal mask are discussed below.
Figure 3.23: Scanning electron micrograph of nanopillars after being etched in three
stages, interspersed with two pump cycles at 55 W table power and 100 W chamber power,
in ICP-RIE, using 100 nm thick Ti/Au etch mask. The profile is shown for radial dimensions
(a) Rp = 50 nm, p/2Rp = 10 (b) Rp = 100 nm, p/2Rp = 5 (c) Rp = 200 nm, p/2Rp = 2.5.
3.5.3.5 Etch in RIE using a metal mask: results
A major convenience of the Plasmatherm (SLR 720) RIE system is that it offers a load-lock
system. This reduces the pumping time from 12 hours to 0.5 hours, allowing multiple
etches in a 24 hour period and faster optimization of the etch conditions. Before mounting
the samples, the chamber is cleaned with SF6/O2 and O2 plasma, which is followed by
conditioning with the desired SiCl4/Ar etch recipe. The samples were mounted onto a 6"
Si wafer with a drop of fomblin oil which rests on an alumina platen. The temperature of
the substrate electrode was maintained at 20oC.
The effect of etch parameters on the rate and quality of etching nanopillars in the load-
locked RIE system is shown in figure 3.24. As the pressure is increased from 10 mTorr to
25 mTorr, the etch rate increases. This is a result of increased ion energy causing enhanced
physical sputtering of the etched species. Therefore, as the pressure is increased, nanopil-
lars start to collapse. At a pressure of 25 mTorr, all nanopillars with radial dimensions Rp
≤ 200 nm collapse after 10 min of etching. Only those nanopillars survive reaction ion
113
3. NANOPILLAR ARRAYS IN RTDS AND QCLS: EXPERIMENTAL REALIZATION
etching whose radial dimensions are larger than Rp = 250 nm. It is only at a pressure of
10 mTorr, where nanopillars of Rp = 50 nm survive plasma etching.
Figure 3.24: Etch rate vs. process parameters to etch nanopillars in an RIE, with a NiCr/Au
etch mask (a) at constant power of 100 W and SiCl4/Ar flowrate of 4/10 sscm (b) at
constant pressure of 10 mTorr and SiCl4/Ar flowrate as 4/10 sscm (c) at constant pressure
of 25 mTorr and constant power of 100 W (d) scanning electron micrograph of nanopillars
etched at a pressure of 10 mTorr and an RF power of 150 W in SiCl4/Ar (4/8 sscm)
chemistry.
At this pressure, the etch rate is only 85 nm/min. Etch rate can be increased by in-
creasing the RF power, which leads to enhancement in the density of ions. We find that the
etch rate increases from 85 nm/min to 100 nm/min by increasing the RF power from 100
W to 150 W, beyond which the rate saturates, see figure 3.24(b). The etch rate can also be
enhanced by increasing the ratio of the SiCl4 to Ar flow rate, figure 3.24(c). However, this
increase must be made with care as a higher SiCl4 content in the gas mixture promotes
anisotropic chemical etching, leading to undercutting of the pillars.
Figure 3.24(d) shows the scanning electron micrograph of GaAs / AlxGa1−xAs nanopil-
lars etched with a NiCr/Au mask, in the Plasmatherm RIE system at a pressure of 10 mTorr,
114
3.5 Fabrication of electrically connected nanopillar arrays
RF power of 150 W and SiCl4/Ar gas flowrate of 4/8 sscm. While the fidelity of pattern
transfer (∆Retch = -32 nm, Rp = 150 nm) is not among the best that has been reported
in this work, the sidewall profile and the aspect ratio are satisfactory (θ = -0.38 o, aspect
ratio = 1:7). The etch depth is limited by mask erosion which starts to play a role be-
yond 2.3 µm. For the pillars shown in figure 3.24(d), except towards the end where the
dimensions of the pillar are 2Retch|z=0 = 214 nm, the variation in the sidewall profile falls
within the acceptable tolerance in radial dimensions calculated for the larger pillars (Rp =
108.2 to 165.5 nm, window R in figure 2.21 and Rp = 107 nm to 162.1 nm, window S in
figure 2.23).
3.5.3.6 Nanopillar etching summary
According to the processing guidelines discussed in section 3.5.1, the etch processes were
developed in view of maximizing the aspect ratio of GaAs / AlxGa1−xAs nanopillars. In
table 3.4, we compare the etch quality of nanopillars that demonstrated the highest aspect
ratio under three different types of processes: (i) nanopillars etched with a dielectric etch
mask in an ICP RIE (ii) nanopillars etched with a metal etch mask in an ICP RIE, and (iii)
nanopillars etched with a metal etch mask in an RIE.
Etch assessment parameters ICP RIE RIE
Dielectric mask Metal mask Metal mask
Highest aspect ratio achieved 1:12 1:15 1:7
Etch depth 3.56 µm 3.4 µm 2.3 µm
Fidelity of transfer, Rmask - Rp ≈ 50 nm ≈ 2 nm ≈ 2 nm
Fidelity of transfer, Retch - Rp ∆Retch = +35 nm
(Rp = 100 nm)
∆Retch = -15 nm
(Rp = 100 nm)
∆Retch = -32 nm
(Rp = 150 nm)
Sidewall inclination +0.53o -1.1o -0.38o
Endurance of etch mask Mask erosion occurs
as etch depth ap-
proaches 3 µm
No mask erosion oc-
curs as etch depth
approaches 3 µm
Mask erosion occurs
as etch depth ap-
proaches 2.3 µm
Suitability for THz QCL High aspect ratio,
but sputtering may
occur upon reaching
end point.
High aspect ratio
and fidelity, but
sputtering may
occur upon reaching
end point.
Medium aspect ratio
and fidelity, sputter-
ing may occur upon
reaching end point,
but permitted in the
chamber.
Table 3.4: Comparison of systems and processes in terms of etch assessment parameters
and suitability for processing laterally confined THz QCLs.
Nanopillars etched with a metal mask in an ICP RIE show the highest aspect ratio
among the three processes. Furthermore, this process demonstrates the highest fidelity
of pattern transfer and resistance to plasma processing. In spite of superior credentials,
115
3. NANOPILLAR ARRAYS IN RTDS AND QCLS: EXPERIMENTAL REALIZATION
nanopillars etched in an ICP RIE with a metal mask may not be appropriate for processing
laterally confined THz QCLs. This is because of micromasking from metal erosion, which
is a risk while etching GaAs / AlxGa1−xAs deep features in an ICP RIE. Etching nanopillars
with an improved dielectric mask in an ICP RIE may also not be suitable for processing lat-
erally confined THz QCLs. From table 3.4, we infer that the highest etch depth, for GaAs /
AlxGa1−xAs nanopillars is limited to 2.3 to 3.56 µm. Such low etch depths impose an addi-
tional constraint on the processing of laterally confined THz QCLs. The laterally confined
THz QCL must use metal-metal waveguide configuration. Therefore, unlike nanopillars in
described in this section, which have a GaAs end point, the nanopillars etched in a lasing
ultrathin THz QCL would have a metal contact as the end point. This increases the risk
of metal exposure to the plasma chamber in an ICP RIE system. Exposure of the plasma
chamber to the metal can be minimized in an ICP RIE by using a hybrid mask and by
stopping the etch a few nanometers before the metal contact by using in-situ monitoring.
However, the feasbility of these processes was beyond the timescale of this work.
Instead of using the ICP RIE for etching THz QCLs, we have attempted to optimize
etching of sub-200 nm pillars on an RIE system. Although, the quality of etch in terms of
the aspect ratio is not the best (1:7) and the etch depth is limited to 2.3 µm, the result
holds promise to investigate lateral confinement.
3.5.4 Insulation
One of the most crucial steps in the fabrication of electrically connected nanopillar arrays
is the insulation of the nanopillars. This involves sidewall insulation and planarization of
the underlying topography to enable top-contact fabrication. The processing guidelines,
discussed in section 3.5.1, require a planarizing material which offers the flexibility of re-
moval post top-contact fabrication. The foremost rationale behind this requirement is that
the insulation adds significant absorption losses, which are detrimental to the observation
of any weak electroluminescence from laterally confined THz QCLs, see section 2.11.5. Be-
sides the optical properties of the active region, the electrical permittivity and conductivity
of the insulation plays a big role in determining the figure of merit of the waveguide. The
second argument behind ensuring flexibility of removal is that the insulating layer may
passivate the charge on the sidewall of the nanopillars defined in RTDs and QCLs [97].
While this is difficult to quantify, the choice of insulation may influence the density of sur-
face states Di t , which in turn affects the Fermi level pinning at the nanopillar surface and
hence the depletion width, see sections 2.5.2 and 2.6. In the next section, we discuss the
qualities of an ideal planarizing material and assess the merits of some of the commercially
116
3.5 Fabrication of electrically connected nanopillar arrays
available insulators.
3.5.4.1 Insulation assessment parameters
From the perspective of device processing it is essential to choose an insulator that shows
planarising properties, i.e. it can non-conformally coat the underlying topography. Fur-
thermore, the insulator must have good gap fill properties, i.e. a small L, see figure 3.25.
Here, L = p - 2Rp, where p is the pitch and Rp is the radial dimension of the nanopillar. Fi-
nally, the thickness of the insulator must be sufficient to planarize nanopillars of etch depth
H, which is typical of ultrathin THz QCLs. The thickness of the insulator after planariza-
tion generally depends on it’s viscosity. While certain materials may planarize a feature
of height H upon a single coat, others may need to be coated multiple times to reach the
desired height.
In general, the percent planarization is evaluated empirically in terms of the equation:
%P =

