Plasmonic Interactions in the
Quantum Tunnelling Regime
Kevin John Savage
Jesus College
University of Cambridge
A thesis submitted for the degree of
Doctor of Philosophy
September 2012
ii
Declaration
The work presented in this thesis was carried out at the Nanophotonics Group
in the Cavendish Laboratory, University of Cambridge between October 2007
and September 2011. This dissertation is the result of my own work and in-
cludes nothing which is the outcome of work done in collaboration except where
specifically indicated in the text. It has not been submitted in whole or in part
for any degree at this or any other university, and is less than sixty thousand
words long.
Kevin. J. Savage
The picture on the title page is a dark-field image of two axially aligned, gold
coated, atomic force microscopy tips. Each tip apex is functionalised with a
spherical ball (effective nanoparticle) supporting a localised surface plasmon
polariton resonance in the visible spectrum, thus giving them their green ap-
pearance. The image is over-exposed to accentuate the scattered light from the
nanoparticle on each tip apex.
iii
To my parents John and Sonia, and my brothers Mark and Gareth.
iv
Acknowledgements
I would like to thank my supervisor Professor Jeremy Baumberg for giving me
the opportunity to be a founding student member of the Cambridge node of
the Nanophotonics Centre. His creativity, inclusive philosophy, ‘patience with
the student’, hard-work, and general zest for exploring the unknown, have been
inspirational and much appreciated. Our regular discussions gave me a deeper
understanding of scientific research and every aspect covered in this thesis. I
am confident that without Jeremy’s support, enthusiasm, understanding and
foresight the research could well have been cut-short leading to a significantly
different conclusion.
Throughout my studies I have thoroughly enjoyed the fact that modern
experimental research is impractical without the close interaction of many peo-
ple. I am therefore indebted to many more people that I can mention individu-
ally. So to everyone as a whole, I would like to say that I am extremely grateful
for your guidance, facilitation, patience, friendship and (usually) constructive
criticism during my time at the Cavendish lab.
The advice, help, insight, knowledge and fun I gained from working with Dr.
Bruno Soares was invaluable throughout the first two years of my studentship,
and we had many enjoyable times both inside and outside the lab. In my final
year I had the pleasure of working closely with Dr. Matthew Hawkeye whose
scientific method, extensive knowledge, ability in the lab (and on the football
pitch), positive attitude and patience taught and helped me a great deal in a
short time. Both Bruno and Matt were great companions around the lab and
helped advance the project enormously. Here I would also like to thank Richard
Hardy, who was the group’s technician for the majority of my studies. His
prompt, high-quality work, presence in the office, and advice on the ‘finer things
in life’ were all greatly appreciated.
I would like to say a huge thank you to all the past and present post-docs,
students, associates and visitors of the Cambridge Nanophotonics group I had
the pleasure to work with, especially: Prof. F. J. García de Abajo (BEMAX!), Dr.
Max Bock, Dr. Gabriel Christmann, Peter Christofolini, Dr. Stavros Christopou-
los, Dr. Robin Cole, Dr. Chris Coulson, Dr. Richard Denk, Petros Farah (com-
puter issues!), Dr. Alex Finnemore, Dr. Nick Gibbons, Cornelius Grossmann,
Dr. Stefan Guldin, Andrew Haines, Lars Herrmann, Dr. Fu Min Huang, James
v
Hugall (JDAP, amongst many other things!), Andreas Kontogeorgos, Dr. Ni-
raj Lal, Ben Michaelis, Dr. David Snoswell, Christian Steuwe, Jason Sussman,
Richard Taylor and Dr. Silvia Vignolini. Additionally, the more recent members
of the group, Lindsey Ibbotson, Matt Millyard, Wendy Niu, Alan Sanders and
Qibin Zhao, who all arrived near the end of my studies, were friendly compan-
ions during my final months in the lab. I can honestly say every single person
in the group provided me with assistance, advice, friendship and some damn
good times during my studies. I will fondly remember (and in some cases try
to forget) our office banter, late nights in the lab, Fountain sessions, frequent
excursions, winter schools, early morning cycles home, and all the ups, downs,
and everything in between.
The continuous support and encouragement from all my friends outside the
lab in Cambridge kept me buoyant through the tough and frustrating times, and
I would very much like to acknowledge Dr. James Birrell, Dr. Ruaidhri Farrell,
Bethan Shaffery, Dr. Ben Tolley and Debbie Toms. I especially thank Chris Coul-
son, who not only was a fellow Nanophotonics group member and contributed
immensely to my studies both hands-on in the lab and through countless hours
of wide-ranging discussion and debate, but also willingly suffered through my
ever-presence as one of his housemates. His eclectic music selections provided a
sometimes surreal soundtrack to many an evening lab session where the topics
of conversation were often far ‘outside the box’.
I would like to extend my whole-hearted gratitude to all the Cavendish me-
chanical and electronic workshop staff who assisted me with the construction
of the experimental rig and various trinkets, with special thanks going to Nigel
Palfrey. Additionally I would like to thank all the secretarial and IT staff who
kept the project running smoothly behind the scenes, especially Angela Camp-
bell and Colin Edwards. Here I would also like to acknowledge the funding
from the EPSRC that made this project possible, and provided me with such a
great opportunity to add my own research contribution.
Outside Cambridge, I would like to thank all my friends from Southampton
and Wokingham, particularly Lucy Collins, David and Paul Dodds, Nick Neish,
Dr. Dean Read, Alex Wilson and Karl Wojna, who have been ever-ready to
provide ample retreats from the world of physics.
Finally, I thank my family for their unwaivering support and encouragement
in every aspect of my life, and it is to them to which this thesis is dedicated.
vi
Abstract
Driven by exciting new research and applications, top-down and bottom-up fabri-
cation techniques are producing ever more intricate, reproducible, plasmonic nano-
architectures with gaps and junctions approaching the single nanometre and atomic
scales. Such atomic-sized features promote the intersection of physics, chemistry and
biology in plasmonics. Consequently, understanding light-matter interactions in such
closely spaced, electromagnetically coupled, metallic nanosystems is of vital impor-
tance to a tremendous variety of current and future nanophotonic technologies. This
thesis describes the first dynamically controlled, optically broadband, experimental in-
vestigations of light-driven plasmonic coupling between two metal nanostructures with
sub-nanometre separation.
A new experimental apparatus and nanosystem alignment technique was devel-
oped to enable the required sub-nanometre inter-nanoparticle geometry to be created
and probed. Two conducting atomic force microscopy tips with nanoparticle function-
alised apices are brought into nanoscale ‘tip-to-tip’ axial alignment with dynamically-
controlled spacing and ultra-wide optical access. Resonant electrical parametric mix-
ing, created by oscillating the electromechanically coupled tips, is utilised to extract
an electronic signal due to nanoscale changes in inter-tip position. Experimental re-
sults match theory confirming the viability of the technique. By functionalising the
tip apices, this unique multi-functional observation platform allows the plasmonic re-
sponse of nanoparticle dimers with sub-nanometre separations to be characterised.
By simultaneously capturing both the electrical and optical properties of tip-
mounted gold nanoparticles with controllable sub-nanometre separation, the first ev-
idence for the quantum regime of optically driven tunnelling plasmonics is revealed
in unprecedented detail. It is demonstrated that quantum mechanical effects are criti-
cally important at approximately the 0.3 nm scale where spatially non-local tunnelling
plasmonics controls the optical response. All observed phenomena are in good agree-
ment with a recently developed quantum-corrected model of plasmonic systems. The
findings imply that tunnelling establishes a quantum limit for plasmonic field enhance-
ment and confinement. Additionally, the work suggests the highly enhanced local
density of photonic states in nanoscale cavities could enable coherent plasmon-exciton
coupling. This thesis prompts new experimental and theoretical investigations into
quantum-domain plasmonic systems, and impacts the future of nanoplasmonic device
engineering, nanoscale photochemistry and plasmon-mediated electron tunnelling.
vii
viii
List of Publications
K. J. Savage, M. M. Hawkeye, R. Esteban, A. G. Borisov, J. Aizpurua and J. J.
Baumberg, "Revealing the quantum regime in tunnelling plasmonics," Nature,
vol. 491, no. 7425, pp. 574-577, 2012.
K. J. Savage, M. M. Hawkeye, B. F. Soares and J. J. Baumberg, "From microns
to kissing contact: Dynamic positioning of two nano-systems," Applied Physics
Letters, vol. 99, p. 053110, 2011.
ix
x
Contents
1 Introduction 1
1.1 Thesis Layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 Theoretical Background 7
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Optical Frequency Response of Metals . . . . . . . . . . . . . . . . . 9
2.2.1 Electromagnetic Waves in Media . . . . . . . . . . . . . . . . 9
2.2.2 The Drude Model . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.3 Plasmonic Excitations . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.3.1 Surface Plasmons and Surface Plasmon Polaritons . . . . . 19
Ideal Surface Plasmon Polaritons . . . . . . . . . . . . . . . . 24
Real Surface Plasmon Polaritons . . . . . . . . . . . . . . . . 26
2.3.2 Localised Surface Plasmon Polaritons . . . . . . . . . . . . . 27
Quasi-static solutions . . . . . . . . . . . . . . . . . . . . . . 31
Mie solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.3.3 Numerical Methods Overview . . . . . . . . . . . . . . . . . 36
2.3.4 Coupled Localised Surface Plasmon Polaritons . . . . . . . 42
Non-contact regime . . . . . . . . . . . . . . . . . . . . . . . 45
Contact regime . . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.4 Open Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3 Experimental Rig 55
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.2 Mechanical Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.2.1 AFM Tip Alignment Unit . . . . . . . . . . . . . . . . . . . . 58
3.2.2 Optical Microscope Unit . . . . . . . . . . . . . . . . . . . . . 62
3.3 Electronic Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.3.1 AFM Tip Alignment Electronics . . . . . . . . . . . . . . . . 66
3.3.2 Nanojunction Conductance Electronics . . . . . . . . . . . . 70
3.4 Optical Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.4.1 Microscope Unit Imaging Optics . . . . . . . . . . . . . . . . 72
3.4.2 Supercontinuum Laser Dark-field Spectroscopy Optics . . . 74
xi
Contents
Chromatic Dispersion Compensation . . . . . . . . . . . . . 79
Minimising the Poisson Spot Background . . . . . . . . . . . 81
Supercontinuum Laser Setup Validation . . . . . . . . . . . 84
3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
4 Dynamic Positioning of Two Nanosystems 87
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
4.2 Theoretical Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . 88
4.2.1 Parametric Mixing: Toy Model . . . . . . . . . . . . . . . . . 89
4.2.2 Numerically Implemented Alignment Simulation . . . . . . 93
4.3 Experimental Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . 100
4.4 Alignment Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
4.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
5 Quantum Tunnelling Plasmonics 107
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
5.2 Theoretical Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . 109
5.2.1 Expected Plasmonic Modes . . . . . . . . . . . . . . . . . . . 109
5.2.2 Quantum Corrected Model . . . . . . . . . . . . . . . . . . . 114
5.3 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
5.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
5.5 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
5.5.1 Tunnelling Plasmonics Regime . . . . . . . . . . . . . . . . . 130
5.5.2 Understanding the Plasmonic Modes . . . . . . . . . . . . . 133
5.5.3 Coherent Plasmon-Exciton Coupling . . . . . . . . . . . . . 138
Unexplained Experimental Results . . . . . . . . . . . . . . . 142
5.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
6 Summary and Outlook 149
A Charge Transfer Model 153
B List of Symbols and Abbreviations 159
Bibliography 171
xii
List of Figures
1.1 Schematic timeline showing the evolution, and combined research
and commercial output, related to the fields of electronics and
plasmonics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2.1 Graphs comparing the experimental values of < [ε˜m(ω)] and
= [ε˜m(ω)] for gold and silver against Drude model fits. . . . . . . . 17
2.2 SPP interface geometry and field line schematic. . . . . . . . . . . . 21
2.3 Graph showing the behaviour of k˜spp for a Drude metal (with
negligible damping) compared with that of the light line k0. . . . . 24
2.4 Graph showing the behaviour of k˜spp for gold and silver com-
pared with the light line k0. . . . . . . . . . . . . . . . . . . . . . . . 27
2.5 Schematic representation of a LSPP on a spherical NP. . . . . . . . 28
2.6 Schematic representations of LSPP excitation in the quasi-static
and Mie regimes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.7 Comparison of scattering cross-sections for an Au NP calculated
using the quasi-static, Mie and BEMAX formalisms. The elec-
tric near-field and representative induced surface charge density
distributions for the dipole LSPP resonance are also shown. . . . . 39
2.8 Schematic of lightning rod effect in a metallic tip system. . . . . . . 40
2.9 Non-contact and overlapping NP dimer configurations used to
represent tip nanosystem. . . . . . . . . . . . . . . . . . . . . . . . . 44
xiii
List of Figures
2.10 NP dimer LSPP coupling configurations in the limiting cases of
parallel and perpendicular polarisation with respect to the dimer
axis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.11 Near-field intensity map and associated scattering cross-section
for an Au NP dimer with d = 1 nm excited with incident light
polarised parallel to the dimer axis. . . . . . . . . . . . . . . . . . . 47
2.12 Near-field intensity map and associated scattering cross-section
for an overlapping Au NP dimer with d = −2 nm excited with
incident light polarised parallel to the dimer axis. . . . . . . . . . . 50
2.13 Evolution of LSPP dimer modes as NP separation d is reduced
through the non-touching to overlapping transition. . . . . . . . . . 51
3.1 Schematics of techniques for NP dimer and nanocavity creation. . 56
3.2 Scaled drawing of the co-axial tip-to-tip geometry in relation to
the light-cone produced by a NA 0.9 microscope objective under
lamp illumination. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.3 CAD drawing of the AFM tip alignment unit. . . . . . . . . . . . . 60
3.4 Photo of the completed tip alignment unit mounted on the trans-
lation stage of an Olympus BX51 optical microscope. The inset
and sub-inset show high magnification DF images of typical AFM
tips used. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.5 Photo of the overall experimental rig including microscope unit. . 63
3.6 Photo of the tip alignment unit mounted ‘side on’ with sample
holder and microscope objective shown. . . . . . . . . . . . . . . . . 64
3.7 Top-down photo of the microscope unit with schematic overlay
of lamp, laser and combined light paths. . . . . . . . . . . . . . . . 65
xiv
List of Figures
3.8 Schematic showing the setup of the main electronic measurement
equipment used for the axial AFM tip-to-tip alignment technique
developed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.9 Schematic showing the setup of the main electronic measurement
equipment used for the tip-to-tip (NP dimer) conductance mea-
surements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.10 Schematic of the halogen lamp imaging optics incorporated into
the custom built microscope unit. . . . . . . . . . . . . . . . . . . . . 72
3.11 Dark-field image of Au NPs deposited on a glass substrate. The
NPs are of different shapes and maximal dimension of approxi-
mately 100 nm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
3.12 Schematic of the supercontinuum laser illumination optics incor-
porated into the custom built microscope unit. . . . . . . . . . . . . 75
3.13 Top-down photo of the DF setup for the supercontinuum laser. . . 76
3.14 Typical laser illumination and scattered light collection geome-
tries, including intensity profile and polarisation distribution of
focussed laser spot. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
3.15 Typical example of chromatic dispersion minimisation by high
precision optical element alignment and Amici prism pair disper-
sion compensation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
3.16 Schematic ray diagrams and CCD images showing how laser light
diffracted by the DF stop is not filtered out by the normal DF laser
setup and hence degrades the DF characterisation capability. . . . 82
3.17 Schematic ray diagrams and CCD images showing how laser light
diffracted by the DF stop is filtered out by the improved DF laser
setup enabling high-quality DF characterisation capabilities. . . . . 84
xv
List of Figures
3.18 Scattering spectra of Au NPs deposited on a glass substrate, with
insets showing corresponding NP images. Both spectra and im-
ages are taken under supercontinuum laser illumination. . . . . . . 85
4.1 Coordinate system used in theoretical analysis of electromechan-
ically coupled dual AFM tip system. . . . . . . . . . . . . . . . . . . 88
4.2 Schematic of axially symmetric tip-tip model used for theoretical
modelling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
4.3 Simplified schematic of experimental apparatus with signal
ground, band-pass filter and voltage amplifier shown. . . . . . . . 101
4.4 Experimentally measured | I˜3ωS | vs. numerical simulation with
the same parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
4.5 Typical alignment scan sequence results for d0 = 460 nm→ 220
nm with V0 ≈ 10 V. Evolution of the FWHM, and uncertainty in
position, of the alignment grid minimum are also shown. . . . . . 103
4.6 Flow diagram of total AFM tip (NP dimer) alignment procedure. . 105
5.1 Geometry of NP functionalised AFM tips used in experiment and
theoretical simulation. . . . . . . . . . . . . . . . . . . . . . . . . . . 110
5.2 Far-field scattering cross-section of a large Au NP (a = 150 nm). . 111
5.3 Comparison of the far-field scattering cross-section of an Au tip
and Au NP functionalised tip. . . . . . . . . . . . . . . . . . . . . . . 112
5.4 Review schematic of experimental setup for measurement of LSPP
mediated optical scattering response in the quantum transport
regime. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
xvi
List of Figures
5.5 Schematic of tip-tip snap to contact process. . . . . . . . . . . . . . 121
5.6 Measured DF scattering spectra from the NP dimer at different
separations d. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
5.7 Optical detection of onset of quantum tunnelling in NP dimer
with sub-nanometre separation. . . . . . . . . . . . . . . . . . . . . . 124
5.8 Simultaneously measured DC conductance and DF scattering ver-
sus the force applied to the inter-NP cavity. . . . . . . . . . . . . . . 127
5.9 Evolution of the LSPP modes through the quantum regime and
the quantum limit of LSPP confinement. . . . . . . . . . . . . . . . 129
5.10 DC conductivity σQT0 of a NGJ, and charge transfer across NGJ,
versus separation d. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
5.11 Near-field magnitude and phase in the vicinity of the AFM tips
apex NPs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
5.12 The NP dimer nanocavity-localised near-field distributions. . . . . 135
5.13 Possible coupled plasmon-exciton mode-splitting phenomena in
plasmonic nanocavity. . . . . . . . . . . . . . . . . . . . . . . . . . . 143
5.14 Possible coupled plasmon-exciton mode-splitting phenomena in
atomic-scale cavity created in using different NPs. . . . . . . . . . . 145
xvii
xviii
Chapter 1
Introduction
Initial investigations into the interaction of light with metals were first carried
out in the early 1900’s and over the following century the field has been re-
discovered under several different guises. Around three decades after the in-
troductory classical plasma model treatments of metallic media, the advent of
quantum mechanics had transformed the description of all materials in the solid
state. Consequently the immediate potential of Semi-Conductor (SC) devices as
switches was realised and, as illustrated by Fig. (1.1), the advances in solid
state physics rapidly overtook the stagnating scope of light-metal interactions.
The experimental study of light-metal coupling was eventually resumed in the
1960’s under the brand of ‘metal optics’ and substantial progress was made
confirming recently developed theories. Over the next four decades investiga-
tions into metal optics provided the basis for what is now known as plasmonics,
analogous to how solid state physics research formed the basis for modern elec-
tronics. Driven forward primarily by chemical sensing applications, e.g. Sur-
face Enhanced Raman Scattering (SERS), this latest re-incarnation of the field
of light-metal interactions emerged strongly towards the end of the 1990’s, and
over the last 15 years has undergone rapid growth and diversification. This ex-
plosion in research output can be attributed to several factors, perhaps the most
prominent being our ever increasing capability to specifically tailor and charac-
terise complex nanosystems [1–3]. Such control, in tandem with continuously
improving theoretical modelling, now enables the creation of novel nanosystems
specifically designed to reveal and exploit new aspects of the plasmonically de-
termined optical response.
However, it is not just progression in technical capacity powering the ac-
tive expansion of plasmonics research and application output. To explore the
1
Chapter 1: Introduction
Figure 1.1: A schematic timeline showing the evolution, and combined research
and commercial output, relating to the fields of electronics and plasmonics.
Note the vertical axis is scaled to accentuate differences in output up to around
the 1970’s. Plasmonics currently lags electronics by around 30-40 years.
full breadth of plasmonic investigation demands collaboration over a range of
disciplines including physics, chemistry, and increasingly biology. Nanoscale
plasmonics thus provides impetus for the coalescence of historically separate
fields into new multi-disciplinary frameworks. The possibilities for scientific
exploration in this cross-over regime are innumerable, exciting, and offer direct
routes to new discoveries and original applications. It is not surprising then that
plasmonic interactions in nanosystems already have a wide-ranging and rapidly
expanding list of research level applications, for example, in metamaterials [4],
solar cells [5], biosystems [6], Surface Enhanced Raman Spectroscopy (SERS) [7],
photonic circuits [8] and cancer research [9].
The intersection of the natural sciences in plasmonics requires that the en-
ergy and information carried by light be transferred into, and extracted from,
truly nano- and molecular scale volumes. In general, the principles behind
the sub-wavelength propagation, localisation and manipulation of electromag-
netic waves dictate the blueprint of almost all plasmonic architectures and de-
vices. The necessary deep sub-wavelength control of light is now beginning to
2
Chapter 1: Introduction
be made possible by plasmonic design and nanoscale construction, and there-
fore it is highly likely that plasmonic research will lead to an expansive range of
potentially revolutionary ‘real-world’ applications, for example in energy pro-
duction [10], medicine [11], computing [12] and optics [13].
To this end, top-down and bottom-up nanofabrication techniques continue
to produce ever smaller, more intricate, nanosystems driving towards the molec-
ular and atomic size regimes. It is therefore imperative that plasmonic interac-
tions on the 2 nm→‘0’ nm ‘quantum length’ scale are investigated and under-
stood. Almost every present and future utilisation of plasmonic systems could
benefit from an improved understanding of this critical regime, with specific
examples including, molecular-opto electronics, single-emitter SERS, optical fre-
quency domain metamaterials, high-efficiency solar cells, active plasmonics, ex-
treme non-linear interactions, and quantum optics and computation.
The description of the majority of plasmonic phenomena can be initialised by
considering the specific type of surface excitations known as Surface Plasmons
(SPs) and Surface Plasmon Polaritons (SPPs), where SPPs describe a mixed state
of propagating light and metallic matter [14]. When SPPs are confined locally on
an extended system, or on an isolated nanosystem, they are commonly referred
to as Localised Surface Plasmons Polaritons (LSPPs). Understanding LSPP and
SPP properties and interactions in innovative nanosystems is therefore key to
advancing the field and hence creating new, or extending existing applications.
As LSPPs are often highly radiative in nature a copious amount of infor-
mation on the plasmonic response of a nanosystem is found from using broad-
band, far-field, optical elastic scattering spectroscopy as a non-destructive probe.
Therefore the aim of the work described in this thesis becomes clear: the first
broadband optical scattering measurements revealing the plasmonic response of
a single nanosystem where the effects due to physical features on the quantum
length scale are expected to be prevalent.
The type of nanosystem engineered to expose this intriguing quantum reg-
ime was a NanoParticle (NP) pair, commonly referred to as a NP (homo-) dimer
in analogy to a molecular dimer. In this case each individual NP is created on
the vertex of a commercially available Atomic Force Microscopy (AFM) tip by
functionalisation of the apex. Two such AFM tips were then maneuvered into
axial tip-to-tip alignment and the inter-NP (apex) separation dynamically con-
trolled from microns to less than 1 nm. The NP dimer system created was exam-
3
Chapter 1: Introduction
ined by simultaneous measurement of the optical scattering and electronic con-
duction characteristics. Investigation of the non-classical plasmonic response,
introduced by the sub-nanometre gaps created, was the primary goal of the
present work.
1.1 Thesis Layout
The thesis is organised into six chronologically ordered chapters. The first three
chapters describe the motivation for the work, the current physical understand-
ing of the phenomena, and the experimental framework developed. This foun-
dation shows how the work and results described in the last three chapters are
relevant, viable and ultimately a useful contribution to the field of plasmonics.
Firstly, Chapter (2) provides a brief theoretical overview of the concepts in
electrodynamics and solid state theory relevant to plasmonics in general. More
specific attention is devoted to SPPs on semi-infinite, planar, metal-dielectric
interfaces, and LSPPs on isolated NPs and electromagnetically coupled NP
dimers. Recent developments in experiments and theory describing LSPPs in
nanosystems with features on the order of the quantum length scale are high-
lighted and discussed in the context of the work undertaken. By combining
the well established theoretical basis for plasmonic nanosystems in the classi-
cal regime, and current experimental and theoretical state-of-the-art work by
the wider community, some open questions are posed. Finally, with these back-
ground components in place, a summary of current understanding and research
intentions is presented.
A new system was developed to create NP dimers with sub-nanometre sep-
aration as current methods cannot provide the required combination of low-
background ultra-wide optical access, electronic measurement capability, dy-
namic position control, and appropriate NP size and shape characteristics. The
experimental rig built to create and interrogate the required NP dimer systems,
formed by the functionalised apices of two AFM tips, is described in Chap-
ter (3). Here the mechanical, electronic and optical design characteristics of
the developed rig are described in detail to provide the basis for all subsequent
experimental work. An overview of the versatile observation platform produced
concludes the chapter.
4
Chapter 1: Introduction
The creation of the required NP dimer systems demanded the experimental
and theoretical validation of a new, general, nanosystem pair alignment me-
thod. The technique developed is described in Chapter (4) and is based on the
resonant electromechanical actuation of the coupled, dual AFM cantilever, tip
and apex NP composite system. Changes in AFM cantilever (and therefore tip)
oscillation amplitude, i.e. changes in inter-tip position, are measured electroni-
cally as one of the AFM tips is scanned in an alignment grid. This enables the
NP functionalised apices of the AFM tips to be aligned on the nanoscale thus
creating an effective NP dimer with controllable inter-NP separation. First, an
analytical toy model is outlined to provide physical insight into the complex
non-linear coupled system, followed by a more rigorous numerically imple-
mented model. The experimental results are compared against the developed
simulation, and conclusions drawn on the effectiveness of the technique.
In Chapter (5) the preceding work is amalgamated to create NP dimers with
sub-nanometre inter-NP separation. The plasmon mediated optical response
and inter-NP electronic conduction are measured simultaneously in the deep
sub-nanometre regime. The experimental results reveal optically driven quan-
tum tunnelling plasmonic coupling phenomena for the first time and are com-
pared against recently developed theory. A simple charge transfer time-scale
model is presented to provide an improved physical understanding of the plas-
monic response in the sub-nanometre regime and conclusions are drawn. The
measurements also show features possibly attributable to coherent plasmon-
exciton coupling in the inter-NP nanocavity, that would provide the first evi-
dence of ‘plexcitonic’ quasiparticles in such NP dimer systems. A basic classical
argument is presented to support the strong-coupling hypothesis.
Finally, Chapter (6) presents a summary of all the work carried out in this
thesis. Based on the motivations, theoretical background, results and analysis,
several possible future investigations are recommended.
5
6
Chapter 2
Theoretical Background
2.1 Introduction
The foundations of what is now known as the field of plasmonics were laid
by Drude in 1900. Inspired by Maxwell’s work on electromagnetism Drude
developed a theory to describe the collective response of conduction electrons
in a material to incident electromagnetic radiation [15]. The Drude model as-
sumes that a material contains immobile positive ions surrounded by a free-
electron plasma in which the electrons can be considered to be classical and
non-interacting with each other. These assumptions imply that the response of
electrons in a material can be modelled by only considering the force exerted
on them by a perturbing electromagnetic field and the damping force present
due to their electromagnetic interactions with scatterers, e.g. impurities and
phonons. Although a relatively simple classical kinetic gas model, it provides
very good explanations of the electrical conductivity and optical response of
metals, as characterised by frequency dependent, complex dielectric and con-
ductivity functions, ε˜m(ω) and σ˜m(ω) respectively. The Drude model also pre-
dicts the existence of collective longitudinal oscillations in the bulk, and surface,
of the free-electron plasma. In the following years a quantum mechanical treat-
ment of the problem defined a Bulk Plasmon (BP) and SP as quasi-particles
describing quanta of these oscillations [16, 17].
In 1902, soon after Drude’s work predicted BPs, Wood unknowingly dis-
covered the first evidence of SPPs while investigating the light reflected from
diffraction gratings [18]. He observed anomalous behaviour in the intensity
of the reflected light as a function of wavelength. At the time these results
7
Chapter 2: Theoretical Background
could not be explained and they became known as Wood’s anomalies. Near
the beginning of 1908, in seemingly unrelated theoretical work, Mie published
the electromagnetic solutions to Maxwell’s equations for the optical response of
spherical particles [19]. Concurrently Zenneck found solutions describing elec-
tromagnetic waves guided along a planar conductor-dielectric interface [20]. It
would not be until almost 70 years later that Drude’s simple model, Wood’s
anomalies, and Mie’s and Zenneck’s solutions could all be interpreted under a
unified theory of electromagnetic surface waves hybridised with spatial oscilla-
tions of the surface electron plasma.
Rayleigh provided a partial explanation for Wood’s anomalies in 1907 [21],
but it was not until important theoretical contributions from Fano in 1941 [22]
and Bohm [16], Pines [17] and Ritchie [23] in the 1950’s, that a theoretical frame-
work was in place to describe collective electron behaviour and the possibil-
ity of light strongly coupling to metallic surfaces. These theories explained
the experimentally measured electron energy loss on transmission through thin
metallic films, most notably conducted by Powell and Swan [24]. The electron
transmission experiments spurred on the first theoretical work by Stern and
Ferrell describing the SP [25]. In the late 1960’s light was first deliberately cou-
pled onto metallic surfaces, most prominently by Ritchie [26], Kretschmann and
Reather [27]1, and Otto [29]. The description of this phenomena was achieved
through a general theory describing the coupling of collective surface elec-
tron oscillations to electromagnetic surface waves, and a new class of quasi-
particle describing this phenomena was introduced as a SPP by Cunningham
in 1974 [30]. Only now could the final aspects of the anomalous reflectance
behaviour observed by Woods be fully understood as a result of SPP excita-
tion. The (Sommerfeld-)Zenneck electromagnetic surface waves are naturally
recovered in the low-frequency limit of the SPP dispersion, while the renewed
interest in SPs also led to the re-interpretation of Mie’s electromagnetic solu-
tions for spherical geometries. The solutions for metallic particles could now be
described intuitively in terms of localised SP modes [31], or more generally as
SPPs confined to a particle, referred to as LSPP modes.
Over the last three decades, SPPs and LSPPs excited on metallic systems
with dimensions and features designed and controlled on the nanoscale, have
1It should be noted that in 1959 Turbader [28] had already used the so called ‘Kretschmann’
experimental configuration for reflectance measurements on aluminium films. The characteristic
reflection dips recorded were due to light coupling to the metallic surface.
8
Chapter 2: Theoretical Background
been extensively studied due to their huge range of plasmonic responses and
associated applications. The present work is concerned with the study of the
electromagnetic interactions between LSPPs confined on individual metallic NP
systems as they are brought into conductive contact. This chapter will therefore
first present a brief theoretical background on the electromagnetic response of
metals under the Drude model framework. From this starting point a basic
theoretical treatment of SPs and SPPs in simple planar systems is presented
followed by an introduction to LSPPs and coupled LSPP interactions in NP
systems. The open questions that the current work has aimed to address are
then presented within the context of the theoretical background.
2.2 Optical Frequency Response of Metals
A full description of how classical optical frequency electromagnetic waves in-
teract with different media is of vital importance to understand and predict
plasmonic behaviour. This can be achieved by finding solutions to the macro-
scopic Maxwell’s equations which fit the boundary conditions and distributions
of charge and current appropriate to the system in question. The fundamental
macroscopic material parameters through which these solutions are expressed
are the main topic of this section.
2.2.1 Electromagnetic Waves in Media
The macroscopic Maxwell’s equations describing electromagnetic waves with
wavevector q˜ and angular frequency ω within an isotropic linear medium take
the form2
∇ ·D = ρext (2.1)
∇ · B = 0 (2.2)
∇× E = −∂B
∂t
(2.3)
2S.I units are used throughout this work. For clarity, in certain cases the tilde annotation ˜ is
used to emphasise a complex quantity. The ‘wide-hat’ annotation ̂ denotes a quantity that has
undergone a Fourier transform.
9
Chapter 2: Theoretical Background
∇×H− ∂D
∂t
= Jext (2.4)
where the displacement field D and electric field E are linked via the consti-
tutive relation D = ε0ε˜rE and the magnetic flux density field B and magnetic
field H are linked via the constitutive relation B = µ0µ˜rH. Here ε0 and µ0 are
the permittivity and permeability of free-space respectively and ε˜r and µ˜r are
the relative dielectric constant (relative permittivity) and relative permeability
respectively. In general ε˜r and µ˜r are tensors, however as the medium is as-
sumed to be isotropic these quantities reduce to complex scalars. Finally ρext is
the external charge density and Jext is the external current density. The physics
of the ‘free’ and ‘bound’ internal charges and currents are described by the dy-
namics of a total internal polarisation (P) of the system via D = ε0E+ P and the
magnetisation (M) of the system via H = 1µ0 B + M. Re-arranging Eqns. (2.3)
and (2.4) to eliminate H leads to a wave equation in terms of E and Jext
−∇ (∇ · E) +∇2E = ε0ε˜rµ0µ˜r ∂
2E
∂t2
+ µ0µ˜r
∂Jext
∂t
(2.5)
Taking the Fourier transform (∇ → iq˜, ∂∂t → −iω) of Eqn. (2.5) and assuming
the medium to be non-magnetic (µ˜r = 1) gives
q˜
(
q˜ · Ê
)
− q˜2Ê = −ε˜rk20Ê− iωµ0Ĵext (2.6)
where k0=(ω/c) is the free-space wavenumber with c equal to the speed of light
in vacuo. For transverse (electromagnetic) solutions q˜ · Ê = 0 3 and assuming
there are no sources of external stimuli present, i.e. ρ̂ext = 0 and Ĵext = 0, Eqn.
(2.6) reduces to
q˜ =
√
ε˜Tr k0 ⇒
√
ε˜Tr =
q˜
k0
(2.7)
where ε˜Tr is the relative permittivity for transverse excitations. For longitudinal
(electrostatic) solutions q˜ · Ê = q˜Ê and assuming no external stimuli Eqn. (2.6)
3It should be stressed that this equation, and similar equations throughout this work, do
not have their usual geometrical meaning because q˜ is a complex quantity. In other words,
transverse (longitudinal) excitations in an absorbing medium are not required to have Ê entirely
perpendicular (parallel) to <[q˜].
10
Chapter 2: Theoretical Background
reduces to
ε˜Lr = 0 (2.8)
where ε˜Lr is the relative permittivity for longitudinal excitations. In general the
approximation ε˜Tr = ε˜Lr = ε˜r is valid, however for large q˜ the difference in
dielectric response has to be taken into account.
Considering the relationship between D and the internal current density
J˜int(= ∂P∂t = σ˜E) yields the fundamental link between ε˜r and the conductivity
σ˜ [32]
ε˜r = 1+
iσ˜
ε0ω
(2.9)
Substituting Eqn. (2.9) into Eqn. (2.7) gives
q˜ = k0
(
1+
iσ˜
ε0ω
) 1
2
(2.10)
The relative dielectric constant and conductivity are in general complex, i.e.
ε˜r = ε1 + iε2 and σ˜ = σ1 + iσ2. These macroscopic material parameters relate
the bulk material electromagnetic response to the averaged response of the con-
stituent atoms and electrons. The complex refractive index of the medium can
be defined as n˜ = n + iκ =
√
ε˜r where n is the real refractive index and κ is the
extinction coefficient4. The complex relative dielectric constant and refractive
index can be linked by equating their real and imaginary parts
ε1 = n2 − κ2 (2.11)
ε2 = 2nκ (2.12)
From Eqns. (2.11) and (2.12) the expressions for n and κ for media without gain
characteristics are
4Strictly n˜ = ±√ε˜rµ˜r. As the complex refractive index is a derived concept (does not appear
in Maxwell’s equations) either sign can be taken. All materials considered in this work are so
called ‘epsilon negative’ materials (over the frequency range of interest) and the positive root is
taken. However for certain metamaterials, e.g. double negative materials (not found in nature),
the negative root should be taken [33]. Note causality is maintained in both cases [34].
11
Chapter 2: Theoretical Background
n =
√√√√√ε21 + ε22 + ε1
2
and κ =
√√√√√ε21 + ε22 − ε1
2
(2.13)
For a monochromatic electromagnetic plane wave travelling in the x-direction
through a medium it is postulated that the electric field varies as
E = E0ei(q˜x−ωt) (2.14)
where E0 is the electric field amplitude with the vectorial character describ-
ing the polarisation (linear polarisation in this case). Substituting the complex
wavenumber q˜ = k0
√
ε˜r = k0n˜ into Eqn. (2.14) gives
E = E0 exp (−k0κx) exp
(
iω
(n
c
x− t
))
(2.15)
This shows how the real and imaginary components of the complex refractive
index (and hence complex relative dielectric constant) of the medium affect the
propagation of an incident electromagnetic wave. The argument of the left hand
exponential is real and describes the attenuation of the wave as it propagates
through the medium. The argument of the right hand exponential is imaginary
and hence describes the oscillatory part of the wave. If |ε1|  |ε2| the real part of
the refractive index n is mainly determined by ε1 and therefore ε1 describes the
change in phase velocity (vp = ωk0n ) of the propagating wave due to polarisation
of the medium. The group velocity vg = ∂ω∂<[q˜] determines the velocity of energy
propagation through the medium.
2.2.2 The Drude Model
Through ε˜r the macroscopic Maxwell’s equations provide the framework to de-
scribe how electromagnetic waves interact with different media. A brief discus-
sion on how the Drude model describes ε˜r for metallic media is now presented
to provide a physical understanding of the origin of this most important param-
eter.
