Undamped energy transport by collective surface plasmon oscillations along metallic nanosphere chain
W. Jacak, J. Krasnyj, J. Jacak, A. Chepok, L. Jacak, W. Donderowicz, D. Z. Hu, D. M. Schaadt
aa r X i v : . [ c ond - m a t . m e s - h a ll ] M a r PACS No: 73.21.-b, 36.40.Gk, 73.20.Mf, 78.67.Bf
Undamped energy transport by collective surface plasmon oscillations along metallicnanosphere chain
W. Jacak , J. Krasnyj , , J. Jacak , A. Chepok , L. Jacak , W. Donderowicz , D. Z. Hu , and D. M. Schaadt Institute of Physics, Wroc law University of Technology,Wyb. Wyspia´nskiego 27, 50-370, Wroc law, Poland; Theor. Phys. Group., International University, Fontanskaya Doroga 33, Odessa, Ukraine; Institute of Applied Physics/DFG-Center for Functional Nanostructures, University of Karlsruhe, Karlsruhe, Germany
The random-phase-approximation semiclassical scheme for description of plasmon excitations inlarge metallic nanospheres (with radius 10–100 nm) is developed for a case of presence of dynamicalelectric field. The spectrum of plasmons in metallic nanosphere is determined including both surfaceand volume type excitations and their mutual connections. It is demonstrated that only surfaceplasmons of dipole type can be excited by a homogeneous dynamical electric field. The Lorentzfriction due to irradiation of e-m energy by plasmon oscillations is analysed with respect to thesphere dimension. The resulting shift of resonance frequency due to plasmon damping is comparedwith experimental data for various sphere radii. Collective of wave-type oscillations of surfaceplasmons in long chains of metallic spheres are described. The undamped region of propagation ofplasmon waves along the chain is found in agreement with some previous numerical simulations.
I. INTRODUCTION
Experimental and theoretical invesitigations of plasmon excitations in metallic nanocrystals rapidly grew mainlydue to perspectives of possible applications in photovoltaics and microelectronics. A significant enhancement ofabsorption of the incident light in photodiode-systems with active surface covered with nano-dimension metallicparticles (of Au, Ag or Cu) with planar density ∼ /cm was observed . This is due to a mediating role in lightenergy transport played by surface plasmon oscillations in metallic nano-compounds on semiconductor surface. Thesefindings are of practical importance towards enhancement of solar cell efficiency especially for thin film cell technology.Hybridized states of the surface plasmons and photons result in plasmon-polaritons which are of high importancefor applications in photonics and microelectronics , in particular for transportation of energy in metallic modifiedstructures in nano-scale .Surface plasmons in nanoparticles were widely investigated since their classical description by Mie . Many partic-ular studies, including numerical modelling of multi-electron clusters, have been carried out . They were develop-ments of Kohn-Sham attitude in form of LDA (Local Density Approach) or TDLDA (Time Dependent LDA) ? addressed, however, to small metallic clusters, up to ca. 200 electrons (as limetted by severe numerical constraints).The random phase approximation (RPA) was formulated for description of volume plasmons in bulk metals andutilised also for confined geometry mainly in numerical or semi-numerical manner . Usually, in these analyses the jellium model was assumed for description of positive ion background in metal and dynamics was addressed to elec-tron system only , and such an attitude is preferable for clusters of simple metals, including noble metals (alsotransition and alkali ones).In the present paper we generalise the bulk RPA description , using semiclassical approach, for a large metallicnanosphere (with radius of several tens nm, and with 10 − − electrons) in an all analytical calculus version .The plasmon oscillations of compresional and traslational type, resulting in excitations inside the sphere and on itssurface, respectively, are analysed and referred to volume and surface plasmons. Damping effects of plasmons viaelectron scattering processes and radiation losses are included, the latter ones, via Lorentz friction force. The shiftof the resonance frequency of dipole-type surface plasmons (only plasmons induced by homogeneous time-dependentelectric field), 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 analysed and wave-type plasmon modes are described. A coupling in near field regime between oscillating dipoles of surface plasmonstogether with retardation effects of energy irradiation lead to a possibility of undamped propagation of plasmon wavesalong the chain in the experimentally realistic region of parameters (of separation of spheres in the chain and theirradii). These effect would be of particular significance for by plasmon arranged transport of energy along