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Kinetic theory of surface plasmon polariton in semiconductor nanowires Y. Yin∗ and M. W. Wu† Hefei National Laboratory for Physical Sciences at Microscale and Department of Physics, University of Science and Technology of China, Hefei, Anhui, 230026, China (Dated: January 18, 2013) Based onthesemiclassical modelHamiltonian ofthesurfaceplasmon polariton andthenonequi- librium Green-function approach, we present a microscopic kinetic theory to study theinfluence of 3 theelectronscatteringonthedynamicsofthesurfaceplasmonpolaritoninsemiconductornanowires. 1 Thedampingof thesurface plasmon polariton originates from theresonant absorption bytheelec- 0 trons (Landau damping), and the corresponding damping exhibits size-dependent oscillations and 2 distincttemperaturedependencewithoutanyscattering. Thescatteringinfluencesthedampingby n introducing a broadening and a shifting to the resonance. To demonstrate this, we investigate the a dampingofthesurfaceplasmonpolaritoninInAsnanowiresinthepresenceoftheelectron-impurity, J electron-phononandelectron-electronCoulombscatterings. Themaineffectoftheelectron-impurity 7 andelectron-phononscatteringsistointroduceabroadening,whereastheelectron-electronCoulomb 1 scatteringcannotonlycauseabroadening,butalsointroduceashiftingtotheresonance. ForInAs nanowiresunderinvestigation,thebroadeningduetotheelectron-phononscatteringdominates. As ] aresult,thescatteringhasapronouncedinfluenceonthedampingofthesurfaceplasmonpolariton: l l Thesize-dependentoscillationsaresmearedoutandthetemperaturedependenceisalsosuppressed a h inthepresenceofthescattering. Theseresultsdemonstratetheimportantroleofthescatteringon - thesurface plasmon polariton damping in semiconductor nanowires. s e PACSnumbers: 73.20.Mf,73.22.Lp,72.30.+q,71.10.-w m . t a I. INTRODUCTION dau damping. These influences can be incorporatedinto m microscopic models based on time-dependent density- - Since the pioneering theoretical work by Ritchie1 and functional theory or semiclassical model Hamiltonian of d the electron-loss spectroscopy measurements by Pow- the SPP.42–44 Calculations based on these models have n ell and Swan,2 physics of surface plasmon polariton shown good agreement with experiments.45–47 o c (SPP) has been extensively studied for more than five In recent years, doped semiconductors, such as SiC, [ decades.3–9 SPPs are electromagnetic (EM) waves cou- GaAs, InAs, Cu2S and Cu2Se, are suggested as promis- pled to the collective excitations of electrons on the sur- ingcandidatestoreplacemetalsinSPPapplications.48–52 2 face of a conductor.10,11 In this coupling, the electrons The SPPs in doped semiconductors are characterizedby v 5 oscillatecollectivelyinresonancewiththeEMwavesand their substantial low losses and tunable frequencies.53–56 5 hence trap the EM waves on the surface. The resonant They are alsoeasierto be manipulatedvia doping or ex- 0 coupling leads to the SPPsand givesriseto their unique ternal electric/magnetic fields and to be integrated into 5 properties, such as the enhancement of the surface elec- complex, functional circuits.57–59 The further develop- 1. tric fields and the slowing down of the group velocity of mentoftheSPPsindopedsemiconductorsrequiresabet- 1 the EM waves.12–20 Applications exploiting these prop- ter understanding of their damping processes. However, 2 erties have been widely studied in biosensing,21 solar the physics involved in doped semiconductors can be 1 cells,22 quantum information processing,23–25 subwave- quite different from that in metals. For doped semicon- : v length optical imaging and waveguiding devices.11,26–31 ductors,althoughtheLandaudampingisstillbelievedto i The SPPs often suffer from dissipative losses.32 Over- betheleadingdampingprocessatlargewavevectors,60,61 X comingthelossesiscrucialfortheimprovementofperfor- the charge depletion/accumulation layer62,63 and the ar manceofmanySPP-baseddevices,suchasthefidelityof electron scattering64,65 are found to have important in- the waveguide and the sensitivity of the single-molecule fluence on the damping. Of particular importance is the sensor.21,33 An effective modulation of the losses is also effectoftheelectronscattering,sincethe typicalscatter- highly desirablefor activeplasmonicdevices proposedin ingratecanbecomparabletotheSPPfrequencyinsemi- recentyears.34–36 Thus,athoroughunderstandingofthe conductors. However, to the best of our knowledge, this dampingprocessesresponsibleforthedissipativelossesis effect has only been discussed by using phenomenologi- essential. Since the SPPs in metals have been the major calrelaxationtimes.64,65 Amicroscopictheoryexploiting focusformanydecades,previousstudiesonthedamping this effect has not been established yet. processeshavetraditionallybeenfocusedonmetals. Itis In this paper, by combining the semiclassical model foundthatthedominantdampingprocessisthedecayof Hamiltonian of the SPP42–45 and the nonequilibrium the SPPs into electrons, i.e., the Landau damping.37–41 Green-function approach,66,67 we present a microscopic The interbandtransitions andthe many-body exchange- kinetic theory to study the damping of the SPP in correlations can have pronounced influences on the Lan- doped semiconductors, within which the relevant elec- 2 tron scatterings are treated fully microscopically. The main purpose of this work