A Universal Self-Amplification Channel for Surface Plasma Waves

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by  Hai-Yao Deng

A universal self-amplificationchannel forsurfaceplasma waves
Hai-Yao Deng1,2, Katsunori Wakabayashi1,3, and Chi-Hang Lam4
∗ † ‡
1Department of Nanotechnology forSustainable Energy, School of Science and Technology,
Kwansei Gakuin University, Gakuen 2-1, Sanda 669-1337, Japan
2DepartmentofPhysicsandAstronomy,UniversityofExeter,StokerRoadEX44QLExeter,UnitedKingdom
3National InstituteforMaterialsScience (NIMS), Namiki 1-1, Tsukuba 305-0044, Japan and
4Department ofAppliedPhysics, TheHongKongPolytechnicUniversity, HungHum, Hong Kong
Wepresent atheoryofsurfaceplasmawaves(SPWs)inmetalswitharbitraryelectroniccollisionrateτ 1.
7 −
1 Weshowthatthereexistsauniversalintrinsicamplificationchannelforthesewaves,subsequenttotheinterplay
0 betweenballisticmotionsandthemetalsurface. Weevaluate thecorresponding amplificationrateγ0, which
2 turnsouttobeasizablefractionoftheSPWfrequencyωs.Wealsofindthatthevalueofωsdependsonsurface
scatteringproperties,incontrastwiththeconventionaltheory.
n
a
J
I. INTRODUCTION interplaybetweenballisticelectronicmotionsandboundaries
4
resultsinanelectricalcurrentthatallowsanetamountofen-

] Collective electronicoscillations on the surface of metals, ergytobedrawnfromtheelectronsbySPWs,whichwouldin-
ll dubbed surface plasma waves (SPWs) [1–3], have emerged evitablyself-amplifyinthecollision-lesslimit,therebydesta-
a asapivotalplayerinnanoscopicmanipulationoflight[4–8]. bilizing the system. The hereby predicted self-amplification
h Thefunctionalityofmanyprototypicalnanophotonicdevices of collision-less SPWs is analogous to the Landau damping
-
s critically relies on the distance SPWs can travel before they in collisionless bulk plasma waves [28–31]. While the latter
e aredampedoutduetoenergylossesviaseveralchannels[7– is caused by slowly moving electrons that strip energy from
m
12]. SPWs canlose energydue to Joule heat, inter-bandab- the wave, the former is attributable to ballistically moving
t. sorption,radiationemissionandindividualelectronicmotions electronsimpartingenergyto the wave when boundariesare
a (Landaudamping). Most of the losses can be efficiently but present.
m
nottotallyreducedunderappropriatecircumstances. Ampli- Inthenextsection,wedescribethesystemunderconsider-
- fiers have been contrived to compensate for the losses [13– ation,stateourmainresultsandconceivetheirpossibleexper-
d
21],whichareallextrinsicandrequireexternalagentssuchas imentalsignature.InSec.III,theSPWtheoryispresented.In
n
o adipolargainmediumtosupplytheenergy. Sec.IV,wedetailthemethodsusedinthetheory.Wediscuss
c ThestandardtheoryofSPWswasformulatedshortlyafter the results and conclude the paper in Sec. V. Some calcula-
[ theirdiscoveryin1957[1]andhasbeencomprehensivelydis- tionsandargumentsnotcoveredinSec.IIIandIVaregiven

1 coursed in many textbooks [3, 5, 6, 22, 23]. In this theory, inappendicesAandB.
v theelectricalpropertiesofmetalsareeffectedbyafrequency-
0 dependent dielectric function ǫ. To analytically treat ǫ, the
6 simple Drude model or the slightly more involved hydrody-
II. RESULTS
0
namicmodel[22,24,25,27]isinvariablyinvoked.Foreither
1
model to be valid, electronic collisions must be sufficiently
0
frequentsothattheelectronicmeanfreepath,l =v τ,where System - We consider a prototypical system, namely, a
. 0 F
1 v is the Fermi velocity and τ the relaxation time, is much semi-infinitemetal(SIM)occupyingthehalfspacez 0and
F ≥
0 shorter than the SPW wavelength or the typical length scale interfacing with the vacuum at a geometrical surface z = 0.
7 of the system. The general case with arbitrary τ, especially ASIMisnotsheerlyofacademicinterest: itcanberegarded
1
the collision-less limit, where τ , defies these models asonehalfofathickmetalfilm. Inthespiritoftheso-called
:
v andhasyettobeentertained.Othe→rm∞odelsbasedonabinitio jelliummodel,[32–34]themetalistreatedasafreeelectron
Xi quantummechanicalcomputations[22,26]arehelpfulinun- gasembeddedinastaticbackgroundofuniformlydistributed
derstandingthecomplexityofrealmaterialsbutfallsshortin positivecharges. Onthewholethesystemisneutral. Theki-
ar providinganintuitiveandsystematicpictureofSPWsunder- netic energy of the electrons is written ε(v) = 1mv2, where
2
pinnedbyelectronsexperiencinglessfrequentcollisions. m and v denote the mass and velocity of the electrons, re-
InthepresentworkweemployBoltzmann’stransportequa- spectively. Inter-bandtransitionsareaccordinglyignoredbut
tiontoderiveatheoryofSPWsinmetalswitharbitraryτ. Our their effects will be discussed in Sec. V. The surface is of a
analysisrevealsauniversalintrinsicamplificationchannelfor hard-walltypeandpreventselectronsfromleakingoutofthe
SPWs. Theexistenceofthischanneldoesnotdependonτbut system. Throughoutwe reserver = (x,y) for planarcoordi-
iswarrantedbyageneralprinciple. Weshowthattheunique nates while x = (r,z) denotes the complete position vector.
Weneglectretardationeffectsintotal.