Until now ε˜r has been referred to as the complex relative dielectric constant
of the medium. However the dielectric and conductive response of a homoge-
12
Chapter 2: Theoretical Background
neous5 medium are functions of the frequency (temporal non-locality) of the
incident electromagnetic wave and the wavevector q˜ (spatial non-locality) it ac-
quires inside the medium due to the scattering with electrons. In certain metal-
lic media the approximate frequency dependence of ε˜m(q˜,ω) and σ˜m(q˜,ω) is
well described by the Drude model6. The q˜ dependence of the dielectric and
conductive response can be described by a more rigorous quantum mechanical
hydrodynamic-Drude model where the electrons can now feel the forces due
to variations in pressure of the electron gas, created mainly due to the Pauli
exclusion principle.
In most situations the wavevector contribution to ε˜m(q˜,ω) and σ˜m(q˜,ω) can
be neglected, as the q˜ dependent terms are small in the optical regime. It is
important to note that this simplification is not always valid, and it has to be
taken into account that not only light, but also electrons in the metal, trans-
port energy. For example, when light interacts with a NP dimer with inter-NP
separation d, evanescent wavevector components k˜E are created with magni-
tude
∣∣∣k˜E∣∣∣ = 2pi/d. These can in turn impart a significant momentum h¯q˜ to the
electrons in the vicinity of the nanogap between the NP dimer. For metals, if
d / 5 nm then q˜ is no longer negligible and can modify the material response
considerably [35].
In this large q˜ regime, the response of an electron at position re now also
depends on the electron excitations at nearby positions r′e. The homogeneity
approximation greatly simplifies the description of the system by allowing the
dependence on electron position to enter ε˜m and σ˜m via ε˜m (re − r′e, t− t′) and
σ˜m (re − r′e, t− t′) respectively. Thus the spatial and temporal non-locality enter
ε˜m and σ˜m via ε˜m(q˜,ω) and σ˜m(q˜,ω) respectively in the Fourier domain [14].
The spatially non-local short-range interactions are contained within a volume
defined by the penetration depth of the induced-surface-charge δe ≈ 0.5 nm,
and the dimensions of the nanogap. Here δe is defined by the 1/e decay length
of the induced longitudinal charge density oscillations (BPs), i.e. δe ∼ 1/= [q˜0],
where q˜0 is defined via the longitudinal dielectric function of the metal under
the required condition ε˜Lm(q˜0,ω) = 0 (Eqn. (2.8)). This effect contributes to the
overall non-local screening response of the nanosystem. It is important to stress
5Homogeneous in the sense that all relevant length scales are substantially larger than the
lattice spacing of the metal, e.g. approximately 0.4 nm for Au.
6Throughout this thesis subscripts ‘m’ and ‘d’ denote metallic and dielectric media respec-
tively.
13
Chapter 2: Theoretical Background
that in the large
∣∣∣k˜E∣∣∣ domain, BPs (and SPs) can be excited not only by particle
impact, but also by light. Finally, it should also be noted that the homogeneity
approximation is perhaps on the edge of validity in this regime. This can be
overcome by the extremely computationally expensive, ab initio, full quantum
mechanical methods described in Sec. (2.3.3).
Similarly, such spatial non-locality also has to be taken into account when
considering individual metal NPs of dimensions less than ≈ 10 nm, substan-
tially less than the mean free path length of the excited electrons around the
Fermi-level7. Under the high symmetry case of a single spherical NP the spatial
non-locality is typically taken into account phenomenologically by increasing
the damping rate associated with the conduction electrons in the material [38].
The increase in damping is attributed to the rate of electron collisions with
the NP surface for the electrons moving close to the Fermi-level, and is there-
fore intuitively understood as a surface scattering phenomena. Recent work
has shown that the physical nature of this effect is more generally described
by considering the role of the large q˜ components created [35]. The large q˜
components describe how the induced-surface-charges (screening charges) now
penetrate inside the NP by a non-negligible amount (≈ 0.5 nm) compared to
the NP dimensions via BP excitation. This is in contrast to the spatially local
description where the induced-surface-charges follow a surface-delta-function.
The excitation of dispersive BPs into the NP’s interior, and the associated pres-
sure variations (due to Pauli-exclusion effect) in the electron gas, describe a
component of the spatially non-local response of the dielectric function.
In the Drude model the electron-electron interaction and lattice potential
specifics are not taken into account. Instead the assumption is made that some
aspects of the metal’s band structure are incorporated into the effective optical
mass m∗e of each electron. It is also assumed that the perturbing field E is
a self-consistent field that includes the mean value of the field produced by
the electrons themselves (mean-field approximation). It should be noted that
the Drude model was extended by Sommerfeld in 1933 to include quantum
mechanical Fermi-Dirac statistics, thus forming the well known free-electron
(Drude-Sommerfeld) model. This model showed that only the electrons near the
7In the single NP case, spatial non-locality is often referred to as the ‘finite-size effect’, ‘sur-
face scattering effect’ or ‘interface damping effect’. This should not be confused with quantum
confinement effects, and additional electron-density ‘spill-out’ mediated non-local screening,
that are only non-negligible in NPs of less than approximately 4 nm diameter [36, 37].
14
Chapter 2: Theoretical Background
Fermi-level contribute as the Pauli-exclusion principle does not allow electrons
occupying lower levels to change their electronic state. However the following
results are the same regardless of which model is used.
Under the long-wavelength (spatially local response) approximation the wa-
velength of all exciting light components is much greater than the metal’s unit
cell dimensions. This simplification ensures that the forces due to the varia-
tions in the pressure of the electron gas are negligible. Thus spatially non-local
quantum mechanical models need not be used and the dielectric function of
the metal ε˜Tm(q˜,ω) = ε˜Lm(q˜,ω) = ε˜m(ω). Following from these assumptions,
the equation of motion for a conduction electron in an isotropic free-electron
plasma can be written as
m∗e r¨e + m∗eγ r˙e = −eE(re, t) (2.16)
where γ is the phenomenological electron collision frequency and e is the charge
of an electron. Moreover, assuming a macroscopic spatially uniform perturbing
field E(re, t) → E(t), with a harmonic time dependence, and a linear response,
gives
E(t) = E0e−iωt (2.17)
r˜e(t) = r˜0e−iωt (2.18)
where r˜0 is the complex amplitude of the electron oscillation. Substituting (2.17)
and (2.18) into (2.16) and solving for r˜0 allows the macroscopic polarisation
of the metal to be determined. This leads to an expression that describes the
frequency dependence of ε˜m(ω) via D= ε0E + P
ε˜m(ω) = 1−
ω2p
ω2 + iγω
(2.19)
where ωp is the bulk plasma frequency defined as ω2p =
nee2
ε0m∗e with ne being the
conduction electron density of the metal. The real and imaginary components
of ε˜m(ω) are given by
15
Chapter 2: Theoretical Background
< [ε˜m(ω)] = ε1(ω) = 1−
ω2pτ
2
1+ω2τ2
(2.20)
= [ε˜m(ω)] = ε2(ω) =
ω2pτ
ω(1+ω2τ2)
(2.21)
where τ = 1/γ is the phenomenological electron relaxation time. Using the
same arguments as above an expression for σ˜m(ω) can also be obtained via
Jint = ∂P∂t = σ˜mE
σ˜m(ω) =
ω2pε0τ
1− iωτ =
σ0
1− iωτ (2.22)
where σ0 = ω2pε0τ is the zero-frequency conductivity. Note that by comparing
Eqns. (2.19) and (2.22) the general result of Eqn. (2.9) is recovered as expected.
A first extension to this model can be made by including the contribution of
the bound electrons to the polarisability of the metal. For example, in the noble
metals, the filled d-band close to the Fermi surface creates a highly polarised
environment. This so called ‘quasi-free’ electron Drude model yields a dielectric
function of the form
ε˜m(ω) = εbk −
ω2p
ω2 + iγω
(2.23)
where εbk is the background contribution to the dielectric function found as
ω → ∞ [14]. The form of the real and imaginary components of ε˜m(ω) are
shown in Fig. (2.1).
In the small damping limit ωpτ  1, and thus Eqns. (2.20) and (2.21) show
ε˜m
(
ωp
) ≈ 0. For a longitudinal oscillation q˜ · Ê = q˜Ê, therefore if there are no
external sources of current present Eqn. (2.6) demands ε˜m (ω) = 0. These results
imply that at frequency ωp transverse electromagnetic modes cannot be sup-
ported in the metal, hence ωp defines the natural frequency of a collective purely
longitudinal oscillation in the metal’s free-electron plasma. These plasma waves
cannot therefore couple to purely transverse electromagnetic waves8 and can
generally only be excited by direct particle impact, for example from external in-
8In the q˜ = 0 (long-wavelength or spatially local) limit Gauss’s law shows that a purely
transverse electromagnetic wave in a bulk medium cannot create charge density fluctuations
(longitudinal oscillations) in a free-electron plasma.
16
Chapter 2: Theoretical Background
cident electrons. In a quantum mechanical treatment it is shown that the quanta
of collective longitudinal charge oscillations are quasi-particles known as BPs.
Spatial non-locality also introduces modifications to the BP dispersion due to
an entire series of collective longitudinal oscillations at higher frequencies with
finite q˜ dependence. To 0th order (only considering a spatially local response)
BPs oscillate with associated discrete plasma frequency ωp. As discussed previ-
ously, this approximation is usually considered to be valid as higher-order terms
in the series only become non-negligible for incident wavenumber k ' 109 m−1,
i.e. far outside the typical free-space and evanescent optical regimes.
12
10
8
6
4
2
0
54321
Energy (eV)
2000 900 600 400 300 250
Wavelength (nm)
 Au Experiment
 Au Drude fit
 Au quasi-Drude fit
 Ag Experiment
 Ag Drude fit
 Ag quasi-Drude fit
-60
-40
-20
0
54321
Energy (eV)
2000 900 600 400 300 250
Wavelength (nm)
 Au Experiment
 Au Drude fit
 Au quasi-Drude fit
 Ag Experiment
 Ag Drude fit
 Ag quasi-Drude fit
(a)
(b)
Figure 2.1: (a) Graph comparing the experimental values of < [ε˜m (ω)] for gold
and silver (see, e.g. Palik Handbook [39]) against Drude model fits, and quasi-
Drude model fits to the region below 2 eV. (b) Same as in (a) but for = [ε˜m (ω)].
Physical model fitting parameters for gold are ωp = 1.34× 1016 rad s−1, τ =
8.5× 10−15 s, εbk = 9.5 and for silver ωp = 1.29× 1016 rad s−1, τ = 9.5× 10−15
s, εbk = 2.7 [40]. Note the poor agreement at visible and shorter wavelengths
due to interband transitions.
17
Chapter 2: Theoretical Background
The Drude model is reasonable at modelling certain metals, e.g. the alkali
and noble groups, especially at low incident photon energies (/ 1.5 eV). The
alkali metals have closed shells of valence electrons that are effectively inert and
conduction bands that are at considerably higher energies. Therefore the ap-
proximately spherical Fermi surface is far from the edges of the first Brillouin
zone and hence the lattice (periodic potential) only very weakly perturbs (scat-
ters) the electrons. For the noble metals the situation is complicated because
the contribution from the electrons occupying the valence d-bands cannot be
neglected. However the effect is found to be moderate and hence the alkali and
noble metals are often referred to as ‘nearly-free-electron’ metals where absorp-
tion of light is assumed to be proceed primarily by intraband transitions. It is
interesting to note, at lower energies, the agreement between the experimen-
tally measured < [ε˜m (ω)] and the Drude model fit is significantly worse than
that between the associated = [ε˜m (ω)] and Drude model fit. This is because the
resonant interband transitions affect < [ε˜m (ω)] over a much broader frequency
range due to the Lorentzian lineshape of the resonant transitions.
As shown in Fig. (2.1), the quasi-Drude Model predicts the optical response
of gold and silver with good experimental agreement at low energies9. However,
at around approximately 1.5 eV optically induced interband transitions from the
occupied states at the top of the d-bands to the empty states around the Fermi-
level begin to occur. Hence, although the quasi-Drude model is effective for
describing the more delocalised (‘nearly-free’) sp-band electron contributions, it
does not superimpose the resonant behaviour of the induced interband tran-
sitions. These resonant phenomena can be modelled under the Drude-Lorentz
formalism where interband transitions are described classically by including ex-
tra oscillators of type I, with mass m∗I , density of charge nI , collision frequency
γI and resonance frequency ω0I , to Eqn. (2.16). This leads to a theoretical ap-
proach that involves solving systems of Lorentz oscillators [41] and yields a
dielectric function of the form
ε˜m(ω) = εbk +
ω2p
ω2 + iγω
+∑
I
nIe2
ε0m∗I (ω
2
0I −ω2 − iωγI)
(2.24)
9Here εbk takes into account not only the background polarisation, but also some of the effects
of interband transitions. This approach is therefore questionable. Ideally the full approach
described by Eqn. (2.24) should be appropriately fitted to the experimentally measured values
of ε˜m(ω).
18
Chapter 2: Theoretical Background
The additional terms are used to represent interband transitions, but can also
be used to describe ‘bound’ electrons (dielectric insulators), impurity atoms and
phonons in the metal. This model can therefore also be used to phenomenolog-
ically describe dielectric functions for insulators and semi-conductors. Despite
these theoretical advances, accurately modelling such complex systems is still
an immense challenge and hence all the subsequent work in this thesis uses di-
electric values found from experiment. For the metals investigated in this work
the values for the real and imaginary parts of ε˜m (ω) are compiled from appro-
priate sources, e.g. the Palik Handbook of Optical Constants of Solids II [39], and
are plotted for Au and Ag in Fig. (2.1). It therefore should be noted that even
though spatial non-locality is not characterised by the experimentally found
ε˜m (ω), the dielectric data used in all theoretical calculations throughout this
work still inherently includes some quantum mechanical (transition) effects.
2.3 Plasmonic Excitations
The most basic systems that can sustain both SPs and SPPs are those that can
be considered as a semi-infinite flat isotropic single interface. Modern exper-
iments and applications have now moved beyond investigation of this simple
system, however this fundamental example still provides excellent insight into
what these phenomena actually are, their properties, and requirements for exci-
tation. A theoretical treatment for a simple SP and SPP system will be described
first, followed by brief overviews of LSPP excitations on NPs and coupled LSPP
interactions between NP pairs.
2.3.1 Surface Plasmons and Surface Plasmon Polaritons
A BP is defined in a semi-infinite three-dimensional (3D) isotropic metal. At an
effectively two-dimensional (2D) metal-dielectric interface a natural collective
longitudinal oscillation of the metal’s surface electrons can be excited, known
as a SP. Due to their purely longitudinal nature, in the spatially local regime,
SPs (like BPs) are non-propagating (vg = 0) excitations and can generally only be
excited by particle impact. It is important to note that BPs and SPs are associated
with electrostatic rather than electromagnetic fields. This can be seen directly by
considering the irrotational nature of a purely longitudinal oscillation
19
Chapter 2: Theoretical Background
iq˜× Ê(ω) = 0 ⇒ ∇× E(t) = 0 (2.25)
Recalling that ∇×∇Φ = 0 it immediately follows that the electric field of a
purely longitudinal oscillation can be expressed as the gradient of an electro-
static potential, E = −∇Φ. Combining this result with Gauss’s law (Eqn. (2.1))
yields Poisson’s equation. Furthermore, assuming the medium to be charge
neutral (ρint = 0) and ρext = 0, the problem is reduced to solving Laplace’s
equation to determine the associated SP resonances. Solving for the SP reso-
nances in the electrostatic limit is often referred to as finding the non-retarded
SP behaviour of the system as the interactions between different parts of the
metal surface is considered to be instantaneous (c = ∞).
The coupling of an incident electromagnetic wave to the interface can be the-
oretically treated by modelling the resulting SPP as an interface bound elec-
tromagnetic wave10. Assuming that the incident electromagnetic wave has no
Ey-component the simplest system geometry can be modelled as shown in Fig.
(2.2). Using Maxwell Eqn. (2.4), under the assumptions of harmonic time de-
pendence and no external sources of charge and current density, one can find
explicit expressions relating the different field components of the Transverse
Magnetic (TM) or p-polarised modes (Ex, Ez and Hy are non-zero) for the sys-
tem defined in Fig. (2.2)
Ex = −i 1
ωε0ε˜r
∂Hy
∂z
(2.26)
Ez = i
1
ωε0ε˜r
∂Hy
∂x
(2.27)
Transverse Electric (TE) or s-polarised modes will not be considered in this
case as they have no longitudinal (Ex) and normal (Ez) E-field components. In
this system purely TE waves cannot be bound to the interface11 and cannot ex-
cite the required longitudinal oscillations in the surface free-electron plasma.
Therefore SPPs are only associated with light that has both normal and longitu-
dinal E-field components with respect to the interface.
10In almost all cases the SPPs excited can be thought of as proxies for the incident light and
thus carry the same information.
11For the system considered boundary conditions demand that the amplitude of the E and H
components of a TE-polarised wave equal zero at the interface, thus confirming this statement.
20
Chapter 2: Theoretical Background
Figure 2.2: (a) Schematic of the metal-dielectric interface showing the surface
bound electromagnetic wave, coupled to longitudinal oscillations of the surface
charge, propagating in the x-direction. (b) Magnitude of the electric field decays
away exponentially on both sides of the interface thus reflecting the excitations
bound nature. The Ez-field penetration depths into the respective media are δm
and δd.
Under the additional assumptions that both the metallic and dielectric medi-
ums are non-magnetic and macroscopically homogeneous (negligible variation
of dielectric function over distances on the order of a wavelength), Eqn. (2.5) re-
duces to the Helmholtz wave equation. Considering transverse electromagnetic
waves propagating along the interface with harmonic time dependence hence
allows the governing wave equation for TM modes to be determined
∂2Hy
∂z2
+
(
k20ε˜r − k˜2spp
)
Hy = 0 (2.28)
where k˜spp is the x-component of wavevector. Seeking a propagating wave so-
21
Chapter 2: Theoretical Background
lution bound to the surface, i.e. with evanescent decay in the perpendicular
z-direction, on each side of the metal-dielectric interface, requires
For z > 0: Hy = Ad e i(k˜sppx+k˜zdz) (2.29)
For z < 0: Hy = Am e i(k˜sppx−k˜zmz) (2.30)
with =[k˜zm] ∧ =[k˜zd] > 0. Here Am and Ad are the field amplitude coefficients
in the metal and dielectric respectively and k˜zm and k˜zd are the z-components
of the wavevector associated with the SPP excitation in the metal and dielectric
respectively. Substituting these solutions into Eqns. (2.26) and (2.27) and solving
the appropriate boundary conditions to ensure the continuity of Hy and Ex at
the interface gives the results
Am = Ad and
k˜zd
k˜zm
= − εd
ε˜m
(2.31)
where εd is the dielectric function of the dielectric medium. Note as the com-
ponents of k˜zm ∧ k˜zm ∧ εd > 0 for a surface bound electromagnetic wave, it is
apparent <[ε˜m] < 0. As described in Section (2.2.2) the Drude model shows
that in the non-transparent regime (ω < ωp) negative permittivity is a result
of collective oscillations in the free-electron plasma. In other words, if an elec-
tromagnetic wave (see Fig. (2.2)) is confined and propagating along the inter-
face it must be coupled to collective longitudinal oscillations in the free-electron
plasma, otherwise the electromagnetic field boundary conditions and the Drude
model would not be simultaneously satisfied.
Finally, implementing conservation of momentum on each side of the inter-
face, i.e. satisfying Eqn. (2.28), yields the complex SPP dispersion relation under
the spatially local response approximation
k˜spp = k0
(
ε˜mεd
ε˜m + εd
)1/2
(2.32)
Several properties of SPPs can be obtained from this equation by decomposing
it into its real and imaginary components and considering momentum conser-
vation on each side of the interface. From this analysis the SPP propagation
22
Chapter 2: Theoretical Background
length Lspp and penetration depths δm and δd into the metallic and dielectric
mediums respectively can all be determined. In the negligible damping limit
< [εm(ω)]  = [εm(ω)] and maintaining that =[k˜zm] ∧ =[k˜zd] > 0 it is found
that
Lspp ≈ 1k0
( < [ε˜m] εd
< [ε˜m] + εd
)−3/2 (< [ε˜m])2
= [ε˜m] (2.33)
k˜zm ≈ k0
(
(< [ε˜m])2
< [ε˜m] + εd
)1/2
with δm =
1∣∣∣= [k˜zm]∣∣∣ (2.34)
k˜zd ≈ k0
(
(εd)
2
< [ε˜m] + εd
)1/2
with δd =
1∣∣∣= [k˜zd]∣∣∣ (2.35)
As Eqns. (2.31), (2.34) and (2.35) show it is required that < [ε˜m] εd ∧ (< [ε˜m] +
εd) < 0 to yield a surface bound solution. Therefore, in general, SPPs only exist
at interfaces where < [ε˜m] is negative and greater in magnitude than εd. The pro-
cess of SPP attenuation in this system is described by the decay of electron-hole
pairs, created by SPP induced interband and intraband transitions of electrons.
The attenuation therefore proceeds via free-electron damping processes, e.g.
scattering with phonons and lattice ions [42].
Due to their longitudinal (electrostatic) nature BP and SP decay can gener-
ally be described by Landau damping. The electrons associated with BPs and
SPs have phase velocities almost equal to that of the longitudinal plasma wave
(vp = ωp/k0n). As a consequence these electrons experience an essentially static
electric field which can be either positive or negative depending on the relative
phase between the electron motion and the plasma wave. After averaging over
all possible phases, an individual electron therefore has an equal chance of being
accelerated or decelerated by the wave. However, due to the Maxwellian distri-
bution of electron velocities, there are more electrons with velocities slightly less
than that of the plasma wave. Therefore, on average, more electrons are being
accelerated than decelerated by the wave. This results in a net transfer of energy
from the plasma wave to individual electrons in the metal, i.e. the plasma wave
is damped [14].
23
Chapter 2: Theoretical Background
Ideal Surface Plasmon Polaritons
The behaviour of k˜spp at the interface between a Drude metal with negligible
damping (and local response) and air is shown in Fig. (2.3). A SP has only a
discrete frequency of resonant excitation (assuming no linewidth of the reso-
nance) in this system. However it is evident that an incident electromagnetic
wave coupling to the surface and exciting longitudinal oscillations in the 2D
free-electron plasma allows resonant excitation of SPPs at a continuum of fre-
quencies. A SPP is a propagating surface bound electromagnetic wave coupled
to longitudinal oscillations in the metal’s surface charge density12. The full elec-
trodynamic solution for the SPP dispersion presented here includes the effects
from retardation, i.e. the finite speed of light is taken into account. This is
therefore often referred to as a solution derived in the retarded limit.
100x106806040200
In-plane wavevector (m-1)
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0
Fr
eq
ue
nc
y 
(ω
/ω
p) 
Sommerfeld-Zenneck surface wave
SPP
RPP Light line
SP
 
 
 
 
RPP
Figure 2.3: Graph showing the behaviour of k˜spp for a Drude metal (with negli-
gible damping) compared with that of the light line k0. The dashed horizontal
lines show the positions of the BP and SP resonances. Note that the in-plane
wavevector component k|| = <[k˜spp] = k0nd sin θi where nd is the refractive in-
dex of the dielectric medium and θi is the angle of an incident electromagnetic
wave.
12The electric field in the dielectric medium that is associated with a SPP is mostly transverse
in nature, i.e. Ez  Ex and is purely transverse (electromagnetic wave in a homogeneous
medium) as <[k˜spp]→ 0. As <[k˜spp]→ ∞ (electrostatic limit) it can be shown that Ez = Ex.
24
Chapter 2: Theoretical Background
The frequency of collective longitudinal oscillation induced in the free-
electron plasma depends on <[k˜spp] with the asymptotic value being the nat-
ural resonance frequency of a SP. The electric field associated with a BP and
SP is electrostatic and hence the characteristic SP resonance frequency of the
metal can be obtained via a straight-forward solution of Laplace’s equation
for the single interface geometry shown in Fig. (2.2). This yields the rela-
tion between ωp and the surface plasmon frequency ωsp which is found to be
ωsp = ωp/
(√
1+ εd
)
, and thus ωsp = ωp/
√
2 in air. For ω < ωsp the SP be-
comes polaritonic in nature (a SPP) and the dispersion strongly deviates from
the dashed horizontal line (Fig. (2.3)) predicted by the electrostatic treatment.
In real metals, damping processes, interband transitions and spatially non-local
effects contribute to corrections to this result and re-definitions of ωp and ωsp
are required [43, 44].
At low-frequencies the light barely interacts with the surface electrons and
hence is not usually referred to as a SPP but rather a grazing incidence Sommer-
feld-Zenneck wave. Between ωsp and ωp there is only a purely imaginary k˜spp
and hence no SPP propagation. In the transparent regime above ωp, electrons
in the metal can no longer respond rapidly enough to screen the incoming elec-
tromagnetic wave. Therefore radiation into the metal occurs and the (bulk)
Radiative Plasmon Polariton (RPP) dispersion is observed. The form of the RPP
dispersion is given by substituting ε˜m(ω) given by Eqn. (2.19) into the transverse
dispersion relation Eqn. (2.7). For negligible damping this yields
q˜ =
(
ω2 −ω2p
c2
)1/2
⇒ ω
ωp
=
(
1+
c2q˜2
ω2p
)1/2
(2.36)
The transition into the transparent regime is clearly shown by Eqn. (2.36) as
when ω > ωp the complex wavenumber q˜ of the transverse excitation has
= [q˜] = 0. In the spatially non-local regime longitudinal excitations in the bulk
and on the surface, i.e. BPs and SPs, feel the forces due to pressure variations
in the electron gas and hence show dispersive character described by the longi-
tudinal dielectric function under the condition ε˜Lm(q˜0,ω) = 0 [32]. This allows
BPs and SPs to be excited at a continuum of frequencies (at <[k˜spp] ' 109 m−1)
rather than the discrete frequencies labelled in Fig. (2.3). These excitations thus
become propagating (vg 6= 0), however they maintain their electrostatic nature.
25
Chapter 2: Theoretical Background
As can be seen from Fig. (2.3) <[k˜spp] is always to the right of the k0 line
because as the coupling between the light and electrons on the metallic surface
increases the light has to ‘drag’ the electrons along the surface to a greater ex-
tent. Therefore free-space electromagnetic waves cannot directly couple to the
flat semi-infinite interface and some intermediate step is required to overcome
the momentum mismatch. Several techniques [45] to do this are; prism cou-
pling, direct emission into SPP via light-emitting diode, scattering from a defect
on a surface and scattering from a periodically corrugated metal surface, tra-
ditionally a diffraction grating. All of these techniques can be used to impart
momentum to the light (increase of the in-plane wavevector component) so that
it can couple (or de-couple) to SPP modes. It can now be understood how some
of the aforementioned Wood’s anomalies, that could not be fully explained by
Rayleigh, arose in Wood’s experiments. The diffraction grating used in the ex-
periment imparted additional momentum to the incident light thus allowing
resonant coupling into SPPs with dispersion similar to that shown in Fig. (2.3).
Therefore resonant SPP excitation produced relatively high absorption at cer-
tain frequencies. These SPP resonances manifested themselves as the drops in
reflectivity that Woods measured at specific wavelengths.
Real Surface Plasmon Polaritons
When considering the experimentally obtained values of ε˜m(ω) the SPP disper-
sion is modified dramatically as shown in Fig. (2.4). It is apparent that SPPs
in real metals approach a maximum finite wavenumber at ωsp. This is because
when damping (real metal) is introduced, purely longitudinal BPs and SPs can
no longer exist. For example in the case of a BP the condition for longitudinal
oscillation, ε˜m(ωp) = 0, can no longer be satisfied. For real metals one can mod-
ify the definitions of ωp and ωsp accordingly. This limitation puts a lower bound
on both the SPP wavelength and the amount of SPP confinement perpendicular
to the dielectric side of the interface, e.g. for gold λspp = 2pi/<[k˜spp] ≈ 470 nm,
δm ≈ 37 nm (skin-depth) and δd ≈ 170 nm at ωsp. The expected current density
J˜int induced by the electric field propagates parallel to the surface and is sig-
nificant up to a depth ∼ O(δm). Therefore for NPs with dimensions ∼ O(2δm)
induced current density is expected to be significant throughout the entire NP
volume.
26
Chapter 2: Theoretical Background
30x10625201510
In-plane wavevector (m-1)
6.0x1015
5.5
5.0
4.5
4.0
3.5
3.0
2.5
Fr
eq
ue
nc
y 
ω
 
(ra
d s
-
1 )
 Au:
 Ag:
.
.
.
.
.
.
.
Fr
eq
ue
nc
y 
ω
 
(ra
d s
-
1 )
Figure 2.4: Graph showing the behaviour of k˜spp for gold and silver compared
with the light line k0. The horizontal dashed lines show the positions of the SP
resonances. The shaded regions (horizontal lines with width equal to |=[k˜spp]|)
describe the FWHM of the SPP resonance in k-space.
The quasi-bound part of the SPP dispersion relation between ωsp and ωp is
also allowed when considering real metals. In this regime <[k˜zm] and <[k˜zd]
are non-negligible and hence the excitation begins to propagate away from the
surface. A definition of ωp can be introduced by considering when =[k˜spp]
crosses the light-line as now SPPs take on a more radiative nature, as shown
in Fig. (2.3). However in the frequency range ω > ωp field penetration into
the metal (δm and δd) does not increase significantly as interband absorption
processes dominate the metals response.
2.3.2 Localised Surface Plasmon Polaritons
In this thesis nearly-touching and touching NP functionalised AFM tips are the
plasmonic structures of interest and hence an understanding of the LSPP modes
that can be excited on these nanostructures is essential. The LSPP mediated re-
sponse of these nanosystems can be understood by separating the response into
two regimes. The LSPP response of an isolated NP and the LSPP response of a
strongly electromagnetically coupled identical NP-NP homo-dimer, referred to
as a NP dimer for brevity. A brief overview of LSPPs on isolated spherical NPs
is first presented.
27
Chapter 2: Theoretical Background
After initial investigations by Faraday [46] in the 1800’s, and Mie’s semi-
nal work in 1908, it was not until the late 1970’s that metallic NPs were once
again heavily studied. The realisation of the role of LSPPs on metallic NPs in
SERS drove the re-interpretation of absorption and scattering by NPs under the
plasmonics framework [31, 47]. Over the past 30 years the quest to create ever
more complex plasmonic nanosystems for surface-enhanced spectroscopies [48],
single molecule detection [49], metamaterials [4] and optical circuitry and com-
ponents [50] has seen the field expand exponentially. Now new synthesis tech-
niques have produced NP morphologies such as spheres [51], cubes [52], cups
[53], rings [54], rods [55] and recently even ‘nanodecahedra’ [56] and ‘nanos-
tars’ [57]. Another interesting class of NP type are ‘nanoshells’ [58]. A nanoshell
consists of a dielectric core surrounded by a thin (thickness < 30 nm) metallic
shell that hence creates a dielectric-metal-dielectric interface. Nanoshells there-
fore exhibit a substantially different plasmonic response to that created by the
single metal-dielectric interface of a homogeneous NP. For simplicity only ho-
mogeneous isotropic spherical NPs are addressed here in further detail, how-
ever the ideas presented here are applicable to the understanding of almost all
plasmonic nanosystems.
As depicted in Fig. (2.5), unlike the semi-infinite planar interface system
discussed in section (2.3.1) the curved surfaces of NPs provide confinement of
all participating electrons to a nanoscale volume via the boundary conditions
on E, D, B and H at the position r = a where a is the radius of the NP.
Figure 2.5: Schematic representation showing the ` = 1 dipole distribution
of participating electrons in a spherical NP (outline not shown) under infinite
plane-wave excitation. The depolarisation field ( yellow ) creates an effective
restoring force and thus resonant electron oscillations can occur in analogy to
a driven damped harmonic oscillator.
28
Chapter 2: Theoretical Background
The electrons all move in response to an incident electromagnetic plane wave
and the electric-field oscillations thus create a time-varying build-up of polar-
isation charges on the surface of the NP. The depolarisation field created acts
as an effective restoring force hence allowing resonant electron oscillations at
specific excitation frequencies. More fundamentally, the boundary conditions
constrain the electron plasma, and naturally, resonance conditions result.
In Fig. (2.6a) the phase of the electromagnetic wave is approximately con-
stant over the volume of the NP as a  λ0/nd = λ, where λ0 and λ are the
wavelength of the incident light in free-space and a dielectric medium respec-
tively.
Figure 2.6: Schematic representations of LSPP excitation in the (a) quasi-static
(ideally a λ) and (b) Mie (a ∼ λ) regimes.
29
Chapter 2: Theoretical Background
The resulting Localised Surface Plasmon (LSP) resonance frequencies can be
calculated via Laplace’s equation under the electrostatic field approximation,
i.e. in the non-retarded limit. Extending this to the quasi-static approximation
yields the associated absorption and scattering cross-sections by allowing for a
time-varying field but neglecting the non-uniform phase (retardation) over the
NP volume.
These simplifications cannot be made for the situation shown in Fig. (2.6b)
where a ∼ λ. The phase of incident electromagnetic field is not constant over
the NP’s volume and the LSPP response has to be obtained including retarda-
tion of the fields via a full electrodynamic solution to Maxwell’s equations. In
broad terms LSPPs can be thought of as propagating SPPs confined to a NPs
geometry that hence interfere to form a standing wave (localised) SPP at spe-
cific frequencies. A rigorous description of LSPPs on spherical geometries was
achieved by Mie in 1908 [19]. Mie showed that the different electromagnetic
eigenmodes of spherical particles are dipolar or multipolar in character and
that their excitation magnitude can be determined by expanding the internal
and scattered fields into a set of normal modes described by vector spherical
harmonic functions. The Mie solutions can therefore model the LSPP response
of any diameter particle (metallic or dielectric) in the classical regime. However,
if a  λ the laws of geometric optics may be more appropriate to describe a
NP’s response to light.
Strictly speaking a LSP cannot be excited by propagating light illumination
because like a SP a truly LSP is a purely longitudinal excitation and must be
associated with an electrostatic field. A LSP can be thought of as the asymp-
totic limit of an LSPP as c → ∞ (non-retarded limit). Equivalently, as SPPs
propagate into more confined regions of a nanosystem they can be adiabatically
transformed toward the LSP limit, i.e. the electric field associated with the SPP
becomes increasingly electrostatic in character (B → 0) [59]. In this thesis SPPs
localised on a nanosystem with critical dimensions much less than Lspp are re-
ferred to as LSPPs. A more detailed discussion on the nature of BPs, SPs, LSPs
and the polaritonic character of SPPs can be found in [44,60] and the references
therein.
30
Chapter 2: Theoretical Background
Quasi-static solutions
A good intuition into the LSPP response of NPs can be gained from the quasi-
static approximations of the multipolar LSPs driven by a general electrostatic
field of infinite extent. This is achieved by expressing Laplace’s equation in a
spherical polar basis and looking for solutions of the form [14]
Φ`,m(r, θ, φ) =
∞
∑
`=0
`
∑
m=−`
[
Am` r
` + Bm` r
−(`+1)
]
Pm` (cos θ)e
imφ (2.37)
where ` = 1, 2, 3... is the orbital angular momentum number, e.g. ` = 1 for a
dipole, m = −`...0...` (in integer steps) is the azimuthal number, e.g. m = 0 for
azimuthally symmetric fields, and Pm` (cos θ) are the associated Legendre Poly-
nomials of order (`, m). The coefficients Am` and B
m
` are determined both inside
and outside the sphere by ensuring continuity of the tangential E-field and nor-
mal D-field components at r = a and enforcing physical boundary conditions
as r → (0,∞). As expected the electrostatic field and charge distributions deter-
mined closely resemble atomic orbitals.
Using Eqn. (2.37) the resonance frequencies ω`np and polarisabilities α` of a
spherical NP under the quasi-static approximation are13 [61]
ω`np = ωp
(
`
εd(`+ 1) + `
)1/2
(2.38a)
α` = 4piεda(2`+1)
`(ε˜m − εd)
` ε˜m + (`+ 1)εd
(2.38b)
The m dependence vanishes due to the symmetry implied by the spatially infi-
nite electrostatic excitation of the spherical NP. As ` increases the polarisability
decreases and hence higher-order modes are generally not evident for small NPs
with a / 40 nm. Typically the quasi-static formalism is only used in the regime
where a / λ/10. In this case, assuming an infinite uniform driving electrostatic
field (` = 1 and m = 0), only the dipole mode can be excited. For non-spherical
NPs with sharp asperities higher-order modes can still be excited due to the
non-uniform electrostatic field created. When ` → ∞ the SP resonance condi-
13It is stressed here that the ` in ω`np and α` is used as an annotation while in the RHS of
Eqns. (2.38) all instances of ` denote a numerical value.
31
Chapter 2: Theoretical Background
tion is recovered as expected because the surface electron oscillations become of
short enough wavelength that they effectively feel a quasi-planar interface. The
resonant nature of α` implies that the surface charge density must be resonant
in character and hence on resonance a strongly enhanced electric near-field will
be present in certain regions just above the NP’s surface.
