Coulomb drag in topological insulator films

0.75 MB·

by  Hong Liu

Coulomb drag in topological insulator films
HongLiu
ICQD,HefeiNationalLaboratoryforPhysicalSciencesattheMicroscale,UniversityofScienceandTechnologyofChina,Hefei230026,Anhui,
China
WeizheLiu,DimitrieCulcer∗
SchoolofPhysics,TheUniversityofNewSouthWales,Sydney2052,Australia
6
1
0
2

n
a
Abstract
J

1 We study Coulomb drag between the top and bottom surfaces of topological insulator films. We derive a kinetic
1
equationforthethin-filmspindensitymatrixcontainingthefullspinstructureofthetwo-layersystem,andanalyzethe

electron-electroninteractionindetailinordertorecoveralltermsresponsibleforCoulombdrag. Focusingontypical
]
l topological insulator systems, with film thicknesses d up to 6nm, we obtain numerical and approximate analytical
l
a results for the drag resistivity ρ and find that ρ is proportional to T2d−4n−3/2n−3/2 at low temperature T and low
D D a p
h
electron density n , with a denoting the active layer and p the passive layer. In addition, we compare ρ with
- a,p D
s graphene, identifying qualitative and quantitative differences, and we discuss the multi valley case, ultra thin films
e
andelectron-holelayers.
m
. Keywords: Electron-electroninteractions,Coulombdrag,Topologicalinsulator
t
a
m
- 1. Introduction tures such as TI/superconductor junctions have been
d
fabricated [39, 40], which are expected to give rise to
n Three-dimensionaltopologicalinsulators(3DTIs)are
topological superconductivity and Majorana fermions
o
a novel class of bulk insulating materials that possess
c [41,42].
conductingsurfacestateswithachiralspintexture[1–
[
9]. Thanks to their topology, these surface states re- Transport experiments and theoretical work have
1 mostlyfocusedonlongitudinal[43–48]andHalltrans-
main gapless in the presence of time-reversal invariant
v port properties [37, 49–51], thermoelectric response
perturbations. Following their initial observation [10–
1
[52–54] and weak antilocalization [55–58], all essen-
9 19], improvements in TI growth have made them suit-
tially single-particle phenomena. The interplay of
2 able for fundamental research [20–23]. Although the
2 reliable identification of the surface states in transport, strong spin-orbit coupling and electron-electron inter-
0 actions in TIs is at present not completely understood
whichremainsthekeytoTIsbecomingtechnologically
. [59–66].
1 important, has remained elusive, a number of experi-
0 mentshavesuccessfullyidentifiedsurfacetransportsig- An interaction effect that can be tested experimen-
6
naturesinisolatedsamples. Thesewereinitiallymostly tally in transport is Coulomb drag, which is caused by
1
singled out via quantum oscillations or in gated thin the transfer of momentum between electrons in differ-
:
v
films [18–22]. Recently, four-point transport measure- ent layers due to the interlayer electron-electron scat-
i
X ments on clean surfaces in an ultrahigh vacuum have tering. Coulomb drag has been used for decades as an
reportedasurface-dominatedconductivity[24]. Acur- experimentalprobeofinteractions[67–69],andhasre-
r
a rent induced spin polarization also constitutes a signa- centlyattractedconsiderableattentioninmasslessDirac
tureofsurfacetransport[25,26]andwasreportedinre- fermion systems such as graphene [70–83]. Our focus
cent experimental studies [27–29]. Magnetic TIs have in this paper is on Coulomb drag in TIs with no mag-
also been successfully manufactured [30–32], and the netic impurities. Unlike graphene, the spin and orbital
anomalous [33] and quantum anomalous Hall effects degreesoffreedomarecoupledbythestrongspin-orbit
[34, 35] have been detected [36–38]. Hybrid struc- interaction, TIs have an odd number of valleys on a
PreprintsubmittedtoPhysicaE January12,2016
singlesurface, andtherelativepermittivityisdifferent, In a two-layer system the indices α ≡ ks l represent
k
whileinknownbandTIsscreeningisqualitativelyand the wave vector, band, and layer indices respectively.
