Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Equation of state wikipedia , lookup
Density of states wikipedia , lookup
Lorentz force wikipedia , lookup
Partial differential equation wikipedia , lookup
Electrostatics wikipedia , lookup
Thomas Young (scientist) wikipedia , lookup
Time in physics wikipedia , lookup
Theoretical and experimental justification for the Schrödinger equation wikipedia , lookup
JOURNAL OF APPLIED PHYSICS 108, 084304 共2010兲 Undamped collective surface plasmon oscillations along metallic nanosphere chains W. Jacak,1,a兲 J. Krasnyj,1,2 J. Jacak,1 A. Chepok,2 L. Jacak,1 W. Donderowicz,1 D. Z. Hu,3 and D. M. Schaadt3 1 Institute of Physics, Wrocław University of Technology, Wyb. Wyspiańskiego 27, 50-370, Wrocław, Poland Theor. Phys. Group, International University, Fontanskaya Doroga 33, Odessa 65-026, Ukraine 3 Institute of Applied Physics/DFG-Center for Functional Nanostructures, Karlsruhe Institute of Technology, Weberstrasse 5, 76-133 Karlsruhe, Germany 2 共Received 14 July 2010; accepted 25 August 2010; published online 19 October 2010兲 The random-phase-approximation semiclassical scheme for describing plasmon excitations in large metallic nanospheres 共with radius ⬃10– 60 nm兲 is developed for the case when a dynamical electric field is present. The spectrum of plasmons in metallic nanospheres is determined including both surface and volume-type excitations and their mutual connections. It is demonstrated that only dipole-type surface plasmons can be excited by a homogeneous dynamical electric field. The Lorentz friction due to irradiation of electromagnetic energy by plasmon oscillations is analyzed with respect to sphere dimension. The resulting shift in resonance frequency due to plasmon damping is compared with experimental data for various sphere radii. Wave-type collective oscillations of surface plasmons in long chains of metallic nanospheres are described. The undamped region for collective plasmon propagation along the metallic chain is found in agreement with previous numerical simulations. © 2010 American Institute of Physics. 关doi:10.1063/1.3493263兴 I. INTRODUCTION Experimental and theoretical invesitigations of plasmon excitations in metallic nanocrystals rapidly grew mainly due to possible applications in photovoltaics and microelectronics. A significant enhancement of absorption of incident light in photodiode-systems with an active surface covered with nanodimension metallic particles 共of Au, Ag, or Cu兲 with planar density ⬃108 – 1010 / cm2 was observed.1–7 This is due to a mediating role in light energy transport carried out by surface plasmon oscillations in metallic nanocompounds on a semiconductor surface. These findings are of practical importance for enhancement of solar cell efficiency, especially for thin film cell technology. On the other hand, hybridized states of surface plasmons and photons result in plasmon-polaritons8 which are of high importance for applications in photonics and microelectronics,9,10 in particular, for transportation of energy in metallic modified structures in nanoscale.11,12 Surface plasmons in nanoparticles have been widely investigated since their classical description by Mie.13 Many particular studies, including numerical modeling of multielectron clusters, have been carried out.14–18 They were mostly developments of Kohn–Sham attitude in form of local density approximation 共LDA兲 or time dependent LDA 共TDLDA兲 共Refs. 14–17兲 for small metallic clusters only, up to approximately 200 electrons 共as limited by numerical calculation constraints that grow rapidly with the number of electrons兲. The random-phase-approximation 共RPA兲 was formulated19 for description of volume plasmons in bulk a兲 Electronic mail: [email protected]. 0021-8979/2010/108共8兲/084304/12/$30.00 metals and utilized also for confined geometry mainly in a numerical or seminumerical manner.16 Usually, in these analyses the jellium model was assumed for description of positive ion background in the metal and dynamics was addressed to the electron system only.16–18 Such an attitude is preferable for clusters of simple metals, including noble metals 共also transition and alkali metals兲. It should be noted, however, that the TDLDA approach was successfully applied to nanoshells with radius above 10 nm and thickness up to a few nanometers20 via acceleration of the numerical procedures. It was achieved by assuming screened effective Coulomb interaction, which enhanced convergence of iterations, thus allowing for effective solution of Kohn–Sham type equations for even more than 104 electrons within a jellium model. For the nanoshell systems also numerical solution of the RPA-type polarization self-consistent equation has been found.21 These analyses showed that plasmon resonances in nanoshells are similar to classical Mie-type ones, redshifted, however, with respect to them, similarly as it was for small clusters 共mostly due to spill-out of electrons beyond the jellium rim兲. The radiation effects were not included, corresponding to not severe contribution of radiation losses even for relatively large radius of nanoshells, because the number of electrons in a shell is much lower than in a full nanosphere of similar radius. One can thus expect that dimension scaling of radiation losses, being of cubic type for nanospheres,22 does not apply to shells, resulting in the overall shift in radiation effects in shells to larger radius region in comparison to full spheres 共likely, above the radii considered in Refs. 20 and 21兲. In the present paper we generalize the bulk RPA description,19 using a semiclassical approach for a large metallic nanosphere 共with radius of several tens of nanometers 108, 084304-1 © 2010 American Institute of Physics Downloaded 09 Dec 2010 to 129.13.72.198. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions 084304-2 J. Appl. Phys. 108, 084304 共2010兲 Jacak et al. and with 105 – 107 electrons兲 in an all-analytical calculus version.22 The electron liquid oscillations of compressional and translational type result in excitations inside the sphere and on its surface, respectively. They are analyzed and referred to as volume and surface plasmons. Damping of plasmons due to electron scattering 共nonlocal processes兲 and due to radiation losses 共retardation effects accounted for via the Lorentz friction force兲 is included. The shift in the resonance frequency of dipole-type surface plasmons 共only such plasmons are induced by homogeneous time-dependent electric field兲, due to damping phenomena, is compared with the experimental data for various nanosphere radii. Collective surface dipole-type plasmon oscillations in the linear chain of metallic nanospheres are analyzed and wave-type plasmon modes are described. A coupling in the near-field regime between oscillating dipoles of surface plasmons, together with retardation effects for energy irradiation, lead to the possibility of undamped propagation of plasmon waves along the chain in the experimentally realistic region of parameters 共separation of spheres in the chain and their radii兲. These effects are of particular significance for plasmon arranged nondissipative transport of energy along metallic chains for application in nanoelectronics. In Sec. II of this paper, the standard RPA theory in the quasiclassical limit, is generalized for a confined system of spherical shape. The resulting equations for volume and surface plasmons are solved in the following section 共with the particularities of calculus shifted to the Appendix兲. Section III contains a description of the Lorentz friction for surface plasmons oscillations of the dipole-type. Analysis of collective wave-type surface plasmon oscillations in a chain of metallic nanospheres is presented in Sec. V. Besides the theoretical model the comparison of the characteristic nanoscale plasmon behavior with available experimental data, including our own measurements, is presented. II. RPA APPROACH TO ELECTRON EXCITATIONS IN METALLIC NANOSPHERE A. Derivation of RPA equation for local electron density in a confined spherical geometry 冋 Ĥe = 兺 − j=1 + ប2ⵜ2j − e2 2m 冕 ne共r0兲d3r0 + e共R + r j,t兲 兩r j − r0兩 册 1 e2 + ⌬E, 兺 2 j⫽j 兩r j − r j⬘兩 共2兲 ⬘ where r j and m are the position 共with respect to the sphere center兲 and the mass of the jth electron, respectively, R is the position of the metallic sphere center, ⌬E represents electrostatic energy contribution from the ion “jellium,” and 共r , t兲 is the scalar potential of the external electric field. The corresponding electric field E共R + r j , t兲 = −grad j共R + r j , t兲. Assuming that the space-dependent variation in E is weak on the scale of the sphere radius a, then E共R + r j , t兲 ⯝ E共R , t兲, i.e., the electric field is homogeneous over the sphere. Then 共R + r j , t兲 ⯝ −E共R , t兲 · R + 1共R + r j , t兲, where 1共R + r j , t兲 = −r j · E共R , t兲. Hence, one can rewrite the Hamiltonian 共2兲 in the form Ĥe = Ĥe⬘ − eNE共R,t兲 · R, where Ne 冋 Ĥe⬘ = 兺 − j=1 + ប2ⵜ2j − e2 2m 冕 共3兲 ne共r兲d3r0 + e共r j,t兲 兩r j − r0兩 册 1 e2 + ⌬E, 兺 2 j⫽j 兩r j − r j⬘兩 共4兲 ⬘ and the corresponding 共to Ĥe兲 wave function can be represented as, ⌿共re,t兲 = ⌿⬘共re,t兲ei共eN/ប兲兰E共R,t兲·Rdt , 共5兲 with iប ⌿⬘ / t = Ĥe⬘⌿⬘, re = 共r1 , r2 , . . . , rN兲. A local electron density can be written as follows:19 共r,t兲 = 具⌿共re,t兲兩 兺 ␦共r − r j兲兩⌿共re,t兲典 = 具⌿⬘共re,t兲兩 兺 ␦共r j j − r j兲兩⌿⬘共re,t兲典, 共6兲 with the Fourier picture Let us consider a metallic sphere with radius a located in vacuum, = 1, = 1 and in the presence of dynamical electric field 共magnetic field is assumed to be zero兲. We will consider collective electrons in the metallic material. The model jellium16–18 is assumed in order to account for the screening background of positive ions in the form of static positive charge uniformly distributed over the sphere ne共r兲 = ne⌰共a − r兲, Ne 共1兲 where ne = Ne / V and ne兩e兩 is the averaged positive charge density, Ne is the number of collective electrons in the sphere, V = 共4a3 / 3兲 is the sphere volume, and ⌰ is the Heaviside step-function. Neglecting ion dynamics within the jellium model, which is adopted in particular for description of simple metals, e.g., noble, transition, and alkali metals, we deal with the Hamiltonian for collective electrons, ˜共k,t兲 = 冕 共r,t兲e−ik·rd3r = 具⌿⬘共re,t兲兩ˆ 共k兲兩⌿⬘共re,t兲典, 共7兲 where the “operator” ˆ 共k兲 = 兺 je−ik·r j. Using the above notation one can rewrite Ĥe⬘ in the following form, in analogy to the bulk case23 Ne 冋 册 冕 Ĥe⬘ = 兺 − j=1 ប2ⵜ2j e2 − 2m 42 冕 冕 + e2 163 + e2 42 d3kñe共k兲 1 ˆ+ 关 共k兲 + ˆ 共k兲兴 k2 d3k˜1共k,t兲关ˆ+共k兲 + ˆ 共k兲兴 d 3k 1 ˆ+ 关 共k兲ˆ 共k兲 − Ne兴 + ⌬E, k2 where ñe共k兲 = 兰d3rne共r兲e−ik·r, ˜1共k , t兲 = 兰d3r共r , t兲e−ik·r. 4 / k2 = 兰d3r1 / re−ik·r, 共8兲 and Downloaded 09 Dec 2010 to 129.13.72.198. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions 084304-3 J. Appl. Phys. 108, 084304 共2010兲 Jacak et al. Utilizing this form of the electron Hamiltonian one can write the second time-derivative of ˆ 共k兲, 1 d2ˆ 共k,t兲 = 兵关ˆ 共k兲,Ĥe⬘兴,Ĥe⬘其, dt2 共iប兲2 ⫻具⌿⬘兩兺 je−ik·r j共ប2ⵜ2j / 2m兲兩⌿⬘典. Performing the inverse Fourier transform, Eq. 共11兲 finally attains the form, 共9兲 ប2ⵜ2j 2 2 2␦共r,t兲 具⌿ 兩 ␦ 共r − r 兲 ⵜ 兩⌿⬘典 = − ⬘ 兺 j ,t2 3m 2m j 再 which resolves itself into the equation 再 2p ⵜ ⌰共a − r兲 ⵜ + 4 d ␦ˆ 共k,t兲 ប បk = − 兺 e−ik·r j − 2 共k · ⵜ j兲2 + 2 ik · ⵜ j 2 dt m m j 2 冎 2 ប 2k 4 e2 + 2 − 4m m22 e − m83 − e m83 e2 − m22 冕 冕 冕 2 2 冕 d qñe共k − q兲共k · q兲˜1共q,t兲 j 再 + 冎 ប 2k 4 e2 2 兩⌿⬘典 − 4m m22 − q兲 冕 2 2 ⫻共k · q兲˜1共q,t兲 − e m83 ⫻共k · q兲˜1共q,t兲 − e2 m22 − q兲␦ˆ 共q兲兩⌿⬘典. 