1− Z
H
abs(γ)
90

100 (3.1)
where Z is the height of the insulator over the nanopillars and γ is the angle that measures
the smoothness of the insulator after planarising, see figure 3.25. For uniform planarization
and insulation it is necessary to reduce Z , the buildup of the insulator above the nanopillar.
At the same time, it is also necessary to reduce γ, the angle subtended by the insulation
with the horizontal surface, to ensure a robust insulation.
Figure 3.25: Planarization schematic
Apart from these processing parameters, the criteria used while evaluating the merit
of each insulator is the ability to form suspended top-contact to the nanopillars. Table 3.5
compares a selection of standard insulators against parameters such as single coat thick-
ness and ease of removal. Table 3.5 also lists the reported refractive index of the insulators
at THz frequencies.
117
3. NANOPILLAR ARRAYS IN RTDS AND QCLS: EXPERIMENTAL REALIZATION
Material Processing parameter Optical parameter
Thickness Ease of Removal Refractive index, ηins
SU8 (SU-8 2) 2.0 µm at 2000 rpm Dry etch, difficult to
remove after cure
1.7 [130]
Polyimide (HD4104) 3 µm at 5000 rpm Removed in oxygen
plasma and HF after
cure
1.8[131]
BCB (Cyclotene 4022-
35)
2.6µm at 5000 rpm Dry etch, difficult to
remove after cure
1.58 [132]
Resist (1828) 2.8µm at 5000 rpm Removed in Acetone -
HSQ (XR 1541-E) 200 nm at 1000 rpm Can be removed in HF
after cure
1.96[133]
SOG (Methylsiloxane
500 F)
640 nm at 2000 rpm Can be removed in HF
after cure
1.96[133]
Table 3.5: A comparison of planarizing materials against single coat thickness on a bare
substrate, ease of removal and refractive index at THz frequencies. Thickness has been
compared against the following product suppliers - SU8 from MicroChem (SU-8 2), Poly-
imide from Dupont (HD4104), Benzocyclobutene (BCB) from Dow Chemicals (Cyclotene
4022-35), Resist from Shipley (1828), Hydrogen siloxane (HSQ) from Dow Corning (XR
1541-E), Spin-on-glass (SOG) from Filmtronix (Methylsiloxane 500 F).
Although the single coat thickness of the three planarizing materials SU8, polyimide
and BCB is in the range of typical ultrathin THz QCLs, they are difficult to remove post top-
contact fabrication. On the other hand, while the single coat thickness of the photoresist is
suitable for ultrathin THz QCLs, it can be easily removed in a common cleanroom solvent
(acetone). Ideally, the planarization layer must be removable by chemistries that do not
interfere with lithographic stages. HSQ and SOG, are planarizing materials that can be
left as is or be removed in acid, if required. However, use of HSQ to planarize features
is experimentally tasking. With a single coat thickness of 200 nm, it requires 10 to 18
spin-coating stages to planarize features of etch depth in the range of 2 to 3.5 µm. On the
contrary, with SOG, it requires 3 to 5 coats to planarize features of the same etch depth.
For these reasons, we have chosen the SOG process to planarise the nanopillars.
3.5.4.2 Trilayer spin-on-glass process
The SOG process necessiates two conformal depositions CVD1 and CVD2, before and after
spin coating, hence the name trilayer process, see figure 3.26(a)-(d). In an SOG trilayer
process, the first conformal layer improves the adhesion of the spin on glass on metallized
contacts [134]. Following the deposition of CVD1, the chip is planarized with SOG. Mul-
tiple spin coating stages may be required to planarize features if the etch depth is larger
than the single coat thickness. Because the planarised SOG is quite smooth, a second thin
118
3.5 Fabrication of electrically connected nanopillar arrays
layer of SiO2 (CVD2) is deposited by PECVD to improve ball bonding of gold contacts on
top of the insulator. Following the trilayer insulation process, the n+ GaAs layer (or self
aligned mask) on top of the nanopillars is exposed by etching the insulator back in a plasma
chamber. This process is called etch back of the planarizing material, see figure 3.26.
Figure 3.26: Trilayer planarisation process schematic showing (a) nanopillars processed by
reactive ion etching (b) conformal coating of first PECVD layer, CVD1 (c) planarisation by
SOG, spin coated in multiple stages (d) conformal coating of second PECVD layer, CVD2
(e) etch back of planarized nanopillars, with exposed top-contact. The arrows show the
cross-section of the planarised and etched back nanopillars.
Figure 3.26(a) illustrates the deposition of a thin layer of SiO2 (CVD1) by plasma
enhanced chemical vapour deposition (PECVD) at a rate of 7 nm/min. Following a pre-
bake at 125oC, SOG is spin coated under nitrogen atmosphere, humidity≤ 30 %, at a speed
determined by the height, H. This is shown in figure 3.26(b). For a longer shelf life, it is
important to store SOG at a temperature of 4oC. However, the solution must be allowed
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3. NANOPILLAR ARRAYS IN RTDS AND QCLS: EXPERIMENTAL REALIZATION
to warm up to the ambient temperature before spin coating to prevent condensation of
water vapour. Another essential requirement while spin coating is to use a plastic syringe,
as spin-on-glass tends to coagulate in a glass syringe or pipette. Following the spin-coating
process, the sample is post baked at 65oC, 115oC, 125oC and 200oC for a minute each,
after which it is cooled to 125oC and 115oC for a minute, bringing it to bake at 65oC for 10
minutes, see figure 3.27(a). The above process is repeated when applying a second coat.
It is important to post bake and cool slowly as SOG can tend to crack. The resist is then
cured under nitrogen gas at a ramp rate of 2.0 oC per min from ambient temperature to
400oC and dwelled for an hour at 400 o C. Figure 3.26(c) illustrates the deposition of the
second layer of SiO2 (CVD2) by plasma enhanced chemical vapour deposition (PECVD) at
a rate of 7 nm/min. It is important to note that to obtain a lower leakage current density,
the insulation has to be free of voids typically associated with a poor or a rough etch
back. After curing the spin-on-glass at 400 o C, and depositing the second PECVD layer,
windows are selectively opened by optical lithography in the photoresist so as to protect
the majority of the insulation, and expose only those regions on the chip that define the
nanopillar array, see figure 3.26(e). The exposed insulation is etched in a CHF3/O2 (30/2
sscm) plasma, at a pressure of 60 mTorr and RF power of 75 W in the plasma pod described
in section 3.4.3.1.
The three parameters that need to be engineered to ensure smooth planarisation and
etch back are: (i) thickness t1, of the first conformal layer CVD1 (ii) SOG spin speed
and number of coats, which determines thickness tSOG , of the spin on glass, and (iii)
thickness t2, of the second conformal layer CVD2. A high t1, has the benefit of reducing
the thickness of SOG required to planarise the nanopillars, thereby reducing the number
of coats required to fill a certain etch depth. On the other hand, a low t1 improves the gap
filling length, which is given as L = p - 2Rp - 2t1. The minimum gap filling length that
can be achieved using SOG 500F is limited by the manufacturer’s specifications to ∼ 200
nm [125]. As discussed in Chapter 3, pitch p determines the effective refractive index of
the nanopillar array, see eqns. 2.37 and 2.38. Therefore, given L and t1, we can estimate
the lower limit to the value of p that is feasible using the SOG process. This lower limit is
given by L + 2t1 + 2Rp. Therefore, the minimum p/2Rp ratio that can be experimentally
achieved by the SOG process is given by p/2Rp = t1/Rp + 200/2Rp + 1, where t1 and Rp
is in nm. Table 3.6, shows the minimum p/2Rp ratio that is constrained by planarization,
for each radial dimension in the case when the thickness of the CVD1 layer t1 is 25 nm and
when it is 130 nm.
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3.5 Fabrication of electrically connected nanopillar arrays
Figure 3.27: Schematic diagram, showing (a) the optimized post baking procedure after
spin coating SOG and, (b) optimized curing procedure of SOG in nitrogen.
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3. NANOPILLAR ARRAYS IN RTDS AND QCLS: EXPERIMENTAL REALIZATION
Minimum Pitch/diameter
(p/2Rp) ratio constrained by
planarization
CVD1 thickness, t1 in nm
25 nm 130 nm
Radius, Rp in nm
50 p/2Rp = 3.5 p/2Rp = 5.6
100 p/2Rp = 2.25 p/2Rp = 3.3
150 p/2Rp = 1.83 p/2Rp = 2.53
200 p/2Rp = 1.63 p/2Rp = 2.25
Table 3.6: Pitch/diameter (p/2Rp) ratio constrained by planarization for each radial di-
mension, when t1 = 25 nm and when t1 = 130 nm.
Figure 3.28 shows the leakage characteristics of the etched back insulation at room
temperature. The trilayer comprises of 130 nm thick CVD1, approximately 500 nm thick
spin on glass and 30 nm thick CVD2 layer. The footprint of the top-contact is the same as
the footprint of a standard 500 x 500 nanopillar array, which is equal to 250000 µm2, see
inset in figure 3.28(a). Sample 1 and 2 represent two reference devices for leakage test on
the same chip. The measured current through the test device is 500 nA, which is flat in
the voltage range 0 to 3 V. This current corresponds to a current density of 0.1 mA/cm2,
which is four-five orders lower than the current density through the nanopillars in RTDs
(hundred to thousand A/cm2). For comparison, the leakage current of the two samples is
plotted together with the room-temperature current through three 500 x 500 nanopillar
arrays, that were contacted in the same manner as the test devices, figure 3.28(b). The
nanopillar characteristics, which show increasing current with applied bias and a distinct
NDR peak, correspond to nanopillars processed from W0021, with radial dimensions Rp =
50 nm, 100 nm and 150 nm. Further details about the experimental set-up are discussed
in section 3.7.
3.5.4.3 Results from trilayer spin-on-glass process
In this section, we discuss the optimization of planarization and etch back process. Fig-
ure 3.29(a) shows the cross-sectional view of nanopillars imaged after deposition of the
first conformal layer (100 nm of SiO2) and planarisation by SOG, spin coated once at
2000 rpm. Since the deposition of the third layer (CVD2) is conformal, the SEM after
spin coating and cure of SOG is representative of the insulation topography. From fig-
ure 3.29(a), we note that the nanopillars have been planarized well (γ ≈ 0). In spite of
good planarization, the etch back of the complete insulation (CVD1 (t1 = 100 nm)/SOG
(2000 rpm, once)/CVD2 (t2 = 100 nm)) leaves it rough, and open to current leakage, see
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3.5 Fabrication of electrically connected nanopillar arrays
Figure 3.28: (a) Leakage current and current density through the trilayer insulation at
room temperature. The size of the devices is 500 µm by 500 µm. The inset shows the
schematic diagram of the device. (b) Leakage current through the insulation in comparison
to an array of nanopillars with a footprint of 500 µm by 500 µm, measured at 300 K.
The array of nanopillars, with radial dimensions Rp = 50 nm, 100 nm and 150 nm were
processed from wafer W0021.
figure 3.29(b).
Two mechanisms play a role in producing this microstructure after etch back. In the
first scenario, the etch rate of SOG is faster than the etch rate of PECVD deposited SiO2
layer, see figure 3.30(a). Therefore, the insulation between the pillars is removed faster
than the insulation over the nanopillar. The roughening of the insulator is a result of phys-
ical sputtering during reactive ion etching. The roughening is exacerbated if the thickness
of the insulator above the nanopillars after removal of CVD2 layer (t1 + Z ) is high, see
figure 3.30(b). It is possible to reduce the etch rate of SOG with respect to PECVD SiO2 by
decreasing the percentage of O2 in the plasma. However, we find that the etch rate of SOG
remains unaffected by decreasing the O2 percentage from 11.4 % to 6.2 %.
A smooth etch back is achieved by reducing both Z and t1. At the same time, t1 and
t2 are chosen to maintain a t1+t2 ≥ 100 nm for a good insulation. We achieve this by
reducing t1 from 100 nm to 25 nm, and by spin coating SOG to a thickness tSOG , which
is less than H+t1, see figure 3.31(b). This is obtained by increasing the spin speed from
2000 rpm to 4000 rpm. Although the thickness of the spin glass decreases with increase in
speed of spin coating [125], in practice it may also vary depending upon the height H and
the pitch p of the nanopillar array. This implies that the trilayer planarization process has
to be optimized separately for each active region.
Note that reducing Z and t1 also reduces γ. For example, in the current example γ re-
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3. NANOPILLAR ARRAYS IN RTDS AND QCLS: EXPERIMENTAL REALIZATION
Figure 3.29: Scanning electron micrograph showing cross-sectional view of (a) nanopillars
imaged after deposition of CVD1 (100 nm of SiO2) and planarisation by SOG (spin coated
once at 2000 rpm) (b) nanopillars insulated by CVD1/SOG/CVD2 (100 nm/2000 rpm/100
nm) and etched back in CHF3/O2 plasma for 6 min. The cross-sectional view of pillars is
obtained by cleaving the pillars through the insulation.
Figure 3.30: Schematic diagram showing a poor etch back (a) when the etch rate of SOG is
higher than the etch rate of SiO2 (b) when the thickness of insulation above the nanopillars
after removal of CVD2 layer (t1 + Z ) is high.
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3.5 Fabrication of electrically connected nanopillar arrays
duces from 0o to -35o, see figure 3.31(b), which shows the scanning electron micrograph
of nanopillar arrays after conformal coating by CVD1 (t1 = 25 nm) and planarization by
SOG (spin coated once at 4000 rpm). Since the deposition of the third layer (CVD2) is
conformal, the SEM after spin coating and cure of SOG is representative of the insula-
tion topography. Following the conformal deposition of CVD2 (t2 = 75 nm) layer, the
nanopillars are etched in CHF3/O2 plasma for 2.5 mins, see figure 3.31(c). Note, that the
top-contact of the nanopillars is only partly exposed after 2.5 mins. For complete etch-
back, the nanopillars are etched for a minute more. Figure 3.31(d) shows the scanning
electron micrograph of nanopillars after 3.5 min of etching. The increase in the apparent
dimensions of the nanopillars measured by the SEM after etch back (diameter = 222 nm
in figure 3.31(d)), as compared to the original dimensions (diameter = 174 nm in fig-
ure 3.31(a)), is most likely due to charging at the insulator surface, which is aggravated
by roughness - a consequence of etch back. However, the roughness of the insulator is not
as severe as compared to figure 3.29, which opens up voids in the insulation.
Figure 3.31: Scanning electron micrograph of nanopillars (a) imaged after reactive ion
etching (b) imaged after deposition of CVD1 (25 nm of SiO2) and planarisation by SOG
(spin coated once at 4000 rpm) (c) etched back in CHF3/O2 plasma for 2.5 min (d) etched
back in CHF3/O2 plasma for 3.5 mins. The cross-sectional view of pillars shown in (b), (c)
and (d) is obtained by cleaving the pillars vertically.
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3. NANOPILLAR ARRAYS IN RTDS AND QCLS: EXPERIMENTAL REALIZATION
3.5.4.4 Device recovery after ineffective planarization
The height of the nanopillars will vary depending upon the active region of choice. For
example, in the case of a double barrier RTD, the height H of the nanopillar would range
from 600 nm - 900 nm. On the other hand, in the case of THz QCLs, the height H of the
nanopillar would range from 2 - 3.5 µm. Therefore, the thickness of the conformal layer,
the spin speed of the spin on glass and the number of coats required, must be engineered
for each active region. However, optimization of the planarization parameters requires
detailed calibration.
The flexibility of the insulation removal provides another huge benefit. It allows device
recovery after an etch back. Both spin on glass and SiO2 can be removed in buffered hy-
droflouric acid. However, buffered hydroflouric acid also etches aluminium, aggressively.
Therefore, this method of device recovery is not suitable for active regions with AlAs bar-
riers. Active regions that contain AlxGa1−xAs barriers with low aluminium content (x ≤
0.7), may be suitable for device recovery as they are resistant to buffered hydroflouric acid
1. While complete device recovery may be possible in theory, certain challenges may need
to be overcome.
Figure 3.32(a) shows the scanning electron micrograph of nanopillars fabricated using
a double barrier RTD structure - test wafer A1700. As mentioned in section 3.2, A1700
comprises of a 10 nm well, sandwiched between two 5 nm thick Al0.33Ga0.67As barriers.
The first barrier is at a distance of 438 nm from the top GaAs surface, and the second
barrier is at a distance of 453 nm from the top GaAs surface. The average position of the
barriers is indicated in figure 3.32(a). The height of the nanopillar is ≈ 629 nm. When pla-
narized by the trilayer process, the combined height of the insulation reaches ≈ 884 nm.
The height of the insulation is inferred from figure 3.32(b), showing the unexposed area of
the planarized chip on the right. The scenario shown in figure 3.32(b) is very similar to the
one discussed in figure 3.30(b). When the planarized nanopillar array is etched back in the
CHF3/O2 plasma for 6 mins, the insulation is left rough and open to voids, see the exposed
area of the chip in figure 3.32(b). To test the removal of insulation, a nominally similar
sample to the one shown in figure 3.32(a-b) is etched in stagnated buffered hydroflouric
acid for 12 hours. What is promising to note, is that the barrier remains unaffected by the
etch. However, in spite of the long etch, we find remnants of SOG and SiO2 between the
nanopillars and on the GaAs surface. This is a consequence of etching a very large-area
of the insulation (standard chip size : 6 x 7 mm) under stagnated buffered hydroflouric
acid. In order to overcome this problem, we etched the planarized nanopillars in magneti-
1http://terpconnect.umd.edu/ browns/wetetch.html
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3.5 Fabrication of electrically connected nanopillar arrays
cally stirred buffered hydroflouric acid. While the insulation was completely removed, we
observed narrowing of the pillars around the Al0.33Ga0.67As barriers. This narrowing also
causesed a few nanopillars to collapse at the neck. This observation leads us to speculate
that agitated buffered hydroflouric acid reacts with the Al0.33Ga0.67As barrier, while the
static acid does not.
Figure 3.32: Scanning electron micrograph of nanopillars (a) imaged after reactive ion
etching (b) imaged after trilayer insulation (CVD1 (50 nm)/SOG (2500 rpm, once)/CVD2
(50 nm)) and plasma etch back for 6 mins (c) imaged after chemical etching in still
buffered hydroflouric acid for 12 hours (d) imaged after chemical etching in buffered hy-
droflouric acid, magnetically stirred for 1 hour. The cross-sectional view of pillars shown
in (b) is obtained by cleaving the pillars through the insulation.
Although the device cannot be recovered completely, we can still contact the remaining
nanopillars. This is important to keep in mind while optimizing the etch back process for a
laterally confined THz QCLs based on GaAs / AlxGa1−xAs material system. Perhaps, a suit-
able device recovery process should comprise of two stages, with the first stage involving a
long etch in stagnated buffered hydroflouric acid and the second stage involving a fast re-
moval of the etch residue in magnetically stirred buffered hydroflouric acid. Optimization
of the device recovery process was beyond the scope of the present work. In the following
section, we summarize the key points necessary for a structurally and an electrically robust
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3. NANOPILLAR ARRAYS IN RTDS AND QCLS: EXPERIMENTAL REALIZATION
insulation.
3.5.4.5 Nanopillar insulation summary
We have demonstrated a planarization technique that can be designed to insulate nanopil-
lars in a range of GaAs / AlxGa1−xA active regions. While meeting the height requirements
imposed by laterally confined THz QCLs (H = 2 - 3.5 µm), this planarization technique is
compatible with the processing of laterally confined RTDs (H = 600 - 900 nm). This insu-
lation technique involves a trilayer process where the thickness of each layer is engineered
to ensure smooth planarization and etch back, which is a requirement for structural and
electrical stability of the device. Among most insulation techniques, this trilayer spin on
glass process has the advantage of removal by wet chemical etching. While this quality
can be exploited to achieve a suspended top-contact if required, it also allows for partial
device recovery after a poor etch back. In the next section we describe the fabrication of
top-contact after the trilayer insulation process.
3.5.5 Fabrication of top-contact and bond pads
Figure 3.33(a) shows the schematic diagram of the nanopillar array after etch back in a
plasma chamber. Windows patterned by optical lithography, allow only the nanopillar ar-
ray to be exposed to reactive ion etching, leaving the rest of the insulation unexposed.
Following the etch back, the optical photoresist, which is baked during the plasma process-
ing is removed in acetone. The next step involves deposition of the top-contact, a rotatilt
evaporation of NiCr/Au, figure 3.33(b). Following the deposition of the top-contact, bond
pads are evaporated over the insulator which electrically connect to the top of the nanopil-
lar array, figure 3.33(c). The bond pads over the top-contact give the flexibility to produce
both insulated and suspended contact. Since the top-contact bond pads do not come in
contact with the hydroflouric acid at any stage of processing, we choose to use Ti as a
wetting layer for gold. Finally, the bottom-contact pads are exposed by chemically etching
the trilayer insulation through an optically defined window, figure 3.33(d). In the next
section, we discuss the fabrication of suspended top-contact to the array of nanopillars.
3.5.6 Suspended top-contact
From table 3.5, we recognized that most insulating materials require plasma processing for
removal. However, plasma processing, being anisotropic in nature does not allow the for-
mation of a suspended top-contact after metal evaporation, see general process flowchart
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3.5 Fabrication of electrically connected nanopillar arrays
Figure 3.33: Schematic diagram showing nanopillars after (a) etch back in a plasma cham-
ber (b) deposition of top-contact (c) deposition of top-contact bond pad, and (d) removal
of insulation over the bottom-contact pads.
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3. NANOPILLAR ARRAYS IN RTDS AND QCLS: EXPERIMENTAL REALIZATION
for fabricating electrically connected nanopillar arrays (figure 3.2). From table 3.5, we
also noted the possibility of using two insulating materials that fulfill the height require-
ments of laterally confined THz QCLs, which can be removed by wet chemical processing.
These two materials were (i) spin on glass, and (ii) standard optical photoresist. Below we
discuss the processing routes of the two insulating materials, and evaluate their feasibility
for forming suspended top-contacts to laterally confined THz QCLs (and the test RTDs).
3.5.6.1 Possible routes to suspended contact
SPIN ON GLASS ROUTE
As discussed previously, spin on glass and SiO2 can be chemically etched using buffered
hydroflouric acid. Since the nature of wet etch is isotropic, the etchant removes the insu-
lation in the lateral direction as well as the vertical direction, which can be exploited to
remove the trilayer insulation between the nanopillars. Note that the top of the insulation
over the nanopillars is protected by the metal and is inaccessible for etching in the vertical
direction, see figure 3.34(a). To overcome this problem, a window defined using optical
lithography is opened in the insulation layer, 20-30 microns away from the nanopillar ar-
ray, see 3.34(b). On a chip with multiple array patterns, this window is approximately
360 µm away from the opposite array.
We note that until the insulation below the window is removed, the etch through the
airgap window is isotropic, i.e. the etch rate in the vertical direction is equal to the etch
rate in the lateral direction. Once the insulation below the window is removed, the etch
progresses at a much faster rate owing to increased surface area available to the etchant.
For a 1.25 µm thick insulation, buffered hydroflouric acid etches at a rate of ≈ 6.5 µm/min
through a 10 µm wide x 250 µm long window as opposed to 80 nm/min in the vertical
direction. Figure 3.34(c) shows the trilayer insulation first etched in the vertical direction,
followed by etching in the lateral direction, see figure 3.34(d).
Note that the length of the airgap window (250 µm) for a 500 x 500 array is inten-
tionally designed to be less than 500 µm inorder to protect the lateral etch from extending
to the bottom-contact and the other arrays on the chip. Furthermore, depending upon the
height of the insulation, the etch has to be calibrated in time to ensure that the etchant
does not attack the insulation near the edge 3, to protect the mechanical stability of the
suspended top-contact. Figure 3.35 shows a rough schematic diagram illustrating the ex-
tent of the lateral etch for each nanopillar array on the chip. This extent (radius = 320
to 360 µm) is limited by the shape of the window, the placement of the arrays and the
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3.5 Fabrication of electrically connected nanopillar arrays
Figure 3.34: Suspended top-contact process schematic (not to scale), showing (a) Ti/Au
bond pads evaporated on planrized (and etched back) nanopillars (b) Optical lithography
to define a window at a distance from the array (c) Trilayer insulation vertically etched at
a rate of ≈ 80 nm/min (d) Trilayer insulation laterally etched at a rate of ≈ 6.5 µm/min
to form air gaps between the nanopillars.
Figure 3.35: Schematic diagram showing the lateral etch of insulation through the airgap
window in relation to all nanopillar arrays on the chip.
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3. NANOPILLAR ARRAYS IN RTDS AND QCLS: EXPERIMENTAL REALIZATION
placement of the bottom-contact. Improvements in the optical mask will help optimize
the extent of the lateral etch and at the same time maintain the mechanical stability of the
top-contact. However, these improvements were beyond the timescale of the current work.
The advantage of this method is that the removal of insulation is the last step in the
process flow. Therefore, this processing route offers the flexibility to obtain electrical mea-
surements with a robust insulation and optical measurements with the insulation removed.
PHOTORESIST ROUTE
In 2010, Offermans et. al demonstrated suspended contacts to individual InAs nanowires,
obtained by using an optical photoresist as a planarizing material [135]. In Offermans
work, a vertical InAs nanowire was grown using a bottom-up technique on an InP sub-
strate. The bottom-contact and the top-contact were electrically isolated from each other
by selective patterning of an insulator after the growth of InAs nanowires.
Here, we evaluate the feasibility of Offermans approach to top-down nanopillar arrays
in THz QCLs and the test RTDs. Akin to Offermans work on InAs wires where the bot-
tom layer is conducting, the end point of a laterally confined THz QCL and the test RTD
structure is also conducting. In the case of metal-metal THz QCLs (section 4.2), this layer
is a gold contact, whereas in the case of RTDs (section 3.6) this layer is an n+ GaAs con-
tact, see figure 3.36(a). Therefore, the first step after reactive ion etching of nanopillars,
would involve deposition and selective pattering of an insulator on the conducting layer,
figure 3.36(b). Following planarization by an optical photoresist, see figure 3.36(c), win-
dows would be patterned by optical lithography that selectively expose the bottom-contact,
the nanopillar array and the oxide, see figure 3.36(d). Finally, a rotatilt evaporation would
allow the formation of top and bottom-contact to the nanopillar arrays, figure 3.36(d).
Note that Offerman’s approach combines top-contact and bond pad fabrication in the
same step. Furthermore, this approach only allows for the formation of suspended con-
tacts, i.e. this process does not have the flexibility to achieve an unsuspended top-contact.
Offermans approach can be extended to THz QCLs using the process flow described in
figure 3.36. Extending the process to RTDs requires a few intermediate steps, which are
briefly discussed in section 3.6.2. While the photoresist route is an appealing alternative,
it requires a redesign of the existing processing flow, which was not within the timescale
of the current work.
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3.5 Fabrication of electrically connected nanopillar arrays
Figure 3.36: Schematic diagram showing fabrication of suspended top-contact using Of-
fermans approach. The process flow involves (a) reactive ion etching of nanopillars in
THz QCLs or RTDs (b) deposition and patterning of oxide (c) spin coating of photoresist
(d) selective exposure with windows for bottom-contact, nanopillar array and oxide (e)
evaporation of metal contact.
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3. NANOPILLAR ARRAYS IN RTDS AND QCLS: EXPERIMENTAL REALIZATION
3.5.6.2 Spin on glass route to suspended contact: results
Figures 3.37 and 3.38 show the scanning electron micrograph of nanopillars imaged after
the formation of suspended contact. The cross-section of the nanopillars is imaged by
cleaving the array in the centre, in a direction perpendicular to the length of the window.
Figures 3.37(a-b) show the scanning electron micrograph of a 500 x 500 array of Rp =
200 nm nanopillars processed from an RTD active region (H = 850 nm). The lateral etch
rate of the trilayer insulation is equal to 3.15 µm/min. For clarity, the 270 nm airgap is
shown at a higher magnification in figure 3.37(b). In section 3.5.4.3, we discussed that the
insulator etched back in the plasma and the underlying topography may not be planar, see
figure 3.31. The shape of the underlying insulator, when etched back gets mapped onto the
top-contact. Therefore, the apparent buckling of the top-contact, shown in figure 3.38(a-
b), is not a consequence of the weight of the top-contact, but a consequence of the insulator
shape derived after plasma etching.
Figure 3.37: Scanning electron micrograph (a-b) imaged after the formation of suspended
top-contact in nanopillar arrays that are planarized and etched back in a reactive ion
plasma. Figure 3.38(b), shows the 270 nm airgap, formed in RTD active region, at higher
magnification for clarity.
Figure 3.37(a) shows the formation of a 1.25 µm high airgap in an array of Rp =
150 nm, H = 1.75 µm pillars. The insulation etches at a rate of 6.5 µm/min in the
134
3.5 Fabrication of electrically connected nanopillar arrays
lateral direction. This was the very first demonstration of a suspended top-contact using
a trilayer process at the Cavendish labs. The 500 x 500 nanopillar array was processed
from a THz QCL active region, with n+ GaAs as the end point during reactive ion etching.
Figure 3.37(a) shows the airgap stretching over 36 µm. The magnified image of the airgap
is shown in figure 3.37(b). The collapse of a few pillars occured partly during the stage
of reactive ion etching (section 3.5.3) and partly during cleaving. Note that in the case
of figure 3.37(a-b), the nanopillars were planarized but not etched back. Therefore, the
top-contact, which follows the topography of the underlying insulator is also planar. Note
that this process is equally compatible with nanopillars etched in both GaAs / AlxGa1−xAs
based RTD and THz QCL active regions, where the end point is an n+ doped GaAs contact.
Therefore, figure 3.38(a-b) is a good representative of a suspended top-contact in a THz
QCL active region using the trilayer process.
Figure 3.38: Scanning electron micrograph (a-b) imaged after the formation of suspended
top-contact in planarized nanopillars. Figure 3.37(b), shows the 1.25 µm airgap, formed
in THz QCL active region, at higher magnification for clarity.
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3. NANOPILLAR ARRAYS IN RTDS AND QCLS: EXPERIMENTAL REALIZATION
3.6 Fabrication of RTDs
In section 3.5, we discussed processing techniques that are common to nanopillars in GaAs
/ AlxGa1−xAs based RTD and THz QCL active regions. In this section, we elaborate on
additional processing stages that are unique to the test structures, i.e. the RTDs. Before we
delve on the additional complexities of fabricating nanopillar arrays in RTDs, it is useful
to discuss the fabrication of large-area RTDs. The fabrication process described below was
used to process large-area RTDs with a Schottky top-contact to ensure consistency with
the non-aligned metal-based NiCr/Au process described in section 3.5.2. Although the
same fabrication process can be used for large-area RTDs with an ohmic top-contact, these
devices already existed in the lab and hence we did not attempt to reprocess them.
3.6.1 Large-area RTDs
The term large-area refers to an area where the effects of surface depletion and hence
quantum confinement in the lateral direction are negligible, i.e. the area of the device is
 R2d . Instead of circular pillars, large-area RTDs were defined as square mesas. The size
of these mesas ranged from 10 µm x 10 µm to 50 µm x 50 µm. Figure 3.39 shows the
various processing steps used to fabricate the large-area RTDs.
Figure 3.39: Process schematic (a) mesa etch (b) lithography and deposition of bottom-
contact (c) deposition of SiO2 (d) opening of windows, deposition of top Schottky contact
and bottom bond pad.
The square mesa was wet etched in a solution of H2SO4 : H2O2 : H2O mixed in a ratio
of 1:8:80 which results in sloping sidewalls caused by lateral chemical etching, see fig-
136
3.6 Fabrication of RTDs
ure 3.39(a). To avoid isotropic etching, the mesa can be dry etched in a reactive ion etcher,
which results in more vertical sidewalls. This is described in section 3.4.3.1. Following
the mesa definition, AuGeNi contacts were evaporated and annealed at 430oC for 80 s in
forming gas atmosphere to form ohmic bottom-contacts, figure 3.39(b). Windows for top
and bottom-contact were opened in an insulating layer of 100 nm thick SiO2 deposited by
plasma-enhanced chemical vapor deposition (PECVD), figure 3.39(c). The evaporation of
the top-contact and the bottom-contact pad were combined in a single evaporation step,
figure 3.39(d). The bond pads to the top-contact were evaporated over the SiO2 insulation
(see section 3.5.5). Before characterization, the device was wire bonded to the top and
bottom bond pads.
3.6.2 Nanopillar arrays in RTDs: bottom-contact dilemma
In a GaAs / AlxGa1−xAs based active region, where the end point after etching nanopil-
lars is an n+ doped GaAs contact, bottom-contact fabrication poses a big challenge. This
is specially true for our test double well RTD structures which terminate in an n+ doped
GaAs contact. As described in section 3.6.1, the bottom n+ doped GaAs layer is contacted
by AuGeNi contacts which are evaporated and annealed at 430oC for 80 s. This annealing
step has two ramifications. First, the metal etch mask, which is in contact with the top of
the nanopillars diffuses down to the GaAs / AlxGa1−xAs layers during the annealing stage
of the bottom-contact, resulting in a short circuit. Second, if the bottom-contacts are evap-
orated after planarization, the spin on glass layer cracks upon rapid thermal annealing.
Figures 3.40(a-b) show the flowchart for the fabrication of electrically connected nanopil-
lar arrays in RTDs. Similar to large-area RTD processing, the bottom-contacts for nanopillar
RTDs, were fabricated before etching the nanopillars, in the case where the etch mask was
a metal, figure 3.40(a). However, inclusion of bottom-contacts before etching exposes the
reactive ion chamber to a high metallic content on the chip for the entire duration of the
etch, restricting this process to RIE systems alone. While the metallic content on the chip as
a result of using a metal etch mask is around 0.02 %, the metallic content on the chip due
to bottom-contacts is around 3.3 %. Consequently, even using an RIE process, we protect
the bottom-contacts and bond pads by SiO2 before e-beam lithography, to avoid a potential
risk of micromasking as a result of etch related debris. However, this process increases the
number of lithographic stages. In the case of a dielectric etch mask, electrical shorting is
not an issue. Therefore, we prefer to deposit the bottom-contact and bond pads after the
etching stage but before the planarization stage, figure 3.40(b).
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3. NANOPILLAR ARRAYS IN RTDS AND QCLS: EXPERIMENTAL REALIZATION
Figure 3.40: Process flowchart to fabricate electrically connected nanopillar arrays in RTDs
(a) in the case of a metal etch mask, and (b) in the case of a dielectric etch mask.
3.7 Experimental set-up for characterizing RTDs
Four terminal (4T) measurements as opposed to two terminal (2T) measurements allow
the true device electrical characteristics of the device to be measured as it eliminates lead
resistance, see figure 3.41. In the 4T measurement, the voltmeter indirectly measures the
voltage drop across the device (rRT D), and the top and bottom doped contacts (rtc , rbc)
that form part of the device, see figure 3.41(a). In the 2T measurement, the voltmeter
measures the voltage drop across the lead resistances rlead , resistance across the device
RT D, and the top and bottom doped contacts (rtc , rbc) that form part of the device, see
figure 3.41(b). Unlike 2T measurements which require only two contacts, 4T measure-
ments require four contacts to allow for the elimination of the lead resistance. While this
is straightforward to implement on a large-area device, these measurements are not trivial
to carry out on electrically connected nanopillar arrays as this necessiates two top-contacts
and two bottom-contacts. To allow for a useful comparison between large-area devices and
the nanopillar arrays, all the Schottky RTD devices discussed in this work were character-
ized by 2T measurements.
138
3.8 Results from RTDs: A1700
Figure 3.41: Experimental set up for characterizing RTDs using (a) four terminal measure-
ments, and (b) two terminal measurements.
These measurements were made using an HP 4155 A semiconductor parameter ana-
lyzer at room temperature (300 K) and at liquid helium temperature (4K). To avoid Joule
heating, the voltage was pulsed at a period of 10 ms with a duty cycle of 10 %.
3.8 Results from RTDs: A1700
A1700 was used as the main test structure for the nanopillars as it is based upon AlxGa1−xAs
barriers, which closely resembles the epitaxial structure of the THz QCLs used in this work.
Results from large-area RTDs processed from ohmic contacts are presented first, as a refer-
ence. This is the ideal configuration of RTDs under the absence of any constraints and in-
volves using a shallow ohmic top-contact such as Pd/Ge and a deep ohmic bottom-contact
such as AuGeNi. Next, we present the electrical characteristics of large-area RTDs fabri-
cated with a Schottky top-contact and an ohmic bottom-contact. This contact geometry is
representative of the contact geometry used for an array of nanopillars. Finally, we discuss
the results from nanopillar arrays processed from A1700.
3.8.1 Large-area RTDs: ohmic contact
Figure 3.42 shows the 4T measurements of a large-area RTD processed from A1700, with
an ohmic top and bottom-contact at 77 K. The large-area device, which has a mesa size of
approximately 70 µm x 70 µm, shows a distinct NDR under positive and negative polari-
ties. From theoretical calculations, we should expect three NDR peaks under both positive
139
3. NANOPILLAR ARRAYS IN RTDS AND QCLS: EXPERIMENTAL REALIZATION
and negative polarities, which correspond to three resonance states in A1700, see sec-
tion 2.2. These NDR peaks should occur at abs(Vp) = 2Ei , where i = 0, 1 and 2 [136].
Here, Vp is the voltage associated with each resonant state at which the current is maxi-
mum. The three NDR peaks should occur at a bias abs(Vp) equal to 0.064 V, 0.254 V and
0.514 V, corresponding to energy levels E0 ∼ 32 meV, E1 ∼ 127 meV and E2 ∼ 257meV
1. In reality, there is a slight discrepancy between the experimentally measured NDR peak
and the NDR peak predicted from simplistic theoretical calculations based on the epitaxial
structure. In a 4T measurement, this deviation occurs primarily because of the depletion
region formed in the GaAs layers between the contact and the barriers [102, 137]. There-
fore, while the theoretical calculations predict the first NDR peak to be at abs(Vp) = 0.064
V, this peak is observed at -0.31 V under negative bias upon alignment with the ground
state, see figure 3.42. The peak current density associated with resonant tunneling of
electrons is Jp = -85 A/cm2. Under positive bias, the first NDR appears at +0.36 V upon
alignment with the ground state and the peak current density associated with the NDR is
Jp = +92 A/cm2. A record of device behavior at 4K is absent, however this should not
influence the discussion of results in section 3.8.2 by a great extent.
Figure 3.42: Electrical characterization of large-area RTD, with mesa size 70 µm x 70 µm
processed from A1700, with an ohmic top-contact. The measurement was carried out at
77K, in a 4T configuration and the voltage was swept from -1 V to +1 V in steps of 5 mV.
The data was provided by Dr. Patrick Sze.
Figure 3.43 shows the 4T electrical characteristics for a range of mesa areas varying
from 70 µm x 70 µm to 200 µm x 200 µm. All the devices exhibit an NDR peak in the
1An approximate numerical solution to 2.2
140
3.8 Results from RTDs: A1700
Figure 3.43: Electrical characterization of large-area RTDs with mesa sizes varying from
70 µm x 70 µm to 200 µm x 200 µm. The devices were processed from A1700, with an
ohmic top-contact. The measurement was carried out at 77K in a 4T configuration and the
voltage was swept from -1 V to +1 V in steps of 5 mV. The data was provided by Dr. Patrick
Sze.
positive and negative polarity. As the mesa area increases from 70 µm by 70 µm to 200
µm by 200 µm, the peak voltage associated with the ground state NDR increases from Vp
= -0.31 V to Vp = -0.46 V under negative bias and from Vp = +0.36 V to Vp = +0.81 V
under positive bias. This voltage drop is associated with the additional series resistance
of the doped contacts, which scales with device area and is independent of whether the
measurement is 4T or 2T [138, 139]. The other parameters which are influenced by an
increase in mesa area include the peak current density and the peak to valley current ratio.
The peak current density associated with the NDR peak decreases from Jp = - 85 A/cm2
to Jp = -33 A/cm2 under negative bias and from Jp = +92 A/cm2 to Jp = +66 A/cm2
under positive bias. The peak to valley current ratio (Jp/Jv) falls from 11.6 to 1.78 under
negative bias (and from 10.8 to 1.66 under positive bias) for the same increase in mesa
dimensions. Even though there exists a slight variability among large-area RTD devices,
the threshold voltage remains nominally identical for all the different mesa sizes, i.e. Vth =
-0.065 V in the case of negative polarity and Vth = +0.05 V in the case of positive polarity.
3.8.2 Large-area RTDs: Schottky contact
Figure 3.44(a) shows a comparison between a large-area RTD device processed from
A1700 with a Schottky contact and a large-area RTD device processed from A1700 with an
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3. NANOPILLAR ARRAYS IN RTDS AND QCLS: EXPERIMENTAL REALIZATION
ohmic contact. The dimensions of the large-area RTDs with a Schottky contact are equal
to 10 µm by 10 µm and the dimensions of the large-area RTDs with an ohmic contact are
equal to 70 µm by 70 µm. At these dimensions, the quantum confinement effects due to
surface depletion are negligible [139].
Figure 3.44: (a) 2T measurements of large-area RTDs (processed from A1700 with a Schot-
tky top-contact) at 4K in comparison with 4T measurements of large-area RTDs (processed
from A1700 with an ohmic top-contact) at 77 K. The dimensions of the large-area RTDs
with a Schottky contact are equal to 10 µm by 10 µm and the dimensions of the large-area
RTDs with an ohmic contact are equal to 70 µm by 70 µm (b) Forward and backward volt-
age sweep during the 2T measurements of the 10 µm by 10 µm RTD at 4K. The applied
voltage was swept in steps of 10 mV.
We note a very clear shift in bias between the first NDR peaks of the two devices.
Considering that the peak current density associated with the NDR peak is similar, Jp = -85
A/cm2 (-126 A/cm2) for the RTD with the ohmic (Schottky) contact, it is highly probable
that the NDR peak seen at abs(Vp) = -1.04 V from the Schottky device corresponds to
alignment with the ground state. The shift in bias between the two peaks may be attributed
to the following causes. First, this shift may just perhaps occur because of a difference in
the resistance of the doped contacts arising from a difference in the mesa size of the two
devices that are being compared. Second, this shift may occur due to the additional lead
resistance picked up in a 2T measurement as opposed to a 4T measurement. Finally, this
shift may occur due to a voltage drop of 0.7 V on the collector end due to the Schottky
barrier. The first reason can be eliminated on the basis that the difference in mesa size
would have resulted in a larger Vp as opposed to a smaller Vp for the 70 µm by 70 µm
device, when compared to the 10 µm by 10 µm device. Furthermore, even if the shift
in the NDR peak was due to a difference in mesa size, the threshold voltage of the two
142
3.8 Results from RTDs: A1700
devices would have been similar (see figure 3.43). While the second reason is a possibility
that can not be discounted, the difference in the two peak voltages (0.74 V) between the
devices suggests that this additional voltage is dropped across the Schottky barrier at the
top-contact. The ground state resonance under negative bias for the large-area RTD device
with a Schottky top-contact is shown in greater detail in figure 3.44(b). Note, that the two
separate curves indicate forward and reverse voltage sweep. The hysteresis observed in
figure 3.44(b) is characteristic of RTD behavior, and can be attributed to a combination of
intrinsic charge bistability [140] and extrinsic biasing effects [141].
Figure 3.45: 2T measurements of large-area RTDs processed from A1700 with a Schottky
top-contact. The device has a mesa size of 10 µm by 10 µm. The voltage was swept in
steps from -5 V to +5 V of 10 mV and the measurement was carried out at 4K and 300 K.
Figure 3.45 shows the 2T measurements of a large-area RTD, with a Schottky top-
contact and an ohmic bottom-contact, under an extended voltage range (-5 to +5 V).
Unlike the 4T measurements of large-area RTDs processed with an ohmic top and bottom
contact (figure 3.43), these measurements show an asymmetry under positive and negative
bias. This asymmetry may arise out of a difference in lead resistance that is not excluded
in a 2T measurement or it may also arise out of a difference in the contact resistance
at the emitter and the collector end. We note that the first NDR peak under the positive
polarity appears at a very high voltage (Vp =+4.29 V) as compared to the negative polarity
(Vp = -1.04 V). The peak current JP = 109 A/cm2 is approximately equal to the peak
current observed in the negative bias when the device is in resonant alignment with the
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3. NANOPILLAR ARRAYS IN RTDS AND QCLS: EXPERIMENTAL REALIZATION
ground state. Therefore, it is probable that the NDR peak at Vp = +4.29 V corresponds to
alignment with the ground state. The shift in bias from the expected Vp at approximately
