Optoelectronic and photonic
control of single quantum dots
Samuel James Dewhurst
Wolfson College
University of Cambridge
A thesis submitted for the degree of
Doctor of Philosophy
April 2010
Declaration
This thesis describes work carried out in the Semiconductor Physics Group at
the Cavendish Laboratory and the Cambridge Research Laboratory of Toshiba
Research Europe Limited, from October 2006 to March 2010.
This dissertation is the result of my own work and contains nothing which is the
outcome of work done in collaboration except where specically indicated in the
text. It has not been submitted in part, or as a whole, for any degree at this, or
any other, university. It does not exceed 60,000 words.
S. J. Dewhurst, April 2010
i
Publications
Slow-light-enhanced single quantum dot emission in a unidirectional photonic
crystal waveguide
S. J. Dewhurst, D. Granados, D. J. P. Ellis, A. J. Bennett, R. B. Patel, I. Farrer,
D. Anderson, G. A. C. Jones, D. A. Ritchie and A. J. Shields
Applied Physics Letters 96, 031109 (2010).
Oxide-apertured microcavity single-photon-emitting diodes: simultaneous confine-
ment of current and light
D. J. P. Ellis, A. J. Bennett, S. J. Dewhurst, P. Atkinson, C. A. Nicoll, D. A.
Ritchie and A. J. Shields
Journal of Physics: Condensed Matter 20, 454212 (2008).
Cavity-enhanced radiative emission rate in a single-photon-emitting diode oper-
ating at 0.5 GHz
D. J. P. Ellis, A. J. Bennett, S. J. Dewhurst, C. A. Nicoll, D. A. Ritchie and A.
J. Shields
New Journal of Physics 10, 043035 (2008).
Single electron-spin memory with a semiconductor quantum dot
R. J. Young, S. J. Dewhurst, R. M. Stevenson, P. Atkinson, A. J. Bennett, M. B.
Ward, K. Cooper, D. A. Ritchie and A. J. Shields
New Journal of Physics 9, 365 (2007).
ii
Controlling the polarization correlation of photon pairs from a charge-tunable
quantum dot
R. J. Young, S. J. Dewhurst, R. M. Stevenson, A. J. Shields, P. Atkinson, K.
Cooper and D. A. Ritchie
Applied Physics Letters 91, 011114 (2007).
iii
Acknowledgements
This work would not have been possible without the help and expertise of a
number of people.
I’d like to thank Andrew Shields and David Ritchie for their supervision and
guidance throughout my PhD. Thanks must also go to EPSRC and Toshiba
Research Europe Ltd. for financial support.
I have been fortunate enough to work with Daniel Granados for the majority
of my time in Cambridge, and I am very grateful for all his help and advice. I
am also particularly indebted to Robert Young for assistance in the early stages
of this work, David Ellis for his help with processing and Anthony Bennett for
many useful discussions. I’d also like to thank Mark Stevenson, Martin Ward,
Raj Patel and Niklas Skold for their assistance at various stages.
Ian Farrer and Christine Nicoll were responsible for the growth of the samples
in this work, David Anderson performed the electron beam lithography, and Ken
Cooper processed the samples used in Chapter 4, my thanks go to all of you.
Warmest thanks go to everyone at Toshiba, the Cavendish and Wolfson Col-
lege who I have had the pleasure to live and work with over the last few years.
Finally, I’d like to thank my parents for their support and encouragement over
the years.
iv
Summary
The area of quantum information promises to deliver a range of new technologies
in the fields of quantum computing and quantum communication. Devices based
on semiconductor quantum dots hold great potential for the practical realisation
of many of the components required in the proposed schemes.
This thesis describes the development of several quantum dot devices. By inte-
grating a quantum dot into a p-i-n diode, it was possible to control the dominant
emission lines in its photoluminescence spectrum and to maximise the degree of
polarisation correlation between the two photons emitted in the biexciton decay.
With the same device under a magnetic field, a digital memory was demonstrated.
The polarisation information of a single photon was stored as the spin of an elec-
tron inside the quantum dot, and was deterministically recovered some time later
by the application of an electrical trigger.
A fabrication process was developed in order to produce high quality two
dimensional slab photonic crystals operating with a photonic band gap at ∼ 900
nm. By placing a quantum dot into an appropriately designed H1 photonic crystal
cavity, strong coupling was achieved between the dot and the monopole mode of
the cavity. The vacuum Rabi splitting was found to be constant for all linear
polarisations due to the unpolarised nature of the far-field of the mode.
Finally, a new kind of cavity based on photonic crystal waveguides was de-
veloped. A Purcell enhancement of the in-plane spontaneous emission from a
quantum dot coupled to a unidirectional photonic crystal waveguide was demon-
strated.
v
Contents
Declaration i
Publications ii
Acknowledgements iv
Summary v
1 Introduction 1
2 Background and Theory 3
2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 Semiconductor quantum dots . . . . . . . . . . . . . . . . . . . . 3
2.2.1 Optical properties of single semiconductor quantum dots . 5
2.2.2 Quantum dots as single photon sources . . . . . . . . . . . 7
2.2.3 Fine structure . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2.4 Quantum dots as sources of entangled photon pairs . . . . 8
2.2.5 Applied magnetic field . . . . . . . . . . . . . . . . . . . . 10
2.3 Quantum dots in cavities . . . . . . . . . . . . . . . . . . . . . . . 10
2.3.1 Quantum dot-cavity coupling . . . . . . . . . . . . . . . . 11
2.3.2 Weak coupling . . . . . . . . . . . . . . . . . . . . . . . . 12
2.3.3 Strong coupling . . . . . . . . . . . . . . . . . . . . . . . . 14
2.3.4 Types of cavities . . . . . . . . . . . . . . . . . . . . . . . 15
2.4 Photonic crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.4.1 Photonic band gap . . . . . . . . . . . . . . . . . . . . . . 19
2.4.2 Photonic crystal cavities . . . . . . . . . . . . . . . . . . . 19
vi
CONTENTS
2.4.3 Two-dimensional photonic crystals . . . . . . . . . . . . . 20
2.4.4 Practical considerations . . . . . . . . . . . . . . . . . . . 23
3 Experimental techniques 25
3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.2 Sample growth and basic preparation . . . . . . . . . . . . . . . . 25
3.2.1 Quantum dot growth . . . . . . . . . . . . . . . . . . . . . 25
3.2.2 Isolating single quantum dots . . . . . . . . . . . . . . . . 26
3.3 Optical techniques . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.3.1 Micro-photoluminescence spectroscopy . . . . . . . . . . . 27
3.3.2 Time-resolved photoluminescence . . . . . . . . . . . . . . 29
3.3.3 Auto-correlations . . . . . . . . . . . . . . . . . . . . . . . 30
3.3.4 Cross-correlations . . . . . . . . . . . . . . . . . . . . . . . 32
3.3.5 Application of a magnetic field . . . . . . . . . . . . . . . 32
3.4 Spectroscopy of quantum dots . . . . . . . . . . . . . . . . . . . . 34
3.4.1 Power dependence . . . . . . . . . . . . . . . . . . . . . . 34
3.4.2 Polarization of excitons . . . . . . . . . . . . . . . . . . . . 36
4 p-i-n diode quantum dot devices 37
4.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.2 Sample design and operation . . . . . . . . . . . . . . . . . . . . . 37
4.3 Controlling the polarisation correlation of photon pairs . . . . . . 40
4.3.1 Degree of polarisation correlation . . . . . . . . . . . . . . 41
4.3.2 Lifetime measurements . . . . . . . . . . . . . . . . . . . . 44
4.3.3 Polarisation correlation control . . . . . . . . . . . . . . . 44
4.4 Single quantum dot memory . . . . . . . . . . . . . . . . . . . . . 45
4.4.1 Quantum information processing . . . . . . . . . . . . . . 45
4.4.2 Device operation . . . . . . . . . . . . . . . . . . . . . . . 46
4.4.3 Experimental set-up . . . . . . . . . . . . . . . . . . . . . 48
4.4.4 Quantum dot electron-spin memory . . . . . . . . . . . . . 49
4.4.5 Summary and outlook . . . . . . . . . . . . . . . . . . . . 53
vii
CONTENTS
5 Designing and fabricating photonic crystals 55
5.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
5.2 Finite-difference time-domain simulation . . . . . . . . . . . . . . 55
5.3 FDTD for photonic crystals . . . . . . . . . . . . . . . . . . . . . 57
5.3.1 Boundary conditions . . . . . . . . . . . . . . . . . . . . . 57
5.3.2 Calculating the spectrum . . . . . . . . . . . . . . . . . . . 57
5.3.3 Q-factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
5.3.4 Mode profile . . . . . . . . . . . . . . . . . . . . . . . . . . 60
5.3.5 Mode volume . . . . . . . . . . . . . . . . . . . . . . . . . 60
5.3.6 Purcell factor . . . . . . . . . . . . . . . . . . . . . . . . . 61
5.3.7 Simulation convergence studies . . . . . . . . . . . . . . . 61
5.4 Fabrication of photonic crystals . . . . . . . . . . . . . . . . . . . 63
5.4.1 Initial process . . . . . . . . . . . . . . . . . . . . . . . . . 64
5.4.2 Electron beam resist . . . . . . . . . . . . . . . . . . . . . 65
5.4.3 AlGaAs sacrificial layer . . . . . . . . . . . . . . . . . . . . 68
5.4.4 Dry etching . . . . . . . . . . . . . . . . . . . . . . . . . . 71
5.5 Summary and outlook . . . . . . . . . . . . . . . . . . . . . . . . 75
6 H1 photonic crystal cavities for strong coupling 76
6.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
6.2 H1 mode structure . . . . . . . . . . . . . . . . . . . . . . . . . . 77
6.3 Design and optimisation . . . . . . . . . . . . . . . . . . . . . . . 79
6.3.1 Bulk hole radii . . . . . . . . . . . . . . . . . . . . . . . . 79
6.3.2 Nearest neighbour hole radius . . . . . . . . . . . . . . . . 80
6.3.3 Slab height . . . . . . . . . . . . . . . . . . . . . . . . . . 81
6.3.4 Optimised cavity . . . . . . . . . . . . . . . . . . . . . . . 82
6.4 Fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
6.5 Optical characterisation . . . . . . . . . . . . . . . . . . . . . . . 83
6.5.1 Mode structure . . . . . . . . . . . . . . . . . . . . . . . . 83
6.6 Weak coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
6.7 Strong coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
6.8 Polarisation dependence . . . . . . . . . . . . . . . . . . . . . . . 92
6.9 Summary and outlook . . . . . . . . . . . . . . . . . . . . . . . . 93
viii
CONTENTS
7 Unidirectional photonic crystal waveguides 94
7.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
7.2 Photonic crystal waveguide mode structure . . . . . . . . . . . . . 95
7.3 Design and optimisation . . . . . . . . . . . . . . . . . . . . . . . 98
7.3.1 Basic parameters . . . . . . . . . . . . . . . . . . . . . . . 98
7.3.2 Hole radius . . . . . . . . . . . . . . . . . . . . . . . . . . 99
7.3.3 Slab height . . . . . . . . . . . . . . . . . . . . . . . . . . 100
7.3.4 Optimised waveguide . . . . . . . . . . . . . . . . . . . . . 103
7.4 Fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
7.5 Optical characterisation . . . . . . . . . . . . . . . . . . . . . . . 105
7.6 Purcell effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
7.7 Further work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
7.8 Summary and outlook . . . . . . . . . . . . . . . . . . . . . . . . 111
8 Conclusions 112
A Photonic band gap 115
References 138
ix
Chapter 1
Introduction
The term ‘quantum information’ refers to physical information which is encoded
in a quantum system. Encoding information in this way opens up new methods
of transferring and manipulating data based on the laws of quantum mechan-
ics. Using light as the quantum system has allowed advances in quantum key
distribution and photonic quantum computing [119].
Quantum key distribution involves sharing a cryptographic key between two
parties in perfect secrecy [47]. By using single photons to transmit the key in
schemes such as the BB84 protocol [16, 139], it is possible to detect the pres-
ence of any eavesdroppers. Through quantum teleportation [23], future quantum
networks could distribute keys over longer distances or through more complex
topologies [25].
Quantum computing takes advantage of quantum mechanical phenomena to
perform operations on data that would not be possible with classical systems.
Certain classes of problems, such as factorising large numbers [122], could be
performed far more efficiently with a quantum computer. Future practical imple-
mentations may rely on quantum states of light, as for example in the linear-optics
quantum computing scheme [69, 70].
These applications will require efficient sources of single photons and entangled
photon pairs [85, 119]. The probabilistic process of parametric down conversion
in nonlinear crystals can be used to produce heralded single photons [53] and
entangled photon pairs [24, 104, 120], although these are not on-demand sources.
Furthermore, there is a finite probability of producing more than one photon/pair
1
at a time, so that the pump power must be very low which degrades the source
efficiency. Several systems have been proposed to create efficient on-demand
true single photon sources, including ensembles of atoms [67], single ions [32],
fluorescent molecules [11, 27], nitrogen vacancies in diamond [76], quantum wells
[64] and nanocrystals [86].
Semiconductor quantum dots have proved to be an excellent source of single
photons [15, 94, 116, 151], and it has recently been shown that they can produce
single pairs of polarisation entangled photons on-demand [130]. They offer com-
pact and robust solid state sources which can be readily integrated with other
semiconductor technology. For example, they can be grown in p-i-n diode struc-
tures to achieve electrically driven sources [151]. Furthermore, it is relatively
straight forward to integrate quantum dots into optical cavity structures which
are essential to improve source efficiencies.
Other quantum dot-based devices may find uses as other components in quan-
tum information schemes [96, 153], such as quantum memories [22], quantum
gates [35, 58] and single photon detectors [13, 21].
In this thesis, a number of quantum dot devices are investigated. In chapter
2, the relevant theory concerning quantum dots and optical cavities is set out.
Chapter 3 contains descriptions of the experimental techniques used throughout
this work. The operation of a device based on a quantum dot embedded in a
p-i-n diode is discussed in chapter 4. Chapter 5 concentrates on the development
of a process to fabricate two-dimensional slab photonic crystals to enhance the
performance of quantum dot devices. In chapter 6, strong coupling between a
quantum dot and a H1 photonic crystal cavity is demonstrated. Chapter 7 focuses
on experiments with unidirectional photonic crystal waveguides. Finally, some
concluding remarks are made in chapter 8.
2
Chapter 2
Background and Theory
2.1 Overview
In this chapter, the relevant background to the rest of the thesis will be set out.
Firstly the properties of semiconductor quantum dots will be discussed. This
will be followed by the details of their integration into optical cavities. Finally a
discussion about photonic crystals and the use of them to create cavities will be
presented.
2.2 Semiconductor quantum dots
A semiconductor quantum dot (QD) is a nanometre sized island of a semiconduc-
tor material embedded in another material which has a larger band gap [19]. It
effectively forms a potential well in which electrons and holes can be trapped in
three dimensions (see figure 2.1). The dimensions of a quantum dot are smaller
than the de Broglie wavelength of trapped electrons and holes, so the carriers are
quantum confined and the energy levels available to them are quantized.
The focus of this work is on InAs QDs embedded in GaAs grown by molecular
beam epitaxy (see section 3.2.1). When InAs is epitaxially grown on GaAs, it
initially experiences a strain due to the lattice mismatch between the two crystals.
After a certain critical thickness of InAs is deposited, this strain cause the InAs
3
2.2 Semiconductor quantum dots
Figure 2.1: Simplified band diagrams of a single quantum dot exciton states;
(a) exciton (X), (b) biexciton (2X), (c) negatively charge exciton (X−), (d) dou-
bly positively charged exciton(X2+), (e) positively charged biexciton(2X+). The
dashed red lines represent the available energy levels for carriers confined in the
dot. Dark gray circles represent electrons and light blue circles represent holes.
The arrows associated with each of the carriers indicate the possible spin states
of the carriers.
4
2.2 Semiconductor quantum dots
to break up into small nano-scale islands. Overgrowth with more GaAs completes
the formation of quantum dots.
A QD can be populated with carriers by electrical injection or by the optical
excitation of an electron into the conduction band leaving a hole in the valence
band. This latter method can be achieved by exciting an electron resonantly
inside the dot, or by exciting carriers in the surrounding material and relying on
the fact that some of them will relax into the lower potential energy dot levels
via phonon interactions [92].
Previous studies have shown that the InAs dots studied here can hold one
or two quantized energy levels for each carrier [133] (e.g. the QDs in Figure 2.1
contain one electron energy level and two hole energy levels), although larger
dots may have more. Since electrons and holes are fermions, the Pauli exclusion
principle requires that each energy level can only be occupied by two carriers with
opposite spins. There are therefore only a small number of possible configurations
of carriers that the dot can accommodate. These configurations are known as the
excitonic states of the QD. Figure 2.1 shows a number of possible excitonic states
including the neutral exciton (Figure 2.1(a)), the biexciton (Figure 2.1(b)) and
the negatively charged exciton (Figure 2.1(c)).
2.2.1 Optical properties of single semiconductor quantum
dots
The recombination of an electron and a hole inside a dot leads to the emission
of a single photon with an energy equal to the difference between the energies of
the initial and final excitonic states [93]. The energy of each excitonic state is, in
turn, determined by the complex Coulomb and exchange interactions between its
constituent carriers [92]. Therefore, the emission from a single QD consists of a
number of separate spectral lines, each corresponding to the decay of a different
excitonic state.
For example the biexciton can decay - by the recombination of an electron and
a hole and the emission of a photon - to the exciton. The energy of the biexciton
photon will differ from the energy of the subsequently emitted exciton photon as
the exciton decays to the ground state (see Figure 2.2).
5
2.2 Semiconductor quantum dots
Figure 2.2: Typical spectrum of the emission from a single quantum dot. Each
spectral line corresponds to the decay of a different exciton state in the QD by
the radiative recombination of an electron and a hole, as indicated.
The optical properties of QDs are in fact very similar to atoms. However,
QDs are much larger and can be integrated into solid state structures to allow a
range of photonic semiconductor devices.
The typical decay rate for e.g. the neutral exciton from the QDs studied in this
work is γX ∼ 1 GHz, which is equivalent to a radiative lifetime of τr,X = 1/γX = 1
ns. This finite lifetime leads to radiative broadening of the resulting spectral line
as compared to the monochromatic ideal. This can be approximately understood
by considering the energy-time uncertainty principle, ∆Uτr & ~, which implies
that
∆ω =
∆U
~
& 1
τr
. (2.1)
Here, ∆U is the uncertainty in the emission energy and ∆ω is the spectral line
width. The emission lines from individual QDs take the form of Lorentzian line
shapes. The radiative line width is the minimum possible width; the emission
6
2.2 Semiconductor quantum dots
may be further broadened by e.g. non-radiative relaxation processes due to in-
teractions with phonons or charges in the vicinity of the QD.
2.2.2 Quantum dots as single photon sources
QDs are ideally suited to being used as single photon sources [93, 133, 152]. When
an electron and a hole radiatively recombine, some finite period of time is needed
before the QD can be repopulated with carriers to reform the initial excitonic
state and then decay again. Therefore only one photon can be emitted at a time.
The single photon properties can be further improved, and deterministic con-
trol can be gained by implementing a pulsed excitation scheme. Since the lifetimes
of the excitonic states in QDs are much longer than the lifetimes of the photo-
excited carriers in the surrounding semiconductor, and given an excitation pulse
that is much shorter than the lifetime of the excitonic state (easily achievable
with both optical [94] and electrical [151] excitation schemes), only one photon
can be emitted per excitation pulse.
2.2.3 Fine structure
Exciton fine structure
The character of the holes confined by QDs is typically heavy [12]. The angular
momentum of a heavy hole is jhh = 3/2 and the spin of an electron is se = 1/2.
Therefore the exciton’s total angular momentum projection in the z direction is
mex = s
z
e + j
z
hh = ±1 or ±2. The angular momentum of a single (circularly polar-
ized) photon is jph = ±1. Thus due to the conservation of angular momentum,
only the exciton states with mex = ±1 can decay radiatively, emitting a single
photon. These exciton states are known as bright excitons, whereas the other
states with mex = ±2 cannot normally couple to single photons and are known
as dark excitons. These two doublets are split in energy by the electron-hole ex-
change interaction; the dark exciton doublet has lower energy since the angular
momentum of the hole and the spin of the electron are aligned in this case (see
left side of figure 2.3).
