Simple Model for the Hybridization of Surface Plasmon Resonances in

Jun 14, 2010 - T. J. Davis,* D. E. Gómez, and K. C. Vernon. CSIRO, Materials Science and Engineering and Future Manufacturing Flagship,. Private Bag ...
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Simple Model for the Hybridization of Surface Plasmon Resonances in Metallic Nanoparticles T. J. Davis,* D. E. Go´mez, and K. C. Vernon CSIRO, Materials Science and Engineering and Future Manufacturing Flagship, Private Bag 33, Clayton, VIC, 3168, Australia ABSTRACT An “electrostatic” eigenmode method based on the coupling of evanescent electric fields is presented for modeling the hybridization of localized surface plasmon resonances in metallic nanoparticles of arbitrary shape. The method yields simple analytical expressions for the hybridized energies and excitation amplitudes of nanoparticle ensembles. Because of its ease of applicability and simple conceptual basis, we anticipate that the method will be of value in understanding and predicting the effects of interacting plasmonic nanoparticles. KEYWORDS Plasmon, nanoparticle, hybridization, eigenmode, Fano resonance, induced transparency

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ocalized surface plasmon resonances (LSPR) in metallic nanoparticles result from the collective oscillation of conduction electrons and are associated with strong evanescent electric fields around the particle. As a consequence, the optical properties of a system of closely spaced nanoparticles differs from that of its isolated constituents due to the mutual interactions arising from these fields. Such coupling provides a means by which the optical properties of nanoparticle ensembles can be manipulated producing a variety of effects, such as the splitting of the frequencies of resonant modes,1,2 the red3,10 or blue shifting5,7,9 of the resonant frequencies, the appearance of plasmon-induced electromagnetic transparency,11,12 and increased sensitivity to variations in the local refractive index.13-16 The coupling is important also in devices such as optical antennas17,18 and plasmonic circuits.19-21 Predicting the effects of the mutual interaction between the nanoparticles can be quite difficult, particularly since the analytical solution to the problem is generally complicated. Although existing numerical methods provide accurate models of LSPR in nanoparticles, it can be difficult to extract parameter relationships from them. In this regard, several approximate analytical methods have been developed to understand the optical effects of nanoparticle coupling. Since the fundamental LSPR mode usually has the characteristics of an electric dipole, the electric coupling between nanoparticles has the same form as the coupling between excitonic modes observed in molecular spectroscopy.22 In this case, the van der Waal’s interaction arises from the electric dipole fields of the molecules. This dipole-interaction model has been applied to the coupling of nanoparticle pairs, or dimers,

and predicts the splitting of each resonance into two modes of differing energies.7 Moreover, this model correctly predicts the observed angle dependence of the interaction as has been demonstrated experimentally using lithographically created nanorods.23 The LSPR coupling has also been modeled using the local density approximation for an electron gas,24 which has led to the hybridization model.1,25-28 In analogy with molecular orbital theory, the coupling between nanoparticles leads to coupled resonant modes, or hybrid states, and is essentially equivalent to the molecular exciton coupling theory. The hybridization theory has been applied to dimers, trimers, and quadrumers using a multipole expansion in terms of spherical harmonics.27,29 While these methods have been useful for describing the interactions between ensembles of nanoparticles, they are either limited to dipole interactions between nanoparticle pairs or involve complex parameter relationships. In contrast, it is possible to represent these interactions in terms of the evanescent electric fields arising from the natural resonant modes (or eigenmodes) of each nanoparticle. In particular, for nanoparticles much smaller than the wavelength of light, the electric and magnetic fields decouple and Maxwell’s equations for the spatial dependence of the electric field take the same form as in electrostatics. In this “electrostatic approximation”, it is relatively straightforward to solve for the LSPR modes, which are found as self-sustained surface-charge (or surface-dipole) distributions.30-33 In this model, each mode is treated mathematically in the same way, independently of the shape of the nanoparticle, which is useful for describing the general properties of ensembles of nanoparticles of any shape. In this paper, we present an electrostatic formulation of the interaction of metallic nanoparticles. A rigorous derivation of this method was given recently34 and can be applied

