[Arxiv]Theory of Coulomb drag for massless Dirac fermions(2012)(spin Coulomb drag, przykład w...

7/27/2019 [Arxiv]Theory of Coulomb drag for massless Dirac fermions(2012)(spin Coulomb drag, przykad w intro)

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Theory of Coulomb drag for massless Dirac fermions

M. Carrega,1 T. Tudorovskiy,2 A. Principi,1 M.I. Katsnelson,2 and Marco Polini1,

1NEST, Istituto Nanoscienze-CNR and Scuola Normale Superiore, I-56126 Pisa, Italy2Radboud University Nijmegen, Institute for Molecules and Materials, NL-6525 AJ Nijmegen, The Netherlands

Coulomb drag between two unhybridized graphene sheets separated by a dielectric spacer hasrecently attracted considerable theoretical interest. We first review, for the sake of completeness, themain analytical results which have been obtained by other authors. We then illustrate pedagogicallythe minimal theory of Coulomb drag between two spatially-separated two-dimensional systems ofmassless Dirac fermions which are both away from the charge-neutrality point. This relies on second-order perturbation theory in the screened interlayer interaction and on Boltzmann transport theory.In this theoretical framework and in the low-temperature limit, we demonstrate that, to leading ( i.e.quadratic) order in temperature, the drag transresistivity is completely insensitive to the preciseintralayer momentum-relaxation mechanism (i.e. to the functional dependence of the scatteringtime on energy). We also provide analytical results for the low-temperature drag transresistivityfor both cases of thick and thin spacers and for arbitrary values of the dielectric constantsof the media surrounding the two Dirac-fermion layers. Finally, we present numerical results forthe low-temperature drag transresistivity in the case in which one of the media surrounding theDirac-fermion layers has a frequency-dependent dielectric constant. We conclude by suggesting anexperiment that can potentially allow for the observation of departures from the canonical Fermi-liquid quadratic-in-temperature behavior of the transresistivity.

I. INTRODUCTION

Electron-electron interactions are a source of couplingbetween closely spaced nano-electronic circuits. Thiscoupling has commanded a great deal of attention duringthe past thirty years or so, since it constitutes a poten-tial alternative to the inductive and capacitive couplingsof conventional electronics. Early on it was realized1,2

that Coulomb mutual scattering between spatially sep-arated electronic systems provides a mechanism to relaxmomentum that tends to equalize drift velocities. Thisintrinsic friction due to electron-electron interactions is

modernly referred to as Coulomb drag

39

. Early ex-perimental work was carried out by Gramila et al.10 andby Sivan, Solomon, and Shtrikman11 in semiconductordouble quantum wells.

In these experiments a constant current is imposed onthe two-dimensional (2D) electron gas in one of the wells(the active or drive layer). If no current is allowedto flow in the other well (the passive layer), an elec-tric field develops whose associated force cancels the fric-tional drag force exerted by the electrons in the activelayer on the electrons in the passive one. The transre-sistance D, defined as the ratio of the induced voltagein the passive layer to the applied current in the drivelayer, directly measures the rate at which momentumis transferred from the current-carrying 2D electron gasto its neighbor. Coulomb drag is ultimately caused by

fluctuationsin the density of electrons in each layer sincetwo-dimensional layers with uniformly distributed chargewill not exert any frictional forces upon each other3.

The study of Coulomb-coupled 2D systems has nowbeen revitalized by advances which have made it pos-sible to prepare robust and ambipolar 2D electron sys-tems (ESs), based on graphene12 layers or on the surfacestates of topological insulators (TIs)13, that are described

by an ultrarelativistic wave equation instead of the non-relativistic Schrodinger equation.

Single- and few-layer graphene systems can be pro-duced, for example, by mechanical exfoliation of thingraphite14 or by thermal decomposition of silicon car-bide15. Isolated graphene layers host massless-Diractwo-dimensional electron systems (MD2DESs) with afour-fold (spin valley) flavor degeneracy, whereastopologically-protected MD2DESs that have no addi-tional spin or valley flavor labels appear automatically13

at the top and bottom surfaces of a three-dimensional(3D) TI thin film. The protected surface states of 3D

TIs are associated with spin-orbit interaction driven bulkband inversions. 3D TIs in a slab geometry offer twosurface states that can be far enough apart to makesingle-electron tunneling negligible, but close enough forCoulomb interactions between surfaces to be important.Unhybridized MD2DES pairs can be realized in grapheneby separating two layers by a dielectric16 (such as Al2O3)or by a few layers of a one-atom-thick insulator suchas BN1720. In both cases interlayer hybridization isnegligible and the nearby graphene layers are, from thepoint of view of single-particle physics, isolated. Iso-lated graphene layers can be also found on the surfaceof bulk graphite21,22 and in folded graphene23 (a nat-ural byproduct of micromechanical exfoliation), or pre-

pared by chemical vapor deposition22. We use the termdouble-layer graphene (DLG) to refer to a system withtwo graphene layers that are coupled only by Coulombinteractions, avoiding the term bilayer graphene whichtypically refers to two adjacent graphene layers in thecrystalline Bernal-stacking configuration12.

DLG and TI thin films are both described at low en-ergies by a Hamiltonian with two MD2DESs12 coupledonly by Coulomb interactions, in the absence of single-particle tunneling. Coulomb drag between two spatially-

arXiv:1203.3386v1

[cond-mat.str-el]

15Mar2012


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separated MD2DESs has recently attracted a great dealof theoretical interest2429. The calculations in Refs. 2429 refer to the regime in which both layers are eitherelectron- or hole-doped. The Coulomb drag transresis-tivity in this case is negative and vanishes like T2 at lowtemperatures. Despite the considerable amount of workpublished on the subject recently2429, no clear consen-sus exists on the dependence of the drag transresistivity

in the Fermi-liquid regime on carrier densities in the twolayers, on the interlayer distance, and on the dielectricconstants of the media surrounding the two layers. Themain analytical results obtained earlier by other authorswill be summarized below in Sect. II.

The Coulomb drag transresistivity between twoMD2DESs in the regime in which one layer is electrondoped and the other is hole doped has been recently cal-culated by Mink et al.30. In this intriguing regime, theauthors of Ref. 30 have found that D grows logarith-mically upon lowering the temperature T towards thecritical temperature Tc for exciton condensation

31 (con-densation of electron-hole pairs in a dipolar condensate).

Coulomb drag between two graphene sheets has beenrecently measured by Kim et al.16. This first experimen-tal study represents an important milestone since the au-thors of this work have shown that the strong-couplingregime, i.e. the regime in which the interlayer distance dis much smaller that the typical separation between twoelectrons in each layer, is easy to achieve experimentallywith two, independently-contacted, graphene sheets16.This study has indeed fueled the recent theoretical inves-tigations of Coulomb drag between two MD2DESs men-tioned above.