7
2.2 Semiconductor quantum dots
An InAs QD is generally elongated along one of the in-plane crystal axes (i.e.
lens shaped) and there is an asymmetry in the inbuilt strain of the dot which
remains after the strain relaxing QD formation. Furthermore GaAs exhibits
piezoelectricity, and differing electric fields are experienced along the different
crystal axes due to its structure. These asymmetries give rise to anisotropic
exchange interactions between the carriers in the dot. This causes each of the
exciton doublets, mex = ±1 and mex = ±2, to hybridize, so that the emission
from each of them becomes linearly polarised with distinct polarisations parallel
to the [110] and [1-10] axes of the crystal [43, 74] (labeled pix and piy respectively
in figure 2.3). The two polarisations are furthermore split in energy, known as
the fine structure splitting and denoted by S0.
Biexciton
The biexciton consists of two anti-aligned electrons and two anti-aligned holes
which have zero total spin and angular momentum respectively. Therefore there
is no electron-hole exchange interaction and the state is non-degenerate. The
biexciton can decay by the emission of a single photon to either one of the bright
mex = ±1 exciton states. Thus the biexciton line also exhibits a splitting in an
equal and opposite sense to the exciton line.
2.2.4 Quantum dots as sources of entangled photon pairs
The decay of the biexciton to the ground state via the emission of a biexciton
photon and an exciton photon can be used to create a entangled photon pair
source [130]. If the fine structure splitting of the exciton level is zero (as in the
left side of figure 2.3), the two possible biexciton decay paths are indistinguishable
and they are entangled. The wavefunction describing the two photons emitted
during the biexciton decay is a superposition of the two decay paths and the
polarisations of the pair are not well defined until they are measured.
8
2.2 Semiconductor quantum dots
Circular
symmetry
Circular
asymmetryX
X
0
m = ± 1
m = ± 2
σ -
σ- σ+
σ+
πx
πx
πy
πy
2
S0
Figure 2.3: Energy level diagram showing the energies of the biexciton and exciton
states for symmetric and asymmetric quantum dots. The solid lines indicate
bright exciton states, and the dashed lines indicate dark exciton levels. Allowed
transitions are indicated by coloured arrows. Asymmetry in the shape of the QD
leads to anisotropic exchange interactions between the carriers in the dot, which
cause the exciton levels to split. The photons emitted during decay become
linearly polarised.
9
2.3 Quantum dots in cavities
2.2.5 Applied magnetic field
A magnetic field applied to a QD introduces Zeeman interactions between the
spins of the electrons and holes, and the field [12]. The effects are highly de-
pendent on the orientation of the field due to the asymmetry of the dot. There
are two special cases: the Faraday geometry in which the field is applied parallel
to the growth direction of the dots, and the Voigt geometry where the field is
applied in the plane of the dots.
Faraday geometry
In this case, the Zeeman interaction simply increases the fine structure splitting.
If this increase is large enough, the hybridization of the bright mex = ±1 states
becomes negligible and the two pure eigenstates of the exciton total angular
momentum are recovered.
Voigt geometry
When a large enough magnetic field is applied in the plane of the dots, the
dark and bright states become mixed. This causes the dark excitons to become
optically active [12, 18].
2.3 Quantum dots in cavities
One of the advantages of working with semiconductor QDs is the relative ease
in which they are integrated into optical cavities. Typically only ∼ 2% of the
light from a QD is emitted out of the top of a bare wafer because of the large
refractive index contrast between the semiconductor and air [153]. This fraction
can be substantially increased by directing the emission by coupling it to a mode
of a cavity [14, 135]. Furthermore, the spontaneous emission rate of the QD
can be enhanced by the Purcell effect [111]. If it is made to be much larger
than the dephasing rate of the QD transition, the emitted photons will become
indistinguishable - a property which is essential for many quantum information
processing applications.
10
2.3 Quantum dots in cavities
2.3.1 Quantum dot-cavity coupling
The properties of a cavity mode are defined by its resonant frequency, ωcav, and
its photon decay rate, κ. The decay rate is a measure of how quickly light stored
in the mode is lost. The total optical energy, U , in a mode will decay with time,
t, like
U = U0e
−t/τcav , (2.2)
where τcav = 1/κ is the cavity mode photon lifetime and U0 is the energy in the
mode at t = 0. The spectral width of the mode is controlled by the photon decay
rate, ∆ωcav = 1/τcav ≡ κ. The quality factor, or Q-factor, of the mode is defined
by
Q = ωcav/∆ωcav. (2.3)
Higher Q-factors indicate a lower rate of energy loss relative to the energy stored
in the mode.
If a QD is surrounded by a cavity with a resonant frequency equal to a transi-
tion frequency of the dot, the QD and the cavity will be able to exchange photons
in a resonant way. The strength of the interaction between the QD transition
and the electric field in the cavity, E, is quantified by the QD-cavity coupling
parameter, g, and is given by the electric dipole interaction,
~g = |〈d.E〉|, (2.4)
where d is the electric dipole moment of the transition. 〈d.E〉 is the matrix
element of the transition between the initial and final states. Assuming that the
QD is placed at the maximum of the electric field in the cavity, the QD-cavity
coupling parameter can be expressed in terms of the oscillator strength, f , as
g =
(
e2f
40rm0V0
)1/2
, (2.5)
11
2.3 Quantum dots in cavities
where r and 0 are the relative and vacuum permittivity respectively, m0 is the
free electron mass and V0 is the volume of the cavity mode [112]. Since the
oscillator strength is a fixed property of the QDs, for a well positioned QD the
coupling parameter is controlled by the volume of the mode. By designing cavities
with very small mode volumes, the strength of the coupling can be enhanced.
2.3.2 Weak coupling
When the strength of this interaction is smaller than the losses from the system,
g < (κ, γnr), (2.6)
the QD/cavity is said to be weakly coupled. Here, γnr is the non-resonant decay
rate, which takes account of photons which do not couple to the cavity and non-
radiative decay processes in the QD. In this case, the spontaneous emission rate
of the transition can be strongly modified. The SE rate from a QD transition, γ,
is governed by Fermi’s Golden Rule [40]:
γ =
1
τr
=
2pi
~
|〈d.E〉|2ρ(ω), (2.7)
where ρ(ω) is the optical density of states experienced by the QD. |〈d.E〉|2 is
proportional to the probability that there will be a transition. In free space, the
optical density of states is given by
ρ(ω) =
ω2V0
pi2c3
. (2.8)
The presence of a cavity increases the density of states at frequencies close
to the cavity resonance and restricts it far from resonance. The density of states
then takes the form of a normalised Lorentzian function:
ρ(ω) =
2
pi∆ωcav
∆ω2cav
4(ω − ωcav)2 + ∆ω2cav
(2.9)
12
2.3 Quantum dots in cavities
This equation in conjunction with equation 2.7 implies that the spontaneous
emission rate of a weakly coupled QD transition is enhanced for ω ≈ ωcav, and is
suppressed for ω far away from ωcav, a phenomenon known as the Purcell effect
[111].
The factor by which the spontaneous emission rate is enhanced/suppressed
is quantified by the Purcell factor, FP , which is equal to the ratio between the
spontaneous emission rate of the QD transition with and without a cavity. This
gives
FP =
γcav
γfree
≡ τ
free
r
τ cavr
=
3Q(λ/n)3
4pi2V0
ξ2
∆ω2cav
4(ω − ωcav)2 + ∆ω2cav
, (2.10)
where λ is the free-space wavelength of the QD emission and n is the refractive
index of the material within the cavity. ξ, the normalised dipole orientation
factor, is given by
ξ =
|E(r)|
|Emax| .
|d.E(r)|
|d||E(r)| , (2.11)
where E(r) is the electric field experienced by the QD at position r in the cavity
and Emax is the maximum of the electric field in the cavity. ξ quantifies how well
the electric dipole of the transition is positioned and orientated with respect to
the electric field of the cavity mode. At exact resonance, with the QD positioned
at the maximum of the cavity electric field and with its dipole orientated in the
same direction as the field, this formula reduces to
FP =
3Q(λ/n)3
4pi2V0
. (2.12)
For FP > 1, the spontaneous emission rate is enhanced by the cavity and for
FP < 1, it is suppressed. The ratio of Q/V0 for the cavity must be maximised to
achieve large Purcell factors.
The fraction of the total emission from a QD that is emitted into the mode
of a cavity is called the β-factor. The total emission rate will be the sum of
the resonant and non-resonant emission rates. Assuming that the non-resonant
13
2.3 Quantum dots in cavities
emission rate is not affected by the cavity, then the total emission rate will be
γfree + γcav, and it is easy to see that
β =
γcav
γfree + γcav
=
FP
1 + FP
. (2.13)
For large Purcell factors, the emission into the cavity dominates the non-resonant
emission and the beta factor approaches unity.
2.3.3 Strong coupling
When the strength of the QD-cavity coupling dominates over the losses, the cavity
and the QD transition are strongly coupled. In this regime, when a photon is
emitted it is trapped in the cavity for long enough so that it can be reabsorbed by
the QD. This reversible process, known as Rabi flopping, may occur many times
before the photon is lost from the cavity irreversibly.
The Jaynes-Cummings model describes the QD-cavity interaction [5]. For an
uncoupled system one would expect the two distinct states of one photon plus the
unexcited QD, |g, 1〉, and the excited QD with no photon in the cavity, |e, 0〉, to
be degenerate. However, for a strongly coupled system these states are mixed by
the electric dipole interaction between the QD exciton and the photon, and the
degeneracy is lifted. The new states look like 1√
2
(| g, 1〉± | e, 0〉), and are split in
energy by the vacuum Rabi splitting, given by
∆Uvac = 2~
√
g2 − (κ− γ)
2
16
. (2.14)
These states can be considered as quasi-particles resulting from mixing of the
photon and the QD exciton and are known as the cavity polaritons [112, 142].
Transition from weak to strong coupling
For weak coupling, the second term in equation 2.14 dominates the square root,
so that ∆Uvac is imaginary and there is no splitting. The linewidth of the exciton
is broadened by the Purcell effect like γ = FPγ
free. Close to strong coupling, the
14
2.3 Quantum dots in cavities
linewidth of the exciton is broadened beyond this, and the linewidth of the cavity
mode is reduced substantially below the uncoupled value.
In practice, the decay rate of the cavity is typically much bigger than that of
the QD transition, κ γ. For strong coupling, the first term in equation 2.14 is
larger than the second term, i.e.
g > κ/4. (2.15)
At this point the splitting becomes real and the linewidths of the two states
collapse to become equal at roughly (κ + γ)/2. To resolve the vacuum Rabi
splitting experimentally, these linewidths have to be narrower than the splitting.
By examining equations 2.3, 2.5 and 2.15, is is apparent that the ratio Q/
√
V0
must be maximised to achieve strong coupling [112].
2.3.4 Types of cavities
A number of designs of cavities have been conceived of for integrating with semi-
conductor QDs. Each of them has its own advantages and disadvantages as will
be discussed below.
Planar microcavities
A planar microcavity is formed by surrounding a cavity region by two distributed
Bragg reflector (DBR) mirrors (see figure 2.4 (a)). Q-factors of up to 10000 have
been achieved [118], however since the optical confinement exists only in one
dimension, the mode volume is very large.
Micropillars
Micropillars are made by epitaxially growing a DBR cavity before etching the
chip to form pillars (see figure 2.4 (b)). Light is confined in one direction by the
DBR mirrors and in the other two directions by total internal reflection at the
semiconductor-air interface.
The Q-factors of micropillars are limited by the smoothness of the side walls,
which is limited by etching technologies. Furthermore, there is a trade-off between
15
2.3 Quantum dots in cavities
(c) (d)
(b)(a)
Figure 2.4: Schematic diagrams of (a) planar microcavity, (b) micropillar, (c)
microdisk, (d) two-dimensional photonic crystal slab cavity.
16
2.3 Quantum dots in cavities
Q-factor and the proportion of light emitted out-of-plane depending on the num-
ber of DBR periods in the top mirror [14]. Nevertheless, Q-factors of > 20000
[33, 118] have been demonstrated, with typical modal volumes of ∼ 5(λ/n)3 [135].
Microdisks
Another type of cavity is the microdisk. This consists of a flat, disk-shaped piece
of high refractive index semiconductor suspended from the bulk crystal by a small
tip (see figure 2.4 (c)). Whispering-gallery type modes can be confined by total
internal reflection in the edges of the disk. These can have very high Q-factors
over 100000 [126], with mode volumes of the order ∼ 6(λ/n)3 [135]. However, the
emission is not as strongly directed as in the case of micropillars and complicated
excitation/collection schemes are often required.
Photonic crystal cavities
Finally, photonic crystals can be used to make cavities as discussed in the fol-
lowing section (see figure 2.4 (d)). Three-dimensional photonic crystals offer
the ideal prospect of strong confinement in all three dimensions, however they
are limited by the difficulties of fabricating such complicated structures [101].
Two-dimensional photonic crystal slab cavities - in which light is confined in two-
dimensions by a photonic band gap and in the third by total internal reflection
at a semiconductor-air interface - have therefore received a lot of attention. They
offer very high Q-factors along with tiny mode volumes (∼ (λ/n)3), and are the
cavity of choice for observing strong coupling [112].
Furthermore, the small mode volume of photonic crystal cavities make them
preferable for applications in which a low Q-factor is required alongside a strong
QD-cavity interaction. For example, by coupling both the exciton and biexciton
to a cavity, the efficiency of entangled photon pair sources could be enhanced via
the Purcell effect. Since the lines are typically a few meV apart, a low Q cavity is
required. This leaves the mode volume as the only parameter available to increase
the strength of the interaction. Alternatively, it may be possible to couple each
transition to a separate mode of the same cavity since photonic crystal cavities
usually support a number of modes [121] (see section 6.2).
17
2.4 Photonic crystals
2.4 Photonic crystals
This section will begin by discussing the origins of photonic band gaps. It will
then go on to describe photonic crystal cavities and waveguides.
a
a
r
GaAs
Air
(b) (c)
GaAs, n≈3.5AlGaAs, n≈2.8
λ/4n
(a)
Figure 2.5: Schematic diagrams of examples of photonic crystals. (a) A dis-
tributed Bragg reflector (one-dimensional photonic crystal) is formed by alter-
nate λ/4n layers of two materials with a high refractive index contrast. (b) A
two-dimensional photonic crystal consisting of a periodic array of air holes in a
high-refractive index material. (c) A three-dimensional photonic crystal formed
by a woodpile arrangement of high-refractive index bars.
A photonic crystals is a material with a periodic variation in its refractive
index [60, 146]. For example, the well known Bragg mirror is a one-dimensional
photonic crystal (see figure 2.5 (a)). Alternate λ/4n thick layers of two materials
with a large refractive index contrast give rise to Bragg reflection at the material
interfaces which constructively interfere, so that any light with a wavelength close
to λ is reflected.
A two-dimensional photonic crystal has a refractive index which varies pe-
riodically in two dimensions. An example of this is shown in figure 2.5 (b), in
which a periodic array of holes is formed in a high refractive index material. In
this case, Bragg reflection can occur along the two in-plane dimensions.
Finally, an example of a three-dimensional photonic crystal is illustrated in
figure 2.5 (c). In this case, a three-dimensional periodic variation in the refractive
18
2.4 Photonic crystals
index is achieved by a so-called woodpile arrangement of high refractive index
bars.
2.4.1 Photonic band gap
The periodicity in the refractive index of a photonic crystal causes Bragg reflec-
tions of photons which gives rise to forbidden energy bands at which photons
cannot propagate through the crystal. This is analogous to the periodic potential
in solid state physics which causes Bragg reflection of electrons and gives rise
to electronic band gaps. Light with a frequency inside the band gap which is
incident on a photonic crystal will decay exponentially inside the crystal and will
be reflected. Appendix A examines the phenomena in more detail.
2.4.2 Photonic crystal cavities
GaAs
AlGaAs
0
1
3.0
3.2
3.4
3.6
No
rm
ali
se
d |
ele
ctr
ic 
fie
ld|
Re
fra
cti
ve
 in
de
x 
Figure 2.6: The electric field profile (black line) of a mode confined in a DBR
cavity formed by a λ/n-thick layer of GaAs surrounded by DBR mirrors consisting
of alternate λ/4n-thick layers of GaAs and AlGaAs. The refractive index of the
cavity is shown by the red line.
By introducing a defect into a photonic crystal, it is possible to create an
optical cavity. For example, in the case of a DBR the thickness of one of the
layers could be increased. The DBR surrounding the defect will act like a pair of
19
2.4 Photonic crystals
frequency selective mirrors. If the defect is thick enough, it will be possible for it
to support a localised mode with a frequency inside the band gap. The mode will
be confined in the direction parallel to the periodicity and will exponentially decay
when it enters the photonic crystal. Figure 2.6 compares the normalised electric
field of a mode confined in a DBR cavity to its refractive index distribution.
Because of the finite size of the confinement, defect modes are quantized into
discrete frequencies. By increasing the thickness of the defect, more and more
higher order modes can be supported.
2.4.3 Two-dimensional photonic crystals
A two-dimensional photonic crystal could be made, for example, by a triangular
array of holes in a high refractive index material. In this case there is a periodic
variation in the dielectric function in two dimensions, so a photonic band gap can
exist in those two dimensions.
Because of the mirror symmetries, all the modes can be classified into one
of two polarisations: transverse electric (TE) or transverse magnetic (TM) [59].
The former has non-zero field components (Ex, Ey, Hz) and the latter has (Hx,
Hy, Ez), where x and y are the two in-plane directions, and z is perpendicular
to the periodic plane. The band structures for these two polarisations can be
completely different. For example, a square lattice of rods only supports a band
gap for the TM polarisation, whereas a triangular lattice of holes in a dielectric
can support a complete band gap for both polarisations, as shown in figure 2.7.
A two-dimensional photonic crystal can be approximated in practice by a two-
dimensional photonic crystal slab [36, 41]. This consists of a slab of high refractive
index material perforated by an array of air holes. A cavity can be realised in
this system by removing one (or more) of the holes [100, 105, 106]. For example,
figure 2.8 shows a cavity formed by removing one of the air holes in a triangular
lattice photonic crystal slab, the so-called H1 cavity. Light is confined by the
photonic band gap in the two in-plane directions, and by total internal reflection
(TIR) at the material/air interfaces in the out-of-plane direction. Other methods
of producing a cavity include the double heterostructure concept [52, 75, 124,
131, 141].
20
2.4 Photonic crystals
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
TE
TM
N
o
rm
a
li
s
e
d
 F
re
q
u
e
n
c
y
 (
a
/ λλ λλ
)
ΓΚMΓ
Wavevector
TE Bandgap
TM Bandgap
DOS
Γ K
M
Air band
Dielectric band
(a) (b)
(c)
Figure 2.7: (a) Band structure of a two-dimensional photonic crystal with a trian-
gular lattice of holes in a dielectric with normalised radius r/a = 0.45, calculated
by a scattering matrix method. (b) Finite-difference time-domain calculation of
the density of states for the same structure. Courtesy of D. Gevaux.
d
a
a
r
GaAs
Air
(a) (b)
Figure 2.8: (a) Schematic diagram of a 2D photonic crystal nanocavity formed
by removing one of the air holes in a triangular lattice of holes in a dielectric. (b)
Three-dimensional view of an equivalent cavity formed in a 2D photonic crystal
slab.
21
2.4 Photonic crystals
Figure 2.9 shows the calculated spectrum for an unmodified H1 cavity with
normalised radius of holes r/a = 0.3 and normalised slab height d/a = 0.7. There
is a photonic band gap stretching from λ/a ∼ 3.2 to λ/a ∼ 4.2, with a single
resonant mode confined inside it. The field distributions of the TE-like mode are
shown, which is known as the dipole mode because of the two anti-nodes in its
magnetic field distribution.
3.0 3.5 4.0 4.5
0.0
Mode
Band gap
Normalised wavelength (λ/a)
Band gap
(a) (b)
(c)
(d)
Mode
Ele
ctr
ic 
fie
ld 
am
pli
tud
e (
arb
. u
nit
s)
Figure 2.9: (a) Spectrum for an unmodified H1 cavity with r = 0.3a, d = 0.7a and
n = 3.53, calculated by finite-difference time-domain simulation. The black line
shows the spectrum inside the defect. The red line shows a similar calculation
with no defect, so that the extent of the photonic band gap is made clear. A
mode is present near the centre of the band gap, which is called the dipole mode
because of the two anti-nodes in its magnetic field distribution. (b)-(d) Ex, Ey
and Hz field distributions of the dipole mode.