* To whom correspondence should be addressed: E-mail: [email protected]. Received for review: 04/16/2010 Published on Web: 06/14/2010 Published 2010 by the American Chemical Society

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FIGURE 1. Examples of the localized surface plasmon modes and their energies for different gold nanoparticles calculated using the “electrostatic” eigenmode method. The nanoparticles are assumed to be in a medium with an electric permittivity εb ) 1.75. The coloring illustrates the relative signs of the LSPR.

resonance condition.48 The orientation of the dipole moment ppj of mode j with respect to the polarization of the incident field is important and we shall use this fact when discussing the interaction of an ensemble of nanoparticles. If the nanoparticle mode does not have a dipole moment, then it cannot be directly excited by the incident field (i.e., a dark mode, which could be excited by the evanescent fields of nearby bright particles). Examples of the LSPR modes of gold nanoparticles, calculated using the electrostatic eigenmode method, are shown in Figure 1. The modes can be labeled consecutively in terms of energy, so that the lowest energy mode is j ) 1. The LSPR energy depends on the mode number, on the electric permittivity of the metal and on the permittivity of the background medium. In the examples shown in Figure 1, the nanoparticles have the permittivity of gold35 and the background permittivity is similar to water, εb ) 1.75. When two or more of these nanoparticles are placed in proximity to one another, the electric fields from the LSPR will influence the surface-charges creating new modes. This electric coupling can be described by a coupling matrix representing the influence of the evanescent electric field of one nanoparticle on another. The coupling depends on the distance between the nanoparticles, on the nature of the resonant mode excited on one nanoparticle and how close this resonance is to that of neighboring nanoparticles. If apj(ω) is the excitation amplitude of a mode of an isolated nanoparticle, then it can be shown34 that the excitation amplitude a˜pj(ω) of the nanoparticle in an ensemble of N nanoparticles is given by a matrix equation

to any number of interacting nanoparticles, provided that the size of the ensemble is smaller than the wavelength of light. As we will show, the results are equivalent to the exciton coupling theory for nanoparticle pairs, including the angular dependence for the dipole modes, and from it we derive some simple rules for determining how resonant nanoparticles couple together. More importantly, this method enables the construction of hybridization diagrams in terms of coupling strengths for systems consisting of many particles and many modes. In this formulation, the relative energy levels of coupled systems (and therefore their resonance frequencies) can be derived without requiring complex computation but using only simple algebra. To illustrate the usefulness of this approach, we will consider typical sample systems such as a dimer of nanorods. Furthermore, we also discuss the applicability of this approximation to explain Fano resonances and the phenomenon of plasmoninduced electromagnetic transparency. We start by considering a single metallic nanoparticle. When illuminated by light, the electric field can excite one or more of the LSPR modes depending on the electric permittivity of the nanoparticle and the frequency of the incident light field.33 By approximating the electric permittivity by a Drude model (which is relatively accurate away from the interband transitions), the resonant frequencies of the LSPR can be determined analytically. Within this approximation, the excitation amplitude of the j-th resonance of an isolated nanoparticle p is given by

apj(ω) ) -

Apj (ω -

ppj · E0(ω) j ωp)

(1) N

a˜pj(ω) )

∑ ∑ (δkjδqp - Cqpkj(ω))-1akq(ω) k

where ω is the frequency of the incident electric field E0(ω), ωpj is the resonant frequency of the nanoparticle with dipole moment ppj and Apj > 0 is a constant that depends on the resonant mode, the plasma frequency of the metal and the permittivity εb of the embedding medium. The amplitude of the mode is large when ω ) ωpj, which represents the Published 2010 by the American Chemical Society

(2)

q)1

The term in brackets represents the inverse of a matrix jk which depend on that contains the coupling coefficients Cpq the relative geometric arrangement between pairs of nanoparticles and on their resonant modes 2619

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jk Cpq ≈-

jk Gpq

(ω - ωpj)

