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J. Phys. Chem. C 2010, 114, 22398–22405
Bright and Dark Plasmon Modes in Three Nanocylinder Cluster Vasily Klimov*,† and Guang-Yu Guo*,‡,§ P. N. LebedeV Physical Institute, Russian Academy of Sciences, 119991 Moscow, Russia, Graduate Institute of Applied Physics, National Chengchi UniVersity, Taipei 116, Taiwan, and Department of Physics, National Taiwan UniVersity, Taipei 106, Taiwan ReceiVed: June 19, 2010; ReVised Manuscript ReceiVed: NoVember 9, 2010
Plasmonic excitations in a three interacting nanocylinder cluster have been investigated by both simple analytical modeling and discrete dipole approximation (DDA) calculations. In our analytical model, each cylinder is regarded as an anisotropic electric dipole with four parameters determined by fitting its optical extinction and scattering spectra to that of DDA simulation. Diagonalization of our analytical model for the three interacting dipoles reveals 10 significant plasmon modes, namely, 5 bright modes and 5 dark eigen-modes. Because of rather long-range longitudinal plasmon coupling, the resonant energies of the two longitudinal modes vary pronouncedly with the intercylinder distance even when the intercylinder distance is about 25 times larger than the cylinder diameter. In contrast, for the transverse modes the splitting of resonant energies becomes apparent when the intercylinder distance falls within about five times of the cylinder diameter. Interestingly, for sufficiently large clusters, each resonance peak contains both magnetic dipole (M1) and electric quadrupole (E2) as well as electric dipole (E1) contributions, and these multipole contributions become rather visible (“bright”) when the distance between the two neighboring cylinders is sufficiently reduced. Finally, the optical plasmon excitation spectra from our simple analytical model are in good qualitative agreement with our DDA calculations. 1. Introduction Successful synthesis of metal nanoparticles of different shapes (see, e.g., refs 1-4) has recently stimulated considerable interest in the investigation of their optical properties, mainly because these nanoparticles support a variety of plasmon resonances.5 For example, plasmonic nanoparticles could help enhance fluorescence and Raman scattering from even single molecules by many orders of magnitude.6-8 Plasmonic nanoparticles also play a crucial role in the development of nanolasers,9 improved solar cells,10 new elements for nano-optoelectronics,11 new metamaterials with negative refraction,12 and super- and hyperlenses.13 Optical spectra of single plasmonic nanoparticles are very rich and have been extensively investigated in the past.5 Optical properties of nanoparticle clusters are also of great interest because they provide us with more degrees of tunability and thus allow more interesting applications than single nanoparticles. Clusters of two nanoparticles of spherical, oblate spheroidal, and disklike shapes have been investigated both theoretically and experimentally.14-23 Clusters of three and more spherical nanoparticles have also been analyzed.24-26 The detailed investigations of clusters with more sophisticated constituents are, however, rather scarce.27 In this paper, we therefore study the electromagnetic responses of a cluster made of three nanocylinders of finite length. We will present a general approach that is a combination of analytical and numerical methods. This general approach would enable us to understand complicated physical nature of plasmon oscillations in arbitrary clusters of nanoparticles. As an example, * To whom correspondence should be addressed. E-mail: (V.K.)
[email protected]; (G.Y.G.)
[email protected] † Russian Academy of Sciences. ‡ National Chengchi University. § National Taiwan University.