Not all of the components of a mode confined inside the cavity will be totally
internally reflected, and there will be losses in the vertical direction [59]. In the
air, far away from the structure, light obeys
ω = c|k| = c
√
|k//|2 + k2z , (2.16)
22
2.4 Photonic crystals
where k// is the wave vector in the direction of the photonic crystal plane, and
kz is the wave vector in the out-of-plane direction. Close to the structure, a
continuum of extended states propagating in the air exists for ω > c|k//|, since
kz is not restricted by the photonic crystal. Therefore, components of the mode
with in-plane wave vector inside the so called light line, |k//| < ωcav/c, are not
confined to the photonic crystal slab and can be easily lost from the cavity.
The photonic crystal slab can support components with larger wave vectors than
the air since its dispersion relation approximately looks like |k| = nω/c. Any
components of the mode outside the light line, |k//| > ωcav/c, cannot couple to
air modes and are strongly confined.
There is an intrinsic vertical radiative loss associated with any confined mode,
since some of its components can couple to light line modes at the same frequency.
Figure 2.10 illustrates this point. These losses can be minimised and the Q-factor
maximised by carefully designing the cavity so that the coupling to the light line
is minimised [2, 3, 124, 127, 137].
Losses TIR
GaAs slab
k1 k2<ωcav/c<
Figure 2.10: Schematic diagram of losses inside the light line. Momentum compo-
nents of the mode with |k//| < ωcav/c are lost from the GaAs slab (k1), whereas
components with |k//| > ωcav/c are totally internally reflected (k2).
2.4.4 Practical considerations
As described in section 2.3, coupling a QD transition to a high-Q, low volume
mode of an optical cavity enhances the spontaneous emission rate of the tran-
sition. By carefully designing the properties of a defect and the surrounding
photonic crystal, it is possible to reduce intrinsic losses and to make very high-Q
cavities. Furthermore, the mode volumes of photonic crystal cavities are tiny,
23
2.4 Photonic crystals
typically of order (λ/n)3. In real life cavities, there are also losses due to fab-
rication imperfections and light scattering in the crystal [44]. The structures
must therefore be well fabricated so that there is little unwanted scattering at
the surfaces.
The equation for the Purcell factor (equation 2.10) includes a spectral overlap
term. This can be maximised by carefully designing the cavity so that the mode
appears at the correct frequency. Appendix A sets out the similarities between
electrodynamics and quantum mechanics with respect to the formation of band
gaps and defect modes, etc. However one major difference between these two
theories is the scaling properties. The quantum mechanical Hamiltonian includes
the constant ~2/2m, which is related to a fundamental length, the Bohr radius
~/mcα. Hence, quantum mechanical systems do not scale in size. However, the
Maxwell equations do not contain any reference to a fundamental length scale, so
they do scale. The solutions to a system which differs from another system only
by an overall scaling factor are simply related by the same scaling factor. Hence
it is straight forward to scale the size of a cavity so that a mode appears at a
particular desired frequency.
Equation 2.10 also includes a spatial overlap term, ξ2. This says that there
must be a strong overlap of the QD’s dipole and the electric field of the mode
for a strong interaction, i.e. the QD should ideally be located at the maximum
of the electric field of the mode. This requirement is currently very challenging
since the self-assembled InAs QDs used in this work form in random positions
when they are grown.
24
Chapter 3
Experimental techniques
3.1 Overview
This chapter will begin by explaining the growth of the wafers used in this work.
It will then go on to discuss the optical techniques used to study the samples.
3.2 Sample growth and basic preparation
3.2.1 Quantum dot growth
All of the samples used in this work were grown by solid source molecular beam
epitaxy (MBE) at the Cavendish Laboratory. MBE is a mature technique which
allows precise growth of high purity material with monolayer precision [51] . In
this work, InAs quantum dots were grown by the Stranski-Krastanov growth
method. Since InAs and GaAs have slightly different lattice constants (∼7.5%
different), when InAs is grown on GaAs, there is a strain in the structure. Ini-
tially, a strained layer of InAs is formed; as more material is deposited, a critical
thickness (between 1.6 and 2.0 monolayers, depending on the exact growth condi-
tions) is reached, at which the strain is relaxed by the formation of small islands
of InAs - quantum dots. A thin layer of InAs remains below the QDs, as depicted
in figure 3.1 (a), called the wetting layer. As even more material is deposited,
the QDs grow larger until they begin to coalesce together. The samples used in
this work consist of a thin layer of InAs, close to the critical thickness, such that
25
3.2 Sample growth and basic preparation
a low density of QDs are formed. The QDs are completed by growing over with
more GaAs. Figure 3.1 (b) shows an AFM image of an uncapped low density
layer of QDs.
Substrate InAs layer below 
critical thickness
InAs layer above critical thickness
(a) (b)
1 µm
Figure 3.1: (a) Schematic diagram of QD growth by the Stranski-Krastanov
growth method. (b) AFM image of uncapped QDs.
Due to the self-assembled nature of Stranski-Krastanov grown dots, their size
and shape vary widely and their positions are random. It is, however, possible to
reproducibly grow wafers with similar distributions of sizes and shapes. A feature
of the MBE system used in this work is that a combination of the indium cell
shape and position causes the deposition to be nonuniform across the wafer. A
larger amount of indium is deposited at the centre of the wafer as compared to
the edge. The growth is typically tailored so that the dot density varies from a
high density in the centre to zero at the edge, so that the ideal low density of
dots forms somewhere in the middle.
3.2.2 Isolating single quantum dots
The density of the randomly positioned QDs typically varies between zero and
a few hundred per square micron, whilst the spot of the exciting laser typically
has an area of around 2 µm2. Therefore, a method to isolate and relocate single
quantum dots is required. A common technique used in parts of this work is
simply to evaporate a shadow mask (e.g. of aluminium) perforated with an array
of small holes. The holes allow optical access to a small portion of the wafer.
26
3.3 Optical techniques
When a QD happens to lie beneath one of these holes, it is easy to relocate it.
Holes with area of the order 1 µm2 were typically used. In other aspects of this
work, the fabrication of small optical cavities was sufficient to isolate single dots.
3.3 Optical techniques
This section will detail the experimental techniques used to study the samples.
3.3.1 Micro-photoluminescence spectroscopy
Normal incidence setup
Mirror
CCD
Laser
Lamp
Microscope 
objective
Sample
Camera
Lens
Spectrometer
Removable 
beam splitter
Cold mirror
50-50 beam 
splitter
Bandpass 
filter
Figure 3.2: Schematic diagram of the experimental setup for micro-
photoluminescence spectroscopy
Normal incidence micro-photoluminescence (µPL) spectroscopy was the main
tool used to study the samples. The basic setup is shown in figure 3.2. A laser
is focused to a spot with area ∼ 2 µm2 by an infinity corrected ×100 microscope
objective lens with a numerical aperture of 0.5, and is used to excite the sample.
27
3.3 Optical techniques
The laser used for the bulk of this work was a mode-locked Spectra-Physics pulsed
picosecond Ti-Sapphire laser, pumped with a 532 nm continuous wave (CW)
diode-pumped Nd:YVO4 laser. This system operates with a repetition rate of 80
MHz, with pulse widths of around 3 ps. The emission wavelength was tuneable
between ∼750 nm and ∼1000 nm, and was set at ∼780 nm for all measurements
except where otherwise stated. At this wavelength, corresponding to an energy
above the GaAs band gap, carriers are excited in the GaAs before some of them
relax into the InAs QDs. The power of the laser excitation was controlled with a
set of variable neutral density (ND) filters.
Photoluminescence (PL) emitted by the sample was collected by the same
objective before being dispersed by a Jobin Yvon HR460 grating spectrometer
(resolution ∼ 50 µeV) and detected by a nitrogen-cooled charge-coupled-device
(CCD) camera. The objective was mounted on an XYZ translation stage which
allowed positioning and focusing of the laser spot. A white light lamp could
be turned on and a removable beam-splitter inserted into the setup to enable
imaging of the sample and alignment using a video camera.
A cold mirror was used which reflects/transmits shorter/longer wavelength
light such that most of the laser (∼ 780 nm) and white light was reflected and the
majority of the sample PL (∼ 900 nm) was transmitted towards the spectrometer.
This ensured that as much as possible of the PL and as little as possible of the
laser light was detected. A bandpass filter was placed in front of the spectrometer
to eliminate any remaining laser light reflected from the sample.
The sample itself was mounted inside a continuous flow helium cryostat which
could be cooled down to around the boiling point of helium (4.2 K). An electri-
cal heater inside the cryostat allowed the temperature of the sample to be set
anywhere between this and room temperature.
A removable half-wave plate (HWP) and a polariser could be placed between
the cold mirror and the spectrometer to allow polarisation resolved PL.
Lateral setup
A micro-photoluminescence was designed and implemented to allow PL collection
lateral to the sample growth direction, specifically for samples studied in this
28
3.3 Optical techniques
work. The setup is shown schematically in figure 3.3. The sample was placed
inside a specially modified cryostat with a window allowing both normal incidence
and lateral optical access. A laser was focused to a spot on the sample surface
by a ×100 microscope objective lens as before. The PL emitted from the side of
the sample was collected by a second ×50 objective lens before being detected
in the same way as above. A second removable beam-splitter and video camera
were used to align the second objective.
Mirror
Laser
Lamp
Microscope 
objective 1
Sample
Camera 1
Removable 
beam splitter
Cold mirror
50-50 beam 
splitter
Removable 
beam splitter
Camera 2
Cold finger
Window
To spectrometer
Cryostat Microscope 
objective 2
Removable 
beam blocker
Figure 3.3: Schematic diagram of the experimental setup for lateral micro-
photoluminescence spectroscopy
3.3.2 Time-resolved photoluminescence
Figure 3.4 shows the setup for time-resolved PL (TRPL). The emission from
the sample was focused into the spectrometer in the same way as in figure 3.2.
However, the dispersed light was then directed to a second output port of the
spectrometer via a slit which acted as a spectral filter. This light was then focused
29
3.3 Optical techniques
onto a silicon avalanche photodiode (APD) which generates electrical pulses when
single photons are detected. The APD was 40% efficient at 900 nm. The pulses
from the APD and a reference signal from the laser’s clock were sent to a Becker
& Hickl GmbH SPC-630 single photon counting module (SPCM). The SPCM
measures the time difference between the two pulses and builds up a histogram
of the photon arrival times with respect to the laser pulse. The decay time of
emission lines can be calculated from this histogram.
Lens
Spectrometer
PL
APD
SPCM
Start
Stop
Laser’s clock
Figure 3.4: Schematic diagram of the experimental setup for time-resolved pho-
toluminescence spectroscopy
3.3.3 Auto-correlations
Auto-correlation measurements were performed using a Hanbury-Brown and Twiss
set-up as illustrated in figure 3.5. An emission line was spectrally selected by the
spectrometer and was directed towards a non-polarising 50:50 beam splitter. Two
separate APDs detect the photons from each of the beam splitter’s outputs. One
of the APDs provides the ‘start’ pulse to the SPCM and the other provides the
‘stop’ pulse and a histogram of the time difference, τ , between the events is
built up over time. This histogram approximates to the second order correlation
function, g2(τ), when τ is much smaller than the mean time between detection
events.
The second-order correlation function of the emission can be used to quantify
the quality of a single photon source:
30
3.3 Optical techniques
Lens
Spectrometer
PL
APD 1
SPCM
Start
Stop
APD 2
50-50 beam 
splitter
Figure 3.5: Schematic diagram of the experimental setup for auto-correlation
measurements
g(2)(τ) =
〈I(t)I(t+ τ)〉
〈I(t)〉〈I(t+ τ)〉 , (3.1)
where I(t) is the intensity of the light at time t, and 〈I(t)〉 is the time-averaged
intensity over a long period of time. It is a measure of the correlations between
the intensities of the light beam at different times, t and t+ τ .
For a coherent continuous wave (CW) source obeying Poissonian statistics,
the probability of a photon being emitted at time τ after a first photon is equally
likely for all times, so g(2)(τ) ≈ 1 for all τ . In the case that the source is operated
in pulsed mode, g(2)(τ) will be modulated by the pulses into a series of peaks
whose areas have a Poissonian variation about unity.
In the case of a thermal source, the characteristic intensity fluctuations or
bunches of photons give rise to correlations at small values of τ , so that g(2)(τ =
0) > 1.
For the opposite case of an anti-bunched photon source, g(2)(0) < 1. For a
true single photon source, only one photon is emitted at a time, so the intensity
correlations at τ = 0 fall to zero, i.e. g(2)(0) = 0. In the pulsed scenario, this is
manifest as the absence of a peak at τ = 0. In practice, the measured g(2)(0) will
31
3.3 Optical techniques
not be exactly zero even for a perfect single photon source due to dark counts in
the APDs and the finite time response of the system.
3.3.4 Cross-correlations
Correlations between different excitonic lines were measured using the setup in
figure 3.6. A non-polarising 50-50 beam splitter was placed directly after the
cold mirror and the outputs were focused onto two separate spectrometers. Each
spectrometer spectrally selected a different emission line in the PL. The light se-
lected by the first spectrometer (S1) was resolved into its two rectilinear polarized
components (H and V ) by a polarising beam-splitter before being detected by
two APDs, APD1 and APD2. The light selected by the second spectrometer (S2)
was detected by APD3. The signals from the APDs were used as the start and
stop pulses for two SPCMs.
In this way, the correlation function between the light selected by spectrometer
S2 and the H or V component of the light selected by spectrometer S1 could be
measured. A histogram of the time delay between photons arriving at APD1/2
and APD 3 was built up. The form of the histogram is approximately equivalent
to the correlation function between the two lines.
3.3.5 Application of a magnetic field
A magnetic field was used in some aspects of this work. To apply the field to a
sample, a cryostat with a cold finger extending into the bore of a superconducting
magnet was used. The magnet had a range of up to 5 T. At the centre of the
bore, the magnetic field was parallel to the axis of the bore. The samples could be
mounted on top of the cold finger with the field parallel to the growth direction
of the QDs - the Faraday geometry (figure 3.7 (a)) - or a copper mount and a
prism could be used to achieve the Voigt geometry (figure 3.7 (b)), with the field
in the plane of the dots.
32
3.3 Optical techniques
APD 1
SPCM 1
Start
Stop
APD 2Spectrometer
2
APD 3
PL
Polarising 
beam splitter
SPCM 2
Start
Stop
Spectrometer 1
50-50 beam 
splitter
Delay
Figure 3.6: Schematic diagram of the experimental setup for cross-correlation
measurements
B
Window
Cryostat
Sample Prism
Copper mount
Laser/
white light/PL
Cold finger(a) (b)
Figure 3.7: Schematic diagram of the experimental setup used for applying a
magnetic field in (a) the Faraday geometry, and (b) the Voigt geometry.
33
3.4 Spectroscopy of quantum dots
3.4 Spectroscopy of quantum dots
A typical spectrum of the emission from a large number of quantum dots is shown
in figure 3.8. An emission peak at around 830 nm is associated with excitons
bound to impurities in GaAs [1]. Another peak at around 870 nm is due to
emission from the wetting layer, which behaves like a quantum well [19]. Finally,
the sharp features at longer wavelengths are a result of emission from quantum
dots. Figure 2.2 shows PL from a single QD isolated underneath a hole in an
aluminium mask. Four lines from different excitonic states are present, as labeled.
There are a number of spectroscopic methods used to give information about
individual excitonic lines, which will be detailed in the following subsections.
Wetting layer
Quantum dots
GaAs impurities
800 850 900 950
0
PL
 in
ten
sit
y (
arb
. u
nit
s)
Wavelength (nm)
Figure 3.8: Typical spectrum of the emission from a large number of InAs quan-
tum dots.
3.4.1 Power dependence
Measuring the relationship between laser pump power and the integrated inten-
sity of a QD emission line can give information about which excitonic state the
34
3.4 Spectroscopy of quantum dots
line originates from. Each photon absorbed by the semiconductor will excite
one electron from the valence band into the conduction band, leaving a hole be-
hind. Therefore, one would expect the intensity of the exciton line, which results
from the recombination of one electron and one hole, to vary linearly with laser
power. The biexciton line, resulting from two electrons and two holes, should
vary quadratically.
When exciting with a CW laser at high enough power, repopulation of the
biexciton state from the exciton state will become favourable over the decay of the
exciton to the ground state. Therefore the biexciton will dominate the spectrum
at high powers. When exciting with a pulsed laser whose pulses are much shorter
than the lifetime of the excitonic states, and whose repetition period is much
longer than the state lifetimes, it is likely that the exciton will always decay
following the biexciton decay. In this case, the exciton and biexciton lines will
saturate at high laser powers at roughly the same intensity (as in figure 3.9).
880 888 896
0
X
2X
Int
eg
rat
ed
 in
ten
sit
y
(ar
b. 
un
its)
Laser power (arb. units)
0.9±0.1
2.0±0.2
X
PL
 in
ten
sit
y (
arb
. u
nit
s)
2X
Figure 3.9: Identification of an exciton and a biexciton by power dependence
under pulsed excitation.
35
3.4 Spectroscopy of quantum dots
3.4.2 Polarization of excitons
As described in section 2.2.3, the exciton state is usually split in energy, resulting
in a polarisation splitting in the PL from the exciton and the biexciton. This
splitting can be observed experimentally and can be used to distinguish between
the neutral excitons and the singly charged excitons, which exhibit no polarisation
splitting.
36
Chapter 4
p-i-n diode quantum dot devices
4.1 Overview
To date, a variety of devices using quantum dots have been developed. These
include single photon sources [15, 29, 34, 80, 93, 117, 151] and detectors [13, 21],
sources of polarisation-entangled photon pairs [4, 17, 130, 150] and quantum dot
lasers [20, 105, 148]. This chapter details experiments carried out with a single
quantum dot embedded in the intrinsic region of a p-i-n diode. The chapter begins
with a description of the device and goes on to describe experiments showing its
use to maximise the polarisation correlation of the biexciton decay, and as a
memory device.
4.2 Sample design and operation
The samples used in this study were grown by MBE, consisting of a GaAs λ/n-
thick planar optical cavity [14], centred at ∼900 nm, containing a layer of InAs
QDs at the centre and 5 nm above an AlAs/GaAs superlattice. The cavity
was produced by growing twelve repeats of GaAs/AlAs DBR mirrors below the
cavity and two repeats above it. The top mirror was p-doped with carbon and
the top two repeats of the bottom mirror were n-doped with silicon, to create
a p-i-n diode. The superlattice, which prohibits electrons from tunneling out of
the QDs, was formed by seven repeats of 2 nm-thick layers of Al0.98Ga0.02As and
37
4.2 Sample design and operation
12 repeats 
GaAs/AlAs DBR
AlAs/GaAs 
superlattice 
barrier
InAs QDs
2 repeats p-doped 
GaAs/AlAs DBR
Al shadow maskTop contact
GaAs cavity
Back 
contact
2 repeats 
n-doped
Figure 4.1: Schematic diagram of the p-i-n diode quantum dot device structure
in profile.
1.20 1.25 1.30 1.35 1.40
896
900
904
X+
2X
X
Bias (V)
Wa
ve
len
gth
 (n
m)
X-
Figure 4.2: Plot showing the evolution of the spectrum from a QD in the device
as the forward bias applied across the diode is swept. Blue indicates high PL
intensity, white indicates low intensity. The neutral exciton (X), biexciton (2X)
and charged excitons (X+, X−), identified by power dependence and observations
of the fine structure, are labeled. The other weak lines in the spectra are most
likely from more highly positively charged states.
38
4.2 Sample design and operation
Al0.5Ga0.5As. Figure 4.1 shows a schematic diagram of the sample design.
An aluminium shadow mask of ∼2 µm diameter circular apertures was evap-
orated onto the surface of the device to allow optical access to individual QDs.
Pits were etched down to the n-doped region so that titanium and gold could
be evaporated to electrically contact the region. More titanium and gold was
evaporated to contact the p-doped region via the shadow mask. The contacts
allowed an electric field to be applied across the QDs in the cavity.
