Polarizability of a metallic nanosphere: Local random-phase approximation (LRPA)

Transcription

1 Sri Lankan Journal of Pysics, Vol. 1(1) (01) Institute of Pysics - Sri Lanka Researc Article Polarizability of a metallic nanosere: Local random-ase aroximation (LRPA) Prabat Hewageegana * Deartment of Pysics, University of Kelaniya, Kelaniya 11600, Sri Lanka Abstract We develo a metod for calculating te olarizability of a serical nano article by taking in to account te temoral and satial disersion were disersion due to te Landau daming. To describe tese enomena, we develoed analytical teory based on local random-ase aroximation. Our teory is very general in te sense tat it can be alied to any material wic can be caracterized by a bulk dielectric function of te form ( k, ). Te teory is alied to calculate te olarizabilities of dielectric and metallic nanoseres. Keywords: metallic nanosere; olarizability; random-ase aroximation 1. INTRODUCTION Recently, tere as been exlosive growt of nanoscience and nanotecnology. Nanosystems ossess unique roerties different from tose of macroscoic materials wen caracteristic lengts govern teir roerties. Terefore, te satial disersion becomes muc more imortant were te caracteristic size of te article or distances between tem becomes comarable to te caracteristic scale of te system 1-. One examle of suc a scale is te lattice constant in metals wic is on te order of te electron wavelengt at te ermi surface ~ 1Å. Anoter imortant scale in nanootics: wic is a modern branc of otical science tat exlores ow otical frequency radiation can be confined on te nanoscale, i.e.,1-100 nm (muc smaller tan te otical wavelengt) is te Debye screening radius rd E /(6ne ) * ( or Tomas-ermi screening radius) were is te background dielectric constant of te * Corresonding Autor E mail: s@kln.ac.lk

2 Prabat Hewageegana /Sri Lankan Journal of Pysics, Vol. 1(1), (01) metal tat is due to te core (valence) electrons and ion motion (onons), n is te concentration of electrons in c.g.s units and E is te electron ermi energy * E k / m, were k ( n) 1/ is te electron wave vector at te ermi surface * and m is electron effective mass in te metal. * Using te Bor radius, a B / m e and k, one can rewrite r D for most 6 metals, including noble metals, as r.9 10 ( 1/ D n ) 1 Å. One of te oter lengt is te correlation lengt, l c : tat an electron at te ermi surface travels during a eriod of te otical radiation, is anoter imortant scale for otical interactions in a metal system, wic is on te order of ~ v /, were v is te electron seed at te ermi surface; 8 for metals v ~ 10 cm s -1 1 and ~ 10 s -1 is te otical frequency. Tis yields an estimate lc ~ 1nm wic is te largest satial scale comared wit te oter two scales. Terefore, wen a caracteristic size a of te nanosystem becomes small, it may become comarable to l c wic will make imortant for te nonlocality in te otical resonses of te electron system. Terefore, macroscoic descrition of nanostructured system may not be alicable even on te order of magnitude. It is very clear now tat te otical roerties of nanoseres, wose size ~ 10 nm or less, are very different from tose of te corresonding bulk materials,6. Tis is due to te surface effect: large surface to volume ratio. One of te consequences of suc a small size of te system is tat te electric field E magnetic induction B, magnetic field H and te dislacement vector D are related by a nonlocal relationsi instead of te usual local relation. Assuming tat fields are weak enoug tat dislacement vector D and magnetic induction B can be obtained by erturbation teory terefore, one can obtain integral linear-resonse relation in terms of te corresonding fields (i.e., relations between D and electric field E, and B magnetic filed H) D( r, t) dt ( r r, t t) E( r, t) dr (1) v B( r, t) dt ( r r, t t) H( r, t) dr () v ere ( r r, t t) (dielectric function) and ( r r, t t) (magnetic ermeability) denote te resonse functions in sace and time. Te dislacement D at time t deends on te electric field at time trevious to t (temoral disersion or frequency disersion). In addition to tat te dislacement at oint r also deends on te values of te electric field at neigboring oints r' (satial disersion). A satially disersive medium is terefore also called a non-local medium. Tis effect can be observed at interface between different media or in metallic objects wit sizes comarable wit te mean-free at of electrons. In most cases of interest te effect of te satial disersion is very wee terefore we can assume tat te materials of te system are isotroic. Oterwise, bot and μ would ave been tensors, wic would make no rincial difficulty but make formulas somewat more comlicate. However, on te oter and temoral disersion, is a widely encountered enomenon and it is imortant to take it accurately into account. As we discussed above relations (1) and () are non-local bot in sace and time.