quantitativelydifferent,sinceitdoesnotinvolvethein- Thebandindexs =±with+representingtheconduc-
k
terplayofthelayerandvalleydegreesoffreedom. All tionbandand−thevalenceband,whilethelayerindex
these features impact the drag current. We introduce a l = (a,p) with ‘a’ the active layer and ‘p’ the passive
density matrix method to calculate the Coulomb drag layer. Thetwo-particlematrixelementVee inabasis
αβγδ
currentintopologicalinsulatorfilms,whichfullytakes spanned by a generic set of wave functions {φ (r)} is
α
into account the spin degree of freedom and interband givenby
coherence. The central result of our work is the drag
resistivity,whichanalyticallytakestheform (cid:90) (cid:90)
Vee = dr dr(cid:48)φ∗(r)φ∗(r(cid:48))Vee φ (r)φ (r(cid:48)), (4)
(cid:126) ζ(3) (k T)2 αβγδ α β r−r(cid:48) δ γ
ρ =− B , (1)
D e2 16π 3 3
A2r2n2n2d4
s a p whereVee = e2 istheunscreenedCoulombin-
where kB is the Boltzmann constant, A is the TI spin- teractionr.−r(cid:48) 4π(cid:15)0(cid:15)r|r−r(cid:48)|
orbit constant, r is the Wigner-Seitz radius (effective
s Theone-particlereduceddensitymatrixisthetrace
finestructureconstant)whichrepresentstheratioofthe
electrons’ average Coulomb potential and kinetic en-
ρ =tr(c†c Fˆ)≡(cid:104)c†c (cid:105)≡(cid:104)Fˆ(cid:105) , (5)
ergies, d is the layer separation and na,p are the elec- ξη η ξ η ξ 1e
tron densities in the active and passive layers, respec-
tively. For a single-valley system rs = e2/(2π(cid:15)0(cid:15)rA), whichsatisfies[59]
with(cid:15) therelativepermittivity. Theintralayerresistiv-
r
ityρa,p = e2A4kπFa(cid:126),p2τa,p withkFa,p theFermiwavevectors. dρξη + i[Hˆ ,ρˆ] = i(cid:104)[Vˆ ,c†c ](cid:105), (6)
Theoutlineofthispaperisasfollows. InSec.2the dt (cid:126) 1e ξη (cid:126) ee η ξ
interlayer electron-electron scattering matrix is given,
including the interlayer screened Coulomb interaction. where the many-electron averages such as (cid:104)[Vˆ ,c†c ](cid:105)
ee η ξ
In Sec. 3 we derive the kinetic equation of topological
arefactorizedas
insulators for spin density matrices of top and bottom
surfaces with the full scattering term in presence of an
(cid:104)c†c†c c (cid:105)=(cid:104)c†c (cid:105)(cid:104)c†c (cid:105)−(cid:104)c†c (cid:105)(cid:104)c†c (cid:105)+G . (7)
arbitraryelasticscatteringpotentialtolinearorderinthe α β γ δ α δ β γ α γ β δ αβγδ
impuritydensity. InSec.4, wecalculatetheanalytical
andnumericalexpressionsofdragresistivity. Ourfind- inwhichweintroducetheG asthematrixelements
αβγδ
ingsaresummarizedinSec.5,andwealsodiscussesthe of the two-particle correlation operator Gˆ. G give
αβγδ
broaderimplicationsofourresultsandpresentsacom- risetotheelectron-electronscatteringterminthekinetic
parison with graphene. Sec. 6 discusses extensions of equation [84]. The first two terms on the right side of
our theory to treat the multi-valley case and ultra-thin theEq.(7)whichrepresenttheHartree-Fockmean-field
films, and briefly touches upon exciton condensation. part of the electron-electron interactions have been in-
Finally,Sec.7containsourconclusions. vestigatedinRef.[59].InRef.[59]itwasdemonstrated
that the electrical current and nonequilibrium spin po-
larization undergo a small renormalization due to the
2. Electron-electroninteraction
mean-fieldpartofelectron-electroninteractionsandare
Thesystemisdescribedbythemany-particledensity consequently slightly reduced as compared with their
matrix Fˆ, whichobeysthequantumLiouvilleequation non-interactingvalues. Wearenotincludingthisweak
[84] renormalization here, so the right-hand side of Eq. (6)
dFˆ + i[Hˆ,Fˆ]=0, (2) onlygivestheelectron-electronscatteringterm Jˆee(ρˆ|t)
dt (cid:126) whichhastwocontributions,representingintralayerand
whereHˆ = Hˆ1e+Vˆeewith interlayerelectron-electronscattering. Moreover,since
(cid:88) theintralayerelectron-electronscatteringdoesnotcon-
Hˆ1e= H c†c ,
αβ α β tributetothedragcurrent,weconcentrateontheinter-
αβ layerelectron-electronscattering,forwhichthescatter-
Vˆee=1 (cid:88)Vee c†c†c c . (3) ingtermisdenotedbyJInter(ρˆ|t).Weusebelowthebasis
2 αβγδ α β γ δ oftheeigenstateproblemandaccountforonlydiagonal
αβγδ
2
partofthedensitymatrixJInter(f )=(cid:104)k|JInter(ρˆ|t)|k(cid:105) theorderofk T/(cid:126). WeapproximateCoulombinterac-
k B
1 (cid:88) (cid:90) ∞ (cid:104) tionwithkFd (cid:29)1as
JInter(fk)=(cid:104)k|(cid:126)2L4 vqvq1 dt1eλt1 e−iq·r, (cid:18) e2 (cid:19)2 q2
Sˆ(t,t1)(1−ρˆt1)eiq1·rqρˆqt11Sˆ+(t,t10)(cid:8)(cid:8)eiq·rSˆ(t,t1)e−iq1·rρˆt1 |Vq|2 = 2(cid:15)0(cid:15)r 4kT2FakT2Fpsinh2(qd), (13)
Sˆ+(t,t1)(cid:9)(cid:9)−Sˆ(t,t1)ρˆt1eiq1·r(1−ρˆt1)Sˆ+(t,t1)(cid:8)(cid:8)eiq·r wherekTFl = rs2kFl istheThomas-Fermiwavevectorand
r istheWigner-SeitzradiusintroducedinEq.(1).