In this Article we present in a pedagogical fashion theminimal theory of Coulomb drag between two spatially-

separated MD2DESs in the regime in which both layersare either electron- or hole-doped. We will be only con-cerned with the so-called Fermi-liquid regime in whichboth layers are away from the charge neutrality point.Our theory relies on second-order perturbation theoryin the screened interlayer interaction and on Boltzmanntransport theory. In this theoretical framework and inthe low-temperature limit, we demonstrate that, to lead-ing (i.e. quadratic) order in temperature, the drag tran-sresistivity is completely insensitive to the precise in-tralayer momentum-relaxation mechanism (i.e. to thefunctional dependence of the scattering time on energy).This is in contradiction with the findings reported inRefs. 27 and 28. We also provide new analytical resultsfor the low-temperature drag transresistivity in bothcases of thick and thin spacers, correcting in the lat-ter case a mistake contained in Ref. 26. At odds with allthe previous literature, our results hold true for arbitraryvalues of the dielectric constants of the media surround-ing the two Dirac-fermion layers. Finally, we presentnumerical results for the low-temperature drag transre-sistivity in the case in which one of the media surround-ing the two MD2DEs has a strongly-frequency-dependentdielectric constant. We conclude by suggesting an ex-

d

z

1

2

3

FIG. 1: (Color online) A side view of the double-layer system,which explicitly indicates the dielectric model used in thesecalculations. The two layers hosting massless Dirac fermionsare located at z= 0 and z= d. In a Coulomb-drag transportsetup a constant current flow is imposed in one layer (thebottom one, say). If no current is allowed to flow in the otherlayer, an electric field develops whose associated force cancelsthe frictional drag force exerted by the electrons in the bottomlayer on the electrons in the top one.

periment with a DLG deposited on SrTiO332 that can

pave the way for the observation of departures from thecanonical Fermi-liquid quadratic-in-temperature behav-ior of the transresistivity.

Our manuscript is organized as follows. In Sect. II wereport a summary of the main analytical results whichhave been obtained by other authors. In Sect. III wepresent the model Hamiltonian and the most importantbasic definitions. In Sect. IV we present the Kubo formal-ism approach to the calculation of the drag conductivity,while in Sect. V we present a series of simplifications thatlead to the Boltmann-transport expression for the dragconductivity and resistivity. These two Sections do notcontain original results but make the paper completelyself-contained. Experts can skip Sects. IV-V and go di-

rectly to Sects. VI-VII, which contain the most impor-tant results of this work and all our original results. InSect. VIII we summarize our main findings and draw ourmain conclusions.

II. SUMMARY OF THE MAIN ANALYTICAL

RESULTS OBTAINED EARLIER BY OTHER

AUTHORS

Despite the large body of theoretical work ded-icated to Coulomb drag between two unhybridizedMD2DESs2429, no consensus appear to exist among dif-ferent authors. In what follows we summarize the mainanalytical results that can be found in the existing liter-ature.

1) Tse et al.24 studied Coulomb drag between twounhybrydized graphene sheets separated by a dielectricby employing Boltzmann transport theory. They ne-glected the spatial dependence of the dielectric constantin the z direction (see Fig. 1) and assumed a momentum-independent scattering time. In the weak-coupling limit,i.e. in the limit in which the interlayer distance d is


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much larger than the average distance between two elec-trons in each layer, Tse et al.24 demonstrated that thelow-temperature drag resistivity is given by

D he2

(3)

32

(kBT)2

F,1F,2

1

(qTF,1d)(qTF,2d)

1(kF,1d)(kF,2d)

. (1)

In Eq. (1) qTF, is the Thomas-Fermi screening wave vec-tor, which is proportional to kF,, the Fermi wave numberin each layer. In the symmetric n1 = n2 case the previousequation yields

D he2

T2

n3d4. (2)

2) We now move on to summarize the main resultsby Peres et al.27. The authors of this work used Boltz-mann transport theory, took into account the momentumdependence of the scattering time and also the spatial

dependence of the dielectric constant in the z direction.While the main approach followed by these authors is nu-merical, they do also provide an analytical expression forthe Coulomb drag transresistivity in the weak-couplingregime. For an intralayer scattering time that dependslinearly on momentum, Peres et al.27 found

D he2

T2

n4d6, for kFd 1 . (3)

3) Katsnelson26 used Boltzmann transport theory,took into account, at least partially, the spatial depen-dence of the dielectric constant in the z direction, but

did not take into account the momentum dependence ofthe scattering time. The main results of Ref. 26 are:

D he2

T2| ln(nd2)|

n, for kFd 1

D he2

T2

n3d4, for kFd 1

. (4)

Note that the weak-coupling result by Katsnelson is inagreement with that of Tse et al.24.

4) Hwang et al.28 used Boltzmann transport theory,took into account the momentum dependence of the scat-tering time but neglected the spatial dependence of the

dielectric constant in the z direction. For a momentum-independentintralayer scattering time they find:

D he2

T2

n2d2, for kFd 1

D he2

T2

n4d6, for kFd 1

. (5)

Note that the weak-coupling result in Eq. (5) differs fromthe result by Tse et al.24.

For an intralayer scattering time that depends linearlyon momentum, Hwang et al.28 find instead

D he2

T2| ln(nd2)|

n, for kFd 1

D he2

T2

n3d4, for kFd 1

. (6)

By comparing Eq. (5) with Eq. (6), Hwang et al.28

con-cluded that the functional dependence of D on n andd is very sensitive to the functional dependence of theintralayer scattering time on momentum.

5) Narozhny et al.29 have recently presented a sys-tematic study of Coulomb drag between two MD2DESswhich is based on perturbation theory in the dimension-less coupling constant ee = e

2/(hv), v being the Diracvelocity. The authors of Ref. 29 did not consider thespatial dependence of the dielectric constant along thez direction and discussed mostly the case of low dop-ing (i.e. the regime in which the chemical potential iscomparable with or smaller than kBT). Since the focusof our Article is on the opposite regime, i.e. the Fermi-liquid regime, we now provide a short summary of theresults of Narozhny et al.29 in this regime only. The au-thors of Ref. 29 have demonstrated that the energy de-pendence of the relaxation time is completely irrelevantin relation with the low-temperature drag transresistiv-ity. In the weak-coupling limit Narozhny et al.29 foundfor the drag transresistivity the same doping- and inter-layer separation-dependence as in Refs. 24 and 26. Asdensity decreases, Narozhny et al.29 found that the dragcoefficient acquires logarithmic corrections see Fig. 3 inRef. 29. In particular, in the limit kFd 1, Eq. (41) inRef. 29 reads

D he2

T2

nln

1Nfee

. (7)

As we will see below, our results agree with those ofNarozhny et al.29, and represent their generalizationto the case of a finite Thomas-Fermi screening length( Nfee) and spatially-dependent dielectric constantsalong the z direction.

III. MODEL HAMILTONIAN AND BASIC

DEFINITIONS

We consider two unhybridized layers of massless Diracfermions, each one described at the noninteracting levelby the following single-channel Hamiltonian (h = 1):

H = vk,,

k,, ( k) k,, . (8)

Here v is the bare electron velocity, k is the k p mo-mentum, , are (sublattice) pseudospin labels, and = (

x,

y) is a vector of Pauli matrices which

act on the sublattice pseudospin degree-of-freedom. The


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field operator k,, (k,,) creates (destroys) an elec-tron with momentum k, pseudospin , and layer index = 1, 2.

In each layer, the Hamiltonian (8) can be easily diag-onalized by the matrix

U(k) = 12

eik/2 eik/2

eik/2 eik/2

, (9)

where k is the polar angle of the vector k.The i-th Cartesian component of the current-density

operator (i = x, y) in the -th layer is given by

Jiq, =k,,

kq,,(v

i)k,, . (10)

The Fermi wave number in the -th layer is defined by

kF,

4nNf,

, (11)

where n > 0 is the excess electron density in the -thlayer33 and Nf, is a degeneracy factor (Nf, = 4 for agraphene layer, accounting for spin and valley degenera-cies, while Nf, = 1 for a TI surface state).

In the absence of disorder, the Greens function corre-sponding to the Hamiltonian in Eq. (8) in the imaginaryfrequency axis is given by the following 2 2 matrix

G(k, i) =( + i)11 + vk

( + i)2 v2k2 , (12)

where 11 is the 2 2 identity matrix in the sublattice-pseudospin representation and is the chemical po-tential in the -th layer. In the zero-temperature limit F, = vkF,, where F, is the Fermi energy in the-th layer.