Figure 4.2 shows PL from a single QD in the device as a function of the bias
applied across the diode. The device was operated in forward bias and no electro-
luminescence was observed in the range of biases used in this study. A number of
lines are present in the spectra, originating from different excitonic states [48, 77],
as labeled. The existence of emission from different states is a result of the random
trapping of electrons and holes by the QD before radiative decay. By changing
the bias across the device, the dominant excitonic states present in the spectrum
can be changed. This is due to the changing hole tunneling rate from the dot, as
the bias is varied, illustrated in figure 4.3.
(a) (b)
GaAs
AlAs
InAs
Valence band
Conduction band
Figure 4.3: Simplified diagrams of the band structure of the device with a low
applied forward bias (a) and a higher applied bias (b). The electron tunneling
barrier, grown 5 nm underneath the QD layer, is shown by a higher potential
barrier to the right of the InAs QD. The blue arrows represent the hole tunneling
rates.
39
4.3 Controlling the polarisation correlation of photon pairs
4.3 Controlling the polarisation correlation of
photon pairs
The intensities of the emission lines in figure 4.2 were integrated and plotted as
a function of bias in figure 4.4. Below ∼1 V, no PL was measured from the QD,
indicating that the rate at which the holes tunnel from the dot is much greater
than the radiative decay rate of any of the QD optical transitions. For a bias of 1-
1.25 V, emission from the neutral exciton (X) is very dim since the hole tunneling
rate is faster than its radiative decay rate. The biexciton (2X) is brighter since
its radiative decay rate is faster than that of X. The negatively charged exciton
(X−) is dominant in the spectrum since the most likely decay path from 2X is
for a hole to tunnel out, forming X−. If the second hole tunnels out before the
radiative decay of X−, no emission will be seen.
1.20 1.25 1.30 1.35 1.40
0
X
2X
Int
eg
rat
ed
 in
ten
sity
 (a
rb.
 un
its)
Bias (V)
X
-
X
+
Figure 4.4: Integrated intensities of the emission lines in figure 4.2 as a function
of bias applied across the diode.
As the bias is increased, the hole tunneling rate is decreased below the radia-
tive decay rates of X and 2X, so that they become dominant in the spectrum. At
higher biases, the positively charged exciton (X+) is dominant since holes may
drift into the dot from the p region, while electrons may not drift in from the n
40
4.3 Controlling the polarisation correlation of photon pairs
region due to the tunneling barrier. There are also some other weak lines present,
most likely resulting from more highly positively charged states.
Figure 4.2 shows that it is possible to control the particular excitonic states
which are present in the emission from a QD with this device [39, 140]. This
property could be exploited, for example, by tuning out the unwanted lines from
the QD emission for a given application. The associated increase in the intensity
of the desired line(s) and reduction in background light would benefit many QD
applications such as the generation of triggered photon pairs by the biexciton
decay [130].
4.3.1 Degree of polarisation correlation
-20 0 20
0
1
2
3
-20 0 20
2X, X
2X, X
1.29 V
Se
co
nd
 or
de
r c
orr
ela
tio
n g
(2)
1.27 V
-20 0 20
1.31 V
PeriodPeriod (12.5 ns)
Figure 4.5: Second order correlation of the H polarised biexciton photon with
the H exciton photon (black) and the V exciton photon (red) at different biases.
The period of the laser is 12.5 ns.
This particular QD has a considerable fine structure splitting [43] (S0 >
20µeV). Hence the exciton and biexciton photons are expected to have identical
rectilinear polarisation of H or V [115, 129, 134] (see figure 2.3). Cross-correlation
measurements were carried out to study the degree of rectilinear polarisation cor-
relation, C, between the biexciton and exciton photons as the bias was varied.
41
4.3 Controlling the polarisation correlation of photon pairs
The second order correlation of the H polarised biexciton photon with H
exciton photons and with V exciton photons were simultaneously measured using
the setup described in section 3.3.4. Figure 4.5 shows results for three different
biases. It is clear that the co-polarised photons are correlated and the oppositely
polarised photons are anti-correlated at zero delay, to varying degrees. This
implies that the emission of a H polarised exciton photon is much more likely
than the emission of a V polarised photon following the emission of a H polarised
biexciton photon, as expected.
The results were used to calculate the degree of rectilinear polarisation corre-
lation, C, using
C =
g
(2)
2X,X − g(2)2X,X¯
g
(2)
2X,X + g
(2)
2X,X¯
, (4.1)
where g
(2)
2X,X is the second order correlation function at zero delay between the
H biexciton photons and the H component of the exciton emission, and g
(2)
2X,X¯
is
the correlation between the H-polarised biexciton photons and the V -polarised
exciton photons. The second order correlation function is given by equation
3.1. C is therefore expected to be 100% for an ideal QD emitting polarisation
correlated photons in the rectilinear basis, and 0% for an uncorrelated source.
The red data in figure 4.6 shows the measured C as a function of applied bias.
The error is the smallest when the exciton and biexciton emission are at their
strongest, and increases as they fade away. This is as expected because when the
count rate is small there is a large error. It can be seen that for high biases, C
is high as predicted, e.g. 68 ± 6% at 1.33 V. However at low biases, it tails off
unexpectedly, to 13± 5% at 1.25 V.
Previous studies of nominally uncharged QDs have shown that the degree of
correlation can be limited by background light due to emission from other layers
of the device, and by exciton spin scattering effects [150]. The former is clearly
not the dominant effect since it would be minimal when the emission is at its
maximum at ∼1.31 V. However C increases from 56± 4% at 1.31 V to 68± 6%
at 1.33 V. This suggests that the sharp drop in C at low biases is due to exciton
spin scattering.
42
4.3 Controlling the polarisation correlation of photon pairs
1.20 1.25 1.30 1.35 1.400
25
50
75
100
2
3
4
C (
%)
Bias (v)
Lif
eti
me
 ra
tio
τ X/
τ 2X
Figure 4.6: The ratio of the biexciton and exciton states’ lifetimes (blue) and
the degree of rectilinear polarisation correlation, C, between the biexciton and
exciton photons (red). Error bars span two standard deviations of the random
error. Where not shown, the random error is smaller than the symbol size.
43
4.3 Controlling the polarisation correlation of photon pairs
4.3.2 Lifetime measurements
Further evidence that spin scattering is responsible for the drop in C is provided
by lifetime measurements of the exciton and biexciton which were taken at the
same time as the correlation measurements. The ratio between the exciton and
biexciton lifetimes was calculated for each bias and is plotted as the blue line in
figure 4.6.
The ratio between the lifetimes is expected to be ∼2 for slow exciton de-
phasing [99, 143] which is the typical result for InAs QDs. This result can be
approximately understood by considering the fact that the biexciton has two dif-
ferent decay paths (the m = ±1 excitons), whereas the exciton only has one. It is
therefore expected that the radiative lifetime of the biexciton will be around half
that of the exciton. The measured value is close to 2 at high biases. However at
lower biases, as the degree of correlation drops off, the ratio increases and peaks
at 3.62 ± 0.02. The increase is due mainly to an increase in the lifetime of the
exciton. The spectral lines shift in energy by a very small amount over this range
of biases and therefore changes in the lifetimes due to the quantum confined Stark
effect are expected to be insignificant.
In the case of a fast spin flip rate, theoretical studies have predicted a value
of ∼ 4 for the lifetime ratio [99]. If the spin of a carrier in an exciton is flipped,
a dark exciton is formed which has a very long lifetime. For a fast spin flip rate,
the exciton spends half of its time in the dark state. This would have the effect of
doubling the exciton lifetime and would also explain the drop in C. The lifetime
ratio drops off at lower biases as the hole tunneling time becomes comparable to
the exciton’s radiative lifetime.
4.3.3 Polarisation correlation control
This device allows control over which excitonic state is dominant in the spectrum
from a QD. When light from the charged states is minimised, the emission from
the neutral states is maximised as expected. For low biases the degree of polar-
isation correlation between the biexciton and exciton photons is limited by spin
scattering.
44
4.4 Single quantum dot memory
At low biases, there will be an excess of electrons in the vicinity of the dot
which can cause the exciton’s electron spin to be flipped. Excess electrons may
be confined at the heterojunction at the tunneling barrier. The polarisation
correlation does not drop off at biases where an excess of holes would be expected,
since there is no hole tunneling barrier to trap holes.
It is also possible that a hole spin flip is responsible for the exciton dephasing,
for example due to interactions with other holes from the acceptor level of the p
region.
Choosing a bias to maximise the emission from the biexciton decay and the
degree of polarisation correlation would benefit QD devices based on the biexciton
decay, such as an entangled photon pair source.
4.4 Single quantum dot memory
The experiments in this section detail the operation of the same device as in the
previous section, acting as a single electron-spin memory.
4.4.1 Quantum information processing
The basic unit of quantum information is a qubit, which is the quantum analogue
of the classical bit [40]. Various physical systems have been proposed to encode
a qubit, including the polarisation of a single photon [69, 70], the spin of an
electron or an atomic nucleus [26, 31], and the charge state of a quantum dot or
superconducting island [84, 98]. The use of the polarisation of a photon makes
single qubit operations straight forward, i.e. using appropriate wave plates. Fur-
thermore photons have the advantage that they are easy to transmit between two
distant points, as required for the ‘flying qubit’ in, for example, quantum cryp-
tography [16] and photonic quantum computing [69]. However photons are hard
to store for a long time and do not interact strongly with one another, hindering
simple multiple qubit operations. On the other hand, electron spins are easy to
store and manipulate in a suitable solid-state environment but are difficult to
transmit over long distances.
45
4.4 Single quantum dot memory
The experiments described in this section show that the QD in a p-i-n diode
device can act as an interface between these two systems, so that the advantages
of both can be accessed. The polarisation information of a single photon is stored
as the spin of a single electron trapped in a QD and is deterministically recovered
some time later.
4.4.2 Device operation
1.0 1.2 1.4
894
897
900
X+
X
2X
Bias (V)
Wa
ve
len
gth
 (n
m)
X-
Figure 4.7: PL from a single QD in the device with a magnetic field of 2 T in the
Faraday geometry, as a function of applied bias.
Figure 4.1 shows the structure of the device, as before. Figure 4.3 show a
simplified versions of the band structure of the device around a single QD. In the
diagram, the electron tunneling barrier, grown 5 nm underneath the QD layer, is
represented by a higher potential barrier to the right of the QD.
The operation of the device shares many similarities with the device presented
in reference [73], in which an ensemble of InAs QDs are embedded in the intrinsic
region of a p-type GaAs Schottky diode. The use of a p-i-n diode in this work
allows the integration of an optical cavity into the design (see figure 4.1), to
46
4.4 Single quantum dot memory
enhance the intensity of the emission. Furthermore, by addressing a single QD, a
significant step has been made towards the quantum information processing goal
of storing the polarisation information of a single photon.
The device is operated with an applied magnetic field of 2 T in the Faraday
geometry. This field is big enough so that the Zeeman interaction is much larger
than the fine structure splitting (i.e. gµBB  S0) so that the two m = ±1 exciton
pure spin states are recovered (see section 2.2.5). The photons emitted in the
decay of these states are therefore circularly polarised. Figure 4.7 shows PL from
a single QD in the device with the field turned on, as a function of bias applied
across the diode. As before, in figure 4.2, the bias controls which excitonic state
is dominant in the spectrum. However, the magnetic field introduces a Zeeman
interaction and splits the lines in energy into well resolved doublets.
σ -
σ
Write mode
Readout mode
(a) (b) (c)
(d) (e) (f)
GaAs
AlAs
InAs
Valence band
Conduction band
Figure 4.8: Simplified band diagrams illustrating the operation of the device.
47
4.4 Single quantum dot memory
When a circularly polarised photon is absorbed by the device, an electron is
promoted to the conduction band whose spin (up or down) is deterministically
determined by the polarisation of the initial photon (left or right), according to
the conservation of angular momentum selection rules (see section 2.2.3). In the
write mode of the device (figure 4.8 (a)-(c)), it is biased so that the hole tunneling
rate is much faster than the radiative lifetime of any of the optical transitions.
Carriers are excited into the wetting layer by a circularly polarised laser pulse.
The pulse is weak so that on average only one electron and one hole will relax into
the QD. The hole will tunnel out from the dot and drift towards the p region,
whereas the tunneling barrier prohibits the electron from leaving the dot. A
single electron is left in the quantum dot whose spin state is determined by the
polarisation of the initial photon.
In readout mode (figure 4.8 (d)-(e)), a higher positive bias is applied to the
device to return two holes to the dot. Radiative recombination of the stored
electron with a hole results in the emission of an X+ photon with identical polari-
sation to the initial one. It can be ensured that two holes are returned to the dot
by using the correct positive bias so that emission from X+ is dominant. The use
of two holes ensures that there is always a hole with the correct angular momen-
tum to annihilate with the electron. This means that long lived dark excitons
are not formed which would slow down the operation of the device and reduce its
efficiency.
4.4.3 Experimental set-up
The simplified experimental arrangement used to test the device’s memory op-
eration is shown in figure 4.9. Section 3.3 contains more details. A multimode
diode laser emitting at 869 ± 5 nm which had pulse lengths of ∼ 100 ps and
a variable repetition rate was used to excite carriers in the wetting layer of the
device. The laser was linearly polarised before passing through a quarter wave
plate (QWP) with its fast axis orientated at ±45◦ to the angle of polarisation.
Thus the beam was left or right circularly polarised when it excited the device.
PL from the device was passed through the same QWP which rotated circularly
polarised components back to the linear basis. Finally a combination of a half
48
4.4 Single quantum dot memory
Laser
Sample
Spectrometer
Polariser
Polariser
QWP ±45° HWP 0/45°
DC +
AC bias
Cold 
mirror
Figure 4.9: Simplified schematic diagram of experimental set-up for the memory
experiment (see figure 3.2 for more detail).
wave plate (HWP) and a linear polariser selected one of the linear polarisations
(H or V ) to be transmitted to the spectrometer. It was therefore possible to
excite with and detect either of the circular polarisations.
An AC pulse generator was synchronised with the laser’s clock and applied
an AC pulse to the device some time (shorter than the period of the laser) after
the laser pulse. The circularly polarised laser light populated the quantum dot
with an electron. During the delay time, it was anticipated that the electron’s
spin would be preserved. The application of the AC pulse switched the device to
readout mode, and light of the original polarisation was expected to be emitted
from the device at the wavelength of the positively charged exciton. A small DC
bias was also applied which was used to optimise the emission.
4.4.4 Quantum dot electron-spin memory
Figure 4.10 (a) shows time resolved PL from the X+ state of a single QD in the
device. A DC bias of 1.4 V was applied so that emission from the state was
triggered by the laser pulse. Figure 4.10 (b) shows time-resolved PL from the
same QD, with the laser pulsed at 1 MHz and with an AC pulse applied to the
device on top of a small negative DC bias. The delay between the laser pulse and
49
4.4 Single quantum dot memory
0 5
QD PL
Laser
PL
 in
ten
sit
y (
arb
. u
nit
s)
Delay (ns)
(a)
0.0 0.2 0.4 0.6 0.8 1.0
(b)
La
se
r p
uls
e
1µs
0.8µs
0.6µs
0.4µs
Delay (µs)
0.2µs
Figure 4.10: (a) Time-resolved PL of the laser (red) and the positively charged
exciton of a single QD (black) with an applied DC bias of 1.4 V. (b) Time-resolved
PL from the same dot with a DC bias of -0.05 V to remove the holes from the
dot. 0.2 to 1.0 µs (as labeled) after the laser pulse, an AC pulse of 2.7 V was
applied to return two holes to the dot. Emission from the X+ state is delayed
and correlated with the AC pulse.
50
4.4 Single quantum dot memory
the AC pulse is varied from 0.2 to 1 µs. The emission is clearly triggered by the
AC pulse and not the laser pulse.
When the laser light was removed, no luminescence was measured. This
demonstrates that the emission is due to an electron being stored in the QD
and is not due to electroluminescence induced by the AC pulse. The variation
in the area of the peaks in figure 4.10 (b) is due to fluctuations in the laser
pulse intensity, caused by small lateral movements of the device in the cryostat.
Impedance mismatches between the circuitry used to deliver the AC pulses and
the device meant that the pulses were somewhat attenuated before they reached
the device. Therefore a higher voltage AC pulse was required to put the device
into a given state as compared to the DC bias equivalent. An electron storage
of up to 1 µs was measured with no observable decay in the intensity, indicating
that longer storage times should be possible.
Figure 4.11 shows the polarised PL emitted from the delayed X+ state with
a magnetic field of 2 T in the Faraday geometry, when exciting the sample with
differing polarisations. The AC pulse was delayed by ∼ 1µs with respect to the
laser pulse. With unpolarised excitation, each pair of lines shown in figure 4.7
were polarised in opposite circular polarisations, as expected. When the laser light
was horizontally linearly polarised (top panel), two equal peaks corresponding to
the two pure spin states of X+ were measured. This occurs because the linearly
polarised laser light excites equal numbers of up and down spin electrons in the
device so that the spin of the electron confined in the dot is random.
The bottom two panels of figure 4.11 show that when the laser light was
circularly polarised, the vast majority of the delayed emission had the same po-
larisation. This demonstrate that the spin of the electron is preserved during the
process of non-resonant excitation, relaxation into the dot, removal of the hole
from the dot and the re-injection of two holes.
The fidelity of the polarisation memory, defined as
F = IL,R/(IL + IR), (4.2)
was measured to be 90 ± 5% for both laser polarisations and constant over the
measured delay range of 0.2-1 µs. IL is the integrated intensity of the left-hand
51
4.4 Single quantum dot memory
894 895
R
LLPL
 in
ten
sity
 (a
rb.
 un
its)
Wavelength (nm)
R RL
H
V
Detection
polarisation:
Excitation
polarisation: H
Figure 4.11: PL from X+ delayed by ∼ 1µs in a single QD with a magnetic field
of 2 T applied in the Faraday geometry. The pump polarisation and detection
polarisation are different in each of the panels - horizontal (H), vertical (V), right
hand circular (R) or left hand circular (L) - as labeled. The black dotted line is
shown as a guide to the eye.
52
4.4 Single quantum dot memory
circularly polarised X+ emission peak and IR is that of the right-hand circularly
polarised peak. The value is not 100% due to polarisation errors in the experi-
mental set-up and a degree of spin-scattering during the relaxation of the electron
into the dot after the non-resonant excitation, i.e. a non-unity fidelity of the write
operation.
The asymmetric line shapes of the emission in figure 4.11 are a result of a
variable Stark-shift due to the varying electric field from the AC pulse. The
turn on time of the AC pulse, which was limited by the equipment used, was of
the same order as the radiative lifetime of the X+. Therefore, the energy shift
induced by the Stark effect varied throughout the readout cycle. The slow turn
on time also allowed some light to be emitted from the neutral exciton line which
was not collected in the experiment. This caused a factor of 2 difference in the
intensity of the electrically triggered and laser triggered X+ emission.
4.4.5 Summary and outlook
Storage of the polarisation information of a photon as the spin state of an electron
in a QD and its electronically triggered readout has been demonstrated. The
difference in energy between the input and output photons could allow the transfer
of information between systems operating at different wavelengths.
A storage time of up to 1 µs was demonstrated, although this was limited by
the repetition rate of the device, which was in turn limited by the intensity of the
measured emission. It is expected that the spin memory will persist longer than
this. Similar experiments using multiple QDs have shown storage times of over 1
ms [73].
The readout time was limited by the radiative lifetime of the X+ transition.
Coupling the transition to a high-Q mode of an optical cavity (for example a
photonic crystal cavity - see chapters 6 and 7) would decrease the lifetime of the
state and therefore increase the possible speed of operation.
The non-resonant excitation in the wetting layer limited the degree of polar-
isation memory. The use of larger QDs with more confined energy levels could
allow quasi-resonant excitation through a fast relaxing excited state. This would
53
4.4 Single quantum dot memory
improve the transfer of polarisation information from the input laser to the stored
electron’s spin.
For a true quantum memory, resonant excitation would be required to enable
writing of superposition states, and storage of the hole would be necessary. This
could be done, for example, with a quantum dot molecule.
54
Chapter 5
Designing and fabricating
photonic crystals
5.1 Overview
Quantum dot based devices, like the one discussed in the previous chapter, can
benefit from an optical cavity. In section 2.3, it was shown that cavity effects can
enhance the spontaneous emission rate from a QD and can drastically increase
the collection efficiency of the emission from the device. In the same section, it
was shown that photonic crystal cavities are the best choice for many applications
because of the high Q-factors achievable, along with their tiny mode volumes.
