Plasmon coupling of gold nanorods at short distances and in different geometries.

Abstract

The experimentally determined scattering spectra of discrete, crystalline, gold nanorod dimers arranged side-to-side, end-to-end, at right angles in different orientations and also with longitudinal offsets are reported along with the electron micrographs of the individual dimers. The spectra exhibit both red- and blue-shifted surface plasmon resonances, consistent with the plasmon hybridization model. However, the plasmon coupling constant for gold dimers with less than a few nanometers separation between the particles does not obey the exponential dependence predicted by the Universal Plasmon Ruler equation. The experimentally determined spectra are compared with electrodynamic calculations and the interactions between the individual rod plasmons in different dimer orientations are elucidated.

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Letter
Plasmon Coupling of Gold Nanorods at
Short Distances and in Different Geometries
Alison M. Funston, Carolina Novo, Tim J. Davis, and Paul Mulvaney
Nano Lett., Article ASAP • DOI: 10.1021/nl900034v • Publication Date (Web): 09 March 2009
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Plasmon Coupling of Gold Nanorods at
Short Distances and in Different
Geometries
Alison M. Funston,*,†Carolina Novo,†TimJ. Davis,‡and Paul Mulvaney†
School of Chemistry and Bio21 Institute, UniVersity of Melbourne,
ParkVille, Victoria, 3010, Australia, and CSIRO Materials Science and Engineering,
Clayton South, Victoria, 3169, Australia
Received January 6, 2009; Revised Manuscript Received February 13, 2009
ABSTRACT
The experimentally determined scattering spectra of discrete, crystalline, gold nanorod dimers arranged side-to-side, end-to-end, at right
angles in different orientations and also with longitudinal offsets are reported along with the electron micrographs of the individual dimers.
The spectra exhibit both red- and blue-shifted surface plasmon resonances, consistent with the plasmon hybridization model. However, the
plasmoncouplingconstant for golddimerswithlessthanafewnanometersseparationbetweentheparticlesdoesnot obeytheexponential
dependence predicted by the Universal Plasmon Ruler equation. The experimentally determined spectra are compared with electrodynamic
calculations and the interactions between the individual rod plasmons in different dimer orientations are elucidated.
The localized surface plasmon resonance (LSPR) of metallic
nanoparticles is the result of the collective oscillation of
conduction electrons within the particle upon interaction with
light. The resonance energy of the LSPR is highly sensitive
to the size and morphology of the particle.1-3The close
approach (i.e., within 2.5 times the particle diameter) of two
nanoparticles leads to interaction of their localized surface
plasmon resonances. This interaction has been exploited in
a number of applications, including surface-enhanced Raman
spectroscopy (SERS) to allow the detection of single
molecules,4-8the Universal Plasmon Ruler which has been
used to measure the distance between two metal nanoparticles
in biological systems,9-12and in optoelectronics where the
near field coupling of nanoparticles spaced less than two
diameters apart results in the transmission of light energy
down a nanoparticle chain13-17or through an array.18
This near field interaction between nanoparticles is highly
distance-dependent and has been described using an elec-
tromagnetic analogue of molecular orbital theory, the plas-
mon hybridization model, which highlights the asymmetry
introduced by dimer formation.19-22When the electric field
is oriented along the interparticle axis (for a given particle
pair), the near fields couple in a manner analogous to a
bonding interaction, resulting in a significant red shift of the
plasmon.9,10,23-28When the polarization is oriented perpen-
dicular to the interparticle axis on the other hand, their near
fields couple in a nonbonding type of interaction and a very
small blue shift of the plasmon band is observed.10,26,27For
each of these arrangements the other possible interaction
mode is a dark mode which cannot be excited by light due
to the symmetry of the system. For spheres, the distance
dependence of the coupling has been calculated using the
plasmon hybridization model20and approximated as either
a (d-3)26or exponential function for distances between 2 nm
and 2.5 times the particle diameter, the coupling limit. The
Universal Plasmon Ruler utilizes an exponential function to
model the distance dependence of nanoparticles, enabling
the measurement of distances between nanoparticles by
measuring the plasmon resonance of two coupled par-
ticles.9,10,26-28However, until now, the degree of near field
coupling at extremely small interparticle separation (less than
2 nm) has not been experimentally investigated.