here we will apply this general approach to a cluster of three circular parallel cylinders which centers are on the corners of an equal triangle. The geometry of the cluster under consideration is illustrated in Figure 1. This problem is interesting from both fundamental and practical view points. For example, in a cluster of two nanocylinders, the electric dipole moments may have either parallel or antiparallel coupling. In a cluster of three nanocylinders, however, the antiparallel coupling of the dipole moments would result in a frustration, i.e., the third cylinder would be “unhappy” no matter in whatever ways it is coupled to its two neighboring cylinders. The frustration of this kind in magnetic systems could give rise to a number of interesting phenomena such as spin glass, spin ice, and even Dirac monopole-like behaviors.28-30 In the plasmonic case, odd number of identical single nanoparticles in a cluster would analogously result in complicated and rich structures of resonances that need good physical understanding. To clarify this behavior, the experimental study of plasmonic properties of three cylinder clusters using an electron beam is planned in National Taiwan University. To fabricate these clusters in a controllable way, an anodic alumina substrate method will be used.31 A specific goal of this paper is to provide theoretical understanding of rich physics of the planned experiments, and that is why we will take specific parameters for the nanocylinders in our numerical calculations presented below. The rest of the present paper is arranged as follows. In section 2 we will present the calculated optical properties of single cylinder of finite length and propose an analytical anisotropic dipole model by fitting the model parameters to the numerical results. In section 3 we will first establish a simple analytical description of a cluster of 3 interacting nanocylinders based on the anisotropic dipole model for the single cylinder presented in section 2, and then report the plasmonic eigenmodes and optical properties of the cluster derived from the simple
10.1021/jp105661a 2010 American Chemical Society Published on Web 12/06/2010
Bright and Dark Plasmon Modes in Three Nanocylinder Cluster
J. Phys. Chem. C, Vol. 114, No. 51, 2010 22399
Figure 1. Geometry of the three parallel cylinder cluster.
Figure 2. Extinction (red lines) and scattering spectra (blue lines) of single cylinder with diameter D ) 41.6 nm and aspect ratio R ) H/D ) 2. Cylinder is made of silver. Solid lines show our DDA calculations, while dashed lines represent the results given by our analytical model. Propagation direction of incident wave is perpendicular to the cylinder axis while the incident wave polarization is along the axis (left panel) and perpendicular to the axis (right panel).
analytical model. In particular, we will analyze how the plasmonic excitations of the cluster evolve as the intercylinder distance varies. Section 4 is devoted to the investigation of optical properties of a cluster of three silver nanocylinders with discrete dipole approximation (DDA) numerical approach.32-35 Although we consider cylinders made of silver here, our analytical results can be easily extended to cylinders made of other materials too. In section 5 our conclusions are given. 2. Optical Properties of Single Cylinder: Analytical Model versus DDA Simulation There is no analytical solution for the Maxwell equations in the presence of a circular cylinder of finite length. Nevertheless, several numerical investigations of this geometry by using different computational methods have been reported.16,36 Here the extinction spectra of a single Ag cylinder with aspect ratio R ) H/D ) 2 from our own DDA calculations are shown in Figure 2, where H and D are the height (or length) and diameter of the cylinder, respectively. The experimental dielectric data of Ag from ref 37 are used. Figure 2 indicates that for the longitudinal oscillation (Figure 2a) there is only one plasmon resonance while there are two pronounced plasmon resonances for the transversal polarization (Figure 2b). These qualitative features should also be present in other single cylinders with a different aspect ratio and made of a different material. Of course, one would need new numerical calculations to describe the extinction spectra of another cylinder of a different material with a different aspect ratio. To obtain a fundamental understanding of the plasmonic properties of a nanocylinder, let us first establish an approximate
analytical dipole model for the nanocylinder. We note that the optical properties of a nanoparticle can be described by their dipolar plasmon oscillations or dipolar polarizabilities R. For a fixed polarization, the extinction σext and scattering σsca cross sections can be described by the following formulas
σext ) 4π
2π 8π 2π 4 2 |R| ImR, σsca ) λ 3 λ
