Plasmon Resonances in Metal Nanoparticles with Sharp Edges and

Article pubs.acs.org/JPCC

Plasmon Resonances in Metal Nanoparticles with Sharp Edges and Vertices: A Material Independent Approach Vasily Klimov,*,† Guang-Yu Guo,*,‡ and M. Pikhota† †

P.N. Lebedev Physical Institute, Russian Academy of Sciences, 119991 Moscow, Russia Department of Physics and Center for Theoretical Sciences, National Taiwan University, Taipei 10617, Taiwan

‡

ABSTRACT: By examples of nanoparticles with superellipsoidal shapes, we investigate the optical properties of the entire family of metallic nanoparticles of cubic symmetry. In particular, we find drastic changes in the plasmonic spectrum when the nanoparticle varies its form from the spherical to the cubic one. This would allow one to monitor, e.g., the nanoparticle growth in situ by purely spectroscopic methods. We also find that the dipolar “vertex” modes dominate the spectrum of the cubic nanoparticles.

1. INTRODUCTION At the present time, metal nanoparticles are intensively investigated because of their unique optical properties. These unique properties are associated with localized plasmon resonances which exist in metal nanoparticles and have become a basis for a variety of applications.1,2 For the theoretical consideration of the optical properties of plasmon nanoparticles, one mostly uses the approximation by a spherical particle for which the analytical solution of the optical and plasmon properties is known.3 However, synthesis methods of nearly monodisperse nanoparticles with almost atomically sharp corners have now been developed. In particular, good quality nanocubes were synthesized recently (see, e.g., refs 4-5). Therefore, one definitely should understand the plasmonic properties of nanoparticles with sharp corners like cubes or octahedrons, etc. Though in the real world there may be no sharp edges and corners, in numerical simulations it is often impossible to smooth all boundaries because several domains made of different materials may contact at one point or along a line (see Figure 1) which leads to a sharp corner of one or more domains. As a result, the consideration of nanoparticles with sharp corners is very important also for numerical simulations of complex systems.

The description of the objects with sharp edges is known to be very complicated from the mathematical point of view, even in the case of a two-dimensional (2D) wedge, where the analytical solutions are available.6,7 Recently, some progress was achieved in understanding of the plasmonic spectra in 2D nanowires with sharp edges.8,9 In these previous papers, it was demonstrated that in 2D nanowires the main contribution to the dipole modes are from the edges, and the corresponding eigenvalues of permittivity should depend only on the edge opening angle. However, three-dimensional (3D) nanoparticles are of more interest, and as we will show, the 2D analysis of refs 8 and 9 does not give much insight on the behavior of the plasmonic spectra of 3D nanoparticles with sharp corners. Plasmonic properties of the cube and related particles with sharp corners were already considered in several papers.10−16 However, these papers (except of the pioneering work of ref 10) dealed with the specific materials with substantial losses, and thus, the problem of sharp edges and corners in fact was not considered there. Thus, the goal of this paper is to shed light on this problem. In doing so we consider the generalized eigenvalue approach17 which is independent of the constituent material of the nanoparticles and, in particular, of the Joule losses inside the particles. Unfortunately, such 3D particles cannot be treated analytically, and thus, we will attack this complicated problem with different numerical methods. Our methodology is based on the consideration of an infinite set of nanoparticles with their shapes evolving from the sphere to the cube (see Figure 2).

Figure 1. Illustration of the appearance of sharp edges at the contact of three materials. In this geometry it is impossible to smooth all contacting areas. © XXXX American Chemical Society

Received: December 17, 2013 Revised: May 24, 2014

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dx.doi.org/10.1021/jp412349f | J. Phys. Chem. C XXXX, XXX, XXX−XXX