(3)

jk The geometric factor Gpq describes the influence of LSPR mode k of particle q on the mode j of particle p. Its precise evaluation requires a knowledge of the eigenfunctions of each mode and the relative orientation of the nanoparticles, which are relatively straightforward to compute.32,34 This is important when describing the coupling of higher-order resonances or when the nanoparticles are close together. However, we can deduce many of the properties of coupled systems without a detailed knowledge of this coupling, particularly for the lowest-order modes that usually have strong dipole moments. For two nanoparticles that have modes with dipole moments pjp and pkq, the geometric factor takes the approximate and familiar dipole-dipole coupling form

jk ≈ Gpq

Apj 3 4πεbdpq

(3(ppj · dˆpq)(pkq · dˆpq) - ppj · pkq)

(4) FIGURE 2. The hybridization of surface plasmon resonances arising from the interaction of two nanoparticles. Top: the relative geometry of two nanoparticles showing their polarization vectors as would be jk used in estimating the coupling coefficient Gpq . θ1 and θ2 are the angles formed by the nanoparticles with the horizontal line. Bottom: the hybridization diagram for the two nanoparticles showing the relative shift in energies for different orientations. The solid curve is for nanoparticle 2 horizontal (θ2 ) 0°) and the dashed curve is for nanoparticle 2 vertical (θ2 ) 90°). The arrows in each nanoparticle indicate the direction of the dipole moment pqk used in the calculation.

that depends on the distance dpq between the nanoparticle centers and on the unit vector dpq pointing from the center of particle q to particle p. By convention, we take the jk ) 0. For all coupling of a nanoparticle with itself as zero, Gpp particle pairs, it is a general result from eq 4 that the geometric term in the dipole approximation is symmetric, jk kj ) Gqp and these two coupling terms are equal so that Gpq irrespective of the relative orientation of the two particles. The dipole approximation is useful for estimating the relative signs of the coupling coefficients, particularly when the single nanoparticle modes are predominantly dipolar. However, we will show that it is not essential to make this dipole approximation since the general form of the coupling equations eqs 2 and 3 remains essentially the same and the geometric factor can be left as an unknown constant. This is all we need to determine the hybridization of complex nanoparticle systems exhibiting multiple resonant modes. For accurate evaluations of the coupling we use combinations of analytical and numerical methods. Below we provide examples of applications of this theory to systems consisting of two, three, four and eight nanoparticles. For a nanoparticle pair (each labeled here with subscripts 1 and 2) exhibiting one resonant mode (superscript 1) the coupling formula eq 2 simplifies to a simple 2 × 2 matrix34 (see Supporting Information for details). The inverse of the 11 matrix involves the two-particle determinant ∆2 ) 1 - C11 12C21 and the excitations of the coupled nanoparticles depend on 1/∆2. This means that the amplitudes of the coupled nanoparticles are very large when ∆2 ) 0. This is the condition for a resonance of the coupled system and we use this to Published 2010 by the American Chemical Society

determine the resonances, or hybridized energies. The 11 11 and C21 are found by choosing a coupling coefficients C12 direction for the dipole moment of each nanoparticle and then estimating the geometric coupling terms using eq 4. An example is shown in Figure 2, which depends on the relative orientations of the nanoparticles. Since the geometric term in the dipole approximation is symmetric, we write 11 11 ) G21 ) Gf(θ1,θ2) with G a constant and where f(θ1,θ2) ) G12 3 cos θ1 cos θ2 - cos(θ1 - θ2) results from the vector dotproducts in eq 4. This factor has the same angle dependence as derived from molecular exciton coupling theory.23 If the nanoparticles have identical resonances ωR when isolated, then the resonance condition ∆2 ) 0 with eq 3 leads to (ω - ωR)2 - G2f2(θ1,θ2) ) 0 with solutions ω ) ωR ( |Gf(θ1,θ2)|. There are two solutions so that the coupling of the nanoparticles causes the resonances to split into two modes, one with a higher frequency ω ) ωR + |Gf(θ1,θ2)| and another with a lower frequency ω ) ωR - |Gf(θ1,θ2)|. The splitting depends on the strength of the geometric coupling and leads to the hybridization diagram in Figure 2 which depends on the relative orientations of the nanoparticles. This simple formula accounts for the splitting of the LSPR modes,1,2 the red shift for nanoparticles coupling end-to-end3-9 and the blue shift for nanoparticles coupling 2620

