Photoabsorp tion of Na n C lusters and Vo lum e P lasm on:Theory Rev iew

(Am erican Go ldm anSachs G roup,N ew Yo rk 10018-1405,U SA)

Photoabsorp tion of Na n C lusters and Vo lum e P lasm on:Theory Rev iew

X ia Chun lei

(Am erican Go ldm anSachs G roup,N ew Yo rk 10018-1405,U SA)

The vo lum e p lam son isan im po rtan telem en tary excitation,it cou ld no t coup le to ligh tw aves in the bu lk and thus cou ld no t be investigated by op tical spectro scopy,how ever the situation is d ifferen t fo r clusters due to the p resence of the boundary.W hat ism o re,it is fundam en tal to study the in teraction betw een ligh t andm atter,bu t the pho toabso rp tion of sim p lem etal clusters in the UV region has rem ained unexp lo red experim en tally.W e have carried ou t experim en tson the pho toabso rp tion o f N a20 and N a92 covering the nearUV part asw ell as visib le range(2.0～5.5 eV)em p loying the pho todep letion technique.The abso rp tion cross sections fo r clusters o f som e o ther sizes in the near UV range have also been stud ied.The experim en tal data fo r N a20 in the v isib le part ag reew ith ex isting data very w ell. In addition to the strong su rface p lasm on abso rp tion peaks in the visib le,w e observed app reciab le pho toabso rp tion in the UV region.These cross section spectra have been fitted by Lo ren tzian line shapes and revealed b road peaks sligh tly above 4 eV.The frequency and w eigh t of these peaksm atch the theo retical p red ictions and are iden tified as the nanocluster"vo lum e p lasm on"resonance.The pho toabso rp tion spectra p rovide the first experim en talobservation o f op tically excited"vo lum e p lasm on"co llective electronic states inm etal nanoc luster particles,a phenom enon unique to finite system s.

nanoc luster;vo lum e p lasm on;su rface p lasm on;pho toabso rp tion

1 M ie theory

In a sim p lified D rudem odel,the free valence electrons cou ld oscillate w ith respect to the ionic background unifo rm ly. Fo r spheres that abso rb and scatter ligh t,th is classical p rob lem w as first so lved in fu ll generality by M ie[3].Fo r bu lk,the m o tion equation resem b les that o f a sp ring-m ass system and the resonan t frequency is g iven byω2p=w herenis the num ber density o f conduction electrons.Fo r a finite spherical free electron gas system (clusters fo r exam p le),the so called M ie resonance frequency is

2 Surface p lasm on

The first observation o f p lasm ons in size reso lved free nanocluster o f sim p le m etals happened abou t tw en ty years ago[4]. Em p loying beam dep letion spectroscopy, the abso lu te pho toabso rp tion cross section w as m easu red fo r neu tral sod ium clusters con tain ing 2 to 42 atom s.The data cover a w aveleng th range o f 452 to 635 nm in the visib le. The in tense pho toabso rp tion peaks in the visib le part o f the spectrum are comm on ly referred to as su rface p lasm a resonances because fo r larger particles they co rrespond to translational electron oscillations leading to su rface-localized chargem odes.

The experim en tal pho toabso rp tion cro ss section differ from classicalM ie theo ry in several fo lds:1)The resonan t frequencies o f N an(n=4～42)clustersare low er than theM ie frequency of 3.41 eV;2)Instead o f a sing le sharp peak,the peak s are b roadened and som e spectra have m u ltip le peaks(even fo r som e clustersw ith c losed shells);3)The to tal oscillato r streng th is less than one w ou ld expect from the sum ru le.

The first d iscrepancy (this low er resonance frequency thanωMieis o ften referred to as"red sh ift")is caused by the"sp ill ou t"of the valence electrons beyond the boundary o f the ion co res.A s the sizeof the cluster tends larger,thiseffectp lays a less im po rtan t ro le and the resonan t frequency app roaches the classicalM ie value.

