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Russian-Armenian (Slavonic) University, 123 Hovsep Emin Street, Yerevan, Armenia ... Scientific Laboratory, 2 Alikhanian Br. Street, 0036 Yerevan, Arm...
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Radiation Damping of Surface Plasmons in a Pair of Nanoparticles and in Nanoparticles near Interfaces K. K. Madoyan,† A. H. Melikyan,† and H. R. Minassian*,‡ †

Russian-Armenian (Slavonic) University, 123 Hovsep Emin Street, Yerevan, Armenia A. Alikhanian National Scientific Laboratory, 2 Alikhanian Br. Street, 0036 Yerevan, Armenia

‡

S Supporting Information *

ABSTRACT: A theory of surface plasmon radiation damping in a system of coupled spherical metallic nanoparticles is developed, and a simple formula for the radiation line width is obtained for the first time. It is shown that as a result of surface plasmon frequency redshift, a notable reduction of the radiation damping rate takes place. For small separations, the radiation line width narrows by a factor of 3 as compared to large separations. The theory is expanded to the case of a spherical metallic nanoparticle placed near an interface of two dielectric media. The dependence of the redshift of surface plasmon frequency and the radiation damping rate on the particle−interface separation are calculated. It is revealed that in both cases, a decrease of refractive index of the surrounding media also leads to a decrease of the damping rate.

1. INTRODUCTION

Generalized multiparticle Mie theory can be used to analyze the problem of electromagnetic scattering by aggregates of an arbitrary number of arbitrarily configured spheres.8 Solutions are obtained by applying an asymptotic iteration technique to a truncated subset of an infinite set of linear equations. In the particular case of gold nanosphere dimers, this theory is applied to investigate the spectral shape of the emission of fluorescent molecules9 and field enhancement on broadband SERS spectra.10 Note that the approach developed in ref 8 does not provide explicit analytical expression for the RDR even in sphere dimers. Therefore, to analyze the experimental data on the resonance frequencies in coupled MNPs or in MNPs near the interface, the RDR numerical methods are applied.2,11−14 It was proved recently15−18 that coupled nanospheres and the system “nanosphere near an interface” can be described analytically with good accuracy using eliminated quadrupole moment approximation (EQMA). It was demonstrated particularly that the SP frequencies determined in the frame of EQMA agree very well with known experimental data and numerical resuts.1,19 In this communication, we develop based on EQMA an approximate analytical approach to the theory of radiation damping of SP oscillations in coupled nanospheres, which is expanded to the case of a nanosphere near an interface.

Metallic nanoparticles (MNPs), especially noble metal particles, demonstrate narrow optical resonances due to surface plasmon (SP) oscillations. Depending on the dielectric permittivity of the matrix and the metal as well as the shape of the nanoparticles, the resonance frequency noticeably shifts, which makes MNPs excellent for biophysical applications. From this point of view, the coupled MNPs, the MNP near an interface, and ensembles of MNPs are of special interest because of a strong dependence of SP frequencies on the geometry of the system.1−3 This interest is conditioned by the possibility of using them in biosensing and labeling of macromolecules4 and using them as “nanorulers” for measuring distances at the nanoscale.1 Another possible field of application is connected with the prominent field enhancement at the point of contact in touching nanospheres.5 It is clear that high enough spectral sensitivity of MNP-based optical devices can be achieved only in the case of sufficiently narrow SP resonances. There are three mechanisms of SP resonance line broadening, namely, the Drude relaxation mechanism: electron−phonon and electron-impurity scattering, electron scattering on the MNP surface, and radiation damping. In the case of an isolated nanosphere with size less than 20 nm the second mechanism (the so-called 1/R mechanism, R is the sphere radius) dominates (see p.81 in ref 6). However, for larger particles, the main size dependent mechanism of line broadening becomes radiation damping because its rate is proportional to R3.7 Since in biophysical applications the MNPs (both isolated and coupled) with R ≥ 20 nm are commonly used,4 the problem of calculating the radiation damping rate (RDR) in coupled nanospheres and in nanospheres near an interface becomes important. © 2012 American Chemical Society