+1.04 V to +4.29 V may then perhaps occur primarily due to an additional voltage drop
across the lead resistance, which may be eliminated with a 4T measurement.
Going back to the device characteristics under negative polarity between 0 to -5 V, we
observe three distinct NDR peaks at 4 K (Vp = -1.04 V, -1.8 V and -3.75 V). Although weak,
these peaks are also observed when the device is at room temperature. The observation of
three NDR peaks suggest that they correspond to alignment with the ground state (E0) and
the two excited states (E1 and E2), which is in agreement with the theoretical calculations
discussed in section 3.8.1. We also note an increase in peak current and the peak current
density associated with resonant alignment with the subsequent higher energy states. For
example, the peak current density Jp associated with the third NDR peak is approximately -
5018 A/cm2, which is much larger than the peak current density Jp associated with the first
and second NDR peaks (Jp = -126 A/cm2 and -532 A/cm2, respectively). This behavior is
expected from large-area RTDs with multiple energy states [142].
We also tested multiple devices processed from A1700 with a Schottky top-contact and
an ohmic bottom-contact. The mesa dimensions of these devices ranged from 10 µm by 10
µm to 50 µm by 50 µm, see figure 3.46. All the devices exhibit an NDR peak in the positive
and negative polarity. As the mesa area increases from 10 µm by 10 µm to 50 µm by 50
µm, the peak voltage associated with the ground state NDR increases from Vp = -1.04 V to
Vp = -2.37 V under negative bias and from Vp = +4.28 V to Vp = +4.54 V under positive
bias. This increase is associated with the additional series resistance of the doped contacts,
which scales with device area and is independent of whether the measurement is 4T or
2T [138, 139]. All the devices exhibit similar peak current density upon alignment with
the ground state, i.e. abs(Jp) = 134 ± 27 A/cm2. We further note that even though the
threshold voltage is different under opposite polarities due to asymmetry in lead resistance,
i.e. Vth = - 0.77 V under negative bias and Vth = +1.9 V under positive bias, it is nominally
identical for all the large-area devices that were measured. Similar to large-area RTD
devices processed with an ohmic top-contact, section 3.8.1, we note that the device with
the largest mesa size, i.e. mesa length equal to 50 µm, exhibits the smallest peak to valley
current ratio (Jp/Jv = 1.82 under negative bias and Jp/Jv = 1.61 under positive bias). The
device with the smallest mesa dimensions, i.e. mesa length equal to 10 µm, exhibits a peak
to valley current ratio (Jp/Jv) that is equal to 23.3 under negative polarity and 2.54 under
positive polarity.
It is encouraging to note that the Schottky barrier on the top-contact of a large-area
RTD does not impede the observation of a resonance. This allows us to test the electrical
144
3.8 Results from RTDs: A1700
Figure 3.46: 2T measurements of large-area RTDs processed from A1700 with a Schottky
top-contact. The mesa dimensions vary from 10 µm by 10 µm to 50 µm by 50 µm. The
voltage was swept from -5 V to +5 V in steps of 10 mV and the measurement was carried
out at 4K
robustness of an array of insulated (unsuspended) nanopillars, processed from A1700 with
a Schottky top-contact, see below.
3.8.3 Nanopillar arrays
3.8.3.1 Results from 500 x 500 array, Rp = 100 nm
Figure 3.47 shows the first result of a successful top-contact to an array of insulated
nanopillars. Processed from A1700, this chip consisted of three 500 x 500 array of nanopil-
lars, with pillar sizes defined as Rp = 50 nm, Rp = 100 nm and Rp = 150 nm. Out of these
three arrays, pillars with Rp = 100 nm and Rp = 150 nm survived the etch and processing.
Here, we discuss the results from pillars with Rp = 100 nm, see figure 3.47 inset. The
necking around the quantum well is a result of agitated stirring in buffered hydroflouric
acid during the planarization recovery process. We note distinct NDR peaks at 4 K and at
300 K. These peaks appear at Vp= -2.6 V in the negative bias, and at Vp = +5.52 V, Vp =
+6.66 V in the positive bias.
Figure 3.48 shows the peak at Vp= -2.6 V , in more detail. The appearance of NDR
during forward and the reverse voltage sweeps confirms transport through the nanopillars.
Unlike pA to nA current flowing through single nanopillars processed from RTDs, we ob-
serve a current in the range of µA flowing through nanopillars connected in parallel. This
is at least three orders of magnitude improvement over the previously reported values of
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3. NANOPILLAR ARRAYS IN RTDS AND QCLS: EXPERIMENTAL REALIZATION
Figure 3.47: 2T measurements of 500 x 500 nanopillar array processed from A1700, with
radial dimension Rp = 100 nm measured at 4K and 300 K. The voltage was swept in steps
of 10 mV. The inset shows the scanning electron micrograph of the nanopillar array, before
final insulation and top-contact processing.
Figure 3.48: Negative differential peak corresponding to the ground state of 500 x 500
array of nanopillars processed from A1700, at negative bias, and at a temperature of 4K.
The radial dimensions of the pillars was defined as Rp = 100 nm. The voltage was swept
in steps of 10 mV and the measurement was carried out in 2T configuration.
146
3.8 Results from RTDs: A1700
nanopillar RTDs [89, 94, 102, 108].
3.8.3.2 Challenges associated with NDR mapping
While it is relatively straightforward to attribute NDR peaks to energy states (Ei) for large-
area RTDs, it is not trivial to map NDR peaks to energy states in the case of an array
of nanopillars. This complexity arises due to discretization of energy states from lateral
quantization in a nanopillar. We can however safely assume, that the strongest NDR peak
in an array of nanopillars occurs due to strong confinement in the longitudinal direction,
and hence can be attributed to alignment with any of the longitudinal states Ei .
Besides the mapping of NDR peaks in an array of nanopillars, it may also be non-trivial
to estimate the number of pillars Nt taking part in transport and the electrostatic radius
Re simultaneously. This ambiguity arises from the problem of mapping NDR peaks to the
energy states. The position of the fine feature superimposed upon the NDR peak can be
related to the electrostatic radius Re. In the absence of fine features, we can estimate Re
and Nt from the peak current Ip that is experimentally observed during a measurement.
This current is a product of the current density Jp associated with the resonance, the
electrostatic area of a single nanopillar (piR2e ) and the number of pillars taking part in
transport Nt , see eqn. below.
Ip = JppiR
2
e Nt (3.2)
To estimate Re, we assume that all the nanopillars in the 500 x 500 array are in contact
and take part in transport. This assumption is true when the processing preserves the
number of pillars defined at the e-beam stage. Eqn. 3.2 then gets modified to:
Ip = 250000JppiR
2
e (3.3)
Since previous attempts only focused on single nanopillars [89, 94, 102, 108], it is
useful to gain an insight into how many nanopillars in a 500 x 500 array take part in the
transport. This estimate requires the knowledge of Re. In order to estimate the minimum
number of pillars that are in contact Nt , we ignore sidewall depletion and assume that Re
= Rp. Therefore, eqn. 3.2 gets modified to:
Ip = JppiR
2
pNt (3.4)
In reality, as Re is less than Rp, the number of pillars taking part in the transport will
be higher than the Nt estimated using eqn. 3.4. We can now discuss the results shown in
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3. NANOPILLAR ARRAYS IN RTDS AND QCLS: EXPERIMENTAL REALIZATION
figures 3.47 and 3.48 in view of the analysis presented above.
3.8.3.3 Discussion
Mapping of NDR peaks
It is highly probable that the first NDR peak seen under negative bias at Vp = -2.6 V
in figure 3.47, actually corresponds to the ground state E0. However, since the position
of the ground state and the position of the first excited state are quite close (figure 3.45),
this peak can also correspond to the first excited state. Since we observe two peaks that
are close to each other under positive bias, it is probable that the first NDR peak at Vp =
+5.53 V in figure 3.47 corresponds to the ground state E0 and the second NDR peak at Vp
= +6.66 V in figure 3.47 corresponds to the first excited state E1. However, multiple peaks
may also imply multiple devices coming into alignment.
Estimation of number of pillars
Let us assume that the peak at Vp = -2.6 V corresponds to the ground state. From
figure 3.47, we note that the peak current at Vp = -2.6 V is equal to -72 µA. The density
of current flowing through the device upon alignment must be equal to the density of cur-
rent flowing through a large-area RTD , i.e. Jp = -126 A/cm2 (figure 3.46). Assuming
that only one nanopillar with Rp = 100 nm was contacted successfully, the peak current
flowing through the device, in the best-case scenario, i.e. ignoring depletion, would be
39 nA. Clearly, our result implies that multiple nanopillars are conducting in parallel. If
there was no depletion and all the nanopillars were contacted successfully, the current
flowing through the device would be 9.9 mA. Since the current flowing through the array
of nanopillars shown in figure 3.47 is less than 9.9 mA but greater than 39 nA, this implies
that either only a few nanopillars are contacted or the current path is constrained due to
surface depletion. It is likely, that in this particular device both the scenarios hold true.
Using eqn. 3.4 we can estimate the minimum number of pillars that are in contact and
take part in the transport (i.e. Nt). For an array of nanopillars with physical radius Rp
= 100 nm to pass a peak current of 72 µA, at least 1820 nanopillars must be in contact.
From the inset in figure 3.47, which shows the SEM image of the pillars after etching, we
note that the radial dimensions of the nanopillar is less than 100 nm around the quantum
well (for details see section 3.5.4.4). Substituting the radius as 57 nm in eqn. 3.4, we note
that the minimum number of pillars in contact increases to 7730. The number of pillars
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3.9 Results from RTDs: W0021
calculated using this approach is an under estimate and would increase if we account for
surface depletion in the nanopillars.
Estimation of electrostatic radius
Reversing the argument, assuming that all the 500 x 500 nanopillars are connected,
the empirical electrostatic radius Re can be calculated using eqn. 3.3 by equating 250000
pi R2e Jp to the absolute value of the peak current observed at Vp = -2.6 V , which is -72
µA. From here, we derive Re = 8.3 nm.
Absence of fine features
According to Reed et. al, for a 100 nm nanopillar, we expect fine features superim-
posed upon the NDR peak due to lateral quantization which should be equally spaced at
2∆E = 25.2 meV [89]. However, in spite of a very clear NDR peak and an SEM image
indicating reduced dimensionality, we do not observe fine features in the current-voltage
characteristics. The absence of fine features can be attributed to several causes. Perhaps,
the fine features were washed out due to large statistical variations in the radial dimen-
sions of the pillars within the array in this particular batch of processing. It may also be
that the fine features were averaged during the measurement. For nanopillars processed
from A1700 (figures 3.47 and 3.48), the voltage was swept in steps of 10 mV. If the lateral
quantization was weak, i.e. the splitting of energy levels (∆E) was less than 10 meV, these
features were not picked up during the measurement. Finally, if the electrostatic radius of
the doped contact was equal to electrostatic radius of the undoped well, the fine feature
might have been destroyed due to alignment of the subbands in the doped contact with
the subbands in the undoped well, see section 2.6.5.
3.9 Results from RTDs: W0021
Although, A1700 was the main test structure for the nanopillars, a second wafer W0021
was also investigated. The rationale behind this study can be attributed to the peak current
density which is higher in W0021 as compared to A1700 and commensurate with the
operation of the THz QCLs. Unfortunately, a higher current density requires the use of
thin barriers, which is why W0021 is based on AlAs barriers. The transport behavior of
electrons tunneling through an AlAs/GaAs system is significantly different from that of
Al0.33Ga0.67As/GaAs system. This difference originates from a difference in the potential
149
3. NANOPILLAR ARRAYS IN RTDS AND QCLS: EXPERIMENTAL REALIZATION
profile of the double barrier structure. In the case of Al0.33Ga0.67As, the potential profile
is derived from Γ-point energies such that Al0.33Ga0.67As barriers acts as a tunnel barrier
and GaAs acts as a well. In the case of AlAs, the potential profile can also be derived from
the X-point energy states which causes AlAs to serve as a well for the electrons tunneling
through GaAs barrier. It is also possible that the real potential profile is not based purely
on Γ and X-point energies, but exists as a mixture of the two profiles [143]. Since this adds
additional complexity to the analysis, we do not discuss these states in detail. The purpose
of this section is to introduce the electrical measurements of large-area RTDs, followed by
a discussion of results from the nanopillar arrays. In comparing the emprically observed
NDR peaks, we assume that the longitudinal states that take part in the electron transport
through large-area RTDs are the same states that are also involved in the nanopillar arrays.
3.9.1 Large-area RTDs: ohmic contact
Figure 3.49 shows the 2T measurements of large-area RTDs processed from W0021, with
an ohmic top and bottom-contact at 4 K. The large-area device, with a mesa size of approx-
imately 20 µm x 20 µm, shows distinct NDR peaks under positive and negative polarities.
Assuming a Γ profile of the AlAs/GaAs structure, the first NDR peak should occur at a bias
abs(Vp) = 2Ei = 0.1 V corresponding to alignment with the energy states E0 ∼ 51.7 meV 1.
Experimentally however, the position of the measured NDR peaks deviates slightly from the
theoretically predicted NDR peaks. In a 4T measurement, this deviation occurs primarily
because of the depletion region formed in the low-doped GaAs layers between the contact
and the barriers [102, 137]. In a 2T measurement, an additional contribution comes from
the voltage that is dropped across the leads. Therefore, while the theoretical calculations
predict the ground state NDR peak at abs(Vp) = 0.10 V, the first peak is observed at Vp
= +1.08 V under positive polarity and at Vp = -0.56 V under negative polarity, when the
voltage is sweept from negative to positive. What is interesting to note is that the peak
current density associated with these two peaks is quite different. The peak current den-
sity associated with the NDR at Vp = +1.08 V is Jp = 690 A/cm2 as compared to Jp = -245
A/cm2, which is associated with the peak at Vp = -0.56 V. This suggests that the two peaks
correspond to different bound states. The behavior is reversed when the direction at which
the voltage is swept is reversed. In the reverse voltage sweep, the first peak is observed at
Vp = +0.60 V with Jp = 266 A/cm2 under positive polarity, and at Vp = -1.03 V with Jp =
675 A/cm2 under negative polarity. The similarity in the peak current density between the
peak at Vp = -0.56 V (forward bias) and the peak at Vp = +0.60 V (reverse bias) suggests
1An approximate numerical solution to 2.2
150
3.9 Results from RTDs: W0021
that the two peaks occur due to alignment with the same state, let’s call it E0. In the same
way, the similarity in peak current densities between the peak at Vp = +1.08 V (forward
bias) and the peak at Vp = -1.03 V (reverse bias) suggests that the two peaks occur due to
alignment with the same state, let’s call it E1. Here, we refrain from specifying the origin
of E0 and E1 states, which may belong to Γ-point energy, X-point energy, a mixture of the Γ
and X-point energies, or arise due to accumulation of electrons in the GaAs layer adjacent
to the barrier [137, 143]. We also note the appearance of a small peak at Vp = -0.81 V
( Jp = -124 A/cm2 ) and at Vp = +0.86 V ( Jp = +127 A/cm2 ), which is observed in
conjunction with the peak attributed to state E0, i.e. at Vp = -0.56 V and at Vp = +0.60 V.
We call this peak as E
′
0, since it always appears with E0.
Figure 3.49: Electrical characterization of large-area RTD, with mesa size 20 µm x 20 µm
processed from W0021, with an ohmic top-contact. The measurement was carried out
at 4K in a 2T configuration and the voltage was swept in steps of 5 mV. The device was
fabricated by Ken Cooper and Melanie Tribble.
Figure 3.50 shows the 2T measurements of the 20 µm x 20 µm RTD under an extended
voltage range. At higher voltages, we note the appearance of an NDR peak at Vp = +4.06
V (Jp = +2488 A/cm2) and at Vp = -3.76 V (Jp = -2321 A/cm2). Since the peak current
density associated with the two peaks is similar, it implies that both the peaks correspond
to the same state, which we label as E2.
Figure 3.51 shows the electrical characteristics for a range of mesa areas varying from
20 µm x 20 µm to 50 µm x 50 µm, as the voltage is swept from negative bias to positive
bias. All the devices exhibit NDRs corresponding to state E0 (negative polarity), E
′
0 (neg-
ative polarity) and E1 (positive polarity). Although the peak voltage Vp corresponding to
state E0 increases with mesa area due to an increase in the contact resistance (Vp = -0.56
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3. NANOPILLAR ARRAYS IN RTDS AND QCLS: EXPERIMENTAL REALIZATION
Figure 3.50: Electrical characterization of large-area RTD, with mesa size 20 µm x 20 µm
processed from W0021, with an ohmic top-contact. The measurement was carried out at
4K in a 2T configuration and the voltage was swept from -5 V to +5 V in steps of 10 mV.
The device was fabricated by Ken Cooper and Melanie Tribble.
Figure 3.51: Electrical characterization of large-area RTD, with mesa size varying from 20
µm x 20 µm to 50 µm x 50 µm. The devices were processed from W0021, with an ohmic
top-contact. The measurement was carried out at 4K in a 2T configuration and the voltage
swept in steps of 10 mV. The inset shows the NDR peaks under negative bias in more detail.
The device was fabricated by Ken Cooper and Melanie Tribble.
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3.9 Results from RTDs: W0021
V to -0.81 V), the threshold voltage remains nominally identical (Vth = -0.23 V). We also
note a decrease in the peak current density from Jp = -245 A/cm2 to -84 A/cm2, as well
as a decrease in the peak to valley current ratio, which falls from Jp/Jv = 5.83 to Jp/Jv =
1.86, as the mesa size is increased from 20 µm x 20 µm to 50 µm x 50 µm. This behavior
also holds true for the first NDR peak observed under positive polarity (state E1), where
the peak voltage increases from Vp = +1.08 V to +4.18 V as the area is increased. The
threshold voltage for all the devices remains nominally identical (Vth = +0.26 V). The
peak to valley current ratio tends to decrease with mesa area (Jp/Jv = 7.26 to Jp/Jv =
1.37), and the devices exhibit a peak current density of 654 ± 36 A/cm2.
3.9.2 Nanopillar arrays
Although, we only compared the electrical behavior of A1700 with ohmic and Schottky
contacts (section 3.8.2), it is reasonable to assume that the Schottky barrier on the top-
contact would introduce similar changes to W0021. We can now discuss the results ob-
tained from nanopillar arrays processed from W0021.
3.9.2.1 500 x 500 array, Rp = 150 nm
Results
Figure 3.52 shows the electrical characteristics of a 500 x 500 nanopillar array with
pillar size defined as Rp = 150 nm. We note distinct NDR peaks at 4 K and at 300 K,
indicating successful top-contact to the nanopillar array. Under negative bias, we observe
NDR peaks at Vp = -0.52 V, -0.92 V, -1.36 V, -2.2 V, -2.38 V, -2.69 V. Under positive bias,
the NDR peaks are observed at Vp = +0.48 V, +0.80 V, +1.77 V, +3.68 V. The peaks
observed under negative polarity are shown in detail in figure 3.53. The appearance of
NDR peaks under forward as well as reverse voltage sweeps confirms transport through
the nanopillars, see figure 3.53. What makes the analysis challenging is the appearance
of multiple NDR peaks, which do not necessarily correspond to the states observed in the
large-area RTDs (figure 3.49).
Mapping of NDR peaks
In large-area RTDs processed with an ohmic contact, the first NDR peak was observed
at Vp = -0.56 V (figure 3.49). This state was labelled as E0. Assuming a 0.7 V drop due to
the Schottky top-contact, the first NDR peak due to alignment should occur at an absolute
bias of 1.24 V. However, from figure 3.52, we note two peaks (Vp = -0.52 V and Vp = -0.94
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3. NANOPILLAR ARRAYS IN RTDS AND QCLS: EXPERIMENTAL REALIZATION
Figure 3.52: (a) 2T measurements of 500 x 500 nanopillar array processed from W0021,
with radial dimension Rp = 150 nm measured at 4K and 300 K. The inset, on the top
left shows the scanning electron micrograph of the nanopillar array, before final insulation
and top-contact processing. (b) 2T measurements of the 500 x 500 nanopillar array at 4K,
shown separately at low-voltage. The voltage was swept from -5 to +5 V in steps of 10 mV.
Figure 3.53: 2T measurements of 500 x 500 nanopillar array processed from W0021 at 4K,
during forward and reverse voltage sweeps. The pillar size was defined as Rp = 150 nm.
The voltage was swept in steps of 10 mV.
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3.9 Results from RTDs: W0021
V) that appear before reaching this bias. These peaks are also symmetric with polarity,
i.e. we note two similar peaks in the positive polarity (Vp = +0.48 V and at Vp = +0.80
V) which appear at approximately the same position and exhibit the same peak current.
Perhaps, these peaks appear due to subband states in the contact and do not necessarily
reflect bound states in the well. It is probable that the peak observed at Vp = -1.36 V
corresponds to alignment with the state E0. Similarly, it is probable that the peak at Vp
= +1.77 V corresponds to alignment with the state E1 and the next highest peak at Vp =
+3.68 V corresponds to E2. However, it is difficult to identify the origin of the three peaks
at Vp = -2.2 V, -2.38 V or -2.69 V, and attribute a suitable cause to the multiplicity of the
peaks.
Estimation of number of pillars
When the device is in resonance with the state E0 at Vp = -1.34 V, the nanopillar array
supports a peak current of -154 µA. The density of current flowing through the device
upon alignment must be equal to the density of current flowing through a large-area RTD.
i.e. Jp = -245 A/cm2. Assuming that only one nanopillar with Rp = 150 nm was contacted
successfully, the peak current flowing through the device, ignoring depletion would be -
173 nA. This current is much less than the peak current that we observe, i.e. Ip = -154 µA.
This implies that multiple nanopillars are indeed in contact and take part in the current
transport. The ratio of the two currents (154/0.173), see eqn. 3.4, leads us to believe that
at least 890 nanopillars are in contact. Note that the number of pillars calculated using
this approach is an under estimate and would increase if we account for surface depletion
in the nanopillars.
Estimation of electrostatic radius
If we reverse the argument and assume that all 500 x 500 nanopillars are in contact,
the electrostatic radius Re can be calculated by equating 250000piR
2
e Jp to the peak current
at Vp = -1.34 V. Substituting -154 µA as the peak current and Jp = -245 A/cm2 in eqn. 3.3,
we estimate the electrostatic radius to be 8.9 nm.
Probing for fine features
In section 2.3, we discussed that the splitting between the energy levels due to quanti-
zation should lead to fine features superimposed upon the NDR peak. According to Reed
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3. NANOPILLAR ARRAYS IN RTDS AND QCLS: EXPERIMENTAL REALIZATION
et. al, these features should be equally separated by 2∆E = 16.8 meV for a 150 nm pillar. It
is likely that these features were not picked up in figure 3.53, where the voltage was incre-
mented in steps of 10 mV. To probe this further we remeasured the electrical characteristics
at a higher resolution, i.e. in steps of 5 mV and 2 mV.
Figure 3.54: (a) Electrical characteristics of 500 x 500 nanopillar array, processed from
W0021 with Rp defined as 150 nm. The electrical characteristics were measured at a
voltage step of 5mV and 2mV in a 2T configuration at a temperature of 4K. For clarity, the
current voltage characteristics at 2 mV voltage step have been displaced vertically by 100
µA. The NDR peaks corresponding to E0 and E1 have been shown separately in (b) and
(c), respectively.
The 2T measurements at a voltage step of 5 mV and 2 mV (figure 3.54) are consistent
with the measurements carried out earlier (figure 3.53), in that the same NDR peaks are
observed under nominally identical positions. The position of these NDR peaks only shifts
slightly because of the inherent hysteresis in RTD devices [140, 141]. In addition to these
peaks (highlighted through red arrows), we observe four peaks (highlighted through green
arrows) that are weak.
Let us look closely at the peaks at Vp = -1.36 V and Vp = -1.38 V. The difference
between the two peaks is 20 meV. Perhaps, this additional shoulder peak is a consequence
of resonant alignment with Eo + (m+1/2)∆E state, where m=1 is an integer and∆E is the
energy splitting due to lateral quantization, see eqn. 2.24. The peaks corresponding to Eo
and Eo+ (
3
2
∆E) are shown separately in figure 3.54(b) for greater clarity. The additional
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3.9 Results from RTDs: W0021
peaks at Vp = -1.71 V, Vp = -2.24 V and at Vp = -2.61 V, shown clearly in figure 3.54(b-c),
can not be easily mapped onto the corresponding energy states. Perhaps, these peaks occur
as a consequence of different pillars coming into alignment. It may also be that these peaks
appear as a result of lateral mode mixing, which is not accounted by Reed’s simplistic model
[144]. While it is challenging to map all the NDR peaks to the corresponding states, due
to the complexity introduced by multiple transport pathways it is certain that a 500 x 500
nanopillar array processed from W0021 with Rp = 150 nm can support parallel conduction.
Our results clearly indicate an electrically robust contact to multiple nanopillars, with a
peak current of -154 µA flowing through the device upon alignment with the E0 state. As
the bias is increased, the array can support a maximum peak current of -1565 µA flowing
through the nanopillars. To verify the electrical robustness of smaller pillars, we reduced
the dimensionality of the pillar from 150 nm to 100 nm. We describe these results below.
3.9.2.2 500 x 500 array, Rp = 100 nm
Results
Figure 3.55 shows the electrical characteristics of a 500 x 500 nanopillar array with
pillar size defined as Rp = 100 nm, measured at 4K and 300 K. While the NDR peak at 300
K is relatively weak, we observe a distinct NDR peak at 4K under negative bias (Vp = -2.72
V). This NDR peak is observable in both forward and reverse voltage sweeps, confirming
resonant tunneling as the dominant transport mechanism through the nanopillar array, fig-
ure 3.56. However, the main NDR peak in this device appeared to shift with voltage with
each measurement. This may be a consequence of a large hysteresis due to intrinsic charge
instability [140, 141].
Mapping of NDR peaks and estimation of number of pillars
Even though the peak at Vp = -2.72 V is the first distinct NDR peak that we observe,
the high peak current associated with the NDR implies that this peak corresponds to an
excited state and not the ground state. A similar peak is also observed when measuring the
nanopillar arrays processed with Rp = 150 nm. For a comparison, see figure 3.57. Since
the position of the peak is lower than -3.76 V, it is unlikely that the peak at Vp = -2.72 V
corresponds to alignment with E2 (figure 3.49), seen in the large-area RTDs. The absence
of a clear ground state in this device, as well as ambiguity in assignment of the observed
peak makes it difficult to directly estimate the minimum number of pillars in contact.
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3. NANOPILLAR ARRAYS IN RTDS AND QCLS: EXPERIMENTAL REALIZATION
Figure 3.55: 2T measurements of 500 x 500 nanopillar array processed from W0021, with
radial dimension Rp = 100 nm measured at 4K and 300 K. The voltage was swept from -5 V
to 0 V in steps of 5 mV. The inset shows the scanning electron micrograph of the nanopillar
array, before final insulation and top-contact processing.
Figure 3.56: 2T measurements of 500 x 500 nanopillar array processed from W0021 at 4K,
during forward and reverse voltage sweeps. The pillar size was defined as Rp = 100 nm.
The voltage was swept in steps of 5 mV.
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3.9 Results from RTDs: W0021
However, we can put a lower limit to the number of pillars in contact by correlating the
peak current tunneling through the 100 nm array at Vp = -2.72 V to the peak current
tunneling through the 150 nm array at Vp = -2.38 V.
Figure 3.57: A comparison of electrical characteristics between a 500 x 500 array (Rp =
100nm) and a 500 x 500 array (Rp = 100nm), measured in a 2T configuration at 4K. The
voltage was swept from -5 V to 0 V in steps of 5 mV. For clarity, the main NDR peak shown
by the 100 nm array is indicated with an red arrow and some of the peaks corresponding
to the 100 nm array are highlighted with a green arrow.
We know from our analysis of the 150 nm array, that at least 890 nanopillars take part
in the transport (see section 3.9.2.1). This array supports a peak current that is equal to
-1562 µA at Vp = -2.38 V. Substituting these values of Nt and Ip in eqn. 3.4, we obtain
a peak current density that is equal to -241 A/cm2. Since this peak current density can
be equated to the peak current density associated with the NDR at Vp = -2.72 V, we can
estimate the number of 100 nm pillars that take part in transport. Substituting Jp = -241
A/cm2 and Ip = -2720 µA in eqn. 3.4, we obtain Nt = 2388.
Analysis of fine features
According to Reed’s simplistic model, we should observe equally separated fine fea-
tures superimposed upon each NDR peak corresponding to the bulk RTD states. For a 100
nm pillar, the features would be separated by 2∆E = 25.2 meV. From figure 3.57, we do
observe a fine structure in the electrical characteristics. One might dismiss the few peaks
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3. NANOPILLAR ARRAYS IN RTDS AND QCLS: EXPERIMENTAL REALIZATION
shown in green occurring as a consequence of parasitic pathways through the insulation,
which can open upon application of a very high bias. However, since we reproducibly
observe the main NDR peak (shown in red) upon multiple measurements, there is merit
in studying the energy distribution of these additional peaks. Since every NDR peak is
accompanied by an increase in the conductance, it is easier to identify the peaks by plot-
ting the derivative of current with respect to voltage. Figure 3.58 shows the differential
conductance of the 100 nm and 150 nm arrays, plotted as a function of bias. The position
of the suspected NDR peaks corresponding to the 100 nm array, is highlighted in blue.
Figure 3.58: A comparison of differential conductance at 4K, between a 100 nm pillar size
500 x 500 array, with a 150 nm pillar size 500 x 500 array processed from W0021. The
voltage was swept in steps of 5 mV. For clarity, the main NDR peak shown by the 100 nm
array is indicated with an red arrow and the other peaks corresponding to the 100 nm
array are highlighted with a green arrow.
If the peaks were distributed randomly, i.e. if the energy separation between consec-
utive peaks was not a constant, then these peaks would occur as a result of unwanted
parasitic effects. However, if Reed’s model was to hold true, the energy distribution of the
peaks would be biased around 2∆E. If we denote the difference between two consecutive
peaks shown in figure 3.58 as∆E’, we can plot a histogram that counts the number of peaks
with respect to ∆E’. The bin size is limited by the voltage step of the measurement, which
is 5 meV in this case. In an ideal scenario, we should have observed multiple instances
(counts > 1) where the peaks are separated by ∆E’ = 2∆E = 25.2 meV and one instance
(count = 1) where the peaks are separated by ∆E’ = 2E1- 2(E0 + (m+1/2)∆E). Here E0
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3.9 Results from RTDs: W0021
Figure 3.59: Histogram showing the energy distribution of peaks at 4K corresponding to
a 500 x 500, 100 nm nanopillar array processed from W0021. The inset shows the ideal
energy distribution of peaks at 4K, evaluated from Reed’s simplistic model.
+ (m+1/2)∆E is the highest energy level that is a result of splitting E0. This distribution
holds true if all the nanopillars in the array have the same dimensions. The ideal distri-
bution according to Reed’s model is shown as an inset to figure 3.59. Our observations
agree partially with Reed’s model, i.e. we do observe that some of the peaks are equally
separated, see figure 3.59. In the histogram shown, we note that there are four instances
where the peaks are separated by ∆E’ = 40-45 meV. We also note that there are three in-
stances where the peaks are separated by ∆E’ = 30-35 meV. Perhaps, the bias around two
energy splittings (∆E’ = 30-35 meV and ∆E’ = 40-45 meV) indicates bimodal distribution
of nanopillar sizes within the array. We do note that the experimentally observed energy
separation (∆E’) is slightly higher than the energy splitting estimated from Reed’s model
(2∆E). The single instance at ∆E’ = 200-205 meV may correspond to the difference be-
tween Ei and Ei−1 + (m+1/2)∆E. The other peaks, which show a count of 1 or 2, suggest
that the energy distribution is not entirely governed by the Reed’s simplistic model. For
an accurate explanation of energy distribution of peaks other effects such as estimation of
depletion, lateral mode mixing due to non-uniform potential, mixed valley through AlAs
barriers and multimodal distribution of dimensions should be taken into account.
Below, we describe the electrical characteristics of another array of nanopillars, where
the radial dimension of the pillar is reduced from 100 nm to 50 nm.
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3. NANOPILLAR ARRAYS IN RTDS AND QCLS: EXPERIMENTAL REALIZATION
3.9.2.3 500 x 500 array, Rp = 50 nm
Results
Figure 3.60 shows the electrical characteristics of a 500 x 500 nanopillar array with
pillar size defined as Rp = 50 nm. In agreement with earlier samples, we note distinct
NDR peaks at 4 K and at 300 K, indicating successful top-contact to the nanopillar array.
We observe several NDR peaks at 4K. Under negative bias, these peaks are observed at Vp
= -1.23 V, -1.57 V, -1.83 V, -2.85 V, -3.02 V, -5.91 V, -6.32 V and -7.1 V. Under positive bias,
these peaks are observed at Vp = 0.71 V, 1.17 V, 5.52 V and 7.59 V. We also observe that
the negative differential resistance peaks appear during both forward and reverse voltage
sweeps, which confirms transport through the nanopillars, see figure 3.61. The appear-
ance of multiple peaks makes it extremely challenging to map the NDR peaks to the bound
states.
Figure 3.60: 2T measurements of 500 x 500 nanopillar array processed from W0021, with
radial dimension Rp = 50 nm measured at 4K and 300 K. The inset shows the scanning
electron micrograph of the nanopillar array, before final insulation and top-contact pro-
cessing. (b) 2T measurements of the 500 x 500 nanopillar array at 4K shown separately at
4K. The voltage was swept from -8 V to +8 V in steps of 10 mV.
Mapping of NDR peaks
162
3.9 Results from RTDs: W0021
Figure 3.61: 2T measurements of 500 x 500 nanopillar array, pillar size 50 nm processed
from W0021 at 4K, during forward and reverse voltage sweeps. The voltage was swept in
steps of 10 mV.
Under negative bias, the first peak is observed at Vp = -1.23 V. After accounting for the
voltage drop across the Schottky contact, this peak corresponds to the E0 state observed in
the large-area device. Similar to the large-area device, an additional peak is observed at Vp
= -1.57 V, which perhaps corresponds to the E’0 state. It is difficult to identify the origin of
the peak at Vp = -1.83 V. The 2T measurements of the 50 nm array also indicate a peak at
around Vp = -2.85 V and at Vp = -3.02 V. Perhaps these peaks corresponding to the 50 nm
array occur due to the same bound state responsible for the NDR peak at Vp = -2.72 V in
100 nm arrays and the NDR peaks around Vp = -2.38 V in 150 nm arrays. It is encouraging
to note fine features superimposed upon main NDR peaks. However, this feature appears
at a very high voltage. Perhaps, these NDR peaks at Vp = -5.91 V, Vp = -6.32 V and Vp =
-7.1 V corresponds to the E2 state seen in large-area devices. Similar NDR peaks at higher
voltages under positive bias at Vp = +5.52 V and Vp = +7.59 V suggest that these peaks
can also be associated with the E2 state seen in large-area devices. The low-voltage peaks
at Vp = +0.71 V and at Vp = +1.17 V may not correspond to a bound state within the
quantum well, but occur as a result of quantization within the contacts.
Estimation of number of pillars
When the device is in resonance with the state E0 at Vp = -1.24 V, the nanopillar array
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3. NANOPILLAR ARRAYS IN RTDS AND QCLS: EXPERIMENTAL REALIZATION
supports a peak current of -13 µA . The density of current flowing through the device upon
alignment must be equal to the density of current flowing through a large-area RTD. i.e.
Jp = -245 A/cm2. Assuming that only one nanopillar with Rp = 50 nm was contacted
successfully, the peak current flowing through the device, ignoring depletion would be -
19 nA. This current is much less than the peak current that we observe, i.e. Ip = -13µA.
Clearly, multiple nanopillars are in contact and participate in resonant tunneling of elec-
trons in parallel. The ratio of the two currents, i.e. the experimentally observed Ip and
the expected current through a single nanopillar (13/0.019), provides an estimate of the
minimum number of pillars in contact, see eqn. 3.4. Here, we believe that at least 623
nanopillars are in contact and take part in transport. Note that the number of pillars calcu-
lated using this approach is an under estimate and would increase if we account for surface
depletion in the nanopillars.
Analysis of fine features
Figure 3.62 shows the differential conductance of the 50 nm array plotted as a function
of applied bias. The position of the main NDR peaks is highlighted with red arrows and the
fine features are highlighted with green arrows. Assuming that Reed’s model is applicable,
the energy distribution of the peaks shown in figure 3.62 would be biased around 2∆E
= 50.2 meV, for Rp = 50 nm. This distribution can be obtained from a histogram, which
counts the number of peaks with respect to ∆E’, where ∆E’ is the difference between two
consecutive peaks. The bin size is limited by the voltage step, which was equal to 10 mV in
this case. In an ideal scenario, we should have observed multiple instances (counts > 1),
where the peaks are separated by∆E’ = 2∆E = 50.2 eV. Also, we should have observed one
instance (count = 1), where the peaks are separated by ∆E’ = 2E1- 2(E0 + (m+1/2)∆E).
Here E0 + (m+1/2)∆E is the highest energy level that is a result of splitting E0. The
ideal distribution for an array of pillar size 50 nm is shown as an inset to figure 3.63(a),
assuming that all the nanopillars in the array have the same radial dimensions.
The energy distribution of peaks shown in figure 3.63 agrees in part with the Reed’s
model. While we do observe that some of the peaks are equally separated and the energy
distribution is biased, it is spread across ∆E’. We see that there are four instances where
the peaks are separated by ∆E’ = 45-55 meV. We also note that there are four instances
where the peaks are separated by ∆E’ = 25-35 meV and three instances where the peaks
are separated by ∆E’ = 15-25 meV. The single instance at ∆E’ = 1.25 V corresponds to
the difference between the bulk states 2E1 and 2E0, figure 3.63(b). Perhaps a bias around
multiple energy bands implies a multimodal distribution of nanopillar sizes within the
164
3.9 Results from RTDs: W0021
Figure 3.62: Differential conductance of a 50 nm pillar size 500 x 500 array processed
from W0021, and measured at 4K. The voltage was swept in steps of 10 mV.
array. Appearance of peak’s in the histogram which show a count of unity or 2 suggest
that factors such as mixed valley transport in AlAs barriers and lateral mode mixing due
to non-uniform potential need to be taken account while calculating the estimated energy
distribution.
To conclude, we have demonstrated successful electrical contact to a 500 x 500 array
of nanopillars processed from double barrier RTDs. Clear NDR signatures for a range of
radial dimensions (50 nm - 150 nm) and wafer structures (A1700/W0021) implies that the
process is robust. From our measurements, it is apparent that multiple nanopillars take part
in electron transport allowing us to reach currents that are at least four orders of magnitude
higher than what has been achieved so far in laterally confined RTDs [89, 94, 102, 108].
For example, when the applied bias is in resonance with the ground state, the nanopillar
array processed from W0021 can carry a current of 13 µA (Rp = 50 nm) to 154 µA (Rp =
150 nm). At higher bias, i.e. when the device is in resonance with the excited states, the
array can support a current of up to 2000-4000 µA. We acknowledge that not all NDR peaks
can be easily mapped onto corresponding energy states, owing to the inherent complexity
introduced by factors such as multiple transport pathways within a single nanopillar, non-
uniform lateral potential and parallel conduction of pillars. However, for a test structure,
appearance of NDR peaks is sufficient proof to scale the process to nanopillar arrays in THz
QCLs.
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3. NANOPILLAR ARRAYS IN RTDS AND QCLS: EXPERIMENTAL REALIZATION
Figure 3.63: (a) Histogram showing the energy distribution of peaks at 4K corresponding
to a 500 x 500, 50 nm nanopillar array processed from W0021. For clarity, the energy
scale is limited to 0-250 meV. The inset shows the ideal energy distribution of peaks at
4K, evaluated from Reed’s simplistic model (b) Complete histogram showing the energy
distribution of peaks at 4K corresponding to a 500 x 500, 50 nm nanopillar array processed
from W0021.
166
3.10 Fabrication of nanopillar arrays in metal-metal THz QCLs
3.10 Fabrication of nanopillar arrays in metal-metal THz QCLs
In section 3.5 we discussed processing techniques that were common to nanopillars in GaAs
/ AlxGa1−xAs active regions. We then validated the processes on test RTD structures. The
next step - i.e. scaling the process to nanopillar arrays in THz QCLs, requires us to address
some additional challenges that are unique to QCL active regions. In section 3.5, we noted
that the highest etch depth for GaAs / AlxGa1−xAs structures was limited to 2.3 to 3.56
µm. At this height, the gain is low and the only waveguide configuration that allows an
ultrathin THz QCL to lase is the metal-metal waveguide configuration, 4.2. This implies
that nanopillars etched in an ultrathin THz QCL have a metal contact at the bottom.
Typically, in a THz QCL, a metal contact at the bottom comprises of a titanium wetting
layer and a thermocompressed gold layer 4.3. Note that post etching, the nanopillars ad-
here to the gold contact and the host substrate through the titanium layer. This effects any
subsequent stage of processing, which involves buffered hydroflouric acid. As mentioned
in section 3.5, buffered hydroflouric acid is necessary to remove etch residue as well as the
insulation for a suspended contact. However, buffered hydroflouric acid etches titanium
at a rate faster than the etch residue or the insulation. This problem was identified at a
much later stage of this doctoral work, and could not be rectified within the time limit.
Therefore, for successful scaling of our process to metal-metal THz QCLs, it is essential to
investigate the possibility of using NiCr as the wetting layer for gold, while evaporating the
bottom-contact. In the next section we summarize the key points discussed in this chapter.
3.11 Summary and conclusions
In this chapter, we first discussed techniques that allow a top-down fabrication of nanopil-
lar arrays. We then demonstrated fabrication of pillars in GaAs / AlxGa1−xAs systems with
sub-200 nm radius and a high aspect ratio (1:15) nanopillars. This involved a careful opti-
mization of etch mask, etching system and choice of insulation. Each processing stage was
designed in view of structural and electrical stability of the array, with due consideration
to compatibility with the final laterally confined THz QCL device for improved tempera-
ture performance. Using test RTD structures, which show similar current densities as THz
QCLs, we then validated the electrical robustness of the nanopillar arrays. Finally, using
test RTDs, we demonstrated parallel conduction through an array of 500 x 500 sub-200
nm nanopillars.
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3. NANOPILLAR ARRAYS IN RTDS AND QCLS: EXPERIMENTAL REALIZATION
168
"One doesn’t discover new lands
without consenting to lose sight of the
shore for a very long time."
Andre Gide
Chapter 4
Ultrathin QCLs
4.1 Introduction
In the previous chapter we noted that in order to implement the top-down nanofabrication
approach to THz QCLs, the thickness of the active region needs to be scaled down to 3.5
µm or less. We refer this scaled down THz QCL, as an ‘ultrathin’ THz QCL. A THz QCL
typically contains a cascade of more than 100 repeat periods. Each period consists of an
injection stage, upper lasing level, lower lasing level and an extraction stage. Depending
upon the design, the thickness of a THz QCL can range from 6 - 14 µm. This implies that in
general, ultrathin QCLs are low-gain active regions. This is because the peak modal gain,
GM is proportional to the number of periods, Np and inversely proportional to length of
the period, Lp.
Detailed derivation of the gain can be found in [10, 38]. A simplistic description is
provided below.
4.1.1 Effect of active region thickness on gain
In a single period of the active region, the intersubband radiative transition between the
upper lasing level ‘u’ and the lower lasing level ‘l’ can be quantified in terms of a parameter
called the material gain, Gp. This material gain, Gp is calculated by solving the rate-
equations describing the radiative transition in terms of (a) the injection efficiency, ηi , of
the charge carriers into the upper lasing level (b) the charge carrier lifetime in the lower
lasing level, τl , and, (c) the charge carrier lifetime in the upper lasing level, τu. The charge
carrier lifetime in the upper lasing level can be expressed as τ−1u = τ−1ul +τ−1ug +τ−1escp. Here,
scattering of charge carriers from u to l is defined by a time constant of τul . This is the
direct path to non-radiative relaxation of electrons in the upper lasing level. The electrons
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4. ULTRATHIN QCLS
in the upper lasing level can also parasitically scatter into the ground state with a time
constant indicated by τug . Another contributor to the parasitic leakage is the tunneling
of electrons through the barriers from the upper lasing level into the extractor state. This
process is associated with a time constant defined by τescp. All of these processes are
illustrated in figure 4.1.
Figure 4.1: Schematic diagram showing radiative (curly arrow) and non-radiative (straight
arrows) processes across the lasing levels in a THz QCL.
The material gain, Gp, is a function of the dipole matrix element