This chapter details efforts made in designing and fabricating photonic crystal
structures with the goal of enhancing the properties of quantum dot devices. It
begins with a discussion on the simulation technique used to design the structures
and goes on to discuss the problems encountered when trying to fabricate them.
5.2 Finite-difference time-domain simulation
Finite-difference time-domain (FDTD) simulation was used to design the photonic
crystal structures studied in this work. FDTD is a simulation technique which
provides a numerical solution to Maxwell’s equations for the electromagnetic fields
in a material with arbitrary (r, ω) and µ(r, ω). It relies on the fact that in
55
5.2 Finite-difference time-domain simulation
Maxwell’s equations, a change in the electric field in time is dependent on the
change in the magnetic field in space, and vice versa (see equation A.1).
The simulation domain is split into a grid with grid points spaced ∆x, ∆y and
∆z apart. At each grid point (i, j, k) the electric and magnetic field components
are calculated at the points on a Yee cell, as shown in figure 5.1. Time is also
discretised into steps of ∆t. The calculation is carried out in an iterative manner:
the electric field at each grid point is calculated at time t = n∆t, followed by
the magnetic field at time t = (n+ 1/2)∆t, followed by the electric field at time
t = (n+1)∆t, where n is an integer, and so on. The electric field at time t = n∆t
is equal to the electric field at t = (n − 1)∆t plus an additional term computed
from the spatial variation of the magnetic field at time t = (n− 1/2)∆t.
Ey
Ey
Hz
Ex
Ez
Hx
Ey
Ez
Ex
Hy
Ez
Ex
x
y
z ∆y
∆z
∆x
(i, j, k)
Figure 5.1: Yee cell used in FDTD simulation. The electric and magnetic field
components are computed at points shifted half the grid spacing from one another.
In practice the material parameters, the computational domain, and details
of the electromagnetic field excitation are input into a simulation. The spatial
grid is defined, which should be small enough to resolve the smallest feature of
the simulated field, and the time step is chosen, which must obey the Courant
condition,
56
5.3 FDTD for photonic crystals
c∆t <
1√
1/∆x2 + 1/∆y2 + 1/∆z2
, (5.1)
to obtain a stable simulation. The boundary conditions at the edges of the
computational domain must also be specified. The most common choice in this
work is the perfectly matched layer (PML), which absorbs the incident radiation
without producing back reflections.
5.3 FDTD for photonic crystals
A commercial FDTD package, RSoft FullWAVE, was used for simulations in this
work. This section will describe the particular techniques used to calculate the
spectrum of a given photonic crystal structure and to find the Q-factor, profile
and volume of its confined modes.
5.3.1 Boundary conditions
PML boundary conditions were used for most of the boundaries as shown in
figure 5.2. However, a symmetric mirror boundary condition was used to cut the
simulation domain in half through the centre of the slab. This was possible since
all the TE-like modes of interest in this work are symmetric about the centre of
the slab, with only one anti-node in the field profile.
5.3.2 Calculating the spectrum
To calculate the spectrum, the structure of interest is first drawn out in a CAD
environment, with the relevant refractive index assigned to each component. A
number of electric dipole-like sources are then positioned throughout the cavity.
Electric field monitors are placed inside the cavity and just outside the cavity.
Figure 5.3 shows a typical FDTD layout for a spectrum calculation.
The sources are made to emit a TE-polarised broadband impulse field at
time zero. As the simulation progresses in time, this field is allowed to dissipate
throughout the simulation domain. Most frequencies of the impulse field will pass
57
5.3 FDTD for photonic crystals
PML
PML
PM
L PML PML
Symmetric
Figure 5.2: Planar view (left) and cross-sectional view (right) of the typical
boundary conditions used for photonic crystal simulations. The purple lines
represent the simulation domain edges with the boundary conditions, perfectly
matched layers (PML) or symmetric mirrors, labeled.
3.0 3.5 4.0 4.5
0.00
Monitors inside cavity
Outside cavity
Normalised wavelength (λ/a)
(b)(a)
Ele
ctr
ic 
fie
ld 
am
pli
tud
e (
arb
. u
nit
s)
Figure 5.3: (a) Typical layout for an FDTD calculation of the spectrum of a
photonic crystal cavity. Grey represent GaAs regions and red represents air re-
gions. The purple line is the simulation boundary. The small yellow boxes are
electric dipole-like sources and the small green boxes are electric field monitors.
(b) Typical result from a spectrum calculation of an unmodified H1 cavity with
r/a = 0.3, d/a = 0.7, n = 3.53 and 5 photonic crystal periods surrounding the
defect. The black and red lines are the sums of the spectra from the monitors
inside and outside the cavity respectively.
58
5.3 FDTD for photonic crystals
relatively undisturbed throughout the structure and will be picked up by all the
monitors, inside and outside the cavity. Those frequencies within the photonic
band gap will be inhibited from passing through the photonic crystal and will
not be picked up by the monitors outside the cavity. The resonant frequencies of
the TE cavity modes will be confined in the cavity and will be picked up mainly
by the monitors inside the cavity.
The monitors are set to monitor the electric field as a function of time. After
the simulation period has ended, a Fourier transform of the data is calculated to
provide the frequency response at each monitor point, in which cavity modes are
manifested as narrow peaks. A number of monitors are used and their spectra are
summed together to give an approximate total spatially averaged cavity response.
Care must be taken to avoid the situation in which one monitor is positioned at
a point of high field intensity for a particular resonant mode. In this case, the
spectrum may be dominated by the frequency of that mode, and other features
may be swamped.
Figure 5.3 (b) shows a typical result for a H1 cavity. The black line is the
summed spectra from five monitors placed inside the cavity, clearly showing a
mode at λ/a ∼ 3.6. The red line is from five monitors placed outside the cavity,
which makes the extent of the photonic band gap clear. A small peak at the
wavelength of the mode appears in this spectrum, showing that the use of five
photonic crystal periods is insufficient to strongly confine the mode (see section
5.3.7).
5.3.3 Q-factor
An intuitive way to calculate the Q-factor of a mode would be to monitor the
decay of the field energy stored in the mode as a function of time. By fitting the
result to an exponential decay, Q can be extracted. However, this method can
require prohibitively long simulation times to achieve accurate results.
Instead, a method based on Fourier analysis is implemented in RSoft Full-
WAVE. In this method, the field picked up by a monitor is approximated as a
sum of complex exponentials:
59
5.3 FDTD for photonic crystals
f(t) =
∑
j
cje
−iω˜jt, (5.2)
where the complex frequency ω˜j = ωj + iγj contains the oscillatory frequency ωj
and the loss rate γj, which is simply related to Q. The entire field is represented
by a number of terms in this sum; even the parts of the field which radiate away
quickly can be considered to behave like this but with a very high loss rate. This
changes the problem to a fit of the parameters cj and ω˜j and allows accurate
values of ω˜j to be extracted from relatively short simulation times.
5.3.4 Mode profile
Mode profiles were calculated using two steps. Firstly, randomly positioned
sources inside the cavity were set to emit a spectrally narrow CW field for the
duration of the simulation, with a frequency identical to that of the mode of in-
terest. A large area spatial field monitor was positioned at the centre of the slab
and was set to return the values of Ex, Ey and Hz at the end of the simulation.
The simulation time was set to be long enough so that a steady state situation
was reached by its end. Stopping the simulation part way through an optical
cycle ensured that energy was contained in both the electric and magnetic fields,
so that both their profiles could be obtained from one simulation.
Following this, a second similar calculation was carried out with a single source
positioned at a maximum in the electric field of the mode, to obtain a more precise
profile, without any source induced distortions.
5.3.5 Mode volume
The mode volume is defined as
V0 =
∫
V
UE(r)
max[UE(r)]
dV, (5.3)
where
60
5.3 FDTD for photonic crystals
UE(r) =
1
2
(r)|E(r)|2, (5.4)
is the electric energy density of a mode and V is the whole simulation domain. It
can be calculated directly from the mode profile. Since the electric energy density
varies in time, for consistency the mode volume is evaluated at the point in the
optical cycle where the electric energy is maximum.
5.3.6 Purcell factor
The maximum possible Purcell factor for a well positioned QD can be calculated
by inputing the calculated values into equation 2.12.
5.3.7 Simulation convergence studies
4 6 8 10 12 14 16 18
0.70
0.75
3.30
3.35
3.40
2000
3000
N
V 0
(λ/
n)3
Q
λ/a
Figure 5.4: Results of simulations for the monopole mode of a H1 photonic crystal
cavity as the size of the grid, ∆x = ∆y = ∆z = a/N , is decreased. The cavity
parameters are r = 0.36a, r’ = 0.2a, d = 0.7a with 5 photonic crystal periods
surrounding the defect. The dashed blue line indicates the value used in the rest
of this work.
61
5.3 FDTD for photonic crystals
Before any useful calculations could be performed, it was necessary to find
suitable simulation conditions. Figure 5.4 shows the results of calculations in
which the grid size of the simulation was progressively made smaller. For each
grid size, the wavelength, Q-factor and mode volume were calculated for the
monopole mode in a modified H1 cavity with r/a = 0.36, r′/a = 0.2, d/a = 0.7
with 5 photonic crystal periods surrounding the defect. r′ is the radius of the six
nearest neighbour holes which are modified to increase the Q-factor of the mode,
as discussed in section 6.3.
For large grid dimensions, the features of the photonic crystal are not resolved
properly and the resulting Qs are artificially low. As the grid size is made smaller,
the structure is better resolved in the simulation, and the calculated Q converges
towards the real value. The wavelength and mode volume follow similar patterns.
This result motivated the choice of ∆x = ∆y = ∆z = a/15 as the grid size used
in the rest of this work.
2 3 4 5 6 7 8 9
0.8
0.9
3.278
3.280
0
2000
4000
6000
Number of periods
V 0
(λ/
n)3
λ/a
Q
Figure 5.5: Results of simulations of the monopole mode of a H1 photonic crystal
cavity as the number of periods is increased. The cavity parameters are r = 0.36a,
r’ = 0.2a and d = 0.7a. The grid size is ∆x = ∆y = ∆z = a/15.
Figure 5.5 shows a similar set of calculations as the number of photonic crys-
tal periods surrounding the defect is increased. For small numbers of periods,
62
5.4 Fabrication of photonic crystals
the mode is not well confined in-plane, and the calculated Q is low. At around 8
periods, Q saturates as the out-of-plane losses become the dominant loss mech-
anism. The wavelength and mode volume on the other hand reach stable values
at around 5 periods. The rest of this work uses at least 5 periods surrounding
the defect. The number is chosen for each simulation with the trade off between
precision and long simulation times in mind.
5.4 Fabrication of photonic crystals
The task of fabricating photonic crystal structures is a considerable one. To
produce a photonic band gap at a wavelength of around 900 nm, where the QDs
emit, air holes with radii around 100 nm are required. Any small errors in the
fabrication of the structure can lead to a dramatic loss in the quality factor of the
cavity modes [6, 44]. For example, the walls of the holes must be both straight
and smooth. Roughness can lead to optical scattering at the surfaces, and non-
straight side walls leads to a non-optimum photonic band gap.
In this work, great effort was put into finding processing conditions to produce
high quality structures. Several improvements were made to an initial processing
method [46]. The quality of the processing was evaluated by examining the
structures with a scanning electron microscope (SEM) and by comparing their
measured Q-factors to the simulated values. In the following sections, the quoted
Q-factors are for the monopole mode of an optimal modified H1 cavity (see section
6.3). The cavity is formed by shifting outwards and reducing the size of the six
nearest neighbour holes surrounding a single defect, and has a simulated Q-factor
of ∼ 30000.
A number of the changes made in the processing had knock-on effects on other
areas. In particular, many of the changes required that the etching processes
be re-optimised. For the sake of brevity and clarity, the details of these re-
optimisation steps are not included.
63
5.4 Fabrication of photonic crystals
PMMA
InAs QDs
GaAs
AlGaAs
(a) (b) (c) (d)
Figure 5.6: Schematic diagram of the initial method to produce photonic crystals.
A layer of electron-beam patterned PMMA (b) is used as an etch mask for a
SiCl4/Ar based RIE etch of the GaAs slab (c). HCl is used to selectively etch the
underlying AlGaAs layer (d).
5.4.1 Initial process
Figure 5.6 shows the initial scheme to fabricate two-dimensional slab photonic
crystals. A wafer is grown comprising a 100 nm GaAs buffer layer on a GaAs sub-
strate followed by a 500 nm AlGaAs sacrificial layer and a 150 nm GaAs slab with
a layer of InAs QDs at its centre. An 80 nm thick layer of polymethyl methacry-
late (PMMA) resist is spun onto the wafer and is patterned using electron-beam
lithography with a current dose of ∼450 µC/cm2. The resist is developed and
holes are etched into the chip using a reactive ion etching (RIE) process with
SiCl4 and Ar. Finally, the remaining resist is removed and the AlGaAs layer be-
neath the etched regions is removed by a selective wet etch with dilute HCl. This
results in a suspended photonic crystal slab with air regions above and below.
After optimisation, the best Q-factors achieved with this method were around
2000. SEM images showed that the hole walls had a tapered profile (see figure
5.7). In the best case the size of the large holes was increased by around 16% at
the top and were around 10% too small at the bottom, giving a taper angle of
∼ 12◦. The small holes typically were increased by around 10% at the top and
were ∼5% too small at the bottom, giving a taper angle of ∼ 6◦. The tapering
was caused by the PMMA erroding very quickly at the edges of the holes during
the etch.
64
5.4 Fabrication of photonic crystals
a) b)
Figure 5.7: SEM images showing a photonic crystal cavity produced using the
initial process. (a) Planar view, the tapered walls appear as bright rings around
the holes. (b) Oblique angle view of the same cavity.
5.4.2 Electron beam resist
Silicon nitride approach
In an attempt to remove the tapering, a harder etch mask of silicon nitride (Si3N4)
was used. Figure 5.8 shows a schematic diagram of the process. A ∼150 nm thick
layer of Si3N4 is deposited onto the wafer by plasma-enhanced chemical vapour
deposition (PECVD). A layer of PMMA is spun on top of this and is exposed
by electron beam lithography and developed. A trifluoromethane (CHF3) based
RIE process transfers the pattern into the Si3N4 layer, followed by a SiCl4 based
RIE etch into the GaAs using the Si3N4 layer as an etch mask. The remaining
PMMA is removed before the Si3N4 is etched away by a combination of a second
CHF3 based RIE etch and a HF wet etch. Finally, the AlGaAs sacrificial layer is
removed with dilute HCl.
After optimisation of the CHF3 RIE process, SEM images revealed that the
pattern transfer into the Si3N4 layer was very good, with smooth and straight
side walls (see figure 5.9). The subsequent GaAs etch was much improved over
the original process, with smoother and straighter side walls, indicating that a
harder etch mask was beneficial to the RIE process (see figure 5.10). Furthermore,
the size distribution of the holes was generally much narrower and more repro-
ducible than for samples produced using PMMA. However, some undercutting
65
5.4 Fabrication of photonic crystals
PMMA
InAs QDs
GaAs
AlGaAs
Si3N4
(a) (b) (c) (d)
(e) (f)
Figure 5.8: Schematic diagram of the photonic crystal fabrication method employ-
ing a layer of silicon nitride. A CHF3 based RIE process transfers the photonic
crystal pattern from a layer of PMMA (b) into a Si3N4 layer (c). A SiCl4/Ar
RIE process then transfers the pattern into the GaAs slab using the Si3N4 layer
as an etch mask (d). The remaining Si3N4 is removed with a second CHF3 etch
(e), before etching away the AlGaAs layer with HCl (f).
Si3N4
GaAs
AlGaAs
Si3N4
(a) (b)
Figure 5.9: (a) SEM image of the Si3N4 surface after the CHF3 RIE etch and
removal of the PMMA. (b) Cross-sectional SEM image of the air holes etched
into the Si3N4 layer showing straight and smooth side walls.
66
5.4 Fabrication of photonic crystals
Si3N4
GaAs
Figure 5.10: Cross-sectional SEM image of a photonic crystal cavity after etch-
ing the pattern from the Si3N4 layer into the GaAs. Slight undercutting and
roughness are observed. The dashed blue and white lines are guides to the eye to
emphasize the profile of the etched hole in GaAs and to show a straight reference,
respectively.
and roughness were still present.
These improvements did not translate into a significant increase in the Q-
factors, the maximum achieved was around 3000. It was found that this was
due to the imperfect removal of the Si3N4 layer at the end of the process, which
degraded the device Q.
ZEP520A
ZEP520A is an electron beam resist which has a heavier molecular weight than
PMMA and is therefore more resistant to etching. It also has a higher sensitivity
than PMMA but with an equivalent resolution, so that a lower electron beam
dose is required to expose it. This resist was used in place of PMMA in the hope
of achieving better photonic crystals.
A process was developed in which a 250 nm layer of ZEP was spun onto the
wafer, which was then baked at 180◦C for one hour. A thin layer of E300 anti-
charge coating was spun on top of this layer, to reduce the build up of charge in
the ZEP during the exposure. The optimum electron beam dose was found to
be 120 µC/cm2 for the photonic crystal pattern and 200 µC/cm2 for the large
67
5.4 Fabrication of photonic crystals
ZEP
GaAs
AlGaAs
(a) (b)
Figure 5.11: (a) SEM image of a completed photonic crystal cavity produced using
ZEP520A resist. (b) Cross-sectional SEM image of a photonic crystal structure
before removing the ZEP from the surface and under-etching the AlGaAs layer.
The air holes are somewhat rough and barreling is present.
aligning features. After the exposure, the anti-charge coating was removed by
spinning with a droplet of de-ionised water and the ZEP was developed with
xylene. Following the RIE etch, the remaining ZEP was removed by exposing it
to ultra-violet light for 5 minutes and immersing it for 10 minutes in hot (∼ 80◦C)
N-methyl-2-pyrrolidone (NMP).
The use of this resist, after re-optimisation of the processing, led to much
straighter and smoother side walls as compared to PMMA because of the greater
resistance to dry etching. Furthermore, there were no problems with its removal
since exposure to ultra-violet light decomposes the polymer. Figure 5.11 shows
SEM images of the photonic crystals produced using ZEP520A resist.
5.4.3 AlGaAs sacrificial layer
The AlGaAs layer beneath the GaAs slab greatly affects the quality of the pho-
tonic crystals. A number of changes to its composition were made, which will be
detailed in this section.
68
5.4 Fabrication of photonic crystals
Layer thickness
Firstly, the thickness of the layer was increased from 500 nm to 900 nm. After
the removal of the AlGaAs layer, the GaAs surface beneath it acts as a mirror. It
has been shown that the position of this bottom mirror can affect the Q-factors
of the cavity modes [66]. According to the literature, the choice of 900 nm (i.e. λ)
is close to optimum and will decrease the possibility of light confined in a mode
evanescently coupling to the GaAs layer below.
AlGaAs aluminium content
ZEP
GaAs
AlGaAs
Si N3 4
GaAs
AlGaAs
(a) (b)
Figure 5.12: (a) Cross-sectional SEM image of a photonic crystal with a high Al
content (98%) AlGaAs layer after the RIE etch. Hollow regions have opened up
in the AlGaAs layer beneath the GaAs slab, due to the high reactivity of the
AlGaAs. The profiles of the GaAs air holes are undercut as a result. Note that
the sample shown has been processed using a Si3N4 mask. (b) Cross-sectional
SEM image of a photonic crystal after etching with a 70% AlGaAs layer. The
dashed lines indicate the locations of the material boundaries.
The aluminium content of AlxGa1−xAs is easily controllable during the MBE
growth. AlGaAs with higher Al content is more reactive and will therefore etch
faster during both the dry and wet etches. Initially, the Al content of the AlGaAs
was high at 98% (i.e. Al0.98Ga0.02As), which caused problems during the dry
etching stage. When the etch reached the AlGaAs layer, the etching rate increased
and the AlGaAs was etched sideways. This caused hollow regions to open up
beneath the GaAs slab through which the reactive ions could flow. These ions
69
5.4 Fabrication of photonic crystals
GaAs
AlGaAs
GaAs
Figure 5.13: Cross-sectional SEM images of a photonic crystal after under-etching
showing a step-like profile in the digital alloy AlGaAs layer. The AlGaAs has not
been fully etched to the GaAs beneath. The dashed line indicates the location of
the lower AlGaAs/GaAs boundary.
could then etch the bottom of the GaAs air holes, leading to undercut hole profiles,
as shown in figure 5.12 (a).
