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, 1–2 1 ELECTROMAGNETIC RESPONSE OF SYSTEMS OF METALLIC NANOPARTICLES ∗ Tomáš Váry∗ , Peter Markoš∗ Department of Physics, INPE FEI STU Bratislava, Slovakia, Ilkovičova 3, 812 19 Bratislava, Slovakia E-mail: [email protected], [email protected] ABSTRACT We study numerically the response of periodic arrays of nanosized metallic particles to the incident electromagnetic waves. We identify localized plasmons excited in individual particles and describe how the resonance frequency depends on particle size and the structure geometry. Numerical data for the absorption enable us to distinguish plasmon resonance from interference effects observed outside the resonance region. Keywords: plasmons, transfer matrix, effective permittivity 1. INTRODUCTION Most of optical properties of metals are determined by free electrons which are very sensitive to the external electric field. Oscillations of electrons due to the incident electromagnetic (EM) field determine the frequency-dependent metallic permittivity of metal which can be expressed by the Drude formula [1] εm = 1 − f p2 , f ( f + iγ) structure of a typical “atom” is shown and described in Fig. 1. Antiresonant coupling of two plasmonic oscillations was used for the experimental proof of the possibility to construct material with negative frequency dependent effective magnetic permeability at THz frequencies [11]. (1) with the plasma frequency f p and the absorption parameter γ. Since f p ∼ 2000 THz for typical metals, the real part of the permittivity (1) is negative for the visible light. The model of the metal as a material with negative permittivity enables us to describe and explain various optical properties of metal without discussing microscopic processes inside the metal [2, 3]. Typical textbook examples are the reflection amplitude of the EM field incident to the metallic surface, the skin depth or the transmission of the field through the thin metallic slab [4]. The negative permittivity model enables us also to describe quantitatively the plasmonic excitations in metals. At the interface between the metal and the dielectrics the surface plasmon [5] can be excited, which concentrates the entire energy of the incident electromagnetic field to the thin two-dimensional layer parallel to the metallic surface [6,7]. While the surface plasmon cannot be excited by the incident electromagnetic wave (the dispersion relation of the surface plasmon lies outside the light cone [4]), localized plasmons in small metallic nanoparticles [7] are excited directly by incident light. Therefore, localized plasmons could be used for the manipulation of the electromagnetic response of nano composites. The size and concentration of the particles influences the frequency dependent reflectivity and absorption, and opens a wide variety of technical and commercial applications [8]. Physically, the most interesting is the use of plasmonic excitations for the design and construction of metamaterials, which are structures with required frequency-dependent effective electric permittivity and magnetic permeability. Metamaterial [9] consists of a periodic lattice of artificial “atoms” (unit cells of the size much smaller than the wavelength of the interacting EM field). Each “atom” is constructed from metallic components which create resonant structure [10]. Schematic Fig. 1 Left: A typical structure of the metamaterial “atom” consists of two metallic elements (two identical nanorods are shown in the figure). Normally incident EM fields induces the oscillations of electrons inside metallic nanoparticles. Owing to the coupling of the field induced in two neighboring nanoparticles, the system possesses two resonant frequencies similar to the two eigenmodes of coupled mechanical pendulums shown in the right panel. The symmetric resonance creates parallel currents in nanorods, the antisymmetric resembles the circular current in loop with two cuts at the ends of nanorods. When periodic lattice of identical “atoms” is fabricated, then two resonances create frequency bands with negative permittivity and permeability [10]. Thus, local currents inside each individual “atom” determine the effective permittivity and permeability of the metamaterial. If two bands overlap, the macroscopic metamaterial has negative effective refractive index. Metallic nanosphere represents the simplest shape of nanoparticle. Thanks to the spherical symmetry, the scattering process of the EM wave on single nanosphere can be analyzed exactly with the use of the Mie theory [3, 12]. However, scattering on the more complicated systems of nanoparticles must be analyzed numerically. In this paper we present results of our numerical simulations of scattering of the EM plane wave incident on various geometrical configurations of metallic nanospheres. The numerical program is based on the transfer matrix method described in [13]. Here