Previous investigations of plasmon coupling have pre-
dominantly utilized electron beam lithography (EBL) to
synthesize nanoparticle arrays, chains, and periodically
repeating sets of nanoparticles organized as interacting pairs.
However, the resolution limit of modern EBL fabrication is
around 10 nm. While theoretical investigation of nanoparticle
pairs less than 2 nm apart has been carried out,29experimental
fabrication and investigation of such particle pairs have
remained challenges. It has, in part, been overcome by the
investigation of single-pair studies of nanoparticles comple-
mented by electron microscopy images of the single
pair23,27,27and the use of laser techniques to form a small
gap between touching particles.24However, it is apparent in
* Corresponding author, afunston@unimelb.edu.au.
†The University of Melbourne.
‡CSIRO.
NANO
LETTERS
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images of the fabricated particles that they contain not one
but many crystalline domains and their perimeter is not well-
defined, resulting in protrusions and inhomogeneities in the
region of closest approach. In such cases it becomes difficult
to determine whether the particles themselves are actually
touching. Chemically synthesized nanoparticles on the other
hand are discrete crystals of highly defined shape. The
challenge for investigating these nanoparticles is the ar-
rangement of the nanoparticles into well-defined configura-
tions.
In this paper we report our investigation into the coupling
between gold nanorods, with interparticle distances smaller
than 2 nm. The shape anisotropy of nanorods leads to
different possible orientations of the nanorods within the
dimer, leading to different orientational modes of coupling.
Coupling of the rod longitudinal modes when the dimer is
arranged end-to-end or side-to-side leads to bonding and
antibonding interactions. Hence upon coupling either an
increase (side-to-side) or a decrease (end-to-end) in the
resulting LSPR mode energy is expected. A similar picture
exists for interaction of the rod transverse modes. We
measure here for the first time the interactions between
individual gold rod pairs in both side-to-side and end-to-
end orientations at extremely close approach and compare
their spectra to those determined by electrodynamic simula-
tions. The nanorod anisotropy also gives rise to many other
possible dimer geometries, including L-shaped and T-shaped
dimers as well as dimers laterally and longitudinally dis-
placed with respect to one another, and these are also
measured at the single-dimer level and characterized in terms
of the active coupled modes. All the orientations investigated
are characterized by electron microscopy images of the single
dimers and are measured experimentally for extremely close
approaches. These data show that the design and assembly
of plasmonic superstructures from nanocrystals will require
incredibly precise tailoring of position and orientation.
The particles were chemically synthesized and are single
crystals. The investigation was carried out utilizing the
recently reported focused ion beam registration method,
allowing correlation of the scanning electron microscopy
(SEM) image of the particle pairs with their scattering
spectrum.30The gold nanorods were synthesized by standard
protocols.31,32The rods had aspect ratios that varied between
2 and 4. A representative transmission electron microscopy
(TEM) image of the rods used along with the absorption
spectrum of the rod ensemble and aspect-ratio frequency
distribution are shown in parts a and c of Figure 1. The rods
were dispersed on indium tin oxide (ITO) coated glass slides
and markers were etched in the substrate using a focused
ion beam (FIB) as displayed in Figure 1b. The etched
markers were located and imaged, and the scattering spectra
of the particles around the markers were collected using a
dark field microscope. The microscope setup consisted of a
0.8-0.95 NA dry dark field condenser and Nikon Plan Fluor
ELWD 40×/0.60 NA objective coupled to a MicroSpec
2150i and Pixis 1024B Acton thermoelectrically cooled
charge-coupled device (CCD) (Princeton Instruments). It is
important to note that the excitation light in this setup was
not polarized. The SEM images of the same particles were
also collected, allowing direct correlation of the size, shape,
and arrangement of the dimers with the scattering spectrum.