s
Sˆ(t,t1)ρˆt1e−iq1·rSˆ+(t,t1)(cid:9)(cid:9)+Sˆ(t,t1)[ρˆt1,eiq1·r]
Sˆ+(t,t1)(cid:8)(cid:8)eiq·rSˆ(t,t1)ρˆt1e−iq1·rρˆt1Sˆ+(t,t1)(cid:9)(cid:9)(cid:105)|k(cid:105), 3. KineticEquations
(8)
Atthisstageonemayincludeexplicitlydisorderand
where v = e2 , 1 is the identity matrix, L2 the area
q 2(cid:15)0(cid:15)rq drivingelectricfieldintheoneparticleHamiltonianand
of the 2D system, Sˆ(t,t1) the time evolution operator writeHˆ1e = Hˆ0+HˆE+Uˆ,whereHˆ0isthebandHamilto-
and {{Aˆ}} ≡ Aˆ − trAˆ [84]. The momentum transfer nianofTIs,Hˆ =eEˆ·rˆistheelectrostaticpotentialdue
E
q = q1 = k − k1 = k(cid:48)1 − k(cid:48). Following a series tothedrivingelectricfliedwith rˆ isapositionoperator
of simplifications, the interlayer Coulomb interaction and Uˆ is the disorder potential. According to Sec. 2,
eventually takes the form v(|kp−a)k1|. Without screening thequantumLiouvilleequationforthereduceddensity
v(pa) = v e−qd. To account for screening, we employ operatorρˆ satisfies
|k−k1| q
the standard procedure of solving the Dyson equation
dρˆ i
for the two-layer system in the random phase approxi- + [Hˆ1e,ρˆ]+Jˆ (ρˆ|t)=0 (14)
mation(RPA)discussedinRef.[68]. Inthisapproach, dt (cid:126) ee
v(pa) inEq.(8)becomesthedynamicallyscreenedin-
|k−k1| and will be projected onto the time-independent basis
terlayerCoulombinteraction
|k,s ,l(cid:105). Below we do not write the band indices ex-
k
v e−qd plicitly. Since the current operator is diagonal in wave
V(q,ω)= q . (9) vector,thequantityofinterestindeterminingthecharge
(cid:15)(q,ω)
current is the part of the density matrix which is diag-
Thedielectricfunctionofthecoupledlayersystemis onal in wave vector [6], which here we denote by f .
k
Intheabsenceofinterlayertunneling,the4×4matrix
(cid:15)(q,ω) =[1−v Π (q,ω)][1−v Π (q,ω)]
q a q p f ,whichdescribesthetwo-layersystem,isassumedto
k
(10)
beapproximatelydiagonalinthelayerindexandcanbe
−[vqe−qd]2Πa(q,ω)Πp(q,ω), reducedtotwo2×2densitymatrices f(a)and f(p). The
k k
quantum Liouville equation can then be broken down
in which the polarization function is obtained by sum-
intotwoseparateequations,onefor f(a)andonefor f(p),
mingthelowestbubblediagramandtakestheform k k
withdifferentdrivingterms. Theelectricfield,whichis
Π(q,ω)=− 1 (cid:88) F(slk)sk(cid:48)(f0(lk),s− f0(lk)(cid:48),s(cid:48)), (11) oganrldy aEpqp.li(e8d) atostthheeadcrtiivvienglatyeerrm, ifsoErathe=pEaassxˆi.veWlaeyreer-,
l L2 ε(l) −ε(l) +(cid:126)ω+i0+
kss(cid:48) k,s k(cid:48),s(cid:48) since the only quantity coupling the two layers is the
interlayerelectron-electronscatteringterm. Thekinetic
with f(l) ≡ f(l)(ε )theequilibriumFermidistribution
0k,s 0 ks equationstaketheform
function,andF(l) = (cid:104)s l|s l(cid:105)(cid:104)s l|s l(cid:105)thewavefunc-
tionoverlap. Instkospk1ologickalikn1sulakto1rwkithnotunneling df(a) i i
k + [H(a), f(a)]+Jˆ(f(a))=− [HE, f(a)], (15a)
1 k+qcosφ dt (cid:126) 0k k 0 k (cid:126) k 0k
F(l) = (1+ss(cid:48) ). (12)
sksk1 2 |k+q| df(p) i
In analytical calculations, the dynamical screened k + [H(p), f(p)]+Jˆ(f(p))=−JInter, (15b)
dt (cid:126) 0k k 0 k k,p
CoulombinteractionV(q,ω)isusuallyreplacedbythe
staticscreenedV atlowtemperatures,Π(q,0) = −kFl wherethebandHamiltonianH(l) = Aσ·(k×zˆ)≡−Akσ·
q l 2πA 0k
withk theFermiwavevector. Thisisbecausethetyp- θˆ, withθˆ thetangentialunitvectorcorrespondingto k.