metallicchains for application in nanoelectronics.The paper is organised as follows. In the next section the standard RPA theory in quasiclassical limit, is generalisedfor the confined system of spherical shape. The resulting equations for volume and surface plasmons are solved inthe following section (with particularities of calculus shifted to the Appendix). The next section contains descriptionof the Lorentz friction for surface plasmons oscillations of the dipole-type. The analysis of the collective wave-typesurface plasmon oscillations in the chain of metallic nanospheres is presented in the last section. Besides the theoreticalmodel the comparison of the characteristic nano-scale plasmon behaviour with available experimental data, includingown measurements, is presented. II. RPA APPROACH TO ELECTRON EXCITATIONS IN METALLIC NANOSPHEREA. Derivation of RPA equation for local electron density in a confined spherical geometry
Let us consider a metallic sphere with a radius a located in the vacuum, ε = 1 , µ = 1 and in the presence ofdynamical electric field (magnetic field is assumed to be zero). We will consider collective electrons in the metallicmaterial. The model jellium is assumed in order to account for screening background of positive ions in the formof static uniformly distributed over the sphere positive charge: n e ( r ) = n e Θ( a − r ) , (1)where n e = N e /V and n e | e | is the averaged positive charge density, N e the number of collective electrons in the sphere, V = πa the sphere volume, and Θ is the Heaviside step-function. Neglecting the ion dynamics within jellium model,which is adopted in particular for description of simple metals, e.g. noble, transition and alkali metals, we deal withthe Hamiltonian for collective electrons,ˆ H e = N e X j =1 " − ¯ h ∇ j m − e Z n e ( r ) d r | r j − r | + eϕ ( R + r j , t ) + 12 X j = j ′ e | r j − r j ′ | + ∆ E, (2)where r j and m are the position (with respect to the dot center) and the mass of the j th electron, R is the position ofthe metallic sphere center, ∆ E represents electrostatic energy contribution from the ion ’jellium’, ϕ ( r , t ) is the scalarpotential of the external electric field. The corresponding electric field E ( R + r j , t ) = − grad j ϕ ( R + r j , t ). Assumingthat space-dependent variation of E is weak on the scale of the sphere radius a then E ( R + r j , t ) ≃ E ( R , t ), i.e theelectric field is homogeneous over the sphere (it holds for | R | ≫ a ). Then ϕ ( R + r j , t ) ≃ − E ( R , t ) · R + ϕ ( R + r j , t ),where ϕ ( R + r j , t ) = − r j · E ( R , t ). Hence, one can rewrite the Hamiltonian (2) in the form:ˆ H e = ˆ H ′ e − eN E ( R , t ) · R , (3)where ˆ H ′ e = N e X j =1 " − ¯ h ∇ j m − e Z n e ( r ) d r | r j − r | + eϕ ( r j , t ) + 12 X j = j ′ e | r j − r j ′ | + ∆ E, (4)the corresponding wave function can be represented as,Ψ( r e , t ) = Ψ ′ ( r e , t ) e i eN ¯ h R E ( R ,t ) · R dt (5)with i ¯ h ∂ Ψ ′ ∂t = ˆ H ′ e Ψ ′ , r e = ( r , r , ..., r N ).A local electron density can be written as follows : ρ ( r , t ) = < Ψ( r e , t ) | X j δ ( r − r j ) | Ψ( r e , t ) > = < Ψ ′ ( r e , t ) | X j δ ( r − r j ) | Ψ ′ ( r e , t ) >, (6)with the Fourier picture: ˜ ρ ( k , t ) = Z ρ ( r , t ) e − i k · r d r = < Ψ ′ ( r e , t ) | ˆ ρ ( k ) | Ψ ′ ( r e , t ) >, (7)where the ’operator’ ˆ ρ ( k ) = P j e − i k · r j .Using the above notation one can rewrite ˆ H ′ e in the following form, in analogy to the bulk case :ˆ H ′ e = N e P j =1 h − ¯ h ∇ j m i − e π R d k ˜ n e ( k ) k (cid:16) ˆ ρ + ( k ) + ˆ ρ ( k ) (cid:17) + e π R d k ˜ ϕ ( k , t ) (cid:16) ˆ ρ + ( k ) + ˆ ρ ( k ) (cid:17) + e π R d k k h ˆ ρ + ( k )ˆ ρ ( k ) − N e i + ∆ E, (8)where: ˜ n e ( k ) = R d rn e ( r ) e − i k · r , πk = R d r r e − i k · r , ˜ ϕ ( k ) = R d rϕ ( r , t ) e − i k · r ,t .Utilizing this form of the electron Hamiltonian one can write the secod time-derivative of ˆ ρ ( k ): d ˆ ρ ( k , t ) dt = 1( i ¯ h ) hh ˆ ρ ( k ) , ˆ H ′ e i , ˆ H ′ e i , (9)which resolves itself into the equation: d δ ˆ ρ ( k ,t ) dt = − P j e − i k · r j n − ¯ h m ( k · ∇ j ) + ¯ h k m i k · ∇ j + ¯ h k m o − e m π R d q ˜ n e ( k − q ) k · q q δ ˆ ρ ( q ) − em π R d q ˜ n e ( k − q )( k · q ) ˜ ϕ ( q , t ) − em π R d qδ ˆ ρ ( k − q )( k · q ) ˜ ϕ ( q , t ) − e m π R d qδ ˆ ρ ( k − q ) k · q q δ ˆ ρ ( q ) , (10)where δ ˆ ρ ( k ) = ˆ ρ ( k ) − ˜ n e ( k ) is the ’operator’ of local electron density fluctuations beyond the uniform distribution.Taking into account that: δ ˜ ρ ( k , t ) = < Ψ ′ ( t ) | δ ˆ ρ ( k ) | Ψ ′ ( t ) > = ˜ ρ ( k , t ) − ˜ n e ( k ) we find: ∂ δ ˜ ρ ( k ,t ) ∂t = < Ψ ′ | − P j e − i k · r j n − ¯ h m ( k · ∇ j ) + ¯ h k m i k · ∇ j + ¯ h k m o | Ψ ′ > − e m π R d q ˜ n e ( k − q ) k · q q δ ˜ ρ ( q , t ) − em π R d q ˜ n e ( k − q )( k · q ) ˜ ϕ ( q , t ) − em π R d qδ ˜ ρ ( k − q , t )( k · q ) ˜ ϕ ( q , t ) − e m π