The Computational Studies of Plasmon Interaction
Demchuk et al. Nanoscale Research Letters
The Computational Studies of Plasmon Interaction
Antonina Demchuk 0
Ivan Bolesta 0
Oleksii Kushnir 0
Ihor Kolych 0
0 Department of Radiophysics and Computer Technologies, Ivan Franko National University of Lviv , Generala Tarnavskoho Str. 107, 79017, Lviv , Ukraine
In this paper, an interaction of metal nanoparticles that appears in the extinction spectra was investigated. The mutual coupling between the nanoparticles, the effect of size difference, and the interparticle separation in silver nanoparticle dimers are studied by computer discrete dipole approximation methods. The obtained results show that nanoparticle interaction results in the distinct collective modes, known as the low-energy bonding modes and the higher-energy antibounding modes. The spectral position of the modes is analyzed as a function of the ratio of interparticle distance to particle size that reduces the dependency on the particle size itself. The optical spectra of nanoparticles that form the fractal cluster were investigated. It was found that the number of spectral bands increase with the growth of the number of nanoparticles in the fractal cluster, which are described within the plasmon hybridization model.
Nanoparticle; Surface plasmon resonance; Optical spectra; Extinction cross section; Hybridization model; Dimer; Fractal cluster; Discrete dipole approximation
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Background
Metal nanoparticles have attracted a great attention due to
their strong interaction with light. The interesting optical
properties of metal nanoparticles, such as bright intense
colors, are the result of interaction of free carriers with
the incident electric field. In the presence of the
oscillating electromagnetic field of the light, the free electrons
of the metal nanoparticle undergo a collective coherent
oscillation with respect to the positive metallic lattice [1].
This process is resonant at a particular frequency of the
light and is called the localized surface plasmon resonance
(LSPR) oscillation.
In a single metal nanoparticle, frequency, strength,
and quality of the LSPR depends on the size,
geometry, the metal composition, and the refractive index of
the local environment. Furthermore, the LSPR of a metal
nanoparticle is sensitive to the presence of other nearby
metal nanoparticles, their sizes, interparticle distance, and
materials. In an assembly of metal nanoparticles with
small interparticle distance compared to the size of
particle, the LSPR is strongly affected by the near-field coupling
of the individual particles [2].
Another aspect of the study is the plasmon spectra
of metal nanoparticles associated with the formation
of metal-dielectric nanocomposites. Nanocomposites can
cause significant influence on linear and nonlinear
susceptibilities of matrix [3], radiation recombination
processes [4], and giant surface-enhanced Raman scattering
(SERS)[5]. Practical interest in dielectric materials with
metal nanoparticles is connected with perspective for
development of optical switches on their basis with
ultrashort time response, limiters of optical laser beam
intensity, for synchronization of the laser modes, etc.
Progress in the investigation of plasmonic spectra of
the metal nanoparticles has been greatly assisted by
developments in technology, experimental techniques,
and numerical modeling of their extinction spectra. This
allows to study the impact of different factors on the
extinction spectra of nanoparticles and their composites.
Pairs of nanoparticles called ?dimers? are the simplest
model systems to study as only two directions of incident
polarization required to explore the interparticle coupling
completely. It is shown [6] that dimers with an
interparticle distance smaller than the diameter of the nanoparticles
can generate high plasmonic enhancements.
The optical spectra of individual particle, dimers and the
chains consisting of particles of the same size, the
dependence on the medium host, the interparticle separation,
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and the direction of incident polarization were studied in
[7]. The authors of [8] explored the extinction spectra and
plasmon hybridization schemes of nanoparticle dimers of
various sizes and materials. In [9], the field enhancement
spectra were investigated as a function of the particle size
difference and the interparticle spacing, and also,
localized surface plasmon resonances in the chains of different
particle sizes were analyzed depending on the particle size
and interparticle separation [10].
The general case of the spherical nanoparticle coupling
is fractal clusters. Since fractal clusters are characterized
by local anisotropy of each particle surroundings, the
supplements compensation to local field (including the
dominant surroundings contribution) do not occur. This
leads to the emergence of strong local electromagnetic
fields, which will vary in different parts of fractal cluster
(field fluctuation)[11]. Local fields, in turn, enhance the
optical susceptibilities of particles that reflects in the
linear and nonlinear optical characteristics of fractal clusters
[12]. As a result, the extinction spectra of fractal clusters
differ significantly from the spectra of individual material
nanoparticles.
This paper studies the nanoparticle interaction that
shows up in the extinction spectra. The spectra of the
simple nanoparticle system (dimers with particles of
similar and dissimilar sizes) were analyzed. The changes of
spectra are investigated to depend on the interparticle
distance and the interaction of particles as the part of fractal
cluster.