2. RDR OF SP OSCILLATIONS IN COUPLED SPHERES Consider SP oscillation damping in a system of coupled identical metallic nanospheres with the centers separated by the distance a and placed on the Z-axis. We show below that the damping rate of SP oscillations in coupled nanospheres can be Received: May 27, 2012 Revised: July 13, 2012 Published: July 17, 2012 16800

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where ε(ω) is the real part of the dielectric function of the tot metal, Φtot 1 and Φ2 are the total internal and external potentials on the surfaces of spheres 1 and 2 correspondingly (see Figure 1), E⃗ 1 and E⃗ 2 are corresponding electric field strengths, and n⃗ is

calculated analytically using the well-justified EQMA approximation developed in ref 16. It was particularly shown in the mentioned paper that for the interparticle center-to-center distances as small as 1.1 of the sphere diameter, the deviation of analytically determined SP frequencies from the results of numerical calculations19 is less than 1%. The energy loss caused by radiation is described by the rate equation

dW 1 =− W dt τ

(1)

where W is the energy of a system and 1/τ is the damping constant, which is the sought quantity of the problem under consideration. The radiation power N = |dW/dt| of a system with a total dipole moment p⃗tot = p⃗ cos ωt is given by the formula N = 2/(3c3) (d 2ptot ⃗ /dt 2)2 (c is the speed of light, the bar denotes averaging over the period of oscillations) (see exp. 67.7 on page 189 in ref 20), which can be expressed in the following form: ω4p ⃗ 2 dW =− 3 dt 3c

ε0

Figure 1. Dark arrows are the dipole moments of surface charge distributions in the case of longitudinal in-phase oscillations (a), and transverse in-phase oscillations (b).

the unity vector of the outer normal of the sphere. We do not take into account the imaginary part of the dielectric function in the boundary conditions (eq 5) in order to solely reveal the contribution of the radiative losses. It is well-known that even in this simplified case, the problem does not allow analytical solution, and time-consuming numerical calculations are usually employed.19 Instead we apply the EQMA in order to analytically calculate σ and φ as a function of the amplitude of the total dipole moment of the system of coupled identical spheres p⃗ = 2p⃗0 (p⃗0 is the amplitude of the dipole moment of one sphere) and eventually W.22 Here we take into account the fact that, due to the symmetry of the problem, the dipole moments of both spheres are equal. First we consider the evolution of the interaction between the particles versus interparticle separation. In the case of large separations (a ≫ R), the oscillating SP field on sphere 1 can be considered uniform at the position of sphere 2 (and vice versa). Consequently, the electric field of the second sphere coincides with the field of a point dipole placed in its center.23 Because of symmetry, the problem is now reduced to the case of interacting point dipoles separated by the distance a. However, when the spheres come closer to each other, the nonuniformity of the dipole field induces higher order multipoles in the surface charge distribution that should be properly taken into account. According to EQMA, instead of calculations involving higher order multipoles, the electric field of each sphere in the outer space is replaced by the field of a point dipole shifted from the center of the sphere on the distance δ determined below. As it was demonstrated in ref 16, these replacements adequately reflect the charge redistribution caused by the Coulomb interaction between the spheres. It is clear physically that, in the case of longitudinal in-phase oscillations (because of attraction), the introduced dipoles must be placed not in the centers but closer to each other, i.e., shifted toward the interparticle gap. In analogy, for transverse in-phase oscillations (because of repulsion) the introduced dipoles must be placed away from the centers and from each other (see Figure 1). The position of each introduced dipole in EQMA is determined from the condition that the quadrupole moment of the surface charge distribution with respect to the chosen location vanishes. For in-phase longitudinal oscillations taking place along the Zaxis, the value of the shift δ as the calculation shows is determined by the formula δ = DZZ/4pZ,16 where pZ and DZZ are correspondingly the components of dipole and quadrupole moments of the sphere with respect to its center. In both cases presented in Figure 1, the oscillations can be excited optically; however, we consider only case (a), meaning that the SP frequency shift is significant only for this type of