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FIGURE 3. The three coupled nanorods that give rise to plasmon-induced electromagnetic transparency. (a) A three-dimensional rendering of the structure; (b) the geometry used in determining the hybridization, including the directions of the dipole moments p chosen for each nanorod; (c) the three lowest-energy resonant modes of the coupled structure, where the color represents the localized surface plasmon resonance; (d) the hybridization diagram where the numbers on the levels refer to the three modes; (e) the fractional energy shift in units of (ω - ωA)/G as a function of the displacement of the central antenna nanorod, parametrized by the angle θ. In this example the difference between the resonances of the parallel nanorods and the antenna is (ωP - ωA)/G ) 0.2. The upper curve is associated with the quadrupole resonance that is not excited when θ ) 0; (f) the absolute value of the excitation amplitude of the dipole antenna as a function of the energy detuning p(ω - ωA), where the asymmetry observed on the feature at 0.1 eV is characteristic of a Fano profile. For this example ωP - ωA ) 0.1 eV and a complex resonance was used (pγA ) 0.015 eV, G ) 0.02) to demonstrate the loss of the Fano profile with excessive damping.

side-to-side.5,7,9 The relative magnitude of the splitting is exact in the dipole limit where the contributions of higherorder multipoles are negligible and where the nanoparticles are sufficiently far apart. However, at small distances, the dipole approximation fails and higher-order multipoles need to be taken into account in order to accurately predict the frequency shifts. The splitting of the LSPR modes for nanoparticle pairs forms the basis of the plasmonic ruler36-38 and it is the coupling of higher-order modes that leads to the observed exponential dependence of the mode splitting on separation. We have shown that the electrostatic eigenmode method describes the coupling between nanoparticle dimers. However, the advantage of the method is that it can be applied to more complex systems, such as the coupling of three nanoparticles, by means of simple matrix algebra. To illustrate this point, we now consider a system of three nanoparticles exhibiting one dominant mode each. In this case, the coupling matrix is relatively straightforward, and 11 it has an inverse with a determinant given by ∆3 ) 1 - C11 12C21 11 11 11 11 11 11 11 11 11 11 - C13C31 - C23C32 - C12C23C31 - C13C32C21. Note that the determinant consists of combinations of coupling coef11 11 ficients that form closed cycles. For example, C12 C21 represents the coupling from nanoparticle 1 to nanoparticle 2 and then from nanoparticle 2 back to nanoparticle 1. The com11 11 11 C23C31 represents the coupling from particle 1 bination C12 Published 2010 by the American Chemical Society

to particle 3, 3 to 2 and finally, 2 to 1. Such closed cycles represent all the paths in which energy can be trapped between the nanoparticles in the ensemble, which is characteristic of a resonance. It is not surprising therefore that the determinant is associated with resonances. The resonances occur when ∆3 ) 0 and the amplitudes of the LSPR oscillations in the coupled nanoparticles are very large. As an example of three-particle coupling, we consider the nanoparticle system shown in Figure 3. This is the basic element in the metamaterial of Liu et al12 which demonstrates the plasmonic analogue of electromagnetic-induced transparency (PEIT). The structure consists of two identical nanoparticles that are bridged by a third which acts as an optical dipole antenna. Note that the parallel nanoparticles have the same resonant frequency ωP whereas the optical antenna is slightly longer so that it has a slightly lower resonant frequency ωA. For simplicity, we can assume that the coupling between nanoparticles 1 and 2 is small com11 11 pared with the coupling to the antenna, so that G12 ) G21 ≈ 11 0. Then the remaining terms in the coupling matrix are G13 11 11 11 ) G31 ) G sin 2θ and G23 ) G32 ) -G sin 2θ, with G > 0. The displacement of the antenna is described by θ (see Figure 3b). The matrix determinant ∆3 ) 0 leads to the expression (ω - ωP)(ω - ωA) - 2G2 sin2 2θ ) 0, so that the resonant frequencies of the coupled system are 2621

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ω1,2 )