Methods
We use the discrete dipole approximation (DDA) method
to simulate the distribution of the electromagnetic field
for the composition of silver nanoparticles. Extinction
cross sections are calculated from the resulting
polarizations of DDA method, using the solving package
?EMSimulation? [13]. The dielectric function of silver
nanoparticles is taken as a bulk silver from the Johnson and
Christy table [14] with phenomenological correction for
size reduction effects. The particles are assumed to be
embedded in a vacuum, which is described by a relative
permittivity ? = 1 or in dielectric media with ? = n2,
where n is the refractive index.
The DDA is a method for calculating scattering and
absorption of electromagnetic waves by
nanocomposites. The DDA was proposed in 1973 by Purcell and
Pennypacker [15], who used it to study interstellar dust
grains. The main idea of the method is that the target is
approximated by an array of N point dipoles at positions
ri with polarizabilities ai. The polarization
Pi = ai ? E(ri)
of each dipole responds to the total electric field at its
position; E(ri) is the sum of an incident plane wave
The medium of the target is characterized by its
complex dielectric function ?. The dipole polarizabilities ai
can be given by the Clausius-Mosotti polarizability [16]:
where d is the diameter of the dipole. Draine [17] showed
that for finite wavelengths, the optical theorem requires
that polarizabilities include also the radiative-reaction
correction of the form
?(nr)
where ?(nr) is the nonradiative polarizability.
The DDA method allows to get an oscillating dipole
moments Pj for every monochromatic incident wave;
from these Pj the absorption and scattering cross sections
are computed [18]:
and a contribution from all the other dipoles
Eother,i = ?
i=j
Cext =
Cabs =
|E0|2 j=1
|E0|2 j=1
Qext = Cext/A,
Qabs = Cabs/A,
Im Ei?nc,j ? Pj
To compare cross sections of dipoles with different
radiuses, we use effective cross sections, which are
calculated as
where A is the area of overlap between the incident beam
and the target object.
The scattering problem for the array of point dipoles,
represented by the system of linear equation, can be
solved with arbitrary accuracy. We use the fast Fourier
transform (FFT) techniques together with the conjugate
gradient method to obtain solutions for targets.
We calculated extinction spectra of fractal clusters with
increasing degree of aggregation of nanoparticles in the
clusters. Simulations of fractal clusters growth conducted
in a ?cluster-cluster? model [19].
To decompose the optical spectra into peak functions,
we performed two steps. Firstly, the method of second
derivative spectroscopy [20] was used to find the hidden
peaks for the initial definition of the peak position. Then,
L-BFGS (Broyden-Fletcher-Goldfarb-Shanno) [21]
algorithm was used to reduce the standard deviation between
the peak superposition and the simulated spectrum of
fractal cluster.
Results and Discussion
Calculation of extinction spectra of dipoles shows that
the interaction between the nanoparticles, which depends
on the distance between them, manifested in the
appearance of two components (Fig. 1). Spectral distance
? between components depends on the distance (D)
between nanoparticles (for fixed radius r), on the size of
nanoparticles (for fixed distances), and on the permittivity
of the environment.
In the case of dielectric medium other than vacuum, the
behavior of the spectra remains qualitatively the same, but
the band shifted to long wavelength region (for isolated
particles and dimers) and splitting between the bands
increases. Analysis of the results shows that the ratio
d = D/r is more universal because it minimizes the
dependency on the nanoparticle radius r.
Figure 1 shows effective cross section extinction
spectra of dimer in vacuum with similar size particles with
radius r = 10 nm dependent on interparticle separation
d, where d = D/r depends on the radius of a
nanoparticle r and the interparticle distance D (Fig. 1a). Cross
section extinction was calculated under the longitudinal,
transverse, and oblique incident field polarization. Both
the longitudinal and transverse plasmon coupling bands
for the homodimer can be recognized under the oblique
polarization.
Calculation results show that at the interparticle
separation d > 2, only one strong plasmon coupling absorption
band can be observed. After decreasing the interparticle
distance, the single plasmon resonance band is split into
two plasmon bands forming two different peaks (Fig. 1b).
When the gap distance between nanoparticles is large
enough, every metal nanoparticle exists as an individual,
so the interaction between nanoparticles is very weak.
When the gap is quite small, the system of nanoparticles
consider to be as a whole. So in both cases, it is
impossible to cause the energy splitting and the resonance peak
position shifting.
Moreover, the plasmon coupling of a dimer shows a
large red-shift as the interparticle distance decreases for
incident polarization parallel to the dimer axis, due to the
applied and induced electric fields that are added to each
other. For incident polarization perpendicular to the axis,
a destructive interaction between the applied and induced
electric fields is predicted, leading to a small blue spectral
shift [22].