zul

which describes
the probability of the electron extending in the growth direction. The material gain, Gp, is
adversely effected by the length of the period, Lp and the broadness of the energy transi-
tion, 2γ, which can lead to electron scattering when large.
Mathematically, the material gain is expressed as:
Gp =
4piJe


zul
2
εoεr2γλLp
ηiτu

1− τl
τul

. (4.1)
where, J is the current density describing the number of electrons injected into the
upper lasing level and λ is the wavelength associated with the energy transition. Dividing
the material gain Gp with the current density J , one obtains the so-called gain-coefficient
g = Gp/J associated with the intersubband transition.
The peak modal gain, GM is proportional to the material gain Gp, which measures
the strength of the intersubband transition, and the spatial overlap of the optical mode
with the gain medium. Therefore, the peak modal gain GM can be written in terms of the
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4.1 Introduction
waveguide confinement factor Γ as:
GM = ΓGp = ΓpNpGp = ΓpNp gJ (4.2)
Here, Γ = ΓpNp where Γp is the overlap factor per period.
As is seen from eqn. 4.2, there is no threshold to achieve population inversion. The gain
is present from the first electron injected into the device as long as the condition τu ≥ τl
is satisfied and population inversion is achieved.
The threshold current density, Jth is related only to the optical cavity losses, which
occurs due to waveguide losses (αw) and mirror losses (αm). While the waveguide loss
αw depends upon the scattering and absorption inside the resonator, the mirror loss αm =
−ln(R1R2)/2L depends upon the reflectivity of the facet (R1, R2) and the cavity length L.
The threshold condition can be obtained by balancing the gain GM with the total losses
(αw+αm). Therefore, we obtain the condition for threshold current density, Jth as:
Jth =
αw +αm
gΓpNp
(4.3)
It is easily seen that as Np reduces, Jth increases.
4.1.2 Ultrathin QCLs: finding the right active region
Practical implementation of top-down nanopillars in QCLs necessitates the use of very thin
active regions to ensure a pillar thickness of less than 3.5 µm, which can be etched in a
reactive ion etching system (see Chapter 3 for processing details). The key to achieving
lasing in a thin QCL is to maximize the total gain, GM in the active region and minimise the
total waveguide loss (αw+αm). Let us first focus on the gain medium. To maximize GM ,
it is necessary to choose an active region with a higher number of repeat periods per unit
height, i.e. a small Lp. As described in the introductory chapter, active regions based on
resonant LO phonon designs typically require 3-4 quantum wells per period as opposed to
bound-to-continuum (BtC) designs, which require 9-10 quantum wells for efficient injec-
tion and extraction within each period. Therefore, resonant LO phonon active regions do
appear likely candidates for implementing the top-down approach to lateral confinement.
Another point to note is that BtC designs are not ideal to observe lateral confinement
effects. This is because we expect the radial potential to split the energy levels by ∼ 5-
10 meV. In absorption or an electroluminescence spectra, energy states associated with
the lateral potential would be superimposed upon the energy states associated with the
longitudnal confinement. These small differences may not be easy to recognize particularly
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4. ULTRATHIN QCLS
in a superlattice structure that relies on minibands where the energy levels are separated
by ≤ 5 meV.
Interestingly, there exists a third alternative that lies in between the two mechanisms
that we have discussed (LO phonon and BtC) - a hybrid that uses both resonant LO phonon
and a miniband of energy states for electron extraction. This 4 quantum well design, is
robust in terms of growth fluctuations, and relies on a BtC transition in which a miniband
is coupled to the extraction stage by resonant LO mechanism [6]. We refer this design as
3 THz 4 QW ETH design throughout this work.
4.2 Ultrathin QCLs in literature
Prior work on achieving THz lasing from ultra-thin (≤ 3.5 µm) active regions has been
primarily based on resonant LO phonon designs, processed in a metal-metal waveguide
configuration. The two reports that demonstrate lasing from ultrathin THz QCLs use (i) a
3 quantum-well (QW) design [3], and (ii) a 4 quantum-well (QW) design [4].
Figure 4.2 shows the IV and spectral characteristics of the 3 QW design, as a function
of active region thickness. Note that the laser shows observable emission between 3 to 3.5
THz as the active region thickness is reduced from 10 µm to 1.75 µm. It is important to
point out that three of the five lasers compared in figure 4.2 were processed from three
distinct wafers with active region thickness 10 µm, 5 µm and 2.5 µm. All these lasers
retained the top n+ doped layer. Lasers with active region thickness 2 µm and 1.75 µm
were derived from the 2.5 µm active region, by chemically thinning the wafer.
Figure 4.2: (a) IV characteristics of 3 quantum-well LO phonon design at 10 K, as a function
of active region thickness. The inset plots the electric field as a function of current density.
(b) Spectral characteristics of the 3 quantum-well LO phonon design showing blue shift
with decreasing active region thickness. Adapted from [3].
The effect of reducing the active region thickness is seen noticably in the threshold
172
4.2 Ultrathin QCLs in literature
current density, which reduces from 925 A/cm2 to 680 A/cm2 at 10K, as the active region
thickness is reduced from 10 µm to 1.75 µm, see figure 4.3. This behavior follows from
eqn. 4.3, which predicts that the threshold current density is inversely proportional to the
number of periods. Another important change brought about by reduction in the active
region thickness is on the maximum operating temperature of the laser. The maximum
operating temperature reduces from 146 K in the case of 10 µm thick laser to 79 K in the
case of 1.75 µm laser.
Figure 4.3: Threshold current density as a function of temperature for 3 quantum-well LO
phonon design, reported for active region thickness varying from 10 µm to 1.75 µm. The
table on the right shows threshold current density for lasers with different active region,
at a temperature of 10 K. Adapted from [3].
Figure 4.4 shows the LIV characteristics of the 4 QW design at 10 K for three different
active regions. Also shown in the inset is the spectral behavior of the 4 QW design and
the threshold current density as a function of operating temperature. All three lasers were
derived from the same wafer, i.e. lasers with thickness 5.1 µm and 2.8 µm were obtained
by chemically thinning the 10 µm wafer. Therefore, while the 10 µm thick laser retained
the top n+ doped layer, the other two lasers were processed without it. Similar to the
behavior of the 3 QW design discussed earlier, the threshold current density of the 4 QW
lasers increases with decrease in active region thickness. The threshold current density
increases from 440 A/cm2 to 575 A/cm2, as the thickness of the active region is reduced
from 10 µm to 2.8 µm. The maximum operating temperature of the lasers decreases from
166 K in the case of 10 µm active region to 117 K in the case of 2.8 µm active region.
From the epitaxial structure detailed in references [3, 4, 33, 145] we note that the 2.8
µm-thin 4 QW design shown in figure 4.4 has lower repeat periods, Np = 50, as compared
to Np = 56 in the 2.5 µm-thin 3 QW design shown in figures 4.2 and figure 4.3. Despite
the lower number of repeat periods, the maximum operating temperature of the former
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4. ULTRATHIN QCLS
Figure 4.4: LIV characteristics as a function of temperature shown for 4 quantum-well LO
phonon design with an active region thickness of (a) 2.8 µm (b) 5.1 µm (c) 10.0 µm. Inset
in (a) shows spectral characteristics of the ultrathin device, with emission at 2.9 THz. Inset
in (c) shows the threshold current density Jth varying as a function of temperature. The
table on the right shows threshold current density for lasers with different active region,
at a temperature of 10 K. Adapted from [4].
(166 K) is nearly two times higher than the latter (95 K). One of the reasons for superior
performance is perhaps due to the improved heat sinking of the 2.8 µm thin 4 QW device,
implemented by employing a Cu-Cu waveguide [4, 33]. Therefore, both the gain medium
and the choice of the waveguide play a crucial role in the operating performance of an
ultrathin THz QCL.
4.3 Fabrication of ultrathin QCLs
THz QCLs can be fabricated in single plasmon (SP) waveguide configuration or in double
metal (DM) waveguide configuration. As discussed in the introduction, a single plasmon
waveguide requires a thick (700 nm) n+ doped GaAs layer, which when suitably processed
acts as an ohmic bottom contact, and confines the plasmon mode to the active region. In
a double metal waveguide, this layer is not necessary as the plasmon mode is predomi-
nantly confined between the top and bottom metal contacts. Therefore, to maximize the
number of repeat periods per unit height, the double metal (DM) waveguide configuration
174
4.3 Fabrication of ultrathin QCLs
is preferable for ultrathin THz QCLs. Further more, a major motivation behind this work
is higher temperature performance, and since DM waveguide devices have superior heat-
sinking properties as compared to SP waveguides, we focus on the former in this chapter.
The latter is discussed separately in [5].
4.3.1 Wafer bonding and host-substrate removal
The fabrication of double metal QCLs involves thermocompression bonding of active region
grown via molecular beam epitaxy (MBE) onto a host n+ GaAs substrate. Prior to bonding,
both the MBE wafer and the host wafer are evaporated with a ∼ 500 nm thick layer of
Ti/Au (figure 4.5(a-b)). Ti acts as a wetting layer which improves the adhesion of Au to
the GaAs surface. The thickness of the Au layer, as well as the surface quality before and
after evaporation determines the quality of the wafer bond. To achieve a good surface
quality before evaporation, it is common to clean the wafers in deionized water, acetone
and IPA before evaporation. A 30 sec dip in 1:10 solution of HCl:H2O removes the oxide
layer on the GaAs surface. We find that removal of the oxide layer in HCl introduces stains
on the wafer surface, detrimental to the surface quality. These arise due to drying of water
droplets on the hydrophilic GaAs surface. It is therefore essential to rinse in DI water for
2-3 minutes until the resistance of the water reads ≥ 10 Ohm. Care must also be taken
to blow dry water droplets from the tweezers, which are used to handle the MBE and
host substrate. Any residues that remain may be cleaned gently with a lens wipe. During
the evaporation, it is important to melt the Au before evaporation to avoid spitting of Au.
The best surface finish is obtained at an evaporation rate of ≤ 0.3 nm/sec, and when the
wafer is placed away from the normal axis of the Au boat. It is also possible to evaporate
Pd/Ge (25/75 nm) before the Ti/Au evaporation which forms a shallow ohmic contact to
the bottom n+ GaAs. However, Pd/Ge metallizations are susceptible to degradation owing
to oxidation of Ge and decomposition of PdGe, which is why this step is avoided in our
processing.
The thermocompression bonding (figure 4.5(c)) is performed at Applied Microengi-
neering Limited (AML) where the bonding is carried out under a uniform pressure of 2
MPa across the wafer at a temperature ∼ 300o C [146]. The MBE substrate is thinned to
200µm by mechanical polishing (figure 4.5(d)). Further substrate thinning is achieved
chemically by using a 5:1 solution of anhydrous citric acid (1g:1ml of H2O) and H2O2
[147]. This solution etches GaAs at a rate of 0.33 µm/min until it reaches the Al0.5Ga0.5As
etch stop layer.
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4. ULTRATHIN QCLS
Figure 4.5: Process steps in the wafer-bonding of a double metal QCL (a-c) Au-Au thermo-
compression bonding of MBE grown active region onto n+ GaAs substrate (d) Mechanical
thinning and polishing of MBE substrate (e) Removal of substrate in a selective etchant.
4.3.2 Mesa etch and metallization
Further processing of laser ridges can be performed either using dry etching, which is
anisotropic and enables smooth vertical walls or using wet etching, which is isotropic and
leads to sloping sidewall etch profiles.
4.3.2.1 Dry etching
When the ridges are processed using reactive ion etching, the etch stop layer is removed
with hydroflouric acid to ensure a smooth surface (figure 4.6(b)). Following the etch stop
removal, the etch mask is lithographically defined which also forms the self-aligned top
contact (figure 4.6(c)). In the case of dry etching laser ridges, it is important to deposit
a thick layer of Ti/Au, preferably 300-400 nm to account for mask erosion during plasma
etching. It is good to note that because the thickness of ultra-thin QCLs is small, it is
relatively easy to obtain vertical sidewalls with minimum mask erosion in a reactive ion
chamber. Powerful BtC designs that employ active regions of thickness ≥ 10µm might
require a dielectric mask for vertical etching, in which case the etch mask does not form a
self-aligned top contact. See section 3.5.3 for details about reactive ion etching.
176
4.3 Fabrication of ultrathin QCLs
Figure 4.6: Dry etching of QCL showing (a) wafer after substrate removal (b) removal of
Al0.5Ga0.5As etch stop in hydroflouric acid (c) photo lithography of etch mask (d) reactive
ion etching of active region with the self-aligned etch mask (e) etched mesa with near
vertical sidewall profile.
4.3.2.2 Wet etching
An alternative approach would be to process laser ridges with widths ∼ 100µm using
chemical etching (see, figure 4.7). A primer (hexamethyldisilazane) is ultilized to increase
the adhesion of the photoresist etch mask to the active region. Retaining the etch stop
layer during wet chemical etching prevents the etchant from opening up growth defects in
the active region. Following the mesa etch in a 1:8:80 solution of H2SO4:H2O2:H2O, the
photoresist and etch stop layers are removed in acetone and hydroflouric acid, respectively
(figure 4.7(d)). A Ti/Au layer (20/100 nm) forms a Schottky contact to the mesa and
facilitates wire bonding to the top of the laser ridge (figure 4.7(e)).
The host substrate may be mechanically thinned down from 500 µm to 200 µm to allow
better optical confinement and improved heat sinking. In the single plasmon waveguide
geometry, the optical mode is confined within a top metal contact and a doped n+ GaAs
layer and hence thinning of the host substrate reduces optical losses. Another layer of
Ti/Au is deposited on the backside of the substrate, which acts as a diffusion barrier for
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4. ULTRATHIN QCLS
Figure 4.7: Wet etching of QCL showing (a) wafer after substrate removal (b) photolithog-
raphy of resist etch mask (c) chemical etch of active region and etch stop (d) removal of
resist and etch stop layer (e) lithography and evaporation of top contact.
indium that is used while mounting the chips. The laser ridges are formed into Fabry-Pèrot
cavities of length ∼ 1-2 mm by cleaving along a crystal axis to produce parallel laser facets.
The cavity length determines the mirror losses as well as the current required to achieve
lasing. Since our measurement set-up has a maximum current range of 4 A, this allows
for a maximum current density of 2000 A/cm2 to flow through a 2 mm long, 100 µm
wide cavity, which is sufficient for the typical threshold current density observed for ultra-
thin THz QCLs. Following indium soldering to copper blocks which ensures good thermal
contact, the devices are bonded with gold wires to gold contact pads evaporated on the Cu
blocks. The devices are then mounted on Cu blocks for electrical characterization in the
cryostat.
178
4.4 Characterization of ultrathin QCLs: methods
4.4 Characterization of ultrathin QCLs: methods
4.4.1 Mounting
THz QCL devices are mounted onto the cold finger of a continuous flow liquid helium
cryostat (Janis Research Company, model ST-300). Brass contacts attached to the cryostat’s
cold finger with nylon screws, clamp the copper block to the cold finger and electrically
connect the device. The contacts are electrically connected to a series of BNC connectors
on the outside of the cryostat via micro-coaxial cables. Optical access to the sample space
is achieved using a polyethylene window, which is transparent to THz radiation.
4.4.2 Electro-optic measurements
Figure 4.8: Electro-optic setup for measuring LIV characteristics when QCL is operated in
the pulsed mode
The QCL can be operated in either pulsed mode or in continuous wave (cw) mode. To
avoid Joule heating in the ultra-thin QCLs (as well as nanopillars), the QCLs reported in
this work were operated in the pulsed mode. CW measurements may be carried out with
an HP 1200 DC power supply.
Figure 4.8 shows the experimental set-up employed to measure the electro-optic char-
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4. ULTRATHIN QCLS
acteristics of THz QCLs. The QCL is powered with 10 kHz pulses at 5-10 % duty cycle,
supplied by an Agilent 8114A pulse generator. The 10 kHz pulses are gated externally
by a low frequency 5V square pulse that is supplied by a separate pulse generator. The
same low frequency pulse is used as an external trigger on the Tektronics TCS 2014 digital
oscilloscope to measure the voltage and current across the device. The frequency of this
external gate or trigger is determined by the response time of the photodetector. If a golay
is used as a THz detector, then the reference frequency is 15 Hz. For spectral and low
power measurements a bolometer is used to measure the THz output, and the reference
frequency is 300 Hz 1. The output of the detector is sent as input to EG&G 5210 lock-in
amplifier, which measures the THz signal at the reference frequency. The instruments are
interfaced to the computer through labview.
4.4.3 Spectral measurements
4.4.3.1 Above lasing threshold
Figure 4.9: Experimental set-up used to collect spectra above lasing threshold. Diagram
from [5].
For results presented in this chapter, we measured the emission spectra above lasing
threshold at Manchester using a Bruker VERTEX 80 Fourier Transform Infra-Red (FTIR)
Spectrometer set up in the rapid scan mode. A liquid-He cooled Si-bolometer was used as
1Optimum operating frequency may vary from 80 Hz - 333 Hz between bolometers.
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4.4 Characterization of ultrathin QCLs: methods
the THz detector. This is shown in figure 4.9. The maximum spectral resolution attainable
using this FTIR Spectrometer is 2.1 GHz (0.07 cm−1). The FTIR is essentially a Michelson
Interferometer, where the radiation from the source (THz QCL) is directed to a beam
splitter. Half of the radiation is reflected from a fixed mirror while the other half is reflected
from a mirror moving continuously over a specific distance. When the two beams are
recombined at the detector (Si-bolometer), an interferogram is produced. The beam path
between cryostat and detector is purged with N2 gas in an attempt to minimize losses due
to water vapor absorption. In the rapid scan mode mode, the mirror moves continuously
over a fixed distance. One complete movement comprises a single scan. While the mirrors
are moving, the output detected from the bolometer is sent to the input of the FTIR, which
performs the Fourier transform of the signal. This signal is fed into the computer. To
avoid beating, the QCL is operated without the slow modulation and is pulsed with fast
modulation alone (10 kHz, continuously triggered, duty cycle of 1-2 %).
4.4.3.2 Below lasing threshold
Figure 4.10: Experimental set-up used to collect spectra below lasing threshold. Diagram
from [5].
In the case of below threshold operation, we operated the Bruker VERTEX 80 FTIR at
Manchester in the step-scan mode to allow collection of the weak signal. In the step-scan
mode, the QCL is pulsed with both fast and slow modulation on. The mirror moves in
steps. The signal collected from the bolometer at each step of the mirror is fed into the
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4. ULTRATHIN QCLS
lock-in. The reference signal provided is the same as the slow modulation (300 Hz). The
DC signal from the lock-in is fed into the FTIR, which is collected and processed by the
computer. The experimental set-up is shown in figure 4.10.
4.5 Ultrathin active regions used
The various active regions investigated for suitability as ultrathin THz QCLs during this
doctoral work are summarized in table 4.1. Corresponding to each design (highlighted
in green) and wafer number grown at Cavendish (highlighted in magenta), we list the
reference design published in literature (highlighted in blue). Among all the reference
designs listed in table 4.1, L422 (full-stack reference L421, L420) is the only ultrathin
wafer which emits at lower repeat periods. While all other wafers, i.e. L207, FL178C-M7
and EV1116, are likely candidates for ultrathin devices, a direct comparison between the
performance of these reference designs and the in-house grown QCLs can only be made
for full-stack wafers. While the first three active regions, i.e. 2.7 THz 3 QW Paris, 3 THz
3 QW Paris and 3 THz 4 QW MIT are resonant phonon designs, the final active region
listed in table 4.1 as 3 THz 4 QW ETH design can be categorized as a BtC design. The 3
QW Paris designs emitting at 2.7 THz and 3 THz are similar as they both exploit a vertical
lasing transition and have nominally same anticrossing energies with injection/extraction
levels. With the exception of W0798 (2.7 THz 3 QW Paris design), which was grown in the
W chamber, all wafers in the V series were grown in the Veeco chamber at the Cavendish
Labs. All QCLs labelled in the table as ‘ultrathin’ were grown thin epitaxially. V674T ,
which is mentioned later in this chapter is also an ultrathin QCL. However, unlike its other
counterparts, V674T was chemically etched down to a thickness of 3 µm.
Design Wafer number Reference in literature Previous growths
2.7 THz 3 QW Paris
V671 Ultrathin
L207 in [148] NAW0798 Ultrathin
V692 Ultrathin
3 THz 3 QW Paris
V702 Ultrathin L421 L420, L422 in [3], Also
in [145, 149]
NA
V703 Full-stack
3 THz 4 QW MIT
V704 Ultrathin
FL178C-M7 in [150] V569 Full-stack
V705 Full-stack
3 THz 4 QW ETH V674 Full-stack EV1116 in [6] V696 Full-stack
Table 4.1: Summary of active regions and wafers used in this work, with the corresponding
references to the literature.
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4.6 2.7 THz 3 QW Paris design
Figure 4.11: Energy band diagram of 2.7 THz 3 QW Paris design evaluated using in-house
code at an electric field, E = 11.2 kV/cm. The modulii-squared wavefunctions are shown in
blue for the upper (4) lasing level and in red for the lower (3) lasing level. Starting from the
injection barrier, the growth sequence in nanometers is given as 4.8/9.4/2.4/7.2/4.2/15.7
where Al0.15Ga0.85As barrier layers are in bold. The center 5.5 of the 15.7 nm injection well
is doped with Si to Nd = 5x 1016 cm−2. The dashed lines enclose one period of the active
region.
4.6 2.7 THz 3 QW Paris design
Figure 4.11 shows the energy band structure of the 2.7 THz 3 QW Paris design computed
using the in-house solver [5]. Based upon one of the first 3 QW designs [145], this de-
sign employs two tunnel-coupled wells for lasing and one well for both resonant-phonon
depopulation and carrier injection. A single well for injection also implies lower intersub-
band absorption loss, a parameter crucial to the observation of emission in ultrathin QCLs.
Shown in table 4.2, is a comparison of the lasing parameters reported in [148] with those
calculated using the in-house Poission-Schrodinger solver.
2.7 THz 3 QW Paris design
Literature [148] In-house
Eul (meV) 11.15 11.2
Field (kV/cm) 12.5 12.0

zul

(nm) 5.1 6.25
ful 0.52 0.77
Table 4.2: Summary of the design parameters calculated for 2.7
THz 3 QW Paris design
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4. ULTRATHIN QCLS
The calculated energy difference between upper and lower lasing levels Eul and the
alignment field agree with the corresponding values reported in literature [148]. However,
we overestimate the dipole matrix element


zul

and the normalized oscillator strength ful .
This difference may be explained if we look at


zul

and ful , closely.
The dipole matrix element


zul

for an optical transition can be expressed in terms of
the envelope wavefunction as:


zul

=
∫ +∞
−∞
dzψ∗f (z)zψi(z) (4.4)
The oscillator strength is proportional to the square of the dipole matrix element and
is given as:
ful =
2m ∗ (E f − Ei)
zul2
ħh2
(4.5)
Therefore, both


zul

and ful depend upon the integral of the envelope wavefunction
in the direction of growth, i.e. z-axis. Typically, this relies upon (i) the summation method
used for numerical integration, and (ii) the distance z, to which the wavefunction is trun-
cated. In our solver, the wavefunctions are truncated up to 3 periods. It is probable that
slight differences in the integral method and the domain of integral between solvers, influ-
ences the calculation of dipole matrix element and oscillator strengths.
We next describe the electro-optic characteristics of ultrathin 2.7 THz 3 QW Paris design
grown thrice at the Cavendish labs.
4.6.1 Ultrathin QCL (V671)
V671 was the first ultrathin QCL grown for this work towards the end of a growth campaign
in November 2010. However, none of the devices processed from V671 appeared to lase.
XRD data from this wafer (reproduced in appendix B) showed that this wafer was thinner
than the target thickness by 0.56 %. Table 4.3 lists the target specifications of V671 in
terms of active region thickness, period length Lp, number of periods Np and waveguide
structure, i.e. thickness and doping of the top and bottom n+ doped GaAs. Perhaps,
V671’s failure to lase is a consquence of insufficient gain due to reduced active region
thickness, see section 4.1.1. Apart from the active region and number of periods, the
waveguide structure plays an equally important role in determining some of operating
characteristics. The influence of n+ GaAs layers on the waveguide losses in an ultrathin
THz QCL is discussed in detail in section 4.10.
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4.6 2.7 THz 3 QW Paris design
2.7 THz 3 QW Paris design
Wafer Length per period,
Lp (nm)
Number of re-
peats, Np
Waveguide Thickness without
etch stop (µm)
V671,
W0798,
V692
43.7 45 Top 75 nm (2e18), Bottom 75
nm (2e18)
2.121
Table 4.3: Wafer structure of the ultrathin 2.7 THz 3 QW Paris design.
Figure 4.12: (a) LIV characteristics of ultrathin 2.7 THz 3 QW Paris design (V671) double-
metal ridge, measured at 4K using golay for THz detection (b) Optical image of 2.7 THz
3 QW Paris design (V671) showing appearance of growth defects, at 5x magnification (c)
Optical image of a growth defect at 50x magnification.
Another possible explanation to device failure may rest in the examination of the elec-
trical characteristics. Figure 4.12 shows that the devices processed from V671 appeared
to short-circuit. As this wafer was processed multiple times into 80-150 µm wide ridges,
by chemical and dry etching, it may have been short circuiting due to growth defects that
resulted in the top Ti/Au contact spiking down to the bottom contact during evaporation,
see figure 4.12(b-c). Since spiking occurs over a length of 1-2 µm, it is not a problem that
influences full-stack THZ QCLs, which are 6-14 µm in height. In an ultrathin THz QCL, the
spiking depth approaches the thickness of the active region, which adversely affects their
performance.
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4. ULTRATHIN QCLS
The existence of interface homogeneities and alloy disorders manifests as increase in
the full width half maximum (FWHM) of the satellite peak recorded by XRD measurements.
Therefore, the poor performance of V671 can be attributed to interface inhomegeneities
and alloy disorders through the FWHM of the satellite peaks (appendix B). Unfortunately,
the FWHM of all the peaks, the substrate peak (43 arc.sec) and the satellite peak (40 to 43
arc.sec), is limited by the resolution of the XRD system, which makes it harder to ascertain
the cause of failure for V671.
4.6.2 Ultrathin QCL (W0798)
During the MBE downtime on the V chamber, V671 was repeated in the W chamber as
W0798. XRD data from this wafer showed a deviation of +0.96 % from the target thick-
ness. The FWHM derived from the substrate peak (43 arc.sec) and the satellite peak (40 to
43 arc.sec) suffered from the same resolution issues as V671, making it difficult to assess
the epitaxial growth of W0798.
Figure 4.13 shows the LIV characteristics of a double metal ridge processed from this
wafer. From the LIV characteristics, it appears that the device short-circuits and fails to
lase. After multiple attempts at characterization, this wafer was discarded from analysis as
this was the first THz QCL attempted in the W chamber, which gave rise to many floating
experimental parameters related to growth in a different system. Issues with thermo-
compression bonding may have also affected the performance of the laser. Furthermore,
W0798 suffered from the same issues as V671 with regards to reduced gain due to active
region design, number of repeat periods and waveguide structure (table 4.3).
4.6.3 Ultrathin QCL (V692)
Grown in December 2012, at the beginning of a new growth cycle, devices processed from
wafer V692 did not lase, see figure 4.14. However, unlike V671 and W0798 these devices
did not short. XRD data (reproduced in appendix B) showed that V692 was grown thinner
than the target thickness by approximately 1.99 %. Broadening of the FWHM derived from
the substrate peak (50 arc.sec) and the satellite peak (50-70 arc.sec) suggested inhomo-
geneity in the growth, possibly due to an arsenic drift. Subsequent to the failure of the
previous two attempts, we found that all three wafers V671, W0798 and V692 were grown
in the reverse order as reported by Chassagneux et al.[148], i.e. the injection barrier was
closer to the bottom contact as opposed to the top. This motivated us to test the V692 in
both forward and reverse bias.
We use results previously reported in [148] to understand the electrical characteristics
186
4.6 2.7 THz 3 QW Paris design
Figure 4.13: LIV characteristics of ultrathin 2.7 THz 3 QW Paris design (W0798) double-
metal ridge, measured at 4K using golay for THz detection
of V692, shown in figure 4.14.THz QCLs processed from the full-stack wafer (L207) in
[148] show a Jth ∼ 900 A/cm2. At this current density, the authors in [148] report an
alignment bias of ∼ 12.5 V, which corresponds to an alignment electric field of 12.5 V
kV/cm. At the same current density, i.e. at J = 900 A/cm2, the voltage dropped across
V692 is ≈ 3.1 V in the reverse bias and ≈ 2.7 V in the forward bias, see figure 4.14.
Accounting for the voltage dropped across the top and bottom Schottky contacts (0.7 V+
0.7 V), the measured bias (1.7 V and 1.4 V, respectively) corresponds to an electric field of
8.1 kV/cm in the reverse sense and 6.2 kV/cm in the forward sense.
In the above analysis, we ignored the increase in threshold current density introduced
by reducing the number of periods (eqn. 4.3). In reality, the Jth expected for the ultrathin
V692 is higher than the threshold current density reported for L207 (900 A/cm2). This
implies that the measured electric field corresponding to alignment is actually higher than
the value calculated from figure 4.14, i.e. greater than 8.1 (6.2) kV/cm in reverse (for-
ward) bias. The results from V692 show that the measured electric field associated with
alignment (≥ 8.1 kV/cm) agrees with the 12.5 kV/cm expected from the 2.7 THz 3 QW
Paris design. This result is encouraging in spite of the absence of observable emission or a
clear NDR signature in the LIV.
In short, while we did not observe emission from all three attempts of the ultrathin
2.7 THz 3 QW Paris design, we found that the electro-optic characteristics of V692 were in
closest agreement with the reported electro-optic characteristics [148]. This then leads us
to examine the points of failure of emission from V692. In addition to XRD data, problems
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4. ULTRATHIN QCLS
Figure 4.14: LIV characteristics of ultrathin 2.7 THz 3 QW Paris design (V692) double-
metal ridge, measured on golay.
with growth (more specifically, higher III-V composition, due to Arsenic drift) were further
investigated by comparing the electro-optic characteristics of another design (3 THz 4
QW ETH design, wafer V696) grown under nominally identical conditions as V692 . Any
differences between the LIV characteristics of double metal ridges processed from V696
and V674 (both grown with 3 THz 4 QW ETH design but at different times) may provide
insights to any growth anomaly in V692. This is discussed in detail in section 4.9.3. The
more likely cause behind the device failure appeared to be the choice of active region,
the number of repeat periods and the choice of waveguide geometry, which plagued the
performance of all three attempts of the 2.7 THz 3 QW Paris design. In the next section,
results from double metal ridges processed from 3 THz 3 QW Paris design are discussed.
Very similar in design in terms of anticrossing energies with injection and extractor states,
the only difference between 2.7 THz 3 QW Paris design and 3 THz 3 QW Paris design, is in
the emission frequency.
188
4.7 3 THz 3 QW Paris design
4.7 3 THz 3 QW Paris design
Figure 4.15 shows the energy band structure of the 3 THz 3 QW Paris design computed
using the in-house solver. Reported in 2007, this design was the first THz QCL structure
which employed only three quantum wells per period [145]. This design was later used
by Belkin et al. to break the record in 2008 of the highest operating temperature in THz
QCLs (178 K) by using copper-copper waveguides [149]. In 2011, Strupiechonski et al.
from Université Paris-Sud 11 first published lasing from ultrathin THz QCL using the same
design as [3]. Since the present work relates to ultrathin QCLs, the nomenclature of this
design rests with Paris, although its origins can be traced to Luo et al. from National
Research Council, Canada.
Figure 4.15: Energy band diagram of 3 THz 3 QW Paris design evaluated using in-house
code.at an electric field, E = 12.1 kV/cm. The modulii-squared wavefunctions are shown in
blue for the upper (4) lasing level and in red for the lower (3) lasing level. Starting from the
injection barrier, the growth sequence in nanometers is given as 4.8/9.6/2.0/7.4/4.2/16.1
where Al0.15Ga0.85As barrier layers are in bold. The center 5.5 of the 16.1 nm injection well
is doped with Si to Nd = 5 x 1016 cm−2. The dashed lines enclose one period of the active
region.
Table 4.4 lists the transition energy, alignment bias, dipole matrix element and the
normalized oscillator strength for the 3 THz 3 QW Paris design. Also shown is a comparison
between the design parameters computed in-house and those reported in the literature by
Luo et al. and Belkin et al. For the same epitaxial structure, Luo et al. and Belkin et
al. report slightly different Eul and


zul

values. The normalized oscillator strength ful ,
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4. ULTRATHIN QCLS
reported as 0.51 by Luo et al. is not explicitly reported by Belkin et al. in [149]. For
comparison here, we estimate ful in [149] from eqn. 4.4 using the reported values of
Eul and


zul

. We find that ful = 0.67 in [149]. For the same epitaxial structure, our
calculations of the transition energy between the upper and lower lasing level, Eul = 12.8
meV are closer to Luo et al. (Eul = 12.9 meV) as opposed to Belkin et al. (Eul = 14 meV).
Compared to Luo et al.’s calculations, our solver slightly overestimates both


zul

and ful .
3 THz 3 QW Paris design
Literature [145], [149] In-house
Eul (meV) 12.9
a 12.8
Field (kV/cm) 12.5 12.1

zul

(nm) 4.7b 5.74
ful 0.51
c 0.74
a 12.9 meV in Ref [145], 14 meV in Ref [149]
b 4.7 nm in Ref [145], 6.1 nm in Ref [149]
c 0.51 in Ref [145], 0.67 in Ref [149].
Table 4.4: Summary of the design parameters calculated for 3
THz 3 QW Paris design
Note that in section 4.6, all the attempts at ultrathin QCLs grown using the 2.7 THz,
3 QW Paris design failed to lase, with reasons attributed to number of repeat periods,
choice of active region, anomalous growth, and poor waveguide geometry. For a system-
atic analysis, the ultrathin QCLs are compared with their corresponding full-stack version,
grown under nominally identical conditions. Below, we discuss experimental results ob-
tained from double metal ridges processed from full-stack and ultrathin THz QCLs, grown
at Cavendish with the 3 THz 3 QW Paris design.
4.7.1 Full-stack (V703)
XRD data from the full-stack 3 THz 3 QW Paris design (reproduced in appendix B) showed
that the layer thickness deviated from the target thickness by +1.86 %. The FWHM of
the substrate peak (47 arc.sec) and the satellite peak (47-54 arc.sec) of V703 was found
to have reduced when compared to V692, suggesting a stabilization of the drift observed
in the earlier samples. Grown in-house for the first time, devices processed from V703
demonstrated lasing with a threshold current density Jth of ≈ 910 A/cm2 and a peak
current density Jpeak of ≈ 1250 A/cm2, see figure 4.16. Jpeak is defined as the current
density at which maximum lasing output is obtained. From figure 4.16(a), we note that at
Jth = 910 A/cm2 the voltage dropped across the double metal ridge is≈ 14.5 V. Once again,
accounting for the top and bottom Schottky contacts, the actual alignment voltage dropped
190
4.7 3 THz 3 QW Paris design
across a 10.8 µm active region (Np = 226) is 14.5 - 2(0.7) = 13.1 V, which corresponds
to an alignment electric field of 12.1 kV/cm. This is in agreement with the theoretically
estimated alignment field calculated for the 3 THz 3 QW Paris design using the in-house
solver listed in table 4.4. From figure 4.16(a), we note that Jth increases exponentially
with temperature, and the full-stack operates up to a maximum temperature of 70 K.
Figure 4.16: (a) LIV characteristics of double-metal ridges processed from full-stack 3
THz 3 QW Paris design (V703) at 10 K (b) Threshold current density Jth as a function of
temperature
Figure 4.17 shows the differential resistance of V703 plotted along with the IV charac-
teristics, measured at 10 K in the pulsed mode (figure 4.16(a)). We note from figure 4.17
that the dynamic range, defined as JN DR−Jth
JN DR
can be estimated to be 1420−910
1420
which is ≈
36% of the entire current range.
For lasers with electro-optic characteristics shown in figure 4.16 and figure 4.17, the
frequency spectrum measured using FTIR spectroscopy is shown in figure 4.18. The spec-
tral characteristics are measured as a function of current, which is increased in steps of
100 mA starting from 1500 mA, i.e. J = 862 A/cm2 < Jth to a current of 2300 mA, i.e. J =
1320 A/cm2. We observe that the experimental results deviate slightly from the theoretical
calculations of the design frequency, Eul = 12.8 meV = 3.1 THz, listed in table 4.4. Just
above threshold (J = 920 A/cm2, current = 1600 mA), we note that the first lasing mode
appears at a frequency of 3.22 THz. We also note the appearance of a second lasing mode
at 3.32 THz as the current is increased to 1800 mA (J = 1034 A/cm2). As the current
is raised further to about 2100 mA, corresponding to J = 1206 A/cm2, which infact ap-
proaches the Jpeak shown in figure 4.16, we observe the apperance of a very strong lasing
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4. ULTRATHIN QCLS
Figure 4.17: Differential resistance of double metal ridges processed from 3 THz 3 QW
Paris design (V703) at 10 K
Figure 4.18: Spectral characteristics of double metal ridges processed from full-stack 3
THz 3 QW Paris design (V703) shown as a function of current, increasing from 1500 mA
to 2300 mA, in steps of 100 mA.
192
4.7 3 THz 3 QW Paris design
mode at 3.4 THz.
We next discuss the performance of the full-stack 3 THz 3 QW Paris design (V703) in
comparision to the performance of L421 reported in literature [3, 145, 149].
4.7.2 Comparison of V703 with Literature
In section 4.2, we discussed results from Strupiechonski et al.’s work on 3 THz 3 QW Paris
design, which demonstrated lasing emission from ultrathin active region (L422) and the
corresponding 10 µm thick full-stack wafer (L421), see figure 4.2. From figure 4.2, we
note that the 10 µm thick active region (L421) exhibits a Jth ≈ 700 A/cm2. This is lower
than Jth ≈ 910 A/cm2, which we measure from double-metal ridges processed from V703,
see figure 4.16(b). Another point of comparison is the maximum operating temperature,
which varies from 70 K in our measurements to 146 K reported by Strupiechonski et al. In
spite of these shortcomings, we find that the dynamic range of L421, calculated as JN DR−Jth
JN DR
= 1120−700
1120
= 37.5 % is only slightly better than the dynamic range exhibited by V703
(36 %). While our results from the full-stack 3 THz 3 QW Paris design show suboptimal
performance, it still holds promise for ultrathin THz QCLs based on the same design, and
grown under nominally identical conditions. Below we discuss results from double metal
ridges processed from V702, which is an ultrathin 3 THz 3 QW Paris design, grown just
before V703.
4.7.3 Ultrathin QCL (V702)
XRD data suggests that the layer thickness of V702 deviated from the target thickness by
+1.46 %, see appendix B. Table 4.5 summarizes the target structure of the ultrathin 3 THz
3 QW Paris design (V702) in terms of period length, number of periods and waveguide
geometry. Compared to the symmetric top and bottom n+ GaAs contact used in ultrathin
2.7 THz 3 QW Paris design (V692), see table 4.3, we use asymmetric top and bottom n+
GaAs contacts in V702.
Wafer Length per period,
Lp (nm)
Number of re-
peats, Np
Waveguide Thickness without
etch stop (µm)
V702 44.1 57 Top 75 nm (2e18), Bottom 50
nm (2e18)
2.668
Table 4.5: Wafer structure of the ultrathin 3 THz 3 QW Paris design.
Figure 4.19 shows the LIV characteristics of double metal ridges processed from V702.
Devices processed from V702 were characterized with a very unusual LIV, and an absence
of any simultaneous emission. Given that the alignment electric field is 12.1 kV/cm for the
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4. ULTRATHIN QCLS
3 THz 3 QW Paris design, the alignment bias for a 2.66 µm thick active region works out
to be 4.5 V. At this voltage, the current density J ≈ 50 A/cm2. This is nearly twenty times
less than Jth = 910 A/cm2 observed for the full-stack THz QCL. This may be indicative of
a high resistive layer within the active region, perhaps a problem that may be attributed to
doping within the first few periods, which severely impediments the electrical performance
in a reduced active region device.
Figure 4.19: (a) LIV characteristics of V702 double-metal ridge, measured on golay at 4K
(b) Differential resistance at 4K
FWHM derived from the substrate peak (32 arc.sec) and the satellite peak (36-43
arc.sec) suggest that these values are limited by the resolution of the system, which hints
at absence of inhomogeniety during the epitaxial growth of V702. While doubts over the
choice of waveguide remain, the main question that arises is whether under-doping and
not inhomogeniety plagued the performance of the full-stack V703 wafer, and these issues
also played a role in degrading the performance of the ultrathin V702 wafer. This is harder
to analyze in an active region grown for the first time at Cavendish, with no previous
in-house data to compare. In order to answer this question, we probed the electro-optic
performance of the 3 THz 4 QW MIT design grown at the same time as the 3 THz 3 QW
Paris design. The full-stack 3 THz 4 QW MIT design is a regrowth of V569, a wafer pre-
viously grown and characterized at the Cavendish Labs. The next section discusses results
from double metal ridges processed from full-stack and ultrathin 3 THz 4 QW MIT design.
194
4.8 3 THz 4 QW MIT design
4.8 3 THz 4 QW MIT design
Figure 4.20 shows the energy band structure of the 3 THz 4 QW MIT design. The 3 THz
4 QW MIT design has lower number of repeat periods per unit length as opposed to its
successor - the (2.7/3) THz 3 QW Paris design. However, compared to its 3 QW counter-
part, the performance of the 3 THz 4 QW MIT design, which consists of an additional well,
is much more robust in terms of growth variations. This is because THz QCLs based on
resonant phonon designs, rely crucially on the alignment of energy states between periods.
Therefore, a change of a few monolayers can dramatically influence the performance of a
THz QCL.
Figure 4.20: Energy band diagram of 3 THz 4 QW MIT design evaluated using in-
house code at an electric field, E = 13.0 kV/cm. The modulii-squared wavefunctions
are shown in blue for the upper (5) lasing level and in red for the lower (4) lasing
level. Starting from the injection barrier, the growth sequence in nanometers is given
as 4.1/6.6/2.5/7.9/4.9/9.0/3.3/15.6 where Al0.15Ga0.85As barrier layers are in bold. The
thickest well is doped with Si to Nd = 1.9 x 1016 cm−2. The dashed lines enclose one
period of the active region.
Shown in comparison to lasing parameters reported in [150], is a summary of the
transition energy, alignment field, dipole matrix and oscillator strength calculated for the
3 THz 4 QW MIT design, using the in-house solver, see, table 4.6. Our estimates of the
transition energy and alignment bias agree with that of [150]. However, unlike the (2.7/3)
THz 3 QW Paris design, where we overestimated