is to understand the influence of these scatterings on the Landau damping of the SPP. To demonstrate this, we focus here on the SPPs in InAsnanowires23,28,29,51andconcentrateontheelectron- impurity(ei),electron-phonon(ep)andelectron-electron (ee) Coulomb scatterings. We find that the scattering can have pronounced influences on the Landau damp- ingofthe SPPbymodulatingtheresonancebetweenthe electronsandtheSPPs. Differentscatteringhasdifferent effect on the resonance. The main effect of the ei and ep 50 scatteringsistointroduceabroadeningtotheresonance, (b) whereas the ee scattering can not only case a broaden- n–0 = 5.0 × 1017 cm-3 40 ing, but also introduce a shifting to the resonance. For InAs nanowires,the ep-scattering–inducedbroadeningis V) 30 found to be the dominant effect. These effects can lead me to a pronounced influence on the damping of the SPP, ω ( 20 wdehpicehndceannceboefsteheenSfProPmdabmotphintgh:e(1s)izTehaensdiztee-mdeppeernadtuenret 10 n–0 = 1.5 × 1017 cm-3 R = 423682 nnnmmm oscillations of the SPP damping are smeared out, and 40 nm 0 (2) the temperature dependence of the SPP damping is 0 6 12 18 suppressed by the scattering. These results demonstrate q (10-3 nm-1) the important role of the electron scattering on the SPP damping. FIG. 1: (Color online) (a) Schematic of the structure of the Thispaperisorganizedasfollows. InSec.II,weintro- electronfieldE andcorrespondingsurfacechargeoftheaxial duce the semiclassicalmodelHamiltonianofthe SPPfor symmetric SPP mode in a cylindrical nanowire. ǫ∞1 (ǫ2) is the dielectric constant inside (outside) the nanowire. The semiconductor nanowires and briefly outline the deriva- redcurveswitharrowsrepresenttheelectricfieldsoftheSPP tionofthekineticequations. Wealsopresentananalytic mode. (b)ThedispersiverelationoftheaxialsymmetricSPP solution for the SPP damping which provides a simple mode for nanowires with different electron density n¯0 and and physically transparentpicture to understand the in- wireradiusR. ThethingreylinemarkstheenergyoftheLO fluenceofthescatteringontheSPPdampingprocess. In phonon. Sec. III, we discuss in detail the influence of the electron scatteringonthe SPPdamping by numericalsolvingthe kinetic equations. The importance of the scattering is low frequency and has been found to be important for demonstratedby studying its influences on the tempera- boththe terahertzemissionandquantumsubwavelength ture and size dependence of the SPP damping. The an- optics.23,28,29,51 In this paper, we focus on the dynamics alytic solution is also compared with the numerical ones ofthisSPPmode. ThecorrespondingHamiltoniantakes in this section. We summarize and discuss in Sec. IV. the form H = Ω b†b , (2) SPP q q q q II. MODEL AND FORMALISM X where b (b†) represents the annihilation(creation) boson q q A. Semiclassical model for SPP-electron system operator for the SPP, with Ω being the corresponding q dispersive relation illustrated in Fig. 1(b). Note that we We consider an n-type free-standing cylindrical set ~=1 throughout this paper. nanowire with radius R as illustrated in Fig. 1(a). The The Hamiltonian of electrons can be written as z-axisischosentobealongthewire. Followingthesemi- H =H0 +H +H +H , (3) classical approach developed in previous works,42–45 we el el ei ep ee decompose the Hamiltonian into wherethefree-electronHamiltonianH0 ismodelledbya el mean-field potential. The interaction Hamiltonians H , ei H =HSPP+Hel+HSPP-el, (1) Hep and Hee represent the ei, ep and ee interaction, respectively. where H , H and H are Hamiltonians for the SPP el SPP-el By choosing the mean-field potential to be an infinite SPP, electrons and the SPP-electron coupling, respec- cylindricalpotentialwellwithradiusR,the free-electron tively. Here we only present the Hamiltonian, leaving Hamiltonian H0 can be written as the details to Appendix A. el For nanowires, there exists one fundamental SPP H0 = εnc† c , (4) el k nkσ nkσ mode with axial symmetry, which has no cutoff at nkσ X 3 in which εn = k2+(λmn˜˜/R)2 is the eigenenergy with m∗ The damping of the SPP is obtained by studying k 2m∗ representing the electron effective mass. The composed the temporal evolution of a coherent SPP wave packet indexn=(m˜,n˜)labelstheelectronsubbandwithm˜ and with central wavevector Qs, which can be expressed as tnr˜ieosrnpepeocrfetisvteheneltyi.nfigλrsmn˜˜tthdkeienanndogtJueslat(rhxae)n.n˜d-TtrhahdezieacrlooqrorufeastnphtoeunBmdeinnssguemlefibugenerncs--, f|Bunscit=ionPoqfptQqhsee−w21a|vBeq|p2eabcqkbe†qt|0piQq(sReisf.c7h0o)s.enThteolbinee-psQqhsap=e m˜ sin[(Qs−q)L/2], where L is the wave packet length. Such states read (Qs−q)L/2 wave packet is typical in a Fabry-Perot SPP resonator, ψ (ρ,ϕ,z) = Jm˜(λmn˜˜ρ/R) eim˜ϕeikz. (5) which has been observedin various SPP systems.25,71–73 nk √πRJ (λm˜) The amplitude ofthe wavepacketcanbe describedby m˜+1 n˜ The ei interaction Hamiltonian can be written as Bs = qpQqsBq. The kinetic equation of Bs is obtained from the Heisenberg equation of the SPP annihilation Ni operatPor b , which has the form H = vnn′ρ (q)c† c , (6) q ei q i n′k+qσ nkσ Xi Xkqσ ∂tBs(t) = pQk−sk′gkn−n′k′G<σ(n′k′,nk;tt), (10) with N being the total impurity number and ρ (q) = nn′,kk′,σ i i X e−iqzi. Here we have assumed that the impurities are where G<(nk,n′k′;tt′) is the “lesser” electron distributedonthesurfaceofthenanowirewithanaxially σ symmetric distribution. vnn′ is the matrix element for Green function defined as G<σ(nk,n′k′;tt′) = the ei interaction. The epqinteraction Hamiltonian can ihc†n′k′σ(t′)cnkσ(t)i (Ref. 66). be written as The kinetic equation of the electron Green function G<(nk,n′k′;tt) is derived by using the nonequilibrium Hep = MQnnq′ aQq+a†−Q−q c†nkσcn′k−qσ,(7) Gσreen-function approach, which can be written as XQq nXn′kσ (cid:16) (cid:17) i∂ (εn′ εn) G<(nk,n′k′;tt) whereaQq(a†Qq)representstheannihilation(creation)op- − t− k′ − k σ erator for the LO phonons, with Q and q representing h = (B +iB†) gn¯n′G<(nk,n¯k′ q;tt) the components of the phonon momentum perpendic- −q q q σ − ular and parallel to the nanowire. Here we use bulk Xn¯q h phonons in the present investigation, which is valid for gnn¯G<(n¯k+q,n′k′;tt) +I<σ (t), (11) nanowires with large diameters.68,69 Mnn′ is the matrix − q σ nk,n′k′ Qq i elementfortheepinteraction. Itisnoticedthatalthough where the first term in the right hand side of the equa- surface-optical (SO) phonons69 also exist in nanowires, tions is the coherent driving term of the SPP, while the they are of marginal importance since the correspond- second term is the scattering term consisting the ei, ep ing electron-SO-phonon interaction is rather weak com- and ee scatterings. pared with the electron-LO-phonon interaction for the Within the rotating wave approximation relative to nanowires we consider here. The influence of the SO the SPP central frequency Ω ,66,67 we obtain the kinetic s phonons will be further addressed in Sec. IIIB. The ee equations for the SPP-electronsystem interaction Hamiltonian can be written as † Hee = Vqnn′c†nkσc†n′k′σ′cn′k′+qσ′cnk−qσ,(8) ∂tB¯s(t)=−i pQk−sk′gkn−n′k′ Pσ(nk,n′k′;t) ,(12) kXk′qXnn′ Xσσ′ nXn′σXkk′ h i ∂ P (nk,n′k′;t)=iωnn′P (nk,n′k′;t) where Vnn′ is the matrix element for the ee interaction. t σ kk′ σ The SPPq-electron coupling Hamiltonian HSPP-el can be +igkn−n′k′pQk−sk′B¯s† fnσ(k)−fn′σ(k′) +I¯nσk,n′k′,(13) written as h i with the detuning H = gnn′ b +b† c† c , (9) SPP-el k−k′ k−k′ k′−k nkσ n′k′σ XnkσnX′k′ (cid:16) (cid:17) ωknkn′′ =εnk′′ −εnk −Ωs. (14) where gnn′ is the SPP-electroncoupling matrix element. Ianndthgensneq′ aeqreuagtiivoenns,inmdaettraixil ienleAmpepnetsndviqxnnA′,.MQnnq′, Vqnn′ IPSnPσ(Pnthkae,mnp′aklbi′t;outv)dee=eaqnu−daitGeiloe<σnc(stnr,okn,Bn¯ps′k(ot′l;)atrti)z=eaitΩisotBn,sr(ertpe)resepiΩseescntttivaetnhlyde. q f (k) represents the equilibrium electron distribution nσ whichisconventionallychosentobethespin-unpolarized B. Kinetic equations Fermi-Dirac distribution at temperature T. It should be emphasized that in the derivation, we In this section, we briefly outline the derivation of the have treated the SPP-electron coupling gnn′ perturba- q kinetic equations for the SPP-electron system. The de- tivelyandlinearizedtheequationsbykeepingonlyterms tails can be found in Appendix B. up to the linear order of gnn′. Thus the SPP couples q 4 only to the electron polarization corresponding to the off-diagonal electron Green function with respect to the electron momentum k and subband index n. The scattering term within the rotating wave approximation can be expressed as I¯= I¯ei +I¯ep +I¯ee, where I¯ei, I¯ep and I¯ee are contributions from the ei, ep and ee scatterings, respectively. Their expressions can be found in Appendix B. C. Landau damping process In the kinetic equations, Eq. (13) describes the resonant excitation of the electron polarizations, while Eq.(12)describesthe back-actionofthe polarizationsto the SPP. The summation in Eq. (12) implies that even without anyscattering,the phase-mixingbetween polarizations with different frequencies can also lead to the damping of the SPP, which is the origin of the Landau damping. To further clarify the Landau damping process de- scribed by the kinetic equations, we solve the equations without the scattering term I¯. From Eq.(13), the corresponding polarization can be solved as Pσ(nk,n′k′;t)=−δk′−k−qgqnn′pQqsB¯s†(t) ×[fnσ(k)−fn′σ(k′)][ei(ωknkn′′+i0+)t−1]/(ωknkn′′ +i0+).(15) Note that in the derivation, we have assumed that the SPP amplitude B¯ varies slowly compared to the polar- s ization P. By substituting Eq. (15) into Eq. (12) and taking the long time limit t , one gets →∞ ∂ B¯ /B¯ = (τ−1+iω ), (16) t s s s − where ω represents the frequency shifting, expressed as s 1 ωs = |pQqsgqnn′|2[fn′σ(k′)−fnσ(k)]ωnn′δk′−k−q, nk,n′k′,qσ kk′ X (17) in which the summation is understood as a principal FIG. 2: (Color online) Schematic of the resonant pairs cor- valueintegral. τ−1 isthedampingrateoftheSPP,which responding to the case of (i) strong Landau damping and has the form (ii) weak Landau damping in the electron spectrum (a) and τ1 =π |pQqsgqnn′|2[fn′σ(k′)−fnσ(k)]δ(ωknkn′′)δk′−k−q. ignionthseofetlhecetrroesnondainsttrpibauirtsioanre(ibll)u.strTatheedtbwyothwicakvegvreecetno/rblruee- nk,Xn′k′,qσ curves corresponding to the strong/weak Landau damping (18) regime. The resonant pairs are centralized around the res- NotethatEq.(18)agreeswiththe Landaudampingrate onance corresponding to the SPP central wavevector Qs as derived from the Fermi golden rule in the literature.44,45 indicated by the vertical black dotted curves. The horizon- Theabovesolutionsuggeststhatthe Landaudamping tal dashed line marks the chemical potential of the electrons process can be understood as the resonant absorptionof µ. (c) The electron polarization and electron population dif- theSPPbyelectrons. Thetwoδ-functionsinEq.