The disorder-free Greens function in the eigenstate

representation reads instead

G,(k, i) = 1i + k, , (13)

where k, = v|k| are Dirac-band energies.In what follows we will need the following matrix ele-

ments:

k,k U(k)U(k)

=ei(kk )/2 + ei(kk )/2

2, (14)

and

x

k,k

U(k)xU(k)

=ei(k+k)/2 + ei(k+k )/2

2. (15)

The two MD2DESs described by Eq. (8) are cou-pled electrostatically by long-range Coulomb interac-tions, which are influenced by the layered dielectric envi-ronment (see Fig. 1). The coupling Hamiltonian reads

Hee = 12S

q,=

V(q)q,q, , (16)

where

q, =k,

kq,,k,, (17)

is the density-operator for the -th layer and V(q) (with = ) is the 2D Fourier transform of the interlayerCoulomb interaction

V12(q) = V21(q) = 8e2

qD(q)2 . (18)

Here

D(q) = [(1+2)(2+3)eqd+(12)(23)eqd] . (19)

For future purposes, we introduce the dynamicallyscreened interlayer interaction U12(q, ), which, at therandom phase approximation (RPA) level, is givenby34,35

U12(q, ) =V12(q)

(q, ), (20)

where

(q, ) = [ 1 V11(q)(0)1 (q, )][1 V22(q)(0)2 (q, )] V212(q)(0)1 (q, )(0)2 (q, ) (21)

is the RPA dynamical dielectric function.

In Eq. (21), (0) (q, ) is the well-known

3739 density-density (Lindhard) response function of a noninteractingMD2DEs at arbitrary doping n. The Coulomb interac-tion in the = 1 (top) layer is given by

V11(q) =4e2

qD(q)[(2 + 3)e

qd + (2 3)eqd] , (22)

while the Coulomb interaction in the bottom layer,V22(q), can be simply obtained from V11(q) by inter-changing 3 1.

Eqs. (18), (19), and (22) have first appeared in Ref. 36and their explicit derivation has been reported in Ref. 26.Notice that in the uniform 1 = 2 = 3 limitwe recover the familiar expressions V11(q) = V22(q) 2e2/(q) and V12(q) = V21(q) V11(q) exp(qd). Mostof the previous work on Coulomb drag in DLG has as-sumed this limit, which rarely applies experimentally.

The aim of this Article is to present a theory ofCoulomb drag, which is valid up to second order in the

dynamically-screened interaction U12(q, ), for the sys-tem described by the Hamiltonian

H =

H + Hee . (23)

Note that we are not including in Eq. (23) any term de-scribing intralayer electron-electron interactions. Thesecan be treated in an approximate fashion by invokingLandaus theory of normal Fermi liquids35, i.e. by renor-malizing the microscopic parameters of the intralayer


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FIG. 2: Second-order Aslamazov-Larkin-type diagramscontributing to the Coulomb drag conductivity (finite-

temperature Matsubara formalism). Solid lines denote thesingle-particle Greens function in the presence of disorder.Wavy lines denote the screened interlayer interaction U12 inEq. (20). Black dots in the triangular portions of the dia-grams denote vertices of current operators in the two layers,which are both proportional to the Pauli matrix x, say, ifone is interested in the longitudinal drag conductivity. Fi-nally, k = k q, k

= k

q, in, = in in, and

in, = in in.

Hamiltonian H. For example, in Eq. (8) one can usethe renormalized quasiparticle velocity40, v , instead ofthe bare velocity v. Anyway, a treatment of the impact

of intralayer interactions on the Coulomb drag transre-sistivity is well beyond the scope of this Article.

IV. KUBO-FORMULA APPROACH TO THE

CALCULATION OF THE DRAG

CONDUCTIVITY

Coulomb drag starts at second order in a perturba-tive expansion for the drag conductivity D in powersof the interlayer Coulomb interaction. To avoid infraredpathologies (stemming from the long-range nature of theCoulomb interaction) of certain integrals that appear inthe theory, Coulomb interactions must be screened: fromnow on we will work with the dynamically-screened in-teraction U12 introduced in Eq. (20) rather than with thebare potential V12 in Eq. (18).

The two Aslamazov-Larkin-type diagrams41 that con-tribute to D up to second order in U12 are depicted inFig. 2. In this figure, solid lines represent disorderedpropagators in the sublattice-pseudospin representation,while wavy lines represent the screened Coulomb inter-action U12. Finally, the vertices (black dots in the trian-gular diagrams) are current operators, one in each layer,evaluated at q = 0 from the very beginning since we areinterested in the drag conductivity in the uniform limit.Both current operators can be taken along the x direc-tion since we are interested in the current response of thepassive layer in the x direction to an electric field ap-plied in the same direction in the active layer, i.e. weare interested in the longitudinaldrag conductivity.

Straightforward algebraic manipulations lead to thefollowing compact expression for the sum of the two dia-grams in Fig. 2:

D(im) =e2

2m

d2q

(2)21

n

U12(q,in)U12(q,in + im)1(q, in + im, in)2(q, in, in + im) , (24)

where = (kBT)1 is the usual thermal factor, m and n are bosonic Matsubara frequencies and

(q, i1, i2) =vNf,

n

d2k

(2)2Tr

G(k, in)G(k + q, in + i2)xG(k + q, in + i1)

+ G(k, in)G(k q, in i1)xG(k q, in i2)

(25)

is the so-called non-linear susceptibility. Here n is afermionic Matsubara frequency, the index = 1, 2 refers

to the layer degree of freedom, and the symbol Tr[. . .]denotes a trace over sublattice-pseudospin indices.


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FIG. 3: The contour of integration C chosen to carry outthe Matsubara sum in Eq. (27). Note the branch cuts onthe real axis and in the lower half of the complex plane atm(z) = im. The crosses denote the poles of the Bose-Einstein thermal factor nB(z).

We now need to perform the analytic continuationim + i0+ in Eq. (24) and then take the d.c. 0limit. It is very well known that the analytic continuation

must be carried out after performing the sum over theMatsubara frequencies n. Let us assume that U12(q, z),seen as a function of the complex frequency z, is analytic,i.e. that it has no branch cuts. Under this assumption,the function (q, z , z

) has two branch cuts when z or z

are on the real axis and is analytic elsewhere, includingthe points z = 0 and z = 0. We then introduce

f(q, z , z) = U12(q, z)U12(q, z)1(q, z, z)2(q, z , z) ,(26)

and perform the Matsubara sum in Eq. (24) using a stan-dard procedure:

J(im) 1

n

f(q, in, im + in)

=

C

dz

2inB(z)f(q, z , im + z) , (27)

where nB(z) = (ez 1)1 is the Bose-Einstein occupa-tion factor on the complex plane and

Cis an appropriate

contour which excludes the branch cuts of the integrand(see Fig. 3), located at m(z) = 0, im. Notice thatthe contour C as defined in Fig. 3 includes the pointsz = 0 and z = im, where the integrand is analytic.We obtain

J(im) = P+

d

2inB()

f(q, +, im + ) f(im, , im + )

+ f(im, im, +) f(im, im, ) , (28)

since only the integrals around the branch cuts contributeto J(im), while the integral over the circle vanisheswhen its radius is sent to infinity. Here = i0+and P stands for the Cauchy principal value, i.e. the

point = 0 is excluded from the integration.