The fragm en tation o f co llective streng th in spherical clusters can be attribu ted to an in terference o f specific particle-ho le (o r m o re com p licated)excitation w ith the su rface p lasm on excitation, w h ile fo r non-spherical clusters,fu rther sp litting cou ld be exp lained by phenom eno logical C lem enger-N ilsson m odel.A s fo r the b roaden ing of the peaks,it cou ld be caused by a lo t o f p rocesses,such as em ission o f an electron,evapo ration o f a sing le neu tral atom,in terference of the co llective state w ith specific partic le-ho le states that lie close in energy,etc[5].The m issing o scillato r streng th is one o f the reasons that w e ex tend the investigation of pho toabso rp tion in to UV range,w h ich w ill be add ressed in sections that fo llow.

3 Vo lum e p lasm on

Since the first observation o f su rface p lasm on in clusters,co llective oscillations o f delocalized valence electrons have been ex tensively investigated bo th experim en tally and theo retically(see, e. g., the rev iew s[5-8]and references therein). How ever,the u ltravio let part of the cluster pho toabso rp tion spectrum has rem ained p ractically unexp lo red experim en tally,desp ite a num ber of theo retical calcu lations p redicting an app reciab le fraction o f the d ipo le oscillato r streng th o f the cluster valence electrons shou ld reside there. In a num ber of papers th is pho toabso rp tion region has been qualitatively iden tified as vo lum e p lasm a resonances,associated w ith com p ressional electron o scillations,w h ich w ill be review ed in the nex t section.

In an infin ite so lid(e.g.,m etal o r g raph ite)the descrip tion o f co llectivem o tion is sim p lified by translational invariance,w h ich im po ses a p lanew ave character on the oscillations and enfo rces strict selection ru les on their excitation. The situation becom es very d ifferen t in nanopartic les,even tho se o f the sam e m etal. Indeed,w hen a boundary su rface is p resen t,the w ave vecto r no longer rem ains a good quan tum num ber and the pho toabso rp tion spectrum becom es richer,m ak ing it possib le to access a g reater variety o f co llective m odes by op tical spectroscopy.A s a resu lt,in fin item etal nanoclusters the com p ressional vo lum e p lasm ons are expected to becom e capab le of coup ling to ligh t.

M o tivated by these considerations,w e have carried ou t a m easu rem en t of pho toabso rp tion cross sections of a pair of p ro to typ ical m etal nanoc lusters N a20 and N a92,covering the near-UV,asw ellas the visib le,partsof their resonance spectra.W e have also m easu red the cro ss section of som e o ther clusters in the UV range.

Befo re go ing to the experim en tal resu lts,w e w ill review som e o f the theo ries.

4 Theory on photoabsorp tion

C lusters arem any-body system s consisting of a num ber of nuc lei and electrons w h ich in teract w ith each o ther,thus the H am ilton ian includes m any term s and it is no t p ractical to so lve fo r the exact so lu tion. In o rder to get so lu tions that are good enough, a series of sim p lifications and app rox im ations are requ ired. A lthough it is im p ractical to talk elabo rate the details, it is help fu l to m en tion them ajo r too ls that have used bym ost theo reticians,nam ely jellium m odel,m ean field theo ry (Hartree-Fock and local density app rox im ation),linear response and random phase app rox im ation.

The basic assum p tion o f the jellium m odel is to igno re the specifics o f the ion ic structu rew ith in the cluster and take it as a un ifo rm ly distribu ted positive backg round. Th is leads to a sim p le background po ten tial fo r spherical clusters.Since w e are dealing w ith the op tical response o fm etal clusters of size from 8 up to 92,in w h ich the delocalization o f valence electrons has already been confirm ed experim en tally,hence it is app rop riate to sim p lify the ion-electron in teraction using jellium m odel and concen trate on the electronic partof the Ham iltonian.

Hartree(1928)p roposed am ethod(Hartree A pp rox im ation)to app rox im ate the in teractions assum ing the effective in teraction (po ten tial energy)fo r the electron is determ ined by the average m o tion of the o ther electrons.Hartree-Fock app rox im ation(HFA)includesone add itional term as com pared to H artree app rox im ation in the po ten tial due to an ti-symm etry o f m any-electron w ave function as suggested by Fock,w hich is called exchange.HFA does no t take in to accoun t the co rrelations,w hich are dealt w ith in density functional theo ry (DFT),fo r exam p le. D FT states that the to tal energy o f a system is a functional(NO T function)o f the electron density,p roved by Hohenberg and Kohn (1964).Local density app rox im ation(LDA)is an app rox im ation built upon DFT w hich assum es the exchangeco rrelation energy is a local function o f the electron density.