(18)in- ference as function of the center wavevector K = k+k′ with 2 dicatethatforamonochromaticSPPwavewithwavevec- k′−k=Qs. (i)/(ii) corresponds to thestrong/weak Landau tor q, the absorption occurs between pairs of states nk dampingregime atlow temperaturesand (iii) correspondsto and n′k′ satisfying the energy and momentum con|seri- both the strong and the weak Landau damping regimes at | i high temperatures. The vertical solid line marks the reso- vations nancecorresponding to Qs. ωnn′ = 0, (19) kk′ k′ k = q. (20) − 5 Eachpair ofthe states nk and n′k′ consista resonant trons. In contrast, in the weak Landau damping regime, pair (nk,n′k′) relevant|fori the S|PPidamping. For the the correspondingresonantpeak isweakandliesoutside multi-subband system, there usually exist several such the large δf region [(ii) in Fig. 2(c)], indicating a small resonant pairs, laying between different subbands n and SPP absorption. In addition to the resonant peak, side n′ and being well-separated from each other. For a non- peakscanalsoexistintheoff-resonantregimeduetothe monochromaticSPPwavepacketwithsufficientlynarrow correspondinglargeδf. At high temperatures where the spectrum in q, the two states nk and n′k′ of eachres- Fermisurfaceissmearedout,thepopulationdifferenceis | i | i onant pair become two wavevector regions. Note that rather flat for both the strong and weak Landau damp- in the following discussion, we shall focus on the non- ingregimesandthecorrespondingpolarizationexhibitsa monochromatic SPP wave packet, and the resonant pair strong peak around the resonance for both regimes [(iii) is referred to as the wavevector region unless otherwise in Fig. 2(c)]. specified. The resonantpairsareillustratedin Figs.2(a) and (b). By using the resonant pairs, one can rewrite Eq. (18) into D. Influence of the scattering τ−1 = τi−1, (21) The scattering influences the Landau damping by i changing the resonant excitation of the polarizations. X τi−1 = π |pQqsgqnn′|2[fn′σ(k′)−fnσ(k)] Schpaencinfieclas,llyin,d(u1)cinthgeascdaetctaeyrinogf ctahne pinotlraordizuacteiodni;ss(i2p)attihvee Xqσ (nk,Xn′k′)∈i scattering between polarizations with different preces- δ(ωnn′)δ , (22) × kk′ k′−k−q sion frequencies induces a frequency-mixing, leading to a modification of the polarization precession frequency. withibeingtheindexfortheresonantpaircorresponding These two effects can be further clarified by assuming to the SPP wave packet, whose spectrum is decided by that for each resonant pair, the scattering term has the the line-shape function pQs. (nk,n′k′) i means that thetwostates nk and nq′k′ belongtoth∈ei-thresonant form | i | i pair. ThusonecanseethatthedampingrateoftheSPP I¯iσ = Γ [P (nk q,n′k′ q) P (nk,n′k′)], (23) wave packet is the sum over the absorption rates of all nk,n′k′ i σ − − − σ q the relevant resonant pairs. X Note that the electron population difference δf = whereΓ standsforthephenomenologicalrelaxationrate i f (k′) f (k)ofthecorrespondingresonantpairplays n′σ nσ for the polarization of the i-th resonant pair. Note that − an important role on its contribution to the SPP damp- (nk,n′k′)and(nk q,n′k′ q)belongtothei-thresonant ing. For the degenerate electrons where a well-defined pair. − − Fermisurfaceexistsaroundthe chemicalpotential,there For eachresonantpair, the polarizationP in the scat- exist two regimes of the SPP damping: (i) a strong Lan- tering term I¯ given above can be obtained by treating dau damping regime where states nk and nk′ of a Γ perturbatively and solving Eq. (13) order by order, | i | i i resonantpairlayineachsideofthechemicalpotentialin yielding the electron spectrum, leading to a large population dif- LfearnendcaeuδdfamanpdinhgernecgeimaeswtrhoenrgeSthPePchdeammipcianlgp;o(tiei)ntaiawleliaeks Pσ(nk,n′k′;t) = δk′−k−qgqnn′pQqsB¯s†(t)[fnσ(k)−fn′σ(k′)] outsidealltheresonantpairs,leadingtoasmallδf anda ×[ei(ωknkn′′−Γ¯ai+iΓ¯bi)t−1]/(ωknkn′′ −Γ¯ai +iΓ¯bi), (24) weak SPP damping. This is illustrated in Figs. 2(a) and (b). Note that at high enough temperatures, the Fermi where surface canbe smearedoutand suchdifference vanishes. ωnn′ ItshouldbeemphasizedthataccordingtoEq.(15),the Γ¯b =Γ (1 kk′ ), (25) resonant pair can be visualized as the resonant peak in i i − ωnn′ q k−q,k′−q the polarizationsbetweenthe twosubbands nandn′. In X Fig. 2(c), we illustrate the polarizations P(nk,n′k′) cor- Γ¯ai =πΓi (ωknkn′′ −ωkn−n′q,k′−q)δ(ωkn−n′q,k′−q). (26) respondingto the resonantpairsshowninFigs.2(a) and Xq (b) as function of the center wavevector K = (k+k′)/2 The summation in Eq. (25) is understood as a princi- withk′ k=Q . Onefindsthatthepolarizationexhibits − s pal value integral. Note that we have omitted the k,k′- aLorentzianpeakaroundtheresonancecorrespondingto dependenceofΓ¯a(b) insideeachresonantpairforsimplic- the central wavevectorQs, as indicated by Eq. (15). i ity. The detail of the derivation is given in Appendix C. Note that such resonant peak can show different fea- BycomparingEq.(24)toEq.(15),onecanseethatthe turesinthestrongandweakLandaudampingregimesat lowtemperatures. InthestrongLandaudampingregime, detuning ωknkn′′ in the resonant denominator is modified the corresponding polarization P exhibits a strong reso- intoωnn′ Γ¯a,indicatingthatthepolarizationprecession kk′ − i nantpeakconcentratedintheregionwithlargeδf [(i)in frequency is shifted by the scattering. The scattering Fig.2(c)], indicating a largeSPP absorptionby the elec- also induces a finite imaginary part Γ¯b to the resonant i 6 theenergyshiftmodifiestheenergyconservationEq.