We are now ready to perform the analytical continua-tion to real frequencies. After some algebra we find

J( + i) = P+

d

2inB( + ) nB()

f(q, , + +)

+ nB()f(q, +, + +) nB( + )f(q, , + )

, (29)

We finally take the d.c. limit 0 in Eq. (29) andwe obtain

lim0

J( + i) +

d

2i

nB()

f(q, , +) ,

(30)

since the terms in the last line of Eq. (29) vanish at leastlike 2. Thus the drag conductivity in the d.c. limitreads5


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D =e2

16

d2q

(2)2

+

d|U12(q, )|2

sinh2(/2)1(q,

+, )2(q, , +) . (31)

This is the central result of this Section and was first ob-

tained in the context of Coulomb drag between ordinary2D parabolic-band electron gases see e.g. Ref. 5.

A. The non-linear susceptibility

Let us go back to the definition of the non-linear sus-ceptibility given in Eq. (25). We introduce the follow-ing definition

g(q,i,i, i) = vNf,

d2k

(2)2Tr

G(k, i)

G(k + q, i

)xG(k + q, i)

(32)

so that Eq. (25) reads

(q, i1, i2) =1

n

g(q, in, in + i2, in + i1)

+ g(q, in, in i1, in i2)

Jg(q, i1, i2)+ Jg(q, i2, i1) . (33)

We here remind the reader that 1 and 2 (n) arebosonic (fermionic) Matsubara frequencies. Let us now

concentrate on the first term on the r.h.s. of Eq. (33),i.e. on Jg(q, i1, i2). For the sake of simplicity, in whatfollows we omit to indicate explicitly the q-dependenceof the functions g(q,i,i

, i) and Jg(q, i1, i2) everytime we write equations that involve them. The completenotation will be restored only at the end of the mathe-matical manipulations that we perform on Eq. (33).

We first perform the fermionic Matsubara sum in thefirst term on the r.h.s. of Eq. (33) by the standard pro-

cedure:

Jg(i1, i2) = 1

n

g(in, in + i2, in + i1)

=

C

dz

2inF(z)g(z, z + i2, z + i1) ,

(34)

where nF(z) = (ez + 1)1 is the Fermi-Dirac occupationfactor on the complex plane. The function g(z, z+i2, z+i1) has three branch cuts in the complex plane locatedat m(z) = 0, i1, i2. Choosing a suitable contourof integration C, which encircles only the poles of nF(z)and leaves outside the branch cuts (in complete analogywith the contour drawn in Fig. 3), we find

Jg(i1, i2) =+

d

2inF()

g(

+, + i2, + i1)

g(, + i2, + i1)+ g( i1, + + i2 i1, +) g( i1, + i2 i1, )+ g( i2, +, + + i1 i2) g( i2, , + i1 i2)

. (35)

Here = i0+ and we used that nF( + im) = nF()if m is a bosonic Matsubara frequency. Notice that weare allowed to remove the index from when it issummed to i1 or i2 (since they will be analyticallycontinued to ) but not when it is summed to i1i2.Analytically continuing i1 +1 and i2 2 [toobtain e.g. 1(q,

+, ) in the integrand on the r.h.s.of Eq. (31)] we finally obtain

Jg(+1 , 2 ) =+

d2i

nF()

g(+, + 2, + + 1) g(, + 2, + + 1)

+ g( 1, + 2 1, +) g( 1, + 2 1, )

+ g(+ 2, +, + + 1 2) g(+ 2, , + 1 2)

. (36)

We observe that the second term on the second line andthe first term on the third line of Eq. (36) are exactly zerosince they involve products of three Greens functions

with poles on the same half of the complex plane. Aftersome algebraic manipulations we get


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Jg(+1 , 2 ) =+

d

2i

[nF() nF( + 2)] g(+, + 2, + + 1)

+ [nF( + 1) nF()] g(, + 2, + + 1)

. (37)

We now take the limit 1, 2 and recast Eq. (34) in the following form:

Jg(+, ) = vNf,+

d

2i[nF( + ) nF()]

d2k

(2)2Tr

GA (k, ) GR (k, )

GA (k + q, + )xGR (k + q, + )

, (38)

where we have introduced the retarded (advanced)

Greens function GR(A) (k, ).The second term on the right hand of Eq. (33) can be

treated in an analogous manner. The final expression for

the non-linear susceptibility reads

(q, +, ) = vNf,

+

d

2i

nF( + ) nF()

d2k(2)2

Tr

GA (k, ) GR (k, )

GA (k + q, + )x

GR (k + q, + )

+

(q, ) (q, )

. (39)

Eq. (39) together with Eq. (31) represent the most im-portant results of this Section and are the starting pointfor the calculation of the drag transresistivity D.

V. BOLTZMANN-TRANSPORT LIMIT

Eqs. (31) and (39) need to be simplified for any prac-tical purpose and for a quantitative estimate of the

Coulomb drag transresistivity.

We first switch from the off-diagonal sublattice-pseudospin representation of the Greens functions tothe diagonal representation. In the latter representationEq. (39) reads

(q, +, ) = vNf,+

d

2i d2k

(2)2

,,

nF( + ) nF()GA,(k, ) GR,(k, )

GA,(k + q, + )GR,(k + q, + ) k,k+qxk+q,k+qk+q,k

+

(q, ) (q, )

, (40)

where we used the definitions in Eqs. (14)-(15) for the density and current vertices.


9/20

9

We now show that the terms with = do notcontribute to (q,

+, ).Let us indeed consider all the terms in the triple sum

in Eq. (40) in which = . We first recall thatxk+q,k+q

= i sin(k+q) and

k,k+qk+q,k =i

2sin(k k+q) . (41)

We then consider the term on the third line of Eq. ( 40)and perform the following changes of variables: k k and . Using the fact that sin(kq) = sin(k+q) and sin(kkq) = sin(kk+q), wecan write the off-diagonal (OD) contribution ( = )to OD (q,

+, ) as

OD (q, +, ) = vNf,

+

d

2i

d2k

(2)2

,

2

nF( + ) + nF( ) nF() nF()

GA,(k, ) GR,(k, )GA,(k + q, + )GR,(k + q, + )sin(k k+q)sin(k+q) .(42)

Since nF(z) + nF(z) = 1, the term in square brackets isidentically zero. The OD contribution to (q,

+, )thus vanishes. From now on we can set = in

Eq. (40) and sum only over and .We can further simplify Eq. (40) by assuming that, in

the presence of weak disorder, the Greens function in the-th layer can be well approximated by the expression,

GR(A), (k, )

()k, i

2(k)

1, (43)

where (k) represents a momentum-dependent scatter-

ing time in the -th layer and ()k, v|k| are Dirac-

band energies measured from the chemical potential of the -th layer.

Note that here (k) should not be interpreted as aone-particle scattering time, but rather as the transportscattering time6. This statement can be justified withinthe Kubo formalism by including disorder-related vertexcorrections, or, in a much more transparent way, by usingthe Boltzmann transport equation, see Appendix A.

Below, we will assume that the transport scatteringtime (k) is isotropic, i.e. that it depends only onk = |k|, but allow different scattering times in the twolayers. The specific functional dependence of on k de-pends on a particular model of intralayer impurity scat-

tering. In the simple case of a momentum-independentscattering time we recover the usual relaxation time ap-proximation. A scattering time which depends linearly

on momentum, (k) k, is a very popular model inthe graphene literature. Early on it was understood42

that such functional dependence of on k is needed toexplain the linear-in-carrier-density d.c. conductivitiesthat are experimentally measured in samples on dielec-tric substrates such as SiO2 (and is typically attributed tocharged impurities located close to the graphene sheet).Below, we will not assume any specific functional depen-dence of on k. Our low-temperature analytical resultsfor the Coulomb drag transresistivity do not depend onthe particular scattering model that one chooses.