( )
(1)
where λ is the wavelength of light in the host matrix. In the derivation of eq 1, it was assumed that σext . σsca. Nevertheless, comparison of eq 1 with the full scale DDA simulations (Figure 2) shows that in fact eq 1 has a wider range of applicability. The dipolar polarizability of a cylinder is a tensor which can be written in the form
[ ]
RT 0 0 R ) 0 RT 0 0 0 RL
(2)
Here the transversal RT and longitudinal RL polarizabilities can be expanded in terms of all the corresponding plasmon eigen-oscillations as5,38
R ) L
∑ l
RlL
V(ε(ω) - εH) ) 4π
εL /εH - 1 L l Cl L εl - ε(ω) l
∑
(3)
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R ) T
∑ l
RlT
V(ε(ω) - εH) ) 4π
εT /εH - 1 T l Cl T εl - ε(ω) l
∑
Klimov and Guo
(4)
where ε(ω) is the specific dielectric function of the cylinders, ∑t CLl ) 1, ∑l CTl ) 1, and εTl (εLl ) is the resonant value of the transversal (longitudinal) dielectric permittivity that is independent of the material that the nanocylinder is made of. Generally, εTl and εLl are complex quantities, and their negative real parts would determine the frequencies of the plasmon resonances. However, eq 3 and eq 4 can be also used with the real parts of εt and εl taken from either experimental measurements or numerical simulations. In this case, eq 3 and eq 4 should be improved by taking the radiative losses into account explicitly as39
RT f RT )
RT 1 - 2/3ikH3RT
RL f RL )
RL 1 - 2/3ikH3RL
(5)
Specifically, for the present case of R ) 2, ε1L ≈ -9.3. Transversal oscillations are more complicated because there are two bright modes for this polarization (see Figure 3). To find the values of ε1T, ε2T and C1T, C2T for the transversal polarizations we make use of our DDA calculations for the single cylinder (see Figure 2). The details of the DDA calculations are given in Section 4 below. We found that for the best fit we should use
εL1 ) -9.9, CL1 ) 0.75;
εT1 ) -1.54,
CT1 ) 0.73;
εT2 ) -3.47, CT2 ) 0.27 (7)
One should be reminded that these data are independent of the material that the cylinder is made of. Therefore, the final expressions for the polarizabilities of the single cylinder can be found via eqs 3, 4, and 5 where the nonvanishing terms are given by eq 7. Interestingly, the longitudinal resonant dielectric constant given by eq 7 is slightly different from that given by eq 6. This small difference is due to the fact that Prescott and Mulvaney36 considered the cylinders with diameter D ) 40 nm while we use D ) 41.6 nm here and is a manifestation of a general rule that plasmon resonances shift in the quadratic manner from the quasistatic solution εQuasi5
ε ) εQuasi + S(ka)2 + ...
where kH ) (εH)1/2ω/c is the wave vector in the host medium with the dielectric constant εH. Therefore, the main problem is to determine the negative values of the real parts of εTl and εLl as well as the positive values of the coefficients Ct and Cl for the major plasmonic oscillations. Fortunately the plasmonic oscillations in a cylinder can be adequately described by one longitudinal and two transversal plasmon resonances, and thus we only need ε1L, ε1T, ε2T and C1L, C1T, C2T because the other plasmon modes would give only very minor contributions. The detailed investigation of ε1L was carried out in ref 36 where its dependence on the size and shape of the nanoparticle was studied. For example, the dependence of ε1L on the aspect ratio R for a nanocylinder of D ) 40 nm would have the following form36
where k is wavenumber in free space and a is the dimension of the nanoparticle. The shape of the nanoparticle is taken into account by factor S, which is size-independent. In what follows, we will use eqs 3, 4, 5, and 7 to describe of the plasmonic properties of a single cylinder. The extinction and scattering spectra obtained with eqs 3, 4, 5, and 7 are plotted in Figure 2 as dashed lines. It is important to have in mind that our model describes a cylinder by five dipole oscillators (two transversal modes for the x-polarization, two transversal modes for the y-polarization and one longitudinal mode for the zpolarization), that is, our cylinder has five degrees of freedom.
-εL1 ) 0.8632R2 + 2.6373R + 0.5082
It is impossible to give a full description of the optical properties of a three cylinder cluster even after knowing the
(6)
(8)
3. Analytical Model Analysis of Plasmonic Properties of a Three Cylinder Cluster
Figure 3. Electric potential distribution for two most important transversal dipole modes (in the quasistatic approximation). Left panel corresponds to the high frequency mode, εT1 ) -1.54, while right panel corresponds to the low frequency mode, εT2 ) -3.47. It is important to note that the low frequency mode is localized on the rims of the cylinder. In both cases, the electric potential of the uniform external electric field is equal to (1.0 on the top and the bottom of the figure, respectively.