Key results - We find that the SPW frequency ω and its
s
amplificationrateγcanbewritteninthefollowingform,
[email protected]
∗
†[email protected] 1
‡[email protected] ωs =ωs0β(p), γ=γ0− τ, γ0 >0. (1)
2
Here ω = ω /√2 is the SPW frequency in the hydro- own electric field E(x,t). Here t denotes the time. Thanks
s0 p
dynamic/Drude model, with ω being the characteristic fre- to the linearity and symmetries of the system, we can write
p
quencyofthemetal,p [0,1]isaparameterthataccountsfor ρ(x,t)=Re ρ(z)ei(kx ωt) andE(x,t)=Re E(z)ei(kx ωt) with-
∈ − −
surfacescatteringeffects,β(p) 1weaklydependson pand out loss of ghenerality. Hiere Re/Im takeshthe real/imaiginary
∼
γ0 includesLandaudampingandisexactlyindependentofτ. partofaquantity,k 0denotesthe wavenumberwhile ωis
Both ωs and γ0 are determined by the secular equation (18) the SPW eigen-frequ≥ency to be determined by the equation
giveninSec.IV.Analyticalexpressionscanbefoundforβ(p) of motion. In general, ω can be complex. Instead of ρ(z), it
and γ0 under certain approximations. Exact ωs and γ0 have provesmoreconvenienttoworkdirectlywith
beencomputednumericallyandaredisplayedinFig.1,where
weEoqbusaetrivoent(h1a)tfγu0rn/ωishs0es≈aαn(i1nt+rinps),icwaimthpαlifi≈ca0t.i1o.nchannelγ0 ρq =Z0∞dzρ(z) cos(qz).
for SPWs. It also shows that, in contrast to the hydrody-
In order for the jellium model to be valid, we must impose
namic/Drude model, ω relies on surface properties via the
s that ρ = 0, where q is a cut-off. q 1 roughly gives the
parameter p. In particular, as seen in Fig. 1 (a), ωs is sig- q>qc c −c
microscopiclatticeconstantofthemetal.Wemaychooseq
nificantly – more than 10% – larger than ω , the value ex- c
pectedfromthehydrodynamic/Drudemodel.s0Inotherwords, kF = mvF/~ ∼ n01/3 ∼ ks = ωs0/vF, where kF is the Ferm∼i
β(p)isunityinthesemodelswhereasitcouldreachupto1.2 wavelengthandn0 themeanelectrondensity. Theresultsdo
in our theory. Such a great contrast would be ideal for ex- notdependontheexactvalueofqcaslongasitissufficiently
perimentallyverifyingthetheory. Unfortunately,incommon large.Bydefinition,
materialssuchasnoblemetals,duetopronouncedinter-band
2
transitions,thereisnosimplerelationbetweenω andω . ρ(z)= ∞dqρ cos(qz).
s p πZ q
Experimentalsignature - We conceiveone possible reper- 0
cussion of the self-amplification channel, noting that the net Intermsofρ ,wecaneasilyfind,bysolvingLaplace’sequa-
q
amplification rate γ = γ0 1/τ and its temperature depen- tion,that
−
dence can be directlymeasured in variousways, e.g. by ex-
4kρ
aminingtheSPWpropagationdistanceorthewidthoftheen- E (z)= i ∞dq q 2cos(qz) e kz , (2)
ergyloss peaksin electronenergyloss spectroscopy(EELS) x − Z0 k2+q2 h − − i
orthequalityofthereflectancedipsintheKretschmann-Otto 4kρ
E (z)= ∞dq q 2(q/k)sin(qz) e kz . (3)
configuration. In this paper, our calculation of γ0 is done at z Z k2+q2 − −
0 h i
zero temperature. However, arguably γ could bear a dif-
0
AsnapshotofE(x,t)foratypicalSPWisdisplayedinFig.3
ferent – probably weak – temperature dependence than 1/τ.
In sufficiently pure samples, in which the residual scattering (a)andthemagnitudeofE(z)isplottedin (b). Of course,in
makingtheseplotsρ hasbeendeterminedbytheequationof
is small enough, there might exist a critical temperature T , q
∗
motiontobesetupshortly.
above which γ < 0 while below it γ > 0, i.e. T marks a
∗
Current densities - Under E(x,t), an electrical current, of
transition of the system from the Fermi sea to another state,
seeSec.Vforfurtherdiscussions.InFig.2,wesketchthesit- densityJ(x,t) = Re J(z)ei(kx−ωt) , willflow. Itcan becalcu-
uationdescribedhere,assuming1/τisdominatedbyphonon latedusingBoltzmanhn’stransporitequation,seeSec.IV.Asa
andimpurityscattering. crucialobservation,wefindthatJ(x,t)canalwaysbewritten
To experimentally investigate the self-amplification chan- in two disparate contributions, J(z) = JD(z) + JB(z). What
nelusingthemostexperimentedmaterialslikegoldandsilver, setsthemapartistheirdistinctrelationstoE(z),asillustrated
wemustclarifytheeffectsofinter-bandtransitions,whichnot in Fig. 4. It turnsout that JD(z) follows E(z) almost locally,
onlybringaboutadditionalenergylossesbutalsostronglyaf- asintheconventionalhydrodynamic/Drudemodel–whichis
fectthevalueofω .WediscusstheseeffectsfurtherinSec.V. applicableonlywhenelectronsexecutediffusivemotions,i.e.
s
τisverysmall. Explicitly,J (z)canbewritten
D
i ω2
III. THEORY J (z)= pE(z)+J(z), (4)
D ω¯ 4π ′
Inthissection, wegiveasystematicexpositionofthethe- where ω¯ = ω+i/τ, ω = 4πn e2/m is the characteristic
p 0
orythatsupportsequation(1). Sometechnicalaspectsareleft
plasmafrequencyofthemetapland
to Sec. IV. The overallobjectiveis to set up the equationof
motion for the charge density, analyze its internal structure J(z)= ∞dq8ρqF(k,q;ω¯)cos(qz), (5)
andsolveit. We firstdiscusstherelationbetweenthecharge ′ Z k2+q2
0
density andthe electric field, andthen prescribesthe current
densityandshowthatitpossessesaspectacularstructure.We signifies a non-local contribution that generates dispersive
proceedthencetotheequationofmotion,extractingitssolu- plasmawaves,with
tionsandunveilingthepropertiesofSPWs.
Electrostatics-Weaimforestablishingtheequationofmo- F(k,q;ω¯)= m 3 d3v( e2f )v ∞ kvx+qvz l.
tion for the charge density ρ(x,t) under the influence of its (cid:18)2π~(cid:19) Z − 0′ Xl=2 ω¯ !