(2)

where ε0 is the dielectric constant of the surrounding media. Comparing 1 and 2 we find ωsp4p ⃗ 2 1 = 3 τ 3c W

ε0

(3)

In the case of harmonic oscillations, the ratio p⃗2/W = k does not depend on energy, thus in order to determine 1/τ from eq 3, we need to calculate the SP resonance frequency ωsp and k as functions of interparticle distance a. We consider particles with radii much smaller than the SP radiation wavelength λsp = 2π/ωsp, where retardation effects can be neglected. This allows us to apply quasi-static approximation,6 which implies solving the Laplace equation with appropriate boundary conditions. In the absence of external field, this procedure leads to a homogeneous system of equations for the electric field potential and its normal derivatives at the boundaries of the coupled spheres and the environment. This system allows for a nonzero solution only for certain values of dielectric function of metal. These allowable values along with experimental data on the frequency dependence of the dielectric function of metal provide the SP frequencies. The total energy of the system in the quasi-static approximation is the sum of kinetic and potential energies of the charges. When the velocities of all charges are equal to zero, the maximum value of the potential energy is equal to the total energy: W = ∫ (E⃗ 2/8π) dV, where E⃗ is the electric field strength. In the case when the volume charge density is absent, performing integration by parts reduces the expression for W to (see eq 16 in ref 21, page 107)

W=

1 2

∮ σφ dS

(4)

where φ is the electric field potential and σ is the surface charge density. Note that for the coupled spheres, the integration is to be carried out over the surfaces of both spheres. In order to determine φ and σ, we need to solve the boundary problem for the Laplace equation with the boundary conditions Φ1tot = Φ2tot , ε(ω)E1⃗ n ⃗ = ε0E2⃗ n ⃗

(5) 16801

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oscillation. The derivation of the electric field potential for the coupled spheres in the frame of EQMA and further calculations of the dipole and quadrupole moments of the surface charge distribution are presented in the Supporting Information. Using the symmetry of the system, which implies that the dipole moments of spheres 1 and 2 with respect to their centers are equal (p′0 = p0) and the definition of the shift δ = D′/4p′016 we obtain from eq S3: 2

a(ωsp) = R 3 2

3

δ(ωsp) =

ε(ωsp) − ε0 ⎛ R ⎞3 ⎜ ⎟ = 1, ε(ωsp) + 2ε0 ⎝ a − δ ⎠ ε(ωsp) − ε0 ⎛ R ⎞4 ⎜ ⎟ = δ 3R 2ε(ωsp) + 3ε0 ⎝ a − δ ⎠

⎞ ∂Φtot 1 ⎛ ε0 − 1⎟ ext ⎜ 4π ⎝ ε(ω) ⎠ ∂R

(6)

ε0 − ε(ω) 2 R 8πε(ω)

∫ ∂∂R (Φexttot)2 dΩ

+

(7)

γ0 =

(13)

⎡⎛ R3 2R6 ⎞ ⎢ ⎜1 − − ⎟ 3 ε(ωsp) − ε0 ⎢⎣⎝ (a − δ ) (a − δ)6 ⎠ ε(ωsp)

(14) (8)

(9)

⎛ ωspR ⎞3 ε(ω) ε0 1 = 2ωsp⎜ ⎟ τ ⎝ c ⎠ ε(ω) − ε0 ⎡⎛ R3 2R6 ⎞ × ⎢ ⎜1 − − ⎟ 3 ⎢⎣⎝ (a − δ ) (a − δ)6 ⎠ −1 9 ⎛ 2δ 2 3R8 ⎞⎤ δR3 ⎥ − ⎟ ⎜ 2 − 5⎝ R (a − δ)4 (a − δ)8 ⎠⎥⎦

(12)