R d q k · q q < Ψ ′ | δ ˆ ρ ( k − q ) δ ˆ ρ ( q ) | Ψ ′ >, (11)One can simplify the above equation upon the assumption that δρ ( r , t ) = π R e i k · r δ ˜ ρ ( k , t ) d k only weakly varies onthe interatomic scale, and hence three components of the first term in right-hand-side of Eq. (11) can be estimated as: k v F δ ˜ ρ ( k ), k v F /k T δ ˜ ρ ( k ) and k v F /k T δ ˜ ρ ( k ), respectively, with 1 /k T the Thomas-Fermi radius , k T = q πn e e ǫ F , ǫ F the Fermi energy, and v F the Fermi velocity. Thus the contribution of the second and the third componentsof the first term can be neglected in comparison to the first component. Small and thus negligible is also the lastterm in right-hand-side of Eq. (11), as it involves a product of two δ ˜ ρ (which we assumed small δ ˜ ρ/n e << (note that δ ˆ ρ (0) = 0 and the coherent RPA contribution of interaction is comprised by the second term in Eq. (11)). The lastbut one term in Eq. (11) can also be omitted if one confines it to linear terms with respect to δ ˜ ρ and ˜ ϕ . Next, dueto spherical symmetry, < Ψ ′ | P j e − i k · r j ¯ h m ( k · ∇ j ) | Ψ ′ > ≃ k m < Ψ ′ | P j e − i k · r j ¯ h ∇ j m | Ψ ′ > . Performing the inverseFourier transform, Eq. (11) attains finally the form: ∂ δρ ( r ,t ) ∂,t = − m ∇ < Ψ ′ | P j δ ( r − r j ) ¯ h ∇ j m | Ψ ′ > + ω p π ∇ n Θ( a − r ) ∇ R d r | r − r | δρ ( r , t ) o + en e m ∇ { Θ( a − r ) ∇ ϕ ( r , t ) } . (12)According to Thomas-Fermi approximation the RPA averaged kinetic energy can be represented as follows: < Ψ ′ | − X j δ ( r − r j ) ¯ h ∇ j m | Ψ ′ > ≃
35 (3 π ) / ¯ h m ρ / ( r , t ) = 35 (3 π ) / ¯ h m n / e Θ( a − r ) (cid:20) δρ ( r , t ) n e + ... (cid:21) . (13)Taking then into account the above approximation and that ∇ Θ( a − r ) = − r r δ ( a − r ) = − r r lim ǫ → δ ( a + ǫ − r ) aswell as that ϕ ( R , r , t ) = − r · E ( R , t ), one can rewrite Eq. (12) in the following manner: ∂ δρ ( r ) ∂t = (cid:2) ǫ F m ∇ δρ ( r , t ) − ω p δρ ( r , t ) (cid:3) Θ( a − r ) − m ∇ (cid:8)(cid:2) ǫ F n e + ǫ F δρ ( r , t ) (cid:3) r r δ ( a + ǫ − r ) (cid:9) − h ǫ F m r r ∇ δρ ( r , t ) + ω p π r r ∇ R d r | r − r | δρ ( r , t ) + en e m r r · E ( R , t ) i δ ( a + ǫ − r ) . (14)In the above formula ω p is the bulk plasmon frequency, ω p = πn e e m , and δ ( a + ǫ − r ) = lim ǫ → δ ( a + ǫ − r ). Thesolution of Eq. (14) can be decomposed into two parts related to the domain: δρ ( r , t ) = (cid:26) δρ ( r , t ) , f or r < a,δρ ( r , t ) , f or r ≥ a, ( r → a +) , (15)corresponding to the volume and surface excitations, respectively. These two parts of local electron density fluctuationssatisfy the equations: ∂ δρ ( r , t ) ∂t = 23 ǫ F m ∇ δρ ( r , t ) − ω p δρ ( r , t ) , (16)and ∂ δρ ( r ,t ) ∂t = − m ∇ (cid:8)(cid:2) ǫ F n e + ǫ F δρ ( r , t ) (cid:3) r r δ ( a + ǫ − r ) (cid:9) − h ǫ F m r r ∇ δρ ( r , t ) + ω p π r r ∇ R d r | r − r | ( δρ ( r , t )Θ( a − r ) + δρ ( r , t )Θ( r − a )) + en e m r r · E ( R , t ) i δ ( a + ǫ − r ) . (17)It is clear from Eq. (16) that the volume plasmons are independent of surface plasmons. However, surface plasmonscan be excited by volume plasmons due to the last term in Eq. (17), which expresses a coupling between surface andvolume plasmons in the metallic nanosphere within RPA semiclassical picture. It is in fact a surface tail of volumecompressional-type excitations, while surface traslational-type exctations have no a volume tail.In a dielectric medium in which the metallic sphere can be embedded, the electrons on the surface interact withforces ε (dielectric susceptibility constant) times weaker in comparison to electrons inside the sphere. To account forit, one substitutes Eqs (16) and (17) with the following ones: ∂ δρ ( r , t ) ∂t = 23 ǫ F m ∇ δρ ( r , t ) − ω p δρ ( r , t ) , (18)and ∂ δρ ( r ,t ) ∂t = − m ∇ (cid:8)(cid:2) ǫ F n e + ǫ F δρ ( r , t ) (cid:3) r r δ ( a + ǫ − r ) (cid:9) − h ǫ F m r r ∇ δρ ( r , t ) + ω p π r r ∇ R d r | r − r | (cid:0) δρ ( r , t )Θ( a − r ) + ε δρ ( r , t )Θ( r − a ) (cid:1) + en e m r r · E ( R , t ) i δ ( a + ǫ − r ) . (19)Let us also assume that both volume and surface plasmon oscillations are damped with the time ratio τ which canbe phenomenologically accounted for via the additional term, − τ ∂δρ ( r ,t ) ∂t , to the right-hand-side of above equations.They attain the form: ∂ δρ ( r , t ) ∂t + 2 τ ∂δρ ( r , t ) ∂t = 23 ǫ F m ∇ δ ˜ ρ ( r , t ) − ω p δρ ( r , t ) , (20)and ∂ δρ ( r ,t ) ∂t + τ ∂δρ ( r ,t ) ∂t = − m ∇ (cid:8)(cid:2) ǫ F n e + ǫ F δρ ( r , t ) (cid:3) r r δ ( a + ǫ − r ) (cid:9) − h ǫ F m r r ∇ δ ˜ ρ ( r , t ) + ω p π r r ∇ R d r | r − r | (cid:0) δρ ( r , t )Θ( a − r ) + ε δρ ( r , t )Θ( r − a ) (cid:1) + en e m r r · E ( R , t ) i δ ( a + ǫ − r ) . (21)From Eqs (20) and (21) it is noticeable that the homogeneous electric field does not excite the volume-type plasmonoscillations but only contributes to surface plasmons. B. Solution of RPA equations: volume and surface plasmons frequencies