M ean field theo ry dealsw ith g round states,in o rder to exp lo re p lasm ons w h ich are co llective excitations,it has to be ex tended to excited states and leads to random phase app rox im ation(RPA).Am ong m any w ays to derive RPA,one w ou ld be by so lving the tim e-dependen t Sch rodinger equation in an ex ternal field. In the case that the ex ternal field isw eak enough,the electric dipo le response o fm etal clusters is linear to the ex ternal field.

D ipo lar vib rations o f delocalized valence electrons in nanoclusters o f sim p le s-electron m etals,such as the alkalis,have been ex tensively investigated theo retically (see, e. g., the review s[5-8]references therein) since their first observation tw en ty years ago[4]. It is no t possib le to show all the w o rks in this section,thus on ly som e o f the rep resen tative w o rks are b riefly review ed here.

U tilizing the tim e-dependen t local density app rox im ation(TDLDA,w h ich is the dynam ical ex tension o f the LDA)w ith in the fram ew o rk o f self-consisten t spherical jellium m odel,Ekard t[9]first calcu lated the dynam ical electronic response p roperties of sm allm etal clusters and po in ted ou t the ex istence o f vo lum e p lasm on in som e sodium clusters. TDLDA ex tends the LDA in to tim e dependen t case (although this is theo retical p rob lem atic) under ex ternal pertu rbation and app rox im ate the sing le-partic le w ave functions by the Kohn-Sham o rb itals in the g round state.The nonin teracting response function is first calcu lated,and then the in teracting electron response function is so lved taken in to accoun t the Hartree field and exchange-co rrelation in teraction (w ith in LDA).The com p lex po larizability (and hence the pho toabso rp tion cross section)cou ld be found from the response function.

In the calcu lation by Beck[10],w h ich also em p loyed TDLDA,a p rom inen tpeak show ed up in the pho toabso rp tion spectra at around 4.4 eV fo r N a20.A recen t TDLDA calcu lation carried ou t by H. Chak rabo rty[11]also found sim ilar resu lts:besides them ain peaks in the visib le range,there is a b road peak in the UV range fo r N a20.To take in to accoun t (to som e ex ten t) the granu lar character o f the ion ic back-g round,Rub io et al[12]app lied TDLDA together w ith spherical-averagep seudopo ten tial (SA PS) to investigate the pho toabso rp tion spectra of various clusters.Instead o f assum ing unifo rm positive backg round,SA PS calcu lates the geom etrical structu re and the g round state electron ic structu re o f a cluster by m inim izing the to tal energy w ith respect to the positions o f all the ions. Th is calcu lation also found d istinct peaks in the UV range.

K resin analy tically so lved the co llective resonance frequencieso f the valence electron cloud in m etal clusters based on the random phase app rox im ation,and po in ted ou t that the oscillato r strength is shared by tw o m odes(su rface and vo lum e) of co llective oscillations[6]. The tw o resonan t frequencies are fu lly determ ined by the ground state electron density p rofileg(r),w hich is a general resu lt independen t o f the exact fo rm o fg(r).

The physical m ean ing o f these su rface and vo lum em odes is in terp reted based on the sim ilarity betw een the p lasm ons in m etal clusters and gian t electric dipo le resonance in nuclei. The su rface m ode resem b les Go ldhaber-Teller(GT)m ode of the nuclear gian t resonance,in w hich the p ro ton d istribu tion vib rates against the neu tron distribu tion unifo rm ly,w h ile the vo lum em ode has its coun terpart as Steinw edel-Jensen-M igdal(SJ)m odesw h ich co rresponds to the density vib rations of the neu tron and p ro ton fluidsw ith the boundary su rface fixed.