At this point, it is useful to introduce the one-particlespectral function corresponding to Eq. (43),

A,(k, ) = 1

m GR,(k, )= i

GA,(k, ) GR,(k, ) , (44)and the identity

GR,(k, )GA,(k, ) = (k)A,(k, ) . (45)

Using these definitions in Eq. (40) we obtain the followingapproximate expression for the non-linear susceptibility:

(q, +, ) vNf,

+

d

2i

d2k

(2)2

,

nF( + ) nF()

i(k + q)A,(k, )A,(k + q, + )

1 + cos(k k+q)

2xk+q,k+q,

+

(q, ) (q, )

. (46)

From now on we will introduce the simplified notation Jxk, vxk,k = v cos(k), where in the last equality


10/20

10

we have used Eq. (15). The definition of Jxk, shouldnot be confused with the definition of the current-densityoperator Jiq, in second quantization given in Eq. (10).

In this Article we are interested in the limit in whichintralayer scattering is weak, which is, for example, themost relevant regime for high-quality DLG samples. Inthis limit we can approximate the spectral functions inEq. (46) with functions,

A,(k, ) ( ()k,) . (47)

Straightforward algebraic manipulations of Eq. (46)with the use of Eq. (47) yield the following Boltzmann-transport (BT) expression for the non-linear susceptibil-ity:

BT (q, +, ) BT (q, ) = Nf,

,

d2k

(2)2

(k + q)Jxk+q, (k)Jxk,

m

nF(()k,) nF(()k+q,)

+ ()k, ()k+q, + i0+

1 + cos(k+q k)

.

(48)

We have dubbed Eq. (48) BT expression for the non-linear susceptibility since after inserting it in Eq. (31)one obtains precisely an expression for the Coulomb dragconductivity that can be derived within a Boltzmann-equation approach. This will be shown in Appendix A.

Note that, in the usual relaxation-time approxima-tion ( independent of k), the BT expression for thenon-linear susceptibility is simply proportional to the in-tralayer scattering time, BT (q, ) .

A. Coulomb drag transresistivity in theBoltzmann-transport limit

We now focus our attention on the quantity which isactually measured in experiments, i.e. the drag tran-sresistivity D, which is precisely the ratio between thevoltage drop in the passive layer and the current in theactive layer is indeed D. This quantity can be easily

found by inverting the 2 2 conductivity matrix,

D = D 1det

1 DD 2

D12

, (49)

where is the intralayer conductivity and the last ap-proximation in Eq. (49) holds true only if D .Within BT theory the intralayer conductivity at finitetemperature is given by43

=e2

2v2Nf,

d2k

(2)2(k)

nF(

()k,)

()k,

, (50)

while the drag conductivity D is given by Eq. (31) withthe expression (48) for the non-linear susceptibility. Thefinal expression for the BT drag resistivity reads

BTD =

e2

1612

d2q

(2)2

+

d|U12(q, )|2

sinh2

(/2)

BT1 (q, )BT2 (q, ) . (51)

Note that in the relaxation-time approximation the BTdrag transresistivity does not depend on the transportscattering times in the two layers (even if these are dif-ferent), since, as noted earlier in Sect. V, BT is propor-tional to in this approximation and so is .

The situation turns out to be much more complicatedin the case in which is an arbitrary function of k =|k|. A full numerical treatment of the three-dimensionalintegral in Eq. (51) at any finite temperature is beyondthe scope of the present work and will be the subject ofa forthcoming publication.


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11

In the next Section, however, we demonstrate that adramatic simplification occurs in the low-temperaturelimit. We will indeed prove that when kBT is muchsmaller than the Fermi energies in the two layers, kBT min(F,),

BTD is insensitive to the precise functional

dependence = (k) of the intralayer scattering time.Since for typical values of doping the Fermi energy in agraphene sheet is very large ( 1400 K for a carrier den-sity on the order of 10

12

cm2

), analytical results in thelow-temperature limit are very useful when both layersare sufficiently away from the charge neutrality point.

In the intriguing regime in which one layer (both lay-ers) lies (lie) at the charge neutrality point BT theory isinapplicable. Coulomb drag in this regime is dominatedby unavoidable electron-hole puddles44 and is certainlyvery interesting but well beyond the scope of the presentArticle. For reasons of symmetry, lowest-order Boltz-mann transport theory applied to the regime in whichone of the two layers is at the neutrality point givesBTD = 0

24,25.

VI. THE LOW-TEMPERATURE LIMIT

In this Section we use Eq. (51) to calculate BTD analyt-ically in the low-temperature limit, for arbitrary values

of the dielectric constants i in Fig. 1, and for a genericscattering time (k).

As already anticipated above, the low-temperatureregime is readily identified by the inequality kBT min(F,). In this limit we can:

(i) use the low-temperature expression for the in-tralayer conductivity42,43

limT0

=e2

4Nf,F,(kF,) ; (52)

ii) evaluate the non-linear BT susceptibilitiesBT1 (q, ) and

BT2 (q, ) at zero temperature and only to

lowest order in in the low-frequency / min(F,) 0limit;

and iii) replace the dynamically-screened interlayer in-teraction with the much simpler statically-screened in-teraction U12(q, 0).

Indeed, the thermal factor sinh2(/2) in Eq. (51)

represents an effective cut off on the values of in theintegral in Eq. (51): it must be 2/.

To find the asymptotic behavior of BT (q, ) in thelimit 0 we first re-write Eq. (48) in the form

BT (q, ) = Nf,,

d2k

(2)2


nF(

()k,) nF(()k+q,)

( + ()k, ()k+q,)

1 + cos(k+q k)

.

(53)

Clearly BT (q, 0) = 0. We thus need to calculate thederivative of BT (q, ) with respect to at = 0 and

in the zero-temperature limit. We find

BT (q, )

=0

= Nf,,

d2k

(2)2


nF( + ()k,)

=0

(()

k, ()

k+q,)

1 +

cos(k+q k)T0

= Nf,,

d2k

(2)2


(

()k,)(

()k, ()k+q,)

1 + cos(k+q k) ,(54)

where the last equality is valid only in the zero- temperature limit. More explicitly, we find


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12

limT0

BT (q, 0) = Nf,,

d2k

(2)2


(

()k,)(

()k+q,)

1 + cos(k+q k) . (55)

Eq. (55) is one of the most important results of this Sec-tion. An explicit expression for limT0

BT (q, 0)

will be given below in Eq. (58). Notice that the two-functions in the above expression impose that onlyintraband excitations ( = = +1) contribute toBT (q, 0) in the low-temperature limit. Moreover,the two -functions together pin the absolute values ofk and k + q to be equal to kF,. This implies that inthe low-temperature limit the transport scattering times(k+ q) and (k) inside the square bracket in Eq. (55)must be both evaluated on the Fermi surface of the -thlayer, i.e. |k + q| = |k| = kF,. Since (k) depends onlyon the absolute value of its argument (and not on the

polar angle ofk), both(k+ q) and (k) can be factor-ized out of the integral in Eq. (55). This is precisely thereason why at the end of Sect. V A we claimed that inthe low-temperature limit BTD is completely insensitiveto the precise intralayer scattering mechanism.