Fl
icalfrequenciescontributingtotheintegralareonlyof The projection of JInter(f ) onto the eigenstates of the
k
3
passivelayerisdenotedbyJInter,where equilibrium density matrix for the passive layer f(p) =
k,p k
n(p) + S(p) and the full density matrix of the active
2π (cid:88) k,0 k,0
JIkn,pte,rsk =−(cid:126)L4 |v(|kp−a)k1||2δk+k(cid:48),k1+k(cid:48)1F(spk)sk1 layer fk(a) = n(ka,)0 + S(ka,)0 + δfE(ak) into Eq. (8), where
k1k(cid:48)k(cid:48)1 δf(a) = n(a) + S(a) is a small correction to the distri-
×F(a) δ[ε(p) −ε(p) +ε(a) −ε(a) ] buEtikon funEcktion cEakused by the applied electric field in
sk(cid:48)sk(cid:48)1 k1,sk1 k,sk k1(cid:48),sk(cid:48)1 k(cid:48),sk(cid:48) (16) theactivelayer
×(cid:8)f(p) [1−f(p) ]f(a) [1−f(a) ]
k,sk k1,sk1 k(cid:48),sk(cid:48) k(cid:48)1,sk(cid:48)1 n(a) = eEaτa(k)·kˆ ∂(f0(+a)+ f0(−a)), (19a)
−[1−f(p) ]f(p) [1−f(a) ]f(a) (cid:9). Ek 2(cid:126) ∂k
k,sk k1,sk1 k(cid:48),sk(cid:48) k(cid:48)1,sk(cid:48)1
We do not need to consider the interlayer electron- S(a) = eEaτa(k)·kˆ ∂(f0(+a)− f0(−a))σ , (19b)
electroncollisionintegralJInterintheactivelayer,since Ek,(cid:107) 2(cid:126) ∂k k(cid:107)
k,a
that does not produce a drag current. The electron-
with τ (k) the momentum scattering time [6]. The re-
a
impurityscatteringterminthefirstBornapproximation
sultinginterlayerelectron-electroncollisionintegralbe-
isgivenby[33]
comes the driving term for the passive layer, and we
Jˆ(f(l))=(cid:10)(cid:82)∞ dt(cid:48)[Uˆ,e−iHˆt(cid:48)/(cid:126)[Uˆ, fˆ]eiHˆt(cid:48)/(cid:126)](cid:11) , (17) search for the solution of the kinetic equation for the
0 k 0 (cid:126)2 kk passive layer, which will yield the drag current. It is
wherethenotation(cid:104)(cid:105)denotestheaverageoverimpurity easy to verify that the electron-electron collision inte-
gral vanishes if we replace the density matrix of both
configurations.
miWtiaenwmilaltrwixriwtehfikc(lh) =cann(klb)1e+wrSit(ktle),nwinithteSrm(kl)sao2ft×he2PHaeurl-i lna(0ylk)e=rs(cid:2)bfy0(lk),i+ts+eqf0(ulk)i,−li(cid:3)b/r2iuamndfoSr0(mlk) =f0((cid:2)lk)f0(=lk),+n−(0lk)f0+(lk),−S(cid:3)0(σlk).z/2Hearree
spinmatrices. Everymatrixinthissectioncanbewrit- theequilibriumdistributionforthechargeandspindy-
tenintermsofascalarpart,labeledbythesubscriptn, namicsinlayerlrespectively,where f(l) aretheFermi-
0k,±
andtwospin-dependentpartsσ andσ . Thekinetic Diracfunctionsforthetwoenergyeigenstates.Wesolve
k(cid:107) k⊥
Eq.(15)canbewrittenas thekineticEq.(18b),yieldingthedensitymatrixforthe
passivelayerinthesteadystate
dn(l)
dtk +PnJˆ(fk(l))=D(kl,)n, (18a) S(p) = eπσk(cid:107) (cid:88)(cid:90) dω|V(q,ω)|2
dS(l) k,(cid:107) 4kBTL4 k(cid:48)q sinh2 β(cid:126)2ω
dtk,(cid:107) +P(cid:107)Jˆ(fk(l))=D(kl,)(cid:107), (18b) ×F+k,+k1δ[ε(kp1),+−ε(kp,+) +(cid:126)ω](f0(pk),+− f0(pk)1,+)
dSd(ktl,)⊥ + (cid:126)i[H0(lk),S(kl,)⊥]+P⊥Jˆ(fk(l))=D(kl,)⊥, (18c) ×F+k,+k(cid:48)1δ[ε(ka1(cid:48)),+−ε(ka(cid:48)),+−(cid:126)ω](f0(ak)(cid:48),+− f0(ak)(cid:48)1,+)
where D(kl) is the driving term for layer l. For the ac- ×Ea·(cid:2)τa(k1(cid:48))vk(cid:48)1 −τa(k(cid:48))vk(cid:48)(cid:3)τp(k),
tive layer, D(a) = −i[HE, f(a)]; for the passive layer, (20)
k (cid:126) k 0k wherev = Aakkˆ. Thecurrentoperatorduetotheband
D(p) = JInter, which both have three components D(l) , k (cid:126)