Bright and Dark Plasmon Modes in Three Nanocylinder Cluster
J. Phys. Chem. C, Vol. 114, No. 51, 2010 22401
4d3e-idkH 1 - idkH + d2kH2 -
dipolar properties of the single cylinders. Nevertheless, in this section we will construct a simplest approximate analytical model that can describe reasonably well the optical properties of the three cylinder cluster in a wide range of the parameter values. 3.1. Simple Analytical Model for a Three Cylinder Cluster. Within this model we approximate each cylinder of the cluster by an anisotropic dipole di, i ) 1, 2, 3 with its polarizability given by eq 2. To describe the interaction between the cylinders and the external field E0(r), we use the retarded 5 0(r,r′,ω), which is a solution of the Maxwell Green function G equations
T R1,2 )
2 5 0(r, r', ω)) - ω εHG 5 0(r, r′, ω) ) ∇ × (∇ × G c ω 25 1 δ(r - r') (9) 4π c
R*T ) RT1 ; 1
()
()
The dipole momenta of all the cylinders can now be found by solving the system of linear equations
T(E (r ) + di ) εHR 0 i
∑ G5 0(ri, rj, ω)dj),
i, j ) 1, 2, 3
j*i
(10)
where 5 R is described by eqs 3, 4, 5, and 7. In these linear equations (eq 10), the terms containing the retarded Green functions describe dipole interactions between the cylinders. It is this retarded interaction that gives rise to the magnetic and quadrupole properties of the plasmonic oscillations in the three cylinder cluster. Knowing the solution of this system of linear equations would allow us to evaluate the extinction and scattering cross sections of the cluster. 3.2. Resonant Dielectric Constants and Plasmonic Frequencies versus Intercylinder Distance. To understand the physical nature of plasmon oscillations in the three cylinder cluster, one should first of all understand free plasmon oscillations in the system of linear equations (eq 10), that is, the nontrivial solution of eq 10 with zero external field. This system of linear equations has nontrivial longitudinal oscillations if the radiative loss-corrected longitudinal polarizability of the single cylinders (eqs 3 and 5) is equal to any one of the three eigenvalues of eq 10, that is
RL1 ) -
;
√45 - 90idkH - 51d2kH2 + 6id3kH3 + 13d4kH4 RT3
) -
RT4
)
2d3e-idkH ; -5 + 5idkH + 3d2kH2
2d3e-idkH ; -7 + 7idkH + d2kH2
T R*T 2 ) R2
(12) Corresponding eigen configurations of dipoles in cylinders are related with irreducible representations of point group D3h25 and are shown in Figure 4. From Figure 4 and also direct calculations of the total dipole moment of the system, we find that modes R2L, R1T, R1*T, R2T, and R2*T have a nonzero dipole moment and hence they are “bright” modes. Other modes have a zero dipole moment and thus are “dark” modes. Dark modes R1L, R1*L, and R3T, are especially interesting because they have both an electric quadrupole and a magnetic dipole moments. This means that a three cylinder cluster with such modes would exhibit magneto-electric behaviors. In contrast, dark mode RT4 has only an electric quadrupole moment. It should be noted that this classification of “dark” and “bright” modes is not very strict, because even “dark” oscillations with either an electric quadrupole or a magnetic dipole can give rise to a substantial contribution to the optical properties of a three cylinder cluster of sufficiently large size (see Figures 6 and 8 below).
d3e-idkH ; -1 + idkH + d2kH2
L R*L 1 ) R1 ;
RL2 )
d3e-idkH 1 2 -1 + idk + d2k2 H H
(11)
Analogously, eq 10 has nontrivial transversal oscillations if the radiative loss-corrected transversal polarizability of the single cylinders (eqs 4 and 5) is equal to any one of the six eigenvalues of eq 10, that is
Figure 4. Dipole moments of the eigen-modes of a three cylinder cluster with the intercylinder being 61.6 nm and the wavelength being 500 nm. Each eigen-mode can be associated with an irreducible representation of the point symmetry group D3h of the cluster, as indicated in brackets.
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Figure 5. Calculated plasmonic resonant wavelengths λ of a three silver cylinder cluster as a function of intercylinder distance d. (a) Longitudinal oscillations and (b) transversal oscillations. Solid lines correspond to the “bright” modes with a nonzero dipole moment (see Figure 4). Dashed lines correspond to the “dark” modes with a zero dipole moment. Here only 8 out of 15 modes are distinguishable because the dark longitudinal and bright transversal modes are 2-fold degenerated.