zul

, ful , in the case of 3 THz 4 QW MIT
design, our estimates of


zul

, ful are less than those reported by [150].
195
4. ULTRATHIN QCLS
3 THz 4 QW MIT design
Literature [150] In-house
Eul (meV) 13 13
Field (kV/cm) 10.2 10.2

zul

(nm) 6.1 4.39
ful 0.86 0.61
Table 4.6: Summary of the design parameters calculated for 3
THz 4 QW MIT design
In the next sub-section, we discuss results from double metal ridges processed from the
full-stack 3 THz 4 QW MIT design.
4.8.1 Full-stack V705 and comparison to V569
Figure 4.21 shows the LIV characteristics of the 3 THz 4 QW MIT design (V705), in com-
parison to a 3 THz 4 QW MIT design (V569), grown previously at the Cavendish Labs
under optimal conditions. We note that double metal ridges processed from V569 exhibit a
Jth ≈ 720 A/cm2. However, similar ridges processed from V705 exhibit a Jth ≈ 755 A/cm2.
We also note the difference in alignment bias between V705 and V569. While V569 shows
an alignment bias≈ 9.2 V, V705 shows a much higher alignment bias even after accounting
for the top and bottom Schottky contact, i.e. 13.4 - 2(0.7) = 12 V. Another point of com-
parison is the Jpeak, which departs from ≈ 1000 A/cm2 in V569 to 860 A/cm2 in V705. As
can be seen from figure 4.21(b), the maximum operating temperature falls from 125 K in
THz QCLs processed from V569 to 70 K in the THz QCLs processed from V705. Although
the devices processed from V705 have been wet-etched and not dry-etched unlike V569,
we do not expect the performance to shift this significantly.
Figure 4.21 shows the differential resistance for both V569 and V705, measured at 10
K in pulsed mode. Shown in figure 4.21 is the dynamic range which is seen to reduce from
JN DR−Jth
JN DR
= 1170−720
1170
= 38 % in V569 to 960−755
960
= 21 % in V705. Even though the III-V
composition that affected V692 and V696 was corrected in the growth run of wafers V702
to V705, a comparison between V569 and V705 appears to reconfirm that the wafers with
2.7/3 THz, 3 QW Paris design were grown under suboptimal growth conditions, possibly
due to variations in higher background impurity levels due to start of growth campaign.
The high background impurity levels may have compensated the intentional doping caus-
ing an under-doped structure, thereby adversely affecting the laser performance. From
XRD data shown in appendix B, we observe that the layer thickness of V705 deviated by
+2.91 % from the target structure. The rocking curves of V705 indicate a FWHM of ap-
proximately 50 arc.sec and 60-61 arc.sec measured from the substrate peak and satellite
196
4.8 3 THz 4 QW MIT design
Figure 4.21: Comparison of (a) LIV characteristics of 3 THz 4 QW MIT design (V705)
double-metal ridge at 10 K with 3 THz 4 QW MIT design (V569) (b) Threshold current
density (Jth) as a function of operating temperature of 3 THz 4 QW MIT design (V705)
with respect to 3 THz 4 QW MIT design (V569). The data on 3 THz 4 QW MIT design
(V569) was taken by Anthony Brewer.
peak, respectively.
4.8.2 Ultrathin QCL (V704)
Here, we present results from double metal ridges processed from ultrathin 3 THz 4 QW
MIT design (V704). XRD data suggests that the layer thickness of V704 deviated by + 2.84
% from the target structure, see appendix B. It is however promising to note an absence
of broadening in the FWHM derived from the substrate peak (32 arc.sec) and the FWHM
derived from the satellite peak (36-43 arc.sec).
Table 4.7 summarizes the target wafer structure of the ultrathin 3 THz 4 QW MIT
(V704) design in terms of period length, number of periods and waveguide geometry. Note
that the thickness and doping of the top and bottom n+ GaAs contacts is consistent with
the ultrathin 3 THz, 3 QW, MIT (V702) design (see, table 4.5), i.e. asymmetric waveguide
geometry is employed.
Wafer Length per period,
Lp (nm)
Number of re-
peats, Np
Waveguide Thickness without
etch stop (µm)
V704 53.9 47 Top 75 nm (2e18), Bottom 50
nm (2e18)
2.712
Table 4.7: Wafer structure of the ultrathin 3 THz 4 QW MIT design.
The electro-optic characteristics of 3 THz 4 QW MIT design (V704) are very similar
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4. ULTRATHIN QCLS
Figure 4.22: Comparison of differential resistance characteristics between 3 THz 4 QW
MIT design (V705) and 3 THz 4 QW MIT design (V569), measured at 10 K in pulsed
mode. The data on V569 was taken by Anthony Brewer.
to that of 3 THz 3 QW Paris design (V702). As in the case of V702, double metal ridges
processed from V704 show a very an unusual IV, see figure 4.23. We note from table 4.6,
that the alignment field for the 3 THz 4 QW MIT design is 10.2 kV/cm. For a 2.66 µm thick
active region, alignment is expected at a bias of 2.7 V. At this voltage, the current density
J ≈ 10 A/cm2. This is nearly two orders less than Jth = 755 A/cm2 observed for the full-
stack THz QCL. Similar to 3 THz 3 QW Paris design (V702), this may be indicative of a
high resistive layer within the active region, perhaps a problem that may be attributed to
doping within the first few periods, which severely impediments the electrical performance
in a reduced active region device.
The possibility of an open circuit is excluded as multiple devices processed from 3 THz
4 QW MIT design (V704) show a room temperature resistance between 15-20 kilohm,
which is similar to the 7-10 kilohm resistance obtained from THz QCLs proceesed out of
full-stack 3 THz 4 QW MIT design (V705) wafer. Note that we do not measure any light
emission using both golay and the more sensitive Si-Ge bolometer as a detector from these
devices.
The poor performance of a full-stack 3 THz 4 QW MIT design (V705) which is a re-
growth of previously characterized 3 THz 4 QW MIT design (V569), in addition to the
failure of ultrathin 3 THz 4 QW MIT design (V704) necessitated a reassessment of our
options. It compelled us to review in-house grown full-stack designs, which were known
198
4.9 3 THz 4 QW ETH design
Figure 4.23: (a) LIV characteristics of double-metal ridge processed from 3 THz 4 QW MIT
design (V704), measured on golay at 4K (b) Differential resistance at 4K.
to perform well in terms of electro-optic behavior and had the potential for emission at
reduced repeat periods. One of these designs was a 3 THz 4 QW ETH design, which
employed a BtC transition for lasing, but with the miniband coupled to a resonant phonon
extraction stage. This design is also less susceptible to growth fluctuations as the alignment
occurs over the width of the miniband as opposed to a single state [6].
4.9 3 THz 4 QW ETH design
This section presents results from double metal ridges processed from full-stack 3 THz
4 QW ETH design and also an ultrathin 3 THz 4 QW ETH design, which is derived by
chemical etching the full-stack wafer post growth. These results are the first demonstration
of emission from ultrathin active regions at the Cavendish, and to our knowledge the first
demonstration of emission in a BtC design.
Figure 4.24 shows the energy band diagram of 3 THz 4 QW ETH design. Although
we have discussed this hybrid design in the introductory chapter, we present the salient
features once again. Compared to the 3 THz 4 QW MIT design, where lasing transition
occurs across energy states in the same well (4 → 3 in figure 4.20), the 3 THz 4 QW
MIT design employs a lasing transition (5 → 4 in figure 4.24) between energy states in
adjacent wells, which makes it much more diagonal in the real space. As we can see
from figure 4.24, states 3 and 2 form a miniband which is coupled to a resonant phonon
extraction stage, i.e. state 1.
Table 4.8 lists the lasing parameters of the 3 THz 4 QW ETH design computed with the
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4. ULTRATHIN QCLS
Figure 4.24: Energy band diagram of 3 THz 4 QW ETH design evaluated using in-
house code at an electric field, E = 7.5 kV/cm. The modulii-squared wavefunctions
are shown in blue for the upper (5) lasing level and in red for the lower (4) lasing
level. Starting from the injection barrier, the growth sequence in nanometers is given
as 5.5/11.0/1.8/11.5/3.8/9.4/4.2/18.4 where Al0.15Ga0.85As barrier layers are in bold.
The thickest well is doped with Si to Nd = 2 x 1016 cm−2. The dashed lines enclose one
period of the active region.
in-house solver, in comparison to the values reported by Amanti et al. [6]. We note that
the transition energies and alignment fields agree with each other. Also, the calculated
dipole matrix element


zul

, is consistent with Amanti et al. to within an accuracy of 6 %.
We find that unlike the resonant phonon designs (sections 4.6 - 4.8), our solver is more
robust for BtC designs in calculating


zul

and ful . Note that


zul

and ful depend upon the
length over which the envelope wavefunctions extend in the growth direction (3Lp, where
3 is the number of periods over which the wavefunction is truncated). Besides the actual
number of periods included in the simulation (section 4.6), it is probable that the in-house
solver is more accurate for the BtC design as opposed to the resonant phonon design as
this length is higher in the former as compared to the latter.
200
4.9 3 THz 4 QW ETH design
3 THz 4 QW ETH design
Literature [6] In-house
Eul (meV) 12 12
Field (kV/cm) 7.6 7.6

zul

(nm) 4.4 4.19
ful 0.36 0.38
Table 4.8: Summary of the design parameters calculated for 3
THz 4 QW ETH design
Below, we discuss results obtained from full-stack 3 THz 4 QW ETH design (V674),
which was grown in the 2010 growth cycle, without under-doping and III-V composition
issues. We also compare the performance of full-stack 3 THz 4 QW ETH design (V674) with
devices processed from 3 THz 4 QW ETH design (V696). V696 is a regrowth of V674 grown
in V chamber under nominally similar conditions as V692, V702-V705 (wafers discussed in
sections 4.6 - 4.8). Therefore, a comparison between full-stack 3 THz 4 QW ETH design
(V674) and full-stack 3 THz 4 QW ETH design (V696) should give us the confidence that
the performance of an ultrathin QCL derived from the former wafer would not be plagued
with growth-related problems. Finally, we present results from V674T , with the superscript
T , indicating that it has been obtained by thinning the full-stack 3 THz 4 QW ETH design
(V674) post growth, unlike the other ultrathin wafers mentioned in this thesis which were
epitaxially grown with reduced repeat periods.
4.9.1 Full-stack (V674)
XRD data suggests that V674 was grown thinner than the target structure by 0.33 %. The
rocking curves of V674 indicate a FWHM of approximately 47 arc.sec and 47-50 arc.sec
measured from the substrate peak and satellite peak, respectively (see appendix B). Fig-
ure 4.25 shows the LIV characteristics of double metal ridges processed from full-stack 3
THz 4 QW ETH design (V674). From figure 4.25, we note that the device exhibits a Jth of
155 A/cm2. At 4 K, we obtain an average power of 3.2 µW at the peak current density. We
also note from figure 4.25, that the full-stack 3 THz 4 QW ETH design (V674) operates up
to a maximum temperature of 134 K. In accordance with general THz QCL behavior, the
threshold current density, Jth increases exponentially with temperature as the gain reduces
with temperature due to thermally assisted scattering.
Figure 4.26 plots the differential resistance along with the IV characteristics. From the
figure 4.26, we estimate the alignment bias at Jth = 155 A/cm2 to be 10.8 V. Taking into
account the 2 x 0.7 V voltage drop due to top and bottom Schottky contacts, for a 11.8 µm
thick active region, this implies an electric field ≈ 7.9 kV/cm. This agrees very well with
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4. ULTRATHIN QCLS
Figure 4.25: (a) LIV characteristics of 3 THz 4 QW ETH design (V674) measured on golay,
in pulsed mode at different operating temperatures (b) Threshold current density Jth as a
function of temperature.
the theoretical alignment field of 7.6 kV/cm predicted for the 3 THz 4 QW ETH design,
see table 4.8. From figure 4.26 we note that the NDR occurs at a JN DR ≈ 280 A/cm2.
Therefore, the dynamic range for the full-stack 3 THz 4 QW ETH design (V674) can be
estimated as JN DR−Jth
JN DR
= 280−155
280
= 44% of the entire current range.
Note that among all the full-stack lasers of different designs presented in this chapter,
the 3 THz 4 QW ETH design (V674) shows the maximum operating temperature, and also
the maximum dynamic range.
4.9.2 Comparison of full-stack V674 with literature
To confirm that results from full-stack 3 THz 4 QW ETH design (V674) are indeed rep-
resentative of a good device, we compare our results to those reported by Amanti et al.
from the same design (see figure 4.27). The dynamic range in Amanti et al. can be esti-
mated as JN DR−Jth
JN DR
= 410−180
410
= 56% from figure 4.27. This is more than what we measure,
however we feel that a lower Jth in our devices may be to our advantage when we subse-
quentially reduce the number of repeat periods. We also note that the maximum operating
temperature reported by Amanti et al. is 150 K (figure 4.27), as opposed to a slightly
lower temperature (134 K) observed in our devices, see figure 4.25. However, we expect a
10-20 K improvement in performance of the full-stack 3 THz 4 QW ETH design (V674) by
thinning the host-substrate.
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4.9 3 THz 4 QW ETH design
Figure 4.26: Differential resistance of 3 THz 4 QW ETH design (V674) measured in pulsed
mode, at 4 K.
4.9.3 Comparison of full-stack V674 and full-stack V696
We now compare the electro-optic performance of two wafers, V674 and V696, both of
which are full-stack 3 THz 4 QW ETH designs, and grown in-house at the Cavendish Labs.
We expect V674 to have a superior performance over V696, as the latter was grown un-
der sub-optimal growth conditions. XRD data suggests that the layer structure of V696
deviated from the target structure by +0.44 %.
We note from figure 4.28, that while double metal ridges processed from full-stack 3
THz 4 QW ETH design (V674) shows a Jth ≈ 155 A/cm2, similarly processed ridges from
full-stack 3 THz 4 QW ETH design (V696) show a Jth ≈ 187 A/cm2. Compared to V674,
devices from V696 show an alignment bias of 7.2 V. Accounting for the 2 x 0.7 V drop due
to Schottky contacts, for a 10 µm active region, this corresponds to an electric field of 5.8
kV/cm. This field deviates from 7.6 kV/cm, which is the theoretically predicted alignment
field for this design. The dynamic range of V696, estimated as JN DR−Jth
JN DR
= 212−187
212
= 12%
is nearly one third of the dynamic range exhibited by V674. From figure 4.28(b), we also
observe that the maximum operating temperature reduces from 134 K to 85 K, in moving
from V674 to V696.
A minor point to keep in mind is that, while the full-stack 3 THz 4 QW ETH design
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4. ULTRATHIN QCLS
Figure 4.27: LIV characteristics of 3 THz 4 QW ETH design as reported in the literature
[6]. The inset shows the laser spectra. Adapted from [6] to indicate Jth, JN DR, alignment
bias and dynamic range.
(V674) incorporates 180 repeat periods, the full-stack 3 THz 4 QW ETH design (V696) is
grown with 160 repeat periods. This difference in Np also contributes to higher Jth and
a lower maximum operating temperature. It is useful at this point to remind ourselves
of eqn. 4.3, which states that Jth varies inversely with Np. For an active region thickness
greater than 5 µm, a difference of 20 repeat periods should not introduce a difference of
more than 10% in Jth and the maximum operating temperature for QCLs processed and
grown under nominally identical conditions [3]. Therefore, if 3 THz 4 QW ETH design
(V696) was grown under the exact same conditions as 3 THz 4 QW ETH design (V674),
we should have observed a Jth = 165 A/cm2 and a maximum operating temperature of 120
K. The discrepancy in observed characteristics of wafer V696 can therefore be attributed
to problems with III-V composition. In particular, V696 is rich in Gallium making it more
conductive than V674.
We next discuss our process of obtaining ultrathin QCLs from full-stack 3 THz 4 QW
ETH design (V674), and subsequently present electro-optic results obtained from this
chemically thinned wafer (V674T ).
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4.9 3 THz 4 QW ETH design
Figure 4.28: Comparison of (a) LIV characteristics of 3 THz 4 QW ETH design (V674)
double-metal ridge at 4 K with 3 THz 4 QW ETH design (V696) (b) Threshold current
density (Jth) as a function of operating temperature of 3 THz 4 QW ETH design (V674)
with respect to 3 THz 4 QW ETH design (V696). The data on 3 THz 4 QW ETH design
(V696) was taken by Yash D. Shah.
4.9.4 Chemical etching of ultrathin QCLs
While an ultrathin THz QCL can be grown epitaxially, the conventional route reported
in the literature employs chemically thinning of a full-stack THz QCL (see section 4.2).
In figure 4.29(a), we show the optical image of 3 THz 4 QW ETH design (V674) after
thinning in a solution of H2SO4:H2O2:H2O, diluted to a ratio of 1:8:80. The solution etches
GaAs/Al0.15Ga0.75As at a rate of 500 nm/min. A consequence of chemical thinning is the
roughening of the GaAs surface, which poses a problem for further processing of dry-etched
photonic devices from a thinned THz QCL. Also seen in figure 4.29(b) are defect etch pits
which open up due to isotropic nature of the chemical etch. Careful selection of defect free
areas enable mesa processing via wet chemical routes, as shown in figure 4.29(b).
While previous reports on ultrathin QCLs do not comment on the surface roughness
of the processed THz QCLs, it may be possible to optimize the etch rate to improve the
surface quality of the thinned wafer. However, substrate quality after chemical thinning is
unlikely to compete with the surface quality of as grown THz QCLs.
4.9.5 Ultrathin QCL (V674T )
Table 4.9 summarizes the wafer structure of the ultrathin 3 THz 4 QW ETH design (V674T ).
There are two main differences between epitaxially grown ultrathin THz QCLs and chem-
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4. ULTRATHIN QCLS
Figure 4.29: (a) Optical image of 3 THz 4 QW ETH design (V674) after chemical etching,
showing growth defects opening as a result of etch (b) optical image of processed ridges
showing ridge with top and bottom Ti/Au contacts.
icaly etched ultrathin THz QCL. The first difference relates to the definite absence of the
top n+ doped GaAs contact layer. The second difference relates to number of periods Np
estimated to be contained within the active region. While this is straightforward to know
in an epitaxially grown ultrathin THz QCLs, for a chemically etched ultrathin Thz QCL
this number is determined by the target etch thickness and the roughness of the wafer
post etching. We measure a variation of about ± 0.5 µm across the wafer, due to surface
roughness introduced after thinning the 3 THz 4 QW ETH design (V674) to a thickness of
≈ 3 µm. Therefore, we can only estimate a range for the number of periods Np. As shown
in table 4.9, we estimate the ultrathin 3 THz 4 QW ETH design (V674T ) to contain Np
between 38-53.
Wafer Length per period,
Lp (nm)
Number of re-
peats, Np
Waveguide Thickness without
etch stop (µm)
V674T 65.6 38-53 Top (None), Bottom 100 nm
(5e18)
3.0 ± 0.5
Table 4.9: Summary of the chemically etched ultra-thin QCLs
Figure 4.30(a) shows the LIV characteristics of the double-metal ridges processed from
ultrathin 3 THz 4 QW ETH design (V674T ), as a function of operating temperture. Com-
pared to a threshold current density of Jth ∼ 155 A/cm2 in full-stack 3 THz 4 QW ETH
design (V674), the threshold current density Jth in ultrathin 3 THz 4 QW ETH design
(V674T ) increases to 188 A/cm2 as Np reduces from 180 to 45 ± 7. It is encouraging to
see clear alignment at Jth and breaking in the IV characteristics of the ultrathin device.
The threshold current density Jth follows an exponential increase with operating temper-
ature as shown in figure 4.30(b). From figure 4.30(a), we note that the device shows
an alignment bias of 3.8 V. Accounting for the 2 x 0.7 V voltage drop across the top and
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4.9 3 THz 4 QW ETH design
Figure 4.30: (a) LIV characteristics of ultrathin 3 THz 4 QW ETH design (V674T ) measured
on a Si-Ge bolometer (b) Threshold current density Jth as a function of temperature.
bottom Schottky contacts, this corresponds to an electric field of 8 ± 1.6 kV/cm, for an
active region of height 3 µm ± 0.5µm. The alignment field of the thinned wafer V674T
is consistent with the alignment field observed in the full-stack 3 THz 4 QW ETH design
(V674).
Compared to the full-stack power of 3.2 µW, the power emitted from this device is quite
small ∼ 64 nW. What is promising is that the ultrathin 3 THz 4 QW ETH design (V674T )
operates up to a maximum temperature of 86K.
Figure 4.31 shows the differential resistance of ultrathin 3 THz 4 QW ETH design
(V674T ), measured in pulsed mode at 10 K. If we compare this to figure 4.26, we note that
the ultrathin device breaks earlier than the full-stack, i.e. JN DR = 244 A/cm2 of ultrathin
3 THz 4 QW ETH design (V674T ) is less than JN DR = 280 A/cm2 of full-stack 3 THz 4
QW ETH design (V674). A lower value of JN DR implies reduction in population inversion
between the upper and lower lasing levels. This may perhaps occur as a consequence of
surface roughness over the top of the ridge where different regions are biased differently,
causing instability and forcing the device into NDR. Increase in Jth and reduction in JN DR,
adversely impacts the dynamic range of the ultrathin 3 THz 4 QW ETH design (V674T ),
lowering it from 44% in full-stack to 23%. The dynamic range of ultrathin 3 THz 4 QW
ETH design (V674T ) is calculated from figure 4.31 as JN DR−Jth
JN DR
= 244−188
244
.
Figure 4.32 shows the spectral characteristics of 3 THz 4 QW ETH design (V674T ),
measured on Bruker VERTEX 80/80v in the rapid-scan mode. In agreement with the de-
signed frequency of the 3 THz 4 QW ETH design, the first lasing mode appears at 3.15
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4. ULTRATHIN QCLS
Figure 4.31: Differential resistance of double metal ridges processed from 3 THz 4 QW
ETH design (V674T ).
THz, as the current density approaches Jth which continues to be the most prominent
lasing mode as the current density approaches Jpeak = 235 A/cm2.
In both the reports of emission from ultra-thin QCLs [3, 4], the authors’ chemically
etched a full-stack active region down to ∼ 1.75 - 2.8 µm. At the Cavendish Labs, we
observed lasing from V674T , obtained by chemically etching 3 THz 4 QW ETH design
(V674), grown with Np =180 down to ∼ 3.0 µm. Note that the previous reports on
ultrathin QCLs came from LO phonon designs, whereas in this work we show emission
from a 4 QW BtC design, with a resonant phonon extraction stage [6].
These results are the first demonstration of a working ultrathin QCL at the Cavendish
Labs, which is very promising for studying the effects of lateral confinement in nanopil-
lars fabricated using this material. However, as is seen from the figure 4.29, wet chemical
etching of the full-stack QCL leaves the final surface of the ultra-thin QCL quite rough and
unsuitable for further fabrication detailed in Chapter 3. For use as a material to process
nanopillars, it is important to obtain simultaneous emission from epitaxially grown ultra-
thin THz QCLs. It is also possible that in the early stages of growth, outgassing from the
substrate, cells and manipulator results in dopant migration in the active region, which
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4.10 Waveguide losses in ultrathin THz QCLs
Figure 4.32: Spectral characteristics of double metal ridges processed from ultrathin 3 THz
4 QW ETH design (V674T ) shown as a function of current, increasing from 240 mA to 315
mA, in steps of 5 mA.
comes into play while growing the ultra-thin QCL. In an etched ’full-stack’ structure, this
region is removed from the active region, resulting in better performance. Perhaps, this
explains the difficulty in the growth of ultra-thin THz QCLs.
The next section investigates the effect of waveguide losses on the performance of
epitaxially grown ultrathin THz QCLs. For a low gain medium to overcome losses and
achieve simultaneous emission, it is necessary to identify a waveguide designed for reduced
repeat period QCLs.
4.10 Waveguide losses in ultrathin THz QCLs
4.10.1 Effect of contact thickness and doping on waveguide losses
In chapter 2, we discussed how active region thickness influences the waveguide losses αw
in a Fabry Pèrot ridge. As discussed before, αw depends upon the extinction coefficient,
κ and wavelength of emission, λ as 4piκ/λ. In chapter 2, we showed that the waveguide
losses αw increase with active region thickness. These results, calculated using the Eigen
mode analysis in COMSOL v4.3, hold true for an active region with average doping 5.609
x 1015 cm−2 per period, and a 100 nm thick top and bottom n+ doped GaAs contact with
a doping of 5 x 1018 cm−2.
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4. ULTRATHIN QCLS
The extinction coefficient κ and hence the waveguide losses αw , depend upon the
thickness and average doping of the active region. Consistent with section 2.11, and
unless explicitly stated, our plasmon mode calculations assume an average doping of 5.609
x 1015 cm−2 per period, which is the average doping of the 3 THz 4 QW ETH design.
Equally important, is the thickness and average doping of the top and bottom n+ doped
GaAs contacts. The role of top and bottom contacts is tabulated in 4.10. The losses,
overlap and figure of merit is calculated for a Fabry Pèrot waveguide in an ultrathin THz
QCL of active region thickness equal to 3 µm.
Waveguide losses for different top and bottom contact configurations
Wafer Bottom
Contact
Top Con-
tact
Doping Loss Overlap FOM
(nm) (nm) cm−2 cm−1 cm
100 100 5 x 1018 9.4384 0.9636 0.1021
75 75 5 x 1018 8.6745 0.9632 0.1110
50 75 5 x 1018 8.3675 0.9629 0.1151
3 THz 4 QW ETH
(V674T )
100 0 5 x 1018 8.2749 0.9630 0.1164
75 0 5 x 1018 7.8950 0.9627 0.1219
50 0 5 x 1018 7.5897 0.9624 0.1268
2.7 THz 3 QW Paris
(V671, W0798, V692)
75 75 2 x 1018 10.5091 0.9618 0.0915
3 THz 3 QW Paris
(V702), 3 THz 4 QW
MIT (V704)
50 75 2 x 1018 9.9126 0.9617 0.0970
75 0 2 x 1018 8.7998 0.9620 0.1093
50 0 2 x 1018 8.2127 0.9619 0.1171
Ideal 0 0 None 7.1166 0.9617 0.1351
Table 4.10: Summary of waveguide losses at 3 THz of a 100 µm wide metal-metal waveg-
uide ridge in a 3 µm active region, with an average doping of 5.609 x 1015 cm−2, per
period calculated using COMSOL v4.3. See text for explaination of colors.
Table 4.10 reveals interesting trends in the waveguide losses αw and mode overlap
Γ. Among the tabulated waveguide configurations, a top and bottom contact of 75 nm
thickness, doped to a carrier concentration of 2 x 1018 cm−2 shows the highest waveguide
losses at 10.5091 cm−1 and the lowest figure of merit (FOM) at 0.0921 cm. On closer
inspection, a top and bottom contact of any thickness, doped at 2 x 1018 cm−2 introduces
higher losses, when compared to contacts of same thickness but a higher doping of 5 x
1018 cm−2.
Note that the waveguide configuration highlighted in blue was actually employed in
double metal ridges processed from 2.7 THz 3 QW Paris designs (V671, W0798, V692),
see table 4.3. It is therefore very likely that the higher optical losses influenced the emission
from these wafers. While the exact values indicated in table 4.10 would vary from design
to design, i.e. it would depend upon the average doping per period, the general trend
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4.10 Waveguide losses in ultrathin THz QCLs
would remain the same. Correcting the results shown in blue for the 2.7 THz 3 QW Paris
designs (V671, W0798, V692), with each period doped at 6.264 x 1015 cm−2, we obtain
a waveguide loss of 11.358 cm−1 and a figure of merit (FOM) of 0.0847 for a 3 µm thick
active region.
While the waveguide configuration highlighted in green, employed in double metal
ridges processed from 3 THz 3 QW Paris design (V702) and the 3 THz 4 QW MIT design
(V704) performs better than the one used in 2.7 THz 3 QW Paris designs (V671, W0798,
V692), it is still extremely lossy. As before, correcting the results for 3 THz 3 QW Paris
design (V702) and 3 THz 4 QW MIT design (V704), we obtain waveguide losses αw =
10.1359 cm−1 and 9.7694 cm−1, given an average doping of 5.782 x 1015 cm−2 and 5.499
x 1015 cm−2, respectively.
These losses are higher than the losses incurred in the waveguide configuration used in
3 THz 4 QW ETH design (V674T ), shown in red in table 4.10. These results indicate that
perhaps, apart from growth problems, the n+ doped GaAs contact geometry and doping
influenced the observation of emission from ultrathin THz QCLs discussed in this chapter.
The Eigen mode analysis of plasmon mode solutions listed in table 4.10 further implies
that the best waveguide configuration exists when there are no n+ doped GaAs contacts.
This is highlighted in pink in table 4.10. Compared to the poorest waveguide (75 nm/75
nm, 2 x 1015), the FOM of ultrathin THz QCLs without the top and bottom contacts, im-
proves by ∼ 40 %. Note, that the doped contacts only play a role in the laser performance
by improving the current injection if they are ohmic. For a Schottky-contact on either sides
of the active region, these layers only add to additional losses within the gain medium.
We next discuss the effect of top contact removal on plasmonic mode, in full-stack and
ultrathin THz QCLs.
4.10.2 Effect of removal of top contact on waveguide losses
In table 4.10, we showed that a double metal waveguide in an ultrathin THz QCL, which
consists of a bottom n+ GaAs contact of 100 nm thickness, doped with a carrier concentra-
tion of 5 x 1015 cm−2 and top contact removed, performs better than a waveguide which
consists of a top and bottom n+ GaAs contact of 100 nm thickness, doped with a car-
rier concentration of 5 x 1015 cm−2. Here, we probe how the performance varies over the
thickness of the active region, assuming the average doping of the 3 THz 4QW ETH design.
Figure 4.33 plots the key defining features of the plasmon mode i.e. waveguide losses,
overlap, figure of merit and effective refractive index, varying as a function of active region
thickness. The plasmon modes have been calculated for the case of double-metal ridges,
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4. ULTRATHIN QCLS
which are 100 µm and 150 µm wide, with and without an n+ doped GaAs layer. In the
case of 100 µm (150 µm) wide ridge, with a top n+ contact, the waveguide losses increases
from 8 cm−1 to 10.5 cm−1 as the active region thickness is reduced from 11.8 µm to 2 µm.
Figure 4.33: (a) Waveguide losses (b) Optical overlap (c) Figure of merit (FOM) (d) Real
part of the refractive index, Re(n), as a function of active region thickness with an average
doping of 5.609 x 1015 cm−2 per period, calculated using COMSOL v4.3.
An interesting point to note from figure 4.33 is the nonlinear reduction in waveguide
losses by removing the top n+ doped GaAs contact, as a function of active region thickness.
This decrease in waveguide losses is particularly significant in thin active regions, with αw
reducing by about 22.7 % by etching the top contact in an active region of thickness 2 µm.
In the case of full-stack active region, this decrease in loss is only about 6.2 %. By removing
the top contact, we also increase the overlap of the plasmon mode with the gain-medium,
thus improving the figure of merit (FOM) defined as FOM = Γ/αw . Note that in a low
gain medium such as the ultra-thin QCL, it is crucial to reduce the waveguide loss αw and
increase the FOM for improved laser performance.
From figure 4.33(a), we can calculate the ratio αw(V674T )/αw(V674) to be 1.128,
212
4.11 Measurement of spontaneous emission
where αw(V674T ) is the waveguide losses associated with a 100 µm wide double-metal
ridge without the top n+ doped contact layer processed from 3 THz 4 QW ETH design
and αw(V674) is the waveguide losses associated with a 150 µm wide double-metal ridge
with the top n+ doped contact layer processed from 3 THz 4 QW ETH design. This ra-
tio matches closely with the ratio of the respective threshold current density, defined as
Jth(V674T )/Jth(V674). From figures 4.26 and 4.31, we estimate the ratio of the thresh-
old current density to be 1.2, which agrees with eqn. 4.3 assuming that the mirror losses
αm are constant.
The same analysis can easily extend to a different active region design. While the
general trend remains as shown in figure 4.33, an active region with average doping higher
than the 3 THz 4 QW ETH design, such as the 2.7/3 THz 3 QW Paris design leads to higher
waveguide losses. On the other hand, an active region with average doping less than the
3 THz 4 QW ETH design, such as the 3 THz 4 QW MIT design leads to lower waveguide
losses.
We now move on to discuss the problem of spontaneous emission measurement from
double metal ridges.
4.11 Measurement of spontaneous emission
In this chapter, we discussed various active region designs in an attempt to identify a suit-
able THz QCL structure to study lateral confinement effects. Without an optical cavity
to amplify the emission, any observable electroluminescence from nanopillars in ultrathin
QCLs is expected to be incoherent or spontaneous. To gain an understanding of the energy
spectra of laterally confined THz QCL structures, it is necessary to compare the sponta-
neous emission from an array of nanopillars to a Fabry Pèrot ridge processed from the
same ultrathin THz QCL. As highlighted previously, the functional requirement of ultra-
thin THz QCLs is that they employ 3-4 QW structures, and are processed with a double
metal waveguide. In this section we discuss some of the problems associated with measur-
ing spontaneous emission from THz QCLs employing double metal waveguides, which are
relevant for ultrathin THz QCLs.
Figure 4.34 shows the measurement of simultaneous emission from double-metal ridge
processed from full-stack 3 THz 4QW MIT design (V705), taken in pulsed mode. From the
LIV characteristics presented in section 4.8, we know that the full-stack 3 THz 4QW MIT
design (V705) exhibits lasing at a Jth = 755 A/cm2. The Jth, Jpeak and JN DR are highlighted
in the LIV characteristics shown on the right in figure 4.34. Corresponding to each value of
current, which is increased in steps of 20 mA starting from 1760 mA, i.e. J = 782 A/cm2
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4. ULTRATHIN QCLS
(just above lasing threshold) to a current of 2080 mA, i.e. J = 924 A/cm2 (just above
NDR), the spectra is recorded in an FTIR set-up using rapid-scan mode. Just above the
threshold, we note the appearance of a lasing mode at 3.14 THz, at J = 782 A/cm2. At a
current of 1820 mA, i.e. J = 808 A/cm2, we observe the appearance of a second mode at
3.22 THz, which is the prominent mode at the peak current density. This is in agreement
with the design frequency of 3 THz 4QW MIT design (see, Eul = 13 meV = 3.14 THz in
table 4.6).
Figure 4.34: Frequency spectra of full-stack 3 THz 4 QW MIT design (V705), taken above
lasing threshold using FTIR spectroscopy in rapid-scan mode. Shown on the right, are LIV
characteristics of the same device measured in pulsed mode, at an operating temperature
of 10 K.
The measurement of spontaneous emission from the same double-metal ridge pro-
cessed from full-stack 3 THz 4QW MIT design (V705) can be challenging to observe as it
is much weaker and broader than the simultaneous emission of a laser. Measurement of
spontaneous emission makes use of a more sensitive, lock-in based technique discussed in
section 4.4.3.2. In this technique, the FTIR is operated in the step-scan mode. Figure 4.35
shows the step-scan measurements of the device shown in Figure 4.34. At a current of
1675 mA, i.e. J =744 A/cm2, the device is just at the threshold current density (Jth =
755 A/cm2) and exhibits a broad electroluminescence centered at 3.14 THz with a full-
width-half-maxima (FWHM) of ∼ 100 GHz. Below the threshold, at 1663 mA (J = 739
214
4.11 Measurement of spontaneous emission
A/cm2), any spontaneous emission is hardly discernable from the noise floor. This might
not be very surprising, considering the optical power due to simultaneous emission from
the double metal, full-stack 3 THz 4 QW MIT design (V705) was only 2.5 µW. One of the
reasons associated with reduced power output is inherent in the choice of waveguide - the
double metal configuration. Owing to impedance mismatch and large beam divergence,
only a fraction of the emitted radiation is collected by the detector.
Figure 4.35: Electroluminescence measurements of full-stack 3 THz 4 QW MIT design
(V705), at and below threshold
Figure 4.36 shows a survey of reported peak powers (due to simultaneous emission) in
THz QCLs. The first point to note from figure 4.36, is that at a constant design frequency
of 3 THz, the BtC designs are more powerful than the resonant LO phonon designs. The
second point to note is that at this frequency, THz QCLs processed in an SP waveguide
configuration emit higher power (∼ 100 mW, pulsed, BtC) as compared to THz QCLs
processed in a double metal (DM) waveguide configuration (∼ 2 mW, cw, LO phonon),
indicating that QCLs processed in an SP waveguide are more powerful. The highest record
of power across frequencies is shared by THz QCLs in SP waveguide configuration, i.e. 250
mW in LO phonon [151] and 100 mW in BtC design [152] .
Interestingly, spontaneous emission has only been observed from THz QCLs, processed
in the more powerful SP waveguide configuration [37, 41, 153]. In our survey of previous
work, we have not found reports of spontaneous emission i.e. below threshold measure-
ment of THz QCLs in a DM waveguide configuration. This may perhaps imply that below
threshold emission of DM THz QCLs is weaker than the sensitivity of currently available,
215
4. ULTRATHIN QCLS
Figure 4.36: Survey of reported peak powers in THz QCLs. Adapted from [123].
commercial detectors. Measurement of spontaneous emission is an area, which we iden-
tify as one of the major experimental challenges in (i) the characterization of ultrathin THz
QCLs and (ii) the study of lateral confinement in ultrathin THz QCLs.
4.12 Summary and Conclusions
One of the first building blocks in the study of lateral confinement effects in QCLs is the
demonstration of an ultra-thin QCL, with an active region thickness that is suitable for top
down processing. In chapter 3, we identified this thickness to be ≤ 3.5 µm, to process
nanopillars of a physical radius ≤ 200 nm. In this chapter, we demonstrate simultaneous
emission from an ultrathin THz QCL, which has an active region thickness of ≈ 3 µm. We
achieved this by chemically etching a full-stack, active region (3 THz 4 QW ETH design),
post growth. Compared to the full-stack QCL (thickness ∼ 11.8µm), the threshold current
density Jth increases by 1.22 times as the thickness is scaled down to 3 µm. Also associated
with the decrease in the number of repeat periods, is the proportional decrease in power.
The strict requirements of nanopillar processing require the ultrathin QCL to have a smooth
surface instead of being chemically thinned down. This surface is typically associated with
epitaxially grown layers of III-V materials. However, in spite of lasing from full-stack 3-4
quantum well QCLs, challenges in the growth of ultra-thin QCLs have so far limited this
work. We also note that the top doped n+ contact plays a significant role in waveguide
losses, and an optimum waveguide geometry is necessary to obtain lasing from as grown
ultrathin QCLs.
216
"If you can’t solve a problem, then
there is an easier problem you can
solve: find it."
George Polya
Chapter 5
Graphene based THz optics
5.1 Introduction
Chapters 2 to 4 focussed on achieving a higher temperature performance of THz QCLs,
through lateral confinement. In this chapter we change gears from THz QCLs to THz optics
based on graphene, as an attempt towards closing the THz gap. At the same time, realizing
that THz QCLs are advocated as the only powerful, compact sources of THz radiation, we
position our work on graphene towards optical elements that may in future be compatible
with the frequencies emitted by most common THz QCLs (2-3 THz).
In section 1.5.3, we discussed that the optical response of graphene can be tuned ex-
ternally by changing the carrier concentration. By controlling the electrostatic bias on
external gates, the carrier concentration n in graphene can be pushed to a level of ∼ 1012
- 1013 cm−2. It is at this carrier concentration, the plasmon resonance in graphene appears
in the infrared (IR) and terahertz (THz) range [154, 155]. While this can in principle be
attained at room temperature in graphene, it requires cryogenic capabilities to achieve the
same in two dimensional electron gases [69]. Plasmons in graphene were imaged recently
using near-field microscopy by exciting graphene with an IR pulse [156]. The purpose of
this chapter is to first spectroscopically investigate the plasmonic properties of graphene at
THz frequencies. After a background discussion on the electronic structure and properties
of graphene, we look at THz time domain spectroscopy of graphene under different gating
configurations. The second half of the chapter focusses on two applications of graphene
to THz optics. In the first instance, we attempt to engineer the plasmonic frequency by
patterning graphene into a one dimensional grating. This may have potential for tunable
filters operating in the THz range. In the second scenario, we tune the free-carrier THz
absorption by graphene with an external bias, and exploit this to indirectly modulate the
217
5. GRAPHENE BASED THZ OPTICS
radiation from a THz QCL.
5.2 Electronic structure of graphene
Graphene, a 2D allotrophe of carbon is sp2 hybridized (with one s and two p orbitals)
which leads to a trigonal planar structure. The carbon atoms are held together with a
sigma σ bond, at a seperation of 1.42 Ao. The unaffected p orbital is perpendicular to the
planar structure, which can bind covalently with neighbouring carbon atoms leading to
the formation of a pi bond. Figure 5.1 compares the structure of graphene with graphite,
which is a 3D allotrophe of carbon, carbon nanotubes which can be viewed as a rolled up
sheet of graphene and fullerenes where carbon atoms are arranged spherically.
Figure 5.1: Graphene (top left) is a honeycomb lattice of carbon atoms. Graphite (top
right) can be viewed as a stack of graphene layers. Carbon nanotubes are rolled-up cylin-
ders of graphene (bottom left). Fullerenes (C60) are molecules consisting of wrapped
graphene by the introduction of pentagons on the hexagonal lattice. Figure from [157]
Figure 5.2: Honeycomb lattice and its Brillouin zone. Figure from [157]
The dispersion relation of graphene can be derived from the tight-binding approach.
As shown in figure 5.2, the structure of graphene can be considered to have a triangular
218
5.2 Electronic structure of graphene
lattice with a basis of two atoms per unit cell. The lattice vectors may then be written as
a1 =
a
2
(3,
p
3), a2 =
a
2
(3,−p3) (5.1)
where a = 1.42Ao is the carbon-carbon distance. The reciprocal-lattice vectors are
given by
b1 =
2pi
3a
(1,
p
3), b2 =
2pi
3a
(1,−p3) (5.2)
The two points K and K ′ at the corners of the graphene Brillouin zone are called Dirac
points. Their positions in momentum space are given by:
K=