we only summarize the main ideas. In the program, it is assumed that the sample has a periodic structure lying in the xy plane. Therefore, only one unit c 2010 FEI TUKE ISSN 1335-8243 2 Electromagnetic response of systems of metallic nanoparticles cell, shown in Fig. 2, is considered and periodic boundary conditions are used in two transversal (x, y) directions. An incident monochromatic plane wave with the frequency f propagates along the z direction, scatters at the sample, and the transmission and reflection amplitudes (t and r) are calculated. The absorption is found from the energy conservation requirement, A = 1 − R − T , where R = |r|2 and T = |t|2 are the reflection and absorption coefficients. Fig. 2 Left: The geometry used in the transfer matrix numerical method. The monochromatic EM wave incoming from the left is scattered on the sample and reflected and transmitted waves are calculated. The fields are calculated only in the finite volume a × b × L where a and b are size of the sample in the x and y direction, respectively and L is the length of the sample along the z direction. Periodic boundary conditions are used in the x and y directions. Right: An example of the discretization of the computed area and corresponding shape of the spherical particle used in numerical simulations. (a) (c) particle is much smaller than the wavelength λ , we find that the scattering cannot be described within the static limit of the Mie theory. To demonstrate the role of the interaction of plasmon resonances excited in neighboring particles, we discuss in sect. 3 the shift of the plasmonic resonance excited in the long one-dimensional chain of near-spaced nanospheres and confirm the dipole-dipole character of the interaction between neighboring plasmons. In sect. 4 we demonstrate the split of the absorption peak due to the interaction of plasmons exited in two neighboring linear chains. In sect. 5 we discuss the multilayer structures composed of very small (with diameter d = 6 nm) nanoparticles. We find a broad maximum in the absorption spectrum, which correspond to plasmonic excitation. Conclusion and discussion are given in sect. 6. (b) Fig. 4 Absorption (A), transmission (T ) and reflection (R) coefficients of the light incident on the square lattice of metallic nanoparticles. The diameter of spheres is d = 96.6 nm. The distance of neighboring particles a = 400 nm is large enough to prevent any interaction of plasmon resonances excited on neighboring particles The present data can be interpreted as a spectrum of a single nanoparticle. (d) 2. SINGLE SPHERICAL NANOPARTICLE 2.1. Numerical results Fig. 3 Structures simulated in the paper: (a) Isolated metallic particle of diameter d. Shown is also the size a of the unit cell (equal to the distance between neighboring particles) and discretization of the unit cell, used in the program. (b) Linear chain along the y direction. The parameter b determines the distance between neighboring particles, a b is a size of the unit cell. We assume a b (c) Two parallel chains of nanoparticles. (d) Layered structure of metallic particles coated by thin dielectric layer which prevents conductive contact between neighboring particles. Four structures shown in Fig. 3 will be investigated. In Sect. 2 we study the response of single metallic nanoparticle to the incident EM wave and identify the position of the plasmonic resonance. We discuss the role of the permittivity of metallic components, which differs from the permittivity of the bulk material. Although the diameter d of the As was discussed in Sect. 1, our numerical program assumes that the sample has a periodic structure in the x and y directions. To investigate the scattering of the EM wave on isolated particle the spatial period a must be sufficiently large so that the interaction between plasmons excited on neighboring particles can be neglected. We found numerically that this condition is fulfilled when the distance a ∼ 4d. Figure 4 shows numerical data for the frequency dependence of the transmission, absorption and reflection coefficient of the EM wave incident on a single nanoparticle with diameter d = 96.6 nm. (The structure is shown in Fig. 3a.) Resonant plasmon frequency λr ≈ 570 nm (2) is identified from the position of the deep minima in the transmission coefficient and maxima of the absorption. c 2010 FEI TUKE ISSN 1335-8243 3 Note that outside the resonant region, the sample is almost transparent: the absorption of the EM inside the nanoparticles is negligible and the transmission is close to unity. we use in numerical simulations the experimental data for thin film and fit them to the function 2 f02 , (3) εr = 1 − f 2 + Γ2 with fitting parameters f0 = 1300 THz and Γ = 400 THz. The real part of the permittivity is zero at the frequency f˜p = 1236 THz, which is much smaller than the plasma frequency ( f p = 2160 THz) given by the Drude formula. Fig. 5 The absorption