To find corresponding resonant dielectric constants, one may equate eigenvalues of the polarizabilities from eqs 11 and 12 to the polarizabilities of the single cylinders from eqs 3 and 4, respectively. One should then solve equations
RT(εres) ) RT1 ;
RT(εres) ) RT2 ...
RL(εres) ) RL1 ;
RL(εres) ) RL2 ...
(13)
the three dipole moments on the three cylinders in the cluster from the solution of eq 10 are given by
d1,z d3RL d2,z d3RL
where the right-hand sides are from eqs 11 and 12 to obtain the resonant dielectric constants εres. It is important to note that the transversal modes have two solutions for each mode because eq 13 for those modes results in a quadratic form in ε. Therefore, we have 15 dielectric constant eigenvalues, because the 3 cylinder cluster has 3 × 15 independent modes. Among them, five modes are doubly degenerated, as will become clear below. With obtained resonant dielectric constants εres, one can find the corresponding plasmon resonant frequencies via
εres ) ε(ω)
d3,z d3RL ∆
)
d3eidkH/2+eidkHRL(1 - eidkH/2 + eidkH)∆
)
eidkH(d3 + RL(1 + eidkH/2 - eidkH)∆)
(d3 + RLeidkH∆)(d3 - 2RLeidkH∆) (d3 + RLeidkH∆)(d3 - 2RLeidkH∆) (d3 + RLeidkH∆)(d3 - 2RLeidkH∆)
) -1 + idkh + d2k2h
(15) Here RL should be taken from eq 3 together with eqs 5 and 7. Correspondingly, the extinction cross sections can be evaluated as
(14)
where ε(ω) is the specific dielectric function of the material in which the cylinders are made. Throughout this paper, we consider only the silver cylinders and thus use the experimental dielectric data from ref 37. The calculated resonant wavelengths λ ) 2πc/Re(ω) are plotted as a function of the intercylinder distance d in Figure 5. One can see from Figure 5 that the transversal modes on the different cylinders start to interact significantly with each other only when the cylinders come rather close together (i.e., d < 100 nm for the bright modes and d < 200 nm for the dark modes). On the other hand, the longitudinal modes on the different cylinders start to interact strongly even when the cylinders are more than 1000 nm apart. This can be explained as follows: the effective polarizabilities of the longitudinal modes are proportional to the volume of a fictitious sphere with its radius being in the order of the height of the cylinder, while for the transversal modes, the effective polarizabilities are proportional to the third power of the cylinder diameter. 3.3. Optical Properties versus Intercylinder Distance. To find the optical properties of the cluster within our approach, one should first solve eq 10 and then evaluate the desired cross sections. For the longitudinal exciting field E0 ) (0,0,1)eikHx,
)
d3 + eidkHRL(-1 + eidkH/2 + eidkH)∆
σext )
3
4πk0
√εH |E0 |
2
∑ Im(E0(rj) · dj) ) j)1
4πk0
√εH
Im(d1,z + eidkH/2d2,z + eidkHd3,z)
(16)
To find the scattering cross sections, we should integrate the radiated power from the three dipoles of eq 15 and then normalize it to the incoming radiation intensity. As a result, we obtain the following expression
σsca
[
|d1 | 2 + |d2 | 2 + |d2 | 3 + Re[d1d* 2 ]R(kHR12) + 3 Re[d1d* 3 ]R(kHR13) + Re[d2d* 3 ]R(kHR23) (d1 · R12)(d* 2 · R12) β(kHR12) 8πk4 Re R212 ) 2 |E0 | (d1 · R13)(d* 3 · R13) Re β(kHR13) 2 R13 (d2 · R23)(d* 3 · R23) Re β(kHR23) 2 R23
[ [ [
] ] ]
]
(17)
Bright and Dark Plasmon Modes in Three Nanocylinder Cluster where
R(x) )
J. Phys. Chem. C, Vol. 114, No. 51, 2010 22403 E1 σsca )
(x2 - 1)sin(x) + x cos(x) and x3 (x2 - 3)sin(x) + x cos(x) β(x) ) x3
M1 σsca )
d)
8π 4 k |m| 2, 3 H
k0 m ) -i ( 2
2
(18)
E2 ) σsca
∑d
8π 4 2 k |d| , 3 H
i
i
2
8π 4 kH |D| k , 3 H 120
∑R × d) i
i
i
DRβ )
∑ [3(d
iRRiβ
+ diβRiR) - 2(di · Ri)δRβ]
i
In the present case of the equal triangle, the arguments of all the R and β functions are equal to kHd, where d is the distance between the centers of the neighboring cylinders. Let us now consider the transversal polarizations. In the case of the incoming field E0 )(1,0,0)eikHz, the extinction crosssection is equal to
σext ) 24πkHd3 Im × RT(2d3 + eidkHRT(1 - idkH + d2kH2))
(4d6 - 2d3eidk RT(1 - idkH + d2kH2) - ei2dk RT2 H
H
(19)