The Journal of Physical Chemistry C

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number of shapes, more or less similar in shape to a perfect cube, were considered. Therefore, it is impossible to find any relationship between the plasmon modes of these particles. Moreover, all the intermediate shapes were with edges and corners which additionally complicate the problem. By this reason, the study of a continuous sequence of smooth surfaces, which describe the transition from a sphere to a cube, and vice versa, is of great interest. We propose to describe the surface of the particles by the parametric eq 1. This parametrization is very important, since all the surfaces described by it are smooth. To calculate the plasmonic spectra of such shapes, we will use the BEM implementation of the generalized eigenvalue approach. The BEM approach is especially effective to find the resonant values of permittivity and also the corresponding plasmon frequencies for all the modes simultaneously, i.e., for both the bright and dark ones. More importantly, this approach deals with only the geometrical shape of the nanoparticles while the constituent materials of the nanoparticles are of no importance and can be taken into account when the specific calculations (e.g., cross sections) should be performed. To find the plasmonic spectrum of a nanoparticle within this approach, one should solve its source free quasi-electrostatic equations:

Figure 2. Surfaces defined by eq 1 for different values of parameter N. When N = 2, the shape is spherical and changes to the cubic when N tends to infinity.

To do it, we consider the nanoparticles where the surface equation is |x|N + |y|N + |z|N = r N

(1)

div(ε(r)e(r)) = 0

18

that is, the equation for a super-ellipsoid, as a function of parameter N. When N = 2, one obtains the equation of a sphere with radius r. When N → ∞, one obtains a cube with edges of 2r. The parametrization of eq 1 is very important, as all the surfaces described by it, are smooth for a finite N. This removes a lot of questions in the numerical study of the optical properties of the nanobodies. The transformation of the nanoparticle shape within eq 1 is shown in Figure 2. The plan of the rest of this paper is as follows. In section 2, we will analyze within the quasi-static boundary element method (BEM) the dependence of the plasmonic spectrum of a nanoparticle on its shape as it evolves from the spherical to the cubical one according to eq 1. Here we find the “plasmonic phase transitions” where the sphere like spectrum abruptly changes to the cube like spectrum with typical high frequency features. In section 3, we will calculate the plasmonic spectrum of the nanocube with both the DDA (discrete dipole approximation) and BEM. In section 4, we will discuss the results obtained, and in section 5, the conclusions drawn from this work will be summarized.

rote(r) = 0

(2)

which, in the case of the domains with sharp interfaces, can be reduced to the solution of the Laplace equations by setting en = −▽ϕn Δφnin = 0, Δφnout = 0 φnin|S = φnout|S εn

∂φnin ∂n

= ε̅ S

∂φnout ∂n

(3)

S

In eq 3, ε̅ is the electric permittivity (real) of the space surrounding the nanoparticle, φinn and φout n are the potentials of the eigenfunctions inside and outside the nanoparticle, respectively, and ∂φn/∂n|S denotes the normal derivative on the surface of the nanoparticle. The last equation in eq 3 ensures the continuity of the normal components of the electrical induction. It is very important that (3) is of purely geometrical nature and has no relation to the specific materials from which the nanoparticle is made of. In particular, the eigenvalues εn are real negative numbers. This is a very important feature of our approach and we call it “a material independent approach”. The dependence of the solution on the specific properties of the constituent materials appears when we expand the solution of a real problem in terms of the eigenfunctions of eq 3:

2. PLASMONIC PROPERTIES OF SUPER-ELLIPSOIDS WITH CUBIC SYMMETRY To clarify the issue about the influence of the edge and vertex smoothness, it is useful to consider a sequence of real or hypothetical nanobodies having a smooth surface and changing their shape from the spherical to cubic one. Consideration of the optical properties of the nanoparticles with a continuous change in the shape is important not only from a mathematical point of view but also because this kind of changes in the shape of the nanoparticles could be encountered during the crystallization or melting of the nanoparticles (see, e.g., ref 19). Therefore, the study of the parametric dynamics of the optical spectra for these processes can be an important tool for monitoring them. Recent theoretical studies of a sequence of different nanoparticles which connect shapes of a sphere and cube were reported in refs 11 and 16. However, only a small discrete

E(r) = E0(r) +

∑ en(r) n

0 ε(ω) − 1 ∫V + en·E dV εn − ε(ω) ∫ + en 2 dV V

(4)

where E(r) and E0(r) are the total and excitation fields, respectively, while ε(ω) describes the dependence of the electric permittivity of the constituent material of the particle. The integration is over the volume of the nanoparticle V+. Note B