The extinction spectrum of a symmetric dimer displays
two distinct modes (Fig. 2a), the lower energy one at 374.2
nm resulting from the longitudinal coupling of the particle
plasmons and the higher-energy one at 349.7 nm from the
transverse coupling.
This behavior in electromagnetic coupling of the
plasmon resonance between two nanoparticles can be
explained by the phenomenon of molecular hybridization.
This means when very strong plasmonic enhancement is
in place within a dimer, two plasmons hybridize to form a
lower energy bonding plasmon mode and a higher-energy
antibonding plasmon mode [23].
In order to demonstrate the effect of introducing a size
asymmetry in the coupling scheme, we discuss the LSPR
scattering spectra obtained from an asymmetric dimer
composed of a 4- and 10-nm silver particle.
The extinction spectra were calculated (Fig. 2b) for
two extreme orthogonal polarizations of the asymmetric
dimer, each contains two modes at 350.8 and 358.9 nm
for the transverse and at 347.5 and 367.6 nm for the
longitudinal polarization, respectively. This observation is in
Fig. 1 a The dimer with nanoparticle radius r and interparticle distance D. b The effective extinction spectra of nanoparticle dimer dependent on
the interparticle separation d
Fig. 2 The extinction spectra of the dimer under the transverse and longitudinal incident field polarization and the plasmon hybridization schema for
a dimer with the same nanoparticle size r = 10 nm (homodimer), b dimer with different nanoparticle sizes: r1 = 10 nm and r2 = 4 nm (heterodimer)
direct contrast to the homodimer case (Fig. 2a) and can
be understood by introducing the asymmetry in the
plasmon hybridization model, as depicted in the hybridization
scheme.
When the sizes of nanoparticles in the system are
different, the electric field can no longer be assumed to be
uniform inside the particles, and high-order (quadrupole,
octupole, etc.) plasmon modes can directly couple with
the electric field of the light simply due to the phase
retardation effect. Excitation of multipole plasmon resonances
is the result of the asymmetric distribution of the
surface plasmons caused by the electromagnetic interactions
between the localized modes. This effect may be used
to enhance the nonlinear optical response of an
effective medium composed of particles with engineered size
dispersion and particle placement [10].
Figure 3 shows relation of the longitudinal and the
transverse field enhancement peak to interparticle separation
for different radiuses of nanoparticles in the homodimer.
The figure shows the exponential relationship between
the position of the peak enhancement and the
interparticle distance of dimer with the asymptotical approaching
the wavelength value 354.8 nm that corresponds to the
isolated sphere enhancement.
The spectral distance ? between the enhancement
peaks follows a universal scaling law as a function
of the dimensionless parameter d = D/r
(quasiexponential behavior) and has given birth to the
concept of "plasmon ruler" that may be used to infer the
length of a molecular chain connecting functionalized
nanoparticles.
Figure 4a show the results for a dimer with a different
radius ratio. The radius of the small nanoparticle is
varied while that of the large nanoparticle is kept constant
at 20 nm. The field enhancement shows two resonance
peaks for the lower radius ratio. At a larger volume ratio,
the peak field enhancement has increased. With increase
in interparticle spacing, the peak enhancement shifts to
Fig. 3 The longitudinal and transverse field enhancement peaks
dependent on the interparticle separation for dimers with radiuses
r = 4 nm, r = 10 nm, r = 20 nm, and r = 40 nm
Fig. 4 The internal field enhancement spectra for four silver nanoparticle dimers with different radius ratios, at a fixed center-to-center separation
a s = 40 nm, b s = 45 nm, c s = 50 nm, and d s = 60 nm. The radius of the small nanoparticle is varied while that of the large nanoparticle is kept
constant at 20 nm
lower wavelengths and shows less pronounced peak
splitting. The smaller separation leads to an increased field
enhancement due to the larger near field provided by the
bigger size particle at this distance.
For small size difference or large interparticle
spacing, low-field enhancement peaks are observed. For most
geometries, the field enhancement peak exceeds that of an
isolated silver nanoparticle, with the exception of dimers
with small radius ratio or large interparticle spacing. The
largest field enhancement peaks are observed at small
spacing and large size difference.
We calculated the extinction spectra of nanoparticles
which form the fractal cluster. Simulation of fractal cluster
growth was carried out from 1000 polydisperse
nanoparticles within the model of ?cluster-cluster? (Fig. 5a), which
corresponds to the natural growth of fractal aggregates
in sols. The radius of nanoparticles is normally
distributed with the mean of 5 nm and the dispersion
of 1 nm.
Figure 5b presents the change (dynamics) of the
extinction cross section spectra within the fractal grows stages.
Curve 1 shows the spectrum of non-interacting particles,
curves 2?11 show the spectra of clusters, with
increasing number of particles respectively, curve 12 shows the
spectrum of fractal clusters with all joined nanoparticles.