In the limiting case of extremely large separations γ0 → 1, we have ε(ω)/ε0 → −2, and expression 13, as it is expected, gives the doubled RDR for an isolated sphere.7 We mention here that expressions 13 and 14 contain only the contribution of radiative losses in γ in accordance with eq 5, where the imaginary part of the dielectric function of metal describing the nonradiative losses is not included. Numerical calculations with eqs 10−14 show that the quantity γ0 almost does not depend on geometrical parameters. Namely, when center-to-center distance a varies from 6R to 2.1R, the quantity γ0 changes within the interval 0.710 ± 0.015 (i.e., only by 4%) for 30 a nm particle, whereas γ decreases from 0.25 to 0.20 eV (ωsp is calculated with use of the plasmon nanoruler eq 11 and the corresponding data for Re(ε(ω)) of ref 24). Moreover, it turns out that γ0 does not depend on ε0 at all, which is a consequence of the boundary conditions of eq 5 and is not connected with applied approximation. Thus the behavior of RDR is actually expressed by the simple dependence similar to that of isolated sphere γ = 0.71·ω4spR3(ε0)1/2 with the SP frequency depending on the ratio of interparticle separation and particle radius, as well as on the dielectric constant of surrounding - ωsp = ωsp(a/R, ε0). Using inverse dependence from eq 11, we calculate SP frequencies as a function of interparticle gap for the coupled MNPs with radii 30 and 20 nm embedded in three different dielectric media. The corresponding values of RDR versus the interparticle gap for the same sets of the parameters are calculated according to eqs 10−12. The results for the SP frequencies and the RDR are presented in Figures 2 and 3. It follows that when the gap decreases from 80 to 3 nm, the redshift of SP frequencies in all cases amounts to about 10% of

Since p⃗0 is the amplitude of the dipole moment of one sphere and p⃗ is the amplitude of the total dipole moment of the system of two spheres, we have p⃗20 = p⃗2/4 and obtain from eqs 9 and 3 for the RDR

+

3R ε(ωsp) + 2ε0 1 ε(ωsp) + 2ε0 3 2 2ε(ωsp) + 3ε0 2 ε(ωsp) − ε0

(11)

−1 9 ⎛ 2δ 2 3R8 ⎞⎤ δR3 + ⎜ 2 − − ⎟⎥ 5⎝ R (a − δ)4 (a − δ)8 ⎠⎥⎦

2p0⃗ 2 ε0 − ε(ω) ⎡⎛ R3 2R6 ⎞ ⎢ ⎜1 − − ⎟ 3 3 ε(ω) ⎢⎣⎝ 3R (a − δ ) (a − δ)6 ⎠ 9 ⎛ 2δ 2 3R8 ⎞⎤ δR3 − ⎟⎥ ⎜ 2 − 5⎝ R (a − δ)4 (a − δ)8 ⎠⎥⎦

1 ε(ωsp) + 2ε0 2 ε(ωsp) − ε0

where

where dΩ is the differential of the solid angle. The integration in eq 8 leads to W=−

3R ε(ωsp) + 2ε0 2 2ε(ωsp) + 3ε0

⎛ ωspR ⎞3 γ = 2ωsp⎜ ⎟ ε0 γ0 ⎝ c ⎠

Using eq 4 with eq 7 for the case of identical coupled spheres, we come to the following expression for the total potential energy of the system of coupled spheres: W=

ε(ωsp) + 2ε0

+

Note that eq 11 (nanoruler equation) derived in ref 16 determines the inverse function of ωsp versus the interparticle separation a, allowing us to calculate the values ωsp(a). Substituting δ(ωsp) from eq 12 into eq 10, we obtain SP resonance line width γ = 1/τ as a function of ε(ω)/ε0 and a, which along with eq 11 gives the parametric dependence of the RDR on interparticle separation, the dielectric constant of the surrounding, and the radius of the particles. It is convenient to present γ in the form