冋 ⫻ 1+ 冕 冕 冕 ប2ⵜ2j 兩⌿⬘典 2m 册 5 ␦共r,t兲 +¯ . 3 ne 共13兲 冋 册 2␦共r兲 2 ⑀F 2 = ⵜ ␦共r,t兲 − 2p␦共r,t兲 ⌰共a − r兲 2 3m t d3qñe共k − q兲 − d3q␦˜共k − q,t兲 − d 3q 共12兲 Taking then into account the above approximation and that ⵜ⌰共a − r兲 = − rr ␦共a − r兲 as well as that 1共R , r , t兲 = −r · E共R , t兲, one can rewrite Eq. 共12兲 in the following manner: d3qñe共k k·q e ˜ 2 ␦共q,t兲 − q m83 冎 3 3 ប2 ប2 ⌰共a − r兲 ⯝ 共32兲2/3 5/3共r,t兲 = 共32兲2/3 n5/3 5 5 2m 2m e 共10兲 ␦˜共k,t兲 ប បk = 具⌿⬘兩 − 兺 e−ik·r j − 2 共k · ⵜ j兲2 + 2 ik · ⵜ j 2 t m m j 2 1 ␦共r1,t兲 兩r − r1兩 ene ⵜ 兵⌰共a − r兲 ⵜ 1共r1,t兲其. m 具⌿⬘兩 − 兺 ␦共r − r j兲 d3q␦ˆ 共k − q兲共k · q兲˜1共q,t兲 k·q d3q␦ˆ 共k − q兲 2 ␦ˆ 共q兲, q d 3r 1 According to the Thomas–Fermi approximation19 the RPA averaged kinetic energy can be represented as follows: 3 where ␦ˆ 共k兲 = ˆ 共k兲 − ñe共k兲 is the operator of local electron density fluctuations beyond the uniform distribution. Taking into account that ␦˜共k , t兲 = 具⌿⬘共t兲兩␦ˆ 共k兲兩⌿⬘共t兲典 = ˜共k , t兲 − ñe共k兲, we find, 2 + k·q d3qñe共k − q兲 2 ␦ˆ 共q兲 q 冕 k·q 具⌿⬘兩␦ˆ 共k q2 共11兲 One can simplify the above equation with the assumption that ␦共r , t兲 = 1 / 83兰eik·r␦˜共k , t兲d3k only weakly varies on the interatomic scale, and, therefore, three components of the first term on right-hand-side of Eq. 共11兲 can be estimated as: k2vF2 ␦˜共k兲, k3vF / kT␦˜共k兲, and k4vF2 / kT2 ␦˜共k兲, respectively, with 1 / kT the Thomas–Fermi radius,19 kT = 冑6nee2 / ⑀F, ⑀F is the Fermi energy, and vF is the Fermi velocity. Thus, the contribution of the second and third components of the first term can be neglected in comparison to the first component. Also small, and thus negligible, is the last term on the righthand-side of Eq. 共11兲, as it involves a product of two ␦˜ 共which we assumed small ␦˜ / ne Ⰶ 1兲. This approach corresponds to RPA attitude formulated for bulk metal19,23 关note that ␦ˆ 共0兲 = 0 and the coherent RPA contribution of interaction is contained in the second term in Eq. 共11兲兴. The last but one term in Eq. 共11兲 can also be omitted if one confines it to linear terms with respect to ␦˜ and ˜1. Next, due to spherical symmetry, 具⌿⬘兩兺 je−ik·r j共ប2 / m2兲共k · ⵜ j兲2兩⌿⬘典 ⯝ 共2k2 / 3m兲 2 ⵜ 3m 冋 再冋 册 3 r ⑀Fne + ⑀F␦共r,t兲 ␦共a − r兲 5 r 冎 2 ⑀F r ⵜ ␦共r,t兲 3mr 冕 + 2p r ⵜ 4 r + ene r · E共R,t兲 ␦共a − r兲. m r d 3r 1 1 ␦共r1,t兲 兩r − r1兩 册 共14兲 In the above formula p is the bulk plasmon frequency, and 2p = 共4nee2 / m兲. The solution of Eq. 共14兲 can be decomposed into two parts that are related to the partial domains, ␦共r,t兲 = 再 ␦1共r,t兲, ␦2共r,t兲, for r ⬍ a, for r ⱖ a, 共r → a+兲, 冎 共15兲 corresponding to the volume and surface excitations, respectively. These two parts of local electron density fluctuations satisfy the equations, 2␦1共r,t兲 2 ⑀F 2 ⵜ ␦1共r,t兲 − 2p␦1共r,t兲 = 3m t2 共16兲 and Downloaded 09 Dec 2010 to 129.13.72.198. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions 084304-4 J. Appl. Phys. 108, 084304 共2010兲 Jacak et al. 2 2␦2共r,t兲 ⵜ =− 2 t 3m − + 再 再冋 册 3 r ⑀Fne + ⑀F␦2共r,t兲 ␦共a + ⑀ − r兲 5 r 冎 2 ⑀F r ⵜ ␦2共r,t兲 3mr 2p r ⵜ 4 r 冕 d 3r 1 1 关␦1共r1,t兲⌰共a − r1兲 兩r − r1兩 + ␦2共r1,t兲⌰共r1 − a兲兴 + ene r · E共R,t兲 m r 冎 ⫻␦共a + ⑀ − r兲. 共17兲 The ⑀ = 0+ shift in the delta argument, ␦共a + ⑀ − r兲 is introduced, as a formal character only, to avoid the singular point position on the edge of integration 共for the delta distribution兲. It is clear from Eq. 共16兲 that the volume plasmons are independent of surface plasmons. However, surface plasmons can be excited by volume plasmons due to the last but one term in Eq. 共17兲, which expresses a coupling between surface and volume plasmons in the metallic nanosphere within the RPA semiclassical picture. This is due to the possible influence of volume charges on surface fluctuations. Oppositely, the surface charges cannot influence the interior of the nanosphere. When the metallic sphere is embedded in a dielectric medium, the electrons on the surface interact with forces 共dielectric susceptibility constant兲 times weaker compared to vacuum. To account for this, one substitutes Eq. 共17兲 with the following one: 2 2␦2共r,t兲 ⵜ =− 2 t 3m − + 再 再冋 册 3 r ⑀Fne + ⑀F␦2共r,t兲 ␦共a + ⑀ − r兲 5 r 冕 d 3r 1 冎 冋 and 1 ␦1共r1,t兲⌰共a − r1兲 兩r − r1兩 册 冎 1 ene r · E共R,t兲 ␦共a + ␦2共r1,t兲⌰共r1 − a兲 + m r + ⑀ − r兲. 2␦1共r,t兲 2 ␦共r,t兲 2 ⑀F 2 = ⵜ ␦˜1共r,t兲 − 2p␦1共r,t兲 + 3m t2 0 t 共19兲 2 ⑀F r ⵜ ␦2共r,t兲 3mr 2p r ⵜ 4 r tuations, in opposition to a sharply defined domain of the surface modes within the semiclassical approximation. Moreover, within the semiclassical approach the inner material dielectric properties are not important 关as it follows from Eq. 共18兲兴, what is no case, however, in TDLDA calculus, with fuzzy surface.20,24 This leads to a certain overestimation of the dielectric surroundings induced redshift in the semiclassical treatment in comparison to TDLDA when the domain of surface fluctuation was wider and, therefore, more distant from screening dielectric medium polarization. This manifest itself in stronger redshift for the sharp domain of surface oscillations in semiclassical approximation, which exceeds the Mie-type classical shift and also the TDLDA redshift 共as it is presented in the next subsection for surface plasmon frequencies兲. One can expect, however, that with growing radius of the nanosphere, when spill-out is gradually reducing, the limiting TDLDA would be convergent with the semiclassical estimation. Note that the more thorough inclusion of dielectric missmatch 共as in the TDLDA method20,24兲 would have a particular significance for closely located nanospheres 共in a chain or dimer兲 when multipolar modes would contribute to mutual coupling of the spheres. In the present paper we confine, however, our consideration to simplified dipole-type coupling only, what is justified for not too small distance of nanospheres. Let us also assume that both volume and surface plasmon oscillations are damped with the time ratio 0 which can be phenomenologically accounted for via the additional term, −共2 / 0兲关␦共r , t兲 / t兴, to the right-hand-side of above equations. They attain the form, 共18兲 Simultaneously, Eq. 