3
(a) (b)
0.95 0.2
Numerical (exact)
0.18
0.9
Fitting by 0.09+0.1 p
0.16
0.85
ωp ωp
/ / 0.14
ωs γ0
0.8
0.12
Numerical (exact)
0.75
0.1
Equation (1)
0.7 0.08
0 0.2 0.4 0.6 0.8 1.0 0 0.2 0.4 0.6 0.8 1.0
p p
FIG.1. NumericallycalculatedSPWfrequencyω anditsintrinsicamplificationrateγ asafunctionoftheFuchsparameter p,byequation
s 0
(18).k/k =0.1andq /k =1.5.Theapproximateanalyticalsolutionsarealsoshown.
s c s
whereJ (z)iscontributedbyelectronsdirectlyemerging
B,emg
1/τ at the surface while J (z) by reflected electrons. Here p
B,ref
is the Fuchsparameter, whichgaugesthe probabilitythat an
electron impinging on the surface gets reflected back. Both
contributionsarisefromelectronsmovingawayfromthesur-
face,v 0. Infact,
z
γ ≥
0
m 3
JB,emg/ref(z)=(cid:18)2π~(cid:19) Z d3vΘ(vz)eveiωv˜zzgB,emg/ref(v). (6)
whereω˜ =ω¯ kv and
x
−
4ρ L (v ,v ,k,q;ω¯)
residual scattering rate g (v)= ef ∞dq q emg/ref x z ,(7)
B,emg/ref − 0′Z k2+q2
0
T*
with
Temperature
k(iv v ) 2(q2v2 ω˜kv )
L (v ,v ,k,q;ω¯)= x− z z ± x .
emg/ref x z kv iω˜ − ω˜2 q2v2
z± − z
FIG.2. Possibleinstabilitycausedbytheself-amplificationchannel
uponcoolingdownthesystem. AtthecriticaltemperatureT∗,γ0 = For p ≈ 1 andsmallkvF/ω¯, it iseasy to show that JB,x(z) ≈
1/τ. Herethesketchassumesthat1/τisdominatedbyphononand 0, implying that JB(z) points normal to the surface. This is
impurityscattering. illustratedinFig.4(b).
Positiveness of γ - Equation (6) strongly constrains the
0
value of ω¯. Actually, it requiresthat Im(ω¯) 0; otherwise,
Here f0(ε) denotes the Fermi-Dirac distribution and f0′ its theintegralinthisequationwoulddivergefor≥anyz,because
dpearpievra.tivOen,lywhteicrmhsiswtiothboedtdakleinnatthezesreoriteesmcpoenrtartiubruetei.nOthnies eiωv˜zz divergesexponentially for vz → 0. In Sec. IV and Ap-
pendixA,thispointisfurtherelaborated.Since
can show that J (0) 0. The factor headingE(z) in Eq. (4)
z′ ≡
is recognized as the Drude conductivity. As this conductiv- 1
γ=Im(ω)=γ , γ =Im(ω¯), (8)
ityisnearlyimaginary,JD(x,t)pointsalmostperpendicularto 0− τ 0
E(x,t),asseeninFig.4(a).
In contrast, J (z) – the ’surface-ballistic’ current– has no the fact that Im(ω¯) 0 suggests an intrinsic amplification
simple relationBwith E(z) and reflects genuine surface ef- channelγ0 competing≥withthelosschannelτ−1. Inwhatfol-
fects: JB(z) would totally disappear if the surface were ab- lows, we show thatω¯ is independentof τ−1 by meansof the
sent. While J (z) is essentially a bulk property, J (z) orig- equationofmotion.
D B
inates from the interplay between ballistic motions and the Equation of motion - The equation of motion can be ob-
surface.Itconsistsoftwoparts, tainedbyrelatingρ(z)andJ(z)viatheequationofcontinuity,
J (z)=J (z)+pJ (z), iω¯ρ(z)+ J(z)= J (0)δ(z), (9)
B B,emg B,ref z
− ∇· −
4
(a) x (in units of k-1) (b)
s
20 30 40 50 60 70
40
14
0.8
12
Metal 30
0.4 -1k)s 10 10
of 8
0 s 20
nit 6
u
-0.4 (in 4 5
z 10
-0.8 2 EEzx
0 0 0
0 0.3 0.6 0.9 -6 0 6
Vacuum ρ(z)
FIG.3. SketchofSPWssupportedonthesurfaceofasemi-infinitemetal. k/k =0.1and p=1. (a)Chargedensitymap(color)andelectric
s
fieldmap(arrows). (b)Plotsofρ(z)andE (z),whereµ = x,z. ρ(z)iscalculatedbyEq.(16)whileE (z)by(2)and(3). Bothρ(z)andE (z)
µ µ µ
havebeennormalized.
(a) (b)
14 Diffusive current density 14 Surface - ballistic current density
-1k )s 12 -1k )s 12
of 10 of 10
s 8 s 8
nit 6 nit 6
n u 4 n u 4
z (i 02 z (i 02
20 30 40 50 60 70 20 30 40 50 60 70
x (in units of ks-1 ) x (in units of ks-1 )
FIG.4.Snapshotsof(a)diffusivecurrentdensityJ (x,t)and(b)surface-ballisticcurrentdensityJ (x,t)inSPWs.Thesetwocurrentdensities
D B
arenotdiscriminatedbythevalueofτ,butbytheirdependenceonthesurface.J signifiesgenuinesurfaceeffectsandwouldtotallydisappear
B
withoutthesurface,whileJ isabulkproperty.k/k =0.1andp=1.Im(ω¯)hasbeenneglectedtoemphasizethedifferences.
D s
which is derived and discussed in the next section. Here from J (z) with elements (q,q;ω¯) given in Sec. IV.
B ′
= (ik,∂ ,∂ ). Substituting J(z) into this equation and In Ap∇pe·ndix B, we show thatM(ω¯) M Z, where M
y z 0 0
F∇ouriertransformingit,wearriveat kv /ω isaconstantandZistheMunitym≈atrixwillallelement∼s
F p
being one. It makes only a minor correction to the diagonal
∞dq (q,q;ω¯) ω¯2δ(q q) ρ =S(ω¯), (10) matrixwithelementsΩ(k,q;ω¯)δ(q q′).
′ ′ ′ q −
Z H − − ′ Theequationofmotion(10) cannowbere-castin a com-
0 h i
pactmatrixform,
whereS(ω¯)=iω¯J (0)countsasasourcetermand
z
(ω¯) ω¯2I ρ=S(ω¯)E, (12)
(q,q;ω¯)=Ω2(k,q;ω¯)δ(q q)+ (q,q;ω¯) H −
H ′ − ′ M ′ whereIistheidenhtitymatrix,ρi isacolumnvectorcollecting
aretheelementsofamatrixdenotedby (ω¯),with all ρ and E is a column vectorwith all elements beingone.