Eqs (20) and (21) can be solved upon imposing the boundary and symmetry conditions—cf. Appendix A. Let uswrite the both parts of the electron fluctuation in the following manner: δρ ( r , t ) = n e [ f ( r ) + F ( r , t )] , f or r < a,δρ ( r , t ) = n e f ( r ) + σ (Ω , t ) δ ( r + ǫ − a ) , f or r ≥ a, ( r → a +) , (22)and let us choose the convenient initial conditions F ( r , t ) | t =0 = 0 , σ (Ω , t ) | t =0 = 0, (Ω = ( θ, ψ )—the spherical angles),moreover (1 + f ( r )) | r = a = f ( r ) | r = a (continuity condition), F ( r , t ) | r = a = 0, R ρ ( r , t ) d r = N e (neutrality condition).We thus arrive at the explicit form of the solutions of Eqs (20) and (21) (as it is described in the Appendix A): f ( r ) = − k T a +12 e − k T ( a − r ) 1 − e − kT r k T r , f or r < a,f ( r ) = (cid:2) k T a − k T a +12 (cid:0) − e − k T a (cid:1)(cid:3) e − kT ( r − a ) k T r , f or r ≥ a, (23)where k T = q πn e e ǫ F = r ω p v F . For time-dependent parts of electron fluctuations we find: F ( r , t ) = ∞ X l =1 l X m = − l ∞ X n =1 A lmn j l ( k nl r ) Y lm (Ω) sin ( ω ′ nl t ) e − t/τ , (24)and σ (Ω , t ) = ∞ P l =1 l P m = − l Y lm (Ω) (cid:2) B lm a sin ( ω ′ l t ) e − t/τ (1 − δ l ) + Q m ( t ) δ l (cid:3) + ∞ P l =1 l P m = − l ∞ P n =1 A lmn ( l +1) ω p lω p − (2 l +1) ω nl Y lm (Ω) n e a R dr r l +21 a l +2 j l ( k nl r ) sin ( ω ′ nl t ) e − t/τ , (25)where j l ( ξ ) = q π ξ I l +1 / ( ξ ) is the spherical Bessel function, Y lm (Ω) is the spherical function, ω nl = ω p r x nl k T a are the frequencies of electron volume free self-oscillations (volume plasmon frequencies), x nl are nodes of the Besselfunction j l ( ξ ), ω l = ω p q lε (2 l +1) are the frequencies of electron surface free self-oscillations (surface plasmon frequen-cies), and k nl = x nl /a ; ω ′ = q ω − τ are the shifted frequencies for all modes due to damping. The coefficients B lm and A lmn are determined by the initial conditions. As we have assumed that δρ ( r , t = 0) = 0, we get B lm = 0 and A lmn = 0, except for l = 1 in the former case (of B lm ) which corresponds to homogeneous electric field excitation.This is described by the function Q m ( t ) in the general solution (25). The function Q m ( t ) satisfies the equation: ∂ Q m ( t ) ∂t + τ ∂Q m ( t ) ∂t + ω Q m ( t )= q π en e m (cid:2) E z ( R , t ) δ m + √ E x ( R , t ) δ m + E y ( R , t ) δ m − ) (cid:3) , (26)where ω = ω = ω p √ ε (it is a dipole-type surface plasmon Mie frequency ). Only this function contributes thedynamical response to the homogeneous electric field (for the assumed initial conditions). From the above it followsthus that local electron density (within RPA attitude) has the form: ρ ( r , t ) = ρ ( r ) + ρ ( r , t ) , (27)where the RPA equilibrium electron distribution (correcting the uniform distribution n e ): ρ ( r ) = (cid:26) n e [1 + f ( r )] , f or r < a,n e f ( r ) , f or r ≥ a, r → a + (28)and the nonequilibrium, of surface plasmon oscillation type for the homogeneous forcing field: ρ ( r , t ) = , f or r < a, P m = − Q m ( t ) Y m (Ω) f or r ≥ a, r → a + . (29)In general, F ( r , t ) (volume plasmons) and σ (Ω , t ) (surface plasmons) contribute to plasmon oscillations. However, inthe case homogeneous perturbation, only the surface l = 1 mode is excited.For plasmon oscillations given by Eq. (29) one can calculate the corresponding dipole, D ( R , t ) = e Z d r r ρ ( r , t ) = 4 π e q ( R , t ) a , (30)where Q ( R , t ) = q π q x ( R , t ), Q − ( R , t ) = q π q y ( R , t ), Q ( R , t ) = q π q x ( R , t ) and q ( R , t ) satisfies theequation (cf. Eq. (26)), (cid:20) ∂ ∂t + 2 τ ∂∂t + ω (cid:21) q ( R , t ) = en e m E ( R , t ) . (31) III. LORENTZ FRICTION FOR NANOSPHERE PLASMONS
Considering the nanosphere plasmons induced by the homogeneous electric field, as described in the above para-graph, one can note that these plasmons are themselves a source of the e-m radiation. This radiation takes awaythe energy of plasmons resulting in their damping, which can be described as the Lorentz friction . This e-m waveemission causes electron friction which can be described as the additional electric field , E L = 23 εv ∂ D ( t ) ∂t , (32)where v = c √ ε is the light velocity in the dielectric medium, and D ( t ) the dipole of the nanosphere. According to Eq.