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that in this case, there is no need to find the magnetic fields for the description of the electric plasmon oscillations. To reduce the computation time and required memory, it would be better to transform the 3D problem [eq 3] into a 2D surface integral equation. This procedure is well-known from the theory of potential.20 The integral equation for the cube surface charge σ(r) then acquires the following form: σn(r) =

1 λn 2π

∫S

ds′σn(s′)

n(r) ·(r − s′) , |r − s′|3

r, r′ ∈ S (5)

where n(r) stands for the outward unit normal vector at point r, integration is over the nanoparticle surface S and λn = (εn − 1)/(εn + 1) are the real eigenvalues. Here, again, the permittivity εn of the nanoparticle is considered as an eigenvalue (real and negative). However, eq 5 is not very good for the nanoparticles with sharp edges. To improve the convergence, it is better to use the adjoint equation21,22 τn(r) =

1 λn 2π

∫S

d2r′τn(r′)

n(r′) ·(r′ − r) , |r − r′|3

Figure 4. Spectra of the resonance dielectric constant depending on the shape of the nanoparticles of cubic symmetry and corresponding to bright dipole modes. The horizontal axis indicates the parameter 1/N defining the shape of the surface according to eq 1. The vertical axis indicates the resonance values of the permittivity. Red stars on the ε axis correspond to the values for the perfect cube from the DDA calculations (see section 3 and Table 1).

r, r′ ∈ S (6)

which has the same spectrum as (5), but its numerical solution convergence is much better especially in the case of nonsmooth surfaces. Physically, eq 6 describes the surface density of the dipoles, which results in the same potential, as the surface charges in (5) do. In this case, the discrete analogue of eq 6 has the form: τi =

λ 2π

eigenvalues of the cube. In particular, one can see that triply degenerate bright dipole modes of the sphere for which ε = −2 do not split and transform into a strongly triply degenerate dipole mode of the cube for which ε ≈ − 4.75. Despite the improved BEM algorithms we have used, the convergence was rather slow. To confirm the results for N → ∞, we have also carried out the DDA simulations for a perfect cube, as will be reported in the next section.

N

∑ Uijτj (7)

j=1

where ⎧ n(s) ·(s − si) d2s , ⎪ S Δ |si − s|3 ⎪ j Uij = ⎨ N ⎪ 2 Uil , i = j π − ∑ ⎪ l = 1, l ≠ i ⎩

∫

3. PLASMONIC SPECTRUM OF A PERFECT CUBE First of all, we should remember that this problem was considered for the first time in 1970s in ref 10 and many important results were reported there already. However, due to the weak computers of the time, the accuracy of the results is poor. The goal of this section is to calculate the cube spectrum on the base of modern computing technologies. 3.1. DDA Calculation. The DDA method is a numerical method in which an object is represented as a set of many polarizable cubes located at the nodes of the regular cubic grid.23 Sometimes, this method is called a coupled dipole approximation (CDA) method. Within the context of this method, there are no restrictions to whether a grid node is occupied by a dipole or not. This means that, in principle, the DDA method can be used for an approximate description of bodies of any shape and composition. However, because the cubic symmetry is the basis for the DDA applications, one can expect that DDA will be especially accurate for determining the resonance properties of a cube. The electromagnetic scattering problem for an incident wave interacting with this cubic array is then solved essentially exactly.24−26 We used the computer program DDSCAT 7.0 developed by Draine and co-workers.25 We have calculated the extinction cross-section of two nanocubes of a small size (3 nm) and a large size (30 nm) in order to neglect the retardation effect and to see the influence of the retardation, respectively. The Drude model ε(λ) = 1 − λ2/λpl2 + 0.01i with λpl = 0.217763 μm (Na) was used. The calculated extinction cross

i≠j

(8)

For further accuracy enhancement, eq 7 can be rewritten in a form, which enables the charge conservation independently from the computational accuracy τi =

λ 2π

N

⎛

j=1

⎝

∑ ⎜⎜Uij −

1 N

N

⎞

i=1

⎠

∑ Uij⎟⎟τj

(9)

In Figure 3, one can see examples of the cube surface triangulation to build eq 9. Figure 4 illustrates the results of our calculations. This figure shows that the eigenvalues of the dielectric constant split due to the change of symmetry and are gradually transformed into the

Figure 3. Examples of partition of the particle surface into triangles for the BEM application. C



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sections are shown in Figure 5. We then extracted from the peak positions in the extinction spectrum for the small cube, the values of the resonant dielectric constants of the cube.