H q
Forlateruse,thesourcetermcanberewritten
4πω¯ k F(k,q;ω¯)
Ω2(k,q;ω¯)=ω2 + · , k=(k,q). (11)
p k·k S(ω¯)=Gt(ω¯)ρ=Z ∞dqGqρq, Gq = 4Gk2(k+,qq;2ω¯), (13)
See that Ω(k,q;ω¯) is an even function of ω¯ and depends on 0
thelengthbutnotthedirectionofk. Thematrix (ω¯)arises where (ω¯) is another column vector, t takes the transpose,
M G
5
and which determinesω¯ and hence the SPW eigen-frequencyω.
Uponomitting (ω¯)from (ω¯),thisequationbecomes
M H
G(k,q;ω¯)=G (k)+G (k,q;ω¯),
D B
4G(k,q;ω¯) 1
1= ∞dq . (18)
with Z k2+q2 Ω2(k,q;ω¯) ω¯2
0 −
ω2 Let us write the solution as ω¯ = ω + iγ and hence ω =
p s 0
GD(k)=k 4π ωs +iγ with γ = γ0 1/τ. See thatthe solutions ωs +iγ0
− ±
occur together, in accord with the fact that ρ(x,t) is a real-
arisingfromJ (0)and valuedfield. InEq.(18),Ω(k,q;ω¯)isgenerallycomplexand
D,z
γ automaticallyincludesLandaudamping.
0
GB(k,q;ω¯)=GB,emg(k,q;ω¯)+pGB,ref(k,q;ω¯) Hydrodynamic/Drudelimits - The hydrodynamicmodelis
revisited if we replace in Eq. (18) G(k,q;ω¯) and Ω(k,q;ω¯)
fromJB,z(0). Here withGD(k) and ωb(k) given by Eq. (15), respectively. If we
furtherdisregardthedispersioninω (k),theDrudemodelis
b
m 3
GB,emg/ref(k,q;ω¯)=iω¯ (cid:18)2π~(cid:19) tIhnebnortehcmovoedreelds,,iω¯niwshriecahl.caseweimmediatelyfindωs = ωs0.
d3vΘ(v )( e2f )v L (v ,v ,k,q;ω¯). (14) Approximate solutions - We can solve Eq. (18) approxi-
×Z z − 0′ z emg/ref x z mately. Tothelowestorderinγ /ω ,whichisassumedtobe
0 s
small,wemaydetermineω byapproximatingtherealpartof
s
Equations(10)-(14)areexactanddonotexplicitlyinvolveτ,
(18)asfollows
therebyconcludingthatω¯ isindependentofτ.
Bulkmodesandlocalizedmodes-Withoutthe surfacewe 4Re G(k,q;ω ) 1
have (q,q;ω¯) 0andS(ω¯)=0. Equation(10)reducesto 1 ∞dq s . (19)
Ω2(k,Mq;ω¯) ′ω¯2 =≡ 0 forany ρ , 0, which all are extended ≈Z0 (cid:2)k2+q2 (cid:3) Ω2(k,q;ωs)−ω2s
q
−
modes and therefore describe cosine bulk plasma waves. If Substitutingtheso-obtainedω intheimaginarypartof(18),
s
wediscretizeqinstepdq,intotalthereare Nc = qc/dqsuch wefind
modes. Solvingthisequation,wefindthedispersionrelation
iωimnbtae(kgg)rinafnaordrytihpneatsrhetedmuinoetdeteogsr.aaGlpieonnlveeo,rlalvolelcyda,tiΩend(Fka,(tqkω¯;,qω¯=;)ω¯ck)ov,uxgl+divpqinovgsz,sreoissfsethatnoe ωγ0s ≈−12 ∞dqR04∞Rdeq[G4(kI,mq;ω[kG2s)+](kq,2q;ωs)]1Ω2(k,q;1ωs)−ω2sω2s , (20)
thecelebratedLandaudamping. Neglectingthis,oneobtains R0 k2+q2 Ω2(k,q;ωs)−ω2s Ω2(k,q;ωs)−ω2s
forsmall k which can be brought into a rather simple form if we take
| |
Ω(k,q;ω ) ω2 andω ω . Namely,
3k kv2 s ≈ p s ≈ s0
ω2(k) ω2 1+ · F ω2, (15)
whichwaswellbknow≈nfpromthe5hydωr2podyn≈amicp/Drudemodel. ωγ0s ≈−12R0∞∞ddqq RIme[[kGG2((+kkq,,qq2;;ωωss))]] = 12IRme(cid:2)JJzz((00))(cid:3). (21)
0 k2+q2
In the presence of the surface, equation (10) admits not R (cid:2) (cid:3)
only extended modes, for which S(ω¯) = 0, but also local- In this approximation, Ω(k,q;ω ) could have an imaginary
s
izedmodes,forwhichS(ω¯) , 0. Thenumberoftotalmodes partonlyifq > ω /v . Landaudampingwouldbeexcluded
c s F
cannot change and is still Nc. The extended modes again fromγ0otherwise.