There are generally a large number of different LSP and LSPP modes that can
be excited on nanosystems due to their complex curved surfaces. The curved
geometry also allows LSPP resonances to be excited by direct light illumination,
i.e. none of the momentum matching techniques mentioned in Section (2.3.1)
are required. This can be explained by considering that the wavenumber k of
the exciting light can be enlarged by wavevector components determined by
the spatial Fourier transform of the NP geometry [62]. These relatively large
magnitude wavevector components are contained within the evanescent near-
field in the immediate vicinity of the NP. The LSPP ‘dispersion’ 14 relation can be
determined by comparing the LSPP resonance frequency against the tangential
wavenumber <[k˜lspp] given by
<[k˜lspp] = 2piλlspp =
2pi
(2pia/`)
=
`
a
(2.39)
where λlspp is the wavelength of the LSPP given by the circumference of the
NP divided by the orbital angular momentum number `. Note that there are
2` density nodes in the longitudinal electron standing wave around any given
circumference. In general the LSPP dispersion lies much closer to the light-
line than the dispersion of a well confined SPP (see Fig. (2.3)) [63]. Thus the
additional momenta required to excite LSPPs on closed curved geometries is
reduced considerably from that required to excite SPPs on semi-infinite pla-
nar geometries. Another consequence of the curved nanosystems is that the
boundary conditions associated with the curved interfaces allow both TM and
TE modes to couple to LSPP excitations.
The far-field scattering and absorption cross-sections Cqssca and C
qs
abs respec-
tively, are calculated under the quasi-static approximation by allowing the non-
retarded solutions for the multipole moments determined from Eqn. (2.38b) to
14Non-interacting LSPPs are non-dispersive excitations. Here ‘dispersion’ refers to a plot
showing the evolution of the discrete LSPP resonance energies against the associated discrete
wavevector magnitudes.
32
Chapter 2: Theoretical Background
oscillate in time. The radiation of the oscillating multipole creates scattering of
the incident field. For small spherical NPs a λ hence the dipole mode (` = 1)
dominates the response. In this regime Cqssca and C
qs
abs are given by [64]
Cqssca =
3
2pi
V2np
(ω
c
)4
ε2d
∣∣∣∣ ε˜m − εdε˜m + 2εd
∣∣∣∣2 (2.40a)
Cqsabs = 3Vnp
(ω
c
)
ε
1
2
d=
[
ε˜m − εd
ε˜m + 2εd
]
(2.40b)
where Vnp = (4/3)pia3 is the volume of the NP. The extinction cross-section
Cqsext is given by
Cqsext = C
qs
sca + C
qs
abs (2.41)
It is important to note that for NPs with a / λ/5, Cqsext is approximately pro-
portional to Vnp and hence the coupling strength of light to an isolated nanosys-
tem, or for example NPs in suspension, will be greatly reduced compared to
that for semi-infinite structures. The scattering response in the ` = 1 dipole
limit is sometimes referred to as Rayleigh scattering, as in 1871 Lord Rayleigh
first treated the scattering of light from molecules in the sky under these ap-
proximations. The results demonstrated by Eqns. (2.38) and (2.40) imply that
the major factors influencing the LSP and therefore LSPP resonances are:
• The NP size (and shape, as shown by a more comprehensive treatment of
ellipsoidal NPs [65]).
• The dielectric function of the metal.
• The dielectric function of the dielectric environment.
Although inferred, it should be noted that two other critical factors influ-
encing the LSP and LSPP resonance response are the NP shape and type. De-
viations from spherical symmetry necessitate an asymmetric polarisability and
hence α` becomes a tensor rather than a scalar quantity. In general the LSP and
LSPP resonance wavelengths for light polarised parallel to the long-axis of a
nanorod tend to red-shift with increasing nanorod aspect ratio. The type of NP,
e.g. whether the NP is a homogeneous sphere or a non-homogeneous spherical
33
Chapter 2: Theoretical Background
core-shell structure, can also strongly affect the plasmonic resonance properties,
depending on the exact properties of the inhomogeneity [66]. The influence of
NP shape and type will be discussed in greater detail in Chapter (5).
For small or slowly varying = [ε˜m(ω)] around the dipole resonance, the reso-
nance condition of Eqn. (2.38b) simplifies to < [ε˜m(ω)] = −2εd. This is known as
the Frölich resonance condition and the associated mode of induced oscillating
charge density is the dipole LSP with frequency (in air, εd = 1) ω1np = ωp/
√
3
as given by Eqn. (2.38a). As Eqns. (2.40a) and (2.40b) show, it is also important
to note for a / 20 nm absorption processes dominate while for a ' 30 nm scat-
tering begins to dominate the LSP (and LSPP) mediated optical response of the
NP.
In this work all the NPs investigated have a > 30 nm hence they will have
a relatively large plasmonically (resonantly) enhanced scattering cross-section.
Due to this, Dark-Field (DF) measurements, where only the scattered light is
collected, will be used to directly observe the LSPP response of the NP systems.
The DF optical measurement techniques employed are described in detail in
Chapter (3).
Mie solutions
In this work the tip apex NPs are of too large diameter (a ≈ 150 nm ∼ λ/5)
to be accurately modelled using the quasi-static approximation and hence the
exact Mie solutions are more appropriate to yield further insight into LSPPs
supported on such NPs. The Mie solutions are obtained by first re-expressing
Maxwell’s wave equation in polar coordinates via use of Debye potentials. By
considering a single uniform incident infinite plane-wave, boundary conditions
are imposed to ensure continuity of the fields at r = a and hence solutions
for the Debye potentials are found in terms of spherical harmonics [62]. The
electric and magnetic field solutions are therefore reconstructed in terms of vec-
tor spherical harmonics. The far-field response is described by the scattering
coefficients a˜` and b˜` that are given by
a˜` =
η2ψ`(ηα)ψ
′
`(α)− ψ`(α)ψ′`(ηα)
η2ψ`(ηα)ξ
′
`(α)− ξ`(α)ψ′`(ηα)
(2.42a)
34
Chapter 2: Theoretical Background
b˜` =
ψ`(ηα)ψ
′
`(α)− ψ`(α)ψ′`(ηα)
ψ`(ηα)ξ
′
`(α)− ξ`(α)ψ′`(ηα)
(2.42b)
where η = (n˜m/nd), the size parameter α = 2piand/λ0 and the Riccati-Bessel
functions ψ`(z) = zj`(z) and ξ`(z) = zh˜
(1)
` (z) with j` and h˜
(1)
` denoting spherical
Bessel and Hankel functions respectively. Here the prime ′ denotes the deriva-
tive of the function with respect to the argument. There is no m dependence in
Eqns. (2.42) as for uniform infinite single plane-wave excitation the boundary
conditions at r = a require m = 1 for all `. By considering the Poynting vector
of the total electromagnetic field surrounding the NP the far-field scattering and
extinction cross-sections are given by [67]
Csca =
2pi
k2
∞
∑
`=1
(2`+ 1)
(
|a˜`|2 + |b˜`|2
)
(2.43)
Cext =
2pi
k2
∞
∑
`=1
(2`+ 1)<
[
a˜` + b˜`
]
(2.44)
where k = 2pind/λ0 is the wavenumber of the incident light in the surrounding
dielectric medium. The absorption cross-section Cabs can be obtained via Eqn.
(2.41). The Mie solutions show significant modifications to the quasi-static re-
sults are required for NPs with a ' 20 nm. The three most prevalent corrections
are
• For noble metals, an additional overall red-shift of the dipole and higher-
order modes with increasing NP diameter due to retardation of the excit-
ing and depolarisation fields.
• Retardation creates a non-uniform field over the volume of a NP and hence
higher-order (e.g. quadrupole) LSPP modes can be excited by the incident
light even for spherically symmetric NPs. This effect becomes significant
when a ' 100 nm.
• A new decay channel: radiation damping due to direct radiative decay of
LSPPs into photons. For increasing NP diameter the radiative damping
overwhelms the decrease in absorption and hence the LSPP resonances
broaden significantly.
35
Chapter 2: Theoretical Background
Radiation damping increases the homogeneous LSPP mode energy linewidth
Γ`LSPP (as shown in Fig. (2.7)) and thus reduces the LSPP mode lifetime (dephas-
ing time) τ`LSPP according to
τ`LSPP =
2h¯
Γ`LSPP
(2.45)
The strength of the LSPP mode can hence be expressed in terms of a mode qual-
ity factor Q`LSPP via Q
`
LSPP = E
`
LSPP/Γ
`
LSPP, where E
`
LSPP is the resonant energy
associated with the LSPP mode. The Mie solutions thus give the dependence
of τ`LSPP on particle diameter and dielectric surroundings and yields values be-
tween 2− 10 fs for NPs between 150− 20 nm diameter15 in air, for ` = 1. For
NPs of / 10 nm diameter spatially non-local effects, primarily non-local screen-
ing due to finite penetration of induced-surface-charge into the NP interior, act
to broaden the LSPP modes significantly. Therefore the LSPP mode lifetimes
are substantially reduced compared to those calculated assuming spatially local
response only.
Finally, it is important to note that both the traditional quasi-static and Mie
solutions assume a single infinite incident plane-wave. In experiment this con-
dition is never met. For example, under the DF illumination conditions used in
this work, both the incident and scattered light are defined over certain angular
ranges given by the Numerical Aperture (NA) of the various optical compo-
nents. This can modify the measured optical response considerably [68] and
hence must be acknowledged when comparisons are made between experiment
and theory. It has also been reported that the measured optical response can be
significantly altered when LSPPs are excited with non-homogeneous illumina-
tion, e.g. a focussed Gaussian beam [69].
2.3.3 Numerical Methods Overview
Since Mie’s solution for spherical systems in 1908, analytical electromagnetic
solutions have been found for other regular geometries and for inhomogeneous
incident illumination, e.g. Generalised Lorenz-Mie Theory (GLMT). Such so-
lutions can be used to calculate the LSPP resonances of nanosystems such as
15For NPs with a < 50 nm Γ`LSPP must be extracted from Cext as non-radiative decay becomes
comparable, and eventually dominates, radiative decay as NP size decreases.
36
Chapter 2: Theoretical Background
multi-layered spheres, ellipsoids, infinitely long circular cylinders, infinitely
long elliptical cylinders and spheres with an eccentrically located spherical in-
clusion, all under different illumination conditions [70]. At the present time
determination of the polarisability of NPs with more complicated geometries
usually requires numerical methods, such as the Boundary Element (BEM) [71],
Discrete Dipole Approximation (DDA) [72] and Finite Difference Time Domain
(FDTD) methods [73]. However it is important to note that substantial progress
has been made on analytical solutions for more complicated nanosystems, most
notably spherical and cylindrical NP dimers [74, 75]. Recent analytical theoret-
ical treatments have also begun to take into account non-local screening due
to the finite penetration of induced-surface-charge for just-touching NP dimer
systems via a non-local hydrodynamic dielectric function [76].
In the present work, a Boundary Element Method for AXially (BEMAX) sym-
metric nanosystems [77] is used for all classical theoretical calculations. The
simulation is based on a rigorous result derived from vector diffraction theory,
that is, the electromagnetic field inside each homogeneous region of a compos-
ite structure is unambiguously determined by the fields and their derivatives at
the boundary of that region, or equivalently, by the distribution of charges and
currents on that boundary. The method begins by expressing the electromag-
netic field scattered by a nanosystem in terms of equivalent boundary charges
and currents. Imposing the boundary conditions for the continuity of the paral-
lel components of the electric and magnetic fields in the presence of an external
incident electromagnetic wave, leads to a system of surface-integral equations.
This system is solved self-consistently by discretisation of the integrals using
N representative points distributed on the boundaries, thus transforming the
integrals into a set of linear equations that are solved using standard numeri-
cal techniques. Demanding axial symmetry of the nanosystem allows analytical
evaluation of the azimuthal surface-integrals hence only the contours of the
nanosystem need to be parameterised rather than the entire nanosystem sur-
face. This effectively reduces the nanosystem from 2D to 1D and the number
of discrete points typically required to reach convergence from N ∼ O(104) to
N ∼ O(102). The only assumptions made are that the media are described by
spatially local, frequency dependent dielectric functions ε˜r(ω) that terminate
abruptly at the material boundaries.
A comparison of the scattering cross-sections calculated using the quasi-
37
Chapter 2: Theoretical Background
static, Mie and BEMAX formalisms is shown in Fig. (2.7). The BEMAX simu-
lations were run on the University of Cambridge high performance computing
cluster by submitting an appropriate input file. The general procedure for run-
ning a BEMAX simulation is as follows
1. Setup the experimental dielectric function data input for all materials used
in the simulation and set the maximum azimuthal number m used in the
proceeding calculation.
2. Set the type (e.g. single plane-wave or Gaussian beam), direction, polari-
sation and wavelength range of the incident electromagnetic radiation.
3. Define the material and geometry of each nanosystem using input dielec-
tric functions, appropriate lines and arcs, and number of discretisation
points. Check geometry output.
4. Define the scanning parameters, e.g. the grid size and density for near-
field calculations, and angular range of collection for optical cross-section
calculations.
5. Set the calculations to be performed, e.g. scattering cross-section or in-
duced surface charge density.
6. Submit input file containing all the above information. Run the simulation.
7. Run the simulation again to check original output for convergence in num-
ber of discretisation points and m.
8. Transfer the output data to a lab computer and plot using appropriate data
handling software, in this case IGOR Pro (Wavemetrics).
As can be seen in Fig. (2.7), the agreement between the scattering cross-
sections determined via the electromagnetic Mie solutions and the BEMAX sim-
ulation is excellent. The red-shift and broadening of the LSPP resonance due
to retardation effects (Mie and BEMAX solutions) is clearly evident. Here the
BEMAX simulations were used to explore the expected response of spherical
NP dimers over a range of NP separations as well as axially symmetric tip sys-
tems. The simulations also provided valuable insight into LSPP excitations and
the range of LSPP response that could be expected for different nanosystem
geometries.
38
Chapter 2: Theoretical Background
12x103
10
8
6
4
2
0
Sc
at
te
rin
g 
cr
os
s-
se
ct
io
n 
(nm
2 )
1000900800700600500400
Wavelength (nm)
 
 
(a)
-100 0 100
x (nm)
-150
-100
-50
0
50
100
150
z 
(nm
)
-15
-10
-5
0
5
10
15
Induced Field Intensity (dB)
-60x10-3
-40
-20
0
20
40
60
N
or
m
al
ise
d 
In
du
ce
d 
Su
rfa
ce
 E
le
ct
ro
n 
D
en
sit
y
(b)
kE 0
Figure 2.7: (a) Comparison of scattering cross-sections for a 100 nm diameter
Au NP (in air) calculated using the quasi-static (Cqssca), Mie (Csca) and BEMAX
(CBEMAXsca ) formalisms. (b) Electric near-field intensity and representative in-
duced surface charge density distributions for 100 nm Au NP (in air) excited
by an incident infinite plane-wave (see Fig. (2.5)) on the dipole LSPP resonance
(λ0 = 538 nm) calculated using BEMAX.
Initial BEMAX simulations showed that the field enhancement in the vicin-
ity of the NP on each tip apex can arise from resonant excitation of LSPPs on
the coupled tip-apex NP system, and the ‘lightning rod’ effect [78]. Resonant
LSPP excitation on the coupled tip-apex NP system depends on the NP mate-
39
Chapter 2: Theoretical Background
rial, size, shape, type and the coupling of the NP to the bulk of the supporting
tip structure. The lightning rod effect is non-resonant and depends only on the
geometry of the metallic system, in this case mainly on the radius of the tip apex
NPs. The physical origin of the lightning rod effect is metallic screening (due
to electromagnetic boundary conditions) producing a crowding of field lines at
sharp features, as shown in Fig. (2.8). If the metallic tip was infinitely sharp
the electric near-field magnitude would become singular at the tip apex as an
infinite surface-charge density would be required to screen the incident electro-
magnetic fields [67]. Due to the relatively large radius of curvature of the apex
NPs (and quantum mechanics!) there is not any unphysical singular behaviour,
however the approximately equipotential surface of each NP still produces a
large potential gradient in a nanoscale volume. Therefore the lightning rod ef-
fect contributes a non-negligible near-field intensity enhancement concentrated
between the tip apex NPs.
Figure 2.8: Schematic of lightning rod effect in a metallic tip system. The length
of the red line on the tip apex surface is significantly shorter than that in free-
space, thus demonstrating field line crowding (enhanced electric near-field). On
the flat tip edges no field line crowding occurs (orange lines of equal length).
The focus of the present work was primarily to investigate NP dimers
(formed by two tip apex NPs) with separation d  5 nm, hence, as discussed
in Sec. (2.2.2), spatially non-local screening effects at the NanoGap or nano-
Junction (NGJ) created, modify the nanosystem dielectric response significantly.
This therefore limits the validity of the BEMAX simulations. Additionally, at
40
Chapter 2: Theoretical Background
sub-nanometre separations, the non-local quantum effect of spill-out electron
density enables optical frequency tunnelling currents across the nanogap. Both
of these quantum phenomena are expected to strongly affect the LSPP response
of nanosystems with sub-nanometre NGJ’s.
At present new numerical simulations are being developed to take
into account the spatial non-locality of the dielectric function via non-
phenomenological parameter-free calculations [79, 80]. This enables some ef-
fects of spatial non-locality in complex nanosystems to be investigated, e.g. a
NP dimer with d / 5 nm, and hence quantum mechanical effects to be further
taken into account. However to consider the full range of non-local effects, in-
cluding quantum electron density spill-out, tunnelling and confinement, ab initio
calculations such as Time-Dependent Density Functional Theory (TDDFT) are
required. Unfortunately, even under the approximations made in TDDFT (local-
density approximation, linear response (for frequency-space TDDFT)), such cal-
culations are still extremely time consuming and generally require the use of
supercomputers [81] or further strong approximations (jellium model (no in-
terband transitions)). As of yet, such calculations cannot be practically used
to model nanosystems ' 3 nm in dimension and thus are not suitable for the
modelling of the vast majority of plasmonic architectures. However, the results
found can be used to deduce a semi-quantitative picture of the expected non-
local LSPP response of larger nanosystems [37,82], and are described further in
Chapter (5). Such ab initio numerical methods would also be required to model
the more realistic atomic-scale geometries expected for the NGJ created upon
break-of-contact, or contact between two NPs. It is well known from Scanning
Tunnelling Microscopy (STM) that the exact atomic structure of the NGJ created
modifies quantum tunnelling and confinement effects significantly [83].
Currently numerical methods are also generally required to study quantum
confinement effects in plasmonic systems. The number of participating conduc-
tion electrons in NPs of less than approximately 4 nm diameter is small enough
for quantum confinement effects to become non-negligible. If the absolute num-
ber of participating conductions is small, i.e. Ne = neVnp / 500 where Vnp is
the NP volume, the energy imparted to each electron in the system by an in-
cident photon ∆Ee ≈ h¯ω/Ne, and the spacings of the electron energy levels,
are both comparable to the thermal excitation energy ET = kBT, where kB is the
Boltzmann constant and T is absolute temperature. In this regime the quantised
41
Chapter 2: Theoretical Background
nature of the electron energy levels can no longer be discarded and generally
comprehensive atomistic theoretical simulations or novel analytical approaches
are required [36, 84].
Even though the NPs considered in this work are relatively large, result-
ing in negligible spacing between electron energy levels, effects due to quan-
tum confinement could still be accessible if quantum emitters, e.g. appropriate
quantum dots or molecules, are present in the inter-NP NGJ’s created. In this
case the interband (or inter-orbital) resonances of the quantum emitters interact
with the broad plasmonic modes highly-localised in the NGJ (see Sec. (2.3.4)).
This results in the hybridisation of the quantum emitters resonant transitions
and the LSPP resonances. The properties, e.g. excitation rate, and radiative
and non-radiative decay rates, of quantum emitters present in NGJ’s are hence
significantly modified, even in the weak-coupling regime [85]. If appropriate
quantum emitters are present in suitable nanoscale NGJ’s strong exciton-photon
coupling is expected to promote Fano interference and vacuum Rabi-splitting
phenomena that can be measured in the broadband, optical, far-field scattering
response. These effects are currently of interest due to the potential applications
in quantum information devices [86,87]. In the current work, the focus is main-
tained on non-local screening and tunnelling phenomena as the exact molecular
content of the NGJ’s created is not investigated in detail.
2.3.4 Coupled Localised Surface Plasmon Polaritons
The electromagnetic coupling of LSPPs in NP dimer nanosystems with different
inter-NP separation d have been extensively studied both experimentally [88]
and theoretically [89] in the d > 1 nm domain. Due to the inherent reduction in
nanosystem symmetry, compared to that of a single nanosphere, analytical solu-
tions for the coupled LSPP modes of non-overlapping NP dimers have primarily
only been determined under the electrostatic approximation for spherical ge-
ometries [90], most notably using the plasmon hybridisation method [74]. New
analytical transformation optics techniques have begun to include some retar-
dation effects (radiation damping) [91] and to consider overlapping cylindrical
NP dimers [75] thus extending the validity of such analytical treatments. While
the analytical theories provide a good intuitive understanding of LSPP modes
present in NP dimers, numerical methods have to be used for precise calcu-
42
Chapter 2: Theoretical Background
lation of the LSPP response. Strong LSPP coupling between NPs with d / a
creates LSPP mode shifting, mixing and splitting with concomitant near-field
enhancement and localisation. These aspects are highly desirable for many ap-
plications, including high-sensitivity chemical and biological sensors [92], op-
tical frequency switching and measurement [93], quantum optics [94], highly
non-linear optics [95] and solar cells [10].
While the LSPP coupling between spherical NPs (a ' 5 nm) with d ' 5 nm
is well understood under a classical framework, controlled experimental mea-
surement of the LSPP response in the combined spatially non-local screening
and electron tunnelling regime (d / 1 nm) has as of yet been intractable. Recent
theoretical work has led the way in beginning to investigate this important and
interesting regime by retarded hydrodynamic model based simulations [80] and
TDDFT simulations [82,96]. In the following, a brief overview of classically cou-
pled LSPP behaviour in a prototypical NP dimer system used to approximate
the NP dimer system under investigation is presented. This provides a founda-
tion for further detail on LSPPs in the non-local screening and quantum regime
given in Chapter (5).
The present work is concerned with the LSPP interactions between two NP
functionalised AFM tips with apex (NP) separation ranging from d ∼ O(µm-
Å) proceeding into d ∼ O(nm) overlapping negative separation. Therefore
understanding LSPP interactions between an AFM tip and a NP, and between
NP pairs is essential. The simplest model to describe both these nanosystems, in
the classical regime, is an identical spherical NP dimer in two different regimes
of separation
1. LSPP coupling between AFM tip apex NPs: represented as two identical
spherical NPs with d ranging from 500 nm to full conductive contact at
d = 0 nm (see Sec. (3.3.2) for the definition of ‘full conductive contact at
d = 0 nm’) and into small overlap d ∼ −1 nm.
2. LSPP coupling between bulk AFM tip and NP apex: represented as two
overlapping (e.g. d = −10 nm) identical spherical NPs where one NP
represents the bulk AFM tip. Note this is a working hypothesis to help
formulate an initial basis of understanding and is not necessarily a good
approximation of the system. A more rigorous treatment of the coupling
between a NP apex and bulk AFM tip body is presented in Chapter (5).
43
Chapter 2: Theoretical Background
Schematics of these two regimes are shown in Fig. (2.9). It should be noted
that more realistic asymmetric NP (hetero-)dimers where the individual NPs
differ in size, shape or composition are also of great interest [97]. Such asym-
metric NP dimer systems can amplify certain physical effects, e.g. plasmonic
Fano resonances and visible LSPP mode splitting, not prevalent in the identical
(homo-)dimer systems assumed throughout this section.
The LSPP mediated optical response of the NP dimer geometries described
in (1) and (2) can be determined using BEMAX in the classical limit. Here the
specifics of the methods used will not be discussed, but an overview of LSPP
behaviour in these two different regimes will be presented.
Figure 2.9: Non-contact and overlapping NP dimer configurations used to rep-
resent tip nanosystem. (a) NP dimer described in regime (1). (b) NP dimer
described in regime (2). The coordinate system implemented in BEMAX is
labelled in (a) with a plane-wave of arbitrary incident direction marked.
44
Chapter 2: Theoretical Background
Non-contact regime
As d ∼ a the electromagnetic near-field interactions become significant and
the NPs become capacitively coupled. Assuming a λ, and a uniform exciting
field, coupled LSPPs on a NP dimer can be described as a system of two interact-
ing, oscillating electric dipoles (quasi-static limit). Depending on the orientation
of the NP dimer and the polarisation of the incident light one can distinguish
between two limiting cases of polarisation, either parallel or perpendicular to
the NP dimer axis. The two possible configurations are shown in Fig. (2.10).
Figure 2.10: NP dimer LSPP coupling configurations. (a) and (b), two limiting
configurations of the NP dimer with respect to the polarisation of the exciting
field ((a)= parallel, (b)=perpendicular).
45
Chapter 2: Theoretical Background
The Coulombic coupling between the individual NPs creates a system analo-
gous to a coupled harmonic oscillator. The Coulombic restoring force acting on
the oscillating electrons inside each particle is either decreased (Fig. (2.10a)) or
increased (Fig. (2.10b)) due to the induced charge distribution of its neighbour-
ing particle. Therefore the LSPP resonances will either shift to lower frequencies
(red-shift) or higher frequencies (blue-shift) respectively. This insight allows an
intuitive understanding of how LSPPs on the individual tip apex NPs will in-
teract with each other. In short, as the inter-tip distance decreases it is expected
that the LSPP resonances of the system will either red-shift or blue-shift depend-
ing on whether the incident exciting light is polarised parallel or perpendicular
to the dimer axis. It is important to note that light polarised parallel to the
dimer axis creates a far greater LSPP coupling strength relative to the case of
perpendicular polarisation. This is due to the hugely increased electric near-
field magnitude created in the vicinity of each particle under these conditions.
As parallel polarisation is required to create a strong plasmonic response, pri-
marily only this orientation of light with respect to the NP dimer axis will be
considered further.
The locally induced electric near-field intensity, and representative induced
surface charge density distribution, on an Au NP dimer excited at the hy-
bridised dipole resonance for incident light polarised parallel to the dimer axis
(d = 1 nm, θ = 90◦ and φ = 0◦) are shown in Fig. (2.11a). The term ‘hybridised’
is used as the LSPP modes of the individual NPs mix and hybridise in analogy
with the theoretical treatment of the quantum states of diatomic molecules in
terms of interacting molecular orbitals. Under the plasmon hybridisation for-
malism, as d is reduced, the dipolar (` = 1) LSPP modes of the two NPs interact
with each other (and higher-order modes when d . 10 nm) to form bonding
(bright) and anti-bonding (dark) hybridised dipolar LSPP modes [74]. Only the
bright dipolar bonding mode has a non-zero dipole moment and hence a sig-
nificant light scattering cross-section, as shown in Fig. (2.11b). It is important
to note the hybridised bonding and anti-bonding modes are only labelled as
‘dipolar’ excitations as the primary interactions are between the ` = 1 LSPP
modes of the individual NPs. However, as d → 0 the hybridised dipolar bond-
ing mode contains interactions with higher-order modes therefore the shift in
mode wavelength does not only have a 1/d3 dependence as higher powers of
1/d also contribute for small d.
46
Chapter 2: Theoretical Background
-100 0 100
x (nm)
-200
-150
-100
-50
0
50
100
150
200
z 
(nm
)
50
40
30
20
10
0
-10
Induced Field Intensity (dB)-2
-1
0
1
2
N
or
m
al
ise
d 
In
du
ce
d 
Su
rfa
ce
 E
le
ct
ro
n 
D
en
sit
y
(a)
kE 0
120x103
100
80
60
40
20
0S
ca
tte
rin
g 
cr
os
s-
se
ct
io
n 
(nm
2 )
1000800600400
Wavelength (nm)
(b)
Figure 2.11: Near-field intensity map and associated scattering cross-section for
an Au NP dimer (in air) with d = 1 nm and incident light polarised parallel to
the dimer axis. (a) Near-field intensity map and representation of the induced
surface charge density at the hybridised dipolar LSPP resonance λ0 = 685 nm.
(b) Associated scattering cross-section CBEMAXsca showing the hybridised dipolar
mode (maroon dotted line). Inset shows schematic of associated surface charge
distribution. Note the higher-order hybridised (‘quadrupole’) mode at λ0 = 550
nm (green dotted line) created by the admixture of different `-modes.
47
Chapter 2: Theoretical Background
Here the NPs remain separated (in the classical sense) hence the large reso-
nant charge build-up at the nanogap must be compensated by the charge distri-
bution around the remainder of the individual NPs surfaces in order to main-
tain charge neutrality. The lateral width of the charge pile-up along the NP
surface is on the order of w ≈ √ad for the hybridised dipole mode [89]. As Fig.
(2.11a) shows, extreme field intensity enhancements are predicted in the vol-
ume of the nanogap, approximately 5× 103 greater than the maximum found
in the single NP case. In reality the hybridised LSPP modes will be slightly
blue-shifted and broadened due to the spatially non-local screening effects of
induced-surface-charge penetration and electron density spill-out around the
vicinity of the nanogap. The electric near-field intensity enhancement at the
nanogap will also be reduced by a factor ranging from ×2 → ×10 because of
the spatially non-local screening increasing the damping and effective d due to
BP excitation.
Following the intuitive description given in Fig. (2.10a), as d decreases to 1
nm the hybridised dipolar bonding LSPP mode is strongly red-shifted from that
of the dipole LSPP resonance of a single NP, i.e. 538 nm to 685 nm. At d ∼ 10
nm the ` = 1 LSPP dipole mode of each NP starts to interact significantly with
higher-order LSPP modes of the other NP, therefore hybridised higher-order
LSPP modes (e.g. labelled hybridised quadrapole mode for strong interaction
between ` = 1 and ` = 2 modes) also begin to be strongly excited (dipole-
active) as shown in Fig (2.11b). These higher-order modes are less dipolar in
nature than the hybridised dipolar bonding LSPP mode and hence their excita-
tion involves yet higher powers of 1/d. All of these modes are associated with
extremely large charge pile-up at the nanogap and hence are highly sensitive
to the geometry and dielectric response of the nanogap itself. As d → 0 the
classical BEMAX simulation predicts that the red-shift of the hybridised LSPP
modes becomes singular, as demonstrated in Fig. (2.13). This non-physical sin-
gular behaviour is evident because spatially non-local dielectric functions, and
non-local electron spill-out density and tunnelling effects, are not considered in
the BEMAX model.
48
Chapter 2: Theoretical Background
Contact regime
In the classical regime conductive contact occurs when at least one atom forms
a conductive bridge between the NP dimer. In this case intraparticle charge
neutrality is no longer required and new LSPP modes are enabled. The polar-
isability of the new combined structure created, primarily determines the LSPP
response for a given incident electromagnetic wave, i.e. the ability of the surface
charges to redistribute over the whole surface of the combined structure. The
conductive nanojunction is not equivalent to a zero-frequency conductor as the
NP dimer system is being driven at optical frequencies. In the optical frequency
regime the approximation σ˜m(ω) ≈ σ0 is no longer valid. The flow of charge
across the nanojunction is therefore considered as a displacement current, given
by I˜d = A ∂P∂t , rather than a conduction current Ic = σ0AE (where A is the effec-
tive cross-sectional area of the nanojunction) that is found in typical electronic
circuits operating at frequencies up to f ∼ O(GHz). The notion of the conduc-
tivity of the nanojunction is hence understood as the magnitude of the effective
dipole (or higher-order) moment of the combined NP dimer structure that the
nanojunction enables, i.e. the magnitude of the effective electric susceptibility
|χe| of the nanosystem as a whole. This ‘plasmonic’ conduction hence depends
on the orientation of the nanosystem with respect to the driving electric field
and thus it can be substantially different to the traditional electronic conduction.
As the dimer NPs begin to overlap a new long-wavelength true (non-
hybridised) dipole mode is formed, as shown in Fig. (2.12b), where each con-
stituent NP has a net charge that oscillates in dipolar fashion over an optical cy-
cle. The locally induced electric near-field intensity, and representative induced
surface charge density distribution, on the Au NP dimer excited at the true
dipole resonance for incident light polarised parallel to the dimer axis (d = −2
nm, θ = 90◦ and φ = 0◦) are shown in Fig. (2.12a). This LSPP reponse can
be understood under the plasmon hybridisation formalism by considering the
hybridisation of the unphysical monopolar (` = 0) LSPP modes16 of the indi-
vidual NPs. Upon contact the unphysical monopolar modes hybridise to create
a physical true dipole mode involving the entire structure.
16Monopolar LSPP modes are unphysical for isolated NPs as charge neutrality within each
NP would not be maintained.
49
Chapter 2: Theoretical Background
-100 0 100
x (nm)
-200
-150
-100
-50
0
50
100
150
200
z 
(nm
)
30
20
10
0
-10
Induced Field Intensity (dB)-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
N
or
m
al
ise
d 
In
du
ce
d 
Su
rfa
ce
 E
le
ct
ro
n 
D
en
sit
y
(a)
kE 0
60x103
40
20
0Sc
at
te
rin
g 
Cr
os
s-
se
ct
io
n 
(nm
2 )
1600140012001000800600400
Wavelength (nm)
(b)
Figure 2.12: Near-field intensity map and associated scattering cross-section
for an overlapping Au NP dimer (in air) with d = −2 nm and incident light
polarised parallel to the dimer axis. (a) Near-field intensity map and represen-
tation of the induced surface charge density at the true dipolar LSPP resonance
λ0 = 1128 nm. (b) Associated scattering cross-section CBEMAXsca showing the true
dipolar mode (black dotted line). Note the strongly hybridised dipolar mode
at λ0 = 611 nm (blue dotted line) created by the admixture of the ` = 1 modes
and higher-order `-modes. Insets show schematics of associated surface charge
distributions.
50
Chapter 2: Theoretical Background
A comprehensive analysis of the LSPP modes through the conductive con-
tact regime was performed using BEMAX [89] and the LSPP mode evolution
is shown in Fig. (2.13). Immediately at contact there is a discontinuous ‘jump’
in the NP dimer system polarisability, and hence also the LSPP modes, and
the true dipole mode appears (labelled C-D). From this singularity, as the NP
overlap increases, all LSPP modes begin to blue-shift.
Figure 2.13: Evolution of LSPP dimer modes as NP (a = 60 nm) separation d is
reduced through the non-touching to overlapping transition (from [89]). Note
the unphysical asymptotic and discontinuous behaviour of the LSPP modes as
d→ 0.
For small overlaps (d/a ≈ −0.05) there is still considerable charge build-up
within the nanojunction (anti-wedge) region, as shown in Fig (2.12a). There-
fore higher-frequency modes created by the admixture of dipolar and multipo-
lar LSPPs are present even after conductive contact (labelled E-G and H). The
transition from a separated to plasmonically connected dimer has been studied
in more detail under a classical framework by considering a NP dimer linked
by bridges of different conductivity and conductance [98]. This has allowed
theoretical estimates of the threshold electrical properties of the nanojunction
required for the onset of the true dipolar charge transfer mode and the blue-
51
Chapter 2: Theoretical Background
shifting of higher-order modes. It is through this −0.5 nm/ d / 2 nm transi-
tion where the most extreme LSPP mediated field-enhancements (localisation),
LSPP environmental sensitivity and LSPP mode shifts, mixing and splitting oc-
cur. As described previously, at these separations d is comparable to the scale of
the electronic wavefunction (at the Fermi energy) and hence spatially non-local
screening effects and quantum transport phenomena are expected to play a vital
role in determining the nature of all the LSPP modes throughout this regime.
For large NP dimer overlap, as depicted in Fig. (2.9b), the true dipolar
LSPP mode begins to de-localise charge from the nanojunction volume as the
NP dimer begins to take on the character of a single elongated NP. The higher-
order multipolar LSPP modes also become increasingly de-localised from the
nanojunction and weaken in oscillator strength rapidly as the NP overlap in-
creases. The true dipolar mode is significantly red-shifted compared to the
hybridised dipolar mode present before contact. As overlap increases the post-
contact LSPP modes settle into the positions expected for a nanorod geometry
(effectively a spherical NP with an anisotropic polarisability). As discussed in
detail in Chapter (5), understanding the complex combined NP-tip dimer sys-
tem is challenging due to the extended nature of the tips, however the overall
tip-NP dimer structure is modelled effectively.
2.4 Open Questions
As introduced throughout this chapter, the LSPP interactions between two iden-
tical spherical NPs (a ' 5 nm) that form a dimer with inter-NP separation d ' 5
nm are well understood when considering infinite uniform planar illumination.
The rapid development of existing and novel fabrication techniques is enabling
a new class of truly engineered nanosystems to be created [88, 99]. Dimensions
and features of these nanosystems are beginning to be created on the single
nanometre or even sub-nanometre scales. The limit for achievable nanogap d is
currently set by several existing methods under various stages of development
that are beginning to create nanogap electrodes with d / 1 nm [99]. How-
ever, as of yet, none of these techniques can produce relatively large (a > 20
nm), individual NP dimer systems with the controlled variable separation −0.5
nm/ d / 2 nm required for investigations of LSPP response in the spatially
non-local screening and quantum transport regime. The existing methods and
52
Chapter 2: Theoretical Background
their relevant limitations are described in detail in Sec. (3.1).
As detailed in Sec. (2.3.4) a huge number of applications are entirely depen-
dent on the properties of the nanogaps created in nanosystems with a strong
plasmonic response, i.e. NP dimers with a > 20 nm and d < 20 nm. Not only
does understanding the LSPP response of nanogaps with d / 2 nm strongly
affect all existing applications but also nanogaps with d / 1 nm are expected
to yield new exciting applications including single-molecule plasmon-assisted
transport, photoelectrochemistry on the sub-zeptolitre scale, optical rectifica-
tion, electrical excitation of plasmons and improved methods for molecular
spectroscopy and sensing. As fabrication techniques improve, and become more
controllable, an understanding of the electromagnetic coupling between LSPP
modes in the −0.5 nm/ d / 2 nm regime is thus essential.
Experimental techniques for creating such relatively large NP dimer systems
with dynamic separation control for −0.5 nm/ d / 3 nm, whilst also allowing
simultaneous optical and electronic characterisation of dimer LSPP response,
are extremely challenging to realise. The challenges involved are mainly due
to all the inherent difficulties of precise truly nanoscale positioning, manipula-
tion and stability, coupled with the requirement for simultaneous high-quality
optical and electronic measurement.