Figure 6. Dependence of the peak positions on the reciprocal of the number of dipoles used in the DDA simulations of a 3 nm cube.

From this figure, we can see that indeed the calculation is converged, and from their interpolation to 1/Nd = 0, we can extract the accurate resonant values of the permittivitie. These values are shown in Figure 4 (on the left) and in Table 1 (on the right column). Table 1. Resonant Dielectric Constants for the Main Dipole Modes line no. 1 2 3 4 5 6 7

Figure 5. Extinction cross-section of a cube with an edge size of (a) 3 and (b) 30 nm. The Drude model is used to describe its properties. The nanocube was modeled with a large number of 48 × 48 × 48 = 110 592 dipoles and the wavelength was changed by a fine step of 0.0001 μm.

A comparison of panels a and b in Figure 5 indicates that increasing the cube edge size by a factor of 10 would result in only insignificant changes of the extinction spectrum. This confirms the applicability of the quasistatic analysis to the nanoparticles presented in section 2. Moreover, our general material-independent approach is found to remain valid even for the larger nanoparticles of edge size at least up to 50 nm. However, in this case, the eigenvalues would acquire a small imaginary part due to the radiation effect, but remain independent of the absorption by the constituent material of the nanoparticles. For a spherical particle in vacuum, for example, we have1

εn [Fuchs10] −3.68 −2.37 ? −1.90 −1.27 −0.78 −0.42

εn [BEM]

εn [DDA]

−4.52 −3.23 −2.62 −1.80 −1.28 −0.75 −0.37

−4.76 −3.57 −2.63 −1.78 −1.25 −0.75 −0.42

3.2. BEM Calculation. Here we apply the BEM method described in section 2 to a perfect cube. Because of the slow convergence, we have carried out a number of simulations with different numbers of triangles and then plot the dependence of the resonant permittivity values as a function of the inverse number of the triangles. The results of the calculations for different grids are shown in Figure 7. From this figure, one can see that simulations are rather slowly converging. Nevertheless, one can extract the final values of the resonant dielectric constants from these BEM calculations using a fitting line ε = ε∞ + α/Nβ, and they are shown in Table 1 (column 3).

4. DISCUSSION OF RESULTS OBTAINED From Table 1, one can see that our results of the BEM and DDA simulations for a perfect cube are in rather good agreement. However, we think that the DDA values are closer to the exact values because the DDA simulations have a better convergence. This phenomenon has a simple physical explanation: the fields near a vertex have a much weakened singularity and can be even finite while the surface charge should be infinite. This conclusion is also in agreement with ref 15 where, by examples of a dielectric cube, it was shown that the convergence of the DDA results with an increasing discretization for a cube and other shapes, which can be exactly described as a set of cubes, is generally faster than for other shapes. In what follows, we will specify the modes with the resonant permittivity values obtained with the DDA approach (column 4 of Table 1).

12 (k 0a)2 − 2i(k 0a)3 + ... 5 3 5 65 i ε2 = − − (k 0a)2 − (k 0a)4 − (k 0a)5 + ... 2 14 392 12 ε1 = −2 −

4 56 11788 469672 − (k 0a)2 − (k 0a)4 − 3 405 601425 95954625 4i (k 0a)6 − (k 0a)7 + ... 2025

ε3 = −

(10)

To get the accurate optical plasmonic excitation spectra, we used a large number of point dipoles by dividing the cube into an increasing number of the cubic cells to check the convergence of the calculations. The convergence dynamics for DDA is shown in Figure 6. D