represent bulk waves and for them equation (12) reduces to Toproceed,weneedtoevaluateG (k,q;ω¯). To thelinear
B
(ω¯) ω¯2I ρ = 0. As shown in Appendix B, because of orderink,wefind
H −
thheconstrainitthatS(ω¯) = 0,thereareintotalatmostN 1
c− m 3
solutionstothisequation. Thenumberoftotalbulkmodesis G (k,q;ω¯) iω¯e2
thenreducedbyonefromthatwithoutthesurface. Theirdis- B ≈ (cid:18)2π~(cid:19)
bpyerTωshiboe(nkmr)e.islasitniognbiuslnkemgloigdiebhlyasaffbeecetnedcobnyveMrte(ωd¯)inatnodasltoilclagliivzeedn ×Z d3vΘ(vz)f0′vz2qω¯22v2z−(1q2+v2zp) + kvz(i1ω¯−p). (22)
modesatisfyingS(ω¯) , 0andrepresentingSPWs,forwhich Carryingouttheintegralgives
equation(12)yields
ω2 k(p 1) 3(1+p)v
G (k,q;ω¯) p − i Fq2 . (23)
ρ=S(ω¯) (ω¯) ω¯2I −1E. (16) B ≈ 4π " 2 − 4 ω¯ #
H −
h i
Thus,Re G(k,q;ω ) k(ω2/4π)(1+ p)/2. Insertingthisin
PluggingthisinEq.(13),weobtain s ≈ p
Eq. (19),(cid:2)we get ω(cid:3) ω 2 p+1. We see that ω gen-
1= t(ω¯) (ω¯) ω¯2I −1E, (17) erally depends on susrf≈ace ss0caqtter−ing2effects, in contrasst with
G H −
h i
6
whatis expectedof the hydrodynamic/Drudemodel. Analo- mann’sequationcanbe uniquelydeterminedupto some pa-
gously, using Eq. (21) and q v ω , we find γ αω , rameters,whichsummarizetheeffectsof–whilewithoutac-
c F s0 0 s0
∼ ≈
withα = 3/2π. Landaudampinghasbeenexcludedhere,as tually knowing – φ (x). With translational symmetry along
s
theapproximationonlytakestherealpartofΩ(k,q;ω ). the surface, only one such parameter, namely, the so-called
s
Numerical solutions - Equation (18) can also be exactly Fuchs parameter p, is needed for the simple specular scat-
solved numerically. A comparisonwith the approximateso- tering picture. Physically, p measures the probability that
lutionis displayedin Fig. 1. Theagreementin the matterof an electronimpinginguponthe surfaceis bouncedback. As
ω is satisfactory, while that for γ is not. The discrepancy usual,wewrite f(x,v,t)= f (ε(v))+g(x,v,t),whereg(x,v,t)
s 0 0
mightbe because the approximatesolution excludesLandau isthenon-equilibriumpartofthedistributionfunctionin the
damping. It should be emphasized that, our numericalsolu- presence of E(x,t). The current density can then be calcu-
tionsdonotdependonthechoiceofq ,providedthelatteris lated by J(x,t) = (m/2π~)2 d3v ev g(x,v,t), where e de-
c
bigenough,i.e. qc ks. notes the electron charge. IRt is worth pointing out that, as
≥
g(x,v,t) is a distribution only for the bulk, the charge den-
sity is not given by ρ˜(x,t) = (m/2π~)2 d3v e g(x,v,t), i.e.
IV. METHODS ρ(x,t) , ρ˜(x,t). Actually,ItiseasytoseRethatρ˜(x,t)satisfies
(∂ +1/τ)ρ˜(x,t)+∂ J(x,t) = 0ratherthanEq.(24). Obvi-
t x
·
Thissectiondiscussesfurthersometechnicalaspectsofthe ously,whatismissingfromρ˜(x,t)isexactlythechargesjust
theory. We set outwith a discussion of the equationof con- localizedonthesurface.
tinuity in the presence of surfaces. Then we describe Boltz- Electronic distribution function - Let us write g(x,v,t) =
mann’sequationandsolveittoobtaintheelectronicdistribu- Re g(v,z)ei(kx ωt) . Intheregimeoflinearresponses, Boltz-
−
tionfunctions.Thencewederivethedynamicequationforthe manhn’sequationreiads
chargedensity.
∂g(v,z) v E(z)
tinEuiqtuy,ati∂otn+o1f/cτonρti(nxu,itt)y+-W∂xesjt(axr,ttw)i=th0th,ewehqiucahtiroenlaotefscothne- ∂z +λ−1g(v,z)+ef0′(ε) ·vz =0, (25)
·
chargedensityρ(x,t)andthecurrentdensityj(x,t)inageneric
(cid:0) (cid:1) where λ = iv /ω˜. In this equation, the velocityv is more of
way. Herethedampingterm ρ(x,t)/τisincludedtoaccount z
− a parameter than an argument and can be used to tag elec-
for electronic collisions that would drive the system toward
tronbeams. Itis straightforwardto solve the equationunder
equilibrium. In the jellium model, ρ(x,t) appears when the
appropriateboundaryconditions(AppendixA).Naturally,we
electrondensityisdisturbedawayfromitsequilibriumvalue
have
n . Thesurfacepreventsanyelectronsfromleakingoutofthe
0
metal. Explicitly,wewritej(x,t) = Θ(z)J(x,t),whereΘ(z)is g(v,z)=g (v,z)+g (v,z),
bulk surface
theHeavisidestepfunction. Insodoing,wehavetreatedthe
surfaceasahardwallandconsideredthefactthatJ(x,t)may where the bulk term would exist even in the absence of sur-
notvanishevenintheimmediateneighborhoodofthesurface faceswhereasthesurfacetermwouldnot. Usingtheexpres-
– which is obviouslythe case with the hydrodynamic/Drude sionsforE(z),weobtain
model.Theequationofcontinuitybecomes
4ρ kv +qv
g (v,z)= ef ∞dq q x z eiqz, (26)
∂t+ 1τ!ρ(x,t)+∂x·J(x,t)=−Jz(x0,t)δ(z), (24) bulk − 0′Z−∞ k2+q2ω¯ −(kvx+qvz)
whichhasthesameformasonewouldexpectforabulksys-
where x0 = (r,0) denotes a point on the surface and δ(z) is tem. Herewehavedefinedρq<0 :=ρ q.
theDiracfunctionpeakedonthesurface.Physically,theright As for gsurface(v,z), we find it w−ith a subtle structure: it
handsideofEq.(24)meansthat,chargesmustpileuponthe can be written as a sum of two contributions, one of which,
surfaceiftheydonotcometoahaltbeforetheyreachit. gD,surface(v,z),hasasingleformforallelectronsirrespective
Equation(24)reducestoequation(9)uponusingtheplane oftheirvelocitieswhiletheother,gB,surface(v,z),doesnot.Ex-
waveformforρ(x,t)andJ(x,t). plicitly,wehave
Boltzmann’sapproach-WeemployBoltzmann’sequation
4ρ k(v iv )
stococpailcculelvateel,Jw(xe,mt)aaysinatrroedspuocensaestuorfEac(xe,pt)o.tenOtinalthφes(xm)icinroto- gD,surface(v,z)=−ef0′Z0∞dq k2+qq2 kvzz−+iω˜x e−kz. (27)
theequationtoaccountforsurfacescatteringeffects.Thecor-
Wemaycombineg (v)andg (v,z)inasingleterm,
bulk D,surface
responding surface field E (x) = ∂ φ (x) is peaked on the
s x s
−
surfaceand,complyingwiththehard-wallpicture,mayhave g (v,z)=g (v,z)+g (v,z),
D bulk D,surface
aninfinitesimalspreadd 0 . Unfortunately,asφ (x)can
s + s
hardlybeknownandvarie→sfromonesampletoanother,this inordertoseparatethemfrom
microscopicmethodisimpracticalandfutile.
g (v,z):=g (v,z).