(11 - 22idkH - 13d2kH2 + 2id3kH313d4kH4) The scattering cross-section is too long to write here and hence is omitted. In the case of the transversal incident field of E0 ) (0,1,0)eikHx, the expressions for all the cross sections are very long too and hence are not displayed here. The calculated extinction cross sections are shown in Figure 6. Remarkably, Figure 6 (left panel) show how the hidden (“dark”) quadrupole mode in a large cluster is transformed into a “bright” one (the right peak) in a small cluster, as the intercylinder distance d is reduced. As already mentioned above in Subsection 3.2, the plasmonic eigen-modes can be either dark or bright in the quasistatic limit. However, as one can see from Figure 6 that “dark” modes R1L, R*L1 become “bright” ones at small distances between the cylinders. It is clear from Figure 4 that these modes have both an electric quadrupole E2 and a magnetic dipole M1 momenta. To understand the role of the different multipole contributions to the optical properties of the cluster, let us calculate the partial scattering cross sections for the three cylinder cluster. The scattering cross sections for any systems can be written in the form M1 E2 σsca ) σE1 sca + σsca + σsca
(20)
where the electric dipole, magnetic dipole and electric quadrupole contributions have the following forms
(21) The contributions from eq 21 for the three cylinder are shown in Figure 7 for the longitudinal excitation E ) [0,0,1]eikHy. One can see from Figure 7 that the right peak is indeed due to the magnetic dipole and electric quadrupole radiations, while the left peak is mainly of the electric dipole origin. This interesting behavior is qualitatively different from that of a single nanoparticle where higher resonances shift to blue side of spectrum. It can also be seen that for a large cluster, all the contributions are converged into one peak because the interaction between the cylinders becomes small. Main characteristics of the plasmonic properties of the three cylinder cluster and also the relative contributions from the different multipoles are summarized in Table 1. 4. Optical Properties of Three Cylinder Cluster from DDA Calculations The results obtained in the previous section are only qualitative in nature. To obtain a quantitative description of the optical plasmonic excitation spectra of the three cylinder cluster and to check our simple analytical model, we have carried out direct numerical calculations by using the DDA method32-35 that allows one to calculate scattering and absorption cross sections of an incident electromagnetic wave by any targets with arbitrary geometries. In the DDA, the target is replaced by a cubic array of point dipoles. The electromagnetic scattering problem for an incident wave interacting with this array is then solved essentially exactly.31-33 We used the computer program DDSCAT 7.0 developed by Draine and co-workers.35 To get accurate optical plasmonic excitation spectra, we used a large number of point dipoles by dividing the cylinder diameter into 35 intervals, that is, we used about 67 000 point dipoles for a single cylinder and about 201 000 point dipoles for the three cylinder cluster. We consider Ag cylinders only and the experimental Ag optical dielectric constants from ref 37 are used. The results of our calculations are shown in Figure 8. As one can see from Figure 8, there is a rather good qualitative agreement between the analytical model and more
Figure 6. Extinction cross sections for the longitudinal (E ) [0,0,1]eikHy, left panel) and transversal (E ) [1,0,0]eikHy, right panel) polarizations of the external field, as a function of wavelength and for several different intercylinder distances d [d ) 61.6 nm (red), 81.6 nm (green), 121.6 nm (blue), and 241.6 nm (black)]. On the left panel, one can clearly see how the hidden (“dark”) quadrupole mode in a large cluster is transformed into the “bright” one (the right peak) in a smaller cluster. On the right panel, one can see only a weak splitting of 2 transversal resonances into 4 resonances due to the interaction between the cylinders.