ωP + ωA ( 2

√(ωP - ωA)2 + 8G2 sin2 2θ

parency” (PEIT) or “Plasmon-induced Electromagnetic Transparency”. Furthermore, the dipole scattering associated with the optical antenna is zero at ω ) ωP (or has a minimum if we include damping). However, the amplitudes of the other two nanoparticles are not zero, as seen in eq 6, indicating that there are still LSPR excitations in these nanoparticles. This is a key feature that identifies PEIT. As demonstrated with this example, the algebraic electrostatic model allows for a systematic approach to achieve this transparency by design. Our eq 7 is equivalent to the coupled-oscillator model, presented by Liu et al,12 for modeling PEIT. It has been suggested that a similar effect occurs with two parallel nanorods provided they have slightly different resonant frequencies.34 The resonances exhibited by the excitation of the dipole antenna, as described by eq 7, are known as Fano resonances, after Ugo Fano39-41 who studied the general form of oscillating systems coupled to a continuum of states. The scattering spectrum of a system exhibiting a Fano resonance is characterized by a strong asymmetry occurring at the resonance of one of the states of the system. Typically, the Fano resonances arise here from the destructive interference between the resonance in the optical antenna A and the resonance of the parallel nanoparticles P when ωA * ωP. The dipole resonance of A needs to be sufficiently broad to drive the resonance of P. In this case A acts like a continuum of states.42 The resonance of P needs to be sufficiently strong to create destructive interference that leads to the asymmetry in the resonance. In the experiment of Liu et al,12 this asymmetry is apparent but is not easily observed because the resonances are relatively broad and the resonance of the parallel nanoparticles is not strong enough. This is probably due to Ohmic losses in the gold nanoparticles. We can show using eq 7 that the system of nanoparticles in Figure 3 gives rise to Fano resonances. To account for damping effects in this model, we have introduced complex resonance frequencies ωP - iγP and ωA - iγA. By assuming that coupling between the antenna and the particle pair dominates over damping of either resonance (i.e., G > γP,A) and that damping on the dipole antenna is stronger than the one for the pair, eq 7 predicts an asymmetric line shape, shown in Figure 3f. This is typical of a Fano profile.40,41 The asymmetric profile can easily disappear when the damping experienced by the particle pair is comparable to that of the dipole antenna (γP ) γA), since in this case the interaction no longer parallels that of a discrete state and a continuum. The Fano resonances associated with coupled nanoparticle systems exhibiting LSPR have been discussed in the literature.2,16,42,43 The electrostatic eigenmode method can be used for even more complex arrangements of nanoparticles. We have shown previously44 that for symmetric arrays of nanoparticles the energy shifts and the resonant modes of the interacting systems can be found by using the machinery of group theory. It can be shown that the eigenfunctions

2 (5)

There is a third resonance, ω3 ) ωP that corresponds to mode 3 in Figure 3c, which is unimportant for our analysis of PEIT. The hybridization diagram shown in Figure 3d is constructed using eq 5. For θ ) 0 the two resonant frequencies correspond to the two uncoupled configurations of nanoparticles (the optical antenna and the parallel pair). The change in position of the optical antenna (θ increasing) increases the coupling and changes both resonances. The resonances split symmetrically about the average of the two individual resonances and the degree of splitting changes with the strength of the coupling (Figure 3e). The excitation amplitudes are obtained from the matrix equation (see Supporting Information for details). In this experiment the incident light was polarized parallel to the antenna but perpendicular to the parallel nanoparticles, so that from eq 1 we have a11 ) a22 ) 0 and we can write a31 ) -A13p13·E0(ω)/(ω - ωA). Then the excitation amplitudes of the coupled nanorods are

a˜11 ) -a˜21 )

a˜31 ) -

G sin 2θ(A31p31 · E0(ω)) (ω - ωP)(ω - ωA) - 2G2 sin2 2θ

(ω - ωP)(A31p31 · E0(ω)) (ω - ωP)(ω - ωA) - 2G2 sin2 2θ

(6)

(7)