Current numerical and analytical theoretical work has yet to take into ac-
count the quantum nature of the transition from non-touching to atomically
linked NP dimers with a ' 2 nm. This is mainly due to the problems involved
in modelling relatively large scale nanosystems on the atomic level because the
fully quantum mechanical methods required, e.g. TDDFT, scale in computa-
tional complexity as N3e .
From this foundation it is clear three vital questions must be addressed as the
field of plasmonics moves into the molecular and atomic-scale regimes. Firstly,
from an experimental basis: what experimental rig and associated methods can
be developed to create the necessary NP dimer systems (a > 20 nm, −0.5 nm/
d / 2 nm) whilst allowing for effective simultaneous optical and electronic
characterisation? The advances made in addressing this question are described
in detail in Chapters (3) and (4).
Secondly, the development of such an apparatus now enables some of the
most critical outstanding questions concerning the LSPP response in NP dimer
systems to be experimentally addressed for the first time: how do the LSPP
53
Chapter 2: Theoretical Background
modes of a NP dimer (a > 100 nm) shift, split and mix in the non-local screening
and quantum transport regime (d / 1 nm)? The progress made in addressing
these important questions is described in detail in Chapter (5).
Finally the experimental work will for the first time enable comparison with
new theoretical models, currently under development. This enables investiga-
tion into what non-local screening and quantum transport effects are of vital
importance to include in theoretical models for a given d, i.e. are the new mod-
els sufficient to allow an understanding of plasmonic phenomena in the complex
quantum-scale regime?
2.5 Summary
Following the open questions that arise from the theoretical background, the
primary goal of the present work is directed at experimental measurement of
the quantum transport mediated LSPP response of NP dimers with a > 100 nm
and d < 1 nm. This enables the experimental resolution of the long-standing
problem of the classically predicted singular nature of the contact transition [89]
described in Sec. (2.3.4). Simultaneously, preliminary experimental investiga-
tions into how non-local screening, described in Sec. (2.2.2), affects the LSPP
response are also carried out in a controlled manner. Additionally, recently pre-
dicted plasmon-exciton coupling phenomena, introduced in Sec. (2.3.3), may
also be measurable in such NP dimer systems, and hence possible evidence is
scrutinised. As all experimental data are taken for individual NP dimers over
the full inter-NP separation range of interest, trends in LSPP response in the
non-local quantum transport and screening regime, are for the first time fully
comparable with classical, and emerging quantum corrected theoretical simula-
tions.
54
Chapter 3
Experimental Rig
3.1 Introduction
A substantial component of the present work was concerned with the concep-
tual and physical creation of a novel, multi-purpose observation platform to
enable study into the open questions described in Sec. (2.4) and beyond. This re-
quired an apparatus that could provide an optically accessible, suitably aligned
metallic NP dimer with dynamically controllable separation from over 500 nm
down to the ångström level. It was essential that the rig allowed high-quality,
high-speed, nanolocalised optical DF measurements, and electronic measure-
ments, to be taken simultaneously at all NP dimer separations.
Schematics of several techniques currently used to create NP dimers and
metallic NGJs are shown in Fig. (3.1). At present, methods for NP dimer cre-
ation include directed self-assembly [100–102], top-down electron-beam lithog-
raphy (EBL) [103] and discrete-step nanomechanical manipulation [104]. Very
recent self-assembly methods [105] and top-down electron-beam lithography
[106] methods enable spectroscopy on NP dimers of fixed separations down to
d ≈ 0.5 nm. However, none of these techniques provide dynamic control of
dimer alignment and separation, limiting their capability for optical investiga-
tions of NP dimer interactions at the quantum scale. For example, the self-
assembled NP dimer shown in Fig. (3.1a) has a minimum separation limited
by the molecular layer used for the self-assembly process. These techniques
also lack the capability for simultaneous measurement of individual NP dimer
optical and electronic response, as making conductive contact to the dimer’s
constituent NPs is challenging.
55
Chapter 3: Experimental Rig
Figure 3.1: Schematics of techniques for NP dimer and nanocavity creation.
(a) Directed self-assembly via molecular layer. (b) Top-down EBL or angle-
evaporation using EBL manufactured mask. (c) Discrete-step nanomechani-
cal manipulation. (d) Electromigration. (e) Mechanically controllable break-
junction. (f) Break-junction nanocavity and NP dimer formed by STM tip and
conductive AFM tip respectively.
In the field of molecular electronics, conductive contact to molecular junc-
tions for simultaneous dynamic optical and electronic measurement has been
achieved using electromigration [107] and Mechanically Controllable Break-
Junction (MCBJ) methods [108] (Figs. (3.1d)-(3.1f)). The use of electromigration
and MCBJ techniques provides the stability required to create long-lived molec-
ular scale junctions. However the techniques lack the dynamic range of align-
ment, separation, NP geometry, and control of contact required to investigate
the full range of plasmonic response. The effective NP dimer created between
a NP and a conductive AFM tip, e.g. in a Tip-Enhanced Raman Spectroscopy
(TERS) setup [109], is shown schematically in Fig. (3.1f). Such a system meets
the general requirements, but suffers from an overwhelmingly large scattering
background and poor optical access, therefore preventing measurement of the
broadband optical (plasmonic) scattering response directly.
56
Chapter 3: Experimental Rig
As described in the previous chapters the technique was conceived by en-
visaging the 3D nanoscale alignment of two conducting AFM probes, with NP
functionalised apices, in an optically accessible, co-axial ‘tip-to-tip’ configura-
tion. As the inter-tip separation is reduced the NP functionalised AFM tip apices
effectively create a NP dimer geometry. A NP can be created on the apex of each
probe by a wide variety of AFM industry standard methods including, metal-
lic coating [110], Electron-Beam Deposition (EBD) [111], mechanically-directed
or chemically-based NP attachment [112], and focused ion beam milling [113].
Creating an effective NP dimer by axially aligning the NP functionalised apices
of conducting AFM tips has several advantages over the existing techniques, as
it allows for,
• Simultaneous optical and electronic measurement in order to characterise
the plasmonic response.
• Dynamic, NP dimer nanoscale alignment and separation control from
d > 500 nm down to the ångström level via an all electronic feedback
mechanism.
• Excellent optical access due to the accessible tip-to-tip geometry thus en-
abling DF spectroscopy techniques under the required light polarisation
conditions.
• A greatly reduced background optical scattering signal compared to that
from NP dimers created by structured substrates and NP dimers formed
in TERS configurations [114].
• A stationary or alternating electric potential to be placed across an indi-
vidual NP dimer and the associated current flow to be measured.
• Potential freedom in selection of NP dimer type and functionalisation.
It should also be noted that by utilising existing AFM tip technology, the cost
and complexity of the rig and associated methods were greatly reduced, whilst
the repeatability and reproducibility of the methods realised were increased.
It is also stressed here that almost all the apparatus was controlled remotely
using IGOR Pro to send Virtual Instrument Software Architecture (VISA) com-
mands via GPIB to the experimental equipment. User friendly software was
57
Chapter 3: Experimental Rig
also created using IGOR Pro to automatically log data and run experimental
procedures thus increasing the efficiency of rig development and experimental
investigations. The rig development and setup can be separated into three cat-
egories; mechanical, electronic and optical design. In this chapter each of these
categories is discussed in detail to provide the necessary background informa-
tion on the experimental setup.
3.2 Mechanical Design
The experimental setup required a mechanical system capable of the nanoscale
alignment of two AFM tips in a tip-to-tip configuration whilst also providing the
necessary framework to allow full access for a multi-purpose optical microscope
with laser illumination DF nano-spectroscopy capabilities. The mechanical con-
struction of the overall rig can therefore be split into two components, the AFM
tip alignment unit and the optical microscope unit. In this section a concise
description of the overall mechanical rig construction is presented.
3.2.1 AFM Tip Alignment Unit
The AFM tip alignment unit was constructed from two lockable manually con-
trolled micron resolution stages (Newport DS25), one computer controlled three-
axis nanometre resolution piezoelectric stage (PI733.3CD), and various specially
designed support structures. Critically, using this combination of positioning
stages allowed the inter-tip position to be moved over a range from millimetres
down to nanometres with nanometre resolution in x, y and z. The tips used for
development were 17 µm long, standard platinum or gold coated, contact mode
AFM tips (≈ 0.2 Nm−1 spring constant), that were formed with a conducting
450 µm long cantilever to an industry standard sized chip holder. The alignment
unit was carefully designed to allow,
• Industry standard AFM chips of dimension 3.4 mm ×1.6 mm ×0.3 mm to
be easily attached and removed from the setup.
• Electrical connections to be made to each AFM chip while electrically iso-
lating the vast majority of the overall rig.
58
Chapter 3: Experimental Rig
• Dark-field microscope objectives of up to 100× magnification (NA 0.9) to
be brought within their short working distance (1 mm) to image the tips
and obtain DF optical measurements.
• Creation of tip-to-tip geometry with maximum stability whilst retaining
the ability to control the inter-tip position in 3D on the nanoscale.
• Ultra-wide optical access to the tip apices to enable a wide range of optical
measurements to be performed effectively, under suitable incident light
polarisation conditions, e.g. DF scattering and transmission measurements
with light linearly polarised parallel and perpendicular to the tip axes.
• Extensions to be added, such as stray electromagnetic field and air current
shielding, and additional electronics.
As discussed in Sec. (2.3.4), for a NP dimer to yield a strong plasmonic response
the incident light needs to be polarised parallel to the dimer axis. The system
geometry shown in Fig. (3.2) is required so that incident light, linearly polarised
parallel to the tip axes (dimer axis), can be directed onto the tip apices (NP
dimer).
Figure 3.2: Scaled drawing (apart from the tips and cantilever thickness, en-
larged for clarity) of the co-axial tip-to-tip geometry in relation to the light-cone
produced by a NA 0.9 microscope objective under halogen lamp illumination.
59
Chapter 3: Experimental Rig
Using this setup, complicated polarisation transformation techniques [115]
used in conventional tip-enhanced spectroscopies [114], are not required to cre-
ate a polarisation state parallel to the dimer axis. This allows the generation of
strong LSPP coupling across the NP dimer when d / a.
A Computer Aided Design (CAD) drawing of the tip alignment unit con-
structed to meet the necessary requirements is shown in Fig. (3.3). The AFM
chips (and therefore tips) are held securely in place by a phosphor bronze ‘chip
clip’ pressing against the underside of the AFM chip thus clamping it into a pre-
cisely machined alignment groove, as shown in the inset in Fig. (3.3). The com-
pressive force on each chip is supplied by tightening two brass screws threaded
between the chip clip and the control rod. The control rods and chip clips were
carefully designed to allow DF objectives (with greater diameter than standard
objectives) to be brought from above to within 1 mm of the tips. The only factor
limiting the objective approach is the width of the AFM chip itself, as shown in
Fig. (3.2). The manual y-axis stage on the left-hand side of Fig. (3.3) is secured
Figure 3.3: CAD drawing of the AFM tip alignment unit. The AFM chips are
firmly secured by the chip clips. The opposing ends of the control rods were
carefully designed to enable maximum optical access. A top-down view of the
AFM chips and cantilevers is shown in the inset.
to the main base plate (green) which in turn is secured to the fixed piezoelec-
tric stage chassis. The manual combined x and z-axis stage on the right-hand
side is connected through to the piezoelectric driven base plate (purple) that
60
Chapter 3: Experimental Rig
is firmly secured to the 3D piezoelectric actuator. Thus only one of the tips
is mounted on the piezoelectric actuator which provides the nanomechanical
facility for inter-tip alignment on the nanoscale.
A photo of the completed mechanical unit is shown in Fig. (3.4). Apart from
the standard 25 nm thickness gold or platinum coated tips, several other types
of AFM probe were used during the system development. Most importantly, for
the simultaneous optical and electronic measurements, nano-indentation AFM
probes with a ‘neck and ball’ apex geometry and an overall 50 nm thick gold
coating were typically used, as they created the desired spherical dimer geome-
try at the apices. These functionalised ‘dimer tips’ were produced by Nanotools
GmbH by performing EBD on standard AFM probes and subsequently evapo-
rating on the required coating layer1.
Figure 3.4: Photo of the completed tip alignment unit mounted on the trans-
lation stage of an Olympus BX51 optical microscope. The inset and sub-inset
(note change of axes) show DF images taken from above of typical AFM tips
used, at maximum (2×100×) magnification.
1Au NP functionalised AFM tips were also produced in-house by attaching an AFM tip to
a nano-manipulator and picking up individual NPs under a SEM (using a small electrostatic
force). However time-constraints ruled this process out as a viable functionalisation method.
61
Chapter 3: Experimental Rig
Typical DF images of standard platinum coated tips, and functionalised
dimer tips, in the tip-to-tip configuration used are shown in the inset and sub-
inset respectively in Fig. (3.4). The specifics of the tip geometries and cantilever
specifications used are described in more detail in Chapters (4) and (5).
The electrical connection to each tip was made by a solder tag secured to
each control rod via one of the chip clip fixing screws. The current signal was
transfered by highly flexible, low capacitance, mini-coaxial ( white ) cable to
standard coaxial connectors shown in the background of Fig. (3.4). The tips
themselves were electrically isolated from the rest of the alignment unit by 5
mm thick nylon washers placed just before the control rods contact the pinch
blocks. The two sides of a control rod were connected by a nylon stud screw to
maintain electrical isolation. The pinch blocks hold each control rod securely in
position at a chosen angle between the tip axes, and are constructed from brass
to avoid possible cold welding to the aluminium alloy control rods.
3.2.2 Optical Microscope Unit
The initial testing and development of the tip alignment unit was carried out
under an Olympus BX51 microscope. As the work progressed it became appar-
ent that a custom built microscope setup would be needed to meet the following
stringent requirements,
• A high level of stability and resistance to external vibration sources, e.g.
vibrations originating from the laboratory floor, optical bench surface and
equipment racks.
• A reasonable level of isolation from air currents, electromagnetic fields
and heat sources.
• Computer controlled 3D translation of the tip alignment unit under the
illumination with > 2 cm travel range and < 50 nm resolution in x, y and
z.
• Facility for a high quality, high magnification, lamp-sourced, Bright-Field
(BF) reflection imaging and DF scattering imaging setup.
• Facility for an ultra-low background laser (supercontinuum) sourced con-
focal DF nano-spectroscopy setup that can be used simultaneously with the
62
Chapter 3: Experimental Rig
halogen lamp sourced imaging system.
• All free-space coupling of light sources between microscope and measure-
ment equipment, i.e. no optical fibers.
A photo of the overall apparatus including the microscope unit is shown in
Fig. (3.5). The whole unit was implemented in a horizontal geometry to allow
shorter, stiffer, more stable supports.
Figure 3.5: Photo of the overall experimental rig including the microscope
unit. The optical microscope unit is built entirely upon an optical breadboard
mounted across two active anti-vibration platforms, all contained within a Fara-
day cage.
All critical posts and customised support structures were built for high
strength and stability. Additionally, where possible, the support structures had
internal passive damping. As shown in Fig. (3.5) the whole microscope setup
was mounted on an optical breadboard connected to two active anti-vibration
platforms (Halcyonics Vario) to provide vibration damping from 1-300 Hz and
additional passive damping at higher frequencies.
The combination of active and passive damping mechanisms implemented
throughout the entire rig and optical bench were designed to complement each
63
Chapter 3: Experimental Rig
other in order to reduce unwanted vibrational resonances of the support struc-
tures. The setup provided a high level of vibration isolation from 1 Hz well into
the kHz regime. The anti-vibration platform was isolated (no direct physical
contact) from the surrounding Faraday cage and air current shielding. All ca-
bles connected to components on the anti-vibration platform were double strain
relieved to avoid transmission of external mechanical vibrations onto the plat-
form and hence to the AFM tips or sample under investigation.
Due to the horizontal setup, the microscope objectives and the tip alignment
unit had to be mounted in an appropriate fashion as demonstrated clearly in
Fig. (3.6).
Figure 3.6: Photo of the tip alignment unit mounted ‘side on’ with sample
holder and microscope objective shown. Sample holder is removed when tips
are to be studied.
The 3D piezoelectric stage was re-calibrated for the given load under the
‘perpendicular’ orientation and tuned to provide relatively high stability but
slow (≈ 50 Hz) response. The custom built sample holder shown in Fig. (3.6)
was mounted on the manual combined x and z-axis translation stage and is thus
connected through to the piezoelectric actuator. This allowed standard samples,
64
Chapter 3: Experimental Rig
e.g. NPs deposited on a microscope slide, to be observed and translated in 3D
with nanometre resolution under the custom microscope illumination. When
the AFM tips were to be investigated the sample holder was removed and the
alignment unit translated along the y-axis towards the objective until the appro-
priate working distance was obtained.
The typical halogen and laser light paths are marked schematically on the
top-down photo of the microscope unit shown in Fig. (3.7). Almost all optical
components were held securely within cage systems (Thorlabs) that acted to
maintain relative translational and angular alignment. The cage rods and plates
also improved the rigidity of the overall apparatus and the resistance to external
vibration. The microscope objective itself was mounted on a custom built high-
stability (no rotating turret) mount securely fastened to the optical breadboard
base.
Figure 3.7: Top-down photo of the optical microscope unit with schematic over-
lay of lamp (green), laser (orange) and combined (purple) light paths.
The halogen lamp (100 W) is not shown in Fig. (3.7) as it protrudes outside
the Faraday cage through a specially cut aperture. This keeps the substantial
heat (and air convection) source shielded from the tips. Temperature variations
65
Chapter 3: Experimental Rig
were kept under ≈ ±0.1 K by the laboratory temperature control system thus
keeping thermal expansion and contraction of all the mechanical structures to a
minimum. The spectrometers were also located outside the shielding and were
not physically connected to the anti-vibration platform. This de-coupled the
vibrations produced by the spectrometer cooling fans from the alignment and
microscope units. The imaging camera and lens tube, labelled in Fig. (3.7), were
mounted on a damped support post. The composite 3D translation unit used to
position the tip alignment unit (hence the tips) with respect to the microscope
objective illumination was formed from three highly stable, low-drift, lockable,
steel translation stages (Newport). The motion of both lateral stages was com-
puter controlled via piezoelectric actuators with 30 nm resolution, while the
vertical stage motion was computer controlled via a heavy load actuator with
50 nm resolution.
3.3 Electronic Design
The electronic systems were designed and developed to enable the axial tip-to-
tip alignment of two AFM tips, i.e. 3D NP dimer alignment, and to perform
conductance measurements on the nanojunction created upon (classical) con-
ductive contact of the NP dimer constituents. The electronic setup can thus be
separated into two distinct components, the AFM tip alignment electronics and
the nanojunction conductance measurement electronics. The electronic setup
for both components is described in this section while the detailed reasoning
for each setup and the physical principles behind the techniques developed are
described in Chapters (4) and (5).
3.3.1 AFM Tip Alignment Electronics
The AFM tip alignment technique, i.e. the 3D dimer alignment mechanism, was
based on applying an alternating potential across the conducting AFM tips (Fig.
(3.4)) to create an oscillating long-range electrostatic force that drives one of the
AFM tips into resonant motion. Alignment feedback was then obtained by mea-
suring the changes in the modulation amplitude of the tip system capacitance
(related to tip oscillation amplitude) as the tip mounted on the piezoelectric ac-
tuator was scanned in an x, y grid, at a given initial tip apex separation d0 in the
66
Chapter 3: Experimental Rig
z-direction. Due to the complexity of the system it was not possible to specify
all the necessary electronic equipment from the outset. By practical develop-
ment, and theoretical modelling of the system, the broad requirement identified
to enable an alignment capability from d0 ≤ 400 nm was an electronic measure-
ment setup with the ability to measure Alternating Current (AC) amplitudes
IAC0 ≈ 0.3− 1± 0.05 pA at frequencies fAC = 10-30 kHz. Such measurements
correspond to capacitance modulation amplitudes of Cm0 ≈ 1− 8± 0.1 aF. Ad-
ditionally each measurement had to be accurately taken within timescales cor-
responding to a 100 ms time-constant to allow a typical 20× 20 point grid scan
to be obtained in a reasonably short time.
A schematic of the AFM tip alignment electronics is shown in Fig. (3.8). The
computer controlled signal generator was used to generate an alternating (kHz
sine wave) electric potential across the high input impedance of the broadband
(0-1 MHz) voltage amplifier. After the amplification stage a current limiting re-
sistor Rcl was used to limit the current flow across the nanojunction that would
be formed if accidental contact occurred during the tip alignment procedure.
The amplified alternating potential amplitude (V0 = 1-15 V) creates an os-
cillating attractive force between the tips that can induce substantial motion of
a cantilever-tip assembly when driven on resonance. The motion thus modu-
lates the tip system capacitance, i.e. (approximately) the capacitance between
the opposing composite control rod, chip clip, AFM chip, cantilever and tip as-
semblies, referred to as C in Fig. (3.8). Critically, the modulation of C creates
harmonic components of IAC that flow across the capacitor formed by the tip
system. These harmonics are related to the cantilever-tip assembly oscillation
amplitude and can be measured by lock-in detection. The optimum harmonic
component was measured and used for alignment feedback, as described in
detail in Chapter (4).
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Chapter 3: Experimental Rig
Figure 3.8: Schematic showing the setup of the main electronic measurement
equipment used for the axial AFM tip-to-tip alignment technique developed.
Note all ground points were connected to a single mains power earth connec-
tion point.
In practice a Digital Storage Oscilloscope (DSO) is placed in parallel with
C to measure the amplitude and phase of the alternating potential across the
AFM tips. This is necessary because the large load impedance |Z˜c| (> 300
MΩ) associated with C is orders of magnitude greater than the impedance |Z˜s|
associated with the stray capacitance of the system Cs. Here the required current
limiting series resistance Rcl ∼ O(|Z˜s|), hence Cs acts to reduce the amplitude of
the potential across the AFM tips VT0 by ≈ 40% compared to V02. The reduction
of VT0 relative to V0 has to be taken into account in both experiment and theory.
A manual Single-Pole Double-Throw (SPDT) switch (orange) was used to
select the nanojunction conductance measurement voltage source and electron-
ics. In a similar fashion a small, low noise, low vibration SPDT reed relay
(green) mounted on the AFM tip alignment unit was used to select (when ener-
gised) the appropriate current characterisation equipment for the nanojunction
2Connecting the DSO (and the required co-axial cable) in parallel with C also loads the
system and reduces the amplitude of the potential across the AFM tips yet further. However it
allows accurate measurement of VT0 and the associated phase.
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Chapter 3: Experimental Rig
conductance measurements. The current flow IAC across C is passed through
a fixed 108 gain, 40 kHz bandwidth transimpedance amplifier with 10 µs rise
time. The resulting output voltage signal VAC is then passed through a dual
channel filtering system appropriately set to form a Band-Pass Filter (BPF) of
≈ 3 kHz bandwidth around the frequency of interest. The pre-filtering prevents
unwanted signals outside the region of interest from overloading the lock-in
amplifier.
Finally, the lock-in amplifier, referenced by the signal generator SYNC (TTL)
output, is used as an exceptionally narrow BPF filter, and accurate phase de-
tector, to measure VAC0 and the associated phase. The current flow amplitude
across C is directly related to VAC0 by I
AC
0 = V
AC
0 × 10−8. Tip alignment feedback
is obtained using the lock-in amplifier to generate an internal reference signal
at the appropriate harmonic frequency of the fundamental, thus enabling mea-
surement of the optimum harmonic component of IAC created by the cantilever-
tip oscillation. Each measurement is taken with a 100 ms lock-in time-constant
(effectively 0.78 Hz filter bandwidth with 24 dB roll-off) and the data transferred
to the experimental control computer via a GPIB interface.
The electronic Signal-to-Noise (SN) ratio was good enough to allow use-
ful alignment measurements to be taken at d0 / 400 nm with a 100 ms time-
constant. To ensure this measurement capability several improvements were
implemented over the course of development of the electronic setup to decrease
the amount of induced current noise:
• Reduced capacitive coupling: all electronic equipment and cables in the
immediate vicinity of the rig were shielded (aluminium foil) or moved
further away from the tips. Additionally all critical components and the
overall rig were electromagnetically shielded in Faraday enclosures.
• Reduced microphonic and triboelectric induced noise currents: the short-
est possible lengths of double strain relieved, low-noise, low-capacitance
co-axial cabling were used, e.g. the cabling between tip 2 and the tran-
simpedance amplifier input was firmly secured and kept to under 30 cm
in length.
• Ground loops: care was taken to reduce the effect of ground loops as far
as practicable using the available equipment, e.g. all electronic equipment
was connected at a single mains power earth connection point.
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Chapter 3: Experimental Rig
The short lengths of low-capacitance cabling used ensured that minimal ad-
ditional loading was introduced and all RC time constants created were kept
below ≈ 5 µs, i.e. well below the required 100 ms lock-in time-constant.
Using the electronic equipment shown in Fig. (3.8), optimised for low-level
current measurement as described above, a noise level of ≈ ±50 fA at fAC is
achieved. This provided a SN ratio that is good enough for alignment to proceed
from d0 ≈ 400 nm, i.e the initial tip apex separation after diffraction-limited
coarse tip-to-tip alignment under the optical microscope unit.
3.3.2 Nanojunction Conductance Electronics
Upon 3D NP dimer alignment d0 is reduced to zero to form full conductive con-
tact between the NPs, while the Direct Current (DC) conductance of the nano-
junction created is continually measured. Stationary voltages VDC ∼ O(µV) are
applied across the nanojunction whilst simultaneously measuring the associated
DC flow IDC ∼ O(nA-µA). The nanojunction DC conductance GDC is calculated
via GDC = IDC/VDC. Full conductive contact (d0 = 0 nm) is defined by the con-
dition GDC ≥ G0 where G0 = (2e2/h) is the quantum of conductance (assuming
ballistic electron transport through a 1D channel). The required potential was
generated by a low-voltage Source Measure Unit (SMU) connected across the
NP dimer, as shown in Fig. (3.9).
When DC conductance measurements were required the circuit shown in
Fig. (3.9) was selected, as described in Sec. (3.3.1) and shown in Fig. (3.8).
The current flow was measured by a low-noise variable gain (103-1011) tran-
simpedance amplifier (with a 10 Hz bandwidth low-pass input filter) and the
resulting voltage signal measured on a DSO connected to the laboratory com-
puter. The SMU maintained a constant potential across the NP dimer regardless
of the contact resistance. The potential was set to compensate for the com-
bined contact potentials (thermoelectric EMF’s), and ground potential differ-
ences, present in the electronic setup. Temperature gradients were kept to a
minimum by the laboratory temperature control system and equipment shield-
ing so that thermoelectric EMF’s were small compared to the offset voltage
created by the difference in ground potentials. These effects typically produced
overall (relatively constant) offset voltages Vo f f ∼ O(µV). The compensating
offset voltage required (−Vo f f ) was found by bringing the base of the AFM tips
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Chapter 3: Experimental Rig
Figure 3.9: Schematic showing the setup of the main electronic measurement
equipment used for the tip-to-tip (NP dimer) conductance measurements. Note
all ground points were connected to a single mains power earth connection
point.
into full conductive contact and using the SMU to adjust the potential across the
AFM tips until IDC = 0 at an appropriate level of gain. This procedure was car-
ried out before each experimental run thus allowing VDC to be set accurately for
the DC conductance measurements. Therefore the quality of DC conductance
measurement was mainly determined by the accuracy of the IDC measurement.
3.4 Optical Design
The optical system was designed to enable good quality, high-magnification
(BF and DF) imaging of various nanosystems, with the facility for laser-sourced
DF nano-spectroscopy measurements to be taken simultaneously. The overall
optical setup can therefore be split into two separate components. Firstly, the
standard halogen-lamp-sourced microscope imaging setup, and secondly, the
confocal DF nano-spectroscopy arrangement utilising supercontinuum laser il-
lumination. The imaging setup was primarily used for nanosystem location and
NP dimer (tip-to-tip) microscopic alignment, while the nano-spectroscopy setup
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Chapter 3: Experimental Rig
was tailored specifically for dynamic (integration times τint ≤ 3 ms) characteri-
sation of plasmonic nanosystems with small optical interaction cross-sections.
3.4.1 Microscope Unit Imaging Optics
The custom built horizontal microscope unit described in Sec. (3.2.2) was popu-
lated with high quality optical components enabling diffraction-limited imaging
of all nanosystems investigated. The imaging optics are based on the traditional
Köhler illumination optical train, as shown in Fig. (3.10). By placing irises at
the appropriate positions within the optical train the range of incident angles
(effective NA) illuminating the nanosystem and the area of illumination on the
sample (focal) plane can be controlled under BF operation.
Figure 3.10: Schematic of the halogen lamp imaging optics incorporated into
the custom built microscope unit. All optics shown are broadband coated (λ0 =
300 nm-1100 nm) and achromatic over the visible spectrum. The red sections
of the objective represent reflective rather than refractive optics.
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Chapter 3: Experimental Rig
As all nanosystems investigated in this work are relatively strong scatter-
ers, as described in Sec. (2.3.2), DF microscopy provides a huge increase in
contrast over traditional BF methods. Because elastic scattering involves a non-
specular change in the propagation direction of the exciting light, the specularly
reflected background signal can be removed by implementing a suitable illumi-
nation and collection geometry. The technique suppresses the illuminating light
using a complementary circular light stop and aperture placed at appropriate
positions within the light-path to form a spatial filter, as shown in Fig. (3.10).
Light is directed onto the nanosystems only at high-angle by the reflective optics
contained within the outer ring of the DF objective. When the nanosystems are
situated on a flat, smooth substrate all the specularly reflected light is blocked
by the DF mirror and therefore only the light that has been scattered by the
nanosystem is measured. This DF optical setup results in isolated nanosystems
appearing as a diffraction-limited bright spots on a dark background. By us-
ing DF imaging the nanosystems on each AFM tip apex were visually aligned
with a high degree of accuracy and repeatability, as the background signal from
the specularly reflective AFM tip body is highly suppressed. General samples
of nanosystems could also be investigated, as shown in Fig. (3.11). Here Au
NPs of different shape with maximal dimension of approximately 100 nm were
deposited onto a glass substrate resulting in excellent background suppression
and high contrast DF imaging.
Figure 3.11: Dark-field image of Au NPs deposited on a glass substrate. The
NPs are of different shapes and maximal dimension of approximately 100 nm.
Obtained using the optical setup shown in Fig. (3.10).
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Chapter 3: Experimental Rig
All characteristic dimensions of the microscope unit and optics, e.g. lens tube
length and tube lens diameter, were designed to avoid the loss of peripheral rays
and thus maintain good imaging quality. The scattered light was split equally
between the CCD and spectrometer by a broadband coated, wedged BeamSplit-
ter (BS). A 50 µm diameter collection pinhole was used as a spatial filter so only
scattered light from the micro-volume of interest is focussed onto the entrance
slit of the spectrometer. The micro-volume itself is defined by the objective,
pinhole and optical train and can be approximately represented by a cylinder
centered on the sample plane with cap area AC, defined by a radius RC = 500
nm, and length LC = 800 nm. The width of the spectrometer entrance slit (50
µm) and focal length of the relevant focussing optic were chosen to optimise
light collection rather than spectral resolution as even higher-order LSPP reso-
nances are generally spectrally broad with wavelength linewidths ∆λlspp > 30
nm. No fibre optic cables were used throughout the setup to avoid additional
coupling loss, and interference fringing effects polluting the optical signal.
3.4.2 Supercontinuum Laser Dark-field Spectroscopy Optics
A 100 W halogen lamp can not provide the necessary intensity or confocal-
ity required for low-background dynamic dark-field spectroscopy, and hence a
broadband laser source is essential. Here a Fianium supercontinuum (450 nm -
1750 nm) laser was incorporated into the microscope unit whilst still allowing
for traditional halogen lamp imaging, as shown in Fig. (3.12). The supercontin-
uum laser light is produced by propagating pulses (τP ≈ 6 ps, centered at 1064
nm) created by a mode-locked Nd:YAG seed laser through a highly non-linear,
single-mode Photonic Crystal Fibre (PCF or holey PCF). The pulse is temporally
and spectrally broadened substantially via a cascade of nonlinear processes in-
cluding self-phase modulation, Raman scattering and four-wave mixing, thus
creating the broadband spectrum required for spectroscopy over the entire visi-
ble regime [116].
Upon exit from the PCF the unpolarised laser pulse has broadened in time
to τP ≈ 700 ps, has an average spectral power density greater than 2 mW nm−1
and an approximately top-hat spectral profile with < 6 dB spectral flatness. A
specially designed broadband optic mounted on the end of the PCF is used to
collimate the beam (φ ∼ O(3 mm)) over the majority of the output spectrum.
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Chapter 3: Experimental Rig
Figure 3.12: Schematic of the supercontinuum laser illumination optics incor-
porated into the custom built microscope unit. All optics shown are broadband
coated (λ0 = 300 nm-1100 nm) and achromatic over the visible spectrum. The
red sections of the objective represent reflective rather than refractive optics.
All acronyms are defined in main text.
Thus a highly collimated supercontinuum laser beam with an approximately
Gaussian intensity distribution is produced at the Fianium laser output. Due to
the effective point-source created by the fibre aperture the resulting collimated
‘white-light’ laser beam has a high degree of spatial coherence. In contrast, the
degree of temporal coherence is low, as the spectral bandwidth is large, and
the correlations between the electric fields corresponding to different pulses are
weak due to pulse-to-pulse variations in the spectral broadening mechanisms.
The laser beam was aligned onto the optical axis defined by the refracting
core (black) of the DF objective. To keep additional aberrations, e.g. chromatic
and spherical, to a minimum, all elements in the laser light path with optical
or diffractive power are precisely aligned in a co-axial manner with the optical
axis3 (±0.2 mm displacement, ±0.3′ angular deviation). The laser beam is first
expanded (DL = 10 mm) to completely fill the back aperture of the DF objective
3Lateral displacements (1 mm-3 mm) of the laser beam by certain optical components e.g.
BS 1 and BS 2 are taken into account.
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Chapter 3: Experimental Rig
core (DOC = 3.24 mm) and to reduce the power density incident on the optical
components to avoid damage. The expansion optics were placed close enough
to the objective so the curvature of the laser field (i.e. the laser spot focal position
along the optical axis) could be matched to that of the halogen lamp source by
a small translation of one of the expansion optics along the optical axis. The
collection pinhole is carefully aligned in 3D to be confocal with the focussed
laser spot.
The light is further attenuated by a factor of 10 using a reflective Neutral
Density (ND 1.0) filter to avoid excessively heating the nanosystem under in-
vestigation and prevent damage to the objective coatings. A (Thorlabs LPVIS)
nanoparticle Linear film Polariser (LP) was found to introduce considerably less
chromatic dispersion than other methods of polarisation (e.g. polarising BS cube
and Wollaston prism) and hence was used to linearly polarise the laser light. A
custom-built DF stop (DS = 2 mm) with smooth edges was cemented onto a
ND 0.1 filter and placed co-axially in the laser beam path to create a symmetric
ring of light that enters into the back aperture of the DF objective refractive core,
as shown in Fig. (3.13).
Figure 3.13: Top-down photo of the DF setup for the supercontinuum laser
with schematic overlay of halogen lamp (green), laser ( yellow ) and scattered
(purple) light paths.
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Chapter 3: Experimental Rig
A custom modified φ0.5′′ BS was cemented in position on a custom-built
glass (BK7) holder connected securely to tip-tilt and translation mounts. This
transparent BS mount allowed the laser and halogen lamp illumination to be
used simultaneously with minimal disruption to the halogen lamp light path.
After all optical elements the total average laser power entering the objective
was typically measured to be P0 ≈ 1 mW thus yielding an average sample
plane intensity I0 ∼ 109 W m−2 and a fluence of Ff lu ≈ 35 J m−2 per pulse. This
relatively large (broadband) intensity can cause considerable sample heating. In
the case of an isolated spherical NP the expected maximum initial temperature
rise can be found by straightforward energy considerations [117]
T =
CabsFf lu
VNPρAucAu
=
CabsFf lu
CNP
(3.1)
where ρAu = 19.32× 103 kg m−3 is the mass density of gold, cAu = 129 J kg−1 K−1
is the specific heat capacity of gold and CNP is the heat capacity of the NP. For
a gold NP in air with a = 50 nm the time required to reach the steady-state
regime (approximately 100 ps [118]) is significantly shorter than τP and hence
Eqn. (3.1) is valid. Appropriately implementing Eqn. (3.1) it is found that
∆T / 250 K. This is several hundred degrees less than the NP melting point of
∆T ≈ 1000 K [119], and hence minimal sample damage is expected. Further-
more the effective NP on the apex of a NP functionalised gold coated AFM tip is
expected to be less susceptible to optical heating as the relatively large AFM tip
body acts as an efficient heat-sink. The expected magnitude of optical heating
in plasmonically coupled NP dimer systems is described in Chapter (5).
The typical laser illumination and scattered light collection geometries used
for investigating a nanosystem are shown in Fig. (3.14a). The maximum col-
lection angle θc is defined by the diameter of the DF iris (1) aperture while the
range of illumination angles are described by θ1 and θ2 and are defined by the
diameter of the DF stop and objective NA respectively. The laser light incident
on the back aperture of the objective (λ0  (DOC ∧DS)) is an annular ring with
approximately planar wavefronts and a homogeneous intensity distribution due
to the beam expansion. Under these conditions an application of scalar diffrac-
tion theory can be used to calculate the ‘obscured’ Airy intensity distribution at
the focal (sample) plane of the diffraction-limited system [62].
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Chapter 3: Experimental Rig
Figure 3.14: Typical laser illumination and scattered light collection geometries,
including intensity profile of focussed laser spot. (a) Incident laser illumina-
tion angles θ1 = 34◦ and θ2 = 64◦. The maximum collection angle θc = 20◦.