This problem was reduced by lowering the Al content of the AlGaAs to 70%,
which lessened the tendency of the AlGaAs to etch sideways (see 5.12 (b)). Fur-
thermore, by aiming to etch only a short way into the AlGaAs layer the formation
of the hollow regions was prohibited, and the best hole profiles were obtained.
AlGaAs can be grown in a number of ways. It was found that AlGaAs grown
as a digital alloy, or with GaAs smoothing layers could not be removed in a
reproducible way during the selective under-etch. The high Ga regions etched
at a slower rate than the high Al regions, behaving as etch stops. The AlGaAs
was therefore etched at a faster rate sideways than it was downwards, so that the
resulting layer had a step-like profile (see figure 5.13). Hence it was necessary to
use non-digitally grown AlGaAs.
Etchant
Initially, dilute HCl was used to selectively etch the AlGaAs layer. However, it
was found that small islands of AlGaAs were left over, both on the bottom of
the GaAs slab, and on the top of the lower GaAs layer, as shown in figure 5.14.
The remains on the bottom of the slab decreased the refractive index contrast
70
5.4 Fabrication of photonic crystals
(a) (b)
Figure 5.14: SEM images of the AlGaAs remains after selective etching with
dilute HCl (a) underneath the GaAs slab, and (b) on the lower GaAs layer.
and introduced optical scattering at the GaAs/air interface, thus reducing the
Q-factor. The remains on the lower GaAs layer had a detrimental effect on its
properties as a mirror.
By switching to using HF for the under-etch, which is more aggressive, these
islands were not produced. Using HF had another beneficial effect of removing any
organic and oxide contaminants from the surface of the samples. This provided
another motivation to use 70% Al content AlGaAs since higher Al contents were
under-etched very aggressively by HF, so that many of the crystals collapsed.
A process was developed in which the RIE etched samples were immersed in
a 48 % HF solution for 10 s followed by 100 s in a 7:1 buffered HF mixture. The
use of buffered HF is preferable over concentrated HF since the etch rate is much
more controllable. However, it was found that a short dip in concentrated HF
was necessary to etch through the thin oxidised layer of AlGaAs which forms
when the samples are exposed to air.
5.4.4 Dry etching
Reactive ion etching
SEM images of the entire process are shown in figure 5.15. The patterning of the
resist was typically excellent, so the accuracy of the pattern transfer into the GaAs
slab had the greatest effect on the final crystals. The use of ZEP520A resist, along
with the necessary re-optimisation of the RIE process, dramatically improved the
71
5.4 Fabrication of photonic crystals
GaAs
ZEP
AlGaAs
GaAs
GaAs
ZEP
AlGaAs
GaAs
ZEP
AlGaAs
GaAs
(a)
(b)
(c)
(d)
Air
Figure 5.15: SEM images showing the steps involved in fabricating photonic
crystals. (a) A layer of ZEP is spun onto the wafer. (b) The photonic crystal
pattern is written by electron beam lithography, and the resist is developed. (c)
The pattern is transferred into the GaAs slab by SiCl4/Ar based RIE. (d) The
AlGaAs layer is removed by a wet etch with HF.
72
5.4 Fabrication of photonic crystals
quality of the crystals produced. This change, along with the others discussed in
this section, brought the maximum Q-factor measured for the monopole mode of
an optimal modified H1 cavity up to ∼7500.
This value, however, was still far from the ideal calculated value of 30000, and
was not high enough for the intended applications (see chapter 6). Typically, the
radii of the large holes were slightly increased by ∼ 3% and the nearest neighbour
holes were increased by ∼ 6%. The holes were very circular with a typical average
circularity, defined as the ratio of the long and short axes of an ellipse, of 93±4%.
The air hole side walls were smooth but tended to exhibit barreling in their profile,
as shown in figure 5.15 (d).
After extensive optimisation of the RIE process, it became apparent that this
barreling was caused by inherent limitations of the RIE etching technology. It
was therefore decided to change to an inductively coupled plasma RIE (ICP)
process, as is discussed in the following section.
Inductively coupled plasma reactive ion etching
ICP is a similar technique to RIE which can achieve much higher plasma densities.
In RIE, a strong radio frequency (RF) field is applied to an electrode in a vacuum
chamber on which the sample to be etched sits, as shown in figure 5.16 (a).
Controlled amounts of gas continually flow through the chamber and the pressure
can be precisely controlled by a butterfly valve in front of the pump. The RF field
ionises the gas molecules, creating a plasma. The free electrons in the plasma are
strongly affected by the field and are accelerated up and down in the chamber.
The electrons which are absorbed by the chamber wall, which is earthed, simply
flow to ground. However, the electrons which are absorbed by the electrode
cannot flow anywhere because it is DC isolated, so that a charge builds up.
The more massive positively charged ions are not as strongly affected by the
RF field. When a negative charge builds up on the electrode, they are acceler-
ated towards it where they collide with the sample. Etching occurs by the ions
chemically reacting with the sample as well as sputtering it by transferring some
of their kinetic energy. Since the ions generally collide with the sample traveling
in the vertical direction, the etch is highly directional.
73
5.4 Fabrication of photonic crystals
(a) (b)Electrode
Sample
Coil
Figure 5.16: Schematic diagrams of the operation of RIE (a) and ICP (b). The
purple shapes represent the confinement of the plasma. The red arrows show the
input of the gases.
(a) (b)
(c)
Figure 5.17: (a) SEM image of a photonic crystal cavity produced using ICP.
(b) Cross-sectional image showing smooth and straight air hole side walls. (c)
Oblique angle view of a cavity.
74
5.5 Summary and outlook
The main disadvantage of using RIE to fabricate photonic crystals is the fact
that both the plasma density and the ion energy are controlled by the same
parameter - the RF power. To achieve higher etching rates, both of these are
increased, leading to a higher sputtering rate which can cause surface damage and
roughness. ICP solves this problem by introducing a separate coil to produce the
plasma, as shown in figure 5.16 (b). In this case the plasma density is controlled
by the RF power applied to the coil and the ion energy is controlled by the RF
power applied to the electrode. It is therefore possible to increase the etching
rate by increasing the plasma density only, and not the ion energy.
A new ICP system was installed and configured as part of this work. After
extensive optimisation of the process, it was possible to achieve very smooth and
straight air hole side walls, as shown in figure 5.17. The inclusion of a laser
interferometer system gave precise control over the depth of the etch, so that
problems associated with the higher etching rate of AlGaAs could be avoided.
These changes led to Q-factors of up to 9900, high enough to see strong coupling
as detailed in chapter 6.
5.5 Summary and outlook
By implementing a number of changes to the processing method, it was possible to
produce high quality photonic crystal structures. The most significant changes
were the use of ZEP520A resist, the use of HF as the selective etchant and
the installation and optimisation of a new ICP system. The fabrication process
developed here is suitable to produce a wide variety of photonic crystal structures.
75
Chapter 6
H1 photonic crystal cavities for
strong coupling
6.1 Overview
This chapter will detail experiments carried out with modified H1 photonic crystal
cavities, with the goal of achieving strong coupling between an excitonic transi-
tion of a QD and the monopole mode of the cavity. The strong coupling regime
offers a route for the practical realisation of future quantum information devices,
such as quantum gates [35, 38, 42, 58], quantum memories [22] or sources of en-
tangled photon pairs [61], and for the exploration of the physics of cavity quantum
electrodynamics [63, 88].
Strong coupling with quantum dots has been observed in microdisks [108],
micropillars [33, 113] and photonic crystal cavities [149]. 2D slab photonic crystal
cavities are particularly suitable for the observation of the strong coupling regime
due to their high Q-factor modes with tiny volumes [112, 138].
A H1 cavity consists of just one missing hole in a triangular lattice photonic
crystal. Strong coupling between a single InAs QD and a mode of an L3 cavity
(a line of 3 missing holes) has recently been achieved by several groups [35, 37,
50, 79, 95, 110, 144, 149]. However, only one group has reported strong coupling
with a H1 cavity, using a nondegenerate polarised dipole mode and operating at
∼ 970 nm [103].
76
6.2 H1 mode structure
The mode volume of H1 modes is typically smaller than L3 modes, so that the
QD-cavity coupling parameter, g, is larger, which is beneficial for achieving strong
coupling. Furthermore, the locally polarised nature of the L3 modes prohibit their
use with QD entangled photon pair sources [28, 121]. If the two neutral exciton
dipoles experience a different LDOS, the indistinguishability of the two decay
paths in the biexciton decay will be broken, and the emitted photons will not be
entangled [128]. In principle, it should be possible to overcome this limitation
using the doubly degenerate dipole modes of a H1 cavity, since the local electric
field in the centre of the cavity has no preferred direction [49, 78, 121].
6.2 H1 mode structure
As described in section 2.4.3, the H1 cavity can support a number of modes
depending on its design [59]. The name of each of the modes is based on the
shape of the field profile, for example the cavity may support monopole, dipole,
quadrupole or hexapole modes. The mode wavelengths, Q-factors and mode
volumes can be controlled by modifying the cavity parameters: the periodicity,
a, hole radius r, and slab thickness, d. Figure 2.9 shows the calculated spectrum
for a H1 cavity, with normalised hole radius r/a = 0.3 and slab height of d/a = 0.7
which supports a single dipole mode.
Furthermore, the positions and sizes of the holes surrounding the defect may
be modified in order to increase the Q-factor of the modes [2, 106]. In the scheme
used in this work, the radius of the six nearest neighbour holes is reduced to a
value of r′ and the positions of their centres are shifted radially outwards by an
amount s = r − r′ as shown in figure 6.1 [46, 56, 107, 114]. Figure 6.2 shows
the effect of this on the mode structure of the cavity: as the size of the cavity
is increased, a series of other modes are pulled down from the air band into the
band gap [46, 121].
The work presented here focuses on the monopole mode of the H1 cavity (M1
in figure 6.2). Although the far-field emission pattern is unpolarised, this mode
is not suitable for use with entangled photon pair sources since the local field at
the antinode has a preferred direction. However, it is ideally suited for observing
strong coupling since the position of the QD should not be as critical as in other
77
6.2 H1 mode structure
d
(b)
a
a
r
GaAs
Air
a
r
(a)
r’
s
Figure 6.1: Schematic diagram of the scheme to modify the H1 cavity.
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35750
800
850
900
950
1000
1050
1100
1150
Q2
Q1
M2
M1
H2
H1
D2
Mo
de
 W
av
ele
ng
th 
(nm
)
Nearest Neighbour Radius (r'/a)
Dipole
QuadrupoleA
QuadrupoleB
Hexapole
Monopole
Air Band
Dielectric Band
D1
Figure 6.2: Simulated position of the modes in a H1 cavity as the radius of the
nearest neighbour holes surrounding the defect, r′, is modified and their centres
are shifted radially outward by s = r− r′. The cavity parameters are r/a = 0.36,
d/a = 0.411 and n = 3.53. Courtesy of D. Gevaux.
78
6.3 Design and optimisation
modes due to the small number of nodes in its field profile. Furthermore, the
light in the mode is confined away from the air holes, which due to fabrication
imperfections can cause optical losses in practice.
6.3 Design and optimisation
This section describes FDTD simulations which were used to find the optimal
parameters to maximise the Q-factor of the monopole mode.
6.3.1 Bulk hole radii
0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45
0.0
0.2
0.4
0.2
0.3
0.4
0.5
No
rm
ali
se
d f
req
ue
nc
y a
/λ
∆ω
/ω c
en
tre
∆ω
Figure 6.3: (a) Contour plot showing the simulated width of the band gap as
the radius of the holes, r, in a hexagonal lattice photonic crystal is changed,
with d/a = 0.7. White/blue regions indicate a low/high density of states. (b)
Gap-midgap ratio of the photonic band gap (∆ω/ωcentre).
Figure 6.3 shows FDTD simulations of the width of the photonic band gap
as the radius of the holes in a hexagonal lattice photonic crystal with d/a = 0.7
is changed. The widest band gap occurs for a value of r/a = 0.4. Since a wide
79
6.3 Design and optimisation
band gap leads to strong confinement [59], it is expected that the optimum value
for the bulk hole radius will be close to this value.
6.3.2 Nearest neighbour hole radius
Figure 6.4: Simulated Q-factor (top) and mode frequency (bottom) of the
monopole mode in a H1 cavity with d/a = 0.411 as the radius of the nearest
neighbour holes, r′, is changed. The trends are shown for three different values of
r/a. Q0 = 12500 is the Q factor of the optimised cavity with d = 0.65a calculated
with 5 periods surrounding the defect. Courtesy of D. Gevaux.
The electric field profile of a mode in an unmodified H1 cavity is terminated
abruptly by the photonic crystal, which gives rise to Fourier components of the
mode which can couple to the light cone (see section 2.4.3). By reducing the
radius of the six nearest neighbour holes and/or shifting their centres outwards,
80
6.3 Design and optimisation
the severity of the boundary is reduced so that fewer of the mode’s Fourier com-
ponents couple to the light cone, and higher Q-factors are achievable [2].
Figure 6.4 shows the simulated Q-factor of the H1 mode as the radius of the
nearest neighbour holes is changed following the scheme shown in figure 6.1. The
radius of the holes must be reduced by some degree for the monopole mode to be
pulled down from the air band, as shown in figure 6.2. By further reducing the
radius, the Q-factor of the mode is increased. The highest Q-factor is achieved
for a value of r′/a = 0.25 and r/a = 0.36.
6.3.3 Slab height
3
4
5000
10000
15000
0.4 0.6 0.8 1.0 1.2 1.4
λ/a
Normalised slab thickness d/a
Q/
(V 0
)1/2
(a.
 u.
)
Figure 6.5: Simulated Q-factor (top) and mode wavelength (middle) of the
monopole mode in a H1 cavity with r/a = 0.36, r′/a = 0.25, n = 3.53 and
with 5 periods surrounding the defect, as the slab thickness, d, is changed. The
blue lines in the centre panel indicate the edges of the photonic band gap. The
bottom panel shows the ratio of Q/
√
V0 which is a figure of merit for strong
coupling.
81
6.3 Design and optimisation
The thickness of the slab also greatly affects the strength of the confinement
(see figure 2.10). Figure 6.5 shows how the Q-factor of the monopole mode in
a H1 cavity with r/a = 0.36 and r′/a = 0.25 changes as the thickness of the
slab is modified. As the thickness of the slab increases, the resonant wavelength
of the mode increases since the cavity becomes larger. The Q-factor increases
until d/a = 1.1 at which point it drops off dramatically due to the decrease in
the size of the photonic band gap causing the monopole to enter the air band
[62, 132]. However, a higher value for the strong coupling figure of merit Q/
√
V0
is achievable for a thinner slab of d/a = 0.65 since the mode volume is strongly
dependent on the slab height.
6.3.4 Optimised cavity
2.5 3.0 3.5 4.0
Ai
r b
an
d
Di
ele
ctr
ic 
ba
nd
Normalized wavelength λ/a
Hexapole
Quadrupole
Monopole
Dipole
(a) (b)
(c)
(d)
+
-
Ele
ctr
ic f
ield
 am
plit
ud
e (
arb
. u
nit
s)
Figure 6.6: (a) Simulated mode structure of the optimised H1 cavity with r/a =
0.36, r′/a = 0.25 and d/a = 0.65. The structure is very similar to that in figure
6.2. The insets show the magnetic field profiles, Hz, for a selection of the modes.
(b)-(d) Ex, Ey and Hz field distributions of the monopole mode.
Figure 6.6 shows the mode structure of the optimised H1 cavity with r/a =
82
6.4 Fabrication
0.36, r′/a = 0.25 and d/a = 0.65. A number of modes are present, as in figure 6.2.
The insets show the magnetic field profiles, Hz, of a selection of the modes. The
monopole in this optimised structure, has a Q-factor of Q = 30000 and a mode
volume of V0 = 0.8(λ/n)
3, calculated with 9 photonic crystal periods surrounding
the defect. Its Ex, Ey and Hz field components are shown in figures 6.6 (b)-(d).
Both the Ex and Ey field profiles of the monopole have an odd mirror sym-
metry about the centre of the cavity. The size of the defect in x and y is around
a ∼ λ/3, so that the positive and negative components of the electric field will
effectively cancel each other out in the far field. Therefore, the photons emitted
from the cavity by coupling to the light cone (or scattering due to fabrication
imperfections) will have no preferred polarisation [66].
6.4 Fabrication
The samples were fabricated as described in section 5.4. Based on the calculation
in figure 6.6, the mode should appear at 900 nm, where the QDs emit, for a
periodicity of around a = 300 nm. Therefore the slab thickness was set at d =
0.65a ∼ 200 nm.
Since the fabrication is never perfect, a square array of cavities with varying
parameters was made. On one axis the periodicity, a, was stepped to compensate
for fabrication imperfection induced shifts in the monopole emission wavelength.
On the other axis, the radius of the nearest neighbour holes, r′, was varied. This
was done to compensate for any uneven etching rate of the different sized holes.
6.5 Optical characterisation
This section will detail PL measurements made on the fabricated structures to
determine their quality.
6.5.1 Mode structure
By measuring cavities with varying values of r′, it was possible to see part of
the cavity mode structure and to determine the location of the monopole mode.
83
6.5 Optical characterisation
860 880 900 920 940 960 980
r'/a = 0.28
r'/a = 0.27
r'/a = 0.26
r'/a = 0.25
r'/a = 0.24
r'/a = 0.23
r'/a = 0.22
r'/a = 0.21
r'/a = 0.20
Hex
apo
le
Qua
dru
pol
e
Qua
drup
ole
Mo
nop
ole
PL
 in
ten
sity
 (a
rb.
 un
its)
 
Wavelength (nm) 
Figure 6.7: PL measured at 5 K of from several H1 cavities with r/a = 0.36,
d = 200 nm, a = 300 nm and 5 photonic crystal periods surrounding the defect.
Each spectrum is from a different cavity with varying r′ as labeled. The black
dashed lines are guides to the eye to show each of the modes shifting in wavelength
between each cavity.
84
6.5 Optical characterisation
Figure 6.7 shows PL from a number of different H1 cavities with r/a = 0.36,
a = 300 nm, d = 200 nm and varying values of r′ as labeled. A number of modes
appear in the spectra, which have been identified based on the wavelengths at
which they appear. Each mode shifts to longer wavelengths as the radius of the
nearest neighbour holes is reduced, as expected.
0.00 0.05 0.10 0.15 0.20 0.25 0.30
3.0
3.2
3.4
3.6
100
1000
10000
Normalised nearest neighbour hole radius r'/a
Experimental 1
Experimental 2
Simulation
Figure 6.8: Q-factors (top panel) and normalised wavelengths (bottom panel) of
the monopole mode as the nearest neighbour hole radius is changed. The red
symbols show data extracted from the data in figure 6.7. The blue symbols show
a similar set of data from a second array of cavities on the same chip. The black
symbols show the calculated theoretical values.
Figure 6.8 show comparisons of the theoretical and experimentally measured
Q-factor and emission wavelength of the monopole mode. An excellent agreement
between the trends of the experimental data and the simulated values can be seen.
The experimental Q-factors are consistently lower than the theoretical values due
to the inevitable fabrication imperfections. The peaks in the two curves do not
exactly match, owing to the uneven etching rates of the different sized holes.
Figure 6.9 shows PL from the monopole mode of a number of nominally
identical H1 cavities from two different chips with r/a = 0.36 and r′/a = 0.25.
85
6.6 Weak coupling
915 920
Chip 1
Chip 2
Q ~ 3000 -
 5000
Q ~ 5000 - 6800
PL
 in
ten
sity
 (a
rb.
 un
its)
Wavelength (nm)
Q = 6800
Chip 2
Chip 1
Figure 6.9: PL measured at 5 K of from the monopole mode from several H1
cavities with nominally identical parameters: r/a = 0.36, r′/a = 0.25, d = 200
nm and a = 300 nm.
The emission wavelengths are remarkably consistent between the cavities on each
chip, typically within a few nanometres. The variation between different chips is
somewhat larger and reflects the inconsistencies in the processing conditions from
chip to chip. The cavities measured here do not represent the optimal processing
as discussed in section 5.4.