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From these plots, one can see that for the two modes with the lowest permittivity values (−4.76 and −3.57), the main charge is localized near the vertices. We will refer to such modes as the vertex modes. The vertices are definitely 3D objects and that is why they cannot be described by a 2D analysis. Thus, we predict that in a 3D nanoparticle with sharp vertices, a substantial contribution to the plasmon spectrum will be due to the vertex modes. Next mode (ε = −2.63) is localized on the wedges of the cube and can be considered as being analogous to the 2D modes in a wire with the edge angle of θ = π/2. This mode should be limited by ε2D = 1 − 2π/θ = −3 and it respects this relation indeed. Let us now consider the peculiarities of the dependence of the plasmon spectrum on the nanoparticle shape (Figure 4). The plasmonic resonances of a sphere (N = 2) εn = −(n + 1)/n, n = 1, 2, 3, ... lie in the interval [-2, −1) and are red-shifted from the bulk plasmon frequency ωpl, if we adopt the Drude dispersion ω = ωpl/(ε + 1)1/2. In contrast, in the cube case, there are also modes in the interval (−1, 0) (e.g., ε = −0.75 and ε = −0.42), which are blue-shifted relative to the bulk plasmon frequency ωpl. We will refer to such modes as high frequency modes. As can be seen from Figure 4, high frequency modes appear at N = 3, and for N > 3, the nanoparticle should be considered as being closer to the cube rather than to the sphere. In fact, a very small change of shape (see Figure 2 for N = 2.72 and N = 3.45) results in a substantial modification of the spectrum. We will refer to this phenomenon as the “plasmonic phase transition”, though other notions (e.g., the spectrum bifurcation) can be used too. To give a physical explanation of this phenomenon, we note that these high frequency cube modes are localized on the edges of the cube, and have opposite signs on the adjacent faces (see Figure 8 for ε = −0.75 and ε = −0.42). These modes are apparently connected with the modes appearing at the edge of an infinite long wedge. The calculations6 for the plasmonic modes in an infinite wedge with an arbitrary angle θ are the argument for this statement. In ref 6, two kinds of modes, namely, odd and even, are found. The odd modes are localized on the edge of the wedge and have opposite signs on the adjacent faces. The even modes are the bulk modes and have the same sign on the adjacent faces. In the case of the infinite rectangular wedge (θ = π/2), the following expressions for ε exist:6

Figure 7. Resonant permittivity values of a perfect cube as a function of 1/N, where N is the number of equal triangles on the surface mesh of the cube. Solid lines are fitting lines.

The results of both the BEM and DDA simulations differ substantially from the first result by Fuchs.10 From our point of view, these differences can be easily understood because Fuchs used a rather rough grid (about 1000 cells) in his BEM calculations which is definitely not enough to find the precise results due to the sharp edges. In our BEM calculations, we have used about 20 000 cells and an improved algorithm. Even under these conditions, we observe a very slow convergence. In any case, we believe that now our results should be reliable ones. It is interesting to compare our results with the results of the previous investigations of the 2D nanowires.8,9 The main results of these previous studies are that the main contribution to the dipole plasmonic spectrum is from the sharp edges, and for the case of the cube with θ = π/2edges, these modes should be limited by ε2D = 1 − 2π/θ = −3. However, in our case, we have definitely a different situation: at least the two modes have the resonant permittivity values greater than −3 in absolute value (−4.76, −3.57) (see also Figure 4). To clarify the situation, we have plotted the distributions of the surface charge of several modes for τ(r) (see eq 6) in Figure 8.

ε=−

tanh(3πμ/4) tanh(πμ/4)

and

ε=−

tanh(πμ/4) tanh(3πμ/4)

(11)

for the even and odd modes, respectively, where μ is a real parameter defining the mode. These expressions show that for the even modes, ε varies in the range −3 < ε < −1/3, and for the odd modes, in the range −1 < ε < −1/3. It is pointed out in ref 6 that the wedge modes with |ε| < 1 are localized on the wedge edge and have opposite signs on the adjacent faces. In contrast, the modes for −3 < ε < −1/3 are the bulk modes and have the same sign on the adjacent faces. Thus, our high frequency modes can be considered as the odd wedge modes. Moreover, the analogous high frequency modes also appear in the clusters of two nanospheres27,28 when the distance between the spheres becomes small in comparison with their diameter. However, in this case, it is difficult to find an explanation similar to one we found for the cube with the help of the edge modes in an infinite wedge. Thus, the phenomenon