Note that the amplitudes of the parallel nanoparticles always have the opposite sign a˜11 ) -a˜12 so that the two nanoparticles resonate in antiphase. This is a quadrupole resonance that does not emit dipole radiation. From eq 7, we observe that for θ ) 0 the excitation amplitude of the optical antenna has the same value a31 as when it is isolated (i.e., no coupling). This is one of the resonances obtained from eq 5 at this angle. The other resonance, eq 6, cannot be directly excited because of its dominant quadrupole-like character (mode 2 of Figure 3c). However, as θ increases and the coupling becomes larger, the quadrupole-like resonance is excited and the splitting of these two resonances (modes 1 and 2 of Figure 3c) increases. An important consequence of our foregoing analysis is that, according to eq 7, the scattering of light at the frequency of the uncoupled resonance, ω ) ωA, should decrease as the coupling increases (i.e., increase in θ). This occurs as a consequence of the shift in the resonance frequency and is a phenomenon that has been called “The Plasmonic analogue of Electromagnetically Induced TransPublished 2010 by the American Chemical Society

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two nondegenerate solutions which is consistent with group theory. The corresponding hybridization diagram based on this equation is the same as that shown in Figure 4d. One of the useful features of the electrostatic eigenmode method is that it can treat multiple nanoparticles as a single particle. That is, it does not require the boundary of the “particle” to be continuous. While we have not discussed this point in detail, the ramifications are that groups of nanoparticles can be treated as a single nanoparticle. For example, we can solve the coupling problem for the eight particle system shown in Figure 4c by treating each pair of nanoparticles as a single particle. The two modes of this “single particle” (Figure 4b) form a symmetric and an antisymmetric combination and correspond to those modes in Figure 2 that have θ1 ) θ2 ) 90°. This allows us to treat this problem as a 4-particle system. It is clear that 4-particle combinations of each of the modes separately have the same symmetry properties as the 4-particle system that was analyzed above, but that it is not possible to form mixed combinations involving the symmetric and antisymmetric modes. This means that the eigenfunctions form two groups with each group represented by A1g + B1g + Eu. The eigenfunctions have been determined numerically, as shown in Figure 4d, and show close agreement with our analysis. In this way, complex arrangements of nanoparticles can be analyzed using a hierarchy of simpler nanoparticle systems.44 Although the electrostatic eigenmode method provides a useful theoretical tool for understanding the interactions between plasmonic nanoparticles and for determining the hybridization of their states, it is an approximate method. The main limitation is that it requires the nanoparticle system to be much smaller than the wavelength of the applied light, so that the associated electric field appears uniform over the nanoparticle ensemble. In other words, the method does not take account of the phase shifts associated with the wave nature of the light field. The effect of the phase shifts is often referred to as retardation and it leads to a red shifting of the LSPR. This means that the electrostatic method does not accurately predict the resonant frequencies and the predictions become less accurate as the nanoparticle ensemble becomes larger. Even so, for moderately sized nanoparticles the errors can be around 5% depending on the nanoparticle shape and on the resonant mode.45 In addition, the coupling between the nanoparticles is described only in terms of the evanescent electric fields and there is no account of the phase effects associated with the wave nature of the fields. Furthermore, there is a broadening of the resonances associated with the radiation of light (radiation broadening), which becomes pronounced as the nanoparticle size approaches that of the wavelength, although it is possible to add corrections for radiation broadening into the electrostatic method.34 Even with these limitations, in many applications the method captures much of the underlying physics of the interactions and in this regard provides a very useful analytical tool.

FIGURE 4. An example of complex configurations of interacting nanoparticles that can be analyzed using group theory. (a) A square array of four nanorods - the arrows show the directions of the dipole moments used to represent the default configuration of the structure; (b) a pair of nanoparticles that is treated as a single nanoparticle with two modes; (c) an ensemble of eight nanoparticles arranged in a square that is equivalent to the configuration of four nanoparticles; (d) the hybridization diagram showing the relative shifts in frequency associated with the coupling of four nanostructures and the corresponding eigenmodes for the parallel and antiparallel combinations of the four pairs of nanostructures. The colors represent the relative signs of the LSPR oscillations, with blue positive and red negative. The eigenmodes were calculated numerically for comparison with theory.