Under the conditions used here, the SEM and scattering
spectra of both single particles and particle pairs were
collected. The dimers were formed largely fortuitously when
the rods were spin-coated onto the ITO; the vast majority of
Figure 1. (a) TEM of gold rod sample, scale bar ) 50 nm. (b) Dark-field image of markers etched in the substrate with gold rods. The gold
rods scatter the incoming light and appear as the white spots in the image. (c) Frequency distribution of aspect ratios for gold nanorod
sample. The inset shows the absorption spectrum of the ensemble in water. (d) Longitudinal plasmon band position as a function of rod
aspect ratio. Each point is one rod for which the spectrum and SEM image were measured. The rods were dispersed on ITO-coated glass
in air.
B
Nano Lett., Vol. xx, No. x, XXXX

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rods in the samples existed as separated, single crystals.
Investigations into the single particles will be reported in
detail elsewhere,33although it is necessary to characterize
the rod sample at a single-particle level for the following
discussion. Figure 1d shows a plot of particle aspect ratio
vs absorption wavelength for the longitudinal band of single
rods. Consistent with earlier work,34,35the peak position of
the longitudinal plasmon mode increases with an increase
in the rod aspect ratio. The line in the figure is a line of best
fit for this sample. There is scatter in the position of the
plasmon band of around 80 nm for rods with a given aspect
ratio.
The scattering spectra and SEM images of particle pairs
arranged side-to-side, end-to-end, and at right angles were
collected and a number of representative examples for each
configuration are shown in Figure 2. All these particle pairs
are arranged as close as possible to one another; from the
SEM image we estimate this spacing is <1 nm. This is an
upper limit as the resolution of the SEM is 1 nm. The
surfactant molecules coating the gold rods are 1-1.5 nm
long, so complete interdigitation would also lead to a
separation of order 1 nm. Within these dimers, the two
particles are interacting; however, the individual resonances
of the two rods in the fully decoupled limit vary slightly
due to small differences in aspect ratio. The individual rods
shown in Figure 2 have aspect ratios varying between 2 and
2.7. From the line of best fit in Figure 1d, their individual
plasmon resonances are expected to vary between 612 and
679 nm and the vertical lines in each part of Figure 2
represent the expected position of the plasmon bands for
noninteracting single particles of identical dimensions to the
rods in that specific dimer. The exact plasmon shift occurring
as a result of the plasmon coupling in the dimers depends
on the aspect ratio of each rod within the dimer; as a result
we report here either the lower limit or a range for this shift.
Rods Aligned End-to-End. For the rods interacting end-
to-end shown in Figure parts a and b of 2, the scattering
spectra are dominated by an extremely intense peak which
is red-shifted by more than 135 nm to >800 nm (for the two
dimers shown λmax ) 816 nm) when compared to the
expected longitudinal plasmon band of the individual rods
in these two dimers, which range from 648 to 679 nm. The
red shift is due to the attractive coupling of the rod
longitudinal plasmon modes, as has previously been observed
for spherical nanoparticles. This trend is consistent with that
observed in ensemble measurements during the consecutive
linear alignment of rods using linkers which have been shown
to selectively bond to the ends of gold rods.36-38The
fractional red shift, ∆λ/λoof the rods shown in Figure 2a is
0.24.
Discrete dipole approximation (DDA) electrodynamic
simulations have been shown to be suitable for the calculation
of scattering spectra of small gold particles39and particle
pairs,27and this method has recently been reviewed and
compared with alternative methods.22We utilized the pro-
gram DDSCAT for the calculations, implemented and made
freely available by Draine and Flatau.40Parameters used in
the calculations are outlined in the Supporting Information.
The DDA simulations for two rods with aspect ratio 2.0,
rod length 60 nm, and hemispherically capped ends in an
end-to-end geometry are consistent with earlier calcula-
tions21,41(Figure 3a). Incident light polarized parallel to the
interparticle axis results in the selective excitation of the rod
longitudinal modes. The longitudinal plasmon modes of the
two rods interact via an attractive coupling with the opposite
poles of the induced dipoles arranged in an alternating
fashion. The induced surface-charge density of this coupled
mode in longitudinally aligned rods is shown in Figure 4.