Figure 8. Distribution of the surface dipole density τ(r) for the main plasmonic modes in a nanocube from the BEM simulation. E



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we have found here is not specific for a superellipsoid of cubic symmetry but also for a whole set of other geometries. The high sensitivity of the new high frequency modes to even a small shape modification (see, e.g., the shapes for N = 2.72 and 3.45 in Figure 2) can give rise to important practical applications. These applications are based on the fact that the set of shapes described by (1) (or analogous one) is not a purely mathematical one but is also related to such a real physical phenomenon as the melting or crystallization. The example of the melting of prisms into perfect spheres under radiation by laser pulses has been demonstrated (see, e.g., ref 19). Usually, to understand if the synthesis or melting is finished, one has to stop the reaction and put the sample under SEM. However, with the information on the plasmonic phase transition, one can probe the shape of the nanoparticles during synthesis in real time by monitoring the presence or absence of high frequency modes in the absorption or scattering spectra.

REFERENCES

(1) Klimov, V. V. Nanoplasmonics, 2nd ed; Fizmatlit: Moscow, 2010 (480 p., in Russian);. Klimov, V. V. Nanoplasmonics: Fundamentals and applications; Pan Stanford Publishing: Singapore, 2014. (2) Chon, J. W. M., Iniewski, K., Ed.; Nanoplasmonics: Advanced Device Applications; CRC Press: Boca Raton, FL, 2013; p 270 . (3) Mie, G. Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen. Ann. Phys. 1908, 330, 377−445. (4) Sun, Y.; Xia, Y. Shape-Controlled Synthesis of Gold and Silver Nanoparticles. Science 2002, 298, 2176−2179. (5) Tao, A.; Sinsermsuksakul, P.; Yang, P. Polyhedral Silver Nanocrystals with Distinct Scattering Signatures. Angew. Chem., Int. Ed. 2006, 45, 4597−4601. (6) Dobrzynski, L.; Maradudin, A. A. Electrostatic Edge Modes in a Dielectric Wedge. Phys. Rev. B 1972, 6, 3810−3815. (7) Davis, L. C. Electrostatic Edge Modes of a Dielectric Wedge. Phys. Rev. B 1976, 14, 5523−5525. (8) Sturman, B.; Podivilov, E.; Gorkunov, M. Universal Plasmonic Properties of Two-dimensional Nanoparticles Possessing Sharp Corners. Phys. Rev. B 2013, 87, 115406−1−11. (9) Sturman, B.; Podivilov, E.; Gorkunov, M. Metal Nanoparticles with Sharp Corners: Universal Properties of Plasmon Resonances. Europhys. Lett. 2013, 101, 57009−p1-p5. (10) Fuchs, R. Theory of the Optical Properties of Ionic Crystal Cubes. Phys. Rev. B 1975, 11, 1732−1740. (11) Noguez, C. Surface Plasmons on Metal Nanoparticles: The Influence of Shape and Physical Environment. J. Phys. Chem. C 2007, 111, 3806−3819. (12) Sherry, L. J.; Chang, S.-H.; Schatz, G. C.; Van Duyne, R. P.; Wiley, B. J.; Xia, Y. Localized Surface Plasmon Resonance Spectroscopy of Single Silver Nanocubes. Nano Lett. 2005, 5, 2034−2038. (13) Hohenester, U.; Krenn, J. Surface Plasmon Resonances of Single and Coupled Metallic Nanoparticles: A boundary Integral Method Approach. Phys. Rev. B 2005, 72, 195429−1−9. (14) Yurkin, M.; Kahnert, M. Light Scattering by a Cube: Accuracy Limits of the Discrete Dipole Approximation and the T-Matrix Method. J. Quant. Spectrosc. Radiat. Transfer. 2013, 123, 176−183. (15) Yurkin, M. A.; Maltsev, V. P.; Hoekstra, A. G. Convergence of the Discrete Dipole Approximation. I. Theoretical Analysis. J. Opt. Soc. Am. A 2006, 2, 2578−2591. (16) Raziman, T. V.; Martin, O. J. F. Polarisation Charges and Scattering Behaviour of Realistically Rounded Plasmonic Nanostructures. Opt. Express. 2013, 21, 21500−21507. (17) Agranovich, M. S.; Katsenelenbaum, B. Z.; Sivov, A. N.; Voitovich, N. N. Generalized Method of Eigenoscillations in Diffraction Theory; Wiley-VCH: New York, 1999; p 378. (18) http://en.wikipedia.org/wiki/Superellipsoid (accessed February 26, 2014). (19) Sun, F.; Cai, W.; Li, Y.; Duan, G.; Nicholas, W. T.; Liang, C.; Koshizaki, N. Laser Morphological Manipulation of Gold Nanoparticles Periodically Arranged on Solid Supports. Appl. Phys. B: Laser Opt. 2005, 81, 765−768. (20) Anderssen, R. S.; Zabrejko, P. P.; Shaposhnikova, T. O.; Michlin, S. G. Integral Equations: A Reference Text; Noordhoff International Publishing: Leyden, 1975 (443 pages). (21) Mayergoyz, I. D.; Fredkin, D. R.; Zhang, Z. Electrostatic (Plasmon) Resonances in Nanoparticles. Phys. Rev. B 2005, 72, 155412−1−15. (22) Mayergoyz, I. D.; Zhang, Z. J. Modeling of the Electrostatic (Plasmon) Resonances in Metallic and Semiconductor Nanoparticles. J. Comput. Electr. 2005, 4, 139−143. (23) Purcell, E. M.; Pennypacker, C. R. Scattering and Absorption of Light by Nonspherical Dielectric Grains. Astrophys. J. 1973, 186, 705− 714. (24) Draine, B. T. Astrophys. J. 1988, 333, 848−872. (25) Draine, B. T.; Flatau, P. J. Discrete Dipole Approximation for Scattering Calculations. J. Opt. Soc. Am. A 1994, 11, 1491−1499. (26) http://www.astro.princeton.edu/∼draine/DDSCAT.html (accessed February 16, 2012).