2pi
3a
,
2pi
3
p
3a

,K′ =

2pi
3a
,− 2pi
3
p
3a

(5.3)
The three nearest-neighbour vectors in real space are given by
δ1 =
a
2
(1,
p
3),δ2 =
a
2
(1,−p3),δ3 =−a(1, 0) (5.4)
while the six second-nearest neighbours are located at δ′1 = ±a1, δ′2 = ±a2, δ′3 =
±(a2,−a1).
Considering that electrons can hop to both nearest and next-nearest-neighbour atoms,
the tight-binding Hamiltonian for electrons in graphene has the form given below.
H =−t ∑
<i, j>,σ
(a†σ,i bσ, j +H.c)− t ′
∑
<<i, j>>,σ
(a†σ, jaσ, j + b
†
σ,i bσ, j +H.c) (5.5)
Here, ai,σ(a
†
i,σ) annihilates (creates) an electron with spin σ (σ = ↑,↓) on site Ri
on sublattice A. Similarly, bi,σ(b
†
i,σ) annihilates (creates) an electron with spin σ (σ =
↑,↓) on site Ri on sublattice B. The nearest-neighbour hopping energy (between different
sublattices A and B) is given as t = 2.8 eV and the next nearest-neighbour hopping energy
(hopping in the same sublattice) is given as t ′ = 0.1 eV. The energy bands derived from
this Hamiltonian can be written as:
E±(k) =±t
p
3+ f (k)− t ′ f (k) (5.6)
where f (k) is given as
f (k) = 2cos(
p
3ky a) + 4cos
‚p
3
2
ky a
Œ
cos