The interaction results in a lowering of the energy of this
resonance (relative to the longitudinal mode of a noninter-
acting rod), and as a consequence a red shift of the resonance
is observed. When the light is polarized perpendicular to the
interparticle axis, the transverse modes of the rods are
excited; however, no appreciable shift is observed due to
the weak coupling in this case. The polarization-averaged
spectra are dominated by the interacting longitudinal plasmon
mode due to its much higher polarizability. When the DDA
data are scaled for the rod length, D, and plotted against the
relative plasmon shift according to the Plasmon Ruler model
(eq 1),9,10,27,28(see inset to Figure 3a) the distance dependence
is approximately exponential for separations greater than 5
nm with A ) 0.12 ( 0.01 and τ ) 0.26 ( 0.04.
Figure 5 shows the experimentally determined and DDA
calculated fractional shifts with interparticle distances 5 nm
and less. Including DDA calculations for interparticle separa-
tions smaller than 5 nm in the determination of the plasmon
Figure 2. Scattering spectrum for two rods aligned (a and b) end-
to-end, (c and d) side-to-side, (e) in a T configuration, and (f) in
an L configuration, all on ITO and in air. Insets show the SEM
images of the particles giving rise to each scattering spectrum. Scale
bar ) 100 nm.
∆λ
λo
≈ Ae-(s/D)/τ
(1)
Nano Lett., Vol. xx, No. x, XXXXC

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decay led to poor fits to an exponential function. The
accuracy of the DDA calculations at close approaches (i.e.,
for s/D < 0.09) is unclear due to the discretization of the
rods as an array of point dipoles with an interdipole spacing
of the order of 1-2 nm as used here. Experimentally, for
the rods shown in Figure 2a, s/D ) 0.015 and ∆λ/λo) 0.24
while for those in Figure 2b, s/D ) 0.013 and ∆λ/λo) 0.21.
DDA simulations of two rods with aspect ratio 2.4 and
separation s ) 1.43 nm give s/D ) 0.020 and ∆λ/λo) 0.2,
and these values are consistent with those obtained experi-
mentally, particularly as the experimental rod separation
cannot be determined with greater accuracy than 1 nm. It
may be concluded that the exponential decay of the plasmon
interaction serves only as an empirical approximation for
the interaction and is valid only if the ratio of interparticle
separation (s) to rod length (or diameter in the case of
spheres) (D) is greater than 5.3 nm/60 nm ) 0.09 according
to our DDA and experimental results.
Additional modes at higher energies are also obvious in
the spectra of the single particles in Figure 2a. A similar
feature, attributed to hybridization of the individual plasmon
modes at small interparticle separations has been predicted
theoretically,20,29,45and was observed following laser-induced
separation of previously touching spheres which resulted in
Figure 3. (a) Polarization averaged extinction of a pair of gold hemispherically capped rods with aspect ratio 2.0 interacting end-to-end as
a function of interparticle separations 56.5, 42.4, 28.2, 14.1, 7.1, 5.3, and 3.5 nm with smaller separations more red shifted. Inset: Fractional
shift of the longitudinal plasmon band as a function of interparticle distance scaled for rod length. The point at Gap/Rod-Length ) 3
represents the plasmon resonance of a fully decoupled rod with the same dimensions as those in the dimer. (b) Polarization averaged
extinction of a pair of hemispherically capped gold rods with aspect ratio 2.5 interacting side-to-side as a function of interparticle separation
from 42.9, 12.9, 5.7, 2.9, and 1.4 nm with smaller separations more blue shifted. Inset: Extinction for polarization parallel to interparticle
axis at d ) 42.9, 12.9, 5.7, 2.9, and 1.4 nm, smaller separations more red shifted. (c) Extinction spectra of a pair of gold hemispherically
capped rods interacting in a T geometry where the polarization is parallel to the interparticle axis (open triangles), perpendicular to the
interparticle axis (open diamonds), and polarization averaged (solid diamonds), the calculated spectrum for a single rod is included for
reference (closed circles). (d) Extinction spectra of a pair of gold hemispherically capped rods interacting in an L geometry with polarization
parallel to the interparticle axis (open diamonds), perpendicular to the interparticle axis (open triangles), and polarization averaged (solid
diamonds), the calculated spectrum for a single rod is included for reference (closed circles). For all calculations, rods are immersed in a
homogeneous medium with a refractive index of 1.5642and the interparticle axis is defined by the black line connecting the nanorods in the
schematic figures of the rod geometry.