Equations 6 allow us to determine the values of δ and ε(ωsp), which provide the SP resonance frequencies of coupled spheres using the experimental data for ε(ω).24 These data will be used below for the determination of the SP RDR in coupled identical spheres as well as in the sphere placed near an interface. Further, we determine σ with the use of the boundary conditions 5 and eqs 4.36, 4.38, and 4.46 of ref 23 as follows: σ=

ε(ωsp) − ε0

(10)

Our ultimate aim is to express the RDR as a function of interparticle separation. This can be achieved by expressing the interparticle separation and the shift from eq 9 in terms of ε(ωsp)/ε0: 16802

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are taken into account. In particular, for the isolated sphere, the value of 0.19 eV of the RDR is obtained. The complete comparison between these numerical data and our results cannot be fully realized since we neglect the retardation effects. As the role of the retardation increases with the increase of size of the system, the presented approach fails at certain size. First order correction to the SP frequency of an isolated sphere conditioned by the retardation is determined by the following equation:25 ε(ωsp) + 2ε0 +

Figure 2. The SP frequencies (dashed lines) and RDR (solid lines) of coupled spheres vs interparticle gap in the system of coupled spheres with 30 nm radii for three different values of refractive index of surrounding media: n0 = 1.33 (a), 1.43 (b), 1.53 (c).

2 12 ⎛ ωspR ⎞ 2 ⎜ ⎟ ε0 = 0 5⎝ c ⎠

(15)

The third term in eq 15 contains the dimensionless parameter μ = ωspR/c determining the SP frequency redshift conditioned by the retardation effects. As the calculations show, the magnitude of the redshift reaches ∼10% when μ = 0.8 (R ≈ 40 nm).22 From eqs 10−12, we obtain a close value of 0.22 eV for the RDR of an isolated R = 40 nm sphere in the media with refractive index 1.5 if we substitute 2.18 eV for ωsp taken from ref 19, where retardation effects are accounted for. It is interesting to mention that (see Figures 2 and 3) in case of n0 = 1.33, the RDR has a flat maximum at the gap values slightly less than 2R, which was also revealed in ref 14.

3. RDR OF SP OSCILLATIONS IN THE SPHERE NEAR AN INTERFACE Due to possible applications in biophysics, the problem of spectral sensitivity of SP resonance in spherical MNP placed near the interface of two adjacent media presents undoubted interest. The above-mentioned system can be used in spectroscopic studies if the mechanisms of SP line width formation, especially those caused by the radiation damping, are well understood. Since the RDR strongly depends on particle− interface distance and on dielectric properties of adjacent media, the system under consideration can be used in studying various biological objects interacting with MNPs.26,27 It can also be used to study metal nanoparticles on thin films and on transmission electron microscopy (TEM) grids. In this section, applying the method developed in section 2, we investigate the “spherical MNP near the interface” system and calculate the RDR dependence on particle-to-interface distance, particle radius, and dielectric properties of adjacent media. Consider a spherical MNP of radius R in the media with dielectric constant ε0, which is placed near the interface of another media with dielectric constant ε1. In order to expand the approach developed in section 2 to this problem, we apply the method of electrostatic images. Owing to this method, the interaction of the sphere with the interface can be reduced to the interaction of the sphere and its electrostatic image with the similar charge distribution. Indeed, if a point charge q is placed in the media (dielectric constant ε0) bordering the other one (dielectric constantε1), the image charge q′ = q(ε1 − ε0)/(ε1 + ε0) is located symmetrically with respect to the interface. Therefore, the surface charge distributions on the spherical MNP and its image are not the same (as in Sec. 2) but differ by the factor β = (ε1 − ε0)/(ε1 + ε0) for every pair of symmetrical points of MNP and its image. This means, for example, that if the dipole moment of the sphere accounting also for the polarization of the dielectric is p⃗0, then the dipole moment of the image is p⃗′ = βp⃗0. Note that, although with decrease of the distance between the dipole and the interface the interaction becomes stronger, increasing the magnitudes of the dipole