coefficient A as a function of the frequency for particles of different size d. To avoid mutual EM interaction between neighboring particles, we keep the same distance/diameter ratio a = (29/7)d. Absorption maxima are red-shifted when the particle size increases. The metallic permittivity is given by the Drude formula (1). Figure 5 shows how the absorption peak depends on the size of the particle. The resonant wavelength decreases from 570 nm for the particle diameter d = 96.6 nm to 545 nm when the diameter decreases to d = 24.1 nm. These results are in good agreement with experiment [14] which reports λr ∼ 606 nm for d = 100 nm and λr ∼ 550 nm for the particle size d = 20 nm. The difference between numerical and experimental data is given by the inaccuracy of both methods. In numerical simulations, the accuracy of obtained results is limited by the discretization of the space which introduces numerical inaccuracy due to approximation of derivatives and deforms the shape of the spheres as is shown in right Fig. 2. Another reason for the inaccuracy of numerical results is that the permittivity of small metallic particles differs from the bulk permittivity given by Eq. (1). On the other hand, experimental data cannot be associated with a given size of a particle, because the size of particles varies considerably within a given sample. Also, particles are not distributed periodically, so that the distance between neighboring particles fluctuates. 2.2. The permittivity of small metallic particles The most important input parameter for numerical simulations is the permittivity of small metallic components. This is different from the Drude formula, given by Eq. (1) since the scattering of electrons at the surface of the particle causes an increase of the imaginary part of the permittivity. To show the importance of the finite size effects on the permittivity, we compare in Fig. 6 the experimental data for the permittivity [15] of metallic films of the thickness 30 nm with the permittivity of the bulk metal. Since there is no analytical model for the permittivity of small particles Fig. 6 Real part of the permittivity of thin silver film [15]. Solid line is the fit given by Eq. (3). Dashed line is the bulk permittivity given by Drude formula, Eq. (1). Inset: Imaginary part of the permittivity [15] compared with Drude formula (dashed line). Thin solid line is the imaginary Drude permittivity increased by +0.15. An increase of the imaginary part of the permittivity for larger frequencies is typical for the absorption peak at to the “plasma frequency” f p ≈ 1236 THz where real part of the permittivity is zero. 2.3. The static limit The scattering of the plane EM wave on spherical objects can be solved analytically within the Mie theory [12]. In the static limit, when the wavelength of the incident EM field is much larger than the diameter of the sphere, λ d, the resonant plasmon frequency can be found from the simple equation [7] εr + 2εd = 0 (4) where εd is the permittivity of surrounding media. Using the Drude permittivity (1) and εd = 1 we obtain that the plasmon is excited at the frequency √ (5) fr ≈ f p / 3 which corresponds to the wave length λr ≈ 236 nm. With the use of the permittivity given by Eq. (3) we obtain that λr ≈ 332 nm, which is still much less than the numerically obtained resonance (2). Therefore, the static limit is not sufficient for the description of plasmonic excitations in nanoparticles. 3. LINEAR CHAIN OF PARTICLES Consider now the one-dimensional linear chain of particles. shown in the Fig. 3b. We formed it by placing the particle in rectangular cell with the unit cell size a × b, where c 2010 FEI TUKE ISSN 1335-8243 4 Electromagnetic response of systems of metallic nanoparticles b is the particle-to-particle distance along the chain, and a is large enough to avoid interchain interactions. The EM wave propagates perpendicularly to the chain. Because of the linear structure of the sample, the polarization of the incident wave is important. The electric field is perpendicular (parallel) to the chain for the p (s) polarized wave, respectively. Figure 7 shows the resonant plasmon frequency as a function of the distance b between particles for both polarizations. For the longitudinal (s) polarization, the resonant frequency decreases when b decreases, while for the transversal (p) polarization we observe an opposite behavior. The shift of resonant frequency exhibits for both polarization the 1/b3 - dependence in agreement with measurements of Maier et al. [16]. This confirms the dipole character of the electric interactions between neighboring particles. Dipoles oscillate with higher frequency when they are oriented perpendicular to chain axis (p polarization). larger side and b for shorter. Distance between two parallel chains is c. Figure 9 shows the absorption spectra for such structure for two different distances c between chains. For