Analysis of the spectra by derivative spectroscopy
method revealed the number of spectral components
(peak functions) in the cluster (Fig. 6a). It is shown that the
number of spectral components increase with the growth
of the number of nanoparticles in the fractal cluster at
different stages of cluster formation.
On the spectrum of non-interacting nanoparticles
(aggregation step 1), one band with a maximum at 3.18 eV
can be observed (curve 5 at Fig. 6a). The spectra of the
initial stages of cluster formation (aggregation steps 2 and
3) show the additional three bands with the maximum at
2.60, 2.83, and 3.36 eV (curves 2, 4, and 6 in Fig. 6a). At
the later stages of aggregation, the additional bands at 2.45
and 3.52 eV are present (curves 1 and 7 in Fig. 6a).
Figure 6a shows that the spectral position of the
component in spectra does not depend on the degree of
aggregation. For most components, the band energy does
not change significantly at different stages of cluster
formation. For the low-energy components (curves 1 and 2 in
Fig. 6a), we can observe large red-shift with the growth of
cluster, and for the high-energy components (curve 6, 7 in
Fig. 6a), small blue-shift is present. This phenomenon can
be explained by interaction of large number of
nanoparticles in cluster and corresponds to the previously obtained
results for the simple system of nanoparticle dimer.
Fig. 5 a The fractal cluster simulated within the model of ?cluster-cluster.? b Extinction spectra of the nanoparticles at different stages (curves 1?12)
of fractal cluster formation
The difference between the extinction spectra of
fractal clusters and non-interacting spectra is shown on
the hybridization scheme on Fig. 6b. Low-energy bands
at 2.60 and 2.83 eV can be associated with bounding
orbitals corresponding to different orientations of
interacting dipoles. High-energy bands at 3.36 and 3.52 eV
are associated with loosening plasmon modes. Moreover,
the most high-energy mode appears in clusters with
sufficiently large number of particles (curve 7 at Fig. 6a).
Dipole-dipole interaction of nanoparticles can cause
splitting in the energy modes, as shown in hybridization
schema (Fig. 6b). The low energy band at 2.45 eV does
not correspond to dipole mode interaction and can appear
at higher levels of aggregation, as a result of high-order
modes interaction (quadrupole, octupole, etc.).
Conclusions
In summary, the optical properties of plasmon resonant
metal nanocomposites were investigated. Plasmon
resonances are numerically evaluated in the silver nanosphere
dimers of similar and different sizes and in the
fractal cluster containing silver nanoparticles embedded in a
vacuum.
We have shown that the interparticle coupling
interaction among the random distribution of silver
nanoparticles can lead to hybridization, splitting, and shifting of the
plasmon energies, as well as the quadrupolar resonances
formed by the interaction between the plasmons of the
metal nanoparticles.
The extinction spectra in dimers with identical
nanoparticle sizes indicate the presence of a dipolar
particle response. However, the interaction of particles with
dissimilar sizes also results in the splitting of the
plasmon resonances into two resonances: the lower energy
bounding plasmon and the higher-energy antibounding
plasmon.
The change of the maximum band as a function of the
particle size and interparticle separation was analyzed. It
is shown that the maximum band depends on the ratio
of the interparticle distance to the size of particles, and
Fig. 6 a The spectral position of components at different stages of cluster formation. The numbers on the horizontal axis correspond to the
numbers of curves in Fig. 5. b The hybridization scheme of plasmon modes in fractal cluster
this relation is more universal because it minimizes the
dependency on the nanoparticle size.
In the fractal cluster consisting of similar nanoparticles
the additional energy modes can appear that indicates the
presence of higher-order plasmon modes (quadrupole or
higher). The spectral position of maximum bands does
not change significantly with the degree of aggregation of
cluster. But the higher modes mostly appears on the later
stages of fractal cluster formation.
The obtained results can be used for analyzing the real
nanocomposites, determining the interparticle distance
between nanoparticles, and creating the nanomaterials
with special properties.
Abbreviations
DDA: Discrete dipole approximation; FFT: Fast-fourier transform; L-BFGS:
Limited-memory Broyden-Fletcher-Goldfarb-Shanno; LSPR: Localized surface
plasmon resonance; SERS: Surface-enhanced Raman scattering
Authors? Contributions
AD implemented the computer simulation methods, calculated the optical
spectra of dimers, performed the data processing, and worked on the
manuscript. IB was responsible for the conception of the work, data analysis,
interpretation of the results, and final approval of the manuscript. OK provided
the data processing and analysis, created the graphical plots of the calculated
spectra, and worked on the manuscript. IK contributed to the implementation
of the DDA method, simulated the fractal cluster structures, and calculated the
optical spectra of the clusters. All authors read and approved the final
manuscript.
Competing Interests
The authors declare that they have no competing interests.
Publisher?s Note
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