Figure 4. Surface charge density of interacting cylindrical, spheri-
cally capped gold nanorods placed 1.5 nm apart and with dimen-
sions 78 nm × 24 nm, calculated using the electrostatic approxi-
mation43,44and where blue represents one charge (for example -ve)
and the red the opposite charge (+ve) of the dipole. The plasmon
resonance for a single rod of these dimensions calculated using
this method is 707 nm. The wavelength given is the resonance
wavelength for the coupled plasmon mode. The general trends in
the resonance wavelengths are identical to those calculated using
DDA.
D
Nano Lett., Vol. xx, No. x, XXXX

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s/D ) 0.03.24The presence of an additional resonance mode
has also been predicted by DDA simulations for the end-
to-end longitudinal plasmon coupling of rods with different
aspect ratios due to symmetry breaking, resulting in the
previously optically inactive transition at higher energy with
net dipole ) 0 gaining a nonzero dipole.21However, the two
rods interacting in Figure 2a have similar dimensions, and
rod dimers with lower symmetry do not display a similar,
higher energy mode (see Figure 2b). From these consider-
ations we conclude it is most likely that the additional high-
energy peak is due to hybridization of the plasmon modes
at the small interparticle separation in this rod pair.
Rods Aligned Side-by-Side. In contrast to the case for
rods aligned end-to-end, the spectra of rod dimers aligned
in a side-by-side configuration display a single peak with
scattering intensity comparable to that for a single rod, at a
wavelength that is blue-shifted by 21-48 nm compared to
single rods (to λmax614 nm). In this arrangement, when the
polarization is parallel to the interparticle distance, the low
intensity transverse plasmon modes interact attractively and
these undergo a red shift as the particles approach. This is
obvious in the DDA calculations (see inset to Figure 3b).
However, the polarization-averaged spectra are dominated
by the longitudinal plasmons of the rods which interact
repulsively, leading to a slight blue shift overall and the red
shift of the transverse bands is washed out. The surface
charge density associated with this mode is shown in Figure
4 and is in accordance with the predictions from the plasmon
hybridization model. These results are consistent both with
ensemble data where the nanorods align side-to-side21and
with previously reported DDA simulations.21,41
Rods Arranged In T and L Geometries. Rod dimers in
which the rods are arranged at right angles introduces the
possibility of coupling between longitudinal and transverse
modes. Two rods arranged in T- and L-shaped configurations
along with their scattering spectra are shown in parts e and
f of Figure 2, and the extinction calculated via DDA for these
two configurations and at different polarizations are shown
in parts c and d of Figure 3. Particles in these configurations
have the potential to form the basis of right-angle and
T-junction structures in plasmonic circuits. For these particle
pairs, intuitively it may be predicted that the transverse mode
of one rod interacts with the longitudinal mode of the other
rod.
Two well-separated plasmon modes are present in the
scattering spectra of rods in an “L” configuration. These
modes are reproduced in the DDA calculations performed
here, which differ somewhat from those reported previously
for two rods interacting at right angles.21Polarization of the
incoming radiation along the interparticle axis leads to
excitation of a lower-energy, attractive mode in which the
two longitudinal plasmons are oscillating in phase with one
another, resulting in dipoles with opposing signs where the
rods approach (see Figure 4). When the polarization is
perpendicular to the interparticle axis, the longitudinal
plasmon bands of the two rods oscillate symmetrically out
of phase and the dipoles interact repulsively. It is interesting
to note that for the L configuration, the coupled modes are
all due to interactions between the longitudinal plasmon
modes of the rods and not longitudinal-transverse coupling.