Black arrows indicate polarisation of incident light. (b) Top-down view of ob-
scured Airy intensity distribution for λ0 = 700 nm with scaled simplified over-
lay of AFM tips with NP functionalised apices (white for maximum intensity).
Schematic of polarisation distribution shown in yellow (strong associated in-
tensity) and green (weak associated intensity). White dashed ring demarcates
collection area AC defined by objective, pinhole and optical train. Dark-red
dashed ring demarcates the laser illumination area AL defined by the FWHM
of the fundamental Airy peak.
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Chapter 3: Experimental Rig
An intensity plot with a scaled simplified overlay of the AFM tips with NP
functionalised apices is shown in Fig. (3.14b) for λ0 = 700 nm. The collection
micro-volume is set to be confocal with the laser illumination and the collection
area circumference is shown by the white dashed line. In this case the incident
light is set to have linear polarisation parallel to the AFM tip axes. In practice,
due to the high objective NA, a more rigorous vectorial diffraction treatment (c.f.
Mie solutions) shows the laser focal spot will have a polarisation distribution
coarsely shown by the overlaid orange and green arrows [120]. However, as
the dominant polarisation over the majority of the fundamental disk of the Airy
distribution is unchanged compared to the original incident polarisation, the
desired optical geometry is achieved with no further modification. Additionally,
it should be noted that in reality the focal spot is slightly elongated (≈ 10%)
along the direction of the original incident linear polarisation (not shown by the
scalar diffraction theory used in Fig. (3.14b)).
Chromatic Dispersion Compensation
An Amici Prism Pair (APP), as shown in Fig. (3.12), is used to compensate for
the chromatic dispersion introduced by all optical components present in the
laser light path before the light enters the objective. By rotating each prism
(circular wedge prism, 30′ wedge angle) individually and monitoring the lateral
chromatic separation of the output light focal spot on a CCD, the dispersion
already present in the laser light is cancelled out by introducing an opposite
dispersion of the same magnitude. A typical example of chromatic dispersion
compensation is shown in Fig. (3.15).
A low-level of chromatic dispersion is desirable for investigations of nano-
systems smaller than the typical illumination area (denoted AL for λ0 = 700
nm) situated on semi-infinite flat substrates. As can be seen from Figs. (3.15a)
and (3.15b) translation of a nanosystem from the white-light chromatic overlap
central regions will create artificial spectral shifts4. The situation is greatly im-
proved in Fig. (3.15c). Here the spectral variation due to chromatic dispersion
is negligible for a large range of non-central nanosystem positions under the
illumination.
4Artificial spectral shifts still result from nanosystem translation even if the lateral chromatic
separation is zero because of the wavelength dependence of the focal spot diameter. However
this will only be non-negligible at the extremities of the illumination area and hence is neglected.
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Chapter 3: Experimental Rig
Figure 3.15: Typical example of chromatic dispersion minimisation by high
precision optical alignment and Amici prism pair dispersion compensation.
Lateral chromatic separation of diffraction-limited supercontinuum laser focal
spot on an Au mirror for (a) Coarse optical element alignment (elements with
displacement  ±0.2 mm and angular deviation  ±0.3′ from optical axis)
and no APP compensation. (b) Optimised optical element alignment and no
APP compensation. (c) Optimised optical element alignment and APP com-
pensation.
A minimum of chromatic dispersion is also essential for systematic inves-
tigations of complex dynamic nanosystems (smaller than AL) mounted in free-
space, approximately at the centroid of the laser illumination, e.g. the NP func-
tionalised apices of the AFM tips. When the measured scattering spectra are
normalised against that from a target with an area much greater than AL, the
spectral response of such nanosystems are highly susceptible to experimental
artifacts introduced by the lateral chromatic separation of the illumination. In
the case depicted in Fig. (3.14b), the area of the overall nanosystem maximal
geometric cross-section under the laser illumination is significantly less than AL
and thus a substantial amount of the illumination power propagates away into
free-space without interacting with the nanosystem. If the lateral chromatic
separation was zero this would cause an approximately spectrally flat reduc-
tion in intensity of the measured response. However if the lateral chromatic
separation was significant (Fig. 3.15b)) then the reduction in intensity will no
longer be spectrally flat. Therefore, purely due to geometrical factors, the lat-
eral chromatic separation produces a significantly different spectral response
to that obtained from the normalisation target. This experimental artifact be-
comes greater with increasing chromatic dispersion and can readily produce
anomalous peaks and troughs of ±40% of the normalisation spectrum magni-
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Chapter 3: Experimental Rig
tude. The chromatic dispersion compensation techniques, demonstrated by Fig.
(3.15c), keep the spectral variation introduced by these experimental artifacts to
under ±2%.
Finally, additional custom made Chromatic Dispersion Compensation Plates
(CDCP’s) are introduced as shown in Fig. (3.12) to correct the chromatic dis-
persion introduced by the wedges of BS 1 and BS 2. The 30′ BS wedge angle
reduces unwanted interference effects and secondary reflections polluting the
laser illumination focal spot, however it chromatically disperses the scattered
light of interest. Both CDCP 1 and CDCP 2 (circular wedge prisms, 30′ wedge
angle) are appropriately orientated following a similar procedure used for the
APP alignment. This ensures the chromatic dispersion of the light is greatly
reduced, thus preventing false images and spectra from being recorded.
Minimising the Poisson Spot Background
The high-degree of spatial coherence of the laser illumination has the unwanted
side-effect of enabling diffraction around the smooth circular DF stop to become
prevalent. The diffracted light substantially diminishes the performance of the
dark-field laser setup as demonstrated in Fig. (3.16). In Fig. (3.16a) no laser light
is diffracted and hence the smooth glass substrate appears completely dark on
the CCD. However, in reality the laser light is strongly diffracted by an angle
−10◦ ≤ θD ≤ 10◦ in the plane defined by the light ray and the optical axis, as
shown in Fig. (3.16b). As every point on the circumference on the DF stop can
be considered to act as a secondary source of light, all the secondary spherical
wavelets constructively interfere on the optical axis thus forming a bright central
spot known as a Poisson (or Arago) spot (in the Fresnel regime, approximately
0-2 m after the DF stop) [121]. After a distance of approximately 6DS from the
DF stop the Poisson spot has the same intensity as the unperturbed laser light.
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Chapter 3: Experimental Rig
Figure 3.16: Schematic ray diagrams and CCD images showing how laser light
diffracted by the DF stop is not filtered out by the normal DF laser setup and
hence degrades the DF characterisation capability. (a) No diffraction at DF stop.
(b) With diffraction at DF stop. (c) With diffraction at DF stop and NP at the
co-axial focal position.
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Chapter 3: Experimental Rig
The diffracted light is specularly reflected off the glass substrate and passes
through the DF iris. The Poisson spot and subsidiary concentric fringes of illu-
mination formed at the focus of the lens are clearly shown by the CCD image in
Fig. (3.16b). The image has been over-exposed to make the subsidiary fringes
formed by off-axis constructive interference clearly visible and the supercon-
tinuum laser is not set to full power (full spectral bandwidth) hence the green
colour of illumination.
The ‘Poisson spot’ background can range in power depending on the experi-
mental parameters but is typically over an order of magnitude greater in power
than the light scattered by the nanosystem (determined by CBEMAXsca ) formed by
the dual AFM tips with NP functionalised apices. In the case of a NP with
approximately 100 nm maximal dimension the Poisson background swamps
the NP scattering signal, as demonstrated in Fig. (3.16c), thus producing poor
quality imaging and spectral measurements. For nanosystems situated on ho-
mogeneous flat substrates a background subtraction may be appropriate. How-
ever, for nanosystems mounted in environments with a complex geometry and
composition, e.g. an AFM tip apex NP, performing a suitable background sub-
traction is often not possible.
Several methods were tested to reduce the Poisson spot background, includ-
ing high-quality axicon illumination, and using relay optics to form images of
the DF stop and DF iris near the objective back focal plane. The optimum solu-
tion was found to involve physically moving the DF stop and a DF iris as close
as possible to the back aperture of the objective, as shown in Figs. (3.12) and
(3.13) and described schematically in Fig. (3.17a). By appropriate placement of
DF iris 1 all the parasitic light diffracted at low-angle is now blocked while DF
iris 2 blocks all the light diffracted at high-angle. The situation shown in Fig.
(3.16a) is thus recovered and high quality supercontinuum laser DF characteri-
sation is enabled, as shown by the CCD image in Fig. (3.17b).
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Chapter 3: Experimental Rig
Figure 3.17: Schematic ray diagrams and CCD images showing how laser light
diffracted by the DF stop is filtered out by the improved DF laser setup en-
abling high-quality DF characterisation capabilities. (a) Improved dark-field
configuration blocks the majority of the diffracted light. (b) Improved dark-
field configuration allows high-quality DF characterisation of all nanosystems.
Note the same colour code as that in Fig. (3.16) is used.
Supercontinuum Laser Setup Validation
To ensure the confocal supercontinuum laser DF nano-spectroscopy system was
working correctly, Au NPs nominally with a = 50 nm were drop-cast onto a
plasma cleaned microscope slide and were imaged and spectrally characterised
using the laser illumination. The images and scattering spectra obtained for
three different NP shapes are shown in Fig. (3.18). The vast majority of the NPs
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Chapter 3: Experimental Rig
were almost identical to NP 1, as expected for nominally spherical NPs (BBI
International). The spectra and images were taken with τint = 3 ms and τint =
0.2 ms respectively. This compares extremely favourably with the integration
times of τint > 1 s and τint ≈ 300 ms (with ×10 gain) for the spectra and image
acquisition respectively, required for the halogen lamp illumination.
To compare with analytical theory the presence of a glass substrate can be
taken into account in the Mie solutions by using an effective refractive index of
the dielectric medium ned = υnd + (1− υ)nglass where nglass = 1.51 is the refrac-
tive index of the glass substrate and υ = 0.58 is an appropriate weighting factor
found from [122]. Under this approximation the agreement of the scattering re-
sponse of NP 1 and that found from the Mie solution is excellent above λ0 = 500
nm for a NP with a = 49 nm. The discrepancy at short wavelength is commonly
found in the literature [123], and in this case can be mainly attributed to the spe-
cific dielectric function of Au used, the experimentally defined DF illumination
and collection geometries and the remaining influence of the glass substrate not
taken into account by the simple model used here.
Figure 3.18: Scattering spectra of Au NPs deposited on a glass substrate, with
insets showing corresponding NP images. Both spectra and images were taken
under supercontinuum laser illumination. RHS image shows corresponding
halogen lamp image of NPs. Black curve is the theoretical scattering response
of a 98 nm Au NP calculated from the Mie solutions.
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Chapter 3: Experimental Rig
3.5 Summary
A purpose-built experimental rig, described in Sec. (3.2), was constructed to
mechanically enable the creation of a relatively large (a ≈ 150 nm) NP dimer
system with dynamically controllable separation from d ≈ 500 nm down to full
conductive contact (GDC ≥ G0). An electronic measurement system, described
in Sec. (3.3), was developed and implemented to facilitate the required NP
dimer alignment by characterisation of the harmonic current flow across the ca-
pacitor formed by the dual, AFM cantilever-tip, composite system. Facility for
dimer separation calibration and conductance measurements is provided by the
DC conductance measurement electronics. Finally, the optical setup, described
in Sec. (3.4), enabled simultaneous DF imaging and optical scattering character-
isation, using standard halogen lamp illumination and a supercontinuum laser
source respectively. Methods were developed to compensate chromatic disper-
sion and reduce contamination by diffracted light. The experimental apparatus
enables electronic, and far-field optical, characterisation of the LSPP response of
NP dimers with sub-nanometre separation.
86
Chapter 4
Dynamic Positioning of Two
Nanosystems
4.1 Introduction
Before the LSPP response of a NP dimer in the quantum regime can be elec-
tronically and optically investigated the NP dimer nano-system first needs to
be aligned and brought towards sub-nanometre separation in a controlled dy-
namic manner. The mechanical and electronic setups described in Secs. (3.2)
and (3.3) enable a compact approach for creating an optically accessible NP dimer
with dynamically controllable separation from d0 ≈ 400 nm to full conductive
contact, without the need for top-down processing or the limited optical ac-
cess of a commercial AFM or STM. As introduced in Chapter (3), the technique
is based on the 3D nanoscale alignment of two conducting NP functionalised
AFM probes in a ‘tip-to-tip’ configuration. To achieve a large dynamic range of
alignment a novel Electrostatic Force Microscopy (EFM) technique was devel-
oped that utilises an alternating potential applied across the conducting AFM
tips to create an oscillating long-range electrostatic force. By exploiting the non-
linear electrical parametric response of the electromechanically coupled AFM
tip system, the NP dimer formed by the two AFM tip apices is aligned with
nanometre-scale precision. This chapter describes the theoretical foundations of
the technique developed and its experimental verification.
87
Chapter 4: Dynamic Positioning of Two Nanosystems
4.2 Theoretical Modelling
The oscillating electromechnically coupled dual cantilever-tip-NP system is
shown in Fig. (4.1). First the governing equations of motion for the system are
presented. These are then solved under a toy model (strong approximations) to
crucially show how parametric mixing creates background-free harmonic sig-
nals that are directly related to the magnitude and phase of a tips oscillation.
Such harmonic signals hence allow axial tip-tip (NP dimer) alignment to be per-
formed. Following from this, a more rigorous numerically implemented model
developed is described in detail. In this model the inter-tip forces and system
capacitance is determined to good accuracy, thus allowing effective comparison
with experiment.
The coordinate system used for modelling is shown in Fig. (4.1a). The
frequency range of interest is around the fundamental flexural mode of the can-
tilevers and therefore the tip system is modelled as a pair of coupled point-mass
harmonic oscillators. The tips are coupled through the superposition of the z-
components of the driving electrostatic attractive force FzEL, and the combined
short-range (< 2 nm) Van der Waals and repulsive tip-tip interaction forces FzTT.
Figure 4.1: Coordinate system used in theoretical analysis of electromechani-
cally coupled dual AFM tip system. (a) Coordinate system used in the analysis.
(b) Over-exposed DF image of 50 nm Au coated AFM tips with NP function-
alised apices via EBD.
The coupled equations of motion for tips i = (1, 2) with apex positions zi
and apex separation d(t) = z1(t)− z2(t) are
88
Chapter 4: Dynamic Positioning of Two Nanosystems
mi
d2zi
dt2
+ βzi
dzi
dt
+ kzi (zi − z0i)= ± (FzEL + FzTT) (4.1)
where mi is the effective mass of the cantilever-tip system, βzi = β
z
0i + β
z
TT is
the z-component of the non-linear damping coefficient, kzi = k
z
0i + k
z
TT is the z-
component of the cantilever spring constant and z0i is the tip apex position when
FzEL = F
z
TT = 0. The sign of the RHS force term is positive for tip 1 and negative
for tip 2. Tip parameters mi = kz0i/ω
2
0i and β
z
0i = k
z
0i/(ω0iQ0i) are derived from
the fundamental cantilever spring constant1 kz0i and experimentally measured
values of the appropriate cantilever resonance frequencies ω0i and Q-factors
Q0i. The βzTT and k
z
TT terms are the air ‘squeeze’ viscous and elastic damping
contributions respectively. The fixed parameters of a typical AFM tip are found
to be
ω0i = 2pi(13) k rad s−1 (4.2a)
Q0i = 90 (4.2b)
kz0i = 0.2 Nm
−1 (4.2c)
βz0i = 2.7× 10−8 kg s−1 (4.2d)
mi = 3× 10−11 kg (4.2e)
4.2.1 Parametric Mixing: Toy Model
To understand the main principles of the NP dimer alignment technique a basic
toy model was developed to elucidate the underlying physical processes. To
allow a simple analytical analysis the AFM tip system is approximated as a
parallel plate capacitor with no fringe fields. Also, d is assumed to always be in
the long-range regime where FzTT = β
z
TT = k
z
TT = 0. In this axially symmetric
system the attractive force (at a defined potential V) due to the capacitance is
given by
1It should be noted the value used for the fundamental cantilever spring constant is calcu-
lated theoretically from simple beam theory and hence can be ≤ 5× different from the true
value [124].
89
Chapter 4: Dynamic Positioning of Two Nanosystems
FzEL(V, d) =
∂U
∂d
=
1
2
∂C
∂d
V2(t) =
−ε0AovV2(t)
2d2
(4.3)
where Aov is the area of cantilever overlap and U is the electrostatic potential
energy stored in the capacitor. By applying
V(t) = V0 cos(ωSt) (4.4)
at a signal frequency ωS ≈ ω01/2, Eqn. (4.3) shows that FzEL and hence C(t)
will primarily oscillate resonantly at ωP = 2ωS ≈ ω01 where ωP is the pump
frequency. Driving at ω01/2 allows only the direct ω01 resonance of the tip
system to be efficiently excited rather than the mechanical parametric resonance
at 2ω01. This is critical to avoid driving the tip system response into a parametric
instability domain [125]. Assuming z02 = 0 and tip 2 is stationary, the tip apex
separation at zero applied force d0 = z01 − z02 = z01 and Eqns. (4.1) reduce to a
single equation of motion
m1
d2z1
dt2
+ βz01
dz1
dt
+ kz01 (z1 − d0) = FzEL(z1, t) (4.5)
Substituting Eqn. (4.3) into Eqn. (4.5) and driving yields
m1
d2z1
dt2
+ βz01
dz1
dt
+ kz01 (z1 − d0) =
(
−ε0AovV20
4(d0 +Λ)2
)
(1+ cos(ωPt)) (4.6)
where Λ = z1 − d0 describes the oscillation of tip 1. Assuming |Λ|  d0 Eqn.
(4.6) can be taken to 1st order, and to good approximation
m1
d2z1
dt2
+ βz01
dz1
dt
+ kze1 (z1 − d0) '
(
−ε0AovV20
4d20
)
(1+ cos(ωPt)) (4.7)
where kze1 is the z-component of the effective spring constant of tip 1 given by
90
Chapter 4: Dynamic Positioning of Two Nanosystems
kze1 = k
z
01 −
(
ε0AovV20
2d30
)
(1+ cos(ωPt)) (4.8)
The simplified Eqn. (4.7) can be recast into an ODE that has the same homoge-
neous form as the Mathieu equation [126]. However, as the mechanical spring
constant parameter kze1 oscillates at ωP ≈ ω01 rather than at 2ω01, the oscil-
lation of kze1 does not drive the system into a parametric resonance and thus
does not affect the overall system behaviour significantly. Therefore only the
time-averaged response of kze1, i.e. k
z′
e1 = k
z
01 − ε0AovV20 /2d30 is considered. The
principle of superposition can now be used to solve the linear differential Eqn.
(4.7) thus yielding the steady-state solution for z1
z1(t) ≈ d0 −
∣∣∣zo f f1 ∣∣∣− zm1 cos(ωPt + ϕ1) (4.9)
where
∣∣∣zo f f1 ∣∣∣, zm1 and ϕ1 are given by
∣∣∣zo f f1 ∣∣∣ ≈ ε0AovV204d20kz′e1 (4.10)
zm1 ≈ ε0AovV
2
0
4d20
((
kz′e1 −m1ω2P
)2
+
(
βz01ωP
)2)1/2 (4.11)
ϕ1 ≈
(
tan−1
(
kz′e1 −m1ω2P
βz01ωP
))
− pi
2
(4.12)
Here
∣∣∣zo f f1 ∣∣∣ is the magnitude of the additional offset in tip 1 position created
because FzEL ∝ V
2, zm1 is the amplitude of tip 1 oscillation and ϕ1 is the phase
of tip 1 oscillation with respect to the driving potential. The capacitance of the
system can hence be obtained as
C(t) ≈ ε0Aov
z1
+ Cbk =
ε0Aov
d0 −
∣∣∣zo f f1 ∣∣∣− zm1 cos(ωPt + ϕ1) + Cbk (4.13)
91
Chapter 4: Dynamic Positioning of Two Nanosystems
where Cbk is the constant background capacitance contribution from the oppos-
ing composite control rod, chip clip, AFM chip system. As
∣∣∣zo f f1 ∣∣∣, zm1  d0 Eqn.
(4.13) can be taken to 1st order to good approximation
C(t) ≈ C0
1+
∣∣∣zo f f1 ∣∣∣
d0
+
zm1
d0
cos(ωPt + ϕ1)
+ Cbk (4.14)
where C0 = ε0Aov/d0. The 1st order current flow across the oscillating capaci-
tance (mechanical varactor) can now determined2 by
I(t) =
dQ
dt
=
d(CV)
dt
= C
dV
dt
+V
dC
dt
(4.15)
Substituting Eqns. (4.4) and (4.14) into Eqn. (4.15) yields
IAC = I(t) ≈− C0V0ωS
1+
∣∣∣zo f f1 ∣∣∣
d0
+
Cbk
C0
 sin (ωSt)
− C0V0zm1
2d0
(ωP −ωS) sin ((ωP −ωS) t + ϕ1)
− C0V0zm1
2d0
(ωP +ωS) sin ((ωP +ωS) t + ϕ1)
(4.16)
Here the difference and sum frequencies are (ωP −ωS) = ωS and (ωP +ωS) =
3ωS respectively. The non-zero complex Fourier coefficients c˜1 and c˜3 of the
current flow, and hence I˜ωS and I˜3ωS at ωS and 3ωS respectively, are extracted
by
c˜1 =
I˜ωS
2
=
1
TωS
∫ TωS
2
−TωS
2
I(t)e−iωSt dt (4.17a)
c˜3 =
I˜3ωS
2
=
1
T3ωS
∫ T3ωS
2
−T3ωS
2
I(t)e−i3ωSt dt (4.17b)
2The capacitance is treated as a time-dependent linear function of V therefore Q(t) =
C(t)V(t), rather than a non-linear function of V, i.e. Q =
∫ V
0 C(V) dV. For more detail the
interested reader is referred to [127].
92
Chapter 4: Dynamic Positioning of Two Nanosystems
where TωS = 2pi/ωS and T3ωS = 2pi/3ωS. Substituting Eqn. (4.16) into Eqns.
(4.17) yields
I˜ωS ≈ C0V0ωS
1+
∣∣∣zo f f1 ∣∣∣
d0
+
zm1
2d0
eiϕ1 +
Cbk
C0
 ei pi2 (4.18a)
I˜3ωS ≈ 3C0V0ωSzm1
2d0
ei(ϕ1+
pi
2 ) (4.18b)
The oscillating AFM tip capacitance at ωP produces electric parametric mix-
ing. This results in power transfer from the fundamental to sum and difference
frequency components that are dependent on zm1. The current component at
(ωP−ωS) = ωS creates the third term on the RHS of Eqn. (4.18a). The constant
pi/2 phase factor in both Eqns. (4.18) is due to the current leading the voltage
by 90◦ after traversing the tip-tip capacitor. Critically, the current component
at (ωP + ωS) = 3ωS (Eqn. (4.18b)) provides a background-free signal directly
related to the magnitude and phase of the tip oscillation. Now if tip 2 is scanned
in the x or y directions, i.e. moved off-axis, FzEL decreases (zm1 decreases) along
with C (C0 decreases) [128]. This therefore allows the AFM tips (NP dimer) to be
precisely aligned in 3D by tracking the current magnitude at 3ωS using lock-in
detection.
4.2.2 Numerically Implemented Alignment Simulation
For effective comparison with experiment the coupled non-linear 2nd order
ODEs (4.1) are solved numerically in IGOR Pro using a standard fifth-order
Runge-Kutta algorithm [129]. The majority of the expressions for the parame-
ters are obtained from analytical theory, and substituted into Eqns. (4.1), thus
allowing accurate simulation of the electromechanically coupled AFM tip sys-
tem.
The position dependent damping and spring constant terms βzTT and k
z
TT
were modelled by the air squeeze damping between two parallel plates of area
chosen by experimental fit. Here the viscous and elastic damping contributions
are determined by solving the Reynolds equation for a compressible gas. The
93
Chapter 4: Dynamic Positioning of Two Nanosystems
viscous component βzTT is created by air being squeezed out of (or pushed into)
the space between the AFM tips and is modelled by [130]
βzTT ≈
3µa AFl2
4d3
(4.19)
where µa = 18.4× 10−6 Pa.s is the coefficient of viscosity of air, AF is the parallel
plate area determined by an experimental fit and l is the side length of the
parallel plate. The elastic component kzTT results from the compression of the
air between the AFM tips and is modelled by [130]
kzTT ≈
4Pa AFσ2s
15d
(4.20)
where Pa = 100× 103 Pa is the ambient air pressure and σs ≈ 10 is the squeeze
number.
Exact analytical evaluation of FzEL and the total capacitance (C) for the AFM
tip system is not possible due to the complex geometry. However, using the
approximations implemented in [131] for an axially symmetric AFM tip, expres-
sions for FzEL and C are obtained. Here the AFM tips are each separated into
three parts; a cantilever, a conical body and a parabolic apex, as depicted in Fig.
(4.2). The cantilevers are treated as infinite parallel-plates in order to calculate
the capacitance per unit area, and hence the appropriate cantilever contributions
to FzEL and C are included. Both the cone and apex components are decomposed
into infinitesimal area elements (dA) that each contribute in the same manner as
an infinite dihedral capacitor with identical relative orientation. As both AFM
tips are good conductors the field lines are perpendicular to the surface at every
point and are well approximated by circular arcs over which the electric poten-
tial decays linearly. Therefore the magnitude of the electrostatic field at tip 2 is
approximately given by
E(x, y, z) ≈ V
larc(x, y, z)
(4.21)
where larc is the length of the circular arc connecting two identical points on tip
1 and tip 2 as shown in Fig. (4.2).
94
Chapter 4: Dynamic Positioning of Two Nanosystems
Figure 4.2: Schematic of axially symmetric tip-tip model used for theoretical
modelling. Examples of approximate electric field lines larc are shown in blue.
Inter-apex distance d is exaggerated for clarity.
Using Eqn. (4.21) expressions for FzEL and C are found via [67]
FzEL(V, d) =
ε0
2
∫∫
A
E2dA · zˆ
≈ ε0V
2
2
∫∫
A
1
l2arc(x, y, z)
dA · zˆ
(4.22)
C = Q/V ≈ ε0
∫∫
A
1
larc(x, y, z)
|dA| (4.23)
where zˆ is a unit vector in the z-direction.
This method is found to yield good agreement (less than 5% error) with the
exact analytical solution for a simple two sphere system. As larc is much greater
than the dimension of each infintesimal surface element, one could expect treat-
95
Chapter 4: Dynamic Positioning of Two Nanosystems
ing each element as a infinite dihedral capacitor of separation larc to be a poor
approximation, as the fringing fields will strongly influence the system. How-
ever, as only the global system quantities FzEL and C are required, it is found that
the vast majority of fringing fields cancel out and hence the approximation does
not introduce significant errors. The expressions derived are hence only valid if
d is not larger than the characteristic physical dimension of the associated AFM
tip component, unless additional corrections are implemented. These expres-
sions are functions of cone height (h1 = h2 = h), apex radius (R1 = R2 = R),
full cone-angle (θtip 1 = θtip 2 = θtip) and Aov. The full expressions derived for
FzEL using Eqn. (4.22) are shown below
FzEL(V, d) = F
z
L + F
z
C + F
z
A (4.24)
where FzL, F
z
C and F
z
A are the contributions of the z-component of force from the
AFM cantilevers, AFM tip cones and AFM tip apices respectively. The individ-
ual contributions are given by
FzL ≈ −
ε0AovV2
2(2h + d)2
(4.25a)
FzC ≈−
(
ε0V2
pi
)
sin(θtip/2)×
(
ln
(
d− δ+ 2h
d + δ
)
− sin (θtip/2) 2 (h− δ)d− δ+ 2h
(
d− δ
d + δ
)) (4.25b)
FzA ≈
−piε0V2 fz
4(1+ fA(d/2R)2)
(
R + d4
R− d
)2
×
 R− d
d
2
(
1+ dR tan
2(θtip/2)
)
+ 2 ln( 2d
d + R + (R− d) cos (θtip)
)
(4.25c)
96
Chapter 4: Dynamic Positioning of Two Nanosystems
where δ defines the height of the parabolic apex and is given by
δ =
R
tan2(θtip/2)
(4.26)
and fz ≈ 0.7 is a correction factor, calculated separately, that is introduced to
take into account that only the z-component of the force is required. Finally, fA
forms part of the empirical correction factor 1/(1 + fA(d/2R)2) that does not
vary the short-range behaviour but gives FzA the correct limiting behaviour for
d R [131]
fA =
ln
(
csc(θtip/2)
)
(1− sin(θtip/2))(3+ sin(θtip/2)) (4.27)
This correction factor is required as the condition d < R is often not satisfied as
coaxial tip-to-tip (NP dimer) alignment is initially carried out in the long-range
(d > 200 nm) regime. Under this formalism, Eqns. (4.25) are valid for d / 10
µm and are hence appropriate for all NP dimer separations considered.
In a similar fashion to FzEL the total capacitance C can be decomposed into
four parallel capacitance contributions describing the total system
C = CL + CC + CA + Cbk (4.28)
where CL is the capacitance associated with the cantilevers, CC is the capaci-
tance associated with the conical sections of the AFM tips, CA is the capacitance
associated with the AFM tip apices and Cbk is the capacitance contribution from
the opposing composite control rod, chip clip, AFM chip system. Using Eqn.
(4.23) expressions for CL and CC are obtained
CL ≈ ε0Aov2h + d + CF (4.29a)
CC ≈ ε0
(
2(h− δ)
cos(θtip/2)
+ (d− δ) tan(θtip/2) ln
(
d− δ+ 2h
d + δ
))
(4.29b)
97
Chapter 4: Dynamic Positioning of Two Nanosystems
where CF is the additional (fringe) capacitance associated with the remaining
non-overlapping sections of the cantilevers, calculated using COMSOL. The z-
component of force associated with this capacitance is assumed to be negligible
(FzF = 0). The approximate d dependence of CF is also found numerically by
COMSOL simulation. The expression for CA is calculated from the corrected
force expression FzA. In general the capacitance of a system is related to the
electrostatic force by
F(V, x, y, z) =
1
2
∇C(x, y, z)V2 (4.30)
Due to the assumed axial symmetry of the AFM tip apex sub-system Eqn. (4.30)
simplifies and an expression for CA is found from the corrected apex force con-
tribution FzA
CA(d) = CA(∞) +
∫ d
∞
2FzA(V, z)
V2
dz =
∫ d
∞
2FzA(V, z)
V2
dz (4.31)
Analytical evaluation of Eqn. (4.31) is awkward and hence the expression is de-
termined numerically using IGOR for fixed values of R and θtip. The numerical
output is fitted to the functional form described in [132] and a typical expression
determined for R = 70 nm and θtip = 45◦ is
CA(d) ≈
(
6× 10−19
)
[F] ln
(
1+
10−7[m]
d
)
(4.32)
Finally the relatively large constant background capacitance contribution Cbk ≈
0.1 pF is measured experimentally. The force associated with this capacitance
has negligible z-component (Fzbk ≈ 0) and hence does not induce significant
cantilever oscillation.
To approximate FzEL and C when the AFM tips are moved off-axis, i.e. when
scanning a Gx × Gy µm2 alignment grid centered on the z-axis at a given d0,
the initial tip separation dx,y0 is used instead of the on-axis AFM tip separation,
where
dx,y0 ≈
(
(d0 + 2R)
2 +
(gx
2
)2
+
(gy
2
)2)1/2
− 2R (4.33)
98
Chapter 4: Dynamic Positioning of Two Nanosystems
where dx,y0 is the distance of closest approach for a given tip 2 position in the
alignment grid, and gx and gy are the positions of tip 2 in the x and y directions
respectively. The range of grid positions for tip 2 are −Gx/2 ≤ gx ≤ Gx/2
and −Gy/2 ≤ gy ≤ Gy/2 in x and y respectively. For the majority of the
experimental values of dx,y0 , the electrostatic driving force F
z
EL ≈ FC, so the FC
component dominates the tip oscillation response. In all cases dx,y0 is much
less than the characteristic dimensions (∼ O(10 µm)) of the conical section.
Therefore, it can be assumed that for a typical Gx = Gy = 1 µm alignment grid,
the maximum geometrically induced reduction in FzEL and C due to tip 2 being
G = ((Gx/2)2 + (Gy/2)2)1/2 ≈ 0.7 µm off-axis is relatively small.
The short-range (< 2 nm), non-capacitative Van der Waals and repulsive tip-
tip interaction forces FzTT, are modelled by the Lennard-Jones force between two
spherical NPs [133]
FzTT(d) = F
z
VdW + HR¯a
6
0/180d
8 (4.34)
where R¯ = R1R2/(R1+R2) = R/2 for identical AFM tips, H is the material
dependent Hamaker constant (H ≈ 25× 10−20 J for gold) and a0 is the material
dependent inter-atomic distance (a0 ≈ 0.2 nm for gold). An expression for
the z-component of the Van der Waals force FzVdW between two AFM tips is
approximated from [134], where the tip apex geometry is taken into account
FzVdW(d) =
HR¯2
(
1− sin(θtip/2)
) (
R¯ sin(θtip/2)− d sin(θtip/2)− R¯− d
)
6d2
(
R¯ + d− R¯ sin(θtip/2)
)2
− H tan(θtip/2)
(
d sin(θtip/2) + R¯ sin(θtip/2) + R¯ cos(θtip)
)
6 cos(θtip/2)
(
R¯ + d− R¯ sin(θtip/2)
)2
(4.35)
Alignment is implemented in the non-contact regime hence contact dynamics,
such as adhesion forces and surface deformation are not treated.
For comparison with experiment all relevant expressions are substituted into
Eqns. (4.1) and the coupled differential equations are solved numerically for the
steady-state behaviour of the system for a given value of dx,y0 . It should be noted
that the additional modifications to the amplitude and phase of V and VAC due
to the stray capacity Cs, and the majority of experimental equipment, were also
modelled by appropriate circuit theory within the simulation. The numerical
99
Chapter 4: Dynamic Positioning of Two Nanosystems
solution found for the steady-state d(t) is used to calculate values for C(t) and
dC/dt that are then substituted into Eqn. (4.15) to give the current flow I(t).
The harmonic component at 3ωS is extracted using a numerical approximation
to the lock-in amplifier by
I˜3ωS ≈ 2
τTC
∫ τTC
0
I(t)e−i3ωSt dt (4.36a)
V˜3ωSL ≈
(
ALI AT√
2
)
I˜3ωS (4.36b)
where V˜3ωSL is the 3ωS voltage signal measured by the lock-in amplifier, τTC =
100 ms is the time-constant used for all lock-in measurements, ALI is the lock-
in amplifier gain, AT = 108 VA−1 is the transimpedance amplifier gain, and
the factor of 1/
√
2 converts peak-to-peak to RMS voltage. Using this combined
analytical and numerical approach allowed | I˜3ωS | to be accurately and efficiently
calculated with only one free parameter AF.
4.3 Experimental Apparatus
The entire apparatus is placed on an active anti-vibration platform and con-
tained within a Faraday cage. The tips are initially aligned to within ±400nm
in x, y, z under a colinear custom-built microscope with ×100, NA=0.9 objec-
tive. The tips used here are 50 nm Au coated nano-indentation AFM probes
with a ‘neck and ball’ apex geometry. As shown in Fig. (4.3), the AFM tip sys-
tem is driven by an amplified signal generator and the current flow across the
AFM tips is passed through a 108 gain transimpedance amplifier (VA). A BPF
centered at 3ωS prevents the current component at ωS overloading the lock-in.
More detail on the experimental rig can be found in Chapter (3).
100
Chapter 4: Dynamic Positioning of Two Nanosystems
Figure 4.3: Simplified schematic of experimental apparatus with Signal Ground
(SG), BPF and VA shown.
The AFM tip system resonance behavior is characterised by performing a
frequency sweep whilst measuring | I˜3ωS |. A typical response curve at dx,y0 =
d0 ≈ 340 nm and V0 ≈ 10 V, measured around half the resonance frequency
of AFM tip 1 ( f01/2), is shown in Fig. (4.4). In this case AFM tip 2 is non-
resonant and hence is almost stationary. The Fano-like lineshape measured is
significantly different to the expected Lorentzian lineshape. This is due to inter-
ference between I3ωS and the relatively large background signal IHD primarily
created by the Harmonic Distortion (HD) present in the output of the electronic
amplifiers. The effect of the HD background on the signal can be understood
by considering the superposition of the I3ωS = <[ I˜3ωS ] and IHD = <[ I˜HD] wave-
forms
<
[
I˜3ωSC
]
= I3ωS + IHD = | I˜3ωS | sin(3ωSt + ϕ3ωS) + | I˜HD| sin (3ωSt + ϕHD)
=
∣∣∣ I˜3ωSC ∣∣∣ sin (3ωSt + ψ)
(4.37)
where <[ I˜3ωSC ] = I3ωSC is the corrected I3ωS signal, ψ is the phase of the I3ωSC
signal, ϕ3ωS is the phase of the I3ωS signal and ϕHD is the phase of the IHD
signal. Here all phases are relative to the driving potential.
101
Chapter 4: Dynamic Positioning of Two Nanosystems
Figure 4.4: Experimentally measured | I˜3ωS | (solid markers) vs. numerical simu-
lation with the same parameters and harmonic distortion (HD) included (solid
line) and absent (dashed line). Here d0 ≈ 340 nm and V0 ≈ 10 V. The frequency
used for alignment scans is fS.