3 Prabat Hewageegana /Sri Lankan Journal of Pysics, Vol. 1(1), (01) However, one can use matematically equivalent descrition in ourier domain wic is local, D( ) ( ) E( ), B( ) ( ) H( ) () Here we ave introduced te corresonding arguments in ourier domain wic is wave vector k and frequency. Terefore, te ourier transform of te electric field is defined as, i( krt ) E( ) E( r, ) e drdt (4) and one can get similar exressions for oter quantities.. MODEL AND EQUATIONS We will emloy local random-ase aroximation (LRPA): suose tat locally te electron ermi liquid everywere as te same roerties as in te bulk metal, a microscoic aroac based on random-ase aroximation (RPA) 7 were te dielectric function ( k, ) ossesses bot te temoral and satial disersion, and dissiation is due to te Landau daming 8. Earlier, aroac based on RPA 9 was develoed for somewat large radii, a r D. In contrast, our teory is valid for te intermediate scale, lc a rd, wic is of ig imortance for nanootics. Suc a nanolocalized fields are due to te interaction rocesses (oscillation of olarization carges) between electromagnetic radiation and conduction electrons at metallic interfaces or in small metallic nanostructures, leading to an enanced otical near field of sub-wavelengt dimension. Suc oscillations on te nanoscale are called surface lasmons. In LRPA, we consider electrons of te metal as a degenerate electron lasma tat ossesses dissiation due to Landau daming and wose dielectric function deends on bot (temoral) and k (satial disersion). Well known Lindard formula 10 is one of te closed solutions in te teory of te ermi systems tat exlicitly gives te nonlocal dielectric resonse function (longitudinal) ( k, ). kv ( ) 1 ln () k v kv kv 4ne were is lasma frequency wic is define as. Te comlex function * m kv in Eq.(), kv kv ln is defined as ln i kv kv ln for kv kv kv 0. Note tat ( k, ) as a non-zero imaginary art, wic describes otical losses, only wen kv. Tese otical losses can be connected to te excitation by a field of incoerent electron-ole airs. Tis enomenon is called Landau daming, wic is described by Eq (). Te Landau daming actually is

4 Prabat Hewageegana /Sri Lankan Journal of Pysics, Vol. 1(1), (01) deasing, were coerent field oscillations are transformed into incoerent electron ole airs, but te total energy of te system is not canged. Landau daming is fulfilled wen te size of te system is comarable to or less tan te correlation lengt, lc ten condition, kv dielectric function can be written as, order of r, is satisfied. In tis limit, te exression, Eq. () for te longitudinal ( ) 1. i (6) k v k v kv Considering LRPA, an external field enetrates a metal mainly to a det on ~ /. In a real metal r D is very small ( ~ 1 nm) ence te contributing wave D v vectors 1 r D k ~ are large. or suc a k te Eq. () as an imaginary art, Im[ ( )] ~ / rd v (7) kv Tis as te same order of magnitude as Re[ ( k, )]. Tis effect (Landau daming) is not a small effect even at its onset. As exlicitly demonstrated in Eq. (7) it is obvious tat te relaxation and losses in te electron system comes only wit a strong satial disersion (deendence on k ). Te Landau daming and te satial disersion are igly imortant for te nanolasmonics, because tey are te most ronounced for te low-frequency. In articular, small-size range of arameters were te nanolasmonic effects are strongest and most interesting. Even toug tese enomena are very imortant for nanootics, it is very difficult to take into account because te exression for te dielectric resonse is relatively comlex in te k sace [see Eq. ()]. Note tat te boundary conditions at te surfaces of te nanostructure are to be imosed in te real r sace. In te limit k / v and taking 1, from Eq. (6) we get te inverse 1 dielectric function, ( k, ) as, 1 1 ( ) (8) 1 i / k were 9 4 / v. Eq. (8) is identical to te corresonding result of te random ase aroximation (RPA), wic is exected to work well in our case wen and k k. Tis formula satisfies general roerties of te satial disersive dielectric resonse, in articular, it describes Landau daming for k / v. Let us assume tat an external electric field E E0 ( ) zˆ is alied to a nanosere wose radius is a and material is describe by longitudinal dielectric function ( k, ). Here retardation is neglected: size of te nanosere is muc smaller tan te wavelengt of te excitation field. Te otential and te radial comonent of te dislacement vector D r for r a is given by r E0r cos cos (9) r