In th is calcu lation,the ground state electron d istribu tion is app rox im ated using the Thom as-Ferm i statisticalm ethod. The su rface resonance frequencyw-as a function o f cluster size has been calcu lated.Specifically,w-fo r N a20 and N a92 are calcu lated to be 2.67 and 2.88 eV.W h ile the vo lum e p lasm on frequencyw+is found to be≈0.75w pfo r clusters con taining from abou t ten up to hund reds o f atom s. The advan tage o f th is m ethod is the requirem en t of m uch less com pu tation and gives an easy overview app licab le to w ide range o f cluster sizes andm aterials.

Yannou leas et al investigated the evo lu tion of the op ticalp roperties o f alkali-m etal clusters using RPA[13,14]. In their pho toabso rp tion spectra fo r N a20,besides them ajo r su rface p lasm on peaks in the visib le range,there are a num ber o f sm all oscillato r streng ths d istribu ted in the UV range,and them ajo r peaks are sp litted in to tw o.

Barack[15,5]ob tained the dipo le spectra app lying the local RPA app roach(the excitation operato rw as restricted to be one-body operato r in th is app roach)from self-consisten t Kohn-Sham g round state densities fo r spherical sod ium c lusters of various sizes.Fo rN a∞(i.e.bu lk),the su rface p lasm on occu rs at the classical M ie resonan t frequency(wMie)exhausts 100%o f the strength,and there ex ists no vo lum e p lasm on.Fo r N a8 and N a440 besides the dom inan t redsh ifted su rface p lasm on peak(w hich cou ld be fragm en ted),there arem any strong ly fragm en ted peaks that take over the rem ainderoscillato r strength.V o lum e p lasm on fo r N a8 hasm o re streng th(16%)than N a440(～10%),and it ism o re redsh ifted.In this calcu lation,the trend of the cen tro id andw eigh t of the vo lum e p lasm on agreesw ith ou r general expectation.

Radu ta et al[16]have ex tended the RPA calcu lation using a p ro jected spherical sing leparticle basis specially targeted at non-spherical clusters. It is found that the vo lum e-type dipo le oscillation state show s up in the pho toabso rp tion cross-section spectra in the UV region,no t on ly fo r spherical clusters,bu t also defo rm ed clusters.Th is resu lt ag reesw ith the conclusions in[6]w h ich p redicted that pho toabso rp tion strength in neu tral sphericalN a clusters is shared betw een a su rfaceand a vo lum e-p lasm a resonance.

A recen t w o rk by Josw ig et al. studied the pho toabso rp tion o f sod ium c lusters based on tim edependen t density-functional theo ry[17]:the spectra w ere calcu lated by p ropagating the tim e-dependen t Kohn-Sham equation in real tim e.Itw as found the spectra o f bo th N a20 isom ers bo th ex tend to UV region and have a d istinct peak at around 4.1 eV.

To summ arize,all these calcu lations expect to see a certain am oun t of pho toabso rp tion in the spectra o f N a clusters in the UV range,w h ile th is has never been done experim en tally.M o tivated by th is,w e carried ou t the pho todep letion experim en t on N a clusters in th is UV range to search fo r the vo lum e-type p lasm ons,w h ich w ill be described in the second parto f th ispaper.

[4] de HeerW A,Selby K,K resin V,et al.Co llective d ipo le oscillations in sm all sod ium c lusters[J].Phys Rev L ett,1987,59:1805-1808.

[1] X ia C,Y in C,K resin V V.Pho toabso rp tion by vo lum e p lasm ons in m etal nanoclusters[J].Phys Rev L ett,2009,102:156802-156805.

[2] Haberland H.C lusters of A tom s and M o lecu lesⅡ[M].Berlin:Sp ringer-V erlag,1994.

[3] M ie G.Beitrige zu r op tik truber m edien,speziell ko llo idalerM etall6sungen[J].A nn der Phys,1908,25:377-452.

[5] B rack M.The physics o f sim p lem etal clusters:selfconsisten t jellium m odel and sem iclassical app roaches[J].RevM od Phys,1993,65:677-732.

[6] K resin V V.Co llective resonances and response p roperties o f electrons in m etal clusters[J].Phys Rep,1992,220:1-52.