As | I˜HD|  | I˜3ωS | it is found that
| I˜3ωSC | ≈ | I˜HD|+ | I˜3ωS | cos (Θ) (4.38a)
ψ = tan−1
(
| I˜3ωS | sin(ϕ3ωS) + | I˜HD| sin(ϕHD)
| I˜3ωS | cos(ϕ3ωS) + | I˜HD| cos(ϕHD)
)
(4.38b)
where Θ =
(
ϕ3ωS − ϕHD
)
. As AFM tip 1 is driven through resonance ∆ (Θ) =
−180◦ hence Eqn. (4.38a) shows that the corrected signal | I˜3ωSC | will demon-
strate a Fano-like lineshape. Therefore | I˜3ωSC | can be directly compared to the
experimentally measured | I˜3ωS |. The HD background was characterised using
a signal analyser and added to the theoretical model by modifying Eqn. (4.4) to
include a V3ωS term
V(t) = V0 cos (ωSt) +V
3ωS
0 cos
(
3ωSt + ϕ3ωS
)
(4.39)
where V3ωS0 ≈ 1 mV is the amplitude of the HD (for V0 = 10 V) and ϕ3ωS ≈ −pi4
is the associated HD phase relative to the fundamental. Inclusion of the HD
background recovers the correct resonance lineshape and yields the correct
102
Chapter 4: Dynamic Positioning of Two Nanosystems
| I˜3ωS | signal offset, as shown in Fig. (4.4). The resonant frequency of tip 1
is observed and the AFM tips then driven at the frequency that maximises sen-
sitivity, in this case fS = f01/2 = 5.75 kHz.
4.4 Alignment Results
For alignment, AFM tip 2 is scanned over a 1× 1 µm2 grid whilst measuring
| I˜3ωS | at each position. Tip 2 is then moved to the optimal alignment position,
the tip apex separation d0 is reduced, and successive alignment scans are taken.
Figure 4.5: Typical alignment scan sequence results for d0 = 460 nm→ 220 nm
with V0 ≈ 10 V. Evolution of the FWHM, and uncertainty in position, of the
alignment grid minimum are also shown. (a) Change in measured | I˜3ωS | with
respect to HD background vs. position of AFM tip 2 in a 1× 1 µm2 grid for
(i) d0 ≈ 460 nm, (ii) d0 ≈ 340 nm and (iii) d0 ≈ 220 nm, all with V0 ≈ 10 V.
(b) FWHM of ∆| I˜3ωS | minimum (from alignment scans) vs. d0. (c) Uncertainty
and magnitude associated with ∆| I˜3ωS | centroid (from alignment scans) vs. d0.
Solid and dotted lines guide the eye, dashed vertical line shows where V0 was
reduced to 0.78V0 to avoid long-range StoC.
103
Chapter 4: Dynamic Positioning of Two Nanosystems
At the same time V0 is reduced to prevent long-range ‘Snap to Contact’ (StoC)
phenomena where ∆FzEL exceeds the change in cantilever restoring force of ei-
ther AFM tip 1 or 2. A typical sequence of alignment scans at d0 ≈ 460 nm, d0 ≈
340 nm and d0 ≈ 220 nm, all with V0 ≈ 10 V, are shown in Fig. (4.5a). Here, the
change in signal with respect to the HD background (∆| I˜3ωS | = | I˜3ωS | − | I˜HD|),
is plotted against tip 2 position. The local minimum (rather than maximum)
observed is again due to the interference between the I3ωS and IHD signals. This
can be understood by further examining Eqn. (4.38a)
∆
∣∣∣ I˜3ωSC ∣∣∣ = | I˜3ωSC | − | I˜HD| ≈ ∣∣∣ I˜3ωS ∣∣∣ cos (Θ) (4.40)
where ∆| I˜3ωSC | is the corrected change in signal with respect to the HD back-
ground. In all cases AFM tip 1 is always driven slightly above resonance because
f01 > f e01 where f
e
01 is the effective resonance frequency due to the electrostatic
force and elastic damping. Therefore using Eqn. (4.38a) it can be shown that
Θ > 90◦ and has a typical value of Θ ≈ 3pi5 . Substituting Θ ≈ 3pi5 into Eqn. (4.40)
shows that the on-axis ∆| I˜3ωSC | ≈ −13 | I˜3ωS | ≈ −0.8 pA (from Fig. (4.4)). This
agrees well with the experimentally measured value, shown in Fig. (4.5a(ii)).
All parameters plotted in Figs. (4.5), and used for experimental analysis, were
extracted from the experimental measurements by fitting an inverted, 2D ellip-
tical Gaussian function to the alignment grid data, assuming zero correlation
between each data point.
As the AFM tips approach, the magnitude of the signal minimum increases
while the associated FWHM and the uncertainty in centroid location decreases
(Figs. (4.5b) and (4.5c)). By thus ‘homing-in’ on the centroid position to d0 <
Ri ≈ 130 nm the FWHM tends to 2R as expected. The numerical simulation
FWHM results are in good agreement with experiment. This is expected as
the required corrections in C and FzEL due to the tips moving off-axis are small.
When d0 < R the uncertainty in centroid position is less than ±3 nm, i.e. 40×
less than R, therefore the NP dimer is axially aligned to a high degree of ac-
curacy. Once in this regime the AC driving potential is set to V0 = 0 and d0
is further reduced, with nanometre precision, at a rate of 2 nm s−1, using the
piezoelectric actuator. An absolute distance scale is obtained by switching to
the DC conductance measurement electronics described in Sec. (3.3.2). A small
104
Chapter 4: Dynamic Positioning of Two Nanosystems
(VDC ∼ O(50 µV)) constant potential is applied across the AFM tips and the
tip separation d0 is controllably reduced whilst observing the current rise (IDC)
upon (mechanically reversible) atomic-scale kissing contact.
A flow diagram showing the total typical alignment procedure is shown in
Fig. (4.6). The frequency sweep and alignment procedures are almost entirely
automated using computer code written in IGOR Pro to send VISA commands
via GPIB to the experimental equipment3. It is found that halting the alignment
procedure when setting V0 ≈ 7 V yields |∆| I˜3ωS || ≈ 1 pA at the grid centroid
typically leaves an inter-tip separation of d ≤ 100 nm.
Figure 4.6: Flow diagram of the total AFM tip (NP dimer) alignment procedure.
The text in blue labels an action that requires an initial human input.
3The AFM tip (NP dimer) alignment software produced, including Graphical User Interface
(GUI), can be obtained from the author on request.
105
Chapter 4: Dynamic Positioning of Two Nanosystems
4.5 Conclusions
The parametric response of an electromechanically-coupled NP functionalised
AFM tip system is utilised as a long-range sensor of 3D AFM tip-tip (NP dimer)
alignment. The technique is experimentally and theoretically demonstrated for
dynamic separation control of metallic NP dimers on the nanoscale. At all sep-
arations the system maintains the facility for simultaneous local optical and
electronic measurement. This enables a versatile experimental apparatus to per-
form time-resolved (ms-µs) dark-field nano-spectroscopy in different plasmonic
interaction regimes. By simultaneously measuring the optical response and
electrical conductivity at d0 < 1 nm, the non-local effect of quantum transport
on the LSPP mediated response of metallic NP dimers is investigated, as de-
scribed in Chapter (5). It is expected that the techniques developed will also
find use in nonlinear and active plasmonics, surface enhanced spectroscopies,
THz quantum transport research and studies of strong plasmon-molecular ex-
citon coupling.
106
Chapter 5
Quantum Tunnelling Plasmonics
5.1 Introduction
The electron correlations supported by quantum tunnelling across the NGJs as-
sociated with relatively large (a > 20 nm) NP dimer systems require a new
description of non-local transport. As introduced by the open questions in
Chapter (2), an improved understanding of the effect of non-local quantum
transport on plasmonic coupling is crucial for a large variety of applications.
Many emerging nanophotonic technologies depend on the careful control of this
plasmonic coupling, including optical nanoantennas for high-sensitivity chem-
ical and biological sensors [135], nanoscale control of active devices [136–138],
improved photovoltaic devices [10] and nanoscale optical trapping [139]. In
general sub-wavelength metallic structures can concentrate light into nanoscale
dimensions well below the diffraction limit [140,141] due to reduced field pene-
tration through a dense electron sea. Nanocavities formed inside a NGJ control
the coupling of LSPPs, thus allowing cavity-tuning [74, 89, 90] targeted to de-
sirable applications which exploit the enhanced optical fields, e.g. nanoscale
optoelectronics and single molecule electronics. However as a NGJ shrinks to
atomic length-scales, quantum effects emerge and standard classical approaches
to describe the photonics of these systems fail.
As discussed in Sec. (2.2.2), one effect of confining electronic wavefunc-
tions inside small metallic nanostructures is to slightly modify the screening
which tunes the plasmons [36]. However tunnelling plasmonics in the Quan-
tum Regime (QR) has more profound effects that cannot be explained through
hydrodynamic models that simply account for quantum effects through smear-
107
Chapter 5: Quantum Tunnelling Plasmonics
ing of the electronic surface-charge [76]. Recent theories show that Quantum
Tunnelling (QT) across the cavity strongly modifies the optical response [82,96],
but computational limits restrict these quantum simulations to very small sys-
tems below a few nanometres. Furthermore, the extreme difficulty in creating
and probing sub-nanometre plasmonic cavities has limited experimental inves-
tigations of plasmonics in the QR.
Self-assembly and top-down nanofabrication achieves gaps of d ≈ 0.5 nm be-
tween plasmonic NPs, but these fail to reach the QT-regime [105,106], and as de-
scribed in Sec. (3.1), are not suitable for optically capturing tunnelling plasmon-
ics. Controllable nanocavities with a sub-nanometre dimension are accessed in
STM and electro-migrated break-junctions where the combined effects of photo-
assisted and photo-induced electron transport can be measured. A significant
amount of investigation has been performed into photo-assisted transport by
measuring the DC photocurrent produced by the rectification of the non-linear
optically driven (optical frequency) QT [142–144]. Photo-induced transport, e.g.
tunnelling of hot electrons created by light absorption processes, has been found
to enhance DC electron tunnelling in STM [145] and electron transport through
metal quantum point contacts [146]. The excitation cross-section and the life-
time of excited electrons in these states (∼ O(fs)) determines the magnitude of
this contribution. Finally, inelastic electron tunnelling has been used to elec-
trically excite LSPPs in nanocavities [147] which in turn can excite propagating
SPPs [148]. The broadband emission due to the radiative decay of the LSPPs and
SPPs is measurable in the far-field and can be used to characterise the nanosys-
tem, e.g. by the visualisation of Fermi’s golden rule [149] and the measurement
of current shot-noise at optical frequencies [150]. Such investigations to date
rely on narrowband illumination conditions and electrically driven LSPP/SPP
excitation to show rectification and emission phenomena, but the effect of QT
on broadband, optically excited plasmon coupling across sub-nanometre cavi-
ties remains unexplored. Here it is shown that optical excitation and detection
of inter-NP optical-frequency tunnelling is highly successful and informative.
By simultaneously capturing both the electrical and optical properties of a
NP dimer, with controllable sub-nanometre separation, the first evidence for
the quantum regime of optically controlled tunnelling plasmonics is observed.
It is also possible initial experimental evidence for coherent plasmon-exciton
coupling in a NP dimer system is obtained for the first time. The experimental
108
Chapter 5: Quantum Tunnelling Plasmonics
apparatus described in Chapter (3), and the NP dimer alignment technique de-
tailed in Chapter (4), provide the specialised setup required to investigate LSPP
interactions in this challenging −0.5 nm/ d / 2 nm regime of extreme non-
locality. By simultaneous dynamic measurement of the DC conductance of the
NGJ, and the LSPP mediated broadband optical scattering response, the first
controlled measurements in this unexplored domain are made possible.
As outlined in Chapter (1), this chapter primarily focuses on the experi-
mental results found due to the culmination of the preceding work presented
in this thesis. A theoretical overview and an outline of the Quantum Corrected
Model (QCM) of BEMAX are presented first, followed by a brief review of the
experimental setup. The remainder of the chapter focusses on experimental
results, comparison with theory, analysis and conclusions.
5.2 Theoretical Modelling
For clarity the LSPP modes expected to be present in the AFM tip mounted NP
dimer system are first briefly introduced using Sec. (2.3.4) as a theoretical basis,
while the specifics of each mode measured are described in detail in Sec. (5.5).
Following from this, an introduction to a newly developed theoretical model
used to compare against the experimentally obtained data is also presented.
5.2.1 Expected Plasmonic Modes
The LSPP modes for a NP dimer created by the axially aligned pair of NP func-
tionalised AFM tips are modified compared with those of a completely isolated
homogeneous NP dimer. The degree of deviation is primarily determined by
the apex NPs plasmonic coupling to the overall neck and body of the host AFM
tip. The typical geometries of the NP functionalised AFM tips used in the ex-
periments are shown by the Scanning Electron Microscope (SEM) and optical
DF images in Fig. (5.1a). The 50 nm Au coated experimental geometry is ap-
proximated by a homogeneous Au geometry in the numerical simulations (Fig.
(5.1b)) to reduce the number of discrete parameterisation points required. This
approximation is valid as the 50 nm Au coating is significantly greater than the
skin-depth δAu ∼ O(35 nm) (Eqn. (2.34)) thus the LSPP modes are only weakly
affected by the underlying dielectric (EBD deposited diamond-like carbon).
109
Chapter 5: Quantum Tunnelling Plasmonics
Figure 5.1: Geometry of NP functionalised AFM tips used in experiment (50
nm Au coating) and theoretical simulation (solid Au). (a) Top: typical optical
DF image of typical NP functionalised AFM tips. Bottom: typical SEM image
(false-colour) of NP functionalised tip apex. (b) Top: typical simplified (axially
symmetric) tip apex geometry used for theoretical simulations. Here αT ≈ 20◦,
DN ≈ 80 nm, aNP ≈ 160 nm and bNP ≈ 140 nm. Bottom: overall truncated
AFM tip geometry used for theoretical simulation with direction (kGB) and
polarisation of the incident Gaussian beam shown.
Following from the discussion in Sec. (2.3.4), it is clear that the plasmon-
ically coupled apex NP and bulk AFM tip will sustain several resonant LSPP
modes of significant scattering cross-section. For example, the particular NP
functionalised tip apices shown in Fig. (5.1a) have a LSPP resonance in the green
(λ0 ≈ 550 nm). The exact nature and spectral position of each mode strongly
depends on the apex geometry, and the plasmonic coupling of the apex NP to
the neck section, labelled in Fig. (5.1a). This plasmonic coupling was intuitively
understood by describing a NP dimer with large overlap (negative d) in Sec.
(2.3.4). Additionally, tightly localised higher-order LSPP modes will also be-
come significant as the apex NPs are moved into closer proximity1 (Fig. (2.13)).
The LSPP interactions in the nanosystem created can hence possibly be under-
stood as the LSPP interactions present in a relatively large NP dimer system
1As these modes are highly-localised they are commonly referred to as LSP modes. However,
they can never be truly LSP modes (see Sec. (5.4)) and are hence referred to as LSPP modes.
110
Chapter 5: Quantum Tunnelling Plasmonics
(a ≈ 150 nm), where the relevant LSPP modes of each NP are red-shifted from
those of an isolated NP due to the anisotropic polarisability created by the plas-
monic coupling of each NP to the associated extended bulk AFM tip structure.
The calculated far-field scattering cross-section for an Au sphere with a = 150
nm is shown in Fig. (5.2).
250x103
240
230
220
210
200Sc
at
te
rin
g 
cr
os
s-
se
ct
io
n 
(nm
2 )
900800700600
Wavelength (nm)
 Au NP: a = 150 nm 
Q
D
Figure 5.2: Far-field scattering cross-section of a large Au NP (a = 150 nm) with
incident wavefronts superimposed to approximate a focussed Gaussian beam.
The dipole and quadrupole LSPP modes are labelled ‘D’ and ‘Q’ respectively.
Scattering collected over 4pi solid-angle.
The relatively large NP promotes optical excitation of the higher-order
quadrupole mode due to the non-uniform field over the volume of the NP
created by the combined effects of retardation and the (approximate) fo-
cussed Gaussian beam illumination. The relative strength of the quadrupole
mode compared to the dipole mode is surprising and is attributed the non-
homogeneous incident illumination [69].
The effect of attaching such a NP to the apex of a substantially larger Au tip
is demonstrated in Fig. (5.3). The incident laser beam, centred at (y, z)=(0, 0)
nm, excites LSPPs highly confined to the apex of both the plain tip (marked by
blue dots) and the NP functionalised tip (marked by red dots). Additionally, the
illumination launches SPPs that propagate up the tips and reflect from the base
[151]. The propagating SPPs thus interfere to form localised quasi-standing-
wave resonances (marked by black dashed lines). These extended LSPP modes
are more apparent in the plain tip geometry due to the reduced loss (higher
reflectivity) at the more accommodating apex structure. It should be noted that
111
Chapter 5: Quantum Tunnelling Plasmonics
while the overall length and cone-angle of the plain and NP functionalised tips
are both equal (to aid comparison), the volume of the plain tip is necessarily
greater, therefore yielding an increased scattering cross-section. In reality the
AFM tips are over an order of magnitude greater in length than those simulated
here, and therefore it is expected that such extended standing-wave LSPP modes
are negligible in experiment.
-1500 -1000 -500 0
z (nm)
-400
0
400
y 
(nm
)
(a)
Au
Air
-1500 -1000 -500 0
z (nm)
-400
0
400
y 
(nm
)
(b)
Au
Air
2.80x106
2.75
2.70
2.65
2.60
2.55Sc
at
te
rin
g 
cr
os
s-
se
ct
io
n 
(nm
2 )
140012001000800600
Wavelength (nm)
3.70x106
3.65
3.60
3.55
3.50
3.45
Sc
at
te
rin
g 
cr
os
s-
se
ct
io
n 
(nm
2 ) Tip + NP
 Tip
(c) (1)
(2)
Figure 5.3: Comparison of the far-field scattering cross-section of an Au tip
and Au NP functionalised tip, simulated using BEMAX. (a) Geometry of plain
Au tip (R = 50 nm), incident wavefronts are superimposed to approximate a
focussed Gaussian beam at the tip apex (y, z)=(0, 0) nm. Incident laser beam
has kGB and polarisation perpendicular and parallel to tip axis respectively.
(b) Same as in (a) but for NP functionalised tip geometry. (c) Scattering cross-
sections associated with the tip (blue) and NP functionalised tip (red) geome-
tries (collected over 4pi solid-angle).
112
Chapter 5: Quantum Tunnelling Plasmonics
The LSPP resonances at λ0 < 830 nm are relatively confined to the apex and
are hence expected to be present even for an infinitely long tip. It is clear the
NP enhances LSPP resonances (1) and (2) relative to the plain tip equivalents. It
is interesting to note that LSPP resonance (1) has approximately the same back-
ground subtracted cross-section as the quadrupole resonance of the isolated NP
(Fig. (5.2)), and it is red-shifted by ∆λ0 = 90 nm. However, more simulations
are required to determine the exact nature of this resonance as it could also be
associated with a higher-order standing-wave like resonance that has been en-
hanced by the NP cross-section. For comparison the green dashed line marks
the position of the theoretically predicted extended standing-wave LSPP reso-
nance. The second enhanced peak, LSPP resonance (2), does not coincide with
a standing-wave resonance and is therefore attributed primarily to the apex NP
and neck region.
The observed resonances (1) and (2) are associated with a significantly en-
hanced surface charge density around the apex region compared to the plain
tip. Only the resonance(s) that localise significant charge density at the NP
apex (rather than, for example, at the neck) will strongly couple (hybridise)
to the LSPPs on the the second tip as the NP dimer is formed. Such a LSPP
resonance will red-shift on approach, and be further enhanced compared to
the LSPP modes more de-localised over the neck and entire tip structure [152].
As d < a it is therefore expected that one LSPP resonance (either (1) or (2))
will dominate the scattering response. At d  a, increasingly higher-order hy-
bridised LSPP modes will become active, in the same manner as that predicted
for a standard NP dimer system (Sec. (2.3.4)). Finally, the rapid decrease in scat-
tering for λ0 < 600 nm is attributed to the modified interaction of the incident
light with the tip structure as ω → ωp (ωp for Au corresponds to λ0 ≈ 520 nm)
where interband transitions begin to dominate the material response.
The LSPP modes between λ = 500 nm - 1000 nm are experimentally mea-
sured and therefore the more localised apex resonances (red dots) shown in Fig.
(5.3c) all lie within the experimental measurement window. Furthermore, the
extended standing-wave LSPPs are not expected to be present. Upon NP dimer
creation (axial tip alignment with d < a) such NP apex LSPP modes begin to
hybridise and will be strongly modified in the regime of extreme non-locality
(d ∼ 1 nm). The experimentally created NP dimer system hence has strong sim-
ilarities to the simple NP dimer system described in Sec. (2.3.4). Importantly,
113
Chapter 5: Quantum Tunnelling Plasmonics
this enables intuitive comparisons with simpler classical NP dimer models to be
made. The LSPP modes experimentally measured, and theoretically simulated,
are described in detail in Sec. (5.5).
5.2.2 Quantum Corrected Model
The full range of non-local response in NP dimer systems with sub-nanometre
separation has only begun to be theoretically treated within the last three years
[82, 96]. However, as described in Sec. (2.3.3), such ab initio approaches are
extremely computationally demanding due to the large number of electrons
involved in the optical response. Recent theoretical work on a QCM of BEMAX
[153], that is experimentally verified for the first time by the work in this thesis,
has made the first step in describing the non-local effect of quantum tunnelling
on the LSPP response of large NP dimer systems with d < 1 nm and a > 3 nm.
Using this theoretical model NP dimer systems with Ne > 107, rather than the
typical Ne ≈ 103 limit for TDDFT simulations, can be described in the non-local
quantum transport regime. Note, all optical scattering and electric near-field
simulations in the remainder of this chapter were performed by R. Esteban and
J. Aizpurua and based on the QCM developed [153].
The QCM approximates the non-local quantum tunnelling effect by repre-
senting the NGJ between two adjacent NPs as a fictitious conducting medium
ε˜QT(d) that mimics electron tunnelling. The calculations employ this fictitious
medium to incorporate quantum effects, derived from coherent electron tun-
nelling and spill-out using Density Functional Theory (DFT), within a classical
electromagnetic framework (in this case BEMAX [77]) to treat large plasmonic
systems [153]. Under this formalism non-local screening effects within the NGJ
(defined by the tunnelling volume) are also taken into account, however this ef-
fect could still be significant immediately outside the tunnelling volume (large
magnitude k˜E components are still present) and therefore could be a source of
discrepancy. Outside the NGJ the relative permittivity εair ≈ 1 and local di-
electric function ε˜Au (taken from experimental data [154]) describe the classical
response of the surrounding air and overall Au tip structure respectively.
First, the general approximations are made that the electric field E in the
NGJ is entirely orientated along the dimer axis (z-axis) and is independent of
position along the z-axis. Furthermore, under the assumption that the transmis-
114
Chapter 5: Quantum Tunnelling Plasmonics
sion of electrons at the Fermi energy (ΩF) through the potential barrier created
by the NGJ occurs at a time-scale much shorter than the period of the optical
field, the tunnelling current adiabatically follows the field in the NGJ. These
assumptions allow the Static Scanning Tunnelling Microscopy (SSTM) model to
be used to calculate the inhomogeneous static (DC) tunnelling conductivity of
the NGJ, σQT0 (d(d0, x, y)), as a function of on-axis separation d0 and lateral po-
sition in x and y for d > 3 Å. Under the SSTM formalism the 1D DC tunnelling
conductivity is given by [155]
σQT0 (d) =
4pim∗e e2
h3
d
∫ ΩF
0
TQT (Ω, d) dΩ (5.1)
where TQT(Ω, d) is the energy-dependent electron tunnelling probability at
each position (d0, x, y) within the NGJ with separation d. It is also assumed
that eVopt  (ΩF ∧ ϕWF) where Vopt = Ed is the optically induced potential
across the NGJ and ϕWF is the work-function of the NP metal. Full quantum-
mechanical treatments (e.g. DFT) can be used to calculate TQT(Ω, d) as the
number of electrons in such simple (assumed flat interfaces as a  d) systems
is relatively small. In this case, for simplicity, a jellium model potential approach
rather than a full atomistic simulation was used to determine TQT(Ω, d) under a
DFT framework, for d > 3 Å [153]. Under this formalism the many-body nature
(local-density approximation) of the problem and long-range image potential
interactions are taken into account [156]. For smaller d, the SSTM approach is
no longer valid as TQT is not small. Therefore an expression for σQT0 (d) over the
full-range of d is estimated by fitting an exponential form to the value at contact
(d → 0) together with the SSTM results at large d. In practice the lateral inho-
mogeneity (x, y dependence) is implemented at the NGJ by considering enough
concentric shells in the x-y plane to provide a converged solution.
From this basis the position dependent tunnelling damping parameter
γQT(d) is determined via σQT0 (d) = σ
QT|ω=0 = ε0ω2p/γQT(d) (c.f. Eqn. (2.22)),
and thus the position dependent spatially local dielectric function of the ficti-
tious tunnelling medium is obtained by
ε˜QT(d(d0, x, y),ω) = εbk(d)−
ω2p
ω (ω+ iγQT(d))
(5.2)
115
Chapter 5: Quantum Tunnelling Plasmonics
where εbk(d) is the background screening contribution. In this case εbk(d = 0) is
the screening contribution of the d-orbital electrons to ε˜Au. For d > 0, εbk(d)→ 0
exponentially with a characteristic length of 1.5 Å, chosen to be close to the
decay of the 5d-orbital of Au [157]. In this treatment γQT is considered to be the
varying parameter, rather than ωp, as this is consistent with the resistive, rather
than capacitive, nature of the NGJ found from full TDDFT calculations [153].
Both ωp and γQT(d = 0) are set equal to their respective free-electron ‘Drude’
contribution found from ε˜Au. Using Eqn. (5.2) to describe the NGJ within the
classical BEMAX framework hence enables the optical response of large-scale
nanosystems with a sub-nanometre NGJ to be well approximated [153].
The physical geometries of the NP functionalised AFM tips are modelled
as shown in Fig. (5.1b). The plasmonic NGJ is formed by two 150 nm radius
spheres connected to ≈ 1.5 µm long cones through rounded 80 nm diameter
necks. In this case each cone is terminated with a planar surface at its base. The
cones are aligned along the z-axis and both combined NP-cone structures are
assumed to be rotationally symmetric.
To model the DF illumination a plane-wave representation of a laser beam
propagating in the −y-direction (perpendicular to the AFM tip axis) is used, as
shown in Fig. (5.1b). The incoming light is described by a plane-wave decom-
position of a focussed Gaussian beam (centered at (x, y, z) = (0, 0, 0)) under the
scalar paraxial approximation2. The fields are thus described by a superposition
of plane-waves propagating with wavevector k. The electric field E at a point
(x, y, z) can be expressed as
E(x, y, z) ∝
∫ ∫
u(kρ)e−i(kxx+kyy+kzz)p(k)dkxdkz (5.3)
where kρ =
√
k2x + k2z, ky =
√
k2 − k2ρ and kρ < k. The function u(kρ) is a
weighting term that is constant for plane-waves incoming at all angles between
θ1 = 34◦ and θ2 = 64◦ with respect to the y-axis, but strongly suppresses other
plane-wave contributions to the integral [158]. This is implemented to better
describe the experimental DF illumination. Finally, p(k) defines the polarisation
of each plane-wave (TM in the corresponding k-z plane). The total scattering
cross-section is obtained by integrating over the full 4pi solid-angle.
2This is a good approximation to the slightly more ‘Airy-like’ intensity distribution produced
in the experiments.
116
Chapter 5: Quantum Tunnelling Plasmonics
Overall a number of immediately obvious assumptions are made in the
QCM, such as the lack of surface irregularities, the presence of perfect rotational
symmetry and the local description of the gold-air interface. For instance, the
theoretical simulations used a DN = 80 nm cone-shaped neck to approximate
the average of the triangular cross-section of the actual neck shown in the SEM
image of Fig. (5.1a). The illumination conditions are mostly described within
the theoretical model, even though spatial filtering and the strong focussing
(high NA) of the laser illumination in the experimental implementation leads
to certain complexities in the field distributions (e.g. polarisation distribution
as described in Sec. (3.4.2) and a tighter elliptical focus) that are not taken into
account. Due to such complexities the collection conditions are captured in a
simplified way within the theoretical model. These combined approximations
likely account for some of the differences in the LSPP mode wavelengths and
scattering intensities between experiment and theory, and are discussed in more
detail in Sec. (5.5.2). The primary effect of changing the illumination and col-
lection conditions in the theory was to alter the signal strength of the different
modes. Nevertheless, even after these approximations, the QCM captures the
essence of the coherent QT process occurring across the NGJ.
5.3 Experimental Setup
The experimental apparatus is described in detail in Chapter (3). Here a brief
review of the setup is presented. The optical setup for the DF scattering mea-
surement is shown in Fig. (5.4a). A custom-built DF microscope designed for
use with supercontinuum (450 nm - 1700 nm) laser illumination is used for all
scattering measurements. The microscope is designed to provide an ultra-low
specular-reflection background and a high degree of mechanical stability, en-
abling DF spectra to be captured over ms timescales with excellent SN levels.
The laser illumination (path shown in red) is expanded by a factor of three with
a Keplerian lens pair, then passed through a reflective ND filter (OD = 1.0),
an APP to correct chromatic dispersion, and a LP (transmission axis oriented
parallel to the tip-tip (z−) axis). As the tip apex NPs are centred under the
laser illumination the linear polarisation along the NP dimer axis is well main-
tained even after strong focussing (Fig. (3.14b)). A 2 mm diameter beam stop
blocks the central portion of the beam thus creating the DF illumination geom-
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Chapter 5: Quantum Tunnelling Plasmonics
etry. The BS 1 directs the laser beam into the objective that focuses the beam to
a diffraction-limited, sub-µm spot at the tip apices, as shown schematically in
Fig. (5.4b).
Figure 5.4: Review schematic of experimental setup for measurement of LSPP
mediated optical scattering response in the quantum transport regime. (a) Re-
view schematic of optical setup. Here the CP has diameter of 50 µm and the
CI has diameter of 1.2 mm. (b) Schematic of illumination of NP functionalised
AFM tip apices with static potential applied and DC current flow measured.
Using the laser and a high magnification, high NA objective provides > 104
times the intensity than that of a typical halogen or xenon lamp setup and
illuminates only the volume of interest. Light scattered from the sample (path
shown in green) is collected with the same objective and a Collection Iris (CI)
blocks the high-NA portion of the beam. The scattered light is split by BS 2,
with half the light passing to a CCD camera for imaging and the other half
focussed onto the spectrometer after spatial filtering with a 50 µm Confocally-
aligned Pinhole (CP). Spectrometer integration times of 3 ms were used and
118
Chapter 5: Quantum Tunnelling Plasmonics
measured spectra were normalised to the reflection from an Au mirror placed
at the sample plane.
The Au-coated, ball-type AFM probes were characterised by optical and elec-
tron microscopy. Typical cantilever spring constants and (lowest-order) reso-
nance frequencies were kz0i ≈ 0.2 Nm−1 and f0i ≈ 13 kHz, respectively. Large
radius of curvature (R = aNP ≈ bNP ≈ 150 nm) tips are selected to minimise
sensitivity to axial tip-tip alignment, to increase the scattering cross-section, and
to support higher-order plasmonic cavity modes in the visible spectrum. There
is approximately < 10% variation in the tip apex geometry parameters for all
AFM tips used, hence additional LSPP coupling effects prevalent in hetero-
dimers are assumed to be small. The AFM tips are mounted on 3-axis piezo-
electric (PZ) actuation stages using closed-loop feedback control with capacitive
positioning sensors. The tips are then axially aligned with nanoscale accuracy
using a recently developed, non-linear EFM technique [159], described in detail
in Chapter (4). Briefly, the alignment is performed by placing an AC potential
across the AFM tips, and measuring the harmonic frequencies generated by the
non-linear electromechanical response of the tip-tip system using lock-in detec-
tion. The tips are brought into nanometre-precise 3D alignment by adjusting
the tip positions to maximise signal harmonics.
The two Au-NP functionalised AFM tips are hence aligned tip-to-tip in 3D,
as shown in Fig. (5.4). The tip apices thus define a NGJ supporting plasmonic
resonances created via strong electromagnetic coupling between LSPPs on each
tip apex NP [160, 161]. This dual AFM tip configuration provides multiple ad-
vantages. First, independent nm-precise movement of both tips is possible with
3-axis piezoelectric stages. Second, conductive AFM probes provide direct elec-
trical connection to the apex NPs enabling simultaneous optical and electrical
measurements. Third, the tips are in free space and illuminated from the side
in a DF configuration. Finally, as the effective NPs at the tip apices are conduc-
tively connected through to bulk AFM tip, cantilever and support structures,
these bodies act as thermal sinks thus reducing heating effects on the stabil-
ity of the NGJ. This arrangement provides for the first time, background-free
broadband spectroscopic characterisation of the tip-tip NP dimer (plasmonic
nanocavity) throughout the sub-nanometre regime. The inter-tip separation d
is initially set to 50 nm and then reduced while recording DF scattering spectra
and DC current flow simultaneously.
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Chapter 5: Quantum Tunnelling Plasmonics
The inter-NP conductance is measured by applying a DC potential (VDC)
across the tips and measuring the resulting current (IDC) with a variable gain
(set to 106 VA−1 in this case) transimpedance amplifier with an integrated 10
Hz low-pass filter. The transimpedance amplifier signal is measured with a
DSO and the junction conductance GDC = IDC/VDC is calculated after taking
into account series contact resistances in the electronic system. The applied
potential VDC has been varied between 50 µV - 500 µV in different experiments
to balance the inter-tip forces with an adequate SN. Typically VDC ∼ 50 µV was
used to keep the inter-tip electrostatic force to a minimum.
After alignment the inter-NP separation is initially set to d ∼ 50 nm and is
then reduced in 1 nm decrements with the PZ stage. A DF scattering spectrum
and DC current measurement are recorded after each step. Scans progress until
Conductive Contact (CC) between the tips is detected, achieved once GDC >
G0 = 2e2/h = 7.748× 10−5 S (a single conductance quantum, corresponding
to an atomic point contact between the Au surfaces [162]). After CC, the tips
are further pressed together with up to an additional 20 nm PZ displacement
(corresponding to approximately 2 nN of additional compressive force) to es-
tablish firm, yet reversible, mechanical contact. As noted in Chapter (4) all
forces calculated using kz0i can have up to a factor of 5 uncertainty in their value
and hence all force measurements quoted throughout this chapter are taken as
approximate (or even guideline) values.
As is commonly seen in conductive AFM and STM experiments a short-range
StoC event is observed when attractive forces between the tips overwhelm the
restoring force of the AFM cantilevers [163]. More specifically the condition for
StoC is when the force gradient is greater than the effective spring constant of
either cantilever
∂ (FzEL + F
z
TT)
∂d
> kzei (5.4)
As the applied potential in all experiments satisfies VDC < 0.5 mV the theoreti-
cal simulation described in Sec. (4.2.2) shows that the electrostatic force (FzEL) is
over 105 times less than the Van der Waals force at 7 nm > d > 1 nm. There-
fore, over the relevant range of d, the attractive Van der Waals component of FzTT
primarily determines when Eqn. (5.4) is satisfied and at what separation StoC
occurs. The theoretical model shows StoC typically occurs at d ≈ 5 nm. Addi-
tionally, because the experiments are performed in ambient conditions, the Van
120
Chapter 5: Quantum Tunnelling Plasmonics
der Waals force causes the water layers (and possibly hydrocarbons) adsorbed
on each Au tip-NP surface to become unstable and liable to StoC. This could
slightly increase the StoC separation predicted, as the meniscus formed creates
a large capillary force that snaps the NP functionalised tips closer together [164].
This situation is shown schematically in Fig. (5.5). It is noted here that as the
applied potential is typically VDC ∼ O(µV), electrostatic field induced meniscus
bridging phenomena [165] are not expected to be present and the Van der Waals
force is therefore the predominant factor controlling meniscus formation.
Figure 5.5: Schematic of tip-tip StoC process. (a) Tip apex NPs are separated
at d ≈ 5 nm. (b) At this point the attractive Van der Waals force cause the
adsorbed water to become unstable and water-to-water StoC occurs forming a
water meniscus. (c) The relatively strong capillary force causes the tip apex NPs
to StoC while the remaining water and hydrocarbon molecular layers (tL ≈ 1
nm thickness) form an insulating barrier between the gold NP surfaces.
Additional evidence for the presence of water is given by the relatively large
pull-off force (FPO) typically required to fully break NP-NP (apex-apex) contact
FPO ' 70 nN. This is commonly found in AFM studies in ambient conditions
as the capillary force can dominate the contact adhesion strength [163]. This
effect remains present when the tip apex NPs are under laser illumination thus
indicating laser heating does not remove the adsorbed water layers or meniscus.
The Joule heating (caused by the absorption of laser light) in the vicin-
ity of the NP tip apices is roughly estimated to result in a local temperature
T ∼ 100 ◦C at the surface layer [117, 118, 166]. As the molecules are adsorbed
(or close) on the Au surfaces it is expected significantly higher temperatures
would be required to remove the water layers. This is consistent with the pull-
off behaviour described above. Finally, it is noted here that the apex NPs are
expected to have a cooling relaxation time τc ∼ O(ns) [118]. Therefore the heat-
ing, and hence thermal expansion and contraction of the NGJ, is expected to
121
Chapter 5: Quantum Tunnelling Plasmonics
approximately follow the laser pulse train (20 MHz repetition rate), which has
a mark-space ratio of 1/50. This will therefore have only a small effect on the
average separation of the apex NP’s over the spectrometer (τint = 3 ms) and
electronic DC amplification (τint = 35 ms) integration periods.