3
2
ky a

(5.7)
219
5. GRAPHENE BASED THZ OPTICS
In eqn. 5.6 the plus sign applies to the upper (pi∗) band and the minus sign applies
to the lower (pi) band. Figure 5.3 shows the complete band structure of graphene. For
t ′ = 0, the spectrum is symmetric around zero energy. Figure 5.3 also shows the band
structure close to the Dirac points (K or K’ on Brillouin zone). Close to the Dirac point,
the dispersion relation can be obtained by expanding the full band structure (eqn. 5.6 and
eqn. 5.7) close to K (or K’ ) vector as k = K+q, where |q|  |K|. This leads to following
relation for energy dispersion in graphene:
Figure 5.3: Electronic dispersion in the honeycomb lattice. Left: energy spectrum (in units
of nearest neighbour hopping energy, t). Right: zoom in of the energy bands close to one
of the Dirac points) Figure from [157]
E±(q)≈±νF |q|+O[(q/K)2] (5.8)
Here, q is the momentum measured relatively to the Dirac points and νF is the Fermi
velocity = 3ta/2, with a value νF = 1x106 m/s.
The dispersion relationship of graphene is remarkably different from traditional semi-
conductor materials, which follow a quadratic response given as E(q) = q2/2m∗e , where m∗e
is the effective electron mass. Unlike graphene where the Fermi velocity is independent of
energy, the velocity in conventional materials varies with energy as ν =
p
2E/m∗e .
The unique bandstructure of graphene leads to many interesting electronic and optical
properties, which is beyond the scope of this work. Some of these properties are reviewed
in references [72, 157]. In the next section, we discuss the Drude model of conductivity
220
5.3 Optical conductivity of graphene
applied to graphene, and compare it to traditional materials.
5.3 Optical conductivity of graphene
According to the free electron (Drude) model for a conventional sytstem (carrier concen-
tration n, mass of electron m∗e , the frequency dependent conductivity can be written as
σ(ω) =
σdc
1− iωτ (5.9)
Here σdc =
ne2τ
m∗e is the Drude dc conductivity and τ is the relaxation time. The above
can be rearranged in terms of the Drude weight D and the scattering constant Γ, where, D
= pine
2
m∗e and the scattering constant Γ is the inverse of the relaxation time τ. This leads to
the following expression for ac conductivity.
σ(ω) =
iDΓ
pi(ω+ iΓ)
(5.10)
The optical conductivity σ(ω) of graphene consists of two contributions. The first
contribution, σint ra(ω) arises from intraband transitions. The second contribution arises
from the interband transitions σinter(ω) [158]. The Drude model can be applied to the
intraband optical conductivity of graphene [159]. The difference in free electron model
for a conventional two dimensional material and graphene arises in the Drude weight. In
the case of graphene the Drude weight has the value D = νF e
2
ħh
p
pin, when electon-electron
interactions are neglected.
Experimental observations in graphene confirm the Drude like response of the intra-
band conductivity. However, the Drude weight extracted from experiments is significantly
lower than theoretical predictions [159, 160]. Figure 5.4 shows the experimentally de-
rived Drude weight for graphene from AC measurements, in comparision to the theoretical
results. Note, that at high bias, i.e. high carrier concentration, the experimentally derived
Drude weight deviates from the theoretical results based on Boltzmann theory. Further-
more, the experimental results show an electron-hole asymmetery that is not explained by
the Boltzmann theory [159].
5.4 Plasmons in graphene
The plasmonic modes in graphene were first calculated by Hwang et. al from a detailed
calculation of the dielectric function ε(q,ω) at an arbitrary wave vector q and frequency ω
221
5. GRAPHENE BASED THZ OPTICS
Figure 5.4: A comparision of experimentally derived Drude weight with the Drude weight
theoretically predicted from Boltzmann theory, as a function of gate bias [159].
[154]. The zeros to the dynamical dielectric function ε(q,ω) yields the plasmonics modes
ωp(q) in graphene.
In the long wavelength limit (q → 0), the plasmon resonance in monolayer graphene
varies relativistically with carrier concentration, n as:
ωp(q→ 0) =
‚
e2νF q
κħh
p
pings gv
Œ1/2
(5.11)
The spin (gs) and valley degeneracies (gv) are both equal to 2. The dielectric constant
of the surrounding medium, κ appears as an average of the top and bottom dielectric
layers. In the case of a two dimension electron gas, the plasmon mode can be calculated
as:
ωp(q→ 0) =
‚
2pine2
κm∗e
q
Œ1/2
(5.12)
Note that the plasmonic frequency has a square root dependence in the case of a con-
ventional two dimensional electron gas (eqn. 5.12). In the case of graphene the plasmonic
frequency varies as n1/4 (eqn. 5.11). Plasmons in graphene were imaged recently using
near-field microscopy by exciting graphene with an infrared pulse [156]. Interestingly, in
the presence of a gate potential, the plasmon waves may be modified by the electric field
to have a linear dispersion relation, ωp = sq, where s is the wave velocity [161, 162].
222
5.5 Growth and processing
5.5 Growth and processing
Historically, graphene was first isolated from graphite by mechanical exfoliation [163].
High-quality graphene films obtained from exfoliation can have mobilities exceeding 15,000
cm2/Vs, however there size is limited up to 100 µm. Graphene flakes obtained from this
method can be very difficult to identify over a large area. For many electronic and optical
device applications, it is necessary to use graphene with an area greater than a few hun-
dred microns. For use in THz optical elements, obtaining an area larger than the beam
waist is a challenge. As described in previous chapters, the most common THz QCLs oper-
ate in the wavelength region of 100-150 µm (2-3 THz). The radiation from these sources
can at best be focussed to a beam size of a few mm. This implies that graphene obtained
through exfoliated means is not suitable for THz applications.
The other alternatives to obtaining graphene include epitaxial growth of graphene and
chemical vapour deposition of graphene. Both these processes allow for a contiguous
sheet of large area of graphene. The first process involves thermal decomposition of SiC,
leaving a layer of carbon behind on the surface [164]. Theoretically, the surface area of
the graphene sheet processed using this annealing method is only limited by the size of the
SiC wafer, which can be as large as 100 mm. The second process involves chemical vapor
deposition of carbon by passing a mixture of methane and hydrogen over a Cu foil in a
heated furnace [165]. The size of the graphene sheet depositied over the foil is controlled
by the size of the Cu foil used in the furnace. In fact, using chemical vapour depsoition Bae
et. al demonstrated graphene sheets that were about 700 mm across [166].
In this work, we use graphene grown by chemical vapour depsoition (CVD) means.
This is described below.
5.5.1 Chemical vapour deposition (CVD) of graphene
Figure 5.5 illustrates the process of chemical vapour deposition over foils composed of
transition metals such as Cu. A mixture of methane and hydrogen flows through a furnace
that is heated to up to 1000oC. The reaction causes the carbon atoms from methane to
chemically adsorb on the Cu surface. Once a carbon atom occupies a position on the
surface, it pushes the other carbon atoms to the side, limiting the thickness of the carbon
atoms to a monolayer. As the temperature is lowered the carbon crystallizes to graphene.
Crystallization of graphene can initiate in parallel at random locations on the surface before
the entire sheet of graphene has formed a lattice. These random locations are referred to
as nucleation sites, from which the crystal grains grow, coalesce and merge to form a
complete sheet of graphene.
223
5. GRAPHENE BASED THZ OPTICS
Figure 5.5: Mechanism of CVD growth of graphene. Reproduced from [167]
Naturally, this kind of growth creates boundaries between different grains that merge.
After the growth, the cystalline grains are called domains and the boundaries between
them are called domain boundaries. These domain boundaries represent defects in the
graphene sheet. This is because, along these boundaries, the bonding of carbon atoms
does not follow the simple Bravaise lattice from a repetition of the unit cell. These grain
boundaries are determined by the crystallinity of the Cu foil and the growth conditions.
While they can adversely influence the electrical and mechanical properties of the graphene
sheet, they can also effect the optical properties of the graphene sheet.
Figure 5.6: Scanning electron micrograph of CVD graphene used in this work. Reproduced
from [7].
In this work, CVD graphene was obtained from Stephan Hoffman’s group, grown by
Dr. Kidambi in the Centre for Advanced Photonics Engineering, Engineering Department,
Cambridge [7]. Kidambi et. al report that that CVD graphene on Cu is not inherently self-
limited to a single layer. Rather, this is dependent upon the CVD conditions and growth
kinetics. These conditions influence the density of nucleation sites, the percentage of mul-
tilayer nuclei and the uniformity of the film. Figure 5.6 shows the scanning electron micro-
224
5.5 Growth and processing
graph of CVD graphene used in this work. The thickness of the copper foil used for growth
was 25 µm, and the purity of the Cu foil was 99.999 %. Note the nucleation and stiching
of grain boundaries in CVD graphene. From the micrograph, Kidambi et. al estimated
the graphene sheet to contain monolayer domains ranging from 30 to 40 µm2, with a few
isolated areas 2-4 µm2 of multilayer graphene.
Graphene deposited over the copper foil can be transferred to any arbitrary substrate
for characterization or post processing. This is discussed below.
5.5.2 Transfer of CVD graphene to arbitrary substrate
Figure 5.7 describes the process flow involved in the transfer of graphene from the Cu foil
to an arbitrary substrate. After the graphene is removed from the CVD chamber, a layer
of PMMA is drop casted on top of it, see figure 5.7(a-b). Following this, the Copper foil is
removed in Ferric Chloride, figure 5.7(c). The film of graphene supported by PMMA floats
on top of the water, which is scooped using the new substrate, see figure 5.7(d). Finally,
the PMMA is removed in acetone, figure 5.7(e).
Figure 5.7: Process schematic: Transfer of as grown CVD graphene onto the substrate.
At this point, it is important to comment on the choice of substrate used for the trans-
fer of graphene. The choice of substrate is dictated by the type of measurements to be
performed. Transport measurements using the backgate require the substrate to be doped
such that the carrier concentration in graphene can be altered, see section 5.6. Optical
measurements on the other hand, require that the substrate be transparent to the incident
signal, see section 5.7. An ideal substrate should meet both the requirements.
For this work, we were provided with graphene samples transferred on p-doped Si
225
5. GRAPHENE BASED THZ OPTICS
wafer covered with a 300 nm thick SiO2 layer. The Si wafer, with 〈100〉 crystal orientation,
had a thickness of 525 µm ± 25 µm and a resistivity of 1 to 100 ohm-cm. This substrate
can be used for both transport as well as the optical measurements.
5.5.3 Characterization of growth quality
The quality of graphene growth was characterized extensively by Kidambi et. al and can
be referred in [7]. Here, we provide the Raman spectra and the atomic force microscope
image of graphene, to illustrate that the CVD graphene used is indeed a monolayer and of
good quality. Figure 5.8 shows the Raman spectra of the CVD graphene obtained from a
532 nm excitation. The spectra is characteristic of monolayer graphene, with a G peak at
1580 cm−1 and a 2D peak at 2700 cm−1 [168]. The absence of a D peak at 1360 cm−1
shows that graphene is free from a large number of defects. The ratio between the 2D peak
and the G peak (I2D/IG) is around 2.35, and the ratio between the D peak and the G peak
(ID/IG) is around 0.1.
Figure 5.8: Raman spectroscopy of monolayer graphene, with 2D peak higher than G peak.
Data was provided by Piran R. Kidambi
Figure 5.9 shows the atomic force microsope (AFM) image of graphene. The bright
specks in the image correspond to PMMA residue after rinsing in acetone. The bright lines
in the AFM are due to wrinkles in the graphene sheet which appear during the transfer of
graphene from Cu foil to the substrate.
226
5.5 Growth and processing
Figure 5.9: Atomic force image of graphene. Data was provided by Piran R. Kidambi.
We next describe the process of device fabrication after the CVD graphene is transferred
from a Cu foil to the substrate of choice, in our case SiO2/Si.
5.5.4 Device fabrication post transfer
Back-gated graphene
The fabrication of a back-gated graphene device is quite straightforward. First, a large
area of graphene (8 mm x 8 mm) is isolated on the wafer. A lithography step, protect-
ing the desired area followed by an etch in the oxygen plasma cleans the edges of any
residual graphene or organic matter, see figure 5.10(a-b). After a gentle rinse in acteone
and IPA, another lithography step is performed which allows the deposition of metal con-
tacts (Ti/Au) on the sides. These two contacts form the source and drain electrodes, fig-
ure 5.10(c). Note, that the contacts overlap partly with graphene and partly with the
underlying SiO2. The gold on the SiO2 surface serves as a bondpad for the wires, so as to
connect the device to the packaging board. The doped Si serves as the back-gate for the
final device.
Top-gated graphene
The carrier concentration in graphene can be controlled with an additional gate. This
gate is implemented on top of the device, and uses chemical gating to achieve control over
the carrier concentration in graphene. While this additional gate can be implemented on
top of the device, it can lead to a large leakage current. To avoid this gate leakage, we
employ a coplanar gate which is separated from the graphene edge by a distance, d ∼ 200
µm, see figure 5.11. Note that in this work we still use the term top-gate to reflect the use
227
5. GRAPHENE BASED THZ OPTICS
Figure 5.10: Schematic diagram showing graphene (a) after transfer on SiO2/Si substrate
(b) after isolation (c) after depsosition of metal contacts.
of an ionic gate geometry. Following the metallization of source, drain and gate contacts,
a solution comprising 2 mg LiClO4 in 9 mg PEO dissolved in acetonitrile, is drop casted
over the device. This kind of gating was first demonstrated by Das et. al in the year 2008
[169]. While drop casting, care is taken to avoid the overlap of the LiClO4 ionic gel with
the source and drain contacts. We find that it is helpful to drop cast the gel while the device
is on a hotplate. Finally, the device is baked overnight (> 12 hours) at 90oC.
5.6 Transport measurements of CVD graphene
Kidambi et. al extracted the transport behavior of their samples from Hall bar measure-
ments. They estimated the sheet resistance of their samples to be in the range of 400-800
ohm and mobilities to be in the range of 2000-3000 cm2V−1s−1. They further observed
that their samples were predominantly p-doped after transfer, with a doping of a few 1012
cm −2 [7, 170]. The origin of this hole doping was attributed to the PMMA residue left on
the surface after transfer. In this section, we independently measure the transport charac-
teristics of CVD graphene in the presence of a back-gate and a top-gate.
Before, we go into the details of the measurement, it is useful to revisit the band struc-
ture of graphene and understand how the position of the Fermi level EF in graphene relates
to the transport behavior. Figure 5.12 plots the bandstructure of graphene under three con-
228
5.6 Transport measurements of CVD graphene
Figure 5.11: Schematic diagram showing graphene (a) after deposition of coplanar gate,
source and drain contacts (b) after drop casting of the ionic gel.
ditions. In the first case, graphene is pristine, i.e the Fermi level EF is at the Dirac point.
Notice, that the charge neutrality point (VCN P), defined as the point where the charge
transport transitions from predominantly hole carriers to predominantly electron carriers
is at 0 V, see figure 5.12(a). In the second scenario, graphene is p-doped. This implies
that the Fermi level EF is below the Dirac point, i.e. it exists in the valence band. At zero
gate bias the transport is mediated by predominantly hole carriers and the charge neutral-
ity point (VCN P) shifts to the right, see figure 5.12(b). In the third scenario, graphene is
n-doped and the Fermi level EF is above the Dirac point. In this case, the predominant
charge carriers at zero gate bias are electrons. The charge neutrality point (VCN P) occurs
at a negative gate bias.
5.6.1 Results from back-gated graphene
The experimental set-up for measuring the transport behavior of graphene is similar to
any field effect transistor measurement, see figure 5.13 Grounding the source contact, we
apply a constant bias on the drain contact. We call this voltage, the drain-source bias (Vds).
The current flowing through graphene between the two contacts is called the drain-source
current (Ids) and is proportional to the width W of the graphene channel and inversely
proportional to the length L of the graphene channel.
229
5. GRAPHENE BASED THZ OPTICS
Figure 5.12: Schematic diagram of the band structure, and the corresponding electrical
characteristics when (a) graphene is pristine (b) graphene is p-doped (c) graphene is n-
doped.
Figure 5.13: Circuit diagram, showing source, drain and gate contacts for transport mea-
surements.
230
5.6 Transport measurements of CVD graphene
The expression for the drain-source current is given by the following expression:
Ids = eµW
∫ Vds
0
n dV
L−µ∫ Vds
0
1/νsat
dV (5.13)
where, µ is the mobility of carriers in graphene and νsat is the saturation velocity [171].
The carrier concentration n, is controlled by the applied bias on the back-gate Vbg . The
relationship between Vbg and n can be evaluated from simple electrostatics, and is given
by the following expression:
Vbg − VCN P = ne/Cbg (5.14)
Here, Cbg is the geometrical capacitance and is equal to
εrεo
d
. In the expression for
capacitance, d is the thickness of the SiO2 layer, which is equal to 300 nm in our device
and εr is the dielectric constant of the SiO2 layer, which is typically equal to ∼ 4. Using
these values, we obtain a back-gate capacitance of 12 nF cm−2 for our device.
Figure 5.14: Drain-source current (Ids) as a function of back-gate bias Vbg , at room tem-
perature 300 K. The drain-source voltage Vds was kept constant at 0.01 V.
Figure 5.14 shows the transport characteristics of graphene at a constant drain-source
bias equal to 0.01 V. The drain-source current was measured at room temperature using a
Keithley SMU. The back-gate leakage current was negligible in this device and was always
below 1 nA. On comparing the electrical behavior of the device to the characteristics shown
231
5. GRAPHENE BASED THZ OPTICS
in figure 5.12, we infer that the graphene is p-doped. Note that the charge neutrality point
is not reached even at a gate bias of +60 V. We do not apply a bias greater than +60 V to
avoid device failure due to Joule heating. On extrapolating the measured characteristics
(blue dashed curve), we estimate that the charge neutrality point lies at around +123 V.
However, this is an overestimate of VCN P as the drain-source current does not reach zero
in practice. Since VCN P lies in the range of +60 V to +123 V, we can evaluate the carrier
concentration n at zero bias using eqn. 5.14. This value lies in the range of 4.5 x 1012 to
9.2 x 1012 cm−2, and agrees with the Hall bar estimates made by Kidambi et. al [7]. The
position of the Fermi level EF at a given carrier concentration can be estimated using the
equation:
EF = ħhνF
p
pin (5.15)
This implies that at zero gate bias the position of the Fermi level EF is approximately
0.24 to 0.35 eV below the Dirac point. The mobility can be estimated using the formula
µ = σsheet/(en), where σsheet is the sheet conductance of graphene. We find that at zero
gate bias, the sheet conductance (Ids/Vds) is equal to 21.8 e2/h. Assuming a carrier con-
centration of 4.5 x 1012 to 9.2 x 1012 cm−2, we estimate the mobility of charge carriers to
lie somewhere in the range of 1190 to 582 cm2V−1s−1 at zero gate bias.
One of the major disadvantages of using a back-gate on graphene is the absence of a
clear VCN P at accessible voltages. While we can still extract the transport parameters (n,
EF and µ) from the device behavior, it gives us a broad range of values. A VCN P to the right
of voltage axis also implies that we can not access the electron transport in graphene using
a back-gate as it requires a bias Vbg > VCN P . Using an ionic gate, the VCN P can be made to
approach zero enabling access to hole and electron transport at very low voltages. This is
described below.
5.6.2 Results from top-gated graphene
In the previous scenario, access to high carrier concentrations as well as access to both
types of carriers, was limited by the extremely small capacitance between the back-gate
and graphene. The capacitive coupling can be enhanced dramatically by using an ionic
(top) gate, see figure 5.15. In our devices we an ionic (top) gate comprising of LiClO4
in Polyethylene oxide matrix. When a bias is applied, free cations (Li+) accumulate near
the negative electrode, creating a positive charge layer and a negative charge near the
interface. This layer of charge around the electrode is called the Debye layer and is char-
acterized by a thickness dt g , which is approximately 1-5 nm. The capacitance between the
232
5.6 Transport measurements of CVD graphene
top gate and graphene is governed by the thickness of this Debye layer and is given by Ct g
= εrεo/dt g . Here, εr is the dielectric constant of the Polyethylene oxide matrix, which is
equal to 5 [169]. This results in a top-gate capacitance of about 2.2 µF cm−2. It is im-
portant to emphasize that even though the thickness of the ion gel varies across the area,
the capacitance remains constant as it is dominated by a monolayer of ions, which forms
the Debye layer. This technique of ionic gating has also been used by other THz groups to
modulate plasmonic modes inside a QCL [172].
Figure 5.15: Schematic diagram illustrating gating using an ion gel.
Figure 5.16 shows the drain-source current (Ids) of the graphene device as a function of
top-gate bias (Vt g). These measurements were carried out using HP 4155 A semiconductor
parameter analyser at room temperature. The drain-source bias (Vds) was constant at 0.1
V. The gate voltage was ramped at a rate of 20 mV/min.
Note that we can access both electrons and holes using the top-gate configuration.
The forward and reverse voltage sweeps show hysterisis in the transconductance behavior.
We note that the charge neutrality point in the forward voltage sweep V fCN P = 0.04 V, is
less than the charge neutrality point in the reverse voltage sweep VrCN P = 0.14 V. This
hysterisis is dependent upon the ramp rate. Transfer of holes (electrons) from the ionic gel
to graphene can result in the right (left) shift of conductance, which can causes negative
hysterisis [173]. In our devices, we observe a positive hysterisis with gate voltage sweep.
Positive hysterisis occurs when the ion changes the local electrostatic potential around the
graphene, thereby pulling more charges from the contacts [173]. Figure 5.17 shows the
transconductance behavior of top-gated graphene at different ramp rates. At large gate
bias (Vt g ≈ 0.8 V), the drain-source current is nearly five times larger when the voltage is
ramped at 50 mV/min as opposed to when it is ramped at 5 mV/ms. The effect of the ramp
rate is also seen on the area under the hysterisis curve, which is larger when the voltage is
ramped at 5 mV/ms, as compared to 50 mV/min.
From figure 5.16 and figure 5.17, we also observe that at large gate bias, the drain-
233
5. GRAPHENE BASED THZ OPTICS
Figure 5.16: Drain-source current (Ids) as a function of top-gate bias Vt g , at room temper-
ature 300 K. The drain-source voltage Vds was kept constant at 0.1 V. The gate voltage was
ramped from -0.5 V to 0.5 V at a rate of 20 mV/min.
Figure 5.17: Drain-source current (Ids) as a function of top-gate bias Vt g , at room temper-
ature 300 K. The drain-source voltage Vds was kept constant at 0.1 V. The gate voltage was
ramped from 0 to 0.8 V at a rate of 50 mV/min, 100 mV/min and 5 mV/ms. For more
clarity, the fastest scan is shown separately in the inset.
234
5.6 Transport measurements of CVD graphene
source current grows slowly and at times shows a dip. Perhaps, this occurs due to the
chemical doping of graphene. This phenonmenon is different from inducing a surface
charge at the interface as it can alter the sheet conductance of graphene.
Figure 5.18: (a) Top-gate leakage current (Igs) as a function of top-gate bias Vt g . The
drain-source voltage Vds was kept constant at 0.1 V. (b) Drain-source current (Ids) as a
function of drain-source bias Vds, at different Vt g .
From figure 5.18(a), we observe that the gate leakage current Igs through an ionic gate
is higher than the leakage through a conventional dielectric. This is because an ionic gel
is not a complete insulator. We note that the leakage current increases with voltage, i.e.
when the top-gate bias is increased from 0 to 0.5 V, the gate-source current increases from
0 to 0.3 µA. However, this increase is nearly two orders less than the increase in the drain-
source current, which changes from approximately 15 µA to 45 µA when the top-gate bias
is increased from 0 to 0.5 V, see figure 5.16. This implies that the transconductance charac-
teristics observed in figure 5.16 is indeed indicative of ionic gating. Figure 5.18(b) shows
the Ids as a function of Vds. The dashed vertical line shown in figure 5.18(b) compares Ids
at Vds = 0.1 V, for different Vt g and agrees with results shown in figure 5.16.
The asymmetry in the drain-source current with respect to the charge neutrality point
agrees with reports of electron-hole asymmetry observed by Horng et al. [159]. However,
contact effects may also introduce an asymmetery in the drain-source current. Note that
in a top-gated device, the ionic gel does not overlap with the source and drain contacts.
Therefore, the ionic gel does not completely cover the graphene sheet, see figure 5.11.
These ungated regions near the contacts add to the resistance and hence lead to the asym-
metry shown in figure 5.16 and figure 5.18(b) [174]. The transport characteristics al-
though similar in form can vary from device to device. Therefore it is useful to quote the
235
5. GRAPHENE BASED THZ OPTICS
gate bias in terms of Vt g - VCN P . In our devices, we typically observe a charge neutrality
point between 0 to +0.5 V, suggesting slight p-doping at zero gate bias.
Figure 5.19: Transport parameters in a top-gated graphene field effect transistor, showing
(a) carrier concentration, n (b) Fermi level, EF and (c) mobility, µ of charge carriers.
Below, we extract the transport parameters (n, EF and µ) from the device behavior,
shown in figure 5.19. The carrier concentration n is related to top-gate bias as:
Vt g − VCN P = ħhνF
p
pin
e
+ ne/Ct g (5.16)
Given the carrier concentration, the Fermi level and the mobility can be calculated
using EF =
ħhνF
p
pin
e
and µ = σsheet/(en), where σsheet is the sheet conductance of graphene.
We note that carrier concentrations approaching 1x1013 cm−2 can be easily achieved
in both directions, on application of bias as low as 0.36 V referred to the charge neutrality
point, figure 5.19(a). At this bias, the fermi level EF is 0.368 eV above or below the
Dirac point, depending upon the polarity (see figure 5.19(b)). A consequence of high
236
5.7 THz time domain spectroscopy (THz TDS) of CVD graphene
carrier concentration is that the mobility of the carriers reduces. In a top-gated graphene
device, the mobility may also be influenced by the disorder introduced by the ions. This
is seen in the reduction of mobility from 1190 (582) cm2V−1s−1 in the case when no top-
gate is implemented (section 5.6.1), to approximately 280 cm2V−1s−1 when a top-gate is
implemented but no voltage on the top-gate is applied (Vt g = 0), figure 5.19(c).
We now move on to discuss the optical response of CVD graphene at THz frequencies.
The optical response of graphene is studied using THz time domain spectroscopy (THz
TDS). In the next section, we first decribe the experimental set-up used for spectroscopy
and then discuss the results obtained from graphene.
5.7 THz time domain spectroscopy (THz TDS) of CVD graphene
THz time domain spectroscopy (THz TDS) uses the coherent detection of broadband pulses,
at room temperature to analyse the spectral response of materials [175]. The THz TDS se-
tups used in this work are based in the Department of Chemical Engineering and Biotech-
nology, Cambridge. These set-ups are routinely used by to characterize biochemical molecules
and pharmaceutical drugs which have sensitivity in the THz spectrum, and can easily be
adapted to make optical measurements on thin films such as graphene as a function of
bias.
THz TDS systems can be operated in the transmission mode as well as reflection mode.
Below we provide description for both the set-ups.
5.7.1 Experimental set-up in transmission mode
The THz TDS system comprises of an ultrafast femtosecond (12 fs) pulse laser, with a
repetion rate of 80 MHz, incident on a photconductive antenna which is used as a broad-
band THz source (see figure 5.20). The ultrafast NIR source (Femtolasers, Femtosource
cM1, Vienna, Austria) operates at a centre wavelength of 800 nm. Using two parabolic off-
axis mirrors the terahertz pulses are focused onto the samples (graphene). The transmitted
pulses are collected using an identical set of parabolic mirrors. The transmitted radiation
is detected using the electro-optic sampling method, which utilizes the birefringent ZnTe
crystal [176]. To increase the speed of data acquisition as well as to improve spectral
resolution and signal-to-noise ratio, a 50 ps rapid optical delay is used. Between 200 and
300 scans are averaged at a scanning frequency of 0.5 Hz for each spectrum. The resulting
timedomain waveform is apodized using a Hamming function. Figure 5.21(a) shows the
THz waveform through dry air in the time domain. The spectrum in the frequency do-
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5. GRAPHENE BASED THZ OPTICS
main is obtained by taking a fast Fourier transform (FFT) of the time domain waveform,
figure 5.21(b).
Figure 5.20: THz TDS set up in transmission mode. The area enclosed by dashed lines is
purged with nitrogen.
Theoretically, the total bandwidth of the FFT signal is determined by the resolution
of the time domain signal, which is 0.03 ps in our measurements [177]. This yields a
total signal bandwidth that is equal to 33 THz. In reality, the total signal bandwidth is
determined by the source and detection optics of the set-up. From figure 5.21(b), we note
that the signal is highest around 0.5 THz and falls down as the frequency moves away,
limiting the total signal bandwidth to less than 3 THz.
The resolution of the THz spectra is inversely proportional to the acquisition length of
the time domain waveform [177]. This holds true for samples investigated using THz TDS
that are more than 1 mm thick. The spectral resolution of the FFT spectra derived from the
time-domain signal for these thick samples can be as small as 30 GHz. In the case where
the samples are thin (< 1 mm), the detector records the reflection of the THz wave from
the substrate. In these samples, the spectral resolution is determined by the thickness of
the substrate.
In this work the thickness of the Si substrate is approximately 525 µm. The reflections
from the substrate restricts the acquisition length to ≈12 ps and the spectral resolution to
238
5.7 THz time domain spectroscopy (THz TDS) of CVD graphene
Figure 5.21: THz waveform through dry air in (a) time domain and (b) frequency domain,
using the TDS set-up in transmission mode.
≈70 GHz.
Figure 5.22 shows the time domain waveform transmitted through a SiO2/Si substrate.
The first pulse transmitted through the substrate is denoted by E1T (t) and has the maximum
amplitude. This is often called the primary pulse. The appearance of a secondary pulse is
marked by E2T (t) and is a consequence of the first reflection from the substrate. This pulse
travels three times through the substrate. The magnitude of the secondary pulse is lower
than the magnitude of the primary transmitted pulse owing to the losses incurred within
the substrate. We also observe a tertiary pulse, called E3T (t). This pulse travels through the
substrate five times and is significantly lower in magnitude. Assuming a refractive index
of 3.42 for Si, the time difference ∆t between the pulses passing through a 525µm thick
substrate can be calculated as 11.9 ps, which agrees with the measured∆t, see figure 5.22.
From figure 5.22, we note that the magnitude of the electric field corresponding to
the primary pulse E1T (t) (1.5 a.u) is less than the strength of the THz signal transmitted
through dry air (2.5 a.u, in figure 5.21) 1. This reduction in the THz signal arises due to
free carrier absorption by the p-doped Si substrate.
In our treatment of the THz signal transmitted through the substrate (figure 5.22), we
ignored the effects of the 300 nm thin SiO2 layer. In reality, the incident broadband THz
signal sees three interfaces - Air/SiO2, SiO2/Si and Si/air. While these interfaces are not a
1THz time domain waveforms shown in figure 5.22 and figure 5.21 were measured one after the other,
and so the experimental conditions are the same.
239
5. GRAPHENE BASED THZ OPTICS
Figure 5.22: THz time domain waveform transmitted through the SiO2/Si substrate. The
schematic diagram shows the incident, transmitted and reflected peaks displaced vertically,
for clarity.
concern here, it is useful to remember these complexities while analyzing the THz signal
from a multilayer system such as a graphene device that is back-gated or top-gated.
5.7.2 Experimental set-up in reflection mode
The THz TDS setup in the reflection mode consists of a modified TPI imaga 2000 supplied
by Teraview Ltd., Cambridge, UK. This set-up was used with a set of 2D linear stages,
controlled at a step of 100 µm to generate a THz map of the graphene device.
Figure 5.23(a) shows the THz waveform through dry air in the time domain. The
spectrum in the frequency domain is obtained by taking a fast Fourier transform (FFT) of
the time domain waveform, figure 5.23(b). From figure 5.23(b), we note that the signal
is highest around 1.1 THz and falls down as the frequency moves away. The source and
detection optics limits the total signal bandwidth to less than 3.5 THz.
Figure 5.24 shows the THz time domain wavefrom reflected from the SiO2/Si substrate.
In the reflection mode, the incided beam is focussed to a spot size of 200 µm , which
is incident onto the surface at angle of 30o [178]. This allows the detector to collect
the primary reflected peak E1R(t) and the secondary reflected peak E
2
R(t). Similar to the
analysis of the THz time-domain waveform obtained using the transmission mode setup,
the difference ∆t between the two peaks, in the reflection mode, can be estimated from
the time it takes for the incident pulse to travel twice the thickness of the Si substrate.
240
5.7 THz time domain spectroscopy (THz TDS) of CVD graphene
Figure 5.23: THz waveform through dry air in (a) time domain and (b) frequency domain,
using the TDS set-up in reflection mode.
For a 525 µm thick Si substrate, ∆t can be estimated as 11.9 ps, which agrees with the
measured ∆t shown in figure 5.24.
Figure 5.24: THz time domain waveform reflected from the SiO2/Si substrate. The
schematic diagram shows the incident and the reflected peaks.
5.7.3 Transmission mode vs. reflection mode set-up
Both the THz TDS systems discussed above have their pros and cons. One of the major ad-
vantages of using the TDS in the reflection mode is for THz spectoscopy of highly absorbing
241
5. GRAPHENE BASED THZ OPTICS
materials (E1T ≈ 0). This includes graphene deposited on highly doped Si substrates that
would otherwise not be suitable for optical measurements.
The transmission mode set-up on the other hand is more suitable for samples (or sub-
strates) which have a thickness less than 1 mm. This is because in the transmission mode,
the beam is incident normal to the surface, figure 5.22. This implies that the primary
transmitted pulse E1T (t) and the secondary transmitted pulse E
2
T (t) carry the spectral in-
formation of the same point in space. In the reflection mode on the other hand, the beam
is incident at an angle of 30o, figure 5.24. This implies that the primary reflected pulse
E1R(t) and the secondary reflected pulse E
2
R(t) carry the spectral information of two different
points in space and can not be compared.
5.7.4 Theory behind THz transmission through graphene
We are now in the position to discuss the THz time domain spectroscopy of graphene. THz
transmission through graphene is a function of it’s conductance, i.e. available density of
states for intraband transitions [159, 179]. Given the frequency dependent conductivity
σ(ω), the THz transmission T(ω) through graphene can be expressed as:
T (ω)/Ts(ω) = |(1+ Zo.σ(ω)/(ns + 1))|−2 (5.17)
Since T(ω) also contains a component of absorption by the SiO2/Si substrate, an es-
timate of transmission through graphene sheet alone is obtained by normalizing T(ω) to
Ts(ω), which is the transmission through the SiO2/Si substrate. In eqn. 5.17 ns is the re-
fractive index of the silicon substrate (∼ 3.42) and Zo is the vacuum impedance (376.7 Ω)
[180]. The conductivity σ(ω) can be tuned by varying the position of the Fermi level. The
relationship between intraband conductivity and the Fermi level EF can be expressed as:
σ(ω)
σo
=
8kB T
piħh ln(e
−EF/2kB T + eEF/2kB T ) 1
1/τ− iω . (5.18)
Here, T is the temperature, kB is the Boltzmann constant, σo = pie2/2h is the quantum
conductance and τ is the relaxation time of carriers. The relaxation time can be estimated
from the mobility of charge carriers µ and the carrier concentration n as τ = µħhppin/eνF .
Note that the expression for intraband conductivity (eqn. 5.18) is similar to the Drude like
behavior discussed in section 5.3 at high carrier concentrations. At high carrier concentra-
tions, i.e. EF  kBT, eqn. 5.18 reduces to:
σ(ω) =
4EF
piħh
1
1/τ− iω (5.19)
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5.7 THz time domain spectroscopy (THz TDS) of CVD graphene
At low carrier concentrations, EF  kBT, eqn. 5.18 can be written as:
σ(ω) =
8kB T
piħh
ln(2)
1/τ− iω (5.20)
Therefore, when the Fermi level is close to the Dirac point, the optical conductivity is
independent of carrier concentration and only depends upon the temperature. We now
discuss the results obtained from THz spectroscopy of monoloayer CVD graphene under
different gating conditions.
5.7.5 Results from back-gated graphene
Figure 5.25 shows the THz transmission of monolayer CVD graphene in time domain as a
function of back-gate voltage, Vbg . The peaks of the primary and secondary transmission
pulses have been plotted separately for clarity. Note that the transmission is highest when
the back-gate voltage on graphene Vbg = + 50 V. At this bias, the p-doped graphene is
closest to the Dirac point. As the Fermi level is moved away from the Dirac point, towards
Vbg = - 50 V, the concentration of holes increases. This causes free carrier absorption
within graphene, which can be observed as reduced transmission at Vbg = - 50 V.
The THz spectrum is obtained by taking a fast Fourier transform of the signal shown
in figure 5.25. If we take the Fourier transform of the entire waveform shown in fig-
ure 5.25(a), the spectrum suffers from spurious oscillations due to etalon effects [181].
While algorithms exist to deconvolve the spectrum from the oscillations, they are harder
to implement in multilayered structures. The other alternative is to take the fourier trans-
form of the primary peak and secondary peak seperately, where the length of the time
domain waveform is equal to ∆t. We follow the latter approach to estimate the spectrum
of graphene on SiO2/Si substrate.
In order to obtain the transmission (or aborption) spectrum of graphene, one of the
most critical step is obtaining a reference. It is common in the literature to refer the trans-
mission of graphene at different applied bias to the transmission of the SiO2/Si substrate
or the transmission of graphene at zero gate bias [179]. However, this is not entirely accu-
rate. In the first case, it is difficult to decovolve the transmission through graphene from
the multiple-stack graphene/SiO2/Si system using this method, especially when trying to
obtain weak resonance features of graphene based metamaterials. In the second case,
transmission at zero-gate bias is not necessarily at maximum. Absorption referred to the
charge neutrality point VCN P eliminates artifacts arising from laser drift, thin-film approx-
imations and slight thickness variations across different substrates [160]. However, this
can be prohibitive in graphene samples that require a very large back-gate bias to reach
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5. GRAPHENE BASED THZ OPTICS
Figure 5.25: THz time domain waveform transmitted through monolayer graphene as a
function of back-gate voltage Vbg (a) Complete waveform (b) Primary transmitted pulse,
E1T (t) (b) Secondary transmitted pulse, E
2
T (t)
244
5.7 THz time domain spectroscopy (THz TDS) of CVD graphene
the Dirac point. For example, in the sample shown in figure 5.25, VCN P lies beyond our
measurement range (+ 50 V).
Since, the THz transmission is maximum at this point, we can assume that the Fermi
level is closest to the Dirac point at Vbg = +50 V. This allows us to calculate the ef-
fect of back-gate on the absorption of graphene with respect to + 50 V. If the Fourier
transform of the THz pulse transmitted through graphene at a bias of + 50 V is given
by T50V (ω), the normalized absorption at each back-gate potential can be calculated as
1− Tbg(ω)/T50V (ω).
Figure 5.26: Normalized absorption spectra of p-doped graphene referred to Vbg +50 V,
(a) derived from primary pulse (b) derived from secondary pulse.
Figure 5.26 shows the normalized absorption through monolayer CVD graphene as
function of gate bias. The first spectrum, figure 5.26(a) is derived from the Fourier trans-
form of the primary transmitted pulse E1T and the second spectrum figure 5.26(b) is derived
from the secondary transmitted pulse E2T . Figure 5.26(a) shows that the absorption is gate
tunable and increases by 5% as graphene is gated away from the Dirac point, into the
valence band. This is consistent with what has been obtained in literature [160, 179].
The absorption spectrum at Vbg = + 30 V and Vbg = - 10 V rolls off to negative values at
frequencies higher than 1 THz. It is likely, that this feature, appearing at voltages close to
+ 50 V, is an artifact introduced by referring the absorption to a bias that is not the charge
neutrality point. The negative absorption vanishes when more signal is collected from
graphene, i.e. when the absorption spectrum is derived from the secondary transmitted
pulse. This pulse traverses through graphene twice. Therefore, the absorption obtained
from graphene, figure 5.26(b) is more than twice the value shown in figure 5.26(a). While
245
5. GRAPHENE BASED THZ OPTICS
interesting features are seen in the absorption spectrum of back-gated graphene at fre-
quencies > 1 THz, the absence of a charge neutrality point in the sample prohibits detailed
analysis.
We can still extract the optical conductivity from the THz transmission. Using eqn. 5.17,
we can evaluate σ(ω) from the transmission T(ω), for each back-gate voltage. Note, that
eqn. 5.17 only holds true when the spectrum is derived from the primary pulse [182].
Figure 5.27 plots the AC conductivity σ(ω) at ω = 1 THz, as a function of back-gate
bias. In agreement with previous reports [179], the frequency dependent conductivity
shown in figure 5.27 varies non-linearly with Vbg . At voltages close to the Dirac point, the
conductivity does not vary significantly with voltage. However, as the gate bias drives the
Fermi level away from the Dirac point, the ac conductivity increases proportionaly. This
is consistent with the theoretical predictions of intraband conductivity under conditions of
low and high carrier density, see eqns. 5.18 to 5.20. Note that when Vbg = 0, the frequency
dependent conductivity is σ(ω) at 1 THz is equal to 21 e2/h. This is in agreement with
the dc conductivity (sheet conductance) extracted from the transport measurements of
back-gated graphene, discussed in section 5.6.1.
Figure 5.27: AC conductivity of graphene as a function of back-gate voltage, Vbg
The absence of a charge neutrality point, can be corrected by using a top ionic gate as
discussed in section 5.6.2. In the next section, we review the results obtained from THz
spectroscopy of top-gated, monolayer CVD graphene.
246
5.7 THz time domain spectroscopy (THz TDS) of CVD graphene
5.7.6 Results from top-gated graphene
Figure 5.28 shows the THz transmission of monolayer CVD graphene in time domain as a
function of top-gate voltage Vt g ≤ 0.5 V. The primary and secondary transmission pulses
have been plotted separately for clarity. Note that the transmission is highest when the top-
gate bias Vt g = + 0.5 V. At voltages below + 0.5 V, i.e. Vt g = 0.4 to 0 V, the transmission
drops, suggesting that the Fermi level has moved away from the Dirac point.
Figure 5.28: THz time domain waveform transmitted through monolayer graphene as a
function of top-gate voltage Vbg < VCN P (a) Complete waveform (b) Primary transmitted
pulse, E1T (t) (b) Secondary transmitted pulse, E
2
T (t)
Decrease in transmission is also observed when the top-gate voltage is increased be-
yond + 0.5 V, i.e. Vt g = 0.6 to 1.0 V. Figure 5.29 shows the THz transmission of monolayer
247
5. GRAPHENE BASED THZ OPTICS
CVD graphene in time domain as a function of top-gate voltage Vt g ≥ 0.5 V. Once again,
the primary and secondary transmission pulses have been plotted separately for clarity.
Figure 5.29: THz time domain waveform transmitted through monolayer graphene as a
function of top-gate voltage Vbg > VCN P (a) Complete waveform (b) Primary transmitted
pulse, E1T (t) (b) Secondary transmitted pulse, E
2
T (t)
Figures 5.28 and 5.29 suggest that the charge neutrality point in this sample exists
when Vt g = + 0.5 V. Below + 0.5 V, graphene is p-doped and the predominant charge
carriers are holes. Above + 0.5 V, graphene is n-doped and the predominant charge carriers
are electrons. Note that this charge neutrality point is higher than 0.14 V observed using
transport measurements for the same sample 4 weeks ago, see section 5.6.2. However,
this is not very surprising as graphene is susceptible to changes in the environment, which
inadvertently dopes graphene. Although the exposure is minimized by storing graphene in
an environment purged with nitogen, it is probable that it gets doped while in transit from
248
5.7 THz time domain spectroscopy (THz TDS) of CVD graphene
cleanroom to the measurement lab. It is therefore essential that the charge neutrality point
for the graphene sample is measured each time an optical or an electrical experiment is
repeated. The device behavior is independent of where the charge neutrality point exists.
Therefore, the gate bias is best referred in terms of Vt g - VCN P . In furture, rebaking the
device before each experiment may yield a reproducible charge neutrality point.
Figure 5.30(a-b) shows the absorption spectrum of graphene derived from the Fourier
transform of the primary pulse E1T . The normalised absorption is calculated using the
expression 1 − Tt g(ω)/TCN P(ω). Note that close to the Dirac point, at Vt g − VCN P = -
0.1 V, graphene is lightly p-doped (n = 5 x 1012 cm−2) and the absorption increases with
frequency by up to about 10% at ω = 2 THz. Similar behavior is obtained at Vt g − VCN P
= + 0.1 V when graphene is doped with electrons. This is in agreement with a Drude like
response of the frequency dependent optical conductivity, see eqn. 5.10.
At higher voltages, the THz absorption is tunable with gate voltage. We observe that
the absorption shows flat response with frequency until ω = 1 THz. However, at 1.6
THz, we observe a distinct absorption feature superimposed on the Drude like response of
graphene as the gate voltage is increased to ± 0.5 V (i.e. carrier concentration n is equal to
± 1.2 x 1013 cm−2). From figure 5.30(a), we note an absorption peak at ∼ 1.6 THz when
graphene is p-doped and EF is below the Dirac point. This feature however reverses in
figure 5.30(b) when graphene is n-doped and EF is above the Dirac point. These features
are not observed in the absorption spectrum of the ion gel, which absorbs 2-4 % incident
THz radiation when referred to the SiO2/Si substrate, shown in Figure 5.31.
Figure 5.30: Normalized absorption spectra of top-gated graphene, derived from primary
transmitted pulse, referred to VCN P = + 0.5 V when graphene is (a) p-doped (b) n-doped.
249
5. GRAPHENE BASED THZ OPTICS
Figure 5.31: THz absorption spectra of ionic gate (LiClO4), derived from the primary
transmitted pulse, referred to the SiO2/Si substrate.
Figure 5.32: Normalized absorption spectra of top-gated graphene, derived from secondary
transmitted pulse, referred to VCN P = + 0.5 V when graphene is (a) p-doped (b) n-doped.
Phase difference shows change in sign at the absorption resonance when graphene is (c)
p-doped (d) n-doped. For clarity, the sign change in phase difference between 1.5 and 1.8
THz is shown in the inset.
250
5.7 THz time domain spectroscopy (THz TDS) of CVD graphene
It is likely that the absorption spectrum derived from the secondary transmitted pulse
E2T would give a stronger signal as it passes through graphene twice. This is indeed what
we observe in the absorption spectrum of graphene shown in figure 5.32. Note, as the
THz pulse travels twice through graphene in the case of Figure 5.32, the 10 % change in
absorption at highest doping is nearly twice the 5 % change in absorption observed at low
frequencies in figure 5.30. Superimposed upon the flat absorption spectrum we observe
peaks evolving with voltage at 1.1 THz and 1.6 THz. The resonance at 1.6 THz appears to
be strong in the case of p-doped graphene. At a doping of 5 x 1012 cm−2 (Vt g - VCN P = ±
0.1 V), the normalized absorption referred to CNP is nearly zero. As the doping increases
to 1.6 x 1013 cm−2 (Vt g - VCN P = ± 0.5 V), the absorption increases by 10 % and the peak
shows 20 % change in absorption at 1.6 THz. Corresponding to the absorption resonance
at 1.6 THz in figure 5.32(a), we see the phase difference, dδ, changing sign from positive
to negative in figure 5.32(c). Interestingly, for the case of n-doped graphene, dδ undergoes
a phase change from negative to positive, see figure 5.32(d).
Similar absorption features are seen in other samples. Figure 5.33 shows the absorption
spectrum of another top-gated graphene device, referred to it’s charge neutrality point. The
absorption spectrum, shown for hole transport is derived from the secondary transmitted
pulse. Similar to figure 5.32, this device also shows a similar gate tunable, absorption
peak at 1.6 THz when graphene is p-doped. The peak at 1.1 THz observed in figure 5.32,
reverses in this device. We do not yet understand the origin of this peak and the reason for
it’s reversal.
We note from our measurements of back-gated graphene (figure 5.26) and those shown
in the literature that the absorption features are unique to top-gated graphene [160, 179,
183]. We note that the absorption spectrum 1− Tbg(ω)/T+50V (ω) is flat until 2 THz at
Vbg = -50 V, with no absorption peak at 1.6 THz which we observe in top-gated, p-doped
CVD graphene.
In the next section, we discuss possible causes that may explain the absorption features
at 1.6 THz.
5.7.7 Role of domains in spectroscopic response
In section 5.4, we discussed that graphene can support THz plasmons. The plasmon fre-
quency ωp is a function of carrier concentration and varies as n
1/4. Ideally, for an un-
patterned, ungated sheet of graphene, the plasmon frequency is given by eqn. 5.11. If
graphene is patterned into circular disks, the patterned two dimensional free electron gas
can act as two dimensional plasmonic grating [184], where the resonance frequency is
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5. GRAPHENE BASED THZ OPTICS
Figure 5.33: Normalized absorption spectra of another top-gated graphene device. The ab-
sorption spectrum, shown for hole transport, is derived from secondary transmitted pulse
and is referred to VCN P
dependent upon the radius of the disk.
We note from our earlier discussion in section 5.5.1, that CVD graphene is inherently
polycrystalline. We suggest that these growth domains can themselves act as a two dimen-
sional plasmonic grating, which causes scattering of plasmons. In our samples, the domain
size is observed to be of the order of 30-40 µm2 from microscopy studies of CVD graphene,
see figure 5.6. Assuming an r = 3.4 µm (which corresponds to an area of 36 µm2), the
plasmon frequency ωp can be calculated using the model for an ungated graphene disk.
The plasmon frequency is then given by the following expression, where the radius of the
domain r is assume to be equal to the radius of the disk [184].
ωp =
r
3νF e2
p
pin
16rħhκ (5.21)
Here, κ is the average dielectric constant of the surrounding medium (PEO and SiO2),
which is equal to (5 + 3.9)/2. For an r = 3.4 µm, eqn. 5.21 predicts that we should observe
resonances at 2.01 THz, 2.20 THz, 2.34 THz, 2.45 THz and 2.55 THz as Vt g−VCN P is varied
from 0.1 to 0.5 V in steps of 0.1 V.
While we observe gate tunability of the absorption peak, we do not measure a shift in
the resonance frequency as the gate bias is swept away from the Dirac point. Note that
eqn. 5.21 ignores the effect of gating on the plasmon frquency. The effect of gate on the
propagation of plasmons was first calculated in conventional semiconductor field effect
252
5.7 THz time domain spectroscopy (THz TDS) of CVD graphene
transistors to predict predict oscillations at THz frequency under asymmetric source-drain
bias due to reflection from the device boundaries [185]. This model was later extended
to suggest plasma oscillations in graphene under aysmmetric boundary conditions [162].
Because the instability conditions require a low drain current, aysmmetric boundary con-
ditions may also be introduced if only one of the contacts is grounded, which was the case
in our experiments. The plasma wave velocity s, under these conditions is higher than vF
and given as:
s =
vFÇ
1−