(30) we arrive at the following: E L = 2 e εv π a ∂ q ( t ) ∂t . (33)Substituting it into Eq. (31) we get (cid:20) ∂ ∂t + 2 τ ∂∂t + ω (cid:21) q ( R , t ) = en e m E ( R , t ) + 23 ω (cid:16) ω av (cid:17) ∂ q ( t ) ∂t . (34)If rewrite the above equation (for E =0) in the form (cid:20) ∂ ∂t + ω (cid:21) q ( R , t ) = ∂∂t (cid:20) − τ + 23 ω (cid:16) ω av (cid:17) ∂ q ( t ) ∂t (cid:21) , (35)thus the zeroth order approximation (neglecting attenuation) corresponds to the equation; (cid:20) ∂ ∂t + ω (cid:21) q ( R , t ) = 0 . (36)In order to solve Eq. (35) in the next step of perturbation, in the right-hand-side of this equation one can subsitute ∂ q ( t ) ∂t by − ω q ( t ) (acc. to Eq. (36)).Therefore, if one assumes the above estimation, ∂ q ( t ) ∂t ≃ − ω ∂ q ( t ) ∂t , then one can include the Lorentz friction in arenormalised damping term: (cid:20) ∂ ∂t + 2 τ ∂∂t + ω (cid:21) q ( R , t ) = en e m E ( R , t ) , (37)where 1 τ = 1 τ + ω (cid:16) ω av (cid:17) ≃ v F λ B + Cv F a + ω (cid:16) ω av (cid:17) , (38)where we used for τ ≃ v F λ B + Cv F a ( λ B is the free path in bulk, v F the Fermi velocity, and C ≃ which corresponds to inclusion of plasmon damping due to electron scattering on other electrons and on nanoparticleboundary. The renormalised damping causes the change in the shift of self-frequencies of free surface plasmons, ω ′ = q ω − τ .Using Eq. (38) one can determine the radius a corresponding to minimal damping, a = √ ω p (cid:0) v F c √ ε/ (cid:1) / . (39)For nanoparticles of gold, silver and copper in air, in water and in a colloidal solution, one can find a ≤ . For a > a damping increases due to Lorentzfriction (proportional to a ) but for a < a damping due to electron scattering dominates and causes also dampingenhancement (with lowering a , as ∼ a , cf. Fig. 1),which agrees with experimental observations . FIG. 1: Effective damping ratio for surface plasmon oscillations, Eqs (37), (38), the upper (blue) curve is the sum of bothterms, ∼ a (red) and ∼ a (green); the minimum corresponds to minimal damping for radius a , Eq. (39), left—for Ag in theair, right—for Au in colloidal water solution Tab. 1. a —nanosphere radius corresponding to minimal dampingrefraction rate of the surrounding medium, n Au, a [nm] Ag, a [nm] Cu, a [nm](air) 1 8.8 8.44 8.46(water) 1.4 9.14 9.18 9.20(colloidal solution) 2 9.99 10.04 10.04Surface plasmon oscillations cause attenuation of the incident e-m radiation where the maximum of attenuation isat the resonant frequency ω = q ω − τ . This frequency diminishes with rise of a , for a > a according to Eq.(38), which agrees with experimental observations for Au and Ag presented in Fig. 2, and Tab. 2 (Au) and Tab 3(Ag). Tab. 2. Resonant frequency for e-m wave attenuation in Au nanospheresradius of nanosheres [nm] 10 15 20 25 30 40 50¯ hω ′ (experiment) [eV] 2.371 2.362 2.357 2.340 2.316 2.248 2.172¯ hω ′ (theory) [eV], n = 1 . hω ′ (theory) [eV], n = 2 2.604 2.603 2.600 2.590 2.565 2.388 1.656Tab. 3. Resonant frequency for e-m wave attenuation in Ag nanospheresradius of nanosheres [nm] 10 20 30 40¯ hω ′ (experiment) [eV] 3.024 2.911 2.633 2.385¯ hω ′ (theory) [eV], n = 1 . hω ′ (theory) [eV], n = 2 2.595 2.591 2.557 2.384 IV. PLASMON-MEDIATED ENERGY TRANSFER THROUGH A CHAIN OF METALLICNANOSPHERES
Let us consider a linear chain of metallic nanospheres with radii a in a dielectric medium with dielectric constant ε .We assume that spheres are located along z -axis direction equidistantly with separation of sphere centers d > a .At time t = 0 we assume the excitation of plasmon oscillation via a Dirac delta ∼ δ ( t ) shape signal of electric field.Taking into account the mutual interaction of induced surface plasmons on the spheres via the radiation of dipoleoscillations, we aim to determine the stationary state of the whole infinite chain. For separation d much shorter thanthe wavelength λ of the e-m wave corresponding to surface plasmon self-frequency, the dipole type plasmon radiationcan be treated within near-field regime, at least for nearest neighbouring spheres. In the near-field region a < R < λ ,the radiation of the dipole D ( t ) is not a planar wave (as for far-field region, R ≫ λ ) but of only electric field type FIG. 2: Extinction spectra for nanospheres of Au (a) and Ag (b) in colloidal water solution for various sphere radii in retarded form (without magnetic field) : E ( R , R , t ) = 1 εR (cid:20) n (cid:18) n · D (cid:18) R , t − R v (cid:19)(cid:19) − D (cid:18) R , t − R v (cid:19)(cid:21) , (40) R —position of the sphere (center) irradiating e-m energy due to its dipole surface plasmon oscillations, R positionof another sphere (center), with respect to the center of the former one, where the field E ( R , R , t ) is given by theabove formula, R < λ , n = R /R , v = c/ √ ε = c/n .When both vectors R and R are along the z -axis (the linear chain) the above equation can be resolved as: E α ( R , R , t ) = σ α εR D α (cid:18) R , t − R v (cid:19) , (41)where α = ( x, y, z ), σ x = σ y = − σ z = 2. Assuming that the z -axis origin coincides with the center of onesphere in the chain, for the l th sphere located in the point R l = (0 , , ld ), an electric field caused by neighbouringspheres, E ( R m , R ml , t ), and the Lorentz friction force caused by self-radiation, E L ( R l , t ), has to be considered. Byvirtue of