More mathematically, we have:

limT0

BT (q, 0) = Nf,(kF,)

+

0

kdk

2(vk F,)

2

0

dk2

Jxk+q,+ Jxk,+ (vk v|k + q|) 1 + cos(k+q k) . (56)

We now employ the following identity,

Jxk+q,+ Jxk,+ =q

kF,cos(q) , (57)

which can be easily derived by using the aforementionedcondition |k + q| = |k| = kF,. The two integrals inEq. (56) can be easily carried out analytically: we find

limT0

BT (q, 0) = NF,qx(kF,)

2vq(2kF, q)

1 q2

4k2F,, (58)

where we have used that qx = qcos(q). Eq. (58) is oneof the most important results of this Section. We stressagain that the applicability of this equation is not lim-ited to the case of a momentum-independent scatteringtime. Rather, Eq. (58) applies to a generic intralayerscattering time = (k). The physical meaning of thisresult is that in the low-temperature limit the non-linearsusceptibility is determined by what happens close to theFermi surface. What matters thus is only the magnitudeof(k) evaluated at the Fermi momentum kF, of the -th layer. This conclusion is in agreement with Narozhnyet al.29.

Using Eq. (58) and the following integral

+

dxx2

sinh2(yx/2)=

82

3y3, (59)

we finally arrive at the desired result for the low-temperature BT drag transresistivity:

limT0

BTD = (kBT)2

6e2F,1F,2v2

qmax0

dq q|U12(q, 0)|2

1 q2

4k2F,1

1 q

2

4k2F,2, (60)

where qmax min(2kF,1, 2kF,2). Once again, we stressthat the result in Eq. (60) does notdepend on the precisefunctional form of (k).

A. Dimensionless variables

It is now useful to introduce dimensionless variables.We scale the wave number q in Eq. (60) with

kF,1kF,2

by introducing x = q/

kF,1kF,2 and the effective in-

teraction U12 with e2/

kF,1kF,2 by introducing U12 =

U12

kF,1kF,2/e2.


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13

In these reduced units Eq. (60) reads (for physical reasons we restore Plancks constant from now on)

limT0

BTD = h

e22ee12

(kBT)2

F,1F,2

xmax0

dx x|U12(x, 0)|2

1 x2

4

kF,1kF,2

1 x

2

4

kF,2kF,1

, (61)

where xmax = min(2

kF,1/kF,2, 2

kF,2/kF,1). InEq. (61) we have introduced the dimensionless couplingconstant ee = e

2/(hv), which has a value 2.2 in DLGand 4.4 in Bi2Te3 TIs if we use the respective Diracvelocities vG 106 m/s and vTI 5 105 m/s. Eq. (61)

is the most important result of this Section.

To proceed further, we write in an explicit mannerthe statically-screened dimensionless interlayer interac-tion U12(x, 0):

U12(x, 0) =8xf12(x)

[x + 2Nf,1eekF,1/kF,2 f11(x)][x + 2Nf,2eekF,2/kF,1 f22(x)] 16Nf,1Nf,22eef212(x), (62)

where we have introduced the crucially-important dimen-sionless parameter

dkF,1kF,2 , (63)and the dimensionless form factors

f11(x) =(2 + 3) ex + (2 3) ex

g(x), (64)

f22(x) = (2 + 1) ex

+ (2 1) ex

g(x), (65)

f12(x) =2

g(x), (66)

with

g(x) = (1 + 2)(2 + 3)ex + (1 2)(2 3)ex . (67)

The only ingredient we have used to write Eq. (62) is thewell-known behavior of the static Lindhard function of adoped MD2DES for wave numbers q smaller than twice

the Fermi momentum kF,:

(0) (q 2kF,, 0) =

Nf,kF,2hv

. (68)

Two asymptotic limits can now be easily inferred fromthe general low-temperature theory in Eqs. (61)-(67):1) the weak-coupling (large interlayer distance and/orhigh density) limit, i.e. the limit in which 1,and 2) the strong-coupling (small interlayer distanceand/or low density) limit, i.e. the limit in which 1.

As discussed briefly above, the applicability of BT the-ory in the low-density regime is questionable: for the BTtheory to remain valid, one should think of entering thestrong-coupling regime by reducing the interlayer separa-tion d while keeping fixed the factor

kF,1kF,2. Note also

that, in the strong-coupling regime, diagrams of higherorder with respect to those presented in Fig. 2 might be-come relevant. The applicability of RPA Eq. (21) isalso questionable in this regime. Only a comparison withexperimental results can tell us about the importance ofthese subtle issues of many-body theory.

B. The weak-coupling limit

In the limit 1, straightforward algebraic manipu-lations of Eqs. (61)-(67) yield the following result for thelow-temperature BT drag transresistivity:

lim

limT0

BTD = h

e2

3

(kBT)2

F,1F,24N2f,1N2f,2

2ee

+

0

dyy3f212(y)

[f11(y)f22(y) 4f212(y)]2,

(69)

where we have introduced a new reduced variable, y =x. The quadrature with respect to the variable y canbe carried out analytically yielding the following result:+

0

dyy3f212(y)

[f11(y)f22(y) 4f212(y)]2

=1

422

+0

dyy3

sinh2(y)=

3

8(3)22 , (70)


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14

where (x) is the Riemann zeta function and (3) 1.2.Note that this quadrature does not depend neither on 1nor on 3.

In summary, we find

lim

limT0

BTD = h

e2(3)

8

22d42eeN

2f,1N

2f,2k

2F,1k

2F,2

(kBT)

2

F,1F,2 h

e222

(kBT)2

n3/21 n3/22 d4.

(71)

The dependence of BTD on temperature, densities andinterlayer distance displayed by Eq. (71) is in agree-ment with the results by Tse et al.24 and by Katsnel-son26, even though the numerical prefactor we find isfour times smaller (we remind the reader that for DLGNf,1 = Nf,2 = 4). It is important to observe that theweak-coupling and low-temperature BT Coulomb dragtransresistivity is sensitive only to the dielectric constant2 between the two MD2DESs (see Fig. 1).

C. The strong-coupling limit

In the limit 0 we can expand all the functionsfij(x) that appear in the effective interaction (62) ispowers of their argument since x is bounded from above,0 x xmax, and 0. Following this procedure wefind the following asymptotic expression for U12(x, 0) inthe limit 0:

lim0

U12(x, 0) =4

(1 + 3)x + 2qTF,1 + qTF,2

kF,1kF,2

, (72)

where we have introduced the Thomas-Fermi screen-ing wave number qTF, = Nf,eekF,. Using this re-sult in Eq. (61) we find the final expression for thelow-temperature BT drag transresistivity in the strong-coupling limit:

lim0

limT0

BTD = h

e24

32ee

(kBT)2

F,1F,2

xmax

0

dx

1 x

2

4

kF,1kF,2

1 x

2

4

kF,2kF,1

x

(1 + 3)x + 2qTF,1 + qTF,2

kF,1kF,2

2 .

(73)

Eq. (73) simplifies considerably if we assume equal den-sities and degeneracies in the two layers: n1 = n2 n,kF,1 = kF,2 kF, F,1 = F,2 F, and Nf,1 = Nf,2 Nf. In this case Eq. (73) reduces to

lim0

limT0

BTD = h

e24

32ee

kBT

F

2F(1 + 3, Nfee) ,

(74)

where

F(1 + 3, Nfee) 2

0

dxx(1 x2/4)

(1 + 3)x + 4Nfee2 .

(75)Note that in the strong-coupling limit limT0

BTD does

not depend on the interlayer distance d and scales like1/n. As far as the dielectric constants i are concerned,

the strong-coupling and low-temperature BT Coulombdrag transresistivity depends only on 1 + 3 but not on2.