6.6 Weak coupling
In the weak coupling regime, the spontaneous emission rate of a QD transition
on resonance with a mode will be enhanced according to equation 2.10. As the
temperature of a cavity is increased above 5 K, the modes redshift due to the
change in the refractive index of GaAs, whereas lines from QDs redshift at a faster
rate, following the change in the GaAs band gap. It is therefore possible to tune
QD lines onto resonance with modes which lie close by at higher wavelengths, by
increasing the temperature of the sample [68]. This process effectively maximises
the spectral overlap term in equation 2.10.
The spatial overlap term, however, cannot be controlled since the QDs are
86
6.6 Weak coupling
randomly positioned throughout the wafer. It is therefore necessary to fabricate
a large number of cavities and to search through them to find one with a suitably
positioned QD with a wavelength close to the mode. The measured Purcell
enhancement is usually limited by imperfect spatial alignment between the QD
dipole and the electric field of the mode.
Figure 6.10 shows PL from the monopole mode of a H1 cavity with a weakly
coupled QD. As the temperature of the sample is increased, the line is tuned onto
and then off of resonance with the mode. As the QD line shifts onto resonance,
its intensity is enhanced due to the Purcell effect. Previous studies have directly
shown that the spontaneous emission rate of QD transitions are enhanced in this
way [46].
907.5 908.0 908.5
M X
X
40 K
30 K
PL
 in
ten
sity
 (a
rb.
 un
its)
Wavelength (nm)
M
Figure 6.10: PL from a monopole mode with Q = 4300 of a H1 cavity with a low
density of QDs at the centre of the slab, as the temperature is increased from 30
K to 40 K. The QD exciton line labeled X shifts onto, and then off of, resonance
with the mode labeled M.
87
6.7 Strong coupling
6.7 Strong coupling
880.0 880.5 881.5
32K
27K
X M
M
20K
40K
PL
 in
ten
sity
 (a
rb.
 un
its)
Wavelength (nm)
29K
X
Figure 6.11: PL of from the monopole mode of a H1 cavity with Q = 8500
(labeled M) as the temperature is increased from 20 K to 40 K. The excitonic
line labeled X comes exactly onto resonance with the mode at 29 K, at which
point two peaks persist in the spectrum. The temerature step between each graph
is smaller between 27 K and 32 K than elsewhere to emphasize the splitting at
resonance.
After extensive optimisation of the processing, as described in section 5.4, it
was possible to achieve very high Q-factors of up to 9900 for the monopole mode.
Figure 6.11 shows PL from the monopole mode of a H1 cavity with Q = 8500
(labeled M) as the temperature of the sample was varied in unequal steps between
20 K and 40 K. The two other lines in the spectra (at 881.15 nm and 880.0 nm at
20 K) are from QD transitions. As the temperature of the sample was increased
88
6.7 Strong coupling
above 20 K, the QD line labeled X was tuned onto resonance with the mode. In
all the measurements, including when the line was exactly on resonance with the
mode at 29 K, two peaks remained in the spectrum. As the temperature was
increased beyond 29 K, the QD line detuned away from the mode.
There is a clear anti-crossing in the peak energies of the exciton and the mode.
This behaviour can be seen more clearly in figure 6.12 in which the energies of
the lines are plotted as a function of their energy detuning. The blue lines in
the figure are the expected peak energies of the QD and mode in the weakly
coupled case. This anti-crossing is a clear signature of the strong coupling regime
as described in section 2.3.3 [108, 113, 149].
Figure 6.12 also shows how the behaviour of the linewidths and integrated
PL intensities of the two lines as a function of their detuning. The linewidths
of the two lines approach each other and converge to the same value when they
are tuned onto resonance. As the QD is tuned away, the linewidths gradually
approach their original values. The integrated intensities of the two lines show
similar behaviour. These observations confirm the existence of strong coupling
[112].
The intensity of the mode was much higher than that of the QD line when de-
tuned. This suggests that other QD lines were contributing to the mode emission,
even though they were not spectrally coupled [8, 57, 97, 145, 147].
Figure 6.13 (a) shows a high resolution PL spectrum from the strongly coupled
QD/cavity at 29 K. At this point, the splitting between the two polariton peaks
is at its minimum. By fitting two Lorentzian lines to the spectrum, a vacuum
Rabi splitting (VRS) of ∆U vac = 115 µeV was extracted. Figure 6.13 (b) shows a
similar plot for a second strongly coupled QD/cavity system, with a larger VRS
of ∆U vac = 141 µeV.
This value for the VRS corresponds to a QD-cavity coupling parameter of
~g = 70 µeV, which exceeds the strong coupling condition g/κ = 0.27 > 1/4 (see
equation 2.15). The maximum possible QD-coupling parameter in the case of a
perfectly positioned QD was calculated from equation 2.5 to be around 135 µeV.
The difference in the measured value is most likely due to the non-ideal position
of the QD, away from the electric field maximum of the cavity mode.
89
6.7 Strong coupling
-300-200-100 0 100 200 300
50
100
150
200
1.40625
1.40650
1.40675
1.40700
X
M
Int
eg
rat
ed
 PL
int
en
sit
y (
a. 
u.)
Energy detuning (µeV)
M
X
En
erg
y (
eV
)
Lin
ew
idt
h (
µe
V)
X
M
X
M
Figure 6.12: Peak energies of the mode and QD line (labeled M and X in figure
6.11) as they are tuned around resonance (top panel). A clear anti-crossing occurs
between their peak energies. The centre and bottom panels show the behaviour
of the linewidths and integrated intensities of the lines respectively.
90
6.7 Strong coupling
1.4085 1.4090 1.4095 1.4100
1.408 1.409
MX
M
52K
51K
50.5K
50K
49.5K
49K
PL
 in
ten
sity
 (a
rb.
 un
its)
Energy (eV)
48K
X
Energy (eV)
∆Uvac = 141 µeV(a) (b)
1.4070 1.4075
PL
 in
ten
sity
 (a
rb.
 un
its)
Energy (eV)
∆Uvac = 115 µeV
Figure 6.13: (a) PL from the strongly coupled QD/cavity in figure 6.11 when
on resonance at 29K, taken by scanning the spectrometer and detecting with an
APD (circles). The blue line shows a double Lorentzian fit to the data, with each
Lorentzian shown in green. A splitting of ∆U vac = 115 µeV is extracted from the
fit. (b) As (a) but for a second strongly coupled QD/cavity with a larger splitting
on resonance of ∆U vac = 141 µeV, and taken with the regular µPL setup. The
inset shows PL from the QD/cavity at various temperatures close to resonance.
91
6.8 Polarisation dependence
These results compare favourably to those recently presented in reference
[103], in which strong coupling was achieved in a H1 cavity for the first time,
using the dipole mode and an InAs QD. Despite having a theoretically higher Q
of ∼ 50000, a smaller mode volume of V0 = 0.43(λ/n)3 and a higher operating
wavelength of ∼ 970 nm, the highest measured Q reported was 7300. This lead
to a VRS of 124 µeV.
6.8 Polarisation dependence
90 180 270
150
155
160
1.3970 1.3975 1.3980
Energy (eV)
PL intensity (arb. units)
∆U
va
c (µ
eV
)
Polarisation angle (degrees)
Figure 6.14: Polarisation dependence of the splitting of a strongly coupled
QD/cavity. The average value is ∆U vac = 157 µeV. The random error results
from errors in fitting the peak energies. The inset shows the raw data, the change
in intensity is due to the in-built polarising nature of the measurement setup.
Figure 6.14 shows the polarisation dependence of the VRS of a third strongly
coupled QD/cavity. The splitting persists for all polarisations with no angle-
dependent variation in its magnitude. This is due to the unpolarised nature of
the far-field of the monopole mode emission. A photon emitted from a QD may
have some polarisation depending on the orientation of the QD axes, which will
92
6.9 Summary and outlook
be closely aligned to the direction of local electric field of the mode for strong
coupling. However, the photon will take the electric field pattern of the whole
mode before it leaks from the cavity. The emitted photons will therefore have no
preferred polarisation.
6.9 Summary and outlook
The Q-factor of the monopole mode in a H1 cavity has been optimised for strong
coupling, principally by modifying the six holes surrounding the defect. Strong
coupling between a single InAs QD and the monopole mode has been observed for
the first time. The VRS persists for all linear polarisations due to the unpolarised
nature of the mode’s far field emission pattern. This result expands on previous
work in solid state cavity quantum electrodynamics and is an important step in
the development of quantum information devices.
Several strongly coupled QD/cavities were found, demonstrating the useful-
ness of the monopole mode for strong coupling with randomly positioned dots,
due to the low number of nodes in the electric field profile. The wavelength of the
first strongly coupled QD/cavity was ∼880 nm, where the neutral exciton fine
structure splitting is zero or very close to zero [130]. This result leads the way
towards the realisation of a strongly coupled entangled photon pair source.
93
Chapter 7
Unidirectional photonic crystal
waveguides
7.1 Overview
This chapter will describe experiments carried out on unidirectional photonic
crystal waveguides with the goal of working towards realising an in-plane quantum
light source.
As explained in section 2.3, by coupling a QD transition to an optical cavity,
it is possible to enhance its spontaneous emission rate and to allow efficient ex-
traction of the light through the Purcell effect [111]. Large Purcell enhancements
have been observed in micropillars [45], microdisks [68] and photonic crystal cav-
ities [10, 29, 34, 46, 72, 82], in some cases entering the strong-coupling regime
[50, 108, 113, 149] (as in chapter 6). However in all of these cavity architectures,
the bulk of the emission is directed out of the plane of the semiconductor chip.
Future photonic integrated circuits would benefit from in-plane quantum light
sources [109].
Two-dimensional photonic crystal slab cavities are attractive because of their
high Qs and small volumes. Light is confined in the plane with a photonic band
gap and in the vertical direction by total internal reflection at semiconductor-air
interfaces. However, to achieve high Q-factors it is necessary to design the cavity
so that out-of-plane emission is suppressed [2], which is incompatible with high
extraction efficiencies in the vertical direction. Nevertheless, recent theoretical
94
7.2 Photonic crystal waveguide mode structure
work has shown that large Purcell factors and high in-plane extraction efficiencies
are achievable for QDs coupled to photonic crystal waveguides (PCWGs) [54].
PCWGs can be made from 2D photonic crystal slabs by removing a line of air
holes [30]. These waveguides can support a number of guided modes which are
confined in the plane by the photonic band gap and in the vertical direction by
total internal reflection. The modes have flat regions in their dispersion relations,
so that their group velocities go to zero for certain frequencies [9, 71]. In particu-
lar, at the edge of the Brillouin zone (kx = pi/a) light is coherently backscattered
by the periodic crystal, and a standing wave forms. Just away from the zone
boundary is the true slow-light regime, where light propagates slowly through
the waveguide. Other standing wave resonances can occur, for example at the Γ
point (kx = 0) if it lies inside the photonic band gap. Large Purcell factors can
be achieved due to the high local photon density of states associated with these
slow-light resonances.
A recent theoretical proposal [91] suggested a design to utilize the properties
of PCWGs to create an efficient in-plane quantum light source. A short waveg-
uide can provide a large Purcell factor, while minimizing the detrimental effects
due to fabrication imperfections [55] and easing the coupling of an individual
QD. Furthermore, for a unidirectional PCWG (i.e. a PCWG with one end termi-
nated by a photonic crystal mirror and at the other end by a semiconductor-air
interface), the majority of the emission can be directed in one in-plane direction.
PCWGs offer other advantages such as broadband large β-factors associated with
the width of the mode beneath the light-line [81, 87, 90, 102], however the work
presented in this chapter is concerned only with the Purcell enhancements due
to the slow-light resonances [136].
7.2 Photonic crystal waveguide mode structure
Figure 7.1 shows a schematic diagram of a unidirectional PCWG, formed by a line
of missing air holes along the Γ-K direction of a triangular lattice photonic crystal
pattern. One end of the waveguide is terminated by a photonic crystal mirror,
the other end by a semiconductor-air interface. The x- y- and z-directions are
95
7.2 Photonic crystal waveguide mode structure
defined as parallel to the length of the waveguide, perpendicular to the waveguide
in the plane of the slab, and perpendicular to the plane of the slab, respectively.
The results of FDTD simulations of the structure with hole radii r/a = 0.3,
slab thickness d/a = 0.7, waveguide width w/a = 1 and length l/a = 12 (pa-
rameters defined in figure 7.1) are shown in figure 7.2(a). The calculation was
carried out with a single x- or y-polarised dipole source placed at the centre of
the waveguide.
There are two main high-Q modes, as in [91], a y-polarised mode (i.e. elec-
tric field orientated in the y-direction, see section 7.3.4) labelled M1, and an
x-polarised mode labelled M2. These are the standing waves formed at the Bril-
louin zone boundary of the two orthogonally polarised waveguide modes present
inside the photonic band gap [102]. The other features in the y-polarised spectrum
are due to Fabry-Perot like resonances between the semiconductor-air interface
and the back photonic crystal mirror. The x-polarised spectrum includes a num-
ber of lower Q features, associated with slow-light resonances of the x-polarised
waveguide mode which are above the light-line [131].
The identification of Fabry-Perot modes alongside the slow-light resonances
was confirmed by repeating the calculation with the waveguide terminated at each
end by PML absorbing boundaries, as shown in figure 7.2(b). In this case, the
Fabry-Perot modes disappear from the spectrum since they are no longer confined
along the length of the waveguide. Furthermore, the spacing between the Fabry-
Perot modes matches well with the free spectral range for a one-dimensional
cavity of the same length, and mode profile calculations show that each mode
has one node more or less than the modes next to it in the spectrum, as expected
for Fabry-Perot modes.
The slow-light resonances, on the other hand, are confined by Bragg reflection
along the length of the waveguide, and therefore persist in the spectrum when the
waveguide is terminated by PML absorbing boundaries. The work presented here
focuses on the y-polarised resonance of the unidirectional PCWG, M1, which is
expected to have a large in-plane collection efficiency [91].
96
7.2 Photonic crystal waveguide mode structure
da
l
r w
GaAs
Air x
y
z
Figure 7.1: Schematic diagram of a unidirectional photonic crystal waveguide.
The relevant parameters are labeled. The coordinates are defined with reference
to the figure on the right.
3.2 3.6 4.0 4.4
x-polarised
y-polarised
Normalised wavelength λ/a
(a)
(b)
M1
M2
Ele
ctr
ic f
ield
 am
plit
ud
e (
arb
. u
nit
s)
Figure 7.2: (a) FDTD simulation of the mode profile of a W1 (i.e. w/a = 1)
unidirectional PCWG with r = 0.3a, d = 0.7a, l = 12a and n = 3.53. The
calculation is done with a single x- or y-polarised dipole source positioned at
the centre of the waveguide. (b) Same as (a) except the PCWG is terminated
by PML absorbing boundaries at each end. The dashed green lines indicate the
positions of the photonic band gap edges.
97
7.3 Design and optimisation
7.3 Design and optimisation
In the same way as for H1 cavities, there are a number of parameters that can be
changed to optimise the Q-factor of the slow-light resonances in a unidirectional
PCWG [83]. These are summarised in figure 7.1 and include the bulk hole radius,
r, the width of the slab, d, the length of the PCWG, l, the width of the waveguide,
w, the position of the cleaved edge and the properties of the nearest neighbour
holes. Using the original theoretical proposals of the device [91] as a starting
point, this section details work undertaken to optimise the Q-factor of M1.
7.3.1 Basic parameters
In reference [89], the effect of the length of a unidirectional PCWG on the max-
imum Purcell factor of the mode M1 was theoretically investigated. For a W1
waveguide with r/a = 0.275 and d/a = 0.5, the maximum Purcell factor was
found to increase with increasing length until it saturated at around l/a = 20
at FP = 252. Since the confinement is due to Bragg reflection along the length
of the waveguide, including more periods of photonic crystal should increase the
confinement. However, this also has the effect of decreasing the in-plane collec-
tion efficiency. In light of this result, a trade-off value of l/a = 12 was chosen for
this work.
In the same study, it was found that terminating the waveguide between
nearest neighbour holes was slightly advantageous over terminating it through
a nearest neighbour hole. Therefore, all the calculations and fabricated samples
were terminated between the nearest neighbour holes, as in figure 7.1.
For simplicity, the width of the waveguide has been fixed at w/a = 1 and the
parameters of the nearest neighbour holes have not been modified throughout this
work. There may well be further scope to optimise the modes by modifying these
parameters [83], however the Q-factors of the design described in the rest of this
chapter are high enough so that the measured values are effectively determined
by fabrication imperfections.
98
7.3 Design and optimisation
7.3.2 Hole radius
The first simulations carried out investigated the effect of changing the radii of
the holes, r. The original theoretical proposal showed that for a 10 unit cell long
PCWG with a hole radius of r/a = 0.275 and d/a = 0.5, a Q-factor of around
1000 was possible for M1, corresponding to a maximum Purcell factor of FP = 59
[91]. Simulations carried out in this work showed that by simply increasing the
radius of the holes, a dramatic increase in the Q-factor is possible.
0.28 0.32 0.36
5001000
1500
3.8
4.0
10000
20000
30000
Q
Normalized hole radius r/a
F P
λ/a
Figure 7.3: Results of FDTD simulations for the y-polarised mode M1 of a W1
unidirectional photonic crystal waveguide with d/a = 0.7, l/a = 12 and n = 3.53,
as the radius of the holes, r/a, is changed.
The simulations were carried out with a W1 unidirectional PCWG with a
slab thickness of d/a = 0.7, with five periods of the crystal on all sides, and with
a length of l/a = 12. The radius of the holes was varied and the normalised
wavelength, the Q-factor and the mode volume were simulated. The maximum
Purcell factor was calculated from these numbers. Figure 7.3 shows the results.
As the radius is increased, the mode’s wavelength decreases as the size of the
waveguide is decreased. It was found that M1 has the largest Q-factor for r/a =
99
7.3 Design and optimisation
0.36, with Q = 25500. The mode volume was calculated to be V0 ∼ 1.4(λ/n)3,
giving a maximum Purcell factor of FP = 1360.
7.3.3 Slab height
0.4 0.6 0.8 1.0 1.2
5001000
1500
3.6
4.0
15000
20000
25000
λ/a
Normalised slab thickness d/a
F P
Q
Figure 7.4: Results of FDTD simulations for the y-polarised mode M1 of a W1
unidirectional photonic crystal waveguide with r/a = 0.36, l/a = 12 and n = 3.53,
as the height of the slab, d/a, is changed.
A similar set of simulations were run to optimise the slab height, with the
results shown in figure 7.4. As the slab height is increased, the wavelength of the
mode increases as the size of the waveguide increases. The Q-factor was found
to increase slightly to Q = 25800 for a slab height of d/a = 0.8. However, the
maximum Purcell factor was slightly lower for this value, at FP = 1310, due to
the mode volume increasing with the slab height. In fact, the highest Purcell
factor was calculated to be FP = 1360 at d/a = 0.7, as before.
100
7.3 Design and optimisation
2.5 3.0 3.5 4.0
Normalised wavelength λ/a
y-polarised
(a)
(b)
x-polarised
M1
M2
M3
Ele
ctr
ic f
ield
 am
plit
ud
e (
arb
. u
nit
s)
Figure 7.5: (a) FDTD simulation of the mode structure of a PCWG with r/a =
0.36 and d/a = 0.7, calculated with a single x- or y-polarised dipole placed at
the centre of the structure and using 5 photonic crystal periods on each side.
The dashed green lines indicate the location of the photonic band gap edges. (b)
FDTD simulation of an equivalent PCWG without mirrors at either end.
101
7.3 Design and optimisation
(a)
Ex
(b)
Ey
(c)
x
y
Figure 7.6: Ex and Ey electric field profiles at the centre of the slab for M1 (a),
M2 (b) and M3 (c).
102
7.4 Fabrication
7.3.4 Optimised waveguide
Figure 7.5 (a) shows the calculated mode structure of the optimised waveguide
with r/a = 0.36 and d/a = 0.7. The larger value of r/a results in a wider band
gap than for r/a = 0.3 as in figure 7.2, so that some modes which were in the
air band appear in the band gap (see figure 6.3). The slow-light resonance M1
appears close to the edge of the photonic band gap, with a calculated Q of 25500.