5. CONCLUSIONS In the present work, the plasmonic properties of the superellipsoid nanoparticles with cubic symmetry have been investigated. The shapes of this set of nanoparticles are described by equation |x|N + |y|N + |z|N = rN, thus including the sphere (N = 2) from one side and the perfect cube (N → ∞) from the other side. First of all, we find that in the plasmon spectrum of the perfect cube the vertex modes with |εres| > 3 dominate. We call these modes as the vertex modes because these modes appear only in the 3D nanoparticles with sharp vertices but never occur in the 2D nanowires with a rectangular cross-section where only the sharp edges with |εres| < 3 can be found. Another important result of our work is the prediction of the drastic change of the particle spectrum even when its shape changes only slightly. In the spectrum of a superellipsoid for N > 3, the high frequency modes localized at the edges of the nanoparticle appear. This interesting result can be useful in optical monitoring of the nanoparticle shape during its synthesis or melting and so on. Finally, our simulations show that for the nanoparticles with sharp vertices and edges, the boundary element methods (which are based on surface integral equations) suffer from slow convergences in comparison with the DDA and related approaches (which are based on the volume integral equations and use the fields rather than the surface currents and charges). This phenomenon has a simple physical explanation: the fields near a vertex have a much weakened singularity and can be even finite while the surface charge should be infinite.

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected]. *E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors would like to thank the Russian Foundation for Basic Research (Grant Nos. 11-02-91065, 11-02-92002, 11-0201272, 12-02-90014, and 14-02-00290), the Presidium of the Russian Academy of Sciences, the Russian Quantum Center and the Skolkovo foundation, and the National Science Council of Taiwan for financial supports of the present work. F



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(27) Klimov, V. V.; Guzatov, D. V. Strongly Localized Plasmon Oscillations in a Cluster of Two Metallic Nanospheres and their Influence on Spontaneous Emission of an Atom. Phys. Rev. B 2007, 75, 024303−1−7. (28) Klimov, V. V.; Guzatov, D. V. Plasmonic Atoms and Plasmonic Molecules. Appl. Phys. A: Mater. Sci. Process. 2007, 89, 305−314.

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Plasmon Resonances in Metal Nanoparticles with Sharp Edges and

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