k k,p k,n Hamiltonianis ˆj= eAσ× zˆ. With j =tr(ˆjS(p))= j =
D(l) andD(l) .TheprojectionoperatorP actsonama- (cid:126) p k,(cid:107) p
k,(cid:107) k,⊥ (cid:107) σDEa,wehave
trixMastr(Mσ ),wheretrreferstothematrix(spin)
k(cid:107)
trace. Analogous definitions hold for the operators P⊥ σ = e2 (cid:80) (cid:82) dω|V(q,ω)|2Im[χ+a+(q,ω)]Im[χ+p+(q,ω)],
and Pn. The operators P(cid:107), P⊥, and Pn single out the D 16πkBT q sinh2β(cid:126)2ω
partsofthedensitymatrixwhichareparalleltoH(l) (in (21)
0k wherethenonlineardragsusceptibilityfortheconduc-
matrixlanguage),orthogonaltoH(l),andscalar,respec-
0k tionbandofonelayerisgivenby
tively.
2π(cid:88)
χ++(q,ω) =− F++ (f(l) − f(l) )
4. CalculationofCoulombdrag l L2 k,k+q k,+ |k+q|,+
k
(22)
To obtain the drag resistivity, the kinetic equation ×[τl(k)vk−τl(|k+q|)vk+q].
for the passive layer must be solved. We feed the ε(l) −ε(l) +(cid:126)ω+i0+
k,+ |k+q|,+
4
Thedragproblemreducestothecalculationofthenon-
linearsusceptibilityofthesystem. Thisfactreflectsthe
physical mechanism behind the drag phenomenon: the
drag current is a result of the rectification by the pas-
sivelayerofthefluctuatingelectricfieldcreatedbythe
active layer [68]. In special cases when the intralayer
(a)
electron-electron correlations are absent, the nonlinear 1
d=10nm
susceptibilityisreducedtotheproductofthediffusion
0.001 d=20nm
constant and the imaginary part of the polarization op-
d=40nm
erator[68]. Atlowtemperaturesthepredominantcon- 2) 10-6
e
tribution to drag is due to intraband processes near the ℏ/
Fermisurface. WhentheFermilevelisfinite,i.e. above ρ(D 10-9
theDiracpoint,electronstakemoreenergytotransition 10-12 d=10nm
from the valence band to the conduction band than to d=20nm
transition within the conduction band or valence band, 10-15 d=40nm
andwithasmallexcitationenergythechannelinvolving 0.1 0.2 0.5 1 2 5
interband transitions becomes inaccessible [73]. The n(1012cm-2)
(b)
small interlayer momentum transfer and excitation en-
ergy, i.e. q < 2kF, (cid:126)ω < Aq is the dominant region 10-4 d=10nm
d=20nm
ofpolarizationcontributingtothedragproblem. Writ-
d=40nm
ing τl(k) = τ0kl, we have for the imaginary part of the ) 10-6
2
susceptibility e
ℏ/
( 10-8
τ q2qˆ ρD
Imχ+l+(q,ω)=− (cid:113)0 d=10nm
8π(cid:126) 1−((cid:126)ω)2 10-10 d=20nm
Aq (23)
×(cid:2)G>(cid:16)2kFlq− (cid:126)Aω(cid:17)−G>(cid:16)2kFlq+ (cid:126)Aω(cid:17)(cid:3), 10-12 2 4 6 d=480nm10
T(K)
(c)
with √ 1.×10-6
G>(x)= x x2−1−arccosh(x). (24) 5.×10-7 Numerical
The susceptibility, which in general is dependent on Analytical
1.×10-7
the electron momentum, reduces to a momentum- 2)
e 5.×10-8
independentformatlowtemperatures,withthescatter- ℏ/
ingtimeevaluatedattheFermilevelτ =τ k . ρ(D
Fl 0 Fl 1.×10-8
5.×10-9
5. Discussion
1.×10-9
10 20 30 40 50 60
To obtain analytical results, we feed the susceptibil-
d(nm)
ity Eq.(23) and the approximate Coulomb interaction
Eq.(13)intoEq.(21). With Figure1: BehaviorofdragresistivityρD asafunctionofelectron
concentrationn,temperatureT,layerseparationd:(a)iselectronden-
(cid:90)0∞ sinh2(ω(cid:126)2ωd/ω2kBT) = 43π2 (cid:32)kB(cid:126)T(cid:33)3, (25) ρρsiDDtywadtietThpe=knFda5eKn=caeknFopdf=kρFDa0.a=5tnTkmFp=;=(5c)0Ki.s5;l(nabmy)e.irsDsteeiepmlaeprcaettrriaioctnucrdeonddseetppaneenntdd(cid:15)reenn=ccee5oo0ff.
