Table Of ContentKinetic 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 influ-
ence 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, re-
spectively.
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 approx-
imation 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 res-
onant 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 polar-
izations 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 corre-
sponding 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 broad-
ening 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 broad-
ening and shifting on the SPP damping rate. The in-
fluence 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, respec-
tively. The corresponding broadenings have the forms
E. Broadening and shifting from a simplified model
v˜nn v˜n′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 =0andv˜nn′
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)= V˜nn¯V˜n¯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, respec-
tively. 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 elec-
tron 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 con-
tributions from different resonant pairs, which are plot-
ted 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