Alternatively, surface scattering effects can be dealt with B B,surface
using boundaryconditions[32, 34–36]. This is possible be-
Insodoing,wehavedecomposed
cause E (x) acts only on the surface. In the bulk, the solu-
s
tions–theelectronicdistributionfunction f(x,v,t)–toBoltz- g(v,z)=g (v,z)+g (v,z)
D B
7
ina diffusiveanda ballisticcomponent. Itisunderlinedthat electroniccollisionrate1/τ. Asakeyconsequenceofthethe-
g (v,z) arises only when the surfaces are present. For bulk ory, we find thatthere exists a self-amplificationchannelfor
B
systems without surfaces, it does not exist even if the elec- SPWs,whichwouldcausethelattertospontaneouslyamplify
tronicmotionsaretotallyballistic, i.e. τ . Assuch, we at a rate γ if not for electronic collisions. Surprisingly, the
0
→ ∞
callit’surface-ballistic’. valueofγ turnsouttobeindependentofτ. Thepresenceof
0
g (v,z) exists only for electrons leaving the surface, i.e. thischannelisguaranteedbythecausalityprinciple.
B
vz 0. Those electronscoulddirectly emergefromthe sur- Whetherthesystemcouldactuallyamplifyornotdepends
≥
face or be those incident on but subsequently get reflected on the competition between γ and 1/τ. If γ > 1/τ, SPWs
0 0
bythesurface. WhatfundamentallysetsgB,surface apartfrom willamplifyandthesystemwillbecomeunstable.Inourthe-
gD(v,z)restwithitssimplezdependence.Actually,wehave ory,thenon-equilibriumdeviationg(v,z)referstotheFermi-
Dirac distribution f (ε); as such, the instability is one of the
gB(v,z)=Θ(vz)eiωv˜zz gB,emg(v)+ pgB,ref(v) , Fermisea. Needless0 tosay,theinstabilitywillbeterminated
h i once the system deviatesfar enoughfromthe Fermisea and
wheretheFuchsparameter pgaugesthefractionofreflected
settlesinastablestate. Clarifyingthenatureofthedestination
electrons and g (v) has been given in Eq. (7). As al-
B,emg/ref stateisasubjectofcrucialimportanceforfuturestudy.
ready remarked, since gB(v,z) ∝ eiω˜z/vz ∝ e−γ0z/vz, we must Onecentralfeatureofourtheoryistheclassificationofcur-
haveγ 0;otherwise,itwoulddivergeeitherwhenz
0 ≥ → ∞ rentdensitiesintoadiffusivecomponentJ (z)andasurface-
or for small v . In Appendix A, we argue that this result is D
z ballisticcomponentJ (z). Thisclassificationisnotbasedon
aconsequenceofthecausalityprinciple,whichstatesthatthe B
thevalueofτbutaccordingtowhetherthecomponentobeys
numberofreflectedelectronsisdeterminedbythatofincident
the (generalized) Ohm’s law or not. Apart from this, these
electrons,ratherthanotherwise.
componentsarealsodiscriminatedinotherways.Firstly,they
Divergence of the current densities - The current density
J(z)=(m/2π~)2 d3vevg(v,z)issplitintwoparts:J (z)and arecontrolledbydifferentlengthscales. Asitlargelyfollows
D the local electric field E(z), the characteristic length associ-
J (z), where J R(z) = (m/2π~)2 d3v ev g (v,z). Using
B D/B D/B atedwithJ (z)isk 1. Ontheotherhand,thelengthforJ (z)
D − B
the expressions for gD/B(v,z), it isR straightforward to obtain isv /γ ,becauseofitssimplez-dependence. Secondly,they
F 0
J (z) given by Eq. (4) and J (z) by Eq. (6). To obtain the
D B are oriented disparately. J (z) is largely oriented normal to
D
equation of motion (10) from the equation of continuity (9),
E(z)locallywhereasJ (z)normaltothesurfaces–especially
B
theFouriertransformof J (z)isneeded.Wefind
∇· D/B for pclosetounity. Thirdly,JD(z)isabulkpropertyandex-
Z ∞dz cos(qz)∇·JD(z)= ω¯i Ω2(k,q;ω¯)ρq, (28) tirsutserseugrafardcleeessffeocftsthaendsuirtfwacoeu;ldOdnisthapepceoanrtwrairtyh,oJuBt(szu)rfraecfleesc.ts
0
If we replace the vacuum by a dielectric ǫ, γ may be re-
0
withΩ(k,q;ω¯)givenbyEq.(11). Additionally, duced by an order of 1/ǫ due to weakened E (z) and J (0).
z z
Roughlyspeaking,ρ(z)presentonthemetalsideinducesmir-
i
∞dzcos(qz) J (z)= ∞dq (q,q)ρ , (29) ror charges amounting to ρ(z) = ρ( z)(ǫ 1)/(ǫ + 1) on
Z0 ∇· B ω¯ Z0 ′M ′ q′ thedielectricside. Ifρ(z) is′highly−loc−alized−abouttheinter-
where (q,q′)isamatrixgivenby face, as is with SPWs, ρ(z) and ρ′(z) will combine to give a
M netchargeof2ρ(z)/(ǫ +1). Asa result, E and J (0)willbe
z z
4ω¯2 m 3 reducedbyafactorof2/(ǫ+1). Thisleadstosmallerγ and
(q,q)= 0
M ′ k2+q′2 (cid:18)2π~(cid:19) smallerωs0 = ωp/√ǫ+1. Studyingdielectriceffectsmaybe
iω˜v importantinapplications.
d3vΘ(v ) e2f z L(v ,v ,k,q;ω¯),(30)
×Z z − 0′ ω˜2 q2v2 x z ′ Anotherproblemthatneedstobeaddressedforexperimen-
(cid:16) (cid:17) − z
tal studies is concerned with the effects of inter-band tran-
where
sitions. In the most experimented materials, such as silver
L(v ,v ,k,q;ω¯)= L (v ,v ,k,q;ω¯)+ pL (v ,v ,k,q;ω¯). and gold, these transitions are known to have dramatic ef-
x z emg x z ref x z
fects. They not only open a loss channel due to inter-band
InAppendixB,weshowthat (q,q′)generallymakesami- absorption, but also significantly shift the SPW frequency.