Realanddotted(dashed)linesrepresentthenumericalandanalytical
wehavethedragresistivity
respectively.
σ (cid:126) ζ(3) (k T)2
ρ ≈− D =− B . (26)
D σ σ e2 16π 3 3
a p A2r2n2n2d4
s a p
In Fig. 1(a) we present numerical results for the de-
pendence of the Coulomb drag resistivity on the elec-
tron number density. The drag resistivity displays a
5
1 dependence with α < 1.5 for d = 10,20,40 nm.
nαnα
Waipthincreasingelectrondensitynthecoefficientαap-
proaches 1.5. The fact that the exact numerical results
showninFig.1(a)disagreemorestronglywiththean-
alytical results for smaller values of n and d is under-
standable, since Eq√. (26) applies only in the kFd (cid:29) 1 (a)
limit with k = 4πn. This trend of an increasing Numerical
F 10-3
quantitative failure of the asymptotic analytical drag
Analytical
formula for small k d has also been noted in graphene
F
[72,73,75]and2DEGsystems[68,69]. Theanalytical ρa 10-5
resultbecomesmoreaccuratewithincreasingkFd. ρ/D
InFig.1(b)weshowtheCoulombdragresistivityasa
10-7
functionoftemperatureT forthreedifferentthicknesses
d = 10,20,40 nm. The overall temperature depen-
dence of the drag resistivity increases nearly quadrati- 10-9
0.1 0.2 0.5 1 2 5
cally and there is no logarithmic correction due to the n(1012cm-2)
absence of backscattering in TIs. The T2 dependence (b)
1.×10-5
stems from the allowed phase space where electron-
5.×10-6
electronscatteringoccursatlowtemperature,andisex-
pectedforanyinteractionstrengthbetweenthetopand
1.×10-6
bottom layers of TIs as Fig. 1(b) shows, provided that ρa 5.×10-7
the carriers can be described using a Fermi liquid pic- /D
ρ
ture. In addition, in TIs the acoustic phonon velocity 1.×10-7
issmallerthaningraphene. Thesefactsmakethecon- 5.×10-8 Numerical
tributionofelectron-phononscatteringprocessestothe Analytical
resistivitymuchmoreimportantinthesurfaceof3DTIs 1.×10-8
2 4 6 8 10
than in graphene. For the surfaces of 3DTIs the effect T(K)
ofelectron-phononscatteringeventsbecomesimportant (c)
10-5
already for T as low as 10K. For this reason we take
Numerical
consider temperatures up to 10K in our numerical cal-
10-6
culations. Ingraphenethiseffectbecomesrelevantonly Analytical
beyond T (cid:38) 200K. It is also evident that Eq. (26) ρa 10-7
becomes increasingly accurate and approaches the nu- ρ/D
mericalresultsasthelayerseparationdincreases. 10-8
The behavior of the drag resistivity ρ as a function
D 10-9
oflayerseparationdisshowninFig.1(c)withT =5K
andkFa = kFp = 0.5nm. Thetrendoftheexactnumer- 10-10 10 20 30 40 50 60
icalresultschangesmoreslowly,afactthatisalsoem-
d(nm)
bodied in Fig. 1(a). Interestingly, in drag experiments
ongraphene[79],theddependenceofthedragresistiv- Figure2: Numerical(bluelines)andanalytical(greenlines)results
ityismuchslowerthanthe1/d4expectedintheweakly ofratiobetweendragresistivityρDandintralayerresistivityρawith
interacting regime, varying approximately as 1/d2 for ρa=ρp:(a)iselectrondensitydependenceof ρρDa atT =5Kandd=
d >4nm,whichiscomparabletoTIs. 20nm;(b)istemperaturedependenceof ρρDa withkFa =kFp =0.5nm
andd=20nm;(c)islayerseparationddependenceof ρD atT =5K
Fig.2showstheratiobetweenρD andtheintralayer andkFa =kFp =0.5nm. Dielectricconstant(cid:15)r =50anρdamomentum
resistivity ρa corresponding to Fig. 1, which illustrates scatteringtimeτa,p=0.8ps.