The quadrature in Eq. (75) can be easily carried outanalytically. The result is

F(1 + 3, Nfee) = 12(1 + 3)4

12Nfee(1 + 3)

(1 + 3)2[3 + 2 ln(2)] + 24N2f 2ee ln(2) + 2[(1 + 3)2 12N2f 2ee]

ln2 +1 + 3Nfee . (76)

Note that F(1 + 3, Nfee) diverges logarithmically inthe weak-screening Nfee 0 limit, in agreement withEq. (41) of Ref. 29.

VII. DEVIATIONS FROM THE

LOW-TEMPERATURE QUADRATIC BEHAVIOR

DUE TO A FREQUENCY-DEPENDENT

SUBSTRATE DIELECTRIC CONSTANT

In this Section we illustrate how deviations from theFermi-liquid quadratic-in-temperature dependence of the

Coulomb drag transresistivity can occur in the situationin which the dielectric constant i of one of the mediasurrounding the two Dirac-fermion layers has a strongfrequency dependence.

Consider, for example, a DLG system on a substratelike SrTiO3. Following Ref. 45, we model the dielectricconstant of this substrate by a frequency-dependent func-tion of the form

3() = + (0 ) 20

20 2 + i. (77)

At room temperature = 5.2 , 0 = 310, 0/(2) =2.7 THz, and /(2) = 1.3 THz.

We would like to understand what is the qualitativerole played by the change 3 3() in the Coulombdrag transresistivity. For simplicity, we assume that thetwo graphene sheets comprising the double layer havethe same density n1 = n2 = n (kF,1 = kF,2 kF). Wealso assume that the Fermi energy F corresponding ton is the largest energy scale in the problem: under thisassumption, we are still allowed to expand the BT non-linear susceptibility BT (q, ) to lowest order in the smallparameter h/F.


15/20

15

Despite h F, we can have two distinct regimessince 3() brings in a new frequency scale, i.e. 0: i) 0 F/h and ii) 0 F/h. In the firstregime what matters is obviously 3( = 0) = 0. Inthe second regime, instead, we have to retain the full fre-quency dependence of3(). The existence of this secondregime ensures the possibility to observe deviations fromBTD T2 above a certain temperature scale.

The low-temperature Coulomb drag transresistivity forthe situation described above is given by Eq. (51) withBT (q, ) given by the expression in Eq. (58) and given by the expression in Eq. (52):

limT0

BTD = h

e2h

163v22F

2kF0

dq q

1 q2

4k2F

+

0

d2 U12(q, 0)|33() 2

sinh2(/2). (78)

Here the notation U12(q, 0)|33() means that from thepoint of view of the electronic system we still have to use

the statically-screened interlayer interaction, while fromthe point of view of the substrate we have to take intoaccount the frequency dependence of 3 through the useof Eq. (77).

The double integral in Eq. (78) can be easily per-formed numerically and some illustrative results are sum-marized in Fig. 4. For temperatures T h0/kB, the low-temperature Coulomb drag tran-sresistivity deviates from the canonical Fermi-liquid be-havior.

Further deviations from the Fermi-liquid temperaturedependence are induced by the explicit (and strong) tem-perature dependence of the dielectric constant of SrTiO3 see, for example, Ref. 32 which here has been ne-glected for simplicity.

VIII. SUMMARY OF OUR MAIN RESULTS

AND CONCLUSIONS

In summary, we have presented in a complete and asmuch pedagogical as possible manner the minimal theoryof Coulomb drag between two spatially-separated two-dimensional (2D) systems of massless Dirac fermions,which are both away from the charge-neutrality point.Our theory relies on second-order perturbation theoryin the screened interlayer interaction and on Boltzmanntransport theory.

In this well-established theoretical framework, whichhas also been adopted by other authors in earlierworks2429, we have clearly demonstrated that the pre-cise functional dependence of the intralayer scattering

5 4 3 2 1

log10(T/TF)

16

15

14

13

12

11

10

9

8

7

6

log10

(BTD

/0

) dkF = 5.0

3()

3( = 0)

5 4 3 2 1

log10(T/TF)

10

9

8

7

6

5

4

3

2

1

0

1

dkF = 5.0dkF = 10.0dkF = 20.0

(T/TF)2

FIG. 4: (Color online) Top panel: the Coulomb drag tran-sresistivity BTD (in units of 0 = h/e2) is plotted as afunction of temperature T (in units of the Fermi temper-ature TF = F/kB). Note that both axes are in logarith-mic scale. Data in this panel refer to: a carrier densityn = 3.1 1014 cm2, an interlayer distance d 2 nm(dkF = 5.0), 1 = 1, and 2 = 7.8. Filled circles label the dataobtained by taking into account the frequency dependence3 = 3() reported in Eq. (77). The solid line labels theresults obtained by setting 3 = 3( = 0) = 0 in Eq. (78).Bottom panel. The Coulomb drag transresistivity BTD (inunits of0a, where the coefficient a changes with changingdkF see text) is plotted as a function of T/TF. Note thatalso in this panel both axes are in logarithmic scale. Curveslabeled by different symbols correspond to different values of

the interlayer distance d (density is fixed at the same valueused in the top panel, i.e. n = 3.1 1014 cm2).

time on momentum plays absolutely no role in determin-ing the low-temperature Coulomb drag transresistivityin the Fermi-liquid regime. This is in contradiction withthe findings reported in Refs. 27 and 28 while it agreeswith the conclusions reached by Narozhny et al.29.

For two layers with identical degeneracies (Nf,1 =


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16

Nf,2 Nf) and densities (n1 = n2 n), we find thefollowing results for the low-temperature Coulomb dragtransresistivity:

1) In the weak-coupling limit Eq. (71) we find

limT0

BTD = h

e2(3)

8

22d42eeN

4f k

6F

k2BT2

h2v2; (79)

The dependence of this result on layer-separation d anddoping n agrees with that found in Refs. 24,26 and 29.2) In the strong-coupling limit Eq. (74) we find

limT0

BTD = h

e24

3

2eek2F

k2BT2

h2v2F(1 + 3, Nfee) , (80)

where the explicit expression of the function F(1 +3, Nfee) is reported in Eq. (76). The independence ofthis result on layer-separation d and the dependence ondoping n ( 1/n) agree with the findings of Ref. 29,even though the authors of this work captured only theweak-screening Nfee 0 asymptotic behavior of thefunction

F(1 + 3, Nfee) and did not take into account

the spatial dependence of the dielectric constant alongthe z direction.

General results for n1 = n2 and Nf,1 = Nf,2 can befound in Eqs. (71) and (74).

Finally, we have shown how deviations from the Fermi-liquid temperature dependence can occur when one ofdielectric constants of the media surrounding the Dirac-fermion layers depends strongly on frequency and tem-perature. Double-layer graphene systems deposited, forexample, on SrTiO3, a very well-known insulator close toa ferroelectric instability32,45, can be used as a testbedfor this idea.

In the future, we plan to present a systematic numeri-

cal study of Eq. (51) at arbitrary temperatures and to in-vestigate more deeply the strong-coupling limit by study-ing i) the impact of diagrams of order higher than two inthe screened interlayer interaction and ii) beyond-RPAcorrections.

Acknowledgments

We wish to thank Andre Geim, Allan MacDonald, andMisha Titov for very useful and stimulating discussions.Work in Pisa was supported by the Italian Ministry ofEducation, University, and Research (MIUR) throughthe program FIRB - Futuro in Ricerca 2010 GrantNo. RBFR10M5BT (PLASMOGRAPH: plasmons and

terahertz devices in graphene). T.T. and M.I.K. ac-knowledge financial support from the Stichting voor Fun-damenteel Onderzoek der Materie (FOM) (The Nether-lands).