Figure 3. The SP frequencies (dashed lines) and RDR (solid lines) of coupled spheres vs interparticle gap in the system of coupled spheres with 20 nm radii for three different values of refractive index of surrounding media: n0 = 1.33 (a), 1.43 (b), 1.53 (c).

the resonance frequency of the pair. At the same time, the RDR decreases more sharply with the decrease of interparticle gap. With decrease of the refractive index of surrounding n0 = (ε0)1/2, the SP frequency and the RDR also decrease. For the same 3 nm value of the gap, the RDR decreases by a factor of 3 when the radius of the particles decreases from 30 to 20 nm. Thanks to this, the sensitivity of the 20 nm nanoruler with respect to the refractive index turns out to be much better as compared to the 30 nm one. Indeed, for the 20 nm particles, when the refractive index changes from 1.33 to 1.53, the SP frequency varies by 0.1 eV (from 2.25 to 2.14 eV), being larger than the radiation line width − 0.06 eV (see Figures 2 and 3). It is important to note that further decrease of the radius does not improve the sensitivity of coupled MNPs because of significant broadening of the SP spectra through electron scattering by the MNP surface (1/R mechanism; see section 2.2.1 of ref 6). For example, in spherical particles with 10 nm radius, 1/R broadening already dominates the radiation line width. It is important to note that, due to a strong redshift (see Figures 2 and 3), the SP frequency falls below the interband transition threshold of Au (2.38 eV).6 This circumstance turns out to be essential in possible applications of coupled MNPs since in this system the nonradiative interband SP line broadening is suppressed. The problem of determination of RDR in the system of coupled spheres is solved numerically in ref 14 by multipole expansion of the fields created by the particles. The boundary conditions imposed on the electromagnetic fields at the spherical surfaces are expressed in an infinite set of linear equations that are truncated. In ref 14, the total size of the system approaches the radiation wavelength (radii of the spheres are 40 nm), and consequently the retardation effects 16803

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moment p⃗0 and its image p⃗′, their ratio remains constant (β). Since the magnitude of the image point charge q′ does not depend on the distance between the real charge and the interface, it follows from the superposition principles that the multipoles of any order of arbitrary charge distribution and its image differ by the same factor β. Thus the problem of finding the SP frequencies is reduced to the problem of two spheres with similar surface charge distributions differing only by the factor β.18 Correspondingly, following eq S1, we substitute the potential of the image sphere acting on the MNP by the potential of a shifted dipole and obtain φ(R⃗) = βp0⃗ ∇

1 ⃗ |R − b ⃗ + δ |⃗

4ε0 1⎛ ⎜⎜1 + 2⎝ (ε0 + ε1)2 2

k=

ε1 ε0

⎞ ⎟⎟ ⎠

(19)

In eq 18 the factor k tends to unity when ε1 → ε0, and, as it is expected, we come to expression 10. Note that instead of eq 11, the dependence of the SP frequency on the parameters is now determined by the modified equation a(ωsp) = β1/3R 3 2

ε(ωsp) − ε0 ε(ωsp) + 2ε0

ε(ωsp) + 2ε0 (16)

2ε(ωsp) + 3ε0

where b⃗ connects the center of the MNP with the center of its image (see Figure 4).

3

+ β −1/3

1 ε(ωsp) + 2ε0 2 ε(ωsp) − ε0

3R 2

(20)

Unlike the case of the coupled spheres, RDR dependence on the parameters is more complicated. Nevertheless, as the calculations show, the main dependence of RDR on the parameters is again expressed by the term γ1 ∼ ω4spR3k(ε0)1/2 with the SP frequency determined by eq 20. In Figure 5 we present the results of calculations of RDR and ωsp versus interparticle gap based on the solution of the system

Figure 4. The spherical MNP near an interface and its image.