5 Prabat Hewageegana /Sri Lankan Journal of Pysics, Vol. 1(1), (01) Dr r Er r E0r cos cos (10) r Te resulting equation for otical olarizability at r a can be written as / E0, were E0a [ 1/ 1] is te induced diole moment of te sere and is te angle between te vectors r and E. Ten te final results for te including nonlocal effects is given by. 1 a a a (11) 1 6a a a r and were ra r 0 J1( kr) J1( ka) dk 0 k J1( kr) J1( ka) dk. k i 1 In te local limit, i.e., k,, a a. (1) Ten reduces to well known exression 11, 1 a. (14) 1. NUMERICAL RESULTS AND DISCUSSION Te inverse dielectric function, 1 ( k, ) is sown in igure 1 for 1, and Im ( k, ), wic is Re ( k, ) (not sown in te gra). ev. Here we can clearly see te significance of te same order of magnitude as (1) igure 1: Absolute value of te imaginary art of te inverse dielectric function 1 ( k, ) lotted 6 against wave vector k for 1, and ev, 9. 4 ev and v 10 m s -1.

6 Prabat Hewageegana /Sri Lankan Journal of Pysics, Vol. 1(1), (01) Te results of numerical comutation from Eq. (1) of te imaginary art of te normalized olarizability for silver nanoseres of radii. nm, nm, 10 nm, 0 nm and 100 nm are sown in igure. Our numerical calculation qualitatively agrees wit Ref. [], it indicates tat te essential ysics of te system is contained in our simle treatment even for article radii as small as. nm. Te sligt variation in te eak osition is resumably due to te different formalism of te used dielectric function. Toward te smaller article size te eak osition as been canged significantly. Tis is due to te Landau daming wic is fulfilled wen te size of te system is comarable to or less tan te correlation lengt, l c, ten condition, k / v is satisfied. Since we used more general dielectric function for te model, two major advantages of te results can be seen: (a) tey can be alied to various different materials and (b) te model is caable of giving quantitatively correct results as sown 1 in igure. arter, one can easily conceder te iger order terms of te ( k, ) in our formalism to sow tat tis agreement would continue for even smaller articles as well. Anoter advantage of our teory is tat one can use tis model to study te satial variations of quantities like electric field, olarization, induced carge density, etc., witin te sere in a simle analytical way. igure : Variation of te imaginary art of / a for silver seres of radii. nm, nm, 6 10 nm, 0 nm and 100 nm. Here 9. 4 ev and v 10 m s -1.

7 Prabat Hewageegana /Sri Lankan Journal of Pysics, Vol. 1(1), (01) To briefly summarize te main results, we ave develoed LRPA teory to describe te olarizability of nanosize meatal sere including nonlocal effects. We oe tat our work will be useful in studying te otical roerties of nanoarticles of oter materials (suc as semiconductors, ionic crystals, etc.) wic we did not consider ere. In articular, tere are two areas of considerable current interest were a teory like LRPA can lay a significant role: It is known tat absortion by onons migt be resonsible for te anomalously large absortion of infrared radiation by very small metallic articles, one could try to see weter LRPA teory would exlain tis large absortion by adding a onon term to te dielectric function, Some of te teories of te surface-enanced Raman scattering are based on an enancement of te effective olarizability due to image effects. As a better aroximation, one can assume te molecules to be dielectric seres and our exressions for can rovide a simle but accurate way for taking into account bot te size as well as te frequency deendence of te olarizability. REERENCES 1. M.J Rice, W.R Scneider, Electronic Polarizabilities of Very Small Metallic Particles and Tin ilms, Pys. Rev. B 8, (197) A.A Lusnikov, A.J Simonov, Te electric olarizability of small metal articles, Pys. Lett. 44, (197) 4.. B.B Dasguta, R. ucs, Polarizability of a small sere including nonlocal effects, Pys. Rev. B 4, (1981) R. ucs,. Claro, Multiolar resonse of small metallic seres: Nonlocal teory, Pys. Rev. B, (1987) 7.. R. Rojas,, Claro, R. ucs, Nonlocal resonse of a small coated sere, Pys. Rev. B 7, (1988) B. Hect Novotny, Princiles of Nano-Otics (Cambridge University Press, Cambridge, 006). 7. G.D. Maan, Many-Particle Pysics (Plenum, New Yor 1990). 8. E.M. Liftsitz, L.P Pitaevskii, Pysical Kinetics (Butterwort-Heinemann, Oxford, 1997). 9. R. Ruin, Otical roerties of small metal seres, Pys. Rev. B 11, (197) J. Lindard, On te roerties of a gas of carged articles, Dan. Mat. Py. Medd. 8, (194) J.D. Jackson, Classical Electrodynamics (Jon Wiley and Sons Inc., New Yor 1999), ed.