The fact that StoC does not coincide with CC is also common in AFM and
STM experiments performed under ambient conditions, as naturally-adsorbed
surface layers (tL ≈ 1 nm - 2 nm thickness [167]), i.e. water and hydrocarbons,
on the tip apices can provide electrical insulation [167–169]. The applied force
required to push through the insulating layer is consistent with other conduc-
tive AFM studies [167, 169]. For simplicity the effect of the surface layer is not
currently incorporated into the QCM. It is expected that such a surface layer
will produce a confinement dependent red-shift in the resonance wavelength of
the LSPP modes [170], and increase the quantum transport mediated conduc-
tivity [96, 171, 172].
5.4 Results
Each PZ scan investigates three interaction regimes: capacitive near-field cou-
pling (50 nm > d > 1 nm), non-local quantum regimes (1 nm > d > 0 nm), and
physical contact with conductive coupling (d ≤ 0 nm). Crucially, this setup en-
ables the resolution of the gradual transition between each regime dynamically.
The measured DF scattering spectra shown in Fig. (5.6), within the capacitive
regime (d ∼ 40 nm), show a single LSPP scattering peak centred near λ0 ≈ 765
nm (mode A). This LSPP resonance is attributed to a hybridised version of one
of the more ‘NP apex localised’ LSPPs described in Sec. (5.2.1). As d is reduced,
this peak red-shifts and strengthens due to increasing near-field interactions be-
tween the LSPPs on each NP. As the cavity shrinks below 20 nm, a second LSPP
scattering peak emerges at shorter wavelengths (mode B, λ0 ≈ 560 nm) and
quickly increases in intensity.
Modes A and B smoothly red-shift, as described in Sec. (2.3.4), until an es-
timated separation d ≈ 5 nm, whereupon attractive inter-tip forces overwhelm
the AFM-cantilever restoring force and snap the tips into close proximity [163].
However no current flow is detectable because this StoC point does not coincide
with CC and the metal surfaces remain separated. StoC reduces d to ∼ 1 nm,
significantly increasing the plasmonic interaction and dramatically changing the
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Chapter 5: Quantum Tunnelling Plasmonics
Figure 5.6: Measured DF scattering spectra (vertically offset for clarity) from
the NP dimer at different separations d. The LSPP resonances are labelled A-C.
LSPP scattering resonances (blue curve, Fig. (5.6)). This increased coupling fur-
ther red-shifts modes A and B and reveals a new higher-order resonance (mode
C) resonant at λ0 ≈ 550 nm. After StoC, increased PZ displacement applies an
additional compressive force (approximately 0.1 nN nm−1 of PZ displacement)
pushing the NPs into closer proximity. In this particular experimental run, after
11.4 nN of externally applied force, current flow is detected through the tips,
indicating metal-to-metal surface contact. Numerical QCM calculations confirm
that the coupled LSPP modes observed are tightly confined in the NGJ and these
modes are described in detail in Sec. (5.5).
As demonstrated in Fig. (5.7), simultaneously monitoring the optical and
electrical properties during approach reveals unprecedented detail about LSPP
evolution through the sub-nanometre regime. The experimentally measured
scattering response displayed in Fig. (5.7a) shows that as the applied force in-
creases and d is thus reduced, all three modes red-shift and modes A and B
weaken while mode C intensifies. It is possible that the experimentally found
linewidths of modes A and B decrease in concert with this reduced scattering
strength while the broadening of mode C is due to increased scattering loss.
These spectral changes are well-reproduced by the QCM simulations that in-
clude quantum tunnelling, shown in Fig. (5.7b).
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Chapter 5: Quantum Tunnelling Plasmonics
Figure 5.7: Optical detection of onset of quantum tunnelling in NP dimer
with sub-nanometre separation. (a) Simultaneously measured electrical con-
ductance and DF optical back-scattering as the force applied to the inter-NP
cavity (after StoC) increases. Conductive contact indicates d = 0, with onset
of the QR at dQR. (b) Theoretical total scattering intensity from the model NP
functionalised tip-tip system incorporating quantum mechanical tunnelling via
the QCM [173]. The threshold (at dQR) indicates where quantum-tunnelling-
induced charge screening overcomes the near-field capacitive interaction be-
tween LSPPs. (c) Selected experimental spectra from the last 1 nm to contact
in (a), shown vertically shifted for clarity. (d) Theoretical scattering intensity as
in (b) but using a purely classical formalism. Solid black lines mark scattering
peaks in (a), (b) and (d).
Inter-tip distance calibration is performed by identifying the characteristic
red-shift to blue-shift LSPP mode behaviour at dQR in the theoretical simulation
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Chapter 5: Quantum Tunnelling Plasmonics
with the same characteristic dQR transition point in the experimental data. Ad-
ditionally, the spectral shifts in plasmonic modes at StoC found in simulation
were matched to experiment, and the CC point was used to set d = 0. This
quantum regime-based optical distance calibration (’plasmon ruler’) using the
QCM simulation is equivalent to a traditional distance calibration using elec-
tronic tunnelling current measurements and STM simulations. A method for
approximately determining distance scales is thus obtained, which agrees with
previous AFM studies showing StoC separations d ∼ O(1 nm).
For d ' 0.4 nm, the LSPP interactions are approximately consistent with the
classical NP dimer picture (Fig. (5.7d)) presented in Sec. (2.3.4). The classical
LSPP interactions account for the rapidly increasing red-shifts as d decreases
and the observed transfer of oscillator strength from modes A and B to mode C.
While these higher-order coupled LSPP modes have been predicted theoretically
[89, 90], they are now clearly revealed dynamically on approach.
Beyond 8 nN however, a new regime deviating strongly from the classical
predictions is seen, with A and B now shifting back to shorter wavelengths in-
stead of red-shifting asymptotically. This crossover is clearly seen in the QCM
simulations at d ≈ 0.31 nm. The quantum and classical predictions diverge at
this crossover point (dQR) because the plasmon interactions enter the quantum
regime when d is sufficiently small to support a critical electron tunnelling rate
between the surfaces. While electron confinement within each NP is minimally
affected, the quantum tunnelling here dramatically modifies the correlations
between electronic fluxes and ideally demands a more sophisticated treatment
based on solving the time-dependent Schrödinger equation [153]. The net re-
sult is that QT charge transfer screens the LSPP surface charge, decreasing the
enhanced fields and reducing plasmonic coupling. For d < dQR, QT increases
exponentially and quickly dominates, reducing the capacitative coupling, and
resulting in Charge Transfer Plasmon Polariton (CTPP) modes that blue-shift as
d→ 0.
It is expected that a combination of plasmon-assisted (plasmon enhanced
photo-assisted) and plasmon-induced tunnelling transport mechanisms de-
scribe the LSPP coupling to electron tunnelling. Plasmon-assisted tunnelling
is taken into account in the QCM, however plasmon-induced tunnelling is not
considered as the contribution of hot electrons with Ω > ΩF to σ
QT
0 is neglected.
In both transport mechanisms, an optically induced (LSPP enhanced) oscillat-
125
Chapter 5: Quantum Tunnelling Plasmonics
ing (∼ 100 THz) potential drives the optical frequency QT. Future investigations
could apply a larger VDC and hence further validate this hypothesis by direct
measurement of a rectified DC photocurrent. It is also noted here that molecules
within the NGJ could also enhance electron transport at specific frequencies un-
related the geometrically defined LSPP resonances.
After conductive contact (d = 0) when the conductance first jumps above
G0 before further increasing, two LSPP scattering peaks are observed, modes D
(800 nm) and E (640 nm). Tracking modes A, B, and C through the QR as d→ 0
shows the gradual nature of this contact transition, in marked contrast to the sin-
gular transition predicted classically where a dense continuum of modes builds
up in the touching limit, as shown in Fig. (2.13) [89, 90]. At dQR mode A weak-
ens dramatically, before a new peak appears which blue-shifts and intensifies
towards CC. While mode B weakens at dQR, mode C strengthens as predicted by
the QCM. In both theory and experiment, modes B and C are replaced by mode
E on contact. Spectra and conductance change minimally for increasing contact
force after CC, indicating a stable final contact. As described in general in Sec.
(2.3.4), mode E is characterised by surface charge highly confined around the
periphery of the contacted surfaces while the QCM simulations approximate
the contact geometry as two smooth gold surfaces. The experimental contact
geometry is unlikely to satisfy this assumption and the contact will be deter-
mined by the asperities of the rough gold coatings. The largest discrepancy is
indeed at the CC point, where the uncertainty in the asperity-mediated contact
geometry is greatest (with less effect at dQR). Experiments on a variety of NP
functionalised AFM tips show repeatable crossover behaviour at d ≈ dQR, with
a typical example shown in Fig. (5.8). This crossover behaviour thus forms an
optical fingerprint of the new tunnelling plasmonics regime.
In addition to the quantum-based simulations provided by the QCM, the
experimental geometry was also modelled using an entirely classical frame-
work, as shown in Fig. (5.7d). These results show that as d → 0, the LSPP
modes (A, B and C) continue to red-shift and higher-order modes continue to
appear in the scattering spectrum. These observations are completely consistent
with the classical picture of plasmonic interactions where, in the limit of tangen-
tially touching spheres, a broadband mode continuum develops [174]. Several
modes are also observed after contact (where d < 0) that blue-shift as the tip
surfaces overlap further, similar to the results for overlapping spheres shown
126
Chapter 5: Quantum Tunnelling Plasmonics
in Sec. (2.3.4). Contrasting the classical simulations with the QCM results (Fig.
(5.7b)) highlights the remarkable effect of QT on plasmonic interactions. Cru-
cially, only by incorporating QT effects via the QCM can the experimental re-
sults be reproduced allowing the clear identification of where the plasmonic
interaction crosses-over into the quantum regime.
For comparison, Fig. (5.8) presents the correlated electrical and optical char-
acterisation of the NP functionalised tip-tip system from another representative
experiment (using a different pair of NP functionalised AFM tips). These re-
sults were obtained under similar conditions as those described in Sec. (5.3),
except the applied DC potential VDC = 0.5 mV was 10 times greater. As de-
scribed above, immediately at StoC (zero externally applied force) three modes
(A, B, and C) are observed, each corresponding to a LSPP resonance of the NP
functionalised tip-tip system. Less externally applied force is required to push
through the insulating layer, likely due to the specifics of the atomic-scale geom-
etry and the 100 times greater electrostatic force created by the larger applied
potential, thus effectively reducing the resolution of the scan.
Figure 5.8: Simultaneously measured DC conductance and DF scattering versus
the force applied to the inter-NP cavity.
Prior to CC, a crossover at d = dQR is observed where the LSPP modes stop
red-shifting and begin blue-shifting. This crossover point is the optical signature
127
Chapter 5: Quantum Tunnelling Plasmonics
of QT modifying plasmonic coupling and corresponds to the threshold point
where QT charge transfer counterbalances the near-field capacitive interactions
between the metal surfaces. An evolution into only two modes after contact (D
and E) is observed, as in all experiments. It should be noted that the relative
strength of mode E is significantly less than that shown in Fig. (5.7a). This is
likely because around CC the strengths of the LSPP modes are highly dependent
on the precise experimental conditions, e.g. specifics of the contact geometry
and molecular species present. Nonetheless, the spectral positions remain rather
systematically robust. The DF scattering spectra and the NGJ conductance are
observed to stabilise after CC, indicating the formation of a stable nanojunction
after contact. Other experimental runs give similar results to those described
here. It is important to note that this consistency also confirms the robustness
of the NP dimer alignment technique.
To understand LSPP evolution through the QR, the cavity field-distribution
is calculated within the QCM accounting for quantum effects, and is shown in
Fig. (5.9). This allows a model to be constructed of tunnelling plasmonics (Fig.
5.9a). For d > dQR (I), spectra are dominated by the near-field interaction of
the cavity-localised surface charges and LSPPs couple according to classical and
hydrodynamic models. Once d ∼ dQR the system enters the QR (II) and QT
opens a conductance channel between the surfaces, modifying the LSPP charge
distribution, screening the electric field, and reducing the interaction strength.
The tunnelling (which is strongly concentrated across the thinnest barrier
region, due to the exponential dependence on separation) pinches off the field
distribution in the centre of the nanogap, thus separating the single field lobe
into two lobes in the contact crevices on either side of the neck (Fig. 5.9b).
Although the light impinges from above, the field in the upper side crevice is
weaker due to the asymmetry introduced by retardation. As d is reduced further
(III) these QT CTPP modes, concentrated around the contact crevices, blue-
shift as their apex becomes blunter. Around dQR the mode strengths become
weakest because the tunnelling increases sufficiently to screen the charges across
the nanogap, but cannot provide sufficient current to drive the charge-transfer
modes into the crevices. Hence at this point the total separated charge localised
to the contact region decreases, reducing the optical cross-section.
The onset of quantum tunnelling fundamentally limits optical field confine-
ment in plasmonic nanocavities, e.g. between NP dimers with sub-nanometre
128
Chapter 5: Quantum Tunnelling Plasmonics
Figure 5.9: Evolution of the LSPP modes through the QR and the quantum
limit of plasmonic confinement. (a) Plasmonic interactions within the three
regimes accessed in experiment. (b) Near-field distributions for modes B→E
from the QCM theory in each regime [173]. Images are of a 40 nm by 5 nm
region, with the same intensity scale. (c) The lateral confinement width w
of each mode, extracted from the simulated near-field distribution (FWHM of
intensity maximum), as the cavity width d is reduced. The dashed line marks
the classical approximation w =
√
Rd. The onset of quantum tunnelling effects
at d = dQR = 0.31 nm sets a quantum limit (wQL) on LSPP mode confinement
in sub-nanometre plasmonic cavities.
separation. The plasmonic surface charge between two spherical surfaces is
confined laterally to w ≈ √Rd [89], as confirmed by the simulations shown
in Fig. (5.9b). However, QT limits w to w ≥ wQL =
√
RdQR. Further reduc-
ing d rapidly increases w, as the surfaces are quantum-mechanically blurred
at this atomic-scale. The tunnelling plasmonics regime thus represents the
quantum limit of compression of light which is plasmonically-squeezed into a
nanogap, as verified in our experiments and QCM calculations. The quantum-
limited mode volume is approximated as Vmin = 14piR d
2
QR and estimated to be
Vmin ≈ 1.7× 10−8λ30 from the experiments at λ0 = 850 nm (mode A at dQR).
Here it is assumed the physical volume is approximately equal to the mode vol-
ume found from Purcell analysis as the vast majority of the energy stored in the
electric near-field is contained within the nanogap rather than the Au NPs [175].
As this limit for plasmonics is six orders of magnitude smaller than the tightest
field confinement observed in photonic crystal cavities [176], it still offers un-
precedented opportunities for directly visualising atomic-scale and molecular
processes with eV-scale photon energies.
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Chapter 5: Quantum Tunnelling Plasmonics
5.5 Analysis
In this section a toy model developed to enable an intuitive explanation of the
quantum tunnelling mediated LSPP behaviour is first presented, followed by a
more in-depth analysis of the character of the LSPP modes A, B and C.
5.5.1 Tunnelling Plasmonics Regime
With the onset of QT charge transfer, the LSPP charge distributions are modi-
fied by the conductive coupling. To provide an intuitive, physical description of
LSPP interactions transitioning into the QT regime, a simple and crude model
of QT and the effect it has on plasmonic charge localisation was developed.
The approach follows that in [98], where the plasmonic properties of NP dimers
linked by a conductive nanojunction were examined in a purely classical formal-
ism. However, here the model is extended to the case where the nanojunction
conductivity is determined by the QT properties of a nanogap.
The LSPP surface charge QSP is given by
QSP = ε0ESP ASP (5.5)
where ESP is the LSPP-enhanced local field, ASP is the surface area of the LSPP,
and a vacuum environment is assumed. When the surfaces are close enough to
allow QT across the nanogap, a charge QQT is transferred between the surfaces
over a half optical cycle (τ1/2 = λ0/2c) given by
QQT = σ
QT
0 (d) EQT AQTλ0/2c (5.6)
where σQT0 (d) is the DC conductivity of the QT junction, EQT is the electric field
driving the QT, and AQT is the QT nanojunction area. Herein it is assumed that
the electron tunnelling time is short compared to the optical half-cycle. The QT
channel transfers this charge across the nanogap, thereby neutralising a fraction
X of the LSPP charge
X =
QQT
QSP
= σQT0 (d)
λ0
2ε0c
(5.7)
where it is assumed that ASP = AQT and ESP = EQT (a good approximation
130
Chapter 5: Quantum Tunnelling Plasmonics
given R is large). In the low-voltage domain (eVopt  (ΩF ∧ ϕWF)), the DC
tunnelling conductivity of a planar metal-insulator-metal junction is calculated
as
σQT0 (d) =
(
4pim∗e e2
h3
)
d
∫ ΩF
0
TQT (Ω, d) dΩ (5.8)
Accurate determination of the realistic tunnelling probability TQT (Ω, d) and as-
sociated σQT0 (d) requires the full theoretical approaches previously mentioned.
However, in the simplest case of a 1D rectangular energy barrier (no image
potential) of height ϕWF the solution to Eqn. (5.8) is analytic (WKB approxima-
tion) [177], and therefore Eqn. (5.8) simplifies to
σQT0 (d) =
3qQTe2
4pih
exp
(
−2qQTd
)
(5.9)
where qQT =
√
2m∗e ϕWF/h¯ is the semi-classical electron tunnelling wavenum-
ber. In Fig. (5.10a), the QT conductivity calculated for this simple model is
compared against that predicted by the appropriate DFT calculations used in
the QCM (using the Au work function as barrier height (ϕWF = 4.8 eV)).
Figure 5.10: (a) DC conductivity σQT0 of a NGJ of width d. The tunnelling
probability is determined using a rectangular barrier (simple model, red), sim-
ple model with image potential correction (green) and DFT simulations (QCM
model, blue). (b) Fraction X of LSPP charge (QSP) transferred by quantum
tunnelling (QQT) at different barrier widths [173].
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Chapter 5: Quantum Tunnelling Plasmonics
At d = 0.31 nm, the quantum crossover point dQR defined in Sec. (5.4), the
simple model predicts σQT0 = 98.6 Sm
−1 which is much smaller than the 7.4×
103 Sm−1 calculated under the QCM model. This discrepancy is unsurprising
given the simplicity of the rectangular barrier model used to derive Eqn. (5.9).
A more complex expression for σQT0 that includes the reduction of the height
and width of the potential barrier due to the image potential is also considered
(Fig. (5.10)) [177]3. When the effect of the image potential is recognised the
agreement is improved dramatically with σQT0 = 5.3× 103 Sm−1 at d = 0.31 nm.
Using Eqn. (5.7) and the calculated σQT0 (d) from the simple, improved and
QCM models, the fraction X of the LSPP charge transported per half-cycle (Fig.
(5.10b) is calculated. These results were obtained for λ0 = 850 nm which is
near the LSPP resonance wavelengths observed in theory and experiment. At
d = 0.36 nm, the full theory predicts that half (X = 0.5) the LSPP charge is trans-
ferred across the nanogap, near the dQR = 0.31 nm crossover point obtained by
the full numerical simulation (Fig. (5.7b)). An intuitive explanation for the QR
red-shift to blue-shift crossover is therefore found as the critical point where
sufficient charge is transported between the NPs to modify the LSPP charge
distribution.
Combining Eqn. (5.7) and Eqn. (5.9), setting X = 0.5, and noting that the
fine structure constant α = e2/2ε0hc ≈ 1/137 gives
d′QR =
1
2qQT
ln
[
3qQTλ0α
2pi
]
=
1
2qQT
ln
[
3qQTαc
ω
]
(5.10)
Evaluated for qQT = 1.1 Å−1 (ϕWF = 4.8 eV) and λ0 = 850 nm, Eqn. (5.10) gives
d′QR = 0.16 nm. Full QCM calculations show that already at dQR = 0.31 nm suf-
ficient screening develops via the quantum transport to overcome the increasing
charge buildup, consistent with the estimate for d′QR above that is based on a
reduced tunnelling conductivity. The expression for σQT0 that includes image
potential effects can be evaluated graphically and yields a value of d′QR = 0.35
nm in good agreement with dQR estimated from the QCM. Therefore properly
including the coherent quantum transport strongly enhances the tunnelling rate,
increases the distance at which tunnelling plasmonics takes over, and hence dic-
3The expression for the image potential used by Simmons in [177] is a factor of two too large
and was corrected accordingly.
132
Chapter 5: Quantum Tunnelling Plasmonics
tates the emergence of QT mediated CTPPs. Thus it is shown that this simple
charge transfer time-scale model provides a reasonable estimate of the onset of
the QR in LSPP interactions. The logarithmically increasing dependence of d′QR
with λ0 shown in Eqn. (5.10) is unphysical, however the expression is approxi-
mately valid. A more detailed analysis is provided in Appendix (A).
5.5.2 Understanding the Plasmonic Modes
An improved understanding of the measured LSPP modes is gained by consid-
ering the simulated near-field distributions in the QCM. First, over the whole
NP at the tip apices (shown schematically in Fig. (5.11a) and Fig. (5.11b)), before
examining the field distribution in the inter-NP nanocavity itself. The calculated
z-component of electric near-field (magnitude |E˜z| and phase ϕz) for modes A,
B, and C at a separation d = 0.6 nm are shown in Figs. (5.11c-5.11h), corre-
sponding to the geometry in Fig. (5.11b). Note that a logarithmic colour-scale
for |E˜z| is used to help visualise the weaker fields away from the nanocavity.
Only the z-component of the field is considered as it is much stronger than the
x- and y-components and more important for the plasmonic interactions in this
geometry since it is parallel to the NP dimer (tip-tip) axis.
Nanocavity localisation is observed for each LSPP mode although the field
enhancement associated with modes B and C is greater than that for mode
A. The maximum field enhancement (normalised to the incident field) in the
nanocavity for modes A, B, and C is approximately 300, 550, and 350, respec-
tively. It is indeed found that mode A is relatively weakly localised to the
nanocavity and is spread out over more of the extended neck-NP structure,
whereas modes B and C are nanocavity resonances with tightly confined sur-
face charge.
While the LSPP modes of simple individual NP homo-dimers are well
known, those for dimers with constituent NPs on the end of long conductively
connected tips remain poorly understood. In contrast to an isolated NP dimer,
it is found that LSPP modes with a certain symmetry in electric field magnitude
over the whole tip-NP system can have a different local symmetry in field mag-
nitude within the nanogap volume. This is the case for modes A, B and C. The
phase distributions in Fig. (5.11) show that the field varies more rapidly around
the NP for modes B and C than for mode A, indicating that modes B and C are
133
Chapter 5: Quantum Tunnelling Plasmonics
higher-frequency LSPP resonances of the whole system. However the modes
cannot be labelled hybridised dipole, quadrupole etc. by just considering their
nature in the nanocavity. The distinction between the field symmetry has to be
qualified whether in the nanocavity or of the entire NP-tip structure.
Figure 5.11: Near-field magnitude and phase in the vicinity of the AFM tips
apex NPs [173]. Schematic showing (a) the simulated dual tip-NP geometry
and (b) the geometry at the NP apices (with neck emphasised). (c,e,g) The
QCM-calculated near-field magnitude (|E˜z|) of the NP dimer shown in (b) at a
separation d = 0.6 nm for modes A, B, and C. (d,f,h) Associated phase (ϕz).
134
Chapter 5: Quantum Tunnelling Plasmonics
Within the nanogap region, defined in Fig. (5.12a), the phase varies smoothly
through the NP-NP nanocavity for modes A and B, suggesting local hybridised
dipolar-like field distributions. This can be seen more clearly in the zoomed in
field distributions of Fig. (5.12).
Figure 5.12: (a) Nanogap geometry. The NP dimer nanocavity-localised near-
field distributions (magnitude |E˜z| and phase ϕz) for modes (b) A, (c) B, and (d)
C at a separation d = 0.6 nm [173]. The white line demarcates the NP surface.
(e) The near-field magnitude of modes A, B, and C along a line midway between
the NP surfaces for different separations, at x = z = 0.
135
Chapter 5: Quantum Tunnelling Plasmonics
The pi phase jump in the nanocavity observed for mode C indicates it is
a higher-order nanocavity resonance consistent with a hybridised quadrupolar
charge distribution in the nanogap arising from the rapid charge oscillation
characteristic of higher-order resonances created via plasmon hybridisation [74,
89].
To compare the localisation of the three LSPP modes in the nanogap, the
electric field confinement factor γC is calculated and is given by
γC =
∫
cavity
∣∣∣E˜∣∣∣2 dy dz∫ ∣∣∣E˜∣∣∣2dy dz (5.11)
where the nanocavity domain is defined by a (∆y, ∆z) = (40 nm, 20 nm) area
centred at (y, z) = (0, 0) and the denominator integral spans the entire simu-
lation region. At a separation of d = 0.4 nm, this yields confinement factors
of 0.57, 0.78, and 0.79 for modes A, B, and C, respectively, showing that the
field intensity distribution associated with modes B and C is more concentrated
within the nanocavity than the mode A field intensity.
The differences in nanocavity confinement and peak field enhancements in-
dicate that the LSPP charge associated with mode A is less-localised to the cavity
and extends over more of the neck-NP structure in comparison to modes B and
C. For these reasons, as the AFM tips (NPs) approach, the experimentally mea-
sured red-shift of mode A is significantly less than that of modes B and C, as
shown in Fig. (5.6). This also partially explains why the quantitative agreement
between experiment and the QCM for the wavelength and linewidth of mode
A is not as close as for modes B and C. The relatively extended nature of mode
A increases the susceptibility of the mode to the geometry of the extended neck
(and possibly AFM tip structure). These geometrical aspects are particularly
challenging to exactly reproduce in the theoretical treatment, and thus a greater
discrepancy between experiment and theory is to be expected.
The electric near-field along a line midway between the NP surfaces as the
nanocavity width is reduced is shown in Fig. (5.12e). For d > 0.3 nm, modes A
and B show a single field maximum at the point on the NP dimer axis (y = 0)
while mode C, being higher-order, exhibits a rapid field oscillation. At d = 0.2
nm, the QT charge transfer has increased sufficiently to suppress the field near
136
Chapter 5: Quantum Tunnelling Plasmonics
the QT channel at y = 0, indicating the emergence of QT charge-transfer modes
before contact. When the NPs eventually touch (d = 0 nm), the field is com-
pletely excluded from the centre point and localised to the anti-wedge regions in
the crevices around the contact. Overlapping the NP surfaces produces contact
modes similar to those seen for classical overlapping NP dimers (Sec. (2.3.4)),
although QT blurs the apex of the anti-wedge regions in contrast to the infinitely
sharp geometries considered in such classical simulations [75, 89, 170].
Finally, it is difficult to determine discrepancies between experiment and the
theoretical QCM that allude to other non-local effects that are not fully taken
into account, e.g. non-local screening phenomena. It is important to consider
the main sources of such possible discrepancies in more detail, as listed below
• Non-local screening and confinement effects not taken fully into account
in the QCM (outside the tunnelling volume) that are expected to blue-shift
(reduce the magnitude of the red-shift of the LSPP modes before dQR) and
slightly broaden the LSPP modes.
• The molecular layer(s) present in the NGJ are not considered in the QCM
and are likely to modify σQT0 and red-shift the LSPP modes. Such layer(s)
could account for the discrepancy in strength of LSPP modes B and C
found experimentally compared to those obtained from the QCM.
• Idealised geometry used in the QCM, most notably the simplifications of
the smooth 50 nm Au coating, neck region and tip length can all modify
the wavelength and linewidth of the LSPP modes.
• Different dielectric properties of the thin-film Au tip coating compared to
those used in the QCM could strongly modify the linewidth of the LSPP
modes. This could be the primary reason for the increased linewidth of all
LSPP modes predicted by the QCM compared to experiment.
• Assumed 1D (smooth interface surfaces) linear (low-voltage domain) tun-
nelling. It is quite possible the large field enhancements at the NGJ and
relatively intense laser illumination drive the system into a non-linear pho-
toemission regime [178]. Additionally, plasmon-induced transport is not
considered, and is likely to increase σQT0 yet further.
137
Chapter 5: Quantum Tunnelling Plasmonics
• Illumination and collection conditions, e.g. light scattering from the base
of the tip modelled in the QCM would not be collected in experiment as
all light scattering from outside the NP apex region is spatially filtered.
All of these points will be addressed in future theoretical work, and the influ-
ence such approximations have on the predictions will be critical to understand.
It is interesting to note that certain relative effects not predicted by the QCM are
observed in the experimental data, e.g. the reduced magnitude of LSPP mode
red-shift before dQR and the significantly increased linewidth of mode E com-
pared to modes B and C (Fig. (5.7a)). The reduced magnitude of red-shift has
been predicted by recent full hydrodynamic simulations and found experimen-
tally [179], while the linewidth discrepancy is predicted by ab initio TDDFT cal-
culations [153]. Both studies include the additional non-local screening effects
not treated in the QCM and therefore might suggest that these particular dis-
crepancies could be due to non-local surface charge smearing. However, these
remaining non-local effects are clearly not the dominant physical processes de-
termining the LSPP mode behaviour in the deep sub-nanometre NGJ regime.
5.5.3 Coherent Plasmon-Exciton Coupling
Evidence of the extremely small mode volumes accessed (Sec. (5.4)) may in-
deed have already been measured experimentally. As introduced in Sec. (2.3.3),
under certain conditions the hybridisation of an active emitting molecular tran-
sition and a LSPP mode is possible. When the hybridisation is sufficiently strong
the coherent coupling forms a mixed-state of a plasmon and molecular exciton,
a plasmon enhanced exciton-polariton. In this case it is possible that the active
species are hydrocarbon molecules adsorbed on the Au NP surfaces due to, e.g.
operating in ambient conditions, SEM contamination or packaging contamina-
tion.
Achieving strong plasmon-exciton coupling depends on the enhancement of
the Local Density Of photonic States (LDOS) at the positions of the molecules,
the oscillator strength of the emitting molecular transition, the spectral over-
lap of the relevant LSPP mode and molecular transition, and the number of
molecules present [180–182]. Assuming the molecules are aligned parallel with
the NP dimer axis an estimate of the maximum LDOS enhancement (γLDOS)
within the NGJ is found from the Purcell factor (Fp) associated with LSPP modes
138
Chapter 5: Quantum Tunnelling Plasmonics
γLDOS ≈ Fp =
3Q`LSPP
4pi2V
(
λ0
nair
)3
(5.12)
For LSPP mode B at V = Vmin this yields a value of γLDOS ≈ 5 × 107 that
is generally 103-104 times greater than the values currently experimentally re-
alised [175, 176, 183, 184]. This extreme enhancement of the LDOS can be at-
tributed to the quantum-limited mode volume achieved, the relatively high
value of Q`LSPP ≈ 15 for LSPP mode B, and the low refractive index of the
NGJ air medium.4 Following from Fermi’s golden rule, the spontaneous radia-
tive decay rate of a quantum emitter is proportional to the number of photonic
states the local environment offers for the decay. Thus the radiative decay rate
associated with the molecules in the NGJ is strongly enhanced.
For plasmonic cavities the Purcell factor can only be considered as an ap-
proximate guideline for the true LDOS value. More rigorous calculations that
include absorption show that Purcell analysis generally significantly underesti-
mates the LDOS in plasmonic NGJs [175]. For plasmonic (lossy) NGJs the sum
of the radiative (ηrm) and non-radiative (ηnrm ) decay rates is proportional to the
LDOS, (ηrm + ηnrm ) ∝ LDOS, rather than the ηrm ∝ LDOS relation found when
assuming the cavity is loss-less. The molecular species expected to be present
in the NGJ are likely to have relatively low free-space quantum efficiency, e.g.
ηr0/(η
r
0 + η
nr
0 ) ∼ 10−2 where ηr0 and ηnr0 are the free-space radiative and non-
radiative decay rates associated with the emitting molecular transition. It is
therefore expected that non-radiative decay is dominant. However, in the large
closely-spaced NP dimer systems used here, the scattering is very strong and
dominates absorption. It is therefore expected that ηrm will be strongly enhanced
compared to ηnrm as the NP dimer does not provide relatively many additional,
fast, non-radiative routes for decay. In the following simple model it is there-
fore assumed that the contribution to the plasmon-molecule coupling from the
enhanced non-radiative decay (quenching) is negligible compared to the contri-
bution from the enhanced radiative decay.
Additionally, the large electric-field in the NGJ (E ∼ 109 Vm−1) maximises
(up to saturation) the absolute number of excitation (and therefore emission)
events. It should also be noted that this plasmonic dimer approach also removes
4Note the LDOS value stated would be reduced by approximately a factor of two if the NGJ
medium was taken to be water.
139
Chapter 5: Quantum Tunnelling Plasmonics
some of the difficulty in placing the quantum emitters at the precise position of
maximum electric field enhancement (maximum LDOS) and strength (maxi-
mum absolute number of interaction events). In this case the cavity (NGJ) is
formed around the quantum emitters rather than having to precisely place the
quantum emitters in the cavity, e.g. at a field anti-node in a planar microcavity.
A simplified classical coupled harmonic oscillator model can be used to ap-
proximately describe the light-exciton coupling [185]. Under the assumptions
that the excitonic molecular absorption transition energy (EAm) and emission
transition energy (EEm) are approximately equal and resonant with the LSPP
mode (zero de-tuning), i.e. Em = E`LSPP ≈ EAm ≈ EEm, the two conditions re-
quired for strong coupling in a plasmonic nanocavity are [181, 185]
g >
∣∣Γ`LSPP − Γm∣∣
4
(5.13)
ΩR >
(
η`LSPP + ηm
)
2
(5.14)
where g is the ‘zero linewidth’ coupling energy between the molecular transi-
tions and the LSPP mode, Γm is the total homogeneous plus inhomogeneous
linewidth of the emitting molecular transition, η`LSPP and ηm are the total decay
rates of the LSPP mode and emitting molecular transition5 respectively, and ΩR
is the Rabi-oscillation frequency.
The first condition (Eqn. (5.13)) is generally satisfied when g is large and the
molecular transitions are ‘impedance’ matched with the LSPP mode, i.e. equal
in linewidth.6 The second condition (Eqn. (5.14)) shows that strong coupling
begins to become evident when the light-exciton energy transfer rate (ΩR) is on
the order of, or greater than, the individual total decay rates associated with
the emitting molecular transition and the LSPP mode. Under this condition,
a periodic exchange of energy at frequency ΩR (vacuum Rabi-oscillations7) can
5Note the total decay rate ηm includes contributions from additional homogeneous broaden-
ing (due to the large number of non-radiative decay routes), and inhomogeneous broadening,
and thus is significantly greater than (ηrm + ηnrm ).
6This condition is not satisfied for quantum dots at room temperature in high-Q cavity sys-
tems. Room temperature strong coupling in high-Q systems can be achieved using relatively
large numbers of organic molecules with large oscillator strengths, such as J-aggregates [186].
7Multi-photon effects are negligible due to low cavity Q-factor (short cavity photon lifetime).
140
Chapter 5: Quantum Tunnelling Plasmonics
occur between the coupled plasmon-exciton modes before the excitation decays.
Under the assumption that Eqn. (5.13) is satisfied, the magnitude of the Rabi-
splitting energy (h¯ΩR) is found to be [185]
h¯ΩR = 2
(
g2 −
(
Γ`LSPP − Γm
)2
16
)1/2
(5.15)
Furthermore, neglecting non-radiative decay effects, an analytical expression for
g is found to be [181]
g2 ≈ 1
2
h¯2ηr0η
`
LSPPγLDOSNm (5.16)
Now making the valid assumption that Γm ∼ Γ`LSPP at room temperature en-
ables an approximate expression for the Rabi-splitting energy to be determined
by substituting Eqn. (5.16) into Eqn. (5.15)
h¯ΩR ≈
(
2h¯Γ`LSPPη
r
0γLDOSNm
)1/2
(5.17)
where Nm is the number of molecules present in the NGJ. Note that ηr0 is propor-
tional to the oscillator strength of the emitting molecular transition. Substitut-
ing Eqn. (5.17) into Eqn. (5.14) yields the condition for the onset of measurable
strong plasmon-exciton coupling
ηr0 '
(
Γ`LSPP + Γm
)2
32h¯Γ`LSPPγLDOSNm
=
(
η`LSPP + ηm
)2
16η`LSPPγLDOSNm
=
ηSC
Nm
(5.18)
where ηSC is the ‘strong coupling’ decay rate (defined for brevity).
The total decay rate associated with the plasmonic nanocavity is typically
η`LSPP = Γ
`
LSPP/2h ≈ 1× 1013 s−1. A standard dye-molecule has ηr0 ∼ 0.25× 109
s−1 and ηm = 6× 1012 s−1 [181], and hence, for LSPP mode B, (ηr0/ηSC)Nm ∼
(103)Nm. Using this analysis for systems where strong coupling to single quan-
tum dots has been observed yields (ηr0/ηSC) ∼ 10 for a typical microcavity
system [182] and (ηr0/ηSC) ∼ 100 for a typical photonic crystal nanocavity sys-
tem [184]. Both experiments are performed at cryogenic temperatures to keep
the total linewidth of the quantum dot’s transitions low compared to that of the
141
Chapter 5: Quantum Tunnelling Plasmonics
high-Q cavity mode linewidth. This implies Eqn. (5.13) is satisfied, and Eqn.
(5.18) is valid to good approximation. Therefore, as expected, the condition
for the onset of strong coupling is satisfied in both cases. For the present NP
dimer system this basic model therefore demonstrates the condition for strong
coupling could be satisfied by at least three orders of magnitude. The extreme
enhancement of the LDOS and relatively large LSPP mode linewidth relax the
general requirement for quantum emitters with strong, narrow absorption and
emission transitions. Analysis shows that even for molecular transitions with
Γm ∼ 0.5 eV (∆λ0 ∼ 200 nm at λ0 = 670 nm) strong coupling could be possible,
i.e. Eqn. (5.13) and Eqn. (5.14) are satisfied. Therefore, even for the weak, broad
absorption and emission transitions expected for the molecules situated in the
present NGJ, measurement of mixed plasmon-exciton modes may be feasible.