Conversely, for the rods in a T-configuration, interactions
do occur between the longitudinal and transverse modes of
the rods. Excitation with light polarized along the interparticle
axis results in the excitation of the longitudinal mode of the
rod forming the stem of the “T”. This excitation interacts
with a mode of the opposing sign in the center of the other
rod, lowering its energy slightly. Conversely, excitation with
light polarized perpendicular to the interparticle axis results
in the excitation of the rod forming the top rod of the “T”,
which is able to interact to a small degree with a dipole-like
mode of the perpendicular rod (see Figure 4). It is possible
these plasmon interactions in the “T” dimer are predomi-
nantly dipole-induced dipole interactions in character. For
this arrangement, the energy difference between the two
modes is calculated to be quite small, although this is not
reflected in the experimental data.
Offset Rods. The experimental scattering spectra for a
number of pairs of rods interacting side-to-side but longi-
tudinally offset with respect to one another are shown in
Figure 6. DDA calculations for this geometry as a function
of rod offset as well as the corresponding geometry and
offsets for rods interacting end-to-end are shown in Figure
S1 of Supporting Information. For the latter, DDA results
show that as one rod is displaced further and further from
the axis of the other, the coupling between the two rods
weakens as expected. This is observed as a blue shift of the
spectrum with increasing offset compared to the highly
coupled, fully aligned rods and a concomitant decrease in
the extinction.
DDA results for rods interacting side-to-side but longitu-
dinally shifted with respect to one another are quite different
from those for the end-to-end geometry. The peak positions
as a function of rod offset are given in Figure 7. Two distinct
effects are important. For both zero and small longitudinal
shifts, the repulsively interacting longitudinal modes domi-
nate the spectra and a weakening of the intensity as the
Figure 5. Fractional shift of longitudinal plasmon band of spheri-
cally capped gold nanorods as a function of interparticle distance
scaled for rod length determined experimentally (purple squares)
and by DDA simulations for rods with aspect ratio 2.0 where s/D
> 0.09 (blue diamonds). The red line is an exponential fit to these
data and the dashed line is the 1/d3fit, s/D < 0.05 with interdipole
spacing 1.76 nm (red circles) and s/D ) 0.020 with aspect ratio
2.4 and interdipole spacing ) 1.43 nm (green triangle). Medium η
) 1.56.
Nano Lett., Vol. xx, No. x, XXXXE

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center-to-center distance of the rods increases is observed.
However, even at small offsets (23%), another lower-energy
mode grows in. This is a result of the lifting of symmetry
within the system. The attractive coupling between the
longitudinal modes becomes allowed as the dipole moment
for this transition becomes nonzero as the rods are offset.
At an offset of 38%, the attractively and repulsively coupled
longitudinal modes give rise to two well-separated peaks in
the spectrum; the repulsive longitudinal coupling has de-
creased considerably as the tips of the rods are further from
one another.
A further increase of the offset between the rods causes
the opposite ends of the two rods to approach one another,
and the repulsive coupling between the longitudinal modes
then becomes attractive due to the opposing signs of the field
at the opposite ends of the rods. This gives rise to a lowering
in energy for this mode as reflected by the strongly red-
shifted resonance. As the separation increases and the fields
at the opposite ends of the rods approach one another, this
resonance becomes more and more red shifted and intense
up to an offset of 61% where maximum coupling occurs.
As the rods move further apart again (>61% center-to-
center separation), attractive coupling of the longitudinal
modes again decreases as the fields move away from one
another and the coupling decreases. When the radiation is
polarized parallel to the interparticle axis and the transverse
modes are excited, a similar effect is observed but in reverse;
this is not obvious in the polarization-averaged spectra. The
experimental data (Figure 6) are consistent with the DDA
results. At smaller rod-to-rod displacements of about 20-50%,
two bands are observed in the experimental scattering spectra.