Eq. (31) the equation for surface plasmon oscillation of the l th sphere is (cid:20) ∂ ∂t + 2 τ ∂∂t + ω (cid:21) q ( R l , t ) = en e m m = ∞ X m = −∞ , m = l, R ml <λ E ( R m , R ml , t ) + en e m E L ( R l , t ) , (42)provided that the dipole field of the m th sphere can be treated as homogeneous over the l th sphere and the sum over m is confined by the distance of m th sphere from l th sphere not exceeding the near-field range ( ∼ λ ). In the case ofthe equidistant chain, R l = ld and R ml = | l − m | d , and using Eqs (41), (30) and 33), one can rewrite Eq. (42) in theform: (cid:20) ∂ ∂t + 2 τ ∂∂t − ω (cid:16) ω av (cid:17) ∂ ∂t + ω (cid:21) q α ( ld, t ) = σ α a d m = ∞ X m = −∞ , m = l, | l − m | d<λ q α (cid:0) md, t − dv | l − m | (cid:1) | l − m | , (43)here α = x, y , which describe the transversal plasmon modes and α = z , which describes the longitudinal one. Theabove equation coincides with the appropriate one from Refs [23,25], if one assumes that π a n e = N = 1 and neglectsthe retardation of the field.Taking into account the periodicity of the infinite chain, one can consider the solution of the above equation in theform q ( ld, t ) = ˜ q ( k, t ) e − ikld . (44)The right-hand-side term in Eq. (43) attains the form m = ∞ P m = −∞ , m = l q α ( md,t − dv | l − m | ) | l − m | = l − P m = −∞ q α ( md,t − dv | l − m | ) | l − m | + m = ∞ P m = l +1 q α ( md,t − dv | l − m | ) | l − m | = 2 e − ikld ∞ P m =1 cos ( mkd ) m ˜ q ( k, t − md/v ) . Thus the Eq. (43) can be written as follows: (cid:20) ∂ ∂t + 2 τ ∂∂t − ω (cid:16) ω av (cid:17) ∂ ∂t + ω (cid:21) ˜ q α ( k, t ) = σ α ω a d ∞ X m =1 , md<λ cos ( mkd ) m ˜ q ( k, t − md/v ) (45)This equation is linear and therefore we look for the solutions of the shape: ˜ q α ( k, t ) = ˜ Q α ( k ) e iω α t , and − ω α + 2 iω α τ α ( ω α ) + ˜ ω α ( ω α ) = 0 , (46)where ˜ ω α ( ω α ) = ω − σ α a d ∞ X m =1 , md<λ cos ( mkd ) m cos (cid:18) ω α mdv (cid:19) (47)and 1 τ α ( ω α ) = 1 τ + ω a v (cid:16) ω α av (cid:17) + σ α ω a d ∞ X m =1 , md<λ cos ( mkd ) m sin (cid:0) ω α mdv (cid:1) ω α . (48)If we confine the sum in the Eq. (47) to m = 1 (the nearest neighbour approximation) we get˜ ω α ( ω α ) ≃ ω (cid:20) − σ α a d cos ( kd ) cos (cid:18) ω α dv (cid:19)(cid:21) (49)and from Eq. (48), 1 τ α ( ω α ) = 1 τ + ω a vd "(cid:18) ω α dv (cid:19) − ( kd − π ) + π , f or α = x, y (50)and 1 τ z ( ω z ) = 1 τ + ω a vd "(cid:18) ω z dv (cid:19) + ( kd − π ) − π , f or α = z. (51)0In the derivation of two above formulae the following summation was performed : ω α ∞ P m =1 cos ( mkd ) m sin (cid:0) ω α mdv (cid:1) = ω α ∞ P m =1 1 m [ sin ( kmd + ω α md/v ) − sin ( kmd − ω α md/v )]= dv h π − π kd + k d + ω α d v i , as the terms in the sum drop quickly to zero then the above formula well approximates the sum with limitation md < λ .Assuming now ω α = ω ′ α + iω ′′ α the Eq. (46) gives the dependence of ω ′ α and ω ′′ α on k . The general solution of Eq.(43) attains the form, q α ( ld, t ) = N s X n =1 ˜ Q α ( k n ) e i ( ω ′ α ( k n ) t − k n ld ) − ω ′′ α ( k n ) t , (52)where k n = πnN s d , L = N s d is assumed length of the chain with N s spheres, and periodic (of Born-Karman type)boundary condition imposed. The components of Eq. (52) describe monochromatic waves with wavelength λ n = πk n = Ln , which are analogous to planar waves in crystals, when damping is not big, i.e., when ω ′′ α ≪ ω ′ α . Providedthis inequality one can approximate:for transversal modes ( α = x, y )( ω ′ α ) = (˜ ω α ) = ω (cid:20) a d cos ( kd ) cos ( ω ′ α d/v ) (cid:21) , (53) ω ′′ α = 1 τ α = 1 τ + ω a vd ω ′ α dv ! − ( kd − π ) + π , (54)and for longitudinal mode ( α = z )( ω ′ z ) = (˜ ω z ) = ω (cid:20) − a d cos ( kd ) cos ( ω ′ z d/v ) (cid:21) , (55) ω ′′ z = 1 τ z = 1 τ + ω a vd ω ′ α dv ! + ( kd − π ) − π . (56)From the Eqs (54) and (56) it follows that ω ′′ α can change its sign. In the case of ω ′′ α < includes also a small nonlinear term with respect to D , aside from the term with ∂ D∂t . Including of it will result in damping of too highly rising oscillations leading to stable amplitude of oscillations.Due to this stabilisation caused by nonlinear effects, undamped wave modes of dipole oscillations will propagate inthe chain in the region of parameters where ω ′′ α ≤ ω ′′ α = τ α = 0, for critical parameters, resolves into: (cid:18) ω α dv (cid:19) = ( kd − π ) − π − vd τ ω a , (57)for α = ( x, y ) and for α = z , (cid:18) ω z dv (cid:19) = − ( kd − π ) + π − vd τ ω a . (58)Obtained from the above equations (cid:0) ω α dv (cid:1) leads to determination of the dependence of wave vector k with respect toparameters d and a , via Eqs (53)-(56). Solution for this equations, found numerically for the chain of Ag nanospheres,is depicted in the Fig. 3.The undamped plasmon waves in the chain appear if d < d max and have k = π/d , d max = 98 . .