Unexplained Experimental Results
The molecules present in the NGJ are expected to have significantly weaker ac-
tive transition oscillator strengths compared to standard dye molecules or quan-
tum dots, however spectral signatures of strong coupling such as mixed LSPP-
exciton mode-splitting or anti-crossing may still be measurable [181]. Anoma-
lous mode-splitting is indeed observed in the same experimental runs shown in
Fig. (5.7a) and Fig. (5.8). The measured scattering spectra taken just after StoC,
i.e. the region below 3 nN of applied force not shown in Fig. (5.7a), are shown
in Fig. (5.13).
At d ∼ 1 nm mode B splits into two distinct peaks before converging back
to a single peak as d is reduced further. At d ∼ 1 nm, ηrmηSC ∼ 50Nm for mode B.
This shows strong coupling might be possible, even for molecules with broad
transitions and relatively low oscillator strength. It is proposed that initially,
as d decreases, the available LDOS in the vicinity of the molecules increases
(and molecular alignments possibly change) and thus the mixed mode-splitting
becomes greater. It is suggested that the disappearance of the the mixed mode-
splitting could be due to the molecules re-orientating, photobleaching, or being
progressively ‘pushed’ out of the NGJ as d is reduced to the atomic-scale.
142
Chapter 5: Quantum Tunnelling Plasmonics
Figure 5.13: Evolution of LSPP modes with increasing applied compressive
force (reducing d) after StoC. Dashed black lines guide the eye. Mode B
has clearly split into two distinct peaks (possibly two mixed plasmon-exciton
modes (red spectra)) at around 1.9 nN of applied force (d ∼ 1 nm).
To fully verify that the strong coupling regime has been reached would re-
quire tuning of either the molecular transition (exciton) energy or the LSPP
mode energy in order to observe anti-crossing behaviour of the mixed plasmon-
exciton modes. However, at around zero de-tuning, the two peaks in the scatter-
ing spectra are expected to be approximately similar in shape and cross-section,
as both correspond to equally mixed plasmon-exciton modes [181,182,187]. For-
tuitously, the mode-splitting shown in Fig. (5.13) shows two approximately
similar peaks around 1.7 nN of applied force (20% difference in height). There-
fore, from the peak-splitting, approximate values for the Rabi-splitting energy
and molecular exciton emission energy can be extracted, and are found to be
h¯ΩR ≈ 150 meV and Em ≈ 1.8 eV respectively. It is expected the true value for
143
Chapter 5: Quantum Tunnelling Plasmonics
ΩR will be slightly less than that determined here, however reasonable conclu-
sions can still be made.
Using the experimentally found value of ΩR and assuming the limiting case
of very broad absorption and emission molecular transition linewidths Γm ≈ 0.5
eV, i.e. Qm ≈ 3.5 (linewidth approximately an order of magnitude greater than
that of a typical J-aggregate), yields g ≈ 125 meV (calculated from Eqn. (5.15))
and (just) satisfies the strong coupling conditions. The free-space spontaneous
radiative decay time of the molecules in the NGJ can now be estimated from g
(Eqn. (5.16))
τr0 ≈
h¯2η`LSPPγLDOSNm
2g2
(5.19)
Evaluating Eqn. (5.19) yields τr0 ∼(5 ns ×Nm). If the NJG contains only one
active molecule then τr0 ∼ 5 ns, however if the NGJ is full with active molecules
of similar size to water then τr0 ∼ 10 µs. These radiative decay times determine
the approximate range of oscillator strength ( fm) that could be associated with
the active molecules within the NGJ. If one molecule is present fm ≈ 4 and if the
NGJ is full fm ∼ 2× 10−3 (Nm ≈ 4000). Typical quantum emitters have oscillator
strengths ranging from fm ∼ 0.5 for weakly emitting organic molecules to fm ∼
50− 100 for large inorganic quantum dots. Therefore it is possible extremely
weak transitions are measurable due to the strong coupling with the nanocavity
LSPP mode.
It is interesting to note the same effect is possibly briefly observed in Fig.
(5.8) around 0.3 nN of applied force. The associated scattering spectra are
shown in Fig. (5.14). Analysis of the data shows that the mixed mode-splitting
is approximately 40% greater in magnitude, and centred slightly to the right
(λ0 = 690 nm), of that shown in Fig. (5.13). This could indicate the same
molecular species is present in both experimental runs, even though a differ-
ent pair of NP functionalised AFM tips were used. In this case the increased
mixed mode-splitting of the disimilar peaks observed could be primarily due
to the different energy of LSPP mode B (different de-tuning) and the reduced
NP separation (modified LDOS). As the peak-splitting occurs within the quan-
tum regime, analysis of the results is further complicated by the strong tuning
(blue-shifting) of LSPP mode B. Therefore, an approximate value for the Rabi-
splitting energy cannot be extracted from these low resolution measurements.
Finally, perhaps due to the weak coupling between the LSPP mode and molec-
144
Chapter 5: Quantum Tunnelling Plasmonics
ular excitons at around 0.2 nN of applied force, it is noted that the lineshape of
LSPP mode B abruptly changes just before mode-splitting occurs.
Figure 5.14: Evolution of LSPP modes with increasing applied compressive
force (reducing d) after StoC. The blue spectrum marks CC while the red dots
show the possible splitting of a mixed plasmon-exciton mode at approximately
0.3 nN of applied force (d ≈ 0.3 nm).
These interesting results invite further controlled experimentation to confirm
the strong coupling hypothesis. For example, the entire tips could be coated
in high damage threshold organic dye molecules with excitonic transitions at
energies and linewidths matched to the LSPP modes. If the mode-splitting
observed is indeed a manifestation of strong plasmon-exciton coupling, this
would be the first experimental evidence for such coupling within a plasmonic
NP dimer nanocavity containing a small number of molecules. Finally, it is
noted here that although the author believes plasmon-exciton coupling is the
most likely explanation of the observed results, LSPP mode splitting phenom-
ena could also possibly result from asymmetries in the NP geometry and com-
position (heterodimer), enabling both bright and dark LSPP mode excitation
and anti-crossing phenomena [74]. Additionally, the double quantum well po-
145
Chapter 5: Quantum Tunnelling Plasmonics
tential formed due to the presence of molecules in the NGJ would permit sym-
metric and anti-symmetric plasmon wavefunction solutions of different energy
and hence could also provide a possible explanation. Both these explanations
are however considered unlikely, as the heterodimer hypothesis predicts split-
ting of LSPP mode C, while the non-degenerate wavefunction hypothesis (at
the least) requires very narrow barrier widths that are not present immediately
after StoC.
5.6 Conclusions
Contrasting the classical simulations with the experimental and QCM results
(Fig. (5.7)), clearly shows the dramatic effect of non-local quantum tunnelling
on plasmonic interactions. Only by incorporating non-local QT effects via the
QCM can the experimental results begin to be properly accounted for, thus en-
abling the clear identification of where the plasmonic interaction crosses-over to
the quantum regime. Other effects, such as non-local screening outside the tun-
nelling region, cannot attribute for the measured LSPP response, and therefore
it is found that QT dominates the plasmonic interactions in nanocavities formed
by NP dimers with ångström scale inter-NP separation.
These experimental and theoretical investigations of plasmonic interactions
in sub-nanometre metal inter-NP cavities demonstrate that quantum mechanics
is highly important at approximately the 0.3 nm scale where non-local tun-
nelling plasmonics controls the optical response. The findings finally experi-
mentally resolve the unphysical nature of the long-standing classical predictions
for the plasmonic response for nearly-touching NPs. The results also imply that
QT establishes a quantum limit for plasmonic field enhancement and confine-
ment. Additionally, it is possible measurements in the sub-nanometre regime
show the first experimental evidence of strong plasmon-exciton coupling in
plasmonic NGJs within NP dimer systems. Understanding both of these as-
pects of plasmonic phenomena is crucial for describing light-metal interactions
down to the atomic-scale. It is also expected stabilising single-atom contacts or
wires will give direct plasmonic access to the quantum transport regime around
1G0 [146].
The present work opens up exciting new prospects that aim to direct
and control the chemistry of single molecules within nanogaps (e.g. for
146
Chapter 5: Quantum Tunnelling Plasmonics
enhanced photocatalysis), exploit single-molecule plasmon-assisted transport
across nanogaps (e.g. for single molecule electronics), enable extreme non-linear
interactions (e.g. attosecond pulse generation), access photoelectrochemistry on
the sub-zeptoliter scale and possibly admit the study of plasmon-exciton (plex-
citon) quasiparticles.
147
148
Chapter 6
Summary and Outlook
In the preceding chapters the theoretical background, experimental apparatus
and techniques, and new experimental results revealing the interaction of light
with metallic nanosystems on the sub-nanometre scale have been presented and
analysed. Here a brief summary of the work carried out for this thesis is pro-
vided, followed by suggested avenues for new or continued investigation.
In Chapter (2) a light-weight overview of the theoretical framework describ-
ing plasmonic interactions propagating on flat planar interfaces was first pre-
sented to introduce the mixed-state of light and metal described by SPPs. From
this foundation, it was described how the control of SPPs via localisation on
isolated sub-wavelength scale NPs is a powerful phenomena that contributed
to the resurrection of metal optics as the now ubiquitous field of plasmonics.
This led onto a more detailed theoretical review of coupled LSPPs in NP dimer
systems and the objectives of the current work were made clear.
As two nanosystems move into single nanometre separation their plasmonic
(and hence optical) response is dominated by the electromagnetic coupling
across the NGJ created. A huge number of future (and current) applications
will rely on the this tunable, non-linear optical response, and the associated
nano-localised, extreme field and LDOS enhancements obtained. The fascinat-
ing sub-nanometre regime is almost unexplored by plasmonics orientated ex-
periments and only now is becoming accessible to viable theoretical treatments.
The absolute necessity to investigate and understand this truly nanoscale regime
where spatially non-local effects, predominantly quantum tunnelling, control
the plasmonic interactions has been the driving force behind the experimental
and theoretical investigations in this thesis.
149
Chapter 6: Summary and Outlook
The experimental apparatus realised to access and interrogate the challeng-
ing sub-nanometre domain between two almost-touching NPs is described in
Chapter (3). The sub-nanometre NGJ was created by the appropriate axial tip-
to-tip placement of two conducting AFM probes with NP functionalised apices.
The description of the overall setup is divided into three sections specifying
the mechanical, electronic and optical components. All three of these systems
were required to work in concert to enable high-quality simultaneous optical
and electronic measurement of the LSPP mediated scattering response of a dy-
namically controlled, atomically separated, NP dimer system.
The detail behind the dynamic AFM tip-to-tip (NP dimer) alignment strategy
used to enter the sub-nanometre regime was presented in Chapter (4). A novel
non-linear EFM technique was implemented by placing an AC potential across
the AFM tips, and measuring the harmonic frequencies generated by the non-
linear electromechanical response of the tip-tip system using lock-in detection.
The tips (NPs) are brought into nanometre-precise 3D alignment by adjusting
the tip (NP) positions to maximise signal harmonics, thus creating an effective
NP dimer system.
Finally, in Chapter (5) the experimental facilities constructed, and techniques
developed, are combined to simultaneously perform low-background super-
continuum confocal laser DF nano-spectroscopy and DC conductance measure-
ments on NP dimer systems with sub-nanometre separation. By simultaneously
capturing both the electrical and optical scattering properties of a NP dimer,
with controllable sub-nanometre separation, the first evidence for the quantum
regime of optically driven tunnelling plasmonics has been observed in exquisite
detail. The experimental results are found to be in good agreement with a very
recently developed QCM where non-local quantum tunnelling effects are taken
into account for large-scale (> 3 nm dimension) plasmonic systems. These
experimental and theoretical investigations of plasmonic interactions in metal-
lic NP dimer nanocavities demonstrate that quantum mechanics plays a critical
role at the 0.3 nm scale where non-local tunnelling plasmonics determines the
optical response. The findings also strongly imply that quantum tunnelling es-
tablishes a limit for plasmonic field localisation and enhancement in nanogaps.
The exceptionally large LDOS within the NGJ created may also enable strong
coupling of light to molecular excitons thus creating mixed plasmon-exciton
(plexciton-polariton) states. Understanding these aspects of plasmonic response
150
Chapter 6: Summary and Outlook
is crucial for describing light-matter interactions down to the atomic-scale, and
is hence highly relevant to a huge variety of existing, and future, research paths
and plasmonic technologies.
The experimental system that has been developed, and the results reported
in this thesis, suggest many interesting opportunities for further investigation.
For example, the work presented here opens up exciting new prospects includ-
ing:
• The possibility to direct and control the chemistry of single molecules
within nanogaps, e.g. for enhanced photocatalysis.
• The exploitation of single-molecule plasmon-assisted transport across na-
nogaps, e.g. for single molecule (optical frequency) electronics.
• Access to photoelectrochemistry on the sub-zeptoliter scale.
• The plasmonic control of non-linear electron tunnelling processes by light.
• Study of optical frequency tunnelling and non-linear optical frequency
rectification phenomena.
• Investigations of the quantum-limit of electric-field mediated enhance-
ment in SERS and optimised sensing geometries.
• The capability to dynamically investigate NP dimer systems bridged by
1D atomic chains formed on the make or break of NP contact. Such dis-
crete steps in nanojunction conductance could lead to associated discrete
plasmonic switching phenomena.
• Extreme classical non-linear phenomena, e.g. high-harmonic generation.
• Ultra-high LDOS enhancement and strong plasmon-exciton coupling with-
in NP dimer systems.
In general, the experimental apparatus and techniques described in this work
may be used for initial experimental investigations relating to fundamental re-
search and applications that rely on, or are optimised by, nanostructures with
sub-nanometre features. The results and interpretations presented in this thesis
are expected to be of interest to the general plasmonics research community and
beyond.
151
152
Appendix A
Charge Transfer Model
The anomalous dependence of Eqn. (5.10) on λ0 can be explained by a more
rigorous treatment of the physical system by including the dielectric properties
of the fictitious quantum tunnelling medium and the Au NPs. These physi-
cal properties are included by considering the displacement field. Following
Gauss’s law, integrating over a Gaussian pillbox containing one of the AFM tip
apex NP surfaces (assumed flat, as a  d) and assuming that the electric field
created due to the localisation of free surface charges is significantly larger than
the incident field gives
∮
A
D˜ · dA = Q˜SP ⇒ ε0
(
ε˜QTESP + ε˜AuEAu
)
=
Q˜SP
ASP
(A.1)
where ESP is the LSPP enhanced electric field in the nanojunction, EAu is the
electric field within the metal surface, QSP is the amount LSPP free surface
charge, ASP is the surface area over which the LSPP charge is situated. Finally
ε˜Au is the dielectric function of Au, and ε˜QT is the dielectric function of the
fictitious tunnelling medium in the nanojunction, given by
ε˜QT = 1− ω
2
p
ω (ω+ iγQT(d))
(A.2)
where γQT(d) is the quantum tunnelling damping parameter with typical val-
ues shown in [153] and ωp is the plasma frequency of Au. Similarly the dielectric
153
Appendix A: Charge Transfer Model
function of Au is given by
ε˜Au = 1−
ω2p
ω (ω+ iγAu)
(A.3)
Now re-arranging Eqn. (A.1) for the magnitude of Q˜SP gives
QSP(ω, d) = ε0ASP
∣∣∣(ε˜QT(ω, d)ESP + ε˜Au(ω)EAu)∣∣∣ (A.4)
The magnitude of electric current through the tunnelling nanojunction INJ is
found by
INJ(ω, d) = σQT(ω, d)AQTEQT (A.5)
where AQT is the surface area of the nanojunction, EQT is the electric field mag-
nitude driving the tunnelling current flow and σQT is the magnitude of the
effective tunnelling conductivity given by
σQT(ω, d) =
γQTσQT0
(ω2 + (γQT)2)
1/2 (A.6)
where σQT0 (d) = ε0ω
2
p/γQT(d) (due to the assumption T(Ω, d) adiabatically
follows the field, hence a DC description of tunnelling can be used). Using
Eqn. (A.5) the amount of QQT transferred across the nanojunction over half an
optical-cycle τ1/2 = λ0/2c is
QQT(ω, d) = σQT(ω, d)AQTEQT
(
λ0
2c
)
= σQT(ω, d)AQTEQT
(pi
ω
)
(A.7)
Hence a general expression for X is found
X =
QQT(ω, d)
QSP(ω, d)
=
σQT(ω, d)AQTEQT
(
λ0
2c
)
∮
A D˜(ω, d) · dA
(A.8)
154
Appendix A: Charge Transfer Model
Under the typical conditions described in Eqn. (A.1), the expression for X re-
duces to
X ≈
σQT(ω, d)AQTEQT
(
λ0
2c
)
ε0ASP |(ε˜QT(ω, d)ESP + ε˜Au(ω)EAu)| (A.9)
Now, under the general assumptions d ' 2Å, i.e. γQT > ωp  γAu, and that
ω is relatively low, i.e. ω around typical LSPP resonance frequencies so that
ESP  EAu, Eqn. (A.9) further reduces to
X ≈
σQT0 (d)AQTEQT
(
λ0
2c
)
ε0ASPESP
(A.10)
as σQT(ω, d) ≈ σQT0 (d) and ε˜QT ≈ 1. Making the assumptions as in the main
text that ASP = AQT and ESP = EQT, finally gives Eqn. (5.7)
X ≈ σQT0 (d)
(
λ0
2ε0c
)
(A.11)
Therefore the λ0 dependence of Eqn. (5.7) and hence that of Eqn. (5.10) is ap-
proximately valid in the combined separation and frequency regime assumed.
The following contains more detail on how taking into account the displace-
ment field modifies the unphysical limiting behaviour (in ω) of the charge-
transfer time-scale model in [98] to give physically plausible results. The condi-
tion for the onset of the blue-shift is given by [98]
τ =
QSP
INJ
≤ pi
2ω
(A.12)
where τ is the necessary time required for the charge transport and the furthest
RHS term is found due to the simple time-scale argument in [98]. Using Eqns.
(A.4) and (A.5) in Eqn. (A.12) gives
ε0ASP
∣∣(ε˜QT(ω)ESP + ε˜Au(ω)EAu)∣∣
σQT(ω)AQTEQT
≤ pi
2ω
(A.13)
155
Appendix A: Charge Transfer Model
The limiting behaviour can be understood as follows. In the case where only the
electric field (not displacement field) is considered (nanojunction is considered
as a vacuum with no tunnelling, as in the main text) the necessary time required
for the charge transport (i.e. for the onset of the blue-shift) is given by Eqn.
(A.12)
τ =
pi
2ω
→ ∞ as ω → 0
→ 0 as ω → ∞
(A.14)
However when the frequency dependent properties of the fictitious tunnelling
junction and Au NPs are taken into account the expression for τ is modified
considerably. Under the general assumptions d > 2Å, i.e. γQT > ωp  γAu,
and ESP = EQT, the low and high frequency behaviour of the time allowed for
charge transfer τ is determined.
For small ω the field inside the Au NPs is negligible compared to that out-
side, ESP  EAu. Additionally σQT ≈ σQT0 . Under these conditions Eqn. (A.13)
reduces to (under the assumption ESP = EQT)
ε0ASP
(
εQT(ω)
)
σQT0 AQT
≤ pi
2ω
(A.15)
An expression for the time allowed for charge transport is found by re-arranging
Eqn. (A.15). As ω → 0 we now have
τ =
piε0
2σQT0
=
(piε0
2
)
ρQT0 (A.16)
therefore in the DC limit the upper bound for the time allowed for the charge
transfer process is not dependent on ω but is proportional to the effective quan-
tum DC tunnelling resistivity ρQT0 of the nanojunction. The greater ρ
QT
0 the
greater the amount of time required to transfer the necessary charge. Likewise,
as ρQT0 decreases, the required amount of time decreases.
For large ω (ω  γQT) it is assumed that ESP ≈ EAu ≈ Einc (where Einc
is the incident field) as the electrons in the Au NPs do not screen the incident
field as effectively because ω > ωp. Therefore the amount of charge localisation
at the nanojunction is relatively small. Therefore (via Eqn. (A.1)) Eqn (A.13) is
156
Appendix A: Charge Transfer Model
modified to (under the assumption ESP = EQT)
ε0ASP
∣∣(ε˜QT(ω)− ε˜Au(ω))∣∣
σQT(ω)AQT
≤ pi
2ω
(A.17)
In this case, from Eqn. (A.17), the time allowed for charge transport is
τ → ∞ as ω → ∞ (A.18)
Therefore the upper bound for the time allowed is no longer zero but is infinity.
This can be understood because as ω → ∞, QSP → 0 rapidly compared to
the decreasing time of an optical cycle. Here QSP → 0 because as ω  ωp
the electrons can no longer respond to the incident field and hence no surface
charge density is induced. Therefore the amount of charge to be transported is
effectively zero and thus the charge transport can take place over any period of
time. The time-scale model is thus not applicable for ω  ωp.
The general condition for the onset of the blue-shift in terms of σQT is
σQT(ω) ≥ 2ωpiAQTEQT
∣∣∣∣(∮A D˜ · dA
)∣∣∣∣ (A.19)
It should be noted the RHS is (for d ' 2Å) a biquartic function of σQT0 and a
high-order polynomial function of ω. An explicit expression for the necessary
value of σQT0 as a function of ω could hence be obtained if required. The value
for σQT0 can be compared against that found from analytical Quantum Mechan-
ical (QM) theory [177] at different d to find the distance required for the onset
of the blue-shift.
In the typical tunnelling regime (3Å≤ d ≤ 6Å, i.e. EQT  EAu and γQT >
ωp > ω  γAu) at optical frequencies. In this case Eqn. (A.19) simplifies
significantly and reduces to
σQT0 '
2ε0ωASP
piAQT
(A.20)
157
Appendix A: Charge Transfer Model
In terms of current density JQT0 we have
JQT0 '
2ε0ωASPESP
piAQT
(A.21)
In the regime of interest 2.5Å≤ d < 6Å Eqn. (A.21) can hence be used to
equate to the tunnelling current density found from analytical QM theory [177]
to determine a value for dQR. For separations 1Å≤ d < 2.5Å the condition
γQT > ωp > ω  γAu no longer holds. Under this situation the full Eqn. (A.19)
needs to be used appropriately. In the 2Å≤ d < 3Å regime it is found that the
limit for JQT0 decreases with increasing ω rather than increasing with increasing
ω.
158
Appendix B
List of Symbols and Abbreviations
Symbols
α Fine structure constant.
α` Polarisibilty of `th localised surface plasmon mode
of spherical nanoparticle.
βzi Total z-component of non-linear damping coefficient
of tip system i.
βz0i z-component of fundamental damping coefficient
of tip system i.
βzTT z-component of air squeez viscous damping coefficient
of tip system i.
χe Effective electric susceptibility.
δ Height of parabolic tip apex.
δAu Skin-depth of gold.
δe Penetration depth of induced surface charge.
δd Penetration depth of surface plasmon polariton electric
field into dielectric.
δm Penetration depth of surface plasmon polariton electric
field into metal.
ε0 Permittivity of free-space.
ε1 Real part of complex scalar dielectric function.
ε2 Imaginary part of complex scalar dielectric function.
εair Scalar relative permittivity of air.
ε˜Au Complex scalar relative permittivity of gold.
εbk Scalar background contribution to dielectric function.
159
List of Symbols and Abbreviations
εd Scalar relative permittivity of a dielectric.
ε˜m Complex scalar dielectric function of a metal.
ε˜QT Complex scalar dielectric function of fictitious tunnelling medium.
ε˜r Relative complex scalar permittivity.
ε˜Lr Relative complex scalar permittivity for longitudinal excitations.
ε˜Tr Relative complex scalar permittivity for transverse excitations.
η Normalised refractive index.
ηr0 Spontaneous radiative decay rate of molecule in free-space.
ηnr0 Spontaneous non-radiative decay rate of molecule in free-space.
ηSC Total decay rate limit for strong coupling.
ηm Total decay rate of molecule.
ηrm Spontaneous radiative decay rate of molecule.
ηnrm Spontaneous non-radiative decay rate of molecule.
η`LSPP Plasmonic cavity total decay rate.
γ Phenomenological electron collision frequency.
γC Electric-field confinement factor.
γI Phenomenological electron collision frequency for the Ith oscillator.
γLDOS Local density of optical states enhancement factor.
γQT Quantum tunnelling damping parameter.
Γ`LSPP Localised surface plasmon polariton `
th mode energy linewidth.
Γm Total linewidth of molecular exciton resonance.
κ Extinction coefficient.
λ Wavelength of light in dielectric medium.
λ0 Wavelength of light in free-space.
λspp Wavelength of surface plasmon polariton.
λlspp Wavelength of localised surface plasmon polariton.
µ0 Permeability of free-space.
µa Coefficient of viscosity of air.
µ˜r Complex scalar relative permeability.
ω Angular frequency.
ω0i Resonance frequency tip system i.
ω0I Resonance frequency of Ith oscillator.
ω`np Localised surface plasmon resonance frequency of `th mode.
ωp Bulk plasma frequency.
ωP Pump frequency.
160
List of Symbols and Abbreviations
ωsp Surface plasmon frequency.
ωS Signal frequency.
Ω Electron energy.
ΩF Fermi energy.
ΩR Rabi-frequency.
φ Spherical azimuthal coordinate/diameter.
Φ Electrostatic potential.
ϕ3ωS Phase of signal at 3ωS with respect to the driving potential.
ϕHD Phase of harmonic distortion signal at 3ωS
with respect to the driving potential.
ϕi Phase of tip i oscillation with respect to the driving potential.
ϕWF Work-function.
ϕz Phase of z-component of electric near-field.
pi Pi (3.14159. . .).
ψ Phase of the superposition of the signal and harmonic distortion
with respect to the driving potential.
ψ` Ricatti-Bessel function of order `.
ρAu Mass density of gold.
ρext External charge density.
ρint Internal charge density.
σ˜ Complex scalar conductivity.
σ0 Zero-frequency scalar conductivity.
σQT Scalar quantum tunnelling conductivity.
σQT0 Zero-frequency scalar quantum tunnelling conductivity.
σ1 Real part of complex scalar conductivity.
σ2 Imaginary part of complex scalar conductivity.
σ˜m Complex scalar conductivity function of a metal.
σs Squeeze number.
τ Phenomenological electron relaxation time.
τ1/2 Period of half an optical cycle.
τint Integration time.
τc Cooling relaxation time.
τ`LSPP Localised surface plasmon polariton `
th mode lifetime.
τr0 Radiative decay time of molecule in free-space.
τP Laser pulse-width.
161
List of Symbols and Abbreviations
τTC Lock-in amplifier measurement time constant.
θ Spherical polar coordinate.
θ1, θ2 Define range of illumination angles.
θc Maximum collection angle.
θi Angle of incidence of electromagnetic wave.
θtip Tip (full) cone-angle.
Θ Difference between signal and harmonic distortion phase.
υ Effective refractive index weighting factor.
ξ` Ricatti-Bessel function of order `.
∧ Logical conjunction (AND).
a Nanoparticle radius.
a0 Inter-atomic distance.
a˜` Mie scattering coefficient.
A Effective cross-sectional area of a nanojunction.
dA Infinitesimal area element vector.
AC Cap area of collection cylinder.
Ad Field amplitude coefficient in dielectric.
AF Parallel plate area determined by experimental fit.
Am` Spherical polar expansion coefficient.
AL Laser illumination area.
ALI Lock-in amplifier gain.
Am Field amplitude coefficient in metal.
Aov Area of cantilever overlap.
AQT Effective quantum tunnelling nanojunction area.
ASP Surface area of localised surface plasmon polariton charge.
AT Transimpedance amplifier gain.
B Magnetic flux density field vector.
b˜` Mie scattering coefficient.
Bm` Spherical polar expansion coefficient.
c Speed of light in vacuo.
cAu Specific heat capacity of gold.
C Total tip system capacitance.
C0 Infinite parallel plate capacitance coefficient.
Cm0 Amplitude of capacitance modulation.
162
List of Symbols and Abbreviations
Cqsabs Quasi-static absorption cross-section.
Cbk Constant background capacitance.
CA Capacitance associated with tip apices.
CC Capacitance associated with tip cones.
Cext Mie extinction cross-section.
Cqsext Quasi-static extinction cross-section.
CF Additional fringe capacitance.
c˜i Complex Fourier coefficient of current at frequency iωS.
CL Capacitance associated with cantilevers.
CNP Nanoparticle heat capacity.
Cs Stray capacitance.
Csca Mie scattering cross-section.
CBEMAXsca Scattering cross-section calculated by BEMAX.
Cqssca Quasi-static scattering cross-section.
d Nanosystem separation.
D Electric displacement field vector.
d0 Separation in z when inter-nanoparticle force is zero.
dx,y0 Separation in x,y,z when inter-nanoparticle force is zero.
dQR Quantum cross-over separation.
DL Laser beam diameter
DOC Diameter of the refracting core of the objective.
dQR Quantum regime crossover distance.
d′QR Simple model estimate of quantum regime crossover distance.
DS Diameter of dark-field beam-stop.
e Electron charge.
E Electric field vector.
E0 Electric field amplitude vector.
Ee Electron energy.
E`LSPP Localised surface plasmon polariton `
th mode resonant energy.
EAm Resonance energy of absorbing molecular transition.
EEm Resonance energy of emitting molecular transition.
EQT Electric field driving quantum tunnelling.
ESP Localised surface plasmon polariton enhanced electric field.
ET Thermal excitation energy.
Ex, Ey, Ez Electric field component amplitudes.
163
List of Symbols and Abbreviations
f Frequency.
F Electrostatic force.
f0i Resonance frequency of cantilever.
f e0i Effective resonance frequency of cantilever.
fA Empirical correction factor.
FzA z-component of driving electrostatic force associated with tip apex.
Fzbk z-component of driving electrostatic force associated the
background capacitance.
FzC z-component of driving electrostatic force associated with tip cone.
FzEL z-component of driving electrostatic force.
FzF z-component of driving electrostatic force associated the
cantilever fringe capacitance.
Ff lu Laser fluence.
FzL z-component of driving electrostatic force associated
with cantilever.
fm Oscillator strength.
Fp Purcell factor.
FPO Pull-off force.
fS Frequency used for alignment scans.
FzTT z-component of Lennard-Jones force.
FzVdW z-component of Van der Waals force.
fz z-component correction factor.
G Distance off tip-tip axis.
G0 Quantum conductance of one dimensional ballistic channel.
GDC Zero-frequency conductance.
gx x position of tip 2 in alignment grid.
gy y position of tip 2 in alignment grid.
Gx Alignment grid length in x.
Gy Alignment grid length in y.
h Planck’s constant/Cone height.
h¯ Dirac constant.
H Hamaker constant.
H Magnetic field vector.
h˜(1)` Spherical Hankel function of the 1
st kind of order `.
Hx, Hy, Hz Magnetic field component amplitudes.
164
List of Symbols and Abbreviations
i Imaginary unit.
I Current.
I0 Average optical intensity.
IAC Alternating current signal.
IAC0 Alternating current signal amplitude.
I˜ωS Complex component of current signal at ωS.
I˜3ωS Complex component of current signal at 3ωS.
I˜3ωSC Corrected complex component of current signal at 3ωS.
Ic Conduction current.
I˜d Displacement current vector.
IDC Zero frequency (DC) current.
I˜HD Complex component of harmonic distortion background.
current signal at 3ωS.
=[...] Imaginary part.
j` Spherical Bessel function of order `.
Jext External current density.
k Wavenumber of light in dielectric medium.
kx,ky,kz Wavevector components of light in dielectric medium.
k0 Free-space wavenumber of light.
kz0i z-component of fundamental cantilever spring constant
of tip system i.
kzi Total z-component of cantilever spring constant
of tip system i.
kB Boltzmann constant.
k˜E Evanescent light complex wavevector in dielectric medium.
kzei z-component of effective cantilever spring ‘constant’
of tip system i.
kz′ei z-component of time-averaged effective cantilever spring
‘constant’ of tip system i.
kGB Wavevector of incident Gaussian beam.
k˜lspp Tangential component of complex wavenumber.
k|| In-plane wavevector component of light in dielectric medium.
k˜spp x-component of complex surface plasmon polariton wavevector.
kzTT z-component of air squeeze elastic damping cantilever
spring constant.
165
List of Symbols and Abbreviations
k˜zd z-component of complex surface plasmon polariton wavevector
in dielectric.
k˜zm z-component of complex surface plasmon polariton wavevector
in metal.
l Side length of parallel plate.
` Orbital angular momentum number.
larc Length of circular arc.
LC Length of collection cylinder.
Lspp Surface plasmon polariton propagation length.
m Azimuthal number.
M Magnetisation field vector.
mi Effective mass of oscillating tip system i.
m∗e Effective optical mass of electron.
m∗I Effective optical mass of an electron contributing to
the Ith oscillator.
n Real part of complex scalar refractive index.
n˜ Complex scalar refractive index.
N Number of discrete points.
nair Refractive index of air.
ne Conduction electron density.
Ne Absolute number of conduction electrons.
nd Refractive index of a dielectric medium.
ned Effective refractive index of a dielectric medium.
nglass Refractive index of microscope slide.
nI Electron density contributing to the Ith oscillator.
Nm Absolute number of molecules.
P Total internal polarisation field vector.
P0 Average optical power.
Pa Ambient air pressure.
Pm` Associated Legendre polynomial.
q˜ Complex wavevector of light within an absorbing medium.
q˜ Complex wavenumber of light within an absorbing medium.
q˜0 Complex wavevector of light at the bulk plasma frequency.
qQT Wavenumber of a tunnelling electron.
Q Charge.
166
List of Symbols and Abbreviations
Q0i Quality factor of tip system i.
Q`LSPP Localised surface plasmon polariton `
th mode quality factor.
QQT Quantum tunnelling charge.
QSP Localised surface plasmon polariton surface charge.
r Position vector.
R Tip apex radius.
R¯ Reduced tip apex radius.
r˜0 Complex amplitude vector of electron oscillation.
RC Collection cylinder cap radius.
Rcl Current limiting resistance.
r˜e Complex electron position vector.
<[...] Real part.
t Time.
tL Thickness of insulating water and hydrocarbon layer(s).
T Absolute temperature.
TωS Period of signal at ωS.
T3ωS Period of signal at 3ωS.
TQT Electron tunnelling probability.
U Electrostatic potential energy.
V Mode volume.
V0 Alternating potential amplitude.
V3ωS0 Amplitude of harmonic distortion voltage signal at 3ωS.
V˜3ωSL Complex voltage signal at 3ωS measured
by lock-in amplifier.
VAC Alternating potential signal after amplification.
VAC0 Alternating potential signal amplitude.
VDC Amplitude of stationary potential.
vg Group velocity.
Vmin Quantum limited plasmon mode volume.
Vnp Nanoparticle volume.
Vo f f Zero-frequency offset potential.
Vopt Optically induced potential.
vp Phase velocity.
VT0 Alternating potential amplitude across atomic force tips.
w Lateral localised surface plasmon polariton confinement width.
167
List of Symbols and Abbreviations
wQL Quantum limit of mode confinement.
x, y, z Global lab frame co-ordinates.
X Fraction of neutralised charge.
zˆ Unit vector in the z-direction.
zmi Amplitude of oscillation of tip i.
zo f fi Additional offset in position of tip i.
Zc Load impedance associated with total tip system capacitance.
Zs Load impedance associated with stray capacitance of the system.
168
List of Symbols and Abbreviations
Abbreviations
1D, 2D, 3D One, two, three-dimensions.
AC Alternating current.
AFM Atomic force microscopy.
APP Amici prism pair.
BEM Boundary element method.
BEMAX Boundary element method for axially symmetric systems.
BF Bright-field.
BP Bulk plasmon.
BPF Band pass filter.
BS Beamsplitter.
CAD Computer aided design.
CC Conductive contact.
CCD Charge coupled device.
CDCP Chromatic dispersion compensation plate.
CI Collection iris.
CP Confocally-aligned pinhole.
CTPP Charge transfer plasmon polariton.
DC Direct current.
DDA Discrete dipole approximation.
DF Dark-field.
DFT Density Functional Theory.
DSO Digital storage oscilloscope.
EBD Electron-beam deposition.
EBL Electron-beam lithography.
EFM Electrostatic force microscopy.
EMF Electromotive force.
FDTD Finite difference time domain.
FWHM Full-width half-maximum.
GLMT Generalised Lorentz Mie Theory.
GPIB General purpose interface bus.
HD Harmonic distortion.
LDOS Local density of photonic states.
LP Linear polariser.
LSP Localised surface plasmon.
169
List of Symbols and Abbreviations
LSPP Localised surface plasmon polariton.
MCBJ Mechanically controllable break-junction.
NA Numerical aperture.
ND Neutral density.
NGJ Nanogap or nanojunction.
NP Nanoparticle.
ODE Ordinary differential equation.
p- Parallel.
PCF Photonic crystal fibre.
PZ Piezoelectric.
QCM Quantum corrected model.
QR Quantum regime.
QT Quantum tunnelling.
RC RC time constant.
RHS Right hand side.
RPP Radiative plasmon polariton.
s- Senkrecht (German: perpendicular).
SC Semiconductor.
SEM Scanning electron microscope.
SERS Surface enhanced Raman scattering.
SG Signal ground.
SMU Source measure unit.
SN Signal to Noise ratio.
SP Surface plasmon.
SPDT Single-pole double-throw.
SPP Surface plasmon polariton.
SSTM Static scanning tunnelling microscopy.
STM Scanning tunnelling microscopy.
StoC Snap-to-contact.
TDDFT Time-dependent density functional theory.
TE Transverse electric.
TM Transverse magnetic.
TTL Transistor transistor logic.
VA Voltage amplifier.
WKB Wentzel, Kramers, Brilluoin.
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