therelativemagnitudesofthedragandlongitudinalre-
sistivities. Theratioisabout10−6 ∼ 10−8,indicatinga
small drag resistivity and reflecting the weak electron-
electron interactions in TIs. In current 3DTIs, the di-
electric constant is as large as 100, indicating weak
electron-impurity and electron-electron Coulomb scat-
6
tering [85]. So the drag resistivity of TIs is much 6.2. Ultra-thinfilms
smaller than that of graphene which the relative static When the TI film is ultra thin tunnelling is enabled
dielectricconstantisabout4. ThisplacesTIssquarely between the top and bottom surfaces. The tunnelling
in the weakly-interacting regime, in which RPA-based between the surface states on the top and bottom sur-
theoriesareapplicable. faces may open an energy gap in the energy spectrum
ForbothFig.1andFig.2, wehaveonlyappliedthe [87]. Inthiscase, themasslessDiracHamiltonian H(l)
0k
approximation k d (cid:29) 1 for TI film thicknesses up to needs to be augmented by a series of tunnelling terms,
F
6nm, with electron densities n ∼ 1012cm−2. This is andisgenerallywrittenas:
a somewhat different regime than graphene, in which
H = Aτ ⊗[σ·(k×zˆ)]+tτ ⊗1, (27)
the condition k d (cid:28) 1 can usually be satisfied. How- k z x
F
ever,forothersystemswithsmallenoughkFdandlarger whereτmatricesrepresentthelayerpseudospinspace,
rs, the interlayer Coulomb interaction is in general not withτz =1symbolisingtheupsurfaceandτz =−1the
small,anditremainstobedeterminedwhetheritisnec- bottomsurface. Here zˆistheunitvectorinthedirection
essarytotakeintoaccounthigher-ordercontributionsin of z,andthetermtrepresentsthetunnellingmatrixele-
the Coulomb interaction. We have found that approx- mentbetweentwooppositetopologicalsurfaces. After
imations applied to |Vq|2 can lead to discrepancies be- thedirectdiagonalization,th√eenergyspectrumoftheTI
tween analytical and numerical results. The numerical thinfilmisgivenby(cid:15) =± t2+A2k2,whichhasagap
resultisafactorof∼102lowerthantheanalyticresult, ofsize2t. WediscusskhowthisgapaffectstheCoulomb
anobservationwhich,inthecaseofgraphene,hasbeen
dragbetweenthetwosurfaces.
confirmedbydragexperiments[76].
We begin with some general results. The scattering
termis:
n(cid:15) (cid:90) 2π
6. Extensionsofthetheory Jˆ(fk)= 4πiA2k(cid:126) dθk(cid:48)|Ukk(cid:48)|2(fk− fk(cid:48))
0
(28)
(cid:18) t2 A2k2cosγ(cid:19)
The focus of the paper up to now has been on thin × 1+ + ,
t2+A2k2 t2+A2k2
films of band TIs in which both layers are doped with
thesametypeofcarrier,thatis,eitherelectron-electron where γ = θk − θk(cid:48) is the angle between the incident
or hole-hole layers. Our theory can straightforwardly and scattered wave vectors. Note that the density ma-
beextendedtotreatstructuresbeyondthoseconsidered trix entering this term is the full density matrix f of
k
thusfar. Inthissectionwepresentthenecessarymod- thedouble-layersystem, ratherthanitsprojectiononto
ifications for treating the cases of multi-valley TIs and eachindividuallayer. Fort = 0Eq. (28)reducestothe
ultra-thinfilms,anddiscussbrieflythepossibilityofex- scattering term introduced in Eq.(17), and we recover
citoncondensation. thewell-knownfactorof1−cosγ,whichensuresthere
is no backscattering in TIs. Based on the ratio of the
interlayer tunnelling t and Fermi energy ε , two limit-
F
6.1. Multi-valleycase ingcasescanbeidentified: weaktunnellingt (cid:28)ε and
F
strongtunnellingt (cid:29) ε . Sothewavefunctionoverlap
F
Certain materials, such as the topological Kondo in- Eq. (12)withtunnellingbecomes
sulatorSmB ,havemorethanonevalley. Inthiscase,a
6 1 t2+A2k(k+qcosφ)
vInaltlheeyfdoelgloewneinragc,ywfeacwtoilrlgfivxwthiellenleeecdtrotondbeenisnittryodnu.cTehde. F(slk)s(cid:48)k(cid:48) = 2(1+ss(cid:48) (cid:112)(t2+A2k2)(t2+A2k(cid:48)2)). (29)
√
polarisation Π ∝ g , while the susceptibility χ ∝ g ,
v v We recalculate the polarization and susceptibility, then
thusthescreeningfunction(cid:15)(q,ω)isproportionaltog
v therelatedG (x)inEq. (24)isrewrittenas
>
[86]. Thedragcurrent,andhencethedragconductivity,
will remain the same as in the single-valley case. For G (x)= x(cid:113)x2−x2−(2−x2)arccosh( x ), (30)
the drag resistivity, based on Eq. (26) we will get the > 0 0 x
√ 0
lineargv dependencebecauseσa,p ∝ 1/ gv.Notethat (cid:113)
therewillbeanintervalleyimpurityscatteringcontribu- wherex0 = 1+ A2q24−t2(cid:126)2ω2.