8) nm fortransversal(longitudinal) modes. For example, for Ag spheres with radius a = 20nm and separation d = 60 nm, theundamped transversal modes appear for 0 ≤ kd ≤ π/ π/ ≤ kd ≤ π and longitudinal for 3 π/ ≤ kd ≤ π/ .1 FIG. 3: Wave vector kd versus sphere separation d , at constant d/a = 3 ( a —sphere radius) (upper), for a = 20 nm (lower) forAg sphere chain, transversal modes—red, longitudinal mode—blue V. CONCLUSIONS
In the present paper we analysed plasmons in large metallic nanospheres induced by homogeneous time-dependentelectric field. Within all-analytical RPA quasiclassical approach the volume and surface plasmons are described and aproof that only dipole-type of surface plasmons can be induced by a homogeneous field (while none of volume modes)is given. An irradiation of energy by plasmon oscillations is described within the Lorentz friction effect. Its scalingwith the nanosphere dimension leads to sphere radius dependent shift of resonant frequency, similarly as observed inexperiments. The description of surface dipole-type plasmon oscillations in single nanospheres is applied to analysis ofcollective oscillation in linear chain of metallic nanospheres. The wave-type collective plasmon oscillations in the chainare also considered. The undamped region of wave energy transport through the chain is found for a certain sphereseparation in the chain with corresponding appropriate wavelength of plasmon waves. This phenomenon confirms asimilar behaviour observed by numerical simulations . Acknowledgments
Supported by the Polish KBN Project No: N N202 260734 and the FNP Fellowship Start (W. J.), as well as DFGgrant SCHA 1576/1-1 (D. S. and D. Z. Hu)
Appendix A: Analytical solution of plasmon equations for the nanosphere
Let us solve first the Eq. (20), assuming a solution in the form: δρ ( r , t ) = n e [ f ( r ) + F ( r , t )] , f or r < a, (A1)2Eq. (20) resolves thus into: ∇ f ( r ) − k T f ( r ) = 0 , ∂ F ( r ,t ) ∂t + τ ∂F ( r ,t ) ∂t = v F ∇ F ( r , t ) − ω p F ( r , t ) , (A2)The solution for function f ( r ) (nonsingular at r = 0) has thus the form: f ( r ) = α e − k T a k T r (cid:0) e − k T r − e k T r (cid:1) , (A3)where α —const., k T = q πn e e ǫ F = r ω p v F ( k T —inverse Thomas-Fermi radius), ω p = q πn e e m (bulk plasmonfrequency).Since we assume F ( r ,
0) = 0, then for function F ( r , t ) the solution can be taken as, F ( r , t ) = F ω ( r ) sin ( ω ′ t ) e − τ t (A4)where ω ′ = p ω + 1 /τ . F ω ( r ) satisfies the equation (Helmholtz equation): ∇ F ω ( r ) + k F ω ( r ) = 0 , (A5)with k = ω − ω p v F / . A solution of the above equation, nonsingular at r = 0, is as follows: F ω ( r ) = Aj l ( kr ) Y lm (Ω) , (A6)where A —constant, j l ( ξ ) = p π/ (2 ξ ) I l +1 / ( ξ )—the spherical Bessel function [ I n ( ξ )—the Bessel function of the firstorder], Y lm (Ω)—the spherical function (Ω—the spherical angle). Owing to the boundary condition, F ( r , t ) | r = a = 0,one has to demand j l ( ka ) = 0, which leads to the discrete values of k = k nl = x nl /a , (where x nl , n = 1 , , ... , arenodes of j l ), and next to the discretisation of self-frequencies ω : ω nl = ω p (cid:18) x nl k T a (cid:19) . (A7)The general solution for F ( r , t ) has thus the form F ( r , t ) = ∞ X l =0 l X m = − l ∞ X n =1 A lmn j l ( k nl r ) Y lm (Ω) sin ( ω ′ nl t ) e − τ t . (A8)A solution of Eq. (21) we represent as: δρ ( r , t ) = n e f ( r ) + σ (Ω , t ) δ ( r + ǫ − a ) , f or r ≥ a, ( r → a + , i.e.ǫ → . (A9)The neutrality condition, R ρ ( r , t ) d r = N e , with δρ ( r , t ) = σ ( ω, t ) δ ( a + ǫ − r ) + n e f ( r ) , ( ǫ → − a R drr f ( r ) = ∞ R a drr f ( r ), a R d rF ( r , t ) = 0, R d Ω σ (Ω , t ) = 0. Taking into account also the continuitycondition on the spherical particle surface, 1 + f ( a ) = f ( a ), one can obtain: f ( r ) = βe − k T ( r − a ) / ( k T r ) and it ispossible to fit α (cf. Eq. (A3)) and β constants: α = k T a +12 , β = k T a − k T a +12 (cid:0) − e − k T a (cid:1) —which gives Eqs (23).From the condition a R d rF ( r , t ) = 0 and from Eq. (A8) it follows that A n = 0, (because of R d Ω Y lm ( ω ) =4 πδ l δ m ).To remove the Dirac delta functions we integrate both sides of the Eq. (21) with respect to the radius ( ∞ R r dr... )and then we take the limit to the sphere surface, ǫ →