Appendix A: Boltzmann transport approach to the

drag conductivity

The Boltzmann transport equation for the distributionfunction f(k, ) of the -th layer reads:

tf(k, )+kf(k, )tk = I(k, )+,(k, ) , (A1)

where I(k, ) is the so-called collision integral, whichdescribes intralayer scattering events, while the term,(k, ) describes momentum transfer between the two

Dirac-fermion layers ( = 2 if = 1 and = 1 if = 2)and is obviously absolutely crucial in the Coulomb dragproblem. We are interested in the steady-state regimein which tf(k, ) = 0 and employ a generalized re-laxation time approximation for the intralayer collisionintegral, i.e.

I(k, ) = f(k, ) f(0) (k, )

(k), (A2)

where f(0) (k, ) = nF(

()k,) is the equilibrium distribu-

tion function. Below we will assume that the transportscattering time (k) is isotropic.

The interlayer collision integral reads

,(k, ) = 2Nf,

,,

d2q

(2)2

d2k

(2)2|U12(q, ()k, ()k+q,)|2(()k, ()k+q, + ()k, ()kq,)

f(k, )[1 f(k + q, )]f(k

, )[1 f(k q, )]

[1 f(k, )]f(k + q, )[1 f(k, )]f(k q, )

1 + cos(k k+q)

2

1 + cos(k kq)2

. (A3)

The degeneracy factor Nf, can be understood from acareful analysis of the conservation of spin and valleydegrees of freedom during the scattering process.

In the semiclassical limit tk = eE, where E is theelectric field in the -th layer. In the regime of smallelectric fields we can linearize the Boltzmann transport


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17

equation by writing

f(k, ) = f(0) (k, ) + f

(1) (k, ) . (A4)

The linearized Boltzmann equation in the steady-stateregime reads

ekf

(0)

(k, )E =

f

(1) (k, )

(k)+

(linear)

,

(k, ) ,

(A5)

where the linearized interlayer collision integral reads

(linear)

,(k, ) = 2Nf,

,,

d2q

(2)2

d2k

(2)2|U12(q, ()k, ()k+q,)|2(()k, ()k+q, + ()k, ()kq,)

f(1) (k, )f

(0)

(k, )[1 f(0) (k + q, )][1 f(0) (k q, )]

f(1) (k + q, )f(0) (k, )f(0) (k, )[1 f(0)

(k q, )]

+ f(1)

(k, )f

(0) (k, )[1 f(0) (k + q, )][1 f(0) (k q, )]

f(1)

(k q, )f(0) (k, )f(0) (k, )[1 f(0) (k + q, )] f(1) (k, )f(0) (k + q, )f(0) (k q, )[1 f

(0)

(k, )]

+ f(1) (k + q,

)f(0)

(k q, )[1 f(0) (k, )][1 f(0) (k, )]

f(1)

(k, )f(0) (k + q,

)f(0)

(k q, )[1 f(0) (k, )]

+ f(1)

(k q, )f(0) (k + q, )[1 f(0) (k, )][1 f(0) (k, )]

1 +

cos(k k+q)2

1 + cos(k kq)2

. (A6)

We now seek solutions of Eqs. (A5)-(A6) of the form

f(1) (k, ) =

(intra) (k, )k E

+ (inter) (k, )k E . (A7)

In the following we will assume that |(intra) (k, )| |(inter) (k, )|. The coefficients (intra) (k, ) and

(inter) (k, ) can be found by calculating the functional

derivative of the linearized Boltzmann transport equationwith respect to E and E, respectively, i.e.

ekf

(0)

(k, )

k

(intra) (k, )

(k)(A8)

and

0 = k(inter) (k, )

(k)+

(linear)

,(k, )

E. (A9)

In writing Eq. (A8) we have neglected the small term

(linear)

,(k, )/E. The solution of Eq. (A8) reads

(intra) (k, ) = ek k()k,

f(0) (k, )

()k,

(k) . (A10)

The second term on the r.h.s. of Eq. (A9) can be cal-culated from Eq. (A6). Solving Eq. (A9) we find

(inter) (k, ) = 2e(k)Nf,

,,

d2q

(2)2

d2k

(2)2|U12(q, ()k, ()k+q,)|2

(()k, ()k+q, + ()k, ()kq,)

f(0) (k, )f

(0)

(k, )[1 f(0) (k + q, )][1 f(0) (k q, )]

(k)k()k, (k q)k()kq,

k


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18

1 + cos(k k+q)

2

1 + cos(k kq)2

. (A11)

To obtain Eq. (A11) we used the identities

f(0) (k, )

()k,

= f(0) (k, )[1

f

(0) (k, )] (A12)

and

f(0) (k, )[1 f(0) (k, )] =

f(0) (k, ) f(0) (k, )

1 exp[(()k, ()k,)].

(A13)The current density in the -th layer reads

J = eNf,

d2k

(2)2f

(1) (k, )k()k,

J(intra) + J(inter) . (A14)

Using Eq. (A10) we find J(intra) :

J(intra) = e

2Nf,

d2k

(2)2(k)

k k()k,

(k E)

f(0) (k, )

()k,

k()k,

= Ee2Nf,

2

d2k

(2)2(k)

()k,

k

2

f

(0) (k, )

()k, . (A15)

In the last equality of Eq. (A15) we used the fact that

k()k, = k

()k,

k, (A16)

(since ()k, depends only on |k|) and that the average

over the angle of (k v)k = v/2 for any constant vector v(in D = 2 spatial dimensions). From Eq. (A15) we find

immediately the longitudinal intralayer conductivity

J(i)

E(i)

=e2Nf,

2

d2k

(2)2(k)

()k,

k

2

f(0) (k, )

()k,

=e2v2Nf,

2 d

2k

(2)2(k)

f(0) (k, )

()k, ,(A17)

where i is a Cartesian index and in the last equality we

used the fact that ()k, = v|k|. Note that Eq. (A17)

coincides with Eq. (50) in the main text.

The interlayer current density reads

J(inter) = eNf,

d2k

(2)2(inter)(k, )(kE)k()k, ,

(A18)where (inter)(k, ) is given in Eq. (A11). After somestraightforward algebra we obtain

J(inter) = E

2Nf,Nf,e2

2

,,,

d2q

(2)2

d2k

(2)2

d2k

(2)2|U12(q, ()k, ()k+q,)|2

(()

k, ()

k+q,

+

()

k

,

()

k

q,

)

f

(0)

(k, )f

(0)

(k

,

)[1 f(0)

(k + q,

)][1 f(0)

(k

q,

)]

(k)k()k, (k + q)k()k+q,

(k)k()k, (k q)k()kq,

1 +

cos(k k+q)2

1 + cos(k kq)2

.

(A19)

Notice that Eq. (A19) has been recasted in a symmetric form by means of a simultaneous exchange of dummy


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19

variables (k k+q, k kq, , and )and thus the integral is multiplied by an additional factor

1/2. From Eq. (A19) we can derive the longitudinal dragconductivity D, which reads as following:

D J(i)

E(i)

=e2

16 d2q

(2)2 +

d|U12(q, )|2sinh2(/2)

BT1 (q, )BT2 (q, ) , (A20)

where the function BT (q, ) coincides with the one de-fined in Eq. (48) of the main text. In writing Eq. (A20)we have introduced an auxiliary variable to disentanglek from k:

( ) =

d( )( ) . (A21)

Electronic address: [email protected]; URL: http://qti.sns.it

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mailto:[email protected]://qti.sns.it/http://qti.sns.it/http://arxiv.org/abs/1202.0735http://arxiv.org/abs/1202.0735http://qti.sns.it/http://qti.sns.it/mailto:[email protected]


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