共16兲 does not change, which reveals the fact that the dielectric surroundings does not influence volume plasmon modes, oppositely to surface modes. One can expect that influence of surrounding dielectric medium would cause a redshift in surface plasmon resonance, what is clear as the polarization of nearby dielectric medium induced by a fluctuating surface charge, reduces this charge due to screening, which results in turn in lowering of oscillation energy. This would be accounted for within even the classical Mie-type approach. It should be noted, however, that in Ref. 20 the stronger redshift induced by dielectric missmatch is found in comparison to the Mie-type classical estimation, for the case of nanoshells. It agrees with similar analyses for small metallic clusters.24 In both these cases spill-out is important, resulting in a widened fuzzy domain of surface fluc- 2␦2共r,t兲 2 ␦共r,t兲 + t2 0 t 再冋 2 ⵜ 3m − 2 ⑀F r ⵜ ␦˜2共r,t兲 3mr + 再 2p r ⵜ 4 r 冕 册 3 r ⑀Fne + ⑀F␦2共r,t兲 ␦共a + ⑀ − r兲 5 r =− d 3r 1 冋 冎 1 ␦1共r1,t兲⌰共a − r1兲 兩r − r1兩 册 冎 1 ene r · E共R,t兲 ␦共a + ⑀ − r兲. + ␦2共r1,t兲⌰共r1 − a兲 + m r 共20兲 From Eqs. 共19兲 and 共20兲 it is noticeable that the electric field homogeneous over the nanosphere does not excite volumetype plasmon oscillations but only contributes to surface plasmons. Downloaded 09 Dec 2010 to 129.13.72.198. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions 084304-5 J. Appl. Phys. 108, 084304 共2010兲 Jacak et al. B. Solution of RPA equations: volume and surface plasmons frequencies Equations 共19兲 and 共20兲 can be solved upon imposing boundary and symmetry conditions—see Appendix. Let us write both parts of the electron fluctuation in the following manner: ␦1共r,t兲 = ne关f 1共r兲 + F共r,t兲兴, for r ⬍ a, ␦2共r,t兲 = ne f 2共r兲 + 共⍀,t兲␦共r + ⑀ − a兲, r ⱖ a, for 共r → a+兲, 共21兲 and let us choose convenient initial conditions F共r , t兲 兩t=0 = 0, 共⍀ , t兲 兩t=0 = 0, 关⍀ = 共 , 兲—spherical angles兲, and 关1 + f 1共r兲兴 兩r=a = f 2共r兲 兩r=a 共continuity condition兲, F共r , t兲 兩r→a = 0, 兰共r , t兲d3r = Ne 共neutrality condition兲. We thus arrive at an explicit form of the solutions of Eqs. 共19兲 and 共20兲 共as it is described in the Appendix兲 f 1共r兲 = − kTa + 1 −k 共a−r兲 1 − e−2kTr e T , 2 k Tr 冋 for 册 for = 冑 4 ene 兵Ez共R,t兲␦m0 + 冑2关Ex共R,t兲␦m1 + Ey共R,t兲␦m−1兴其, 3 m 共25兲 where 1 = 01 = 共 p / 冑3兲 共it is a dipole-type surface plasmon Mie frequency13兲. Only this function contributes to the dynamical response to the homogeneous electric field 共for the assumed initial conditions兲. From the above it follows thus that the local electron density 共within the RPA attitude兲 has the form, 共r,t兲 = 0共r兲 + 1共r,t兲, r ⱖ a, ⬁ ⬘ t兲e−t/ Almn jl共knlr兲Y lm共⍀兲sin共nl 兺 兺 l=1 m=−l n=1 and ⬁ l 兺 Y lm共⍀兲 l=1 m=−l 共⍀,t兲 = 兺 册 冋 ⫻ ⬁ l2p − 共2l + 1兲2nl a 0 dr1 共23兲 l ne关1 + f 1共r兲兴, for ne f 2共r兲, r ⱖ a, r → a+ 冦 0, 兺 兺 Almn rl+2 1 ⬘ t兲e−t/0 , jl共knlr1兲sin共nl al+2 共24兲 where jl共兲 = 冑共 / 2兲Il+1/2共兲 is the spherical Bessel funcis the spherical function, nl tion, Y lm共⍀兲 = p冑1 + 共x2nl / kT2 a2兲 are the frequencies of electron volume free self-oscillations 共volume plasmon frequencies兲, xnl are nodes of the Bessel function jl共兲, 0l = p冑l / 共2l + 1兲 are the frequencies of electron surface free self-oscillations 共surface plasmon frequencies兲, and knl = xnl / a; ⬘ = 冑2 − 共1 / 20兲 are the shifted frequencies for all modes due to damping. The coefficients Blm and Almn are determined by the initial conditions. We assumed that ␦共r , t = 0兲 = 0, so Blm = 0 and Almn = 0, except for l = 1 in the former case 共of Blm兲, which corresponds to homogeneous electric field excitation. This is described by the function Q1m共t兲 in the general solution 共24兲. The function Q1m共t兲 satisfies the equation, 冎 共27兲 for r ⬍ a, 兺 Q1m共t兲Y 1m共⍀兲 for r ⱖ a, r→a+. m=−1 冧 共28兲 In general, F共r , t兲 共volume plasmons兲 and 共⍀ , t兲 共surface plasmons兲 contribute to plasmon oscillations. However, in the case of homogeneous perturbation, only the surface l = 1 mode is excited. For plasmon oscillations given by Eq. 共28兲 one can calculate the corresponding dipole, D共R,t兲 = e Y lm共⍀兲ne for r ⬍ a, 1 ⬁ l=1 m=−l n=1 共l + 1兲2p 冕 0 Blm ⬘ t兲e−t/0共1 − ␦1l兲 sin共0l a2 + Q1m共t兲␦1l + 兺 ⫻ 1共r,t兲 = ⬁ l F共r,t兲 = 兺 再 and the nonequilibrium, of surface plasmon oscillation type for the homogeneous forcing field, 共22兲 where kT = 冑共6nee2 / ⑀F兲 = 冑共32p / vF2 兲. For time-dependent parts of electron fluctuations, we find, 共26兲 where the RPA equilibrium electron distribution 共correcting the uniform distribution ne兲, 0共r兲 = r ⬍ a, k Ta + 1 e−kT共r−a兲 , 共1 − e−2kTa兲 2 k Tr f 2共r兲 = kTa − 2Q1m共t兲 2 Q1m共t兲 + 21Q1m共t兲 + t2 0 t 冕 d3rr共r,t兲 = 4 eq共R,t兲a3 , 3 共29兲 Q1−1共R , t兲 where Q11共R , t兲 = 冑共8 / 3兲qx共R , t兲, = 冑共8 / 3兲qy共R , t兲, Q10共R , t兲 = 冑共4 / 3兲qx共R , t兲, and q共R , t兲 satisfies the equation 关see Eq. 共25兲兴, 冋 册 2 2 ene + 21 q共R,t兲 = E共R,t兲. 2 + t 0 t m 共30兲 III. LORENTZ FRICTION FOR NANOSPHERE PLASMONS Considering the nanosphere plasmons induced by the homogeneous electric field, as described in the above paragraph, one can note that these plasmons are themselves a source of electromagnetic 共e-m兲 radiation. This radiation takes away the energy of plasmons resulting in their damping, which can be described as the Lorentz friction.25 This e-m wave emission causes electron friction which can be described as an additional electric field,25 Downloaded 09 Dec 2010 to 129.13.72.198. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions 084304-6 J. Appl. Phys. 108, 084304 共2010兲 Jacak et al. EL = 2 3D共t兲 , 3v2 t3 共31兲 where v = 共c / 冑兲 is the light velocity in the dielectric medium, and D共t兲 the dipole of the nanosphere. According to Eq. 共29兲 we arrive at the following: EL = 2e 4 3 3q共t兲 . a 3v2 3 t3 册 冉 冊 2 1a ene E共R,t兲 + m 31 v q共R,t兲 . t3 3 3 共33兲 If one rewrites the above equation 共for E = 0兲 in the form, 冋 册 冋 2 + 21 q共R,t兲 t2 冉 冊 2 2 1a − q共R,t兲 + = t 0 31 v 册 q共R,t兲 , t2 3 2 共34兲 thus, one notes that the zeroth order approximation 共neglecting attenuation兲 corresponds to the equation, 冋 册 2 + 21 q共R,t兲 = 0. t2 共35兲 In order to solve Eq. 共34兲 in the next step of perturbation, one can substitute, in the right-hand-side of this equation, 2q共t兲 / t2 by −21q共t兲 关according to Eq. 共35兲兴. Therefore, if one assumes the above estimation, 共3q共t兲 / t3兲 ⯝ −21共q共t兲 / t兲, one can include the Lorentz friction in a renormalized damping term, 冋 册 2 2 ene + 21 q共R,t兲 = E共R,t兲, + t2 t m where 共Air兲 1 共Water兲 1.4 共Colloidal solution兲 2 冉 冊 1 1 1 1a = + 0 3 v 2 2 + + 21 q共R,t兲 t2 0 t = Refraction rate of the surrounding medium, n0 共32兲 Substituting this into Eq. 共30兲, we get, 冋 TABLE I. a0—nanosphere radius corresponding to minimal damping. 共36兲 3 ⯝ Au, a0 共nm兲 Ag, a0 共nm兲 Cu, a0 共nm兲 8.8 9.14 9.99 8.44 9.18 10.04 8.46 9.20 10.04 冉 冊 v F C v F 1 1a + + 3 2B 2a v 3 , 共37兲 where we used for 共1 / 0兲 ⯝ 共vF / 2B兲 + 共CvF / 2a兲, 共B is the free path in bulk, vF is the Fermi velocity, and C ⯝ 1 is a constant兲,26–28 which corresponds to inclusion of plasmon damping due to electron scattering on other electrons and on the nanoparticle boundary. Renormalized damping causes a change in the shift in self-frequencies of free surface plasmons, 1⬘ = 冑21 − 共1 / 2兲. Using Eq. 共37兲 one can determine the radius a0 that corresponds to minimal damping, a0 = 冑3 p 共vFc3冑/2兲1/4 . 共38兲 For nanoparticles of gold, silver and copper in air, in water, and in a colloidal solution, one can find a0 ⱕ 10 nm 共see Table I兲, which corresponds to the experimental data.29,30 For a ⬎ a0 damping increases due to Lorentz friction 共proportional to a3兲 but for a ⬍ a0, the damping due to electron scattering dominates and grows with lowering a 关as ⬃共1 / a兲, see Fig. 1兴, which agrees with experimental observations.26,29 Surface plasmon oscillations cause attenuation of the incident e-m radiation, with maximum of attenuation at the resonant frequency22 1⬘ = 冑21 − 共1 / 2兲. This frequency diminishes with the rise of a, for a ⬎ a0 according to Eq. 共37兲, which agrees with experimental observations for Au and Ag presented in Fig. 2, and Table II 共Au兲 and Table III 共Ag兲. Note that the Lorentz friction, being of the third order with respect to time-retardation shift,25 includes the retardation effects via plasmon damping induced by irradiation.31 The other channel of plasmon damping corresponds to non- FIG. 1. 共Color兲 Effective damping ratios for surface plasmon oscillations for Ag in air 共left and middle兲 and Au colloidal solutions 共right兲 from Eq. 共38兲, respectively. The red curve represents the sum of both terms, the green is the term ⬃共1 / a兲, and the blue is the term ⬃a3; the minimum corresponds to the minimal damping for radius a0 from Eq. 共39兲. Downloaded 09 Dec 2010 to 129.13.72.198. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions 084304-7 J. Appl. Phys. 108, 084304 共2010兲 Jacak et al. FIG. 2. 共Color兲 Extinction spectra for nanospheres of Au 共a兲 and Ag 共b兲 in colloidal water solution for various sphere radii; Lorentzian shapes correspond to light attenuation due to surface plasmons exciting in nanospheres; for large radii 共⬃75 nm for Au and ⬃30– 40 nm for Ag兲 the second attenuation peak appears corresponding possibly to volume mode excited at nonexact dipole approximation regime. local effects including scattering on the nanoparticle surface.32 We have demonstrated that the retardation and the nonlocal effects lead to opposite, redshift and blueshift in the resonance frequency, similarly as it was indicated in Ref. 31. above formula, R0 ⬍ , n = R0 / R0, and v = c / 冑 = c / n0. When both vectors R and R0 are along the z-axis 共the linear chain兲 the above equation can be resolved as, E␣共R,R0,t兲 = IV. UNDAMPED SURFACE PLASMON COLLECTIVE EXCITATION ALONG A CHAIN OF METALLIC NANOSPHERES Let us consider a linear chain of metallic nanospheres with radii a in a dielectric medium with dielectric constant . We assume that spheres are located along the z-axis direction equidistantly with the separation of the sphere centers d ⬎ 2a.26,33 At time t = 0 we assume the excitation of plasmon oscillations via a Dirac delta, ⬃␦共t兲, shaped signal of electric field. Taking into account the mutual interaction of the induced surface plasmons on the spheres via the radiation of dipole oscillations, we aim to determine the stationary state of the whole infinite chain. For a separation d much shorter than the wavelength of the e-m wave corresponding to surface plasmon self-frequency in a single nanosphere, the dipole-type plasmon radiation can be treated within the nearfield regime, at least for nearest neighboring spheres. In the near-field region a ⬍ R0 ⬍ , the radiation of the dipole D共t兲 is not a planar wave 共as it is for far-field region, R0 Ⰷ 兲 but is of only a retarded electric field 共without a magnetic field兲25 E共R,R0,t兲 = 1 再 冋 冉 冊册 冉 冊冎 R30 3n n · D R,t − − D R,t − R0 v R0 v 共39兲 , R is the position of the sphere 共center兲 irradiating e-m energy due to its dipole surface plasmon oscillations, R0 is the position of another sphere 共center兲, with respect to the center of the former one, where the field E共R , R0 , t兲 is given by the 冉 冊 ␣ R0 , 3 D␣ R,t − v R0 共40兲 where ␣ = 共x , y , z兲, x = y = −1, and z = 2. Assuming that the z-axis origin coincides with the center of one sphere in the chain, for the lth sphere located in the point Rl = 共0 , 0 , ld兲, an electric field caused by neighboring spheres, E共Rm , Rml , t兲, and the Lorentz friction force caused by self-radiation, EL共Rl , t兲, have to be accounted for. By virtue of Eq. 