αg
1+αg
‹2 (5.22)
Here αg is the gate-voltage dependent parameter given by the following equation.
αg = 4
s
e3(Vt g − VCN P)
2 ∗ Ct għh2vF (5.23)
These plasmon waves may then reflect from the domain boundaries of monolayer
graphene grown via CVD [186]. In the expression for plasma frequency ωp = sq, the
wavevector q can be replaced as (2m − 1)pi/2r, where r is the radial dimension of the
crystal and m= 1,2, 3....
Assuming an r = 3.4 µm (which corresponds to an area of 36 µm2), the plasmon
frequency ωp is calculated as 1.6 THz at Vt g − VCN P = 0.2 V. This agrees with the experi-
mentally observed resonance in figure 5.32. The plasmon wave velocity s at Vt g = 0.2 V is
calcuated as 4.3 x 108 cm/s. Ideally, the plasmon frequency should shift with electric field,
which is not so in our experiments. Perhaps, at high carrier concentration the plasmon
wave velocity experiences saturation with gate electric field [187].
Perhaps, the absorption and emission at different charge polarity can be explained
within the framework of plasmon scattering. Over the crystalline domain boundary, we
may assume short-circuit conditions at the end closer to the source and open-circuit condi-
tions on the diametrically opposite end closer to the drain. Under these conditions, when
holes are the predominant charge carriers, the current flows towards the source, i.e. to-
wards low impedance. This may cause the plasmon wave to be reflected back with the
opposite polarity thereby destructively interfering with the incident surface plasmon. In
the case where electrons are the predominant charge carriers, the current flows towards
the drain, i.e. towards high impedance. This may cause the plasmon wave to reflect back,
but with same polarity leading to constructive interference and hence absorption.
We therefore observe that in the case where the surface current is carried by holes,
253
5. GRAPHENE BASED THZ OPTICS
the scattering of plasmons due to domain boundaries leads to energy absorption, whereas,
when the surface current in graphene is carried by electrons, the scattering of plasmons
leads to emission. It is probable that the true absorption spectrum and hence the plasmon
resonances, is a superposition of multiple effects, i.e. reflects the statistical distribution of
domains, kinks and wrinkles in CVD graphene.
Interestingly, intrinsic plasmonic resonances have been observed in unpatterned graphene
epitaxially grown on SiC [188], with a background carrier concentration, n ∼ 8 x 1012
cm−2 at no bias. The origin of these resonances corroborated microscopically has been
attributed to uniform defects in epitaxial graphene. While it is known that the applica-
tion of magnetic field splits these intrinsic plasmons into bulk and edge states, there have
been no studies on the effect of charge disorder induced externally on the appearance
of these plasmonic resonance modes. We observe similar plasmonic resonances in CVD
grown, monolayer graphene monolayer. Near-field scanning microscopy of plasmons scat-
tered in graphene at THz frequencies may provide additional insight to the origin of these
resonances [156].
For completeness, we present the THz image of the top-gated graphene device. While
this technique is not suitable for imaging plasmons, it gives us an insight into the conduc-
tivity distribution within the device.
5.7.8 THz imaging of top-gated graphene
Figure 5.34 shows the THz image of a top-gated graphene device measured at a spatial
resolution of 100 µm. The THz image was obtained in the reflection mode by exciting
graphene with a broadband THz source, incident at 30o to the surface normal. In the
image shown, the source is grounded, the drain is left unbiased and the voltage on gate
is equal to VCN P . The incident THz radiation is reflected strongly from regions of highest
conductivity, which in this case are the Ti/Au source, drain and gate contacts. The THz
reflectance E1R in these highly conducting regions is ∼ 0.7 to 0.8. The dashed lines between
the source and drain enclose the area covered by graphene. The position of the drop casted
ionic gel can be seen in the THz image as it takes a coffee ring shape at the center [189].
The THz reflection E1R corresponding to the ionic gel is ∼ 0.5 to 0.6.
On careful inspection of the image, we note that the coplanar gate contact is not uni-
formly conducting. Perhaps, this is a ramfication of the floating potential on the drain,
which adds to the hypothesis that an asymmetric potential distribution across the device
contributes partly to the spectroscopic signatures that we observe.
254
5.8 Graphene ribbons as THz filters
Figure 5.34: THz reflection image of CVD graphene gated at charge neutral condition
5.7.9 Summary
We have demonstrated tunable THz transmission through monolayer graphene grown via
chemical vapour deposition. The transmission is tunable under conditions of back-gating
as well as top-gating. Using a top ionic gate, we obtain a high carrier concentration, n
∼ 1013 cm−2 in graphene and can access both types of carriers. THz time domain spec-
troscopy of such gated monolayer graphene shows plasmonic resonance features around
1.6 THz, which appear as absorption peaks when the graphene is electrostatically p-doped
and change to enhanced transmission when the graphene is n-doped. Superimposed on the
Drude-like frequency response of graphene, these resonance features are related to the in-
herent poly-crystallinity of CVD graphene. An understanding of these features is necessary
for the development of future THz optical elements based on CVD graphene
5.8 Graphene ribbons as THz filters
While unpatterned graphene exhibits a THz response that is dominated by intrinsic plas-
mons, it is theoretically possible to engineer the plasmonic response at THz frequencies by
lithographically defining them into ribbons [71, 190]. Demonstrated first for frequencies
between 3-6 THz by Ju et. al, microribbons in graphene can become particularly exciting
if demonstrated below 3 THz for use as tunable filters with THz sources such as QCLs. The
relationship between the plasmonic frequency and the width w of the ribbon is given by
the following expression.
ωp ∝ n1/4w−1/2 (5.24)
255
5. GRAPHENE BASED THZ OPTICS
Our initial attempts at defining microribbons in graphene were focussed towards de-
signing a THz filter with a plasmonic frequency of 2 THz using the scaling law mentioned
in eqn. 5.24. Assuming the same carrier concentration as Ju et. al (n = 1.5 x 1013 cm−2),
ribbons with width w ∼ 9 µm were patterned with a pitch of 1:1, using photolithographic
techniques over an area equal to the THz beam spot size. However, since the design fre-
quency approached the tail end of the detection bandwidth (see figure 5.21), we failed to
observe a reasonable THz response from these ribbons. Subsequent designs were focussed
at a plasmon frequency centered around ωp = 0.6 THz (width w ∼ 100 µm, n = 1.5 x
1013 cm−2) where the signal to noise ratio of the THz TDS set-up is maximum.
Figure 5.35 shows the absorption spectrum of 100 µm wide graphene ribbons, as a
function of top-gate bias, when the incident THz beam is polarized parallel and perpendic-
ular to the length of the ribbon. The absorption spectrum is derived from the secondary
transmitted pulse and is referred to the charge neutrality point of the ribbons. We observe
that the absorption is tunable with gate voltage. However, the normalized absorption of
the microribbons (2 to 4 %) is less than the normalized absorption of large-area graphene
(10 %), see figure 5.32. This is to be expected since the effective area of graphene is
reduced by nearly half after patterning into ribbons.
Figure 5.35: Normalized absorption spectra of graphene microribbons when the incident
THz radiation is polarized (a) parallel to the length of the ribbons (b) perpendicular to the
length of the ribbons. The absorption spectrum, shown for hole transport, is derived from
secondary transmitted pulse and is referred to VCN P
256
5.9 THz modulator
When the THz radiation is polarized parallel to the length of the ribbons, the absorption
increases with frequency, which is consistent with the Drude like response of the optical
conductivity. We observe weak signatures of an absorption peak at ∼ 1.6 THz, which sug-
gests a contribution due to intrinsic plasmons. From the scaling law (eqn. 5.24), we expect
a resonance feature at ωp = 0.6 THz when the THz radiation is polarized perpendicular to
the length of the 100 µm wide ribbons. The absence of a resonance feature at the design
frequency ωp = 0.6 THz inspite of reaching the required carrier concentration (n = 1.5
x 1013 cm−2 at Vt g - VCN P = -0.8 V), suggests that either the resonance is destroyed by
scattering due to the intrinsic polycrystallinity of graphene or the resonance suffers broad-
ening at longer wavelengths [186, 190]. This implies that scaling the plasmon frequency
of monolayer CVD graphene to longer wavelengths may not be ideal. A better understand-
ing of the resonance features of patterned graphene ribbons can be obtained from FTIR
spectroscopy which offers higher resolution and bandwidth in terms of frequency [71].
However, this was beyond the scope and time limit of this work.
5.9 THz modulator
In this section, we demonstrate a graphene based THz modulator operating at room tem-
perature, which can be used to control the transmission from a QCL emitting at 2.0 THz.
Here, graphene has been used in back-gated geometry. Previous means of modulating
the transmission of a THz QCL relied on direct modulation of the bias voltage by applica-
tion of an RF signal (up to a few GHz) together with a dc bias [191]. External THz QCL
modulators based on electrically-driven active metamaterial structures fabricated on semi-
insulating GaAs have been reported in the past, with modulation depth∼ 60 % [192]. This
work builds upon the work of Rodriguez et. al, who demonstrated a large-area graphene-
based modulator for carrier frequencies ∼ 570-630 GHz [193]. Theoretical predictions
suggest that it is possible to achieve a modulation depth > 90 % by exploiting the gate
tunability of THz transmission through graphene [194].
Rodriguez’s work on broadband modulation relied on Schottky-diode detection of the
amplitude modulated carrier frequency signal from ’high’ to ’low’ by switching the applied
gate voltage on graphene over millisecond time scales [193]. However, in the frequency
range of 1-5 THz, switching of amplitude-modulated THz signals is difficult to detect in
time, primarily because of the slow response (4-300 Hz) of the available THz detectors.
In this work we record the modulated THz transmission by sweeping the frequency of
applied gate voltage ( fmod) on graphene while applying a second, slow modulation to the
QCL ( fre f ). The response is detected by measuring the average THz power as Pavg . The
257
5. GRAPHENE BASED THZ OPTICS
average power can be expressed in terms of the peak power (Ppeak) as follows:
Pavg = Ppeak( fmod). fre f / fmod (5.25)
Note that the peak power measured by the detector is a function of the modulation
frequency on graphene. In eqn. 5.25, this relationship is expressed as Ppeak( fmod). In-
dependent of this dependency, the modulation frequency determines the average power
transmitted through graphene and measured by the detector. The choice of the modula-
tion frequency determines the time over which the THz power is measured.
5.9.1 Experimental setup
The experimental setup used for indirectly modulating the THz transmission from a QCL
using back-gated graphene is shown in figure 5.36. The THz QCL operating at 2.0 THz
(figure 5.37) was grown epitaxially on a semi-insulating GaAs substrate in a bound to
continuum design [195]. A 250 µm x 3mm single plasmon waveguide laser was mounted
on to a copper block and operated at 4.2 K. In pulsed mode, the QCL operates up to a
maximum temperature of 70 K (figure 5.37).
Figure 5.36: Experimental Set-Up. Pulser 1 provides continous pulses at 60 Hz, such
that the THz QCL operates at Jmax in a quasi-cw mode. Pulser 2 provides a burst of fast
modulation to the back-gate of graphene.
The QCL is biased at Jmax ∼ 150 A/cm2 and is operated in a quasi-cw mode by elec-
trically pulsing at fre f = 60 Hz. The duty cycle of this slow pulse is kept at 15%. This
258
5.9 THz modulator
Figure 5.37: (a) Spectral characteristics of THz QCL measured using Bruker IFS/66v FTIR
spectrometer at Jmax ∼ 150 A/cm2 in the pulsed mode (b) Pulsed LIV of THz QCL at
varying temperatures, showing a lasing threshold of 110 A/cm2 at 4.2 K
envelope frequency ( fre f ) is applied because most THz detectors have optimal senstivity
when operating at low frequencies. For completness, we have carried out the experiments
with two different THz detectors: a Golay cell and a composite Si bolometer cryogenically
cooled at 4.2 K. To be consistent we choose fre f = 60 Hz for both measurements. While
the responsivity of a bolometer is flat for fre f < 300 Hz, the responsivity of a Golay cell
falls by a factor of around seven as the fre f is increased from 15 Hz to 60 Hz. However, the
amount of signal detected on the lock-in amplifier for both is significantly above the noise
floor of our measurement set up (SNRdB > 25). The radiation emitted by the QCL source
is collected by a pair of F2 parabolic mirrors and is focussed on to the THz detector. The
graphene based FET is placed directly in front of the THz detector at a maximum spacing
of 5 mm. The source contact on graphene is grounded and the drain contact is left unbi-
ased. The average THz power is modulated by changing the pulse frequency applied to the
back-gate of graphene. The back-gate is pulsed from 0 to +50 V at a frequency which we
denote as fmod . The duty cycle of this pulse is the same as the duty cycle of the reference
pulse, i.e. 15 %. The modulated signal is measured as the average THz power Pavg carried
in the 60 Hz envelope on the lock-in amplifier. For better alignment, the beam spot can be
further reduced by placing an aperture in front of graphene.
5.9.2 Results from indirect modulation of a THz QCL by graphene
Figure 5.38 shows the measured THz response as a function of fmod , detected by a Go-
lay cell. The modulation frequency, fmod is swept from 200 Hz to 80 KHz. The average
259
5. GRAPHENE BASED THZ OPTICS
detected THz power Pavg is normalized to the average detected power at the initial mod-
ulation frequency fmod = 200 Hz. The signal is averaged over 10 scans. We observe that
the average THz power (Pavg) detected by the Golay first falls to a certain minimum at
fmod = fcuto f f , the cutoff frequency. This is where the drain-source current, Ids, given by
eqn. 5.13 (inset, figure 5.38) remains constant, which implies that the peak THz power
Ppeak transmitted through graphene remains constant. However, the time period at which
the peak power is transmitted changes due to an inverse dependence upon fmod . The THz
detector measures the average power, which is directly proportional to the Ppeak/ fmod , see
eqn. 5.25.
Figure 5.38: Average THz power measured as a function of fmod normalized with power
at fmod = 200 Hz using a Golay cell at room temperature. The inset shows drain-source
current Ids measured as a function of gate modulation at Vds = 1 mV.
The drain-source current starts to roll off at f3dB = 10 KHz. This roll off is explained by
the capacitance of the large area SiO2 gate, at which point the carrier density n, induced
by the gate falls. Therefore, at this point, the transmitted THz power through graphene,
i.e. Ppeak increases. Note that eqns. 5.17 and 5.13 suggest that the transmitted power is
proportional to I−2ds . The average THz power Pavg detected by the Golay starts to increase
in spite of the inverse dependence on frequency. For an fmod = 80 KHz, the depth of
modulation is observed to be ∼ 15 %.
Figure 5.39 shows the amplitude modulated THz signal as function of time. The av-
erage THz power is switched from a ’high’ to ’low’ state by switching the frequency of the
back-gate voltage on graphene from high ( fmod > fcuto f f ) to low ( fmod < fcuto f f ). The
260
5.9 THz modulator
back-gate on graphene is switched from Vbg =0 V, fmod = 0 to a 20 sec burst of high fre-
quency pulses (Vbg = 0 to +50 V, fmod = 20 KHz). In the first 20 secs, fmod = 0, the THz
transmission through graphene corresponds to the free-carrier absorption due to holes,
which we call as the ’initial’ state. There is a general drift in the average power detected
over a period of time, which can be attributed to the stability of the QCL source when op-
erated in a quasi-continous mode. It is likely that over a period of time, the Joule heating
of the QCL source causes a drop in the power output. This behaviour is recorded in the
first 20 secs, by Golay (figure 5.39(a)) as well as the bolometer (figure 5.39(b)).
Figure 5.39: Amplitude modulated THz signal recorded with time on (a) golay cell at room
temperature (b) composite Si bolometer cryogenically cooled to 4.2 K. The last pane (c)
shows the switching of fmod from 0 Hz to a 20 sec burst of 20 KHz pulses.
In the next 20 secs fmod = 20 KHz and the transmission increases by ∼ 7%, corre-
sponding to the normalized Pavg( fmod = 20 KHz) (figure 5.38). However, we observe that
261
5. GRAPHENE BASED THZ OPTICS
the measured signal is limited by the response time of the Golay (figure 5.39(a)). When
the same experiment is repeated using a bolometer (figure 5.39(b)), we note an overshoot
which is followed by a decay of the THz signal. The difference in the transient behav-
ior between the two detectors arises possibly because of a difference in the measurement
method. In the case of the Golay, the signal is measured across a capacitor, whereas in the
case of the bolometer, the signal is measured across a resistor 1.
When the back-gate is switched back to Vbg =0 V, fmod = 0, the transmission through
graphene is in the ’low’ state. Ideally, the transmission should be constant when the Fermi
level is held constant. However, the Golay responds to the change in the detected signal
slower than the bolometer. We note that after the Fermi level is swept to and fro, at a high
frequency across the valence band, the transmission in the ‘low’ state attempts to reach the
‘initial ’ state. While we do not yet understand this behaviour, a possible explanation may
come from the inherent hysteresis in the graphene device due to surface charge traps at
the oxide/graphene interface or due to intraband carrier relaxation within graphene itself.
The cycle shown in figure 5.39 is repeated multiple times to generate an amplitude
modulated THz signal from a QCL source. The measurements shown in figure 5.39 describe
the results of THz modulation when the entire area of graphene is exposed to the incident
beam, i.e. the fluence is maximum. The absolute change in transmission remains nominally
identical as the fluence of the incident beam is decreased from 360 mW/cm2 (iris diameter
= 5 mm) to 32 mW/cm2 (iris diameter = 1.5 mm). Figure 5.40 shows that at a fluence of
290 mW/cm2 (iris diameter = 4.5 mm) the difference in the transmitted power between
the ‘high = 0.057’ state and the ‘low = 0.050’ state is 0.007 a.u. At a fluence of 32
mW/cm2 (iris diameter = 1.5 mm), the difference in the transmitted power between the
‘high = 0.01’ state and the ‘low = 0.006’ state is 0.004 a.u. Since the depth of modulation is
expressed as the ratio of change in transmission to the original tranmission, the modulation
efficiency increases with decrease in fluence 2.
With improvements in the modulation efficiency and the speed of modulation, graphene
shows considerable promise as a THz modulator. These improvements are discussed in sec-
tion 6.2.6.
1Note that this explanation for the difference in detector response has been amended after thesis review
and publication of the manuscript [183].
2Note that this is correction to the manuscript [183], where we expressed that instead of the absolute
change in tranmission, the depth of modulation is independent of the fluence.
262
5.10 Summary and conclusions
Figure 5.40: Amplitude modulated THz signal recorded with time on the golay cell at room
temperature as a function of fluence.
5.9.3 Summary
In summary, we show that the THz power from a QCL can be modulated by applying a
gate pulse to large area CVD grown graphene. The average THz power measured on the
detector increases with increase in pulsing frequency fmod after a certain cutoff. The fcuto f f
which is function of the intrinsic RC time constant corresponds to the roll off in graphene
conductance. The THz signal can be amplitude modulated from a ‘low’ to a ‘high’ state,
by switching the backgate voltage on graphene from 0 V DC (or fmod < fcuto f f ) to a high
frequency pulse, with fmod > fcuto f f .
5.10 Summary and conclusions
This chapter focussed on graphene, a material that can operate at room temperature for ap-
plications in THz optics. We showed that THz transmission through wafer scale graphene
grown by chemical vapour deposition (CVD) can be tuned by applying an external bias.
One of the problems with CVD graphene is the inherent polycrystalline nature of the ma-
terial due to random nucleation. We demonstrate that this can effect the spectroscopic
response of graphene and present additional challenges in the design of THz filters. Fi-
nally, we were able to exploit the gate tunability properties of graphene to demonstrate a
graphene based THz modulator for operation with THz QCLs.
263
5. GRAPHENE BASED THZ OPTICS
264
"If I have seen further it is by standing
on ye shouders of Giants."
Isaac Newton
Chapter 6
Conclusions and future work
6.1 Conclusions
This thesis has taken a two pronged approach to close the THz gap. In the first approach,
we looked at improving the operating temperature of THz QCLs. Towards this end, we
developed the building blocks required to implement lateral confinement in THz QCLs. In
the second approach, we investigated the properties of a novel two dimensional material
at THz frequencies and demonstrated it’s applicability to THz optics.
Lateral confinement in THz QCLs through a top-down nanofabrication has been an
elusive dream. The challenge in realizing a top-down array of nanopillars that is electrically
and structurally robust is the least of the problems, as shown in this work. The bigger
challenge lies in the understanding of the radiative transitions within such a structure and
the measurement of THz emission from these structures.
The problem of lateral confinement in THz QCLs was broken into three parts. In the
first part, that is Chapter 2, we attempted to gain an understanding of the electrostatic po-
tential within a lithographically defined nanopillar. Using a simple quantum well structure
as the basis, we found that the electrostatic potential was dependent upon many processing
and growth parameters. Following the discussion on lateral potential, we realized that the
task of fabricating a nanopillar structure was not restricted to obtaining sub-200 nm radial
dimensions. Instead, the challenge of obtaining simultaneous emission from intersubband
states rested on obtaining a few correct radial dimensions. These dimensions exist as win-
dows where light is allowed, and depend upon the active region used. Along with these
constraints, observation of simultaneous emission from such a structure depends upon the
optical loss within the device. We found, almost counter-intutively that the highest packing
density is not the criteria for lowest losses within an array of nanopillars. To obtain the
265
6. CONCLUSIONS AND FUTURE WORK
lowest loss, one needs to implement a suspended top contact to the nanopillar array.
The second part of the problem is demonstrating that such a nanopillar array can be
fabricated. This was dicussed in Chapter 3. The processing routes were validated on sim-
ple resonant tunneling diode structures (RTDs). While our intent was not to improve upon
the current carrying capabilities of zero-dimensional RTDs but to test our processing, we
succeeded in contacting a 500 x 500 array of nanopillars. This allowed vertical transport
through thousands of nanopillars in parallel, pushing the current to four orders of magni-
tude higher than what has been achieved in zero-dimensional RTDs so far. At higher bias,
i.e. when the device is in resonance with the excited states, the array can support a current
of up to 2000-4000 µA. This result improves our confidence in nanopillars processed from
THz QCLs which require high current densities for simultaneous emission.
Implementing nanopillars in THz QCLs requires active regions that are ultrathin. While
this problem has been solved in the literature, it required growth and characterization of
suitable active regions inhouse. Chapter 4 discusses various active region designs and
wafers that were tested for emission in the hope of implementing the top-down approach
for lateral confinement. While challenges in growth and choice of waveguide affected
many of the results discussed in Chapter 4, we were able to demonstrate lasing in an ultra-
thin THz QCl, derived from a full-stack. Since this laser was derived by chemical etching, it
was not found suitable for the extremely surface sensitive processing of nanopillars. How-
ever, by building upon the steps developed in this work towards lateral confinement, we
hope that some of the theories and processing routes can be tested on ultrathin THz QCLs
in future.
Chapter 5 of this thesis looks to graphene for tangible THz optical elements that can
operate at room temperature. Graphene grown by chemical vapour deposition (CVD) tech-
niques was studied using THz time domain spectroscopy. Our measurements of gated-
graphene devices suggested that polycrystallinity in CVD graphene influences the propa-
gation of THz plasmons. This is an important finding to realize, since many applications
of graphene such as THz filters and resonators involve THz plasmons. In the final section,
we exploit the gate tunable properties of graphene to modulate a THz QCL.
In the next section, we present ideas that have originated from this work, that can
provide opportunities for development in future.
266
6.2 Opportunities for future
6.2 Opportunities for future
6.2.1 Optical spectroscopy of nanopillars processed from THz QCLs
Chapters 2-4 have been driven by the quest to achieve electroluminescence from nanopil-
lars etched in ultrathin THz quantum cascade lasers based on 3-4 quantum well (per pe-
riod) designs. In future, the design of an epitaxial structure for nanopillars in THz QCLs
should be independently tailored and not be based on a working ultrathin quantum cas-
cade laser alone. The design of this active region must include the solution to the radiative
states under the influence of a lateral potential. Feedback derived from techniques such
as ultrafast THz spectroscopy [196] and microphotoluminescence [197] should be given
to the epitaxial growth of these designs in order to identify and understand the transition
states, electron relaxation mechanisms and associated relaxation times.
6.2.2 Waveguides for quantum-dot cascade structures
In a THz QCL, a Fabry Pèrot cavity formed by parallelly cleaved laser facets supports the
optical mode. In this case, laser feedback is provided by the two atomically smooth facets
which act as mirrors at the end of the gain medium. The fundamental mode in the laser
spectrum is then determined by the length of the cavity.
The difficulty of achieving electroluminescence in quantum-dot cascade structures has
meant that there has been no research done in the area of identifying a suitable waveguide
for a future quantum-dot cascade based amplifier or a laser. The lack of research in this
direction extends to mid infrared frequencies as well.
This work (Chapter 4) relied on an edge emitter geometry, to observe electrolumines-
cence. There have been suggestions on obtaining laser feedback in quantum-dot cascade
structures fabricated by the top-down approach. This involves etching external mirrors at
the same time as the nanopillars 1. However, planarization of nanopillars, which depends
upon the topography of the underlying structure is likely to be a problem in this design. It
may also prove to be difficult to cleave the mirrors with an accuracy of ≤ 10 µm, which
is the least count of typical scribing and cleaving equipment. Realistic estimates of mirror
thickness lie in the range of 200-500 µm, depending upon experimental skill. This would
add an additional; variable parameter in characterizing the power from these devices.
In the optical mask designed for edge emission in nanopillars we made a provision
to add surface gratings over the top contact. Separated by a period ∧ = λGaAs, where
1Private communication with Prof. A. Tredicucci
267
6. CONCLUSIONS AND FUTURE WORK
λGaAs = 28 µm in GaAs for 3 THz free space radiation, these slits enable surface out-
coupling of the radiation [198, 199]. As described in Chapter 3, the effective refractive
index of the nanopillar array is likely to depend upon the physical radial dimension of the
nanopillars, the pitch of the nanopillar array and the planarizing material. Therefore, the
period ∧ would be governed by the intented quantum-dot cascade structure. The slit width
(typically, 2-3 µm in our design) should be kept larger than the pitch.
Untested on actual nanopillars in THz QCL, these surface gratings also serve as air-gap
windows for lateral etch of the planarizing material which allow for suspended top contact
and hence lower losses. Figure 6.1 shows the proposed schematic diagram for fabricating
suspended top-contact in a surface emitting quantum-dot cascade laser. Note that the
windows opened in the photoresist, subsequent to top metal evaporation are orthogonal
to the emitter gratings, so as to achieve a criss-cross pattern. This minimizes the area open
to the etchant during insulation removal.
Figure 6.1: Process schematic of suspended top contact in surface emitting quantum-dot
cascade laser, showing (a) planarized nanopillar array after top contact evaporation, with
surface gratings (b) definition of air-gap windows in the photoresist (c) equirate lateral
and vertical etch of the insulation through the area shown in dashed lines (d) laterally
etched insulation after removal of the photoresist in acetone. The dimensions are not to
scale.
Note that these suggestions are only speculative and the waveguide design of quantum-
268
6.2 Opportunities for future
dot cascade lasers should take the polarization of the emitted radiation into account, which
in itself is a subject of dedicated research.
6.2.3 Lateral confinement in THz QCLs - Nanofences
The current state of art THz QCLs are based on a cascade of quantum wells and bar-
riers where electrons are confined in the growth direction but free in the other two di-
mensions. This means that traditionally, electrons have in-plane momentum parallel to
the direction of epitaxial growth. This work explored the feasibility of achieving zero-
dimensional confinement in THz QCLs. An intermediate step between the present day THz
QCLs and quantum-dot THz QCLs would be the demonstration of lateral confinement in
one-dimensional THz quantum cascade structures. We coin the term ‘nanofences’ to de-
scribe the design of these intermediary THz quantum cascade structures. Instead of etch-
ing pillars, this proposal is based on etching holes. As described later in this section, our
proposal (nanofences) circumvents the problems of planarization and top-contact faced
during the fabrication of nanopillars.
Just as the history of top-down zero-dimensional structures can be traced back to reso-
nant tunneling diodes [89], the first demonstration of one-dimensional laterally confined
electrons can be traced back to resonant tunneling devices [200]. Shown in figure 6.2, is
the device schematic and the corresponding scanning electron microscope image of a grid
gate which imposes a periodic potential that confines the electrons laterally, governed by
the width of the metal lines. In figure 6.2, this width is t = 50 nm. In work done by Allee
et al. [200], lateral confinement is achieved electrostatically by applying a suitable bias on
the patterned top gate.
While this work has inspired the design of nanofences, it can not be adapted in a
straight forward manner to the THz QCLs. In fact, lateral confinement in THz QCLs is
theoretically impossible to obtain by electrostatic means. This is because the depletion of
charge carriers which needs to occur throughout the height of THz QCLs (3 µm, in ultra-
thin QCLs) can not extend beyond a few nm in the growth direction, depending upon the
doping of epitaxial layers. The nanofence proposal is based on using the metal grid pattern
shown in figure 6.2 as a mask to etch holes in an ultra-thin THz QCL. This etch mask also
forms the self-aligned top contact to the active region.
Depending upon the grid dimensions Lx and L y , the periodic lateral potential can be
classified into two main categories. A pseudo-zero dimensional potential may occur for
metal width t ≤ 300 nm (similar to nanopillar radius, Rp ≤ 150 nm) and Lx = L y ≤
2t. Under this condition, the in-plane moment of electrons would be constrained in both
269
6. CONCLUSIONS AND FUTURE WORK
Figure 6.2: First demonstration of resonant tunneling of one-dimensional electrons in a
3-dimensional array of confined potential. (a) Device schematic of laterally confined res-
onant tunneling field effect transistor (b) Scanning electron micrograph of the top gate.
Adapted from [200]
x an y directions. If t ≤ 300 nm and Lx (and/or L y)  2t, the electrons would have
in-plane momentum in the x (and/or y) direction, which is akin to having an array of
one-dimension electron wires. Under this condition, the electrons at the intersection of the
grid would still be confined in both x and y directions.
The lower limit to t is determined by e-beam dose energy, exposure time, resist thick-
ness and polarity which in our prelimenary work at Cavendish is set at 100 nm. Further-
more, as t approaches 100 nm, a high pattern density, Lx(or L y)−t ≈ t, is typically difficult
to achieve in practice as it causes e-beam proximity effects resulting in significant pattern
distortion and incomplete lift-off of metal. This is shown in figure 6.3(a), where an over
exposure of the e-beam causes reduction in t from 100 nm to 43.8 nm and increase in
pitch, Lx(or L y)− t from 100 nm to 130 nm. Incomplete lift-off is not a problem however
for larger values of t and Lx(or L y)− t as shown in figure 6.3(b), where t shrinks from
300 nm to 260 nm and Lx(or L y)− t increases from 900 nm to 917 nm. Depending upon
whether the defined t is closer to the lower limit ∼ 100 nm or the higher limit ∼ 300 nm,
the periodic potential can be categorized as a weak or strong lateral potential.
It may also be possible to do away with the grid altogether and have a grating structure
instead to obtain pure one-dimensional effects. However, structural stability of a grating
structure for THz QCL emitter structures is likely to be a cause for concern.
Figure 6.4 shows the processing route developed for integrating nanofences with a THz
QCL structure. After the thermocompression bonding and etch stop removal, the first mesa
270
6.2 Opportunities for future
Figure 6.3: Scanning electron micrograph of nanofences etch mask showing (a) incomplete
metal lift off at a lithograhically defined t = 100 nm and Lx − t = 100 nm, for a grid
simulating a strong pseudo-zero dimensional potential (b) metal mask at a lithograhically
defined t = 300 nm and Lx − t = 900 nm, for a grid simulating weak one dimensional
potential.
is chemically etched (figure 6.4(a)). In the second step, the n+ GaAs layer is removed by
dry etching around the second mesa. Note that it is the second mesa that forms the active
region, which finally takes part in the electron transport. Once the second mesa and the
first mesa are isolated, the first mesa naturally acts as the support for the top contact. This
is shown in figure 6.4(c) where the conformally deposited SiO2 layer forms the insulation
for the device. This layer can be thin (∼ 100 nm). The advantage of using the first mesa
for support, insulated by PECVD oxide over the planarization process discussed in Chapter
4 for nanopillars is clear. There is no necessity to spin-coat an insulator.
The next step (figure 6.4(d-e)) is the key to the observation of lateral confinement
effects in nanofences. This step involves opening up of windows in the oxide layer to
enable e-beam lithography of the nanofences (top contact windows) and connection to the
bottom contact (bottom contact windows). The e-beam lithography defines the etch mask
(figure 6.4(f-g)), which at the same time forms a self aligned top contact for the nanofences
(figure 6.4(h)). The window etched into the oxide (top contact windows) must be aligned
to a very high precision with the second mesa. If the window dimensions are larger than
the second mesa dimensions, it is likely that e-beam lithography would result in shorting
of the device. If however, the window dimensions are smaller than the mesa dimension,
say for example by 5 µm on either sides, then these two 5 µm channels would dominate
transport and optical profile over that of the nanofences. To observe lateral confinement
effects, it is therefore necessary to reduce the overlap of the oxide with the second mesa,
and at the same time maintain the electrical stability of the structure.
Theoretically, it is possible to optically align the windows in the oxide layer with the
271
6. CONCLUSIONS AND FUTURE WORK
Figure 6.4: Processing route for nanofences showing (a) wet etch of first mesa (b) reactive
ion etch of second mesa i.e. removal of the n+ GaAs layer (c) conformal deposition of SiO2
via. PECVD (d) optical lithography of windows in the oxide layer (e) device schematic after
removal of oxide via dry etching (f) PMMA spin-coated over device (g) e-beam lithography
of nanofences etch mask and formation of self-aligned top contact (h) reactive ion etching
of active region and formation of natural air gap between laterally confined regions.
272
6.2 Opportunities for future
second mesa, within an accuracy of 200 nm using Moire techniques [201]. Based on an
observation of diffraction patterns (fringes) due to misalignment between a circular grating
etched on the wafer and a grating on the mask plate, we have been able to achieve a high
degree of alignment precision (figure 6.5).
Figure 6.5: Moire patterns showing (a) computer generated image of perfectly circular
fringes (1 and 2) on alignment (b) optical microscope image of fringes indicating slight
misalignment (≤ 0.5 µm) in the direction of the arrow (c) optical microscope image of
fringes indicating misalignment by 2 µm in the direction shown.
While the nanofences approach has it’s advantages over the nanopillars approach to
lateral confinement, it comes at a cost of different but equally important processing issues.
As mentioned before, e-beam lithography and optical alignmnent are likely to be bottle
neck steps in development of the nanofences architecture.
6.2.4 Characterization of graphene domain boundaries by THz TDS
In Chapter 4, we suggested that the absorption features at 1.6 THz, occured as a conse-
quence of scattering by domain boundaries. Understanding of the exact mechanism behind
the absorption features requires a systematic study of the THz absorption with dependence
on the underlying microscopic structure of the grain boundaries. This can be investigated
by performing THz TDS of CVD graphene grown at different growth conditions [7].
Other parameters to explore the origin of the absorption features include a systematic
study of THz absorption as a function of device geometry (gate length, distance of gate
from graphene, thickness and overlap of ionic gel) and device bias conditions (symmetric
and asymmetric).
273
6. CONCLUSIONS AND FUTURE WORK
6.2.5 THz filters from graphene anti-dots
Soon after the demonstration of plasmonic resonances in graphene ribbons[71], disks[184]
and rings[202], Nikitin et. al predicted another novel geometery for plasmon scattering.
This geometry used antidots (or holes), in a sheet of graphene instead of isolated disks or
rings [203]. The advantage of this geometry over the other two dimensional gratings is
that it provides electrical continuity in addition to periodicity.
Over the course of this work, we designed an optical mask to pattern antidot arrays
in graphene based on Nikitin et. al’s proposal. The plasmonic frequency corresponding
to these arrays lies in the range of 0.46 to 2 THz. Although, these gratings were not
implemented owing to a lack of time, they can tested in future for suitability as THz filters.
This assumes that polycrystallinity is not a limitation and large area single crystal sheets of
graphene can be obtained [204].
6.2.6 THz amplitude modulators
The key performance indicators of a modulator are modulation speed and modulation
depth. The modulation speed can be enhanced by reducing the footprint of the graphene
sheet. Modulation speeds as high as GHz can be obtained by using an array of high fre-
quency graphene FETs. [205].
Theoretically, the modulation depth can be increased by employing multiple graphene
layers stacked on top of each other but separated by an insulator [184, 193]. In this type
of device architecture when one layer is biased at the Dirac point and the other layer is
biased away from the Dirac point, it is possible to completely suppress THz transmission
through graphene.
It is also possible to achieve a narrow band metamaterial-based modulator in graphene
[71, 184]. On tuning the graphene in and out of plasmon resonance by changing the
carrier concentration, the transmission can be switched from ‘high ’ to ‘low ’. All these
separate approaches suggest that there still remains an opportunity to optimize and design
a highly efficient, compact graphene based THz modulator.
274
Chapter 7
Appendix A
7.1 Epitaxial structure of RTDs
The epitaxial structure of the two test RTD structures: A1700 and W0021 is given below.
Material Thickness (nm) Doping (cm−3) Mole fraction
GaAs 333 2e18
GaAs 100
AlGaAs 5 0.33
GaAs 10
AlGaAs 5 0.33
GaAs 100
GaAs 333 2e18
Table 7.1: Epitaxial structure of A1700
Material Thickness (nm) Doping (cm−3)
GaAs 1000 2e18
GaAs 100 1e16
GaAs 5
AlAs 2
GaAs 9
AlAs 2
GaAs 5
GaAs 100 1e16
GaAs 500 2e18
Table 7.2: Epitaxial structure of W0021
275
7. APPENDIX A
———————————————————–
276
Chapter 8
Appendix B
Systematic changes in the global or well thickness in the QCL superlattice structure can
directly influence the transport and the frequency of THz emission [5]. The layer thickness,
length of the superlattice period and the interface qualitity of the designed QCL structure
grown by molecular beam epitaxy can be verified using X-Ray diffraction. Below , we
present a brief overview of the X-Ray diffraction (XRD) techique, followed by the rocking
curves (diffraction pattern) for the THz QCLs studied in this thesis. All the rocking curves
shown below were provided by Dr. Harvey E. Beere.
8.1 X-ray diffraction technique
In the X-ray diffraction pattern of a THz QCL, the strongest peak arises from the [004]
reflection of the bulk GaAs crystal. This pattern from the bulk GaAs crystal is modulated
by the diffraction pattern obtained from the artificial supelattice structure.
The superlattice period is much larger than the unit cell spacing. Therefore, the diffrac-
tion pattern exhibits additional closely spaced satellite peaks in the reciprocal space. The
spacing between the satellite peaks is inversely proportional to the length of the period L.
This can be described in a simple formula as:
L =
(Ni − N j)λ
2(sinθi − sinθ j) . (8.1)
Here, λ is the X-ray wavelength and θi , θ j are the angles of diffraction [206]. A
comparison between the observed rocking curve (measured using the PBruker D8 Discover
277
8. APPENDIX B
High Res System) and the simulated curved (on the Philips X’pert software suite) gives an
indication of the wafer quality.
8.2 Rocking curves of THz QCLs
Figures 8.1 to 8.9 show the X-ray diffraction rocking curves measured using the Bruker D8
Discover High Res System of all the THz QCLs presented in this thesis. Measurements made
using the Bruker D8 Discover show a full-width-half-maxima (FWHM) of approximately
35-40 arc.sec for the GaAs substrate peak and approximately 35-50 arc.sec for the satellite
peak, depending on the optics used. The full width half maximum (FWHM) of all the
peaks, i.e. the substrate peak and the satellite peak is limited by the resolution of the
system. Ideally, the substrate peak should show an FWHM of approximately 15-20 arc.sec
and the satellite peak should show an FWHM of approximately 20-30 arc.sec. Therefore,
it is difficult to discern any noticable broadening of the satellite peaks, which would give
insights on inhomogeneity such as flux drift, arising during the epitaxial growth. However,
if there was a big drift, it would be observable in the XRD data measured using the Bruker
system.
Figure 8.1: X-ray diffraction rocking curve from ultrathin 2.7 THz 3 QW Paris design
(V671).
278
8.2 Rocking curves of THz QCLs
Figure 8.2: X-ray diffraction rocking curve from ultrathin 2.7 THz 3 QW Paris design
(V692).
Figure 8.3: X-ray diffraction rocking curve from ultrathin 2.7 THz 3 QW Paris design
(W0798).
279
8. APPENDIX B
Figure 8.4: X-ray diffraction rocking curve from full-stack 3 THz 3 QW Paris design
(V702).
Figure 8.5: X-ray diffraction rocking curve from ultrathin 3 THz 3 QW Paris design
(V703).
280
8.2 Rocking curves of THz QCLs
Figure 8.6: X-ray diffraction rocking curve from full-stack 3 THz 4 QW MIT design (V704).
Figure 8.7: X-ray diffraction rocking curve from ultrathin 3 THz 4 QW MIT design (V705).
281
8. APPENDIX B
Figure 8.8: X-ray diffraction rocking curve from full-stack 3 THz 4 QW ETH design (V674).
Figure 8.9: X-ray diffraction rocking curve from full-stack 3 THz 4 QW ETH design (V696).
282
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