(19) into ωnn′ Γ¯a =0. (27) kk′ − i Thus the corresponding resonant pairs are shifted by the scattering. On the other hand, the energy broadening loosens the energy conservation constraint given by Eq. (19). Thus the corresponding resonant pairs are broadened. Note that the broadening and shifting are usually small andcannot induce overlapsbetweendiffer- ent resonant pairs. Accordingly, the broadening Γ¯b and shifting Γ¯a also i i manifestthemselvesintheSPPdampingrate. Following the same procedure of the derivation of Eq. (18), the SPP damping rate in the presence of the scattering can be written as τ−1 = τ−1, (28) i i X τi−1 = |pQqsgqnn′|2[fn′σ(k′)−fnσ(k)] qσ (nk,n′k′)∈i X X Γ¯b i δ . (29) × (ωnn′ Γ¯a)2+(Γ¯b)2 k′−k−q kk′ − i i By comparing the above equations to Eq. (18), one ob- serves that the δ-function corresponding to the energy conservationEq.(19)isbroadenedintoaLorentzianwith width Γ¯b and shift Γ¯a. i i Itshouldbeemphasizedthatthebroadeningandshift- ing of the resonance pair can also be visualized as the broadening and shifting of the corresponding resonant peakinthepolarizationsasillustratedinFig.3. Thisof- fers asimply wayto interpretthe influence ofthe broadening and shifting on the SPP damping rate. The influence of the shifting depends on the direction of the shift. From the figure, one can see that the shifting re- ducesthe resonantpeak inthe polarizationifthepeakis shifted towards the region with smaller δf, thus the ab- FIG. 3: (Color online) Schematic of the effect of the broad- sorptionof the SPP by the correspondingresonantpairs ening and shifting on the electron polarization for (a) strong isreduced[(iii)inFigs.3(a)and(c)]andthecorrespond- Landau damping regime at low temperatures; (b) weak Lan- ing SPP damping rate is suppressed. Otherwise, if the daudampingregimeatlowtemperaturesand(c)bothstrong peak is shifted towards the region with larger δf [(iii) and weak Landau damping regimes at high temperatures. The thin solid vertical lines represent the resonance without in Fig. 3(b)], the absorption is enhanced and the SPP broadeningandshifting. Theresonanceswiththebroadening damping rate is enhanced. andshiftingarerepresentedbythethicksolidandthindotted Theinfluence ofthe broadeningcanbe differentinthe vertical lines, respectively. For clarification, only theshifting strong and weak Landau damping regimes at low tem- towards the small K direction is illustrated. Note that the peratures as illustrated in Figs. 3(a) and (b). In the populationdifferencesδf becomeflatinbothregimesathigh strong Landau damping regime, the broadening can re- temperatures. duce the sharpresonancepeakofthe polarization[(ii) in Fig. 3(a)] and suppress the absorption of the SPP. Thus thecorrespondingSPPdampingrateissuppressedinthis denominator, representing the decay of the polarization regime. Incontrast,intheweakLandaudampingregime, due to the scattering. the broadeningofthe resonanceincreasesthe absorption Theabovesolutionindicatesthatthescatteringmodi- from the region with larger δf [(ii) in Fig. 3(b)], thus fies the resonance between the polarizationand the SPP the absorption is enhanced and the corresponding SPP byintroducingbothanenergyshiftandanenergybroad- damping rate is enhanced. Note that at high tempera- eningtothecorrespondingresonancepairs. Ononehand, tures, as the polarization exhibits a sharp peak around 7 theresonanceinboththestrongandweakLandaudamp- with V˜ being the screened ee interaction. k¯ = q 1 ing regimes, the scattering tends to suppress the SPP 2(εn εn′+Ω )/Q Q /2andk¯ =2(εn εn′ + damping rate in both regimes [(ii) in Fig. 3(c)]. −Ω )/Qk−Q+sQ−/k2′. Γ¯a(sei),Γ¯sa−(ep)sandΓ¯a(ee2)correskp−ondk′−toQtshe s s s i i i contributions from the ei, ep and ee scatterings, respectively. The corresponding broadenings have the forms E. Broadening and shifting from a simplified model vñn vñ′n′ In order to gain a further understanding of the micro- Γ¯bi(ei) = m∗πni( 0k − 0k′ )(v˜0nn−v˜0n′n′), (37) scopic origin of the broadening and shifting, we discuss | | | | the contributions of the ei, ep, and ee scatterings to the Γ¯b(ep) = πm∗( Mnn Mnn′ ) i Q,−q Q,−q broadeningandshifting withinasimplifiedmodelinthis Xσ n XQ section. [N> f>(k q)+N< f< (k q)]/k q Withinthis model,weassumethatthe scatteringonly × LO nσ − LO nσ − | − | q=q(k) occurs between the polarizations inside each resonant (cid:12) + pair. Under such assumption, all the scattering terms +πm∗( MQnn,−qMQnn,−′q) (cid:12)(cid:12) can be written in an unified form for the i-th resonant XQ pair [N< f>(k q)+N> f< (k q)]/k q × LO nσ − LO nσ − | − | q=q(k) (cid:12) − o I¯iσ = Γa (q)P (nk q,n′k′ q) + nk n′k′ , (cid:12) (38) nk,n′k′ nk,n′k′,i σ − − { ←→ } (cid:12) Xq n Γ¯b(ee) = 2π [Πn(n¯k¯,0) Πn (n¯k¯,0)]/k k¯ −Γbnk,n′k′,i(q)Pσ(nk,n′k′) , (30) i n Xn¯k¯ n − n′ | − |o o + nk n′k′ . (39) whereboth(nk,n′k′)and(nk q,n′k′ q)belongtothe { ←→ } − − i-th resonant pair. Note that the corresponding Γa/b for whereq(k) satisfies[q(k)]2 2kq 2m∗Ω =0andvñn′ eachscatteringmechanismcanbeobtainedbycomparing represen±ts the screen±ed e−i inte±ra±ction. LOnk n′qk′ the scatteringterms[Eqs.