0. It results in the following equation for surface plasmons: ∂ σ (Ω ,t ) ∂t + τ ∂σ (Ω ,t ) ∂t = − ∞ P l =0 l P m = − l ω l Y lm (Ω) R d Ω σ (Ω , t ) Y ∗ lm (Ω )+ ω p n e ∞ P l =0 l P m = − l ∞ P n =1 A lmn l +12 l +1 Y lm (Ω) a R dr r l +21 a l +2 j l ( k nl r ) sin ( ω nl t ) , + en e m p π/ (cid:2) E z ( R , t ) Y (Ω) + √ E x ( R , t ) Y (Ω) + √ E y ( R , t ) Y − (Ω) (cid:3) , (A10)3where ω l = ω p l l +1 . In derivation of the above equation the following formulae were exploited, (for a < r ): ∂∂a p a + r − ar cosγ = ∂∂a ∞ X l =0 a l r l +11 P l ( cosγ ) = ∞ X l =0 la l − r l +11 P l ( cosγ ) , (A11)where P l ( cosγ ) is the Legendre polynomial [ P l ( cosγ ) = π l +1 l P m = − l Y lm (Ω) Y ∗ lm (Ω )], γ is an angle between vectors a = a ˆ r and r , and (for a > r ): ∂∂a p a + r − ar cosγ = ∂∂a ∞ X l =0 r l a l +1 P l ( cosγ ) = − ∞ X l =0 l X m = − l π l + 12 l + 1 r l a l +2 Y lm (Ω) Y ∗ lm (Ω ) . (A12)Taking into account the spherical symmetry, one can assume the solution of the Eq. (A10) in the form: σ (Ω , t ) = ∞ X l =0 l X m = − l q lm ( t ) Y lm (Ω) . (A13)From the condition R σ ( ωt ) d Ω = 0 it follows that q = 0. Taking into account the initial condition σ ( ω,
0) = 0 weget (for l ≥ q lm ( t ) = B lm a sin ( ω ′ l t ) e − t/τ (1 − δ l ) + Q m ( t ) δ l + ∞ P n =1 A lmn ( l +1) ω p lω p − (2 l +1) ω nl n e a R dr r l +21 a l +2 j l ( k nl r ) sin ( ω ′ nl t ) e − t/τ , (A14)where ω ′ l = p ω l − /τ and Q m ( t ) satisfies the equation: ∂ Q m ( t∂t + 2 τ ∂Q m ( t ) ∂t + ω Q m ( t ) = en e m p π/ h E z ( R , t ) δ m + √ E x ( R , t ) δ m + √ E y ( R , t ) δ m − i . (A15)Thus σ ( ω, t ) attains the form: σ (Ω , t ) = ∞ P l =2 l P m = − l Y lm (Ω) B lm a sin ( ω ′ l t ) e − t/τ + P m = − Q m ( t ) Y m (Ω)+ ∞ P l =1 l P m = − l ∞ P n =1 A nlm ( l +1) ω p lω p − (2 l +1) ω nl Y lm (Ω) n e a R dr r l +21 a l +2 j l ( k nl r ) sin ( ω ′ nl t ) e − t/τ . (A16) S. Pillai, K. B. Catchpole, T. Trupke, G. Zhang, J, Zhao, and M. A. Green, Appl. Phys. Let., , 161102 (2006) M. Westphalen, U. Kreibig, J. Rostalski, H. L¨uth, and D. Meissner, Sol. Energy Mater. Sol. Cells , 97 (2000), M. Gratzel,J. Photochem. Photobiol. C: Photochem. Rev. , 145 (2003) H. R. Stuart and D. G. Hall, Appl. Phys. Lett. , 3815 (1998); H. R. Stuart and D. G. Hall, Phys. Rev. Lett. , 5663(1998); H. R. Stuart and D. G. Hall, Appl. Phys. Lett. , 2327 (1996) D. M. Schaadt, B. Feng, and E. T. Yu, Appl. Phys. Lett. , 063106 (2005) K. Okamoto, et al., Nature Mat. , 661 (2004); K. Okamoto, et al., Appl. Phys. Lett. , 071102 (2005) C. Wen, K. Ishikawa, M. Kishima, K. Yamada, Sol. Cells , 339 (2000) L. Lalanne, J. P. Hugonin, Nature Phys. , 551 (2006) A.V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, Phys. Rep. , 131 (2005) S.A. Mayer,
Plasmonics: Fundamentals and Applications , Springer VL 2007 W. L. Barnes, A. Dereux, and T. W. Ebbesen, Nature , 824 (2003) N. Engeta, A. Salandriw, and A. Alu, Phys. Rev. Lett., , 095504 (2005) S. A. Maier and H. A. Atwater, J. Appl. Phys., , 011101 (2005) G. Mie, Ann. Phys. , 329 (1908) M. Brack, Phys. Rev. B , 3533 (1989) L. Serra et al. , Phys. Rev. B , 3434 (1990) M. Brack, Rev. of Mod. Phys. , 677 (1993); W. Ekardt, Phys. Rev. Lett. , 1925 (1984) C.F. Bohren, D.R. Huffman,
Absorption and Scattering of Light by Small Particles , Wiley, New York (1983); U. Kreibig,M. Vollmer,
Optical Properties of Metal Clusters , Springer, Berlin (1995); J. I. Petrov,
Physics of Small Particles , Nauka,Moscow (1984); C. Burda, X. Chen, R. Narayanan, M. El-Sayed, Chem. Rev. , 1025 (2005) D. Pines,
Elementary Excitations in Solids , ABP Perseus Books, Massachusetts (1999) L. Jacak, J. Krasnyj, A. Chepok, Fizika Niskich Temp. , (2009) D. Pines and D. Bohm, Phys. Rev. , 338 (1952); D. Bohm and D. Pines, Phys. Rev. , 609 (1953) L. D. Landau and E. M. Lifshitz,
Field Theory , Nauka, Moscow (1973) (in Russian) M. L. Brongersma, J. W. Hartman, and H. A. Atwater, Phys. Rev. B , R16356 (2000) U. Kriebig and L. Geinzel, Surf. Sci., 156, 678, (1985) S. A. Maier, P. G. Kik, and H. A. Atwater, Phys. Rev. B, 67, 205402 (2003) F. Stietz, I. Bosbach, T. Wenzel, T. Vartanyan, and A. Goldmann, F. Tr¨ager, Phys. Rev. Lett., 84, 5644 (2000) F. Stietz et al , Phys. Rev. Lett. , 5644 (2000) M. Scharte et all. , Appl. Phys. B: Laser Opt. , 305 (2001) I. S. Gradstein, I. M. Rizik, Tables of Integrals, Fizmatizdat, Moscow (1962). V. A. Markel and A. K. Sarychev, Phys. Rev. B, 75, 085426 (2007) E. Hao, R. C. Bayley, G. C. Schatz, J. T. Hupp, and S. Li, Nano Lett. , 327 (2004) B. Lamprecht, A. Leitner, and F. R. Aussenegg, Appl. Phys. B: Lasers Opt.64