Its electric field profile is shown in figure 7.6 (a). The Ex electric field profile has
odd reflection symmetry along y = 0, hence the positive and negative components
will cancel each other in the far field emission [65]. The Ey profile however, has
an even mirror symmetry along y = 0 and the waveguide is asymmetric in the
x-direction, so that the far field of the mode emission will be y-polarised. A
slow-light resonance from a second y-polarised waveguide mode [83] appears in
the spectrum at λ/a ≈ 2.7, with Q = 2200. The other features in the y-polarised
dipole spectrum are Fabry-Perot like resonances between the GaAs-air interface
and the back photonic crystal mirror, all with Q < 200.
The equivalent calculation for x-polarised modes shows M2 further inside the
band gap at λ/a ≈ 3.5 with Q = 4900. The closeby features all have Q < 1700.
The electric field profile of M2, as shown in figure 7.6 (b), confirms that its far-
field is x-polarised. A second x-polarised feature M3, originating from another
waveguide mode [83], appears at λ/a ≈ 3.0, which will be discussed later. Its
electric field profile is shown in figure 7.6 (c).
7.4 Fabrication
The process of fabricating PCWGs is almost identical to the H1 process, with the
addition of a cleaving step to allow in-plane measurement. The sample design
was developed to make this step possible. A large feature was positioned between
two opposing unidirectional waveguides to form the semiconductor-air interface
at one end of the waveguide, as shown in figure 7.7 (a). The sample was cleaved
through this feature, leaving a waveguide on each piece of the cleaved wafer. An
SEM image of a completed device is shown in figure 7.7 (b).
103
7.4 Fabrication
(a) (b)
Figure 7.7: SEM images of (a) the mask design showing the large features be-
tween two opposing unidirectional photonic crystal waveguides, (b) a close up
of a completed device. The waveguide is terminated between nearest neighbour
holes.
220 230 240 250 260
800
900
1000
Wa
ve
len
gth
 (n
m)
Crystal periodicity, a (nm)
M1
M2
Figure 7.8: Calculated wavelengths of the modes M1 and M2 versus the peri-
odicity, a, with a slab height of d = 200 nm and a GaAs refractive index of
n = 3.53.
104
7.5 Optical characterisation
Experimentally, any of the modes can be accessed by choosing a suitable
value of the crystal periodicity, a, so that the mode appears at around 900 nm,
the wavelength at which the InAs QDs emit. Figure 7.8 shows the calculated
wavelengths of M1 and M2 versus the periodicity for a slab height of d = 200 nm.
According to the calculations, the x-polarised M2 should appear at 900 nm for
a waveguide with a = 253nm, and the y-polarised M1 should appear at a = 228
nm.
7.5 Optical characterisation
Figure 7.9 shows polarisation-resolved PL from four different waveguides contain-
ing a low density of QDs at the centre of the slab, with periodicities of a = 230
nm (figures 7.9 (a) and (b)) and a = 255 nm (figures 7.9 (c) and (d)). Figures 7.9
(a) and (c) are measured in-plane, whereas figures 7.9 (b) and (d) are measured in
the out-of-plane setup as indicated. Modes were identified in each of the spectra
and are indicated with green fitted lines. As the temperature of the device is
increased, the modes redshift due to the change in the refractive index of GaAs,
whereas the other sharper lines from QDs redshift at a faster rate, following the
change in the GaAs band gap [68].
The modes can be unambiguously identified as M1 and M2 based on the
values of a for which they appear, i.e. figures 7.9 (a) and (b) show M1 and figures
7.9 (c) and (d) show M2. The mode wavelengths are not exactly as calculated
and are different for each sample because of the inevitable random fabrication
imperfections.
Typically M2 was measured to have Q = 2700, around a factor of 2 lower
than the calculated value. M1 typically had Q = 1200, much lower than the
calculation of 25500. These values are limited by fabrication imperfections in the
structures. The Q-factor of M1 is smaller because the imperfections are more
significant in the waveguide with smaller features.
The mode M2 in figure 7.9 (d) is strongly x-polarised as expected. When
measuring in-plane, the y- and z-polarised components of the field are sampled.
Since Ez is not confined by two-dimensional slab photonic crystals, only Ey is
relevant in this case. The emission of M2 is expected to be unpolarised because
105
7.5 Optical characterisation
894 896 898 9000
0 0
888 890 892 894 0
x-polarizedy-polarizedz-polarized
Wavelength (nm)
PL
 in
ten
sit
y (
arb
. u
nit
s)
(a)
(b)
(c)
(d) z
x
y
a = 230 nm a = 255 nm
Figure 7.9: PL from four separate PCWG structures measured at 45 K, corrected
for the polarisation of the measuring setup. The sharp lines are due to QD
transitions, the broader features are the modes as indicated by the fitted green
lines. These examples are chosen as they all have an excitonic transition on or
near the relevant mode at 45 K. The insets show the two measurement setups:
exciting from the top and collecting from the side (top inset) or from the top
(bottom inset). (a) PCWG with a = 230 nm measured in-plane. (b) PCWG
with a = 230 nm measured out-of-plane. (c) PCWG with a = 255 nm measured
in-plane. (d) PCWG with a = 255 nm measured out-of-plane.
106
7.6 Purcell effect
of the odd reflection symmetry of its Ey profile along y = 0. This is shown in
figure 7.9 (c).
The in-plane PL of the mode M1 is strongly y-polarised as expected from the
even symmetry of its Ey profile, and shown in figure 7.9 (a). The out-of-plane
PL from M1 in figure 7.9 (b) is however, only weakly y-polarised. The presence
of x-polarised photons implies that the Ex field components are not cancelling in
the far-field as expected. This is attributed to the large fabrication imperfections
in the a = 230 nm structure as evidenced by the low Q-factor, since the Ex field
profile has anti-nodes close to the air hole walls.
7.6 Purcell effect
This section focuses on in-plane measurements of M1 in the PCWG shown in
figure 7.9 (a). PL from the same device at 5 K is shown in figure 7.10 (a). As
the temperature of the device is increased, each sharp QD line can be brought
onto resonance with the mode (highlighted by the green line). The line labeled
2X is identified by power dependence measurements as biexcitonic, and lines Xa,
Xb and Xc are identified as excitonic.
Time-resolved measurements of 2X and Xa at 5 K, and when they are on
resonance with the mode are shown in figures 7.10 (b) and (c). Double exponen-
tials were fitted to the curves to extract the lifetimes. This double exponential
decay is typical of the studied samples, the long lifetime component arises from
refilling by another state [123]. When the lines are on resonance, the lifetime of
the transition is enhanced due to the Purcell effect. In contrast, over a similar
temperature range, the lifetime of line Xc increases as it detunes away from the
mode and is suppressed (see figure 7.10 (d)).
Lines 2X and Xa show Purcell enhancements of 1.9 and 2.7 respectively, sug-
gesting that they originate from different QDs. The maximum calculated Purcell
factor for a QD coupled to this mode with Q = 1200 is FP = 65. The difference
with the measured values is most likely due to a mismatch in the position and
orientation of the QDs with respect to the maximum in the electric field of the
mode profiles. In the future, the inclusion of prepositioned QDs could lead to
higher Purcell enhancements [7, 125].
107
7.6 Purcell effect
0
2X
896 897 898 899
Wavelength (nm)
PL
 in
ten
sit
y (
arb
. u
nit
s)
y-polarized
z-polarized
Mode
TRPL at 5K
TRPL T >35K
FitNo
rm
ali
ze
d P
L
int
en
sit
y (
arb
. u
nit
s) 0 5 0 5 0 5 0 5
Delay (ns)
(a)
(b) (c) (d) (e)
Xa XcXb
Figure 7.10: (a) In-plane polarisation-resolved PL from a PCWG with a = 230 nm
at 5 K. The sharp lines are from QD transitions. The broad feature highlighted
by the green line is the y-polarised mode M1. (b) Time-resolved PL from line 2X
at 5 K and when on resonance with the mode at 50 K. (c) Time-resolved PL from
line Xa at 5 K and when on resonance with the mode at 40 K. (d) Time-resolved
PL from line Xc at 5 K and at 40 K. (e) TRPL from a QD exciton located inside
the photonic crystal region.
108
7.7 Further work
Finally, figure 7.10 (e) shows time-resolved PL from a bright QD exciton emit-
ting at 891 nm which has an estimated lifetime of 38.5 ns, a suppression by a
factor ∼30. By moving the position of the laser and peaking up on the QD signal,
it is apparent that this dot is located inside the photonic crystal region surround-
ing the waveguide. Accordingly the radiative lifetime is strongly suppressed due
to the low density of states inside the photonic band gap. Similar lines with very
long lifetimes appear in the spectra of many of the PCWGs measured.
7.7 Further work
0.32 0.34 0.36 0.38
4000
8000
2.83.0
3.2
40000
80000
120000
Q
Normalized hole radius r/a
F P
λ/a
Figure 7.11: Results of FDTD simulations for the x-polarised mode M3 of a W1
unidirectional photonic crystal waveguide with d/a = 0.7, l/a = 12 and n = 3.53,
as the radius of the holes, r/a, is changed.
Figure 7.11 shows the results of FDTD simulations of the Q-factor of the x-
polarised mode M3 as the radius of the holes is changed. The maximum Q-factor
is found to be very high at Q = 125000, with an equivalent maximum Purcell
factor of FP = 6790. In the same way that PL was measured in-plane from
the first x-polarised mode M2, as shown in figure 7.12, it should be possible to
109
7.7 Further work
897 898 899
X
M X 65K
PL
 in
ten
sity
 (a
rb.
 un
its)
Wavelength (nm)
40KM
Figure 7.12: In-plane PL from a QD coupled to the x-polarised mode M2 between
40 K and 65 K. As the temperature is increased, the QD line (labeled X) comes
onto resonance with the mode, before moving off resonance.
couple a QD to this very high Q mode to use as an in-plane quantum light source.
Furthermore, since this mode occurs at a lower normalised wavelength than M1
and M2, a larger periodicity would be required to see the mode at around 900
nm, and it would therefore be much easier to avoid the effects of fabrication
imperfections. Alternatively it may be possible to shift one of the other modes
to be resonant at a lower value of the normalised wavelength by changing other
parameters of the waveguide [83].
110
7.8 Summary and outlook
7.8 Summary and outlook
The operation of a new kind of optical cavity based on a photonic crystal waveg-
uide has been investigated. The design has been significantly improved by FDTD
simulations for a strong Purcell effect. The fabrication of the device led to the
measurement of a Purcell enhancement of FP = 2.7 for the in-plane emission of
a QD for the first time. These results lead the way toward an in-plane quantum
light source for a future photonic integrated circuit.
111
Chapter 8
Conclusions
Quantum dots hold great potential for new technologies based on the use of single
photons. Their usefulness stems from the relatively simple internal dynamics,
which often may be considered as two or three level systems, and the ease of
their integration with other semiconductor techniques. Quantum dots can be
manipulated by controlling either the population of carriers confined by the dot
or the photonic environment that the dot experiences. In this thesis, both of these
methods were used to develop a number of quantum dot based devices which may
find uses as components in future quantum information technologies.
Controlling the polarisation correlation of photon pairs
The biexciton decay may be used to obtain two photons whose polarisations are
either classically correlated in the linear (H or V ) basis, or entangled, depending
on the magnitude of the exciton fine structure splitting. For either of these
applications, emission from other states (e.g. charged excitons) will reduce the
efficiency of the source and may degrade the polarisation fidelity by contributing
unwanted background light. It was demonstrated that by selectively controlling
the hole population of a quantum dot with an electron tunnelling barrier and an
electric field, the particular excitonic species which are present in its emission
can be controlled. By choosing the appropriate conditions, the intensity of the
emission from the biexciton decay (or from either charged exciton, if desired) can
be maximised.
112
Furthermore it was found that the degree of classical polarisation correlation
between the two photons emitted in the biexction decay from this device was
modified as the electric field applied to the quantum dot was changed. Cross-
correlation and time-resolved measurements gave evidence that exciton spin scat-
tering was responsible for this behaviour. By choosing conditions so that the in-
tensity and degree of polarisation correlation are maximised, applications based
on the biexciton decay may benefit.
Single quantum dot memory
The operation of a digital memory device based on a single quantum dot embed-
ded in a p-i-n diode was demonstrated. The angular momentum selection rules of
a photon exciting an electron inside the device when under an external magnetic
field were exploited to populate a single quantum dot with an exciton, whose
electron spin was determined by the circular polarisation of the photon. By ap-
plying an appropriate electric field across the dot, the exciton’s hole was removed,
leaving the electron behind due to the electron tunnelling barrier. In this way,
spontaneous emission from the dot was prohibited for some time. By changing
the electric field it was possible to reintroduce two holes into the dot on demand
so that spontaneous emission could take place. It was found, as expected, that
the emitted photon’s polarisation was strongly correlated with that of the initial
photon. The digital polarisation information of the initial photon was stored as
the spin of an electron inside a quantum dot, and was deterministically recovered
at up to 1 µs later by the application of an electrical trigger.
Fabrication of photonic crystals
Extensive work was carried out in the development of a fabrication process to
produce high quality two dimensional slab photonic crystal structures operating
with a photonic band gap in the near infrared, for integration with InAs quantum
dot devices.
113
Strong coupling in a H1 photonic crystal cavity
The phenomena of strong coupling, in which energy oscillates between an exciton
in a quantum dot and a photon in a surrounding optical cavity, was achieved and
investigated using a single quantum dot and the monopole mode of a H1 photonic
crystal cavity. H1 cavities are appealing because of their tiny mode volume and
because of the properties of the degenerate dipole modes, which are attractive
for use in quantum dot entangled photon pair sources. The design of the cavity
and optimisation of the fabrication process led to sufficiently high monopole Q-
factors so that the photon could be confined in the cavity long enough for it to be
reabsorbed by the quantum dot. The use of the monopole mode meant that the
vacuum Rabi splitting persisted for all linear polarisations, a feature never seen
before. Furthermore, strong coupling was achieved in a H1 cavity at a wavelength
of ∼ 880 nm, where the fine structure splitting of the exciton in InAs quantum
dots is close to zero.
Unidirectional photonic crystal waveguide
Many quantum dot based devices require some kind of optical cavity to direct
the emission in a useful direction, and to speed up the process of spontaneous
emission. In the final chapter of this thesis, the potential of a new type of cavity
based on the use of a photonic crystal waveguide was investigated. A unidirec-
tional photonic crystal waveguide may provide all of the benefits of traditional
optical cavities, with the added benefit of in-plane directed emission (among
others). The design of the cavity was optimised, and its fabrication led to the
measurement of a Purcell enhancement of the in-plane spontaneous emission from
a quantum dot for the first time. This kind of cavity may find applications as an
on-chip quantum light source and in future photonic integrated circuits.
114
Appendix A
Photonic band gap
This Appendix gives a summary of the theory behind photonic crystals, adapted
from the analysis in reference [59]. The macroscopic behaviour of light is described
by the Maxwell equations
∇.B = 0 ∇× E + ∂B
∂t
= 0
∇.D = ρ ∇×H− ∂D
∂t
= J (A.1)
where B is the magnetic field, E is the electric field, D is the displacement field,
H is the magnetic induction field, ρ is the free charge density and J is the current
density. In a medium whose structure does not vary with time, with no free
charges or currents, ρ = 0 and J = 0.
Assuming that the field strengths are small, so that the effects of nonlinear
optics can be ignored, and that the material is isotropic, transparent, and non-
dispersive, then D and E are simply related by a real and positive scalar function
(r). This is the dielectric function, where D(r) = 0(r)E(r), with 0 being
the vacuum permittivity. Similarly H and B are related by a relative magnetic
permeability, µ(r), like B(r) = µ0µ(r)H(r), where µ0 is the vacuum permeability.
µ(r) is in general very close to unity for dielectric materials of interest, so using
B = µ0H, the Maxwell equations can be rewritten as
115
∇.H(r, t) = 0 ∇× E(r, t) + µ0∂H(r, t)
∂t
= 0
∇.[(r)E(r, t)] = 0 ∇×H(r, t)− 0(r)∂E(r, t)
∂t
= 0 (A.2)
Since these equations are linear, the time- and spatial-dependence parts of
their solutions can be separated out like
H(r, t) = H(r)e−iωt
E(r, t) = E(r)e−iωt. (A.3)
These equations describe a harmonic mode of the system as the product of a
mode profile (H(r) or E(r)) and a complex exponential. The real part of the
complex exponential describes a sinusoidally varying field in time. Using Fourier
analysis, any solution can be built out of a combination of these harmonic modes.
Inserting these equations back into the Maxwell equations gives
∇.H(r) = 0 ∇× E(r)− iωµ0H(r) = 0
∇.[(r)E(r)] = 0 ∇×H(r) + iω0(r)E(r) = 0 (A.4)
The equations on the left hand side restrict the mode profiles to be composed of
transverse electromagnetic waves. The other two equations can be combined to
eliminate E(r) to give
∇×
(
1
(r)
∇×H(r)
)
=
(ω
c
)2
H(r) (A.5)
where c = 1/
√
0µ0, the speed of light. This equation is called the master equation
of the system. By solving it subject to the transversality requirement, the allowed
modes H(r) and their frequencies, for a given dielectric function, can be found.
116
Quantum mechanics Electrodynamics
Field Ψ(r, t) = Ψ(r)e−iEt/~ H(r, t) = H(r)e−iωt
Eigenvalue problem HˆΨ = EΨ ΘˆH = (ω/c)2H
Hermitian operator Hˆ = −(~2/2m)52 +V (r) Θˆ = 5× (1/(r))5×
Discrete translational V (r) = V (r + R) (r) = (r + R)
symmetry
Bloch’s theorem Ψkm(r) = ukm(r)e
ik.r Hkm(r) = ukm(r)e
ik.r
Table A.1: Comparison of quantum mechanics and electrodynamics. Adapted
from [59].
The master equation has the form of an eigenvalue problem, and can be
rewritten like
ΘˆH(r) =
(ω
c
)2
H(r) (A.6)
where the operator Θˆ is defined as
Θˆ ≡ ∇×
(
1
(r)
∇×H(r)
)
. (A.7)
In this form, (ω/c)2 are the eigenvalues of the eigenfunctions H(r).
It turns out that the operator Θˆ is Hermitian [59]. The master equation is
therefore analogous to the time independent Schroedinger equation in quantum
mechanics. Many of the interesting results concerning photonic crystals have
analogies in quantum mechanics. Table A.1 summarises the similarities.
The Hermiticity of the operator implies that the eigenfunctions have real
eigenvalues and that they are orthogonal. The eigenfunctions obey a variational
theorem, analogous to the variational principle in quantum mechanics. In vague
terms, this says that a mode tends to concentrate its electric field energy in regions
of high dielectric constant and to minimize the amount of spatial oscillations
in its profile, while remaining orthogonal to the modes below it in frequency.
Furthermore, the spectrum of the system will become discrete if the fields are
spatially bound, either because they are localized or because they are periodic.
117
In a photonic crystal, the structure has a periodic variation in the dielectric
function. i.e.
(r) = (r + R) (A.8)
where R is any linear combination of the primitive lattice vectors of the periodic
structure. As in solid state physics, the modes of this system are Bloch modes -
a plane wave multiplied by a periodic function:
Hkm(r) = ukm(r)e
ik.r, (A.9)
where k is the Bloch wave vector, and ukm(r) = ukm(r + R) is a periodic
function. The behaviour of the modes can therefore be summarised in a band
diagram spanning the Brillouin zone. For each value of k, there is a discrete set
of modes which are labeled with the band index m. For each of these modes,
the frequency varies continuously with k, so it is possible to speak of a frequency
band with band index m.
The periodicity in the dielectric function causes Bragg reflections of photons
which gives rise to band gaps between some of the bands, just as a periodic
potential in solid state physics causes Bragg reflection of electrons which gives
rise to electronic band gaps.
When a periodicity of a is introduced into the dielectric function, the modes
which have a wavelength of 2a are forced into one of two distributions. Either
the nodes of the electric field are positioned in the high- regions, or in the low-
 regions, due to the symmetry of the system. From the variational theorem,
modes with more of their electric field energy concentrated in the high- regions
have a lower frequency (and vice-versa). Therefore there is a frequency difference
between the two cases, and a band gap appears.
The bands below the band gap tend to have more of their energy concentrated
in the high- regions, and those above have more energy concentrated in the low-
regions. In real implementations, often the low- regions are air regions. Hence
the band above the band gap is called the air band and the band below is called
the dielectric band.
118
Light with a frequency inside the band gap cannot propagate through the
photonic crystal. Such light which is incident on a photonic crystal will decay
exponentially inside the crystal and will be reflected.
119
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