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Figure 7. Multipole-decomposed scattering cross sections of the three cylinder cluster in the longitudinal exciting field E ) [0,0,1]eikHy. Red lines correspond to the electric dipole contributions, while green and blue lines correspond to the magnetic dipole and electric quadrupole radiations, respectively. The results from the hypothetical case of silver without losses are also plotted as dashed lines in order to help understand the sophisticated structure of the plasmonic resonances in the three cylinder cluster.
TABLE 1: Main Characteristics of the Plasmonic Properties of the Three Cylinder Cluster (d ) 61.6 nm) λ (µm) 0.3412 0.3544 0.3762 0.404 0.4363 0.5643
mode RT1 RT2 or RT1 RT2 or RL2 R1L
RT3 RT3
E1 , µm2 σsca
M1 , µm2 σsca
E2 , µm2 σsca
0.01125 0.01637 0.01067 0.0412 0.0807 0.008392
0.0002 0.0006896 0 0.003124 0.0051 0.01191
0.001322 0.001225 0.00103 0.001907 0.002681 0.007075
precise numerical calculations for the distance d ) 61.6 nm between the centers of the cylinders. As one can expect, the agreement in the peak position is better for the transversal modes. Of course, nobody expects that the simple 3 dipole model can substitute for the full scale numerical calculations. This model should help gain a fundamental understanding of the main features of the excitation spectrum, and indeed it does offer such an understanding. For example, only from our analytical approach one can tell that the right side peak in the
Figure 8. Extinction (red lines) and scattering (blue lines) cross sections for the longitudinal (left, E ) [0,0,1]eikHy) and transversal (right, E )[1,0,0]eikHy) polarizations of the external fields as a function of wavelength. Solid lines represent the results from the DDA calculations while dashed lines correspond to the results from the analytical model. The intercylinder distance is 61.6 nm.
Bright and Dark Plasmon Modes in Three Nanocylinder Cluster extinction spectrum of the longitudinal polarization is due to the electric quadrupole oscillations in our cluster. This model can also explain that the splitting of the transversal and longitudinal peaks is due to the differences in the interaction between the cylinders. 5. Conclusions In this paper, we have studied plasmonic excitations in a three interacting nanocylinder cluster by both the analytical three anisotropic dipole modeling and also DDA calculations. We find five “bright” plasmonic oscillation modes and five “dark” eigen-modes in the nanocylinder cluster. Furthermore, the resonant energies of the two longitudinal modes are found to change significantly with the variation of the distance between the neighboring cylinders even when the distance is about 25 times larger than the cylinder diameter, due to the long-range nature of longitudinal plasmon coupling. For the transverse modes, on the other hand, the splitting of resonant energies becomes apparent only when the distance falls within about five times of the cylinder diameter. Remarkably, the longitudinal magnetic dipole (M1) and electric quadrupole (E2) modes become clearly observable (“bright”) when the distance between the two neighboring cylinders is sufficiently reduced. All these suggest that plasmon excitations in the three nanocylinder cluster can be manipulated by varying the structure of the cluster. We hope that the interesting results reported here will stimulate further experiments on metal nanocylinder clusters especially electron energy loss measurements in a scanning transmission electron microscope which has been demonstrated recently to be a powerful probe of dark plasmon modes in noble metal nanoparticles.40 Acknowledgment. The authors thank Ming-Wen Chu for stimulating discussions and Bruce T. Drain for helpful e-mail communications on the DDSCAT 7.0 code. This work is supported by Russian Foundation of Basic Researches (Grants 09-02-13560, 11-02-91065, and 11-02-01272) as well as National Science Council and National Center for Theoretical Science of Taiwan. V.K. also thanks NCTS (north) and Department of Physics, National Taiwan University, for hospitality. References and Notes (1) Sun, Y.; Xia, Y. Science 2002, 298, 2176–2179. (2) Kong, X. Y.; Ding, Y.; Yang, R.; Wang, Z. L. Science 2004, 303, 1348–1352. (3) Newton, M. C.; Warburton, P. A. Mater. Today 2007, 10, 50–54. (4) Manoharan, V. N.; Elsesser, M. T.; Pine, D. J. Science 2003, 301, 483–487.
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