[7] de HeerW A.The physics of sim p lem etal clusters:experim en tal aspects and sim p lem odels[J].RevM od Phys,1993,65:611-676.

[8] A lonso J A. Structu re and p roperties of atom ic nanoclusters[M].London:Im perial Co llege Press,2006.

[9] Ekard t W.Size-dependen t pho toabso rp tion and pho toem ission o f sm allm etal particles[J].Phys Rev B,1985,31:6360-6370.

[10] Beck D E.Self-consisten t calcu lation o f the co llective excitations in neu tral and charged jellium m icrospheres[J].Phys Rev B,1991,43:7301-7304.

[11] Chak rabo rty H.p rivate comm unication(2009).

[12] Rubio A,A lonso J A,B lase X,et al.Theo retical m odels fo r the op tical p roperties o f clusters and nanostructu res[J].In tern JM od PhysB,1997,11:2727-2776.

[13] Yannou leasC,B rog lia R A.Size-dependen t pho toabso rp tion and pho toem ission o f sm all m etal particlesCo llective and sing le-partic le aspects in the op tical response ofm etalm icroc lusters[J].PhysRev A,1991,44:5793-5802.

[14] Yannou leas C,V igezzi E,B rog lia R A.Evo lu tion of the op tical p roperties o f alkali-m etalm icroclusters tow ards the bu lk: the m atrix random-phaseapp rox im ation descrip tion[J].Phys Rev B,1993,47:9849-9861.

[15] Reinhard P-G,B rackM,Genzken O.Random-phase app rox im ation in a local rep resen tation[J].PhysRev A,1990,41:5568-5582.

[16] Radu ta A A,Budaca R,Radu ta A l H.Co llective d ipo le excitations in sod ium clusters[J].Phys Rev A,2009,79:023202-023213.

[17] Josw ig J,Tun tu rivuo riL,N iem inen R.Pho toabso rption in sodium clusters on the basis o f tim edependen t density-functional theo ry[J]. J Chem Phys,2008,128:014707-014712.

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钠纳米团簇的光吸收和体等离激元:理论分析

夏春雷
(美国纽约高盛集团,纽约10018-1405)

指出了体等离激元是一种重要的元激发,在固体里它不能与光波耦合所以无法用光波分波来研究,但是在团簇里由于边界的存在,情况不太一样.此外,光与物质之间的相互作用是非常根本的,但简单金属团簇在紫外光区的光吸收从未被实验研究过,尽管这方面的理论计算非常多.采用光倒空技术对N a20和N a92团簇进行了光吸收实验,覆盖了近紫外部分以及可见光范围(2.0～5.5 eV);实验还研究了其它一些团簇在近紫外范围内的吸收截面.在可见光部分N a20的实验数据与现有的数据非常吻合.除了表面等离激元在可见光区域很强的吸收峰外,还观察到紫外区域明显的光吸收.对吸收截面光谱进行洛伦兹拟合揭示了中心略高于4eV宽峰的存在,这个宽峰的频率与权重均与理论预测非常接近,被确定为纳米团簇“体等离激元”共振.这些吸收光谱是金属纳米团簇由光子激发的“体等离激元”集体电子态的首次实验观测与证实,这种实验现象只存在于有限体系.

纳米团簇;体等离激元;表面等离激元;光吸收

Plasm ons(p lasm a w aves)are the co llective dipo le oscillations o f the valence electrons.They cou ld occu r in bu lk, thin film s and m etallic nanoc lusters[1]. Plasm ons have been used to con tro l co lo rso fm aterials back to ancien t tim es[2].Certain m etal nanoparticles im p rove the ligh tabso rp tion in thin-film so lar cells,w h ich is closely related to their localized p lasm on resonance.Since p lasm ons can suppo rt m uch h igher frequencies(in to 100 TH z range,inw h ich rangew ires becom e very lossy),it has been p roposed using p lasm ons to transm it info rm ation on com pu ter chip s instead of conven tionalw ires.

2010-10-14

夏春雷(1980-),男,博士,研究方向:金属纳米团簇的光吸收,E-m ail:x iachun lei@gm ail.com