sufficiently small c = 33.3 nm the absorption peak splits into two peaks (red curve). Interaction between plasmons on particles causes frequency of resonance behave similar to coupled harmonic oscillators. Two resonant frequencies, fr and fr + ∆ f appears, where ∆ f is proportional to the coupling between neighboring particles and converges to zero when the distance c increases. As shown in Fig. 7, coupling between two plasmons excited in two neighboring particles is much stronger when the electric field is parallel to the line connecting particles. Therefore, we expect that coupling is stronger when electric field is parallel to the x axis (Fig. 3c). In this case, the frequency difference ∆ f is negative (Fig. 7). This observation explain our results shown in Fig. 9. For the p polarization (electric field parallel to the x axis), the absorption peak splits into two peaks when c is sufficiently small. The second peak with a larger wavelength corresponds to the interchain interaction. On the other hand, coupling is weak for the s polarized wave (electric field parallel to the x axis). Consequently, we observed only one absorption peak for both values of the interchain distance c. Fig. 7 Resonant frequency for various linear chains of metallic nanoparticles as a function of the particle distance b. Solid lines show fits which confirm the dipole character of particle interactions ∼ 1/b3 . The diameter of the particles d = 50 nm and the distance between chains a = 300 nm. Fig. 8 Representation of the nanoparticle by electric dipoles for the transversal p (top) and longitudinal s (bottom) polarization of the EM field. Fig. 9 Absorption coefficient for two parallel nanoparticle chains. For the s polarization, the absorption peak does not depend on the distance between chains. For the p polarization (the electric field is perpendicular to chains), two absorption peaks occurs due to the dipole-dipole interaction between two neighboring particles. The right peak disappears when the distance between chains increases. Parameters of the structure are as follows: a = 100 nm, b = 33.3 nm, particle diameter d = 29 nm. Drude formula for the permittivity, Eq. (1) was used in simulations. 5. SMALL METALLIC NANOPARTICLES HEXAGONAL LAYERED STRUCTURE 4. TWO PARALLEL CHAINS OF METALLIC NANOPARTICLES Next structure is composed from two parallel linear chains of nanoparticles along the y direction (Fig. 3c). Again the unit cell is rectangular with dimensions a for IN In this section, we simulate the scattering of the EM wave on the multilayer structure of very small nanoparticles. The individual layer is shown in Fig. 3d. Small spheres are self-assembled to the regular hexagonal lattice. To prevent the conductive contact between particles, each c 2010 FEI TUKE ISSN 1335-8243 5 particle is coated by thin dielectric layer. The size of particles d = 6 nm, the thickness of the dielectric layer is only 1 nm. The present structure can be easily prepared experimentally [17] and the reflection coefficient can be measured for various number of layers [18, 19]. Numerical results can be therefore directly compared with available experimental data. This is important for the estimation of the frequencydependent permittivity of nanoparticles. We have namely observed that experimental data for the permittivity shown in Fig. 6 are not more relevant for metallic particles of the size of a few nanometers. To find more realistic frequency dependence of the permittivity, we simulated the scattering of the EM wave at a single layer of particles for various values of the permittivity and compare obtained results with experiment [19]. We found the best (although not perfect) coincidence when the real part of the permittivity follows Eq. (3) and the imaginary part εi = 1 [20]. for 15 layers, and εeff = 8.7 for 25 layers of nanoparticles). Thus, composite medium which contains materials with permittivity εr ∼ 2.25 (coating) and εr ∼ −15 (silver) behaves as weakly absorbing dielectrics with high refractive index. Fig. 11 Absorption and transmission coefficients for multilayer hexagonal nanoparticles structure. Number of layers is 10, 15, 20, 25 and 30. Arrows show the direction of increasing of the number of layers. Note oscillations of the transmission coefficient in the region outside the absorption peak. Reflection coefficient for the structure with 30 layers is also shown. 