tionifshort-rangedisorderispresentinthesystem,but Forweaktunnelling,t(cid:28) Ak ,thecarrierwavefunc-
F
this will simply result in a renormalisation of the scat- tionsareoverwhelminglylocatedinoneofthetwolay-
teringtimetaubyanintervalleyscatteringterm. ers (surfaces), and the notion of Coulomb drag can be
7
retained to a good approximation. The effect of tun- graphene [90], where an exciton condensate phase has
nelling can be taken into account perturbatively. For been identified, with the critical temperature estimated
example,themomentumrelaxationtimebecomes at 10 - 100mK. This estimate was for a sample grown
on a GaAs substrate, in which the effective dielectric
1 1(cid:18) t2 (cid:19)
→ 1+ . (31) constant is expected to be (cid:15)r ≈ 6, whereas in current
τ τ A2k2 TI films the lowest experimentally reported (cid:15) ≈ 30
F r
[91, 92], largely due to screening by the unavoidable
This implies that, as expected, interlayer tunnelling,
bulk of the film. These observations suggest that the
which brings with it interlayer scattering, slightly de-
criticaltemperatureinTIfilmscouldbeatleastanorder
creases the momentum relaxation time. Screening is
ofmagnitudesmallerthaningraphene,placingitinthe
qualitatively different [87], but for weak t it is a good
range1-10mK,whichwouldmakeexcitoncondensa-
approximation to retain the screening function defined
tionratherdifficulttodetectexperimentallyincurrently
in Eq. (10). With the same method of calculating the
availablesamples.
t = 0case,wehavetheanalyticalresultofthedragre-
sistivitywithtunnellingbecomes
7. Conclusion
ρ (cid:20) (cid:18) t (cid:19)2(cid:21)
ρ (t)= D ≈ρ 1−2 . (32)
D [1+(t/Ak )2]2 D Ak We have carried out a detailed analysis of the intra-
F F
layer and inter-layer electron-electron interactions in
We also recalculate the numerical result. Setting t ≈
TIs in order to determine the Coulomb drag resistiv-
0.1Ak which corresponds to an interlayer separation
F ity ρ and devise a complete picture of Coulomb drag
D
d ≈6nm,theresultalsoshowsthatρ (t )ismarginally
D 0 in these materials. We have found that ρ is propor-
D
smaller than the value that would be obtained by ne- tional to T2d−4n−3/2n−3/2 at low temperature and elec-
a p
glectingtunnelling.
trondensity. Wehavecomparedourresultsforρ with
D
When there the interlayer tunnelling is strong, such graphene,concludingthatthedrageffectisexpectedto
that t ∼ AkF, the notion of Coulomb drag is not ap- be weaker in TIs, and that different regimes are acces-
plicabletothethinfilmsystem. Inthiscase,thecarrier
sibleexperimentallyin TIsandgraphene. Thevalidity
wavefunctionsarespreadoverthetwolayers,hencethe
of certain analytical approximations for calculating ρ
D
picture of the Coulomb interaction causing charges in
has also been elucidated. The kinetic equation method
onelayertodragchargesintheotherisnolongervalid.
presentedinthisworkwillbeextendedinafuturepubli-
Themaineffectofelectron-electroninteractionswillbe
cationtodescribemagneticallydopedTIs,inwhichthe
throughtheCoulombrenormalizationoftheconductiv- anomalousHalleffectmakesanimportantcontribution
ity [87]. We note, however, that the film needs to be
toCoulombdrag.
extremely thin for the tunnelling gap to be noticeable
[88], therefore we expect realistic samples to lie in the
weaktunnellinglimit. Acknowledgments
We are grateful to Zhenyu Zhang, Wenguang Zhu,
6.3. Onthepossibilityofexcitoncondensation
ZhenhuaQiao,ChangganZeng,Shun-QingShen,W.K.
Whentheactivelayerisdopedwithelectronsandthe
Tse for enlightening discussions. This work was sup-
passive layer is doped with holes, or vice-versa, exci-
portedbytheInternationalCenterforQuantumDesign
ton condensation may occur. This effect is driven by
of Functional Materials HeFei National Laboratory for
anexchangetermintheinterlayerCoulombinteraction.
Physical Sciences at Microscale University of Science
Inprinciple,forinterlayerexchangetobenonzerotun-
andTechnologyofChina.
nelling also has to be nonzero, but one can think of a
situationinwhichthetunnellingisnegligiblebuttheex-
change is not. The interesting problem concerning the References
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