However, as the ends of the rods approach one another, the
low-energy mode experiences a stronger coupling and red
shifts and increases in intensity.
These experimental results highlight a number of important
aspects about the potential use of plasmon coupling as a
molecular ruler and for the transport of electromagnetic
energy.
In the dipole approximation, the near field of a plasmon
dipole mode is well-known to decay as 1/d3, where d is the
distance from the particle. Analogously, it has been shown
that dipole-dipole coupling (within the dipole approxima-
tion) also decays as 1/d3;26in this case d is the center-to-
center distance between the two coupled particles. However,
as the particles approach to very small separations, the finite
particle size along with interactions of the dipole mode on
one particle with higher modes on its pair become important
and the dipole coupling model becomes invalid. The plasmon
ruler equation10is based upon an exponential function. This
is primarily due to the observation that calculated data sets
which included smaller interparticle distances, s/D ) 0.1,
were described better at smaller separations by an exponential
function compared to the dipole coupling model.
Both functions, that is, exponential and d-3, have been
successfully used to model the plasmon resonance shifts
observed experimentally for larger interparticle separa-
tions.26-28However, we find here that neither function is able
to successfully model the experimentally observed distance
dependence of coupled nanoparticles over the full interaction
range, with significant deviations occurring for both these
models at small interparticle distances where the energy of
the longitudinally coupled plasmon mode red shifts much
more rapidly than that predicted by either model. The
exponential model significantly underestimates the shift in
the plasmon energy due to coupling at very small distances,
s/D < 0.09. As a result of this, it is necessary to use more
sophisticated calculations to completely and correctly de-
scribe the plasmon shift observed at small interparticle
separations, such as the plasmon hybridization method19,20
and extended multipole methods. These methods take into
account the hybridization of the dipole mode of one particle
with higher-order multipoles on its pair, resulting in ad-
ditional, higher-order terms to the plasmon coupling at small
distances. Calculations performed using the plasmon hybrid-
ization method for two 10 nm radius spheres with polariza-
tion of the electric field parallel to the interparticle axis20
replicates the large red shift experimentally observed here
at small interparticle distances.
Figure 6. Scattering spectra for two rods aligned side-to-side on
ITO and in air with various lateral offsets. Insets show the SEM
images of the particles giving rise to each scattering spectrum. Scale
bar ) 100 nm.
Figure 7. Wavelength of peak maximum for spherically capped
gold rods interacting side-by-side as a function of transverse
separation and lateral separation as determined by DDA calcula-
tions. All modes are due to interaction of the longitudinal modes
of the rods. The lines are exponential fits to the data intended as a
rough guide to the eye. Medium η ) 1.56.
F
Nano Lett., Vol. xx, No. x, XXXX

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While the use of rods would improve accuracy for
measurements of distance using the Plasmon Ruler due to
their higher signal and larger shift, their intrinsic advantage,
the particle asymmetry, actually precludes their use in
experiments designed to use plasmon coupling as a measure
of distance. This is a result of the dependence of the coupling
on the geometrical arrangement of the rods, which is
extremely difficult to control. Because of the unpredictability
of the rod arrangement, the interaction between spheres is
more reliable as a method for measuring distances.
The asymmetry of the rods leads to a number of different
trajectories for the approach of two rods into a common
alignment. This is highlighted by the results from the DDA
calculations in Figure 7, where two rods approach one
another differently to align side-to-side. For direct approach
of the rods with no longitudinal displacement (left panel),
the longitudinal coupling results in a gradual blue shift of
the plasmon relative to an uncoupled rod, the energy of which
is given by the horizontal line at 651 nm. The right panel
shows the rods approaching via a longitudinal translation of
the rods. For this trajectory, the symmetry of the system is
lowered compared to the direct side-to-side approach and
this is reflected in the higher number of modes apparent in
the spectrum as the particles approach. Despite the differ-
ent trajectories for the rod alignment, the end result is
ultimately the same.