form bases for the irreducible representations of the symmetries associated with the ensemble. For a symmetric system of four nanoparticles each exhibiting one dominant resonance, shown in Figure 4a, the interactions are described by a 4 × 4 coupling matrix. The symmetry of this system belongs to the D4h point group with an irreducible representation A1g + B1g + Eu. Here A1g is a one-dimensional representation that is symmetric with respect to rotations, B1g is a one-dimensional representation that is antisymmetric with respect to rotation and Eu is a two-dimensional representation symmetric with respect to inversion through the center of the square. Since representations A1g and B1g are scalars then the corresponding modes will have zero dipole moments and group theory predicts there will be two dark modes. Furthermore, the two-dimensional representation Eu is associated with two degenerate “bright” modes. For comparison, we analyze this system using the matrix inversion in eq 2. Assuming all the nanoparticles are identical, each with a resonant frequency ωR, there are two types of coupling: the coupling between adjacent nanoparticles 11 11 ) G21 ) Ga and the coupling between opposite such that G12 11 11 ) Go, as well as the cyclic combinananoparticles G13 ) G31 11 11 ) G34 ,... etc.) The tions of these coefficients (such as G23 condition ∆4 ) 0 leads to a quartic equation that can be put in the form (ω - ωR - Go)2(ω - ωR + 2Ga + Go)(ω - ωR 2Ga + Go) ) 0. This shows the two degenerate solutions and Published 2010 by the American Chemical Society

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Although we have not discussed it here, the coupling method can be applied to other systems. For example, the method was used to determine the effects of a nearby molecule on the localized surface plasmon resonances of a nanoparticle.46 The molecule creates a small change in the local electric permittivity around the nanoparticle that leads to a small shift in the resonant frequency and in the phase of the LSPR relative to the incident light field. This is important in applications in chemical sensing using LSPR. The method has been used to model the influence of a substrate on the LSPR of nanoparticles. In this case the problem was recast into one for the interaction of a nanoparticle with its image charge. This interaction shifts the frequency of the LSPR. The calculated magnitude of these spectral shifts agree with the experimental data.47 In summary, we have presented an electrostatic coupling method for describing the interaction of metallic nanoparticles supporting LSPR. The method provides a formal way of analyzing the coupling between nanoparticles when the ensemble is much smaller than the wavelength of light. With this method it is possible to predict the hybridization of resonant modes in ensembles of nanoparticles by means of simple matrix algebra. This procedure was illustrated for two canonical cases found in the literature, namely a nanoparticle dimer and an interacting system exhibiting the plasmonic analogue of EIT. For this latter case, we also showed how the electrostatic eigenmode method can be extended to account for the observation of Fano spectral profiles. For cases involving more than three particles, the algebra might become more cumbersome. However, in these cases it is possible to exploit the symmetry properties of the interacting configuration of nanoparticles to find the collective resonance modes. Although the method is not accurate in a numerical sense, it leads to mathematical expressions that are able to model the interactions of the nanostructures quite well, yielding insight into the resulting phase shifts and frequency changes. We anticipate that due to its ease of applicability and simple conceptual basis, the electrostatic eigenmode method will be of immense value in understanding and predicting the effects of interacting plasmonic nanoparticles as well as for designing plasmonic systems.

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Supporting Information Available. This material is available free of charge via the Internet at http://pubs.acs.org.

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(47) Vernon, K.; Funston, A.; Novo, C.; Go´mez, D.; Mulvaney, P.; Davis, T. J. accepted Nano Letters 2010. (48) Strictly (as discussed in the Supporting Information), there is an imaginary term iΓ that we have neglected that prevents the amplitude from becoming infinite under this condition. This imaginary component is related to damping in the nanoparticle

Published 2010 by the American Chemical Society

and is associated with the imaginary part of the metal electric permittivity. However, this detail is not important for determining the hybridization properties of coupled nanoparticles, as we discuss below. If required, the resonant frequency ωpj can be replaced by a complex frequency ωpj f ωpj - iΓp/2, where Γp is the loss term in the Drude formula.

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DOI: 10.1021/nl101335z | Nano Lett. 2010, 10, 2618-–2625