共30兲 the equation for the surface plasmon oscillation of the lth sphere is, 冋 册 m=⬁ 2 2 ene + 21 q共Rl,t兲 = + E共Rm,Rml,t兲 兺 t2 0 t m m=−⬁,m⫽l,Rml⬍ + ene EL共Rl,t兲, m 共41兲 provided that the dipole field of the mth sphere can be treated as homogeneous over the lth sphere and the sum over m is confined by the distance of the mth sphere from the lth sphere not exceeding the near-field range 共⬃兲. In the case of the equidistant chain, Rl = ld and Rml = 兩l − m兩d, and using Eqs. 共40兲, 共29兲, and 共32兲, one can rewrite Eq. 共41兲 in the form, 冋 冉 冊 2 1a 2 2 + − t2 0 t 31 v m=⬁ 3 册 3 + 21 q␣共ld,t兲 t3 a3 = ␣21 3 兺 d m=−⬁,m⫽l,兩l−m兩d⬍ 冉 冊 d q␣ md,t − 兩l − m兩 v , 兩l − m兩3 共42兲 where ␣ = x , y, which describe the transversal plasmon modes TABLE II. Resonant frequency for e-m wave attenuation in Au nanospheres. Radius of nanospheres 共nm兲 ប⬘1 共experiment兲 共eV兲 ប⬘1 共theory兲 共eV兲, n0 = 1.4 ប⬘1 共theory兲 共eV兲, n0 = 2 10 2.371 3.721 2.604 15 2.362 3.720 2.603 20 2.357 3.716 2.600 25 2.340 2.702 2.590 30 2.316 3.666 2.565 40 2.248 3.415 2.388 50 2.172 2.374 1.656 Downloaded 09 Dec 2010 to 129.13.72.198. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions 084304-8 J. Appl. Phys. 108, 084304 共2010兲 Jacak et al. 冉 冊 1 1 2a ␣a = + 1 ␣共 ␣兲 0 3 v v TABLE III. Resonant frequency for e-m wave attenuation in Ag nanospheres. Radius of nanospheres 共nm兲 ប⬘1 共experiment兲 共eV兲 ប⬘1 共theory兲 共eV兲, n0 = 1.4 ប⬘1 共theory兲 共eV兲, n0 = 2 10 3.024 3.707 2.595 20 2.911 3.702 2.591 30 2.633 3.654 2.557 40 2.385 3.410 2.384 2 ⬁ cos共mkd兲 a3 + ␣21 3 兺 m3 d m=1,md⬍ 冉 冊 sin ␣md v . ␣ 共47兲 and ␣ = z, which describes the longitudinal mode; = 共2p / 3兲 = 共4nee2 / 3m兲 and v = 共c / 冑兲. The above equation coincides with the appropriate one from Refs. 26 and 33, if one assumes that 共4 / 3兲a3ne = N = 1 and neglects the retardation of the field. Taking into account the periodicity of the infinite chain, one can consider the solution of the above equation in the form, 21 q␣共ld,t兲 = q̃␣共k,t兲e −ikld If we confine the sum in Eq. 共46兲 to m = 1 共the nearest neighbor approximation兲, we get, 冋 ˜ ␣2 共␣兲 ⯝ 21 1 − and from Eq. 共47兲, 1 2a 3 1 = + 1 2 ␣共 ␣兲 0 4 v d 共43兲 . The right-hand-side term in Eq. 共42兲 attains the form, m=⬁ 兺 m=−⬁,m⫽l l−1 兺 = 冉 d q␣ md,t − 兩l − m兩 v 兩l − m兩3 m=−⬁ 冉 d q␣ md,t − 兩l − m兩 v 兩l − m兩3 冉 m=⬁ + 兺 m=l+1 = 2e−ikld 兺 m=1 1 2a 3 1 = + 1 2 z共 z兲 0 2 v d for 冊 冉 冊 3 ˜ ␣2 共␣兲 = 册 3 + 21 q̃␣共k,t兲 t3 2i␣ ˜ ␣2 共␣兲 = 0, + ␣共 ␣兲 = 21 冋 ⬁ 共44兲 共45兲 冉 冊册 cos共mkd兲 2 ␣a 3 ␣md 1− cos 兺 m3 d3 m=1,md⬍ v 共46兲 and zd v 2 + 共kd − 兲2 − 册 2 , 3 ␣ = z. 共50兲 ⬁ This equation is linear and, therefore, we look for the solutions of the shape: q̃␣共k , t兲 = Q̃␣共k兲ei␣t, and we arrive at the condition, where, 冋冉 冊 共49兲 1 1 = 关sin共kmd + ␣md/v兲 − sin共kmd − ␣md/v兲兴 兺 2␣ m=1 m3 cos共mkd兲 a3 2 兺 q̃␣共k,t − md/v兲. m3 d3 m=1,md⬍ − ␣2 + 册 2 , 3 冉 冊 ⬁ ⬁ = ␣21 − 共kd − 兲2 + cos共mkd兲 1 ␣md sin 兺 3 m ␣ m=1 v cos共mkd兲 q̃␣共k,t − md/v兲. m3 2 1a 2 2 − 2 + t 0 t 31 v 2 In the derivation of the two above formulae the following summation was performed:34 Thus, the Eq. 共42兲 can be written as follows: 冋 冋冉 冊 ␣d v 共48兲 ␣ = x,y and 冊 d q␣ md,t − 兩l − m兩 v 兩l − m兩3 ⬁ for 冊 冉 冊册 2 ␣a 3 ␣d 3 cos共kd兲cos d v 冋 册 k2d2 ␣2 d2 d 2 − kd + + . 4 2 2v2 v 6 Because the terms in the sum drop quickly to zero, the above formula approximates well the sum with limitation md ⬍ . Assuming now, ␣ = ␣⬘ + i␣⬙ , the Eq. 共45兲 gives the dependence of ␣⬘ and ␣⬙ on k. The general solution of Eq. 共42兲 attains the form, Ns q␣共ld,t兲 = 兺 Q̃␣共kn兲ei共␣⬘ 共kn兲t−knld兲−␣⬙ 共kn兲t , 共51兲 n=1 where kn = 共2n / Nsd兲, L = Nsd is the assumed length of the chain with Ns spheres, when periodic 共Born–Karman-type兲 boundary conditions are imposed. The components of Eq. 共51兲 describe monochromatic waves with wavelength n = 共2 / kn兲 = 共L / n兲, which are analogous to planar waves in crystals, when damping is not big, i.e., when ␣⬙ Ⰶ ␣⬘ . Provided with this inequality, one can approximate: for the transversal modes 共␣ = x , y兲, 冋 ˜ ␣2 = 21 1 + 共␣⬘ 兲2 = 册 2a3 cos共kd兲cos共␣⬘ d/v兲 , d3 共52兲 Downloaded 09 Dec 2010 to 129.13.72.198. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions 084304-9 J. Appl. Phys. 108, 084304 共2010兲 Jacak et al. FIG. 3. 共Color兲 Wave vector kd for undamped surface plasmon propagation along the metallic nanosphere chain vs sphere separation d at 共top兲 constant d / a = 3, with a the sphere radius and 共bottom兲 for a = 20 nm for Ag sphere chains, respectively; transversal modes are shown in red while longitudinal modes are shown in blue. ␣⬙ = 1 1 2a 3 = + 1 2 ␣ 0 4vd 冋冉 冊 ␣⬘ d v − 共kd − 兲2 + and for the longitudinal mode 共␣ = z兲, 冋 ˜ z2 = 21 1 − 共z⬘兲2 = z⬙ = 21a3 2 1 1 = + z 0 2vd 册 2 , 3 2 册 4a3 cos共kd兲cos共z⬘d/v兲 , d3 冋冉 冊 ␣⬘ d v 2 共54兲 3 2 + 共kd − 兲2 − 共53兲 册 . 共55兲 From Eqs. 共53兲 and 共55兲 it follows that ␣⬙ can change its sign. In the case of ␣⬙ ⬍ 0 the oscillations are destabilized, which could be avoided by inclusions of some nonlinear terms neglected in the expression for the Lorentz friction, which in more accurate form25 includes also a small nonlinear term with respect to D, aside from the term with 3D / t3. Including this will result in damping of too highly rising oscillations and will lead to stable oscillation amplitude. Due to this stabilization caused by nonlinear effects, undamped wave modes of dipole oscillations will propagate in the chain in the region of parameters where ␣⬙ ⱕ 0 共and with fixed amplitude accommodated by the nonlinear term兲. The condi- tion ␣⬙ = 共1 / ␣兲 = 0, for critical parameters, resolves into, 冉 冊 ␣d v 2 = 共kd − 兲2 − 4vd2 2 , − 3 021a3 共56兲 for ␣ = 共x , y兲 and for ␣ = z, 冉 冊 zd v 2 = − 共kd − 兲2 + 2vd2 2 . − 3 021a3 共57兲 共␣d / v兲 obtained from the above equations leads to determination of the dependence of wave vector k with respect to parameters d and a, via Eqs. 