For dimers incorporating rods, a large number of pos-
sibilities for the arrangement of the individual rods dimers
exist, and these display very different plasmon coupling
modes and degrees of coupling. In Figure 8a we have
extended the plasmon hybridization diagrams for rods to
include dimers interacting at right angles. For clarity,
transverse-transverse interactions have been omitted as these
effectively mirror the longitudinal-longitudinal interactions.
The relative energetics shown are qualitative. In contrast to
linearly aligned rods, for which symmetry considerations
result in one of the modes being dark (the higher energy
mode for rods aligned end-to-end and the lower energy mode
for rods aligned side-to-side), lifting of this symmetry results
in all modes being optically active. It is clear that hybridiza-
tion between two longitudinal modes leads to greater
splitting, although the degree of splitting is geometry
dependent and rods aligned with their tips approaching
display greater splitting. In turn, geometries in which the
coupling is between longitudinal and transverse modes
experience less splitting.
These experiments highlight important differences between
coupling for practical waveguiding applications. While both
the “L” and “T” geometries are models for T junctions in
optical circuits, for these geometries to act as T junctions
strong coupling between the two rods must occur. From the
results here, it is obvious that coupling within the T geometry
is not particularly strong as it is a longitudinal-transverse
interaction and excitation of the longitudinal mode of either
rod does not lead to significant transfer of this energy into
the other rod. The L geometry, on the other hand, involves
strong coupling between two longitudinal modes with
transmission of the plasmon resonance throughout the full
structure.
The plasmon hybridization diagram for two rods aligned
side-to-side and then sequentially longitudinally offset with
respect to one another is shown in Figure 8b. Initially, for
the fully aligned rods, hybridization results in the formation
of two energy levels. The longitudinal plasmon modes of
the rods interact symmetrically in the higher-energy, repul-
sive mode while the lower-energy mode is dark. A small
longitudinal shift of the rods reduces the symmetry present
in the previously dark state, resulting in it becoming allowed.
The splitting however is slightly weaker. A further longitu-
dinal shift results in an energetic cross-over of states, with
the symmetric mode becoming attractive due to the proximity
of the oppositely charged rod tips to one another. Further
Figure 8. Plasmon hybridization schemes for (a) rod dimers in different geometric arrangements and (b) rods initially arranged side-to-side
and then increasingly longitudinally offset as a function of the center-to-center offset.
Nano Lett., Vol. xx, No. x, XXXXG

Page 9

displacement lowers the coupling energy for both modes
significantly.
In summary, we have established the viability and versatil-
ity of one-dimensional rod structures for the tuning of the
LSPR in plasmonic structures. This coupling may be
controlled through nanoparticle separation, angle, and inter-
action geometry. We have elucidated new hybridization
interactions which are introduced due to the different possible
interaction geometries of the rods, enabling one to engineer
surface plasmon resonances. The form of the distance
dependence of the plasmon coupling between nanoparticles
has been extended to include small interparticle distances.
These results show that the approximation of the coupling
distance dependence according to the dipole-dipole ap-
proximation or as an exponential does not hold for nano-
particles separated by only a few nanometres or less and
that the plasmon ruler equation, therefore, is not valid for
s/D ratios less than 0.09.
These results highlight the importance of orientation on
the coupling of anisotropic nanoparticles such as rods. Small
changes in the rod orientation lead to relatively large changes
in the plasmon interaction, particularly at close approach.
Acknowledgment. This work was supported through ARC
DP Grant 0451651 and FF Grant 0561486. The authors thank
Sergey Rubanov for assistance with FIB/SEM. C.N. thanks
the University of Melbourne for MIRS and MIFRS post-
graduate scholarships.
Supporting Information Available: DDA calculation
details and figure showing polarization averaged extinction
of a pair of spherically capped cylindrical gold rods. This
material is available free of charge via the Internet at http://
pubs.acs.org.
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