norcorrection,oftheorderofMkvF/ωs,toΩ2(k,q;ω¯)δ(q q′). Including them in our formalism consists of a simple gen-
−
Ultimately, the insignificance of this correction may be as- eralization: in addition to J (z) and J (z), the total cur-
D B
cribed to the factor eiω˜z/vz in gB(v,z), which is extremelyos- rentdensityJ(z)mustnowalso havea componentJint(z) ac-
cillatoryoverzwithaperiodlessthan vF/ωs. counting for inter-band transitions. The equation of motion
∼
SubstitutingEqs.28and(29)intheequationofcontinuity, is obtained by substituting J(z) in Eq. (9). One may write
weimmediatelyobtaintheequationofmotion(10). J (z)= dz σ (z,z;ω)E (z),whereµ,ν= x,y,zand
int,µ ν ′ µν ′ ν ′
theinter-bPandRconductivityσµν caninprinciplebecalculated
usingGreenwood-Kuboformula. Inpractice,calculatingσ
µν
V. DISCUSSIONSANDCONCLUSIONS could be a formidable task even for the imaginablysimplest
surfaces. Nevertheless, one may argue that J (z) primarily
int
Thus, on the basis of Boltzmann’s equation, we have es- affectsthe propertiesofbulkwaves, namely,Ω(k,q;ω¯). The
tablishedarigoroustheoryforSPWsinmetalswitharbitrary causalityprincipleshouldstillprotecttheamplificationchan-
8
nel. Asystematicanalysiswillbepresentedelsewhere. z for any v. For electrons movingaway from the sur-
→ ∞
We remarkthat, γ can also be calculatedby studyingthe face, i.e. v > 0, thisconditionis fulfilledfor anyC(v). For
0 z
temporalevolutionoftheelectrostaticpotentialenergyofthe electrons moving toward the surface, i.e. v < 0, we may
z
system. In particular, equation (21) can be directly derived choose
inthisway. Detailedcalculationsalongthislinewillbepub-
e∂ f
lishedinaseparatepaper. C(v)= v 0 ∞ dz′e−iω˜vzz′E(z′), vz <0, (A2)
To conclude,we havepresenteda theoryforSPWs taking mvz ·Z0
intoaccounttheuniqueinterplaybetweenballisticelectronic
whichleadsto
motions and boundary effects, from which it emerges a
uexnpiveecrtseadltshealtf-tahmepsltiufidcyatiwonillcbheaanrnefalrf-oreracthheinseg wpraavcetisc.alIatnids g(v,z)= em∂vvfz0 ·Zz∞dz′eiω˜(zv−zz′)E(z′), vz <0. (A3)
fundamentalconsequences,tobeexploredinthefuture.
TodetermineC(v)forv >0,theboundaryconditionatz=0
z
Acknowledgement – HYD acknowledges the Interna- hasto be used, which dependson surface properties. In this
tional Research Fellowship of the Japan Society for the paper, we assume that a fraction p (Fuchs parameter)of the
Promotion of Science (JSPS). This work is supported by electrons impinging on the surface are bounced back in the
JSPS KAKENHI Grant Nos. 15K13507and 15K21722and absenceofE (z),i.e.
z
MEXTKAKENHIGrantNo. 25107005.
g (v ,v ,v >0),z=0 = pg (v ,v , v ),z=0
x y z x y z
−
(cid:16) (cid:17) (cid:16) (cid:17)
AppendixA:Electronicdistributionfunctions evaluatedatEz(z)=0. Thenweget
ThegeneralsolutiontoEq.(25)isgivenby C(v)= p e∂vf0 ∞dz′eiω˜(zv+zz′)E(z′), vz >0. (A4)
− mvz ·Z0
e∂ f z
g(v,z)=eiωv˜zz C(v) v 0 dz′e−iω˜vzz′E(z′) , (A1) Bydefinition, pvariesfromzerotounity.Thus,weobtain
− mvz ·Z0 !
g(v,z)=Θ(v )g (v,z)+Θ( v )g (v,z),
z > z <
where C(v) is an arbitrary integration constant to be deter- −
mined by boundary conditions. We require g(v,z) = 0 at where
g>(v,z)= zeiω˜|z−vzz′|+p ∞eiω˜|z+vzz′| ev·E(z′)∂f0 dz′, g<(v,z)= ∞eiω˜|z−vzz′|ev·E(z′)∂f0 dz′. (A5)
−"Z0 Z0 # |vz| ∂ε −Zz |vz| ∂ε
Utilizingequations(2)and(3)forE(z)andcarryingouttheintegraloverz,wefind
′
∂f ε(v) e 4kρ
g (v,z)= 0 ∞dq q g (v,z;k,q), (A6)
>/< − ∂(cid:0)ε (cid:1) vz Z0 k2+q2 >/<
(cid:12) (cid:12)
(cid:12) (cid:12)
where (cid:12) (cid:12)
g>(v,z;k,q)= kv+z−iω˜iv/vx e−kz+ 2((cid:18)ω˜q/vvzqk)2+ω˜qvv2xz(cid:19) cos(qz)+i 2(ω˜(cid:16)q/vvx)+2 ω˜qqk2(cid:17) sin(qz)
z z z
− −
2 ω˜ vx +qv q 2 ω˜ vx qv q
and +kiv+xi−ω˜/vvzz − ((cid:18)ω˜/vvzz)2−qz2k(cid:19)+ pkiv−xi−ω˜/vvzz + ((cid:18)ω˜/vvzz)−2−qz2k(cid:19) eiωv˜zz, (A7)
g<(v,z;k,q)= ki−vxiω−˜/vvzz e−kz+ 2(cid:18)(qω˜(cid:12)(cid:12)(cid:12)v/zv(cid:12)(cid:12)(cid:12)zqk)2+−ω˜q|2vvzx|(cid:19) cos(qz)−i 2(ω˜(cid:16)q/vvzx)+2−ω˜qqk2(cid:17) sin(qz). (A8)
(cid:12) (cid:12)
(cid:12) (cid:12)
Now we can combine the terms with cos(q(cid:12)z)(cid:12)and those with sin(qz) into a single term called g (v,z), while the rest into
bulk
g (v,z).TheirexpressionshavebeengiveninSec.IV.
surface
Ifthesurfaceisnotuniform,weshouldhaveafunction p(r)insteadofaconstant p. Theboundaryconditionshouldthenbe
writteng(x ,(v ,v ,v >0),t)= p(r)g(x ,(v ,v , v ),t). Insuchcase,thetranslationalsymmetryalongthesurfaceisgenerally
0 x y z 0 x y z
−
9
lost and one cannotwork with a single k-componentany more, i.e. there is scattering effects and differentk-componentsare
mixed.Wedonotconsiderthisinthispaper.