The derivation of the electric field potential for the sphere near an interface in the frame of EQMA and further calculations of the dipole and quadrupole moments of the surface charge distribution on the sphere surface and interface are presented in the Supporting Information. It is also proved in the Supporting Information that the total charge and the dipole moment of the interface p⃗int vanishes. Therefore, instead of eq 3, the RDR in the considered case takes the form

Figure 5. The SP frequencies (dashed lines) and RDR (solid lines) of an Au MNP with radius 30 nm near an interface vs particle−interface gap for the refractive indices of surrounding n0 = 1.33 (a), n0 = 1.53 (b), and adjacent media n1 = 4.5.

of eqs S5 for a Au sphere in water bounding with silicon (n1 = (ε1)1/2 = 4.5). As can be seen, when the particle−interface gap decreases from 30 to 1.5 nm for the sphere with radius 30 nm, the SP resonance frequency shifts from 2.41 to 2.27 eV, which is a notable redshift of 6%. In the same range of gap variation, the RDR decreases by 0.014 eV, which can be detected experimentally.

ωsp4psph ⃗2

1 = 3 ε0 τ 3c Wtot

(17)

and the problem is reduced to the determination of p⃗sph and total energy Wtot = Wsph + Wint. Herein, Wsph is the energy of surface charge of the MNP in the total field described by eq S6 and Wint is the energy of the charge induced on the interface in the same field (see Supporting Information). Using formulas S12 and S13 and the definition eq 3, we finally obtain for the RDR γ1 =

ωsp4R3 ε0 k c

3

■

CONCLUSION In summary, the SP RDR in a system of coupled spherical MNPs is calculated analytically, and a simple formula for the radiation line width is obtained. The notable reduction of the RDR due to the strong redshift of SP frequency is predicted, which is independent of the effects caused by nonradiative damping. For small separations, the RDR decreases by a factor of 3 when the radius of the particles decreases from 30 to 20 nm. The theory is expanded to the case of spherical MNP placed near an interface of two dielectric media. The dependence of the redshift of SP frequency and the RDR on particle−interface separation are calculated analytically. In both considered cases, the decrease of the refractive index of surrounding media leads to a decrease of the radiative damping rate. It is also shown that the SP frequencies fall below the interband transition threshold of Au due to strong redshift. This circumstance turns out to be essential in possible

ε(ω) ε(ω) − ε0

⎡⎛ 2R6 ⎞ 9 R3 ⎢⎜1 − β − β2 ⎟+ 3 ⎢⎣⎝ (b − δ ) (b − δ)6 ⎠ 5 ⎛ 2δ 2 3R8 ⎞ δR3 2 − β ⎟ ⎜ 2 −β (b − δ)4 (b − δ)8 ⎠ ⎝R −

−1 ε(ω) 6R3 ⎤ ⎥ (ε0 + ε1)2 ε(ω) − ε0 (b − 2δ)3 ⎦

(ε1 − ε0)ε0

(18)

where 16804

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(25) Bohren, C. F., Huffman, D. R. Absorption and Scattering of Light by Small Particles; John Wiley & Sons, Inc.: New York, 1983. (26) Seelig, J.; Leslie, K.; Renn, A.; K1uhn, S.; Jacobsen, V.; van de Corput, M.; Wyman, C.; Sandoghdar, V. Nano Lett. 2007, 7, 685−689. (27) Wu, Y.; Nordlander, P. J. Phys. Chem. C 2010, 114, 7302−7307.

applications of coupled MNPs since in coupled spheres and in spheres near an interface, the interband SP oscillation damping is suppressed.

■

ASSOCIATED CONTENT

S Supporting Information *

Derivation of the electric field potential for the coupled spheres in the frame of EQMA and further calculations of the dipole and quadrupole moments of the surface charge distribution. Derivation of the total energy of the system and the dipole moments of the surface charge distribution on the sphere surface and interface in the frame of EQMA. This material is available free of charge via the Internet at http://pubs.acs.org.

■

AUTHOR INFORMATION

Notes

The authors declare no competing financial interest.

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REFERENCES

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dx.doi.org/10.1021/jp305144u | J. Phys. Chem. C 2012, 116, 16800−16805