(B11-B13)]toI¯niσk,n′k′ givenin stands for the same term as in the previo{us ←b→ut with} the above equation. the interchange of indices nk n′k′. {} Note that Eq.(30) has a similar structure as Eq. (23), From Eqs. (33-39), one find←s→that different scattering thus one can derive the corresponding broadening and has different contribution to the broadening and shift- shifting following the similar procedure, yielding ing. Only the ee scattering contributes to the shifting, whilethecontributionsfromtheeiandepscatteringvan- Γ¯b = Γb (q) Γa (q) ωknkn′′ , (31) ish. Although all the scatterings can contribution to the i nk,n′k′,i − nk,n′k′,i ωnn′ broadening,theirrelativeimportancecanbequitediffer- Xq (cid:16) k−q,k′−q(cid:17) ent. This is because for the nanowires considered here, Γ¯ai =π Γank,n′k′,i(q)(ωknkn′′ −ωkn−n′q,k′−q)δ(ωkn−n′q,k′−q).(32) the ei, ep and ee matrix elements vqnn′, MQnnq′ and Vqnn′ q arenotsensitivetothesubbandindexn. Thus,fortheei X and ee scatterings, according to Eqs. (37) and (39), the ThebroadeningΓ¯b andshiftingΓ¯a duetoeachscattering termsinthebracketcanlargelycanceleachother,leading i i canbe evaluatedby substituting the correspondingscat- tosmallcontributionsleft. Incontrast,suchcancellation teringtermintoEqs.(30-32). Notethatwehaveomitted is absent for the ep scattering, thus its contribution to the k,k′-dependence of Γ¯a(b) inside each resonant pair the broadening is expected to be larger than the ones i to make the equation simple and physically transparent. from the ei and ee scatterings. For each resonant pair, Γ¯a(b) is evaluated by choosing Equations(28)and(29)withthebroadeningandshift- i (nk,n′k′) corresponding to the SPP central wave vector inggiveninEqs.(33-39)consistthe analyticsolutionfor Qisspe(ia.ke.e,dka′t−qk==QQ.s)ThsiencsehitfthienglinΓ¯ea-schaanpebefuwnrcittitoennpaQqss tshecetiSoPnPoffdearsmapinsigmrpalete.picTtuhree atonaulyntdicersstoaluntdiotnheinintflhuis- s i ence of the scattering on the SPP damping: The SPP Γ¯a(ei) = 0, (33) dampingcomesfromthe resonantabsorptionofthe SPP i by electrons,while the scattering canintroduce a broad- Γ¯a(ep) = 0, (34) ening and a shifting to the resonance and hence affects i Γ¯a(ee) = 2π Πn (n¯k¯ ,0), (35) the damping process. At low temperatures, the broad- i n′ j ening tends to suppress the SPP damping rate in the j=1,2 n¯ X X strong Landaudamping regime. While in the weak Lan- dau damping regime, the broadening tends to enhance where theSPPdamping. Athightemperatures,suchdifference Πn (n¯k¯,q)= Vñn¯Vñ¯n′f> (k¯+q)f< (k¯), (36) vanishesandthebroadeningtendstosuppressthedamp- n′ q q n¯σ n¯σ ing in both regimes. The shifting can suppress the SPP σ X 8 damping if the resonance is shifted towards the region line-shape function of the SPP wave packet is peaked at with smaller δf, but boost it if the resonance is shifted Q . There are four resonant pairs in Fig. 4(b) which lay s toward the region with larger δf. Different scatterings between different subbands: pair (i) is between the sub- can have different contributions to the broadening and bands1and4,pair(ii)isbetweenthe subbands2and6, shifting. From the simplified model in this section, one pair(iii)isbetweenthesubbands3and9andpair(iv)is findsthatthe shiftingisdeterminedbythe eescattering, betweenthesubbands4and8. Inthefigure,thefourres- andthebroadeningismainlydecidedbytheepscattering onantpairs (i-iv) are denoted by the skyblue dots, green for the typical nanowires considered here. squares, brown open circles and yellow triangles, respectively. Thesubbandscorrespondingtoeachresonantpair arealsoplottedwithsolidcurvesinthesamecolor. Note III. NUMERICAL RESULTS thatforR=34nm,onlythe resonantpair(i)isrelevant for the damping. For R = 38.5 and 40 nm, both the resonant pairs (i) and (ii) are relevant. For R = 43 nm, In the numerical investigation, we choose nanowires all the four resonant pairs (i-iv) contribute to the SPP to be free-standing InAs nanowires. The typical electron density n¯ is in the range of 1017 1018 cm−3 damping. 0 ∼ and the wire radius R is around 25 75 nm.74 The LO From Fig. 4(b), one can see that as the radius R in- ∼ phonon energy Ω =29 meV and the electron effective creases,the resonantpairs move from left to right in the LO mass m∗ = 0.023m with m representing the free elec- electron spectrum. When a resonant pair moves across 0 0 tron mass. The dielectric constants of the nanowires are the chemical potential marked by the horizontal black ǫ∞ = 12.3 for high frequency and ǫ0 = 15.5 for low fre- dashed lines, a crossover between the strong and weak 1 1 quency. The dielectric constant outside the nanowire is Landaudamping regimes occurs,whichinduces the size- ǫ = 1.0. Note that for such nanowires, there are 10-20 dependent oscillations shown in Fig. 4(a). Note that 2 electron subbands relevant to the SPP damping. We set the peaks/valleys correspond to the strong/weak Lan- the impurity line density n = 0.5n with n = πR2n¯ dau damping regimes. For example, the oscillation from i e e 0 being the electron line density. R = 32 to 38.5 nm is due to the crossover induced by By numerical solving the kinetic equations Eqs. (12) theresonantpair(i). Whilethecrossoverinducedbythe and (13), one obtains the temporal evolution of the resonant pair (ii) induces the oscillation from R = 38.5 SPP amplitude B¯ . The SPP damping rate τ−1 can to 43 nm. One finds from the figure that R = 34 and s be extracted by fitting the real part of B¯ with a sin- 40 nm correspondto the strong Landaudamping regime s gle exponential decay of the cosine oscillation: Re[B¯ ]= whereas R = 38.5 and 43 nm correspond to the weak s B¯0exp( t/τ)cos(ω t),wheretheinitialvalueoftheSPP Landau damping regime. amsplitu−deB¯0ischossentobereal. Wesetthewavepacket OnealsoobservesfromFig.4(b)thatdue tothe many s length L=100R (Ref. 75). subbands in the nanowires, there usually exist multi- resonantpairsrelevantfortheSPPdamping. Foragiven Q , more and more resonant pairs become involved as s A. Landau damping: Size and temperature the radius R increases. The number of relevant reso- dependence nant pairs are labeled in the R-Q plane in Fig. 4(a). s Note that as the number of the resonantpairs increases, Before we discuss the influence of the scattering, it the magnitude of the size-oscillations become less pro- is helpful to first obtain an understanding of the SPP nounced. Thisismainlybecausetheoscillationsareusu- damping without scattering. In Fig. 4(a), we show a allyinducedby the crossoverdue toone resonantpairas typical behavior of the SPP damping rate as function of R varies. For the system with many resonant pairs, the theSPPcentralwavevectorQ andwireradiusRforthe contribution from one resonant pair becomes less signif- s nanowire with electron density n¯ = 1.5 1017 cm−3. icant. Thus the size-dependent oscillations can be sup- 0 The temperature is chosen to be 100 K. O×ne finds that pressed for nanowires with large R. theSPPdampingrateoscillateswiththe radiusR. Note It should be emphasized that the size-dependent os- that similar size-dependent oscillations have also been cillations can also be suppressed by increasing temper- reportedinmetal nanoparticlesandthin films.43–45,76–78 ature T. This is because the crossover is more pro- Such oscillations are usually attributed to the quan- nounced for strongly degenerate electrons where a clear tized electron states in the nanostructures,43–45 which Fermi surface exists around the chemical potential. In can be understood in terms of the resonant pairs in the high-temperature regime, the crossover is largely sup- nanowires we studied here. To illustrate this, we con- pressed. To show this, we plot the damping rate τ−1 centrate on the damping rate corresponding to a typical as function of the radius R and temperature T for Q = s SPPcentralwavevectorQ =1.66 10−2 nm−1 [skyblue 1.66 10−2 nm−1 in Fig. 4(c). One sees that as temper- s curvesinFig.4(a)]andshowthe co×rrespondingresonant ature×increases, the damping rate τ−1 corresponding to pairs for radii R = 34, 38.5, 40 and 43 nm in Fig. 4(b). the strong Landau damping regime decreases, while τ−1 Each resonant pair can be represented by the resonance corresponding to the weak Landau damping regime in- corresponding to the central wave vector Q , since the creases. Thisleadstoasuppressionofthesize-dependent s 9 (a) (b) RR==3388..55 nnmm R=34 nm R=40 nm RR==3344 nnmm RR==4400 nnmm 0.15 RR==4433 nnmm TT==110000KK V) 0.1 e ω ( 0.05 1100 τττ---111 (((pppsss---111)))00..11 QQss==11..6666 ×× 1100--22 nnmm--11 0.1 05 R=38.5 nm R=43 nm 1100--33 1188 1100--55 3322 33144 3366 33882 4400 4422 44444 4466 1155 1166QQ ..((ss5511--0033 nn--mm11)) ω (eV) 0 0.0.15 RR ((nnmm)) 0 -0.15 0 0.15 -0.15 0 0.15 k (nm-1) k (nm-1) (c) (d) 100 R=34 nm R=40 nm Q=1.66 × 10-2 nm-1 s 1) -ps 100 -1τ ( 10-1 ((iii)) ((iiivi)) nnoosscc((NA)) ττ--11 ((ppss--11))10-1 R=38.5 nm R=43 nm 10-2 10-3 200 300 -1ps) 10-1 46 44 42R 4(0nm 3)8 36 34 32 100 T (K) -1τ ( 10-2 50 100 150 200 250 300 50 100 150 200 250 300 T (K) T (K) FIG. 4: (Color online) (a) The SPP damping rate as function of SPP central wave vector Qs and wire radius R without scattering for n¯0 =1.5×1017 cm−3 andT =100 K.Theskybluecurverepresentsthedampingrate correspondingtotheSPP central wave vector Qs = 1.66×10−2 nm−1. The number of the relevant resonant pairs are labelled in the R-Qs plane. (b) The resonant pairs for R = 34, 38.5, 40 and 43 nm corresponding to Qs = 1.66×10−2 nm−1 in the electron spectrum. For clarification, only the lowest 10 subbands are plotted. (c) The SPP damping rate without scattering as function of radius R and temperature T for Qs =1.66×10−2 nm−1. (d) Temperature dependence of the damping rate τ−1 without scattering for R=34, 38.5, 40 and 43 nm with Qs=1.66×10−2 nm−1. Symbols correspond to thenumerical results. Red curvesrepresent the results from the analytic solution. The contributions of each resonant pair from the analytic solution are also plotted as curves with different colors. The skyblue double-dotted chain, green chain, brown dotted and yellow solid curves correspond to thecontribution from theresonant pair (i-iv), respectively. oscillations in high-temperature regime. pairscanbequitedifferent. ForthestrongLandaudamp- One can also obtain the above results from the ana- ing regime, there usually exists one resonant pair whose lytic solution of the kinetic equations without scattering contributionis muchlargerthanthe other pairs. For ex- [Eqs. (21) and (22)]. To show this, we compare the tem- ample,thedampingrateτ−1ismainlydeterminedbythe peraturedependenceofτ−1fromboththenumerical(red resonantpairs(i)and(ii)forR=34nmandR=40nm, squares)and analytic (red solid curves) solutions for the respectively. For the weak Landau damping regime, the nanowires with R=34, 38.5, 40 and 43 nm in Fig. 4(d). contributions from different resonant pairs can be com- One finds good agreement between each other, indicat- parable. For example, for R = 38.5 and R = 43 nm, al- ing that the analytic solution without scattering offers though the resonant pair (ii) has a large contribution to a good estimation to the numerical results. Note that the damping rate τ−1, the other resonant pairs can also according to the analytic solution, the temperature de- play important roles, especially at high temperatures. pendence of the SPP damping rate originates from the population difference of the resonant pairs. From the analytic solution, one can also identify contributions from different resonant pairs, which are plotted as curves with different colors and line shapes in Fig.4(d). Theskybluedouble-dottedchain,greenchain, Fromthe above results, one finds that the SPP damp- brown dotted and yellow solid curves correspond to the ingexhibitssize-dependentoscillationsanddistincttem- contributions from the resonant pair (i-iv), respectively. perature dependence without scattering, which can be It is clear that the relative importance of the resonant explained by the analytic solution. 10 (a) 100 nosc(N) nosc(A) ep(N) ep(A) 100 1) -s ττ--11 ((ppss--11))1100--21 -1τ (p 10-1 300 R=34 nm R=40 nm 10-3 200 i(A) iii(A) 46 44 42R 4(0nm 3)8 36 34 32 100 T (K) -1ps) 10-1 ii(A) iv(A) (b) 1 ( 100 -τ 10-2 R=38.5 nm R=43 nm 1) 50 100 150 200 250 300 50 100 150 200 250 300 -ps T (K) T (K) 1 ( -τ 10-1 FIG. 6: (Color online) Comparison between the numerical R=34 nm R=40 nm and analytical results of the SPP damping rate for R = 34, 38.5, 40 and 43 nm with the ep scattering. Blue dots repre- -1ps) 10-1 nosc sthenetrtehseulntsumfreormicatlhreesaunltasl,ywtichilseoltuhteiobnl.ueTdhaeshceodnctruirbvuetsiosnhoowf -1τ ( eepe each resonant pair from the analytic solution is also plotted. 10-2 R=38.5 nm R=43 nm aelli The skyblue double-dotted chain, green chain, brown dotted and yellow solid curves correspond to the contribution from 50 100 150 200 250 300 50 100 150 200 250 300 theresonantpair(i-iv),respectively. Forcomparison,thenu- T (K) T (K) merical and analytical results for the damping rate without scattering are also plotted with red squares and solid curves, FIG.5: (Color online) (a) SPPdampingratein thepresence respectively. of all thescattering as function of radius R and temperature T forQs=1.66×10−2 nm−1. (b)SPPdampingrateτ−1cal- culatedfromthenumericalresultsforQs=1.66×10−2 nm−1 In the presence of the scattering, it is seen that the withdifferentscatteringforR=34,38.5,40and43nm. Sym- damping rate τ−1 is markedly suppressed in the strong bols with big blue dots, small green squares, small olive dots represent τ−1 calculated with the ep, ee and ei scatterings, Landau damping regime (R = 34 and 40 nm). In con- respectively. Thebrowntrianglesrepresentτ−1 calculated in trast, for the weak Landau damping regime (R = 38.5 the presence of all the three scatterings and the red squares and 43 nm), the scattering plays different roles in differ- represent τ−1 without scattering. ent temperature regimes: The damping rate is markedly enhanced in the low-temperature regime (T . 150 K), butis largelysuppressedinthe high-temperatureregime B. Influence of scattering (T &150K).Acrossoverexistsattheintermediatetem- perature regime. It is also noted that the damping rate canbeenhanced/suppressedbyalmostoneorderofmag- Now we discuss the influence of the scattering on the nitude by the scattering. SPP damping. In Fig. 5(a), we plot the damping rate τ−1 as function of the radius R and temperature T for To understand these influences, we first identify the Q =1.66 10−2nm−1 inthepresenceofallthescatter- dominant scattering mechanism. To do so, we calcu- s ing. Comp×aringtothecasewithoutscattering[Fig.4(c)], late the damping rates τ−1 with the ep, ee or ei scatter- one finds that the scattering has pronounced influence ing only, and plot them with big blue dots, small green on the SPP damping: (1) The size-dependent oscilla- squaresand smallolive dots in Fig. 5(b), respectively. It tions are effectively smeared out, and (2) the temper- is clear to see from the figure that the damping rate τ−1 ature dependence also becomes weaker compared to the is dominated by the ep scattering.79 case without scattering. We first concentrate on the ep scattering. From the To gain a better understanding of the influence of the analytic solution within the simplified model, we have scattering,in Fig.5(b), we comparethe temperature de- attributed the effect of the ep scattering to the broaden- pendenceofthedampingrateτ−1 withandwithoutscat- ing of the resonant pairs. To see if this picture gives a tering for nanowireswith four typicalradiiR=34,38.5, properdescriptionoftheinfluence oftheepscatteringin 40 and 43 nm. The brown triangles represent τ−1 cal- general case, we calculate the temperature dependence culated in the presence ofall the three scatterings,while of the damping rate τ−1 by using the analytic solution τ−1 without scattering are plotted with red squares for Eqs.(28)and(29)withtheep-scattering–inducedbroad- comparison. Note that R=34 and 40 nm correspond to ening given in Eq. (38). The calculated analytic results the strong Landau damping regime, whereas R = 38.5 are compared to the numerical ones in Fig. 6. and 43 nm correspond to the weak one. Inthe figure,theblue dotsrepresentthedampingrate