Causalityprinciple-InapplyingtheboundaryconditionstoobtainC(v),wehaveimplicitlyassumedIm(ω˜) 0;otherwise,
≥
wewouldfindunphysicalsolutionsthatviolatetheprincipleofcausality,whichstatesthatthenumberofout-goingelectronsis
determinedbythenumberofin-comingelectrons,nototherwise. Itiseasytoshowthat,hadweassumedIm(ω˜)<0,wewould
havefoundtheopposite: thenumberofreflectedelectronswouldbefixedwhilethenumberofincidentelectronswouldgoto
infinityas p 0,whichisunphysical.
→
AppendixB:Thematrix (ω¯)
M
Inthefirstplace,weshowthat (ω¯)/ω2 ikv /ω¯ +...,wheretheellipsisstandsforhigherordertermsinkv /ω¯. Forthis
M p ∝ F F
purpose,letusdiscretizeqinstepdq = 2π/d,i.e. q = l(2π/d),wherekd 1andl = 0,1,2,...takesintegervalues. Equation
l
≫
(10)canthenbebroughtintothefollowingform
Ω2(k,q;ω¯) ω¯2 ρ + (ω¯)ρ =S(ω¯), (B1)
l l l,l l
− M ′ ′
(cid:16) (cid:17) Xl
′
whereMl,l′(ω¯) = (2π/d)M(ql,ql′;ω¯)andρl = ρql. Writing d3vΘ(vz) = 02πdϕ 0π/2dθsinθ 0∞dv2(v/2)andintegratingover
v,wefind R R R R
(ω¯) 3 2π π/2 ω¯2cosθ ω¯kv sinθcosθcosϕ kω¯/v
Ml,l′ =i dϕ dθsinθ − F F L(v sinθcosϕ,v cosθ,k,q ,ω¯).(B2)
ω2 2πkd Z Z (ω¯ kv sinθcosϕ)2 q2v2 cos2θk2+q2 F F l′
p 0 0 − F − l F l′
Tothelowestorderinkv /ω¯,weonlyneedtoretainL(0) intheexpansionLsym = L(m)(kv /ω¯)m. Thus,
F ∞m=0 F
P
2q2v2 cos2θ
L(v sinθcosϕ,v cosθ,k,q,ω¯) F 1+p . (B3)
F F ≈ ω¯2 q2v2 cos2θ
− F (cid:0) (cid:1)
Substitutingthisbackin(B2)andapproximating
ω¯2cosθ ω¯kv sinθcosθcosϕ ω¯2cosθ q2
− F , 1, (B4)
(ω¯ kv sinθcosϕ)2 q2v2 cos2θ ≈ ω¯2 q2v2 cos2θ q2+k2 ≈
− F − F − F
wearriveat
(ω¯) 3(1+p) kv 2π π/2 cos3θ 1
Ml,l′ = i F dϕ dθsinθ . (B5)
ω2p − πkd ω¯ !Z0 Z0 1−(qlvF/ω¯)2cos2θ1−(ql′vF/ω¯)2cos2θ
Obviously,wehave (ω¯)/ω2 kv /ω¯,asstated.
M p ∼ F
WemayproceedfurtherIfwetake
3 2π π/2 cos3θ 1 3 2π π/2
dϕ dθsinθ dϕ dθsinθ cos3θ 1, (B6)
πZ0 Z0 1−(qlvF/ω¯)2cos2θ1−(ql′vF/ω¯)2cos2θ ≈ πZ0 Z0 ∼
from which it follows that (ω¯) M = iω2(1/kd)(kv /ω¯), which is a constant. We then write (ω¯) M Z, where
Z = 1 constitutesa unityMmatrixl,l.′ T≈o sim0 plif−ythpe followingF discussions, let us take Ω(k,q;ω¯) = ω . MEquatio≈n(B01) is now
l,l p
′
written
(ω2 ω¯2)I+M Z ρ=S(ω¯)E. (B7)
p− 0
h i
For bulk modes, S(ω¯) = 0. Note that Z has only one non-vanishingeigenvalueamounting to its dimension N = q d/2π
c c
(ω /v )(kd/2π). Thecorrespondingeigenvectorisρ E, whichobviouslydoesnotsatisfyS(ω¯) = 0andisnotabulkmode∼.
s F
WethereforeconcludethatthereareintotalatmostN∝ 1bulkmodes. Forthesemodes,theeigenvaluesofZareallzeroand
c
−
therefore (ω¯)hasnoimpactonbulkmodes.
WecanMalsoshowthat (ω¯)isnegligibleforthelocalizedmode,forwhichS(ω¯),0. Tothisend,wewrite
M
1 1
ω2 ω¯2 I+M Z − =U 1 ω2 ω¯2 I+M Z˜ − U, (B8)
(cid:20)(cid:16) p− (cid:17) 0 (cid:21) − (cid:20)(cid:16) p− (cid:17) 0 (cid:21)
10
whereU is a similarity transformationthatdiagnolizesZ. We have useda tilde to indicatethe transformedmatrices, e.g. we
writeZ˜ = UZU 1. Assaid,Z˜ hasonlyonenon-vanishingelement. Letitbethel -thelement. ThenZ˜ = N δ δ . Seethat
M 1/N and −˜ (ω¯) i(ω2/2π)δ δ . Introducing ˜t = tU 1 andE˜ = U0E,wecanrewritethel,ls′eculacrel,lq0ula′,tli0onforthe
0 ∼ Ml,l′ ∼ − p l,l0 l′,l0 G G −
localizedmodeas
1
1= ˜t ω2 ω¯2 I+M Z˜ − E˜. (B9)
G (cid:20)(cid:16) p− (cid:17) 0 (cid:21)
Explicitly,
1 1 1 1 1
1= ˜t E˜ = t E + ˜t E˜ t E t E.
Glω2 ω¯2+M Z˜ l Glω2 ω¯2 l Gl0ω2(1+i/2π) ω¯2 l0 −Gl0ω2 ω¯2 l0≈ Glω2 ω¯2 l
Xl p− 0 l,l Xl p−  p − p−  Xl p−
Theterminthesquarebracketmakesonlyacontributionoftheorderof 1/N andcanbeneglectedforlargeN .
c c
∼
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