6. CONCLUSIONS Fig. 10 Reflection spectra for the normal incident plane wave to multilayer of hexagonal close-packed silver nanoparticles. Besides absorption peak located between 400-500 nm, oscillations of the reflection coefficient are observed for larger wavelengths. Real part of permittivity εr of silver is given by Eq. (3 and the imaginary part is constant εi = 1.0. The permittivity of dielectric layer is εd = 2.25. The diameter of nanoparticles d = 6 nm, the thickness of dielectric coating is 1 nm. Reflection coefficient for multilayer structures is shown in Fig. 10. Corresponding data for the transmission and absorption coefficients are given in Fig. 11. Well pronounced maximum of the reflection coefficient and absorption located for 400 < λ < 500 nm corresponds to the excitation of localized plasmons in individual metallic particles. Location of this peak is independent of number of layers. Note that the transmission is very small and decreases when the number of layers increases. Contrary, oscillations of the reflection coefficient, observed for larger wavelengths, are accompanied with oscillations of the transmission coefficient and very low absorption. Therefore, these peaks do not correspond to the excitation of plasmon, but rather with interference of the EM wave in thin layer. Comparing the position of reflexion maxima with that for the homogeneous layer of the same width and permittivity εeff , we find that the multilayer structure of nanoparticles behaves as the dielectric layer with rather high effective permittivity (for instance, εeff = 3.6 Localized plasmons play crucial role in various physical phenomena. It is therefore important to understand their response to incident electromagnetic field. This can be obtained numerically. We demonstrated the application of the numerical transfer matrix method for the calculation of the reflection coefficient and absorption for a few configurations of metallic nanoparticles. Numerical simulation enable us to calculate the transmission, reflection and absorption coefficients therefore can be useful for the interpretation of experiment which usually measured only the reflection coefficient. Our data confirm that plasmonic excitations cannot be studied within the framework of the static limit, in spite of the fact, that the diameter of metallic spheres is order of magnitude smaller than the wavelength of incoming light. The crucial problem of theoretical and numerical analysis of the plasmonic resonances is in the estimation of the permittivity of small metallic particles. Both experimental and numerical data indicate that for relatively large nanoparticles (d > 25 nm, the permittivity of Johnson and Christy [15] can be used [14] and experimental data agree very well with the analytical theory Mie. The estimation of the permittivity of smaller nanoparticles is still an open problem. Physically relevant model must consider the scattering of electrons on the surface of the finite volume of the particle [21]. For extremely small particles, the permittivity is influenced by the spectrum of the quantum eigenstates of the confined electrons [22]. c 2010 FEI TUKE ISSN 1335-8243 6 Electromagnetic response of systems of metallic nanoparticles ACKNOWLEDGEMENT work was supported by the Slovak Research and Development Agency under the contract No. APVV-0108-11. Authors thank J. Chlpı́k for valuable discussions. This [14] C. Sönnichsen et al. Phys. Rev. Lett. 88, 077402 (2002). [15] P. B. Johnson and R. W. Christy, Phys. Rev. B 6, 43704379 (1972). REFERENCES [1] Ch. Kittel, Introduction to Solid State Physics, John [16] S. A. Maier et al., Phys.Rev. B 65, 193408 (2002). Willey and Sons (2005). [2] J. D. Jackson, Classical Electrodynamics, John Wiley [17] M. Weis, private communication. & Sons, Inc. (1999). [18] [3] M. Born and E. Wolf, Principles of Optics, Cambridge University Press (2003). [19] [4] P. Markoš and C. M. Soukoulis, Wave Propagation: [20] From Electrons to Photonic Crystals and Left-Handed Materials, Princeton Univ. Press (2008). [21] [5] E. N. Economou, Phys. Rev. 182, 539 (1968). M. H. Lin, H. Y. Chen, S. Gwo, J. Am. Chem. Soc 132, 11259-11263 (2010). J. Chlpı́k, private communication T. Váry, PhD Thesis, FEI STU Bratislava (2012) (in Slovak). U. Kreibig, L. Grenzel, Surf. Science 156, 675 (1985). [22] J. A. Scholl, A. I. Koh, J. A. Dionne, Nature 483, 421 (2012). [7] S. A. Maier, Plasmonics: Fundamentals and Applications, Springer Science (2007). [6] A. V. Zayats et al., Phys. rep. 408, 131 (2005). [8] E. Hutter and J. A. Fendler, Advanced Materials 16, 1685 (2004). [9] A. K. Sarychev, V. M. Shalaev, Electrodynamics of Metamaterials, World Scientific (2007). [10] V. P. Drachev et al., Laser Phys. Lett. 3. 49 (2006). [11] A. N Grigorenko et al.. Nature 438, 335 (2005) [12] H. C. van de Hulst, Light Scattering by Small Particles, Dover Publications, New York (1957). [13] P. Markoš and C. M. Soukoulis, Phys. Rev. E 81, 036622 (2002). BIOGRAPHIES Tomáš Váry (1984) graduated (MEng) in 2008 at the Department of Physics of the Faculty of Electrotechnical Engineering and Information Technology at Slovak University of Technology in Bratislava. In 2012 he defended the PhD Thesis “Surface electromagnetic waves on interfaces”. He is Assistant at the STU. Peter Markoš (1958) is Assistant Professor at the FEI STU. His research interests include Anderson localization and electromagnetic properties of composite metamaterials. c 2010 FEI TUKE ISSN 1335-8243