共52兲–共55兲. Solutions for these equations, found numerically for the chain of Ag nanospheres, are depicted in Fig. 3. Undamped plasmon waves in the chain appear if d ⬍ dmax and have k = / d, dmax = 98.5共68.8兲 nm for transversal共longitudinal兲 modes. For example, for Ag spheres with the radius a = 20 nm and the separation d = 60 nm, undamped transversal modes appear for 0 ⱕ kd ⱕ / 4 or 3 / 4 ⱕ kd ⱕ 2 and an undamped longitudal mode for 3 / 4 ⱕ kd ⱕ 5 / 4. Let us underline that the determined undamped plasmon oscillation wave modes would explain similar, numerically Downloaded 09 Dec 2010 to 129.13.72.198. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions 084304-10 J. Appl. Phys. 108, 084304 共2010兲 Jacak et al. observed behavior.35–37 Within that numerical analysis two types of collective surface plasmons with distinct propagation along the nanosphere chain were identified, and called as quasistatic 共ordinary兲 and nonquasistatic 共extraordinary兲 surface plasmon modes. The quasistatic modes were damped while nonquasistatic ones were undamped 共and additionally, the latter turn out to be relatively robust against the disorder in the chain兲.35 One can associate these observations with undamped collective plasmon modes with the negative imaginary part of the frequency, accessible in the described above regions of the chain parameters and the wave vector values 共stabilized by nonlinear terms of Lorentz friction at a certain amplitude兲, which are, however, accompanied by ordinary damped modes with the positive imaginary frequency part. f 1共r兲 = ␣ e−kTa −k r k r 共e T − e T 兲, k Tr 共A3兲 where ␣ is a constant, kT = 冑6nee2 / ⑀F = 冑32p / vF2 共kT is the inverse Thomas–Fermi radius兲, p = 冑4nee2 / m; 共bulk plasmon frequency兲. Since we assume F共r , 0兲 = 0, then for function F共r , t兲 the solution can be taken as, F共r,t兲 = F共r兲sin共⬘t兲e−0t , 共A4兲 where ⬘ = 冑2 + 1 / 20. F共r兲 satisfies the equation 共Helmholtz equation兲, ⵜ2F共r兲 + k2F共r兲 = 0, 共A5兲 with k2 = 共2 − 2p / vF2 / 3兲. A solution of the above equation, nonsingular at r = 0, is as follows, V. CONCLUSIONS In the present paper we analyzed plasmons in large metallic nanospheres induced by a homogeneous timedependent electric field. Within an all-analytical RPA quasiclassical approach, the volume and surface plasmons are described. It was verified that only dipole-type of surface plasmons can be induced by a homogeneous field 共while none of volume modes兲. An irradiation of energy by plasmon oscillations is described within the Lorentz friction effect. Its scaling with the nanosphere dimension leads to a sphere radius dependent shift in resonant frequency, similarly as observed in experiments, for nonsmall nanospheres 共with radii 10–80 nm兲. This description of surface dipole-type plasmon oscillations in a single nanosphere is applied to the analysis of collective oscillations in linear chains of metallic nanospheres. The wave-type collective plasmon oscillations in the chain are also considered. The undamped region of wave propagation through the chain is found for a certain sphere separation in the chain with the corresponding wavelength of plasmon waves. This phenomenon confirms similar behavior observed by numerical simulations.35 F共r兲 = Ajl共kr兲Y lm共⍀兲, 共A6兲 where A is a constant, jl共兲 = 冑 / 共2兲Il+1/2共兲 is the spherical Bessel function 关In共兲 is the Bessel function of the first order兴, Y lm共⍀兲 is the spherical function 共⍀ is the spherical angle兲. Owing to the quasiclassical boundary condition, F共r , t兲 兩r→a = 0, one has to demand jl共ka兲 = 0, which leads to the discrete values of k = knl = xnl / a, 共where xnl, n = 1 , 2 , 3 , . . ., are nodes of jl兲, and next to the discretisation of self-frequencies , 冉 2 nl = 2p 1 + x2nl kT2 a2 冊 共A7兲 . The general solution for F共r , t兲 thus has the form, ⬁ l ⬁ ⬘ t兲e− t . Almn jl共knlr兲Y lm共⍀兲sin共nl 兺 兺 l=0 m=−l n=1 F共r,t兲 = 兺 0 共A8兲 ACKNOWLEDGMENTS Supported by the Polish KBN Project No. N N202 260734 and the FNP Fellowship Start 共W.J.兲, as well as DFG under Grant No. SCHA 1576/1-1 共D. Z. Hu and D.S.兲. APPENDIX: ANALYTICAL SOLUTION OF PLASMON EQUATIONS FOR THE NANOSPHERE Let us solve first the Eq. 共19兲, assuming a solution in the form, ␦1共r,t兲 = ne关f 1共r兲 + F共r,t兲兴, for r ⬍ a. 共A1兲 Equation 共19兲 resolves thus into, ⵜ2 f 1共r兲 − kT2 f 1共r兲 = 0, 2F共r,t兲 2 F共r,t兲 vF2 2 = ⵜ F共r,t兲 − 2pF共r,t兲. + 3 t2 0 t 共A2兲 The solution for function f 1共r兲 共nonsingular at r = 0兲 thus has the form, A solution of Eq. 共20兲 we represent as, ␦2共r,t兲 = ne f 2共r兲 + 共⍀,t兲␦共r + ⑀ − a兲, for r ⱖ a, 共r → a + ,i . e . , ⑀ = 0+兲. 共A9兲 The neutrality condition, 兰共r , t兲d3r = Ne, with ␦2共r , t兲 = 共 , t兲␦共a + ⑀ − r兲 + ne f 2共r兲, 共⑀ → 0兲, can be rewritten as follows: −兰a0drr2 f 1共r兲 = 兰⬁a drr2 f 2共r兲, 兰a0d3rF共r , t兲 = 0, and 兰d⍀共⍀ , t兲 = 0. Taking into account also the continuity condition on the spherical particle surface, 1 + f 1共a兲 = f 2共a兲, one can obtain: f 2共r兲 = e−kT共r−a兲 / 共kTr兲. It is possible to fit ␣ 关see Eq. 共A3兲兴 and  constants: ␣ = 共kTa + 1 / 2兲 and  = kTa − 共kTa + 1 / 2兲共1 − e−2kTa兲—which gives Eqs. 共22兲. From the condition 兰a0d3rF共r , t兲 = 0 and from Eq. 共A8兲 it follows that A00n = 0, 共because of 兰d⍀Y lm共兲 = 4␦l0␦m0兲. To remove the Dirac delta functions we integrate both sides of the Eq. 共20兲 with respect to the radius 共兰⬁0 r2dr. . .兲 and then we take the limit to the sphere surface, ⑀ → 0. This results in the following equation for surface plasmons: Downloaded 09 Dec 2010 to 129.13.72.198. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions 084304-11 J. Appl. Phys. 108, 084304 共2010兲 Jacak et al. ⬁ l 2共⍀,t兲 2 共⍀,t兲 2 + Y lm共⍀兲 = − 兺 兺 0l t2 0 t l=0 m=−l ⬁ + 2pne 兺 l 兺 冕 ⬁ 兺 Almn l=0 m=−l n=1 ⴱ d⍀1共⍀1,t兲Y lm 共⍀1兲 l+1 Y lm共⍀兲 2l + 1 冕 a dr1 0 rl+2 ene 1 冑4/3关Ez共R,t兲Y 10共⍀兲 l+2 j l共knlr1兲sin共nlt兲 + a m + 冑2Ex共R,t兲Y 11共⍀兲 + 冑2Ey共R,t兲Y 1−1共⍀兲兴, where 20l = 2p共l / 2l + 1兲. In derivation of the above equation the following formulae were exploited, 共for a ⬍ r1兲, 2Q1m共t兲 2 Q1m共t兲 2 + 01 + Q1m共t兲 t2 0 t ⬁ = 1 al = Pl共cos ␥兲 兺 a 冑a2 + r21 − 2ar1 cos ␥ a l=0 rl+1 1 ⬁ =兺 l=0 la Pl共cos ␥兲, ene 冑4/3关Ez共R,t兲␦m0 + 冑2Ex共R,t兲␦m1 m + 冑2Ey共R,t兲␦m−1兴. l−1 rl+1 1 共A10兲 共A11兲 共A15兲 Thus 共 , t兲 attains the form, ⬁ l B lm ⬘ t兲e−t/ Y lm共⍀兲 2 sin共0l 兺 a l=2 m=−l 共⍀,t兲 = 兺 where Pl共cos ␥兲 is the Legendre polynomial 关Pl共cos ␥兲 ⴱ l Y lm共⍀兲Y lm 共⍀1兲兴, ␥ is an angle between = 4 / 共2l + 1兲兺m=−l vectors a = ar̂ and r1, and 共for a ⬎ r1兲, 0 1 + 兺 Q1m共t兲Y 1m共⍀兲 m=−1 1 2 2 冑 a a + r1 − 2ar1 cos ␥ ⬁ ⬁ ⬁ l 兺 4 l=0 m=−l ⫻ l + 1 rl1 ⴱ Y lm共⍀兲Y lm 共⍀1兲. 2l + 1 al+2 共A12兲 l 兺 qlm共t兲Y lm共⍀兲. 共⍀,t兲 = 兺 共A13兲 l=0 m=−l From the condition 兰共t兲d⍀ = 0 it follows that q00 = 0. Taking into account the initial condition 共 , 0兲 = 0 we get 共for l ⱖ 1兲, qlm共t兲 = Blm ⬘ t兲e−t/0共1 − ␦l1兲 + Q1m共t兲␦l1 sin共0l a2 ⬁ + 兺 Almn n=1 ⫻ 冕 a 0 dr1 共l + 1兲2p l2p − 共2l + 1兲2nl 冕 a 0 Taking into account the spherical symmetry, one can assume the solution of the Eq. 共A10兲 in the form, ⬁ ⬁ 兺兺 l=1 m=−l n=1 +兺 rl1 = 兺 l+1 Pl共cos ␥兲 a l=0 a =−兺 l ne rl+2 1 ⬘ t兲e−t/0 , jl共knlr1兲sin共nl al+2 共A14兲 2 where 0l − 1 / 20 and Q1m共t兲 satisfies the equation, ⬘ = 冑0l dr1 Anlm 共l + 1兲2p l2p − 共2l + 1兲2nl rl+2 1 ⬘ t兲e−t/0 . jl共knlr1兲sin共nl al+2 Y lm共⍀兲ne 共A16兲 1 S. Pillai, K. B. Catchpole, T. Trupke, G. Zhang, J. Zhao, and M. A. Green, Appl. Phys. Lett. 88, 161102 共2006兲. 2 M. Westphalen, U. Kreibig, J. Rostalski, H. Lüth, and D. Meissner, Sol. Energy Mater. Sol. Cells 61, 97 共2000兲; M. Grätzel, J. Photochem. Photobiol. C 4, 145 共2003兲. 3 H. R. Stuart and D. G. Hall, Appl. Phys. Lett. 73, 3815 共1998兲; Phys. Rev. Lett. 80, 5663 共1998兲; H. R. Stuart and D. G. Hall, Appl. Phys. Lett. 69, 2327 共1996兲. 4 D. M. Schaadt, B. Feng, and E. T. Yu, Appl. Phys. Lett. 86, 063106 共2005兲. 5 K. Okamoto, I. Niki, A. Shvartser, Y. Narukawa, T. Mukai, and A. Scherer, Nature Mater. 3, 601 共2004兲; K. Okamoto, I. Niki, A. Scherer, Y. Narukawa, T. Mukai, and Y. Kawakami, Appl. Phys. Lett. 87, 071102 共2005兲. 6 C. Wen, K. Ishikawa, M. Kishima, and K. Yamada, Sol. Energy Mater. Sol. Cells 61, 339 共2000兲. 7 L. Lalanne and J. P. Hugonin, Nat. Phys. 2, 551 共2006兲. 8 A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, Phys. Rep. 408, 131 共2005兲. 9 S. A. Mayer, Plasmonics: Fundamentals and Applications 共Springer, Berlin, 2007兲. 10 W. L. Barnes, A. Dereux, and T. W. Ebbesen, Nature 共London兲 424, 824 共2003兲. 11 N. Engheta, A. Salandrino, and A. Alu, Phys. Rev. Lett. 95, 095504 共2005兲. 12 S. A. Maier and H. A. Atwater, J. Appl. Phys. 98, 011101 共2005兲. 13 G. Mie, Ann. Phys. 25, 377 共1908兲. 14 M. Brack, Phys. Rev. B 39, 3533 共1989兲. 15 L. Serra, F. Garcias, M. Barranco, N. Barberan, and J. Navarro, Phys. Rev. B 41, 3434 共1990兲. 16 M. Brack, Rev. Mod. Phys. 65, 677 共1993兲. 17 W. Ekardt, Phys. Rev. Lett. 52, 1925 共1984兲. Downloaded 09 Dec 2010 to 129.13.72.198. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions 084304-12 18 J. Appl. Phys. 108, 084304 共2010兲 Jacak et al. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles 共Wiley, New York, 1983兲; U. Kreibig and M. Vollmer, Optical Properties of Metal Clusters 共Springer, Berlin, 1995兲; J. I. Petrov, Physics of Small Particles 共Nauka, Moscow, 1984兲; C. Burda, X. Chen, R. Narayanan, and M. El-Sayed, Chem. Rev. 共Washington, D.C.兲 105, 1025 共2005兲. 19 D. Pines, Elementary Excitations in Solids 共ABP, Massachusetts, 1999兲. 20 E. Prodan and P. Nordlander, Nano Lett. 3, 543 共2003兲; E. Prodan, P. Nordlander, and N. J. Halas, ibid. 3, 1411 共2003兲. 21 E. Prodan and P. Nordlander, Chem. Phys. Lett. 352, 140 共2002兲; E. Prodan, A. Lee, and P. Nordlander, ibid. 360, 325 共2002兲. 22 L. Jacak, J. Krasnyj, and A. Chepok, Fiz. Nizk. Temp. 35, 491 共2009兲; W. Jacak, J. Krasnyj, J. Jacak, R. Gonczarek, A. Chepok, L. Jacak, D. Z. Hu, and D. Schaadt, J. Appl. Phys. 107, 124317 共2010兲. 23 D. Pines and D. Bohm, Phys. Rev. 85, 338 共1952兲; D. Bohm and D. Pines, ibid. 92, 609 共1953兲. 24 A. Rubio and L. Serra, Phys. Rev. B 48, 18222 共1993兲. 25 L. D. Landau and E. M. Lifshitz, Field Theory 共Nauka, Moscow, 1973兲 in Russian. 26 M. L. Brongersma, J. W. Hartman, and H. A. Atwater, Phys. Rev. B 62, R16356 共2000兲. U. Kreibig and L. Genzel, Surf. Sci. 156, 678 共1985兲. C. Yannouleas, R. A. Broglia, M. Brack, and P. F. Bortignon, Phys. Rev. Lett. 63, 255 共1989兲; G. Weick, G. L. Ingold, R. A. Jalabert, and D. Weinmann, Phys. Rev. B 74, 165421 共2006兲. 29 F. Stietz, J. Bosbach, T. Wenzel, T. Vartanyan, A. Goldmann, and F. Träger, Phys. Rev. Lett. 84, 5644 共2000兲. 30 M. Scharte, R. Porath, T. Ohms, M. Aeschlimann, J. R. Krenn, H. Ditlbacher, F. R. Aussenegg, and A. Liebsch, Appl. Phys. B: Lasers Opt. 73, 305 共2001兲. 31 F. J. García de Abajo, Rev. Mod. Phys. 82, 209 共2010兲. 32 F. J. García de Abajo, J. Phys. Chem. C 112, 17983 共2008兲. 33 S. A. Maier, P. G. Kik, and H. A. Atwater, Phys. Rev. B 67, 205402 共2003兲. 34 I. S. Gradstein and I. M. Rizik, Tables of Integrals 共Fizmatizdat, Moscow, 1962兲. 35 V. A. Markel and A. K. Sarychev, Phys. Rev. B 75, 085426 共2007兲. 36 E. Hao, R. C. Bayley, G. C. Schatz, J. T. Hupp, and S. Li, Nano Lett. 4, 327 共2004兲. 37 B. Lambrecht, A. Leitner, and F. R. Aussenegg, Appl. Phys. B: Lasers Opt. 64, 269 共1997兲. 27 28 Downloaded 09 Dec 2010 to 129.13.72.198. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions