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Coulomb drag in graphene: perturbation theory B.N. Narozhny,1 M. Titov,2,3 I.V. Gornyi,3,4 and P.M. Ostrovsky3,5 1Institut fu¨r Theorie der Kondensierten Materie and DFG Center for Functional Nanostructures, Karlsruher Institut fu¨r Technologie, 76128 Karlsruhe, Germany 2School of Engineering & Physical Sciences, Heriot-Watt University, Edinburgh EH14 4AS, UK 3Institut fu¨r Nanotechnologie, Karlsruhe Institute of Technology, D-76021 Karlsruhe, Germany 4A.F. Ioffe Physico-Technical Institute, 194021 St. Petersburg, Russia 5L.D. Landau Institute for Theoretical Physics RAS, 119334 Moscow, Russia 2 (Dated: January 23, 2012) 1 0 We study the effect of Coulomb drag between two closely positioned graphene monolayers. In 2 the limit of weak electron-electron interaction and small inter-layer spacing (µ1(2),T ≪ v/d) the n drag is described by a universal function of the chemical potentials of the layers µ1(2) measured in the units of temperature T. When both layers are tuned close to the Dirac point, then the drag a J coefficientisproportionaltotheproductofthechemicalpotentialsρD ∝µ1µ2. Intheoppositelimit 0 of low temperature thedrag is inversely proportional to both chemical potentials ρD ∝T2/(µ1µ2). In the mixed case where the chemical potentials of the two layers belong to the opposite limits 2 µ1 ≪ T ≪ µ2 we find ρD ∝ µ1/µ2. For stronger interaction and larger values of d the drag coefficient acquireslogarithmic correctionsandcan nolonger bedescribedbyapowerlaw. Further ] l logarithmiccorrectionsareduetotheenergydependenceoftheimpurityscatteringtimeingraphene al (for µ1(2) ≫ T these are small and may be neglected). In the case of strongly doped (or gated) h graphene µ ≫ v/d ≫ T the drag coefficient acquires additional dependence on the inter-layer 1(2) - spacing and we recoverthe usual Fermi-liquid result if thescreening length is smaller than d. s e m PACSnumbers: 72.80.Vp,73.23.Ad,73.63.Bd . t a Transportmeasurementsareconceptuallysimplestand temperature ρ T2. Remarkably, this simple argu- D m ∝ by far the mostcommon experimentaltools for studying ment is sufficient to describe the observed temperature - innerworkingsofsolids. Withinlinearresponse,theout- dependence of ρ (deviations from the quadratic depen- D d comeofsuchmeasurementsisdeterminedbytheproper- dence are due to the effect of phonons)2. n ties of the unperturbed system, which are often the ob- The phase-space argument however does not describe o c ject of study. In a typical experiment a current is driven the physics of the effect completely. Indeed, in the pas- [ throughaconductorandthe voltagedropalongthe con- sive layer the momentum is transferred equally to elec- ductor is measured. In most conventional conductors at trons andholes so that the resulting state cancarrycur- 2 lowtemperaturestheresultingresistanceismostlydeter- rent only in the case of electron-hole asymmetry. Like- v 9 mined by disorder (which is always present in any sam- wise,thisasymmetryisnecessaryforthecurrent-carrying 5 ple), while interactions between charge carriers lead to state in the active layer to be characterized by non- 3 corrections that affect the temperature dependence1. zero total momentum. In the Fermi-liquid theory the 6 Considernowadragmeasurementinabi-layersystem electron-hole asymmetry can be expressed5 (assuming . 0 consisting of two closely spaced but electrically isolated either a constant impurity scattering time or diffusive 1 conductors2. Passing a current I through one of these transport6) as a derivative of the single-layer conductiv- 1 1 conductors(“theactivelayer”)isknowntoinduceavolt- ity σ with respect to the chemical potential. In con- 1(2) 1 age V2 in the other conductor (“the passive layer”). The ventional semiconductors3 the asymmetry appears due : v ratio of this voltage to the driving current ρD = V2/I1 to curvature of the conduction band spectrum (leading − i (the transresistivity or the drag coefficient) is a measure to the energy dependence of the density of states (DoS) X of inter-layer interaction. At low enough temperatures and/or diffusion coefficient): ∂σ /∂µ σ /µ . i i i i ar the dragis dominatedby the directCoulombinteraction Theoretical calculations2,3 typically≈focus on the drag between charge carriers in both layers2. conductivity σ . The experimentally measurable drag D The physics of the Coulombdrag is well understood if coefficient ρ is then obtained by inverting the 2 2 D both layers are in the Fermi liquid state3,4. The current conductivity matrix. To the lowest order in the in×ter- in the passive layer is created by exciting electron-hole layer interaction3 (assuming σ σ ) one obtains D 1(2) ≪ pairs(eachpairconsistingofanoccupiedstateabovethe ρ =σ /(σ σ ). (1) Fermisurfaceandanempty state below)inastate char- D D 1 2 acterized by finite momentum. The momentum comes Combining the above arguments (and assuming that the fromthe electron-holeexcitations in the active layercre- single-layerconductivities σ aregivenby the Drude for- i ated by the driving current. The momentum transfer is mula) we arrive at the Fermi-liquid result3 due to the inter-layer Coulomb interaction. Therefore it follows from the usual phase-space considerations that ~ T2 ρ = A , (2) the drag coefficient is proportional to the square of the D e2µ µ 12 1 2 2 where A depends on the matrix elements of the inter- 12 TABLEI:Asymptoticexpressionsforthedragcoefficient (6) layer interaction, the Fermi momenta of the two layers, in the limit of weak inter-layer interaction and small inter- and the inter-layer spacing d (in the diffusive regime, layer spacing µ ,T ≪v/d. where the mean-free path ℓ d, A contains an ad- 1(2) 12 ≪ ditional logarithmic dependence3). The precise form of parameter region drag coefficient A12 iswellknownandcanbeobtainedbymeansofeither ~ µ µ the diagrammatic formalism3 or the kinetic equation7,8. µ1,µ2 ≪T ρD ≈1.41α2e2 T122 Eq.(35) Recently drag measurements were performed in a sys- ~ µ tem of two parallel graphene sheets9. It was shown that µ1 ≪T ≪µ2 ρD ≈5.8α2e2µ1 Eq.(36) 2 this system offers much greater flexibility compared to ~ 8π2 T2 µ prior experiments in semiconductor heterostructures10. T ≪µ1 <µ2 ρD ≈α2e2 3 µ µ ln T1 Eq.(37) 1 2 Thedragcoefficientdependsonthefollowingparameters: (i) temperature T, (ii) chemical potentials of the layers µ , (iii) inter-layer spacing d, (iv) mean-free path ℓ , i i vanishingdisorder,wherethe condition(5)canbe lifted. and (v) the interaction strength. Earlier experiments10 However it is more convenient to perform such calcula- wereperformedonsampleswithlargeinter-layerspacing, tions within the framework of the kinetic equation11,12. such that µ ≫ v/d (where v is the Fermi velocity). In The results of that work are reported elsewhere15. contrast, the graphene-based system allows one to scan Inthe limit ofsmallinter-layerspacingµ ,T v/d 1(2) a wide range of chemical potentials (by electrostatically ≪ the resulting drag coefficient is the “universal” function controlling carrier density) from the Fermi-liquid regime ~ with µ v/d,T to the Dirac point µ = 0 where the drag vain≫ishes due to electron-hole symmetry. ρD =α2e2 r0(µ1/T,µ2/T). (6) Mostexperiments2,10(includingthatofRef.9)areper- The function r (x ,x ) is characterized by the limiting formedin the ballistic regimeℓ d, where ρ does not 0 1 2 i ≫ D cases µ T and µ T, see Table I. In the vicinity explicitly depend on disorder, albeit the conductivities of the Dii≪rac point ρi ≫µ µ /T2, while in the opposite σD and σi do. The graphene-based sample of Ref. 9 is limit ρ T2/(µ µD)∝(wit1h2additional logarithmic fac- characterizedbymuchsmallerinter-layerspacingincom- D ∝ 1 2 tors, see Sec. III). In the “mixed” case µ T µ parison to previous experiments2, with all data taken at 1 ≪ ≪ 2 we find ρ µ /µ . This regime is apparently realized D 1 2 ∝ T <v/d. (3) in the experiment of Ref. 9 when the bottom layer is in proximity of the Dirac point. Therefore in this paper we do not consider temperatures For relatively large inter-layer spacing, µ v/d (and i ≫ larger than the inverse inter-layer spacing, even though thusµ T),anewregimeappears,wherethedragcoef- i ≫ our approach remains valid for T &v/d. ficient acquires additional dependence on the inter-layer In this paper we address the problem of the Coulomb spacing. Here the assumption (4) may be relaxed as the drag in graphene by means of the perturbation theory. perturbation theory remains valid also for intermediate In the leading order, i.e. in the limit of weak interaction values of α (see Sec. IVA where we assume µ = µ ). 1 2 Still, for µ T we find ρ µ2/T2, at µ T the drag α=e2/v 1, (4) ≪ D ∝ ∼ coefficient reaches its maximum, and further decays for ≪ µ T. Thelatterregimeisdescribedbyalongcrossover the drag conductivity σD is described by the standard fro≫m the above logarithmic behavior to the Fermi-liquid (Aslamazov-Larkin-type, see Fig. 1) diagram3. Unlike result which is only achieved in the limit of the small usual metals, the single-layer conductivity in graphene screening length κd 1. Thus ρ (µ & v/d) cannot be comprises two competing contributions: one due to dis- ≫ D described by a single power law. order and one due to electron-electron interaction11–14. Theremainderofthispaperisorganizedasfollows. In We assume that the dominant scattering mechanism is Sec. I we present the perturbative calculation that gives due to disorder. Hence throughout the paper we assume the drag resistivity in the limit α 0 and d 0. While → → α2Tτ 1 Tτ, (5) the resulting expression can only be evaluated numeri- ≪ ≪ cally,weanalyzeallinterestinglimitsanalyticallyforthe where τ is the impurity scattering time (in the case of case of two identical layersin Sec. II and also for the ex- energy-dependent impurity scattering time this parame- perimentallyrelevantcasewherethe twolayersarechar- tershouldbeunderstoodasτ(max[µ,T]),seeSec.IVB). acterized by different chemical potentials, see Sec. III. The latter inequality ensures that the system is in the In Sec. IV we discuss drag at intermediate values of α ballistic regime. Under this condition14 the single-layer and d as well as the role of energy-dependent scattering conductivitiesσ areagaingivenbytheDrude-likefor- time. The paper is concluded by a brief summary and 1(2) mulawiththeimpurityscatteringtime. Aswewillshow, a discussion of the experimental relevance of our results. σ /σ . α2Tτ 1, which allows us to evaluate the Throughout the paper we use the units with ~ = 1 and D 1(2) ≪ drag coefficient ρ using Eq. (1). Our theory can be only restore the Planck’s constant in the results for ρ . D D further extended to the case of stronger interaction or Technical details are relegated to Appendices. 3 B. Non-linear susceptibility in graphene The non-linear susceptibility of electrons Γα(ω) is a ij responsefunctionrelatingavoltageV(r )eiωt toadccur- i rent it induces by the quadratic response: I= dr dr Γ(ω;r ,r )V(r )V(r ), (11) 1 2 1 2 1 2 Z Z FIG. 1: [Color online] The lowest-order diagram for drag conductivity. Solid lines represent the exact single-electrons with I being the induced dc current. From gauge invari- Green’s functions in thepresence of disorder. Spatial coordi- ance it follows that natesarelabeled bythenumbersshown atthecornersofthe triangles and correspond tothesubscriptsin Eq.(13). Wavy lines represent theinter-layerinteraction. dr1Γ(ω;r1,r2)= dr2Γ(ω;r1,r2)=0. Z Z IntermsofexactGreen’sfunctionsofadisorderedcon- I. PERTURBATIVE CALCULATION OF THE ductor the non-linear susceptibility canbe written as3,17 COULOMB DRAG casCeotnhseidderragthceonlidmuicttoivfitwyeσakDicnatnerbaectcioanlcuαla→ted0.wiItnhtthhies Γ=Z 2dπǫ"(cid:18)tanhǫ2−Tµ −tanhǫ+2ωT−µ(cid:19)γ12(ǫ;ω) help of the lowest-order diagram3 ǫ µ ǫ ω µ 1 dω + tanh − tanh − − γ (ǫ; ω) ,(12) σDαβ = 16πT sinh2 ω Γβ1(ω,q)Γα2(ω,q)|D1R2|2, (cid:18) 2T − 2T (cid:19) 21 − # q Z 2T X (7) wherethe argumentsofΓ aresuppressedfor brevity,the where the subscripts 1 and 2 refer to the active and pas- subscripts1and2refertothespatialcoordinatesr and 1 sivelayersrespectively,D1R2 istheretardedpropagatorof r2, respectively, and the triangular vertex γ is given by the inter-layerinteraction,andΓα(ω,q)isthe non-linear a susceptibility (or the rectification coefficient). γ (ǫ;ω)= GR(ǫ+ω) GA(ǫ+ω) GR (ǫ)ˆJ GA (ǫ). 12 12 − 12 23 3 31 h i (13) A. Inter-layer interaction for small α and d Here ˆJ is the current operator and integration over the 3 coordinate r is assumed (see Fig. 1). 3 The bare Coulomb potential has the usual form Thetraditionalapproachcallsforaveragingthevertex γ over disorder. Within the Fermi-liquid theory the av- V =V = 2πe2; V = 2πe2 e−qd. (8) eragedγ doesnotdependonǫ1,3. Thisresultisbasedon 11 22 12 q q theusualapproximationthatthemostimportantcontri- bution comes from electrons near the Fermi surface. In Thedynamicallyscreened(withinRPA)inter-layerprop- contrast,ingraphene-basedsystemstheregimewherethe agator can be written as16 chemical potential is smaller that temperature µ T i ≪ is accessible. Then the vertex γ retains its dependence 1 R = , on ǫ and should be evaluated with care18–20. Details of D12 −ΠRΠR4πe2 sinhqd+ q +ΠR+ΠR eqd 1 2 q 2πe2 1 2 thecalculationareprovidedinAppendicesAandB.The (9) result is (cid:0) (cid:1) whereΠR isthe single-layerretardedpolarizationopera- i tor. Forα 1screeningisineffectiveandtheinter-layer ω2+2ǫω v2q2 ≪ γ(ǫ,ω,q)= 2qevτ(ǫ) − (14) interaction is essentially unscreened. Moreover, in the − ǫv2q2 limit d 0 we may disregard the exponential and write → the interaction propagatoras [2ǫ+ω]2 v2q2 − sgn(ǫ+ω)θ (ǫ,ω,q), 2πe2 ×s v2q2 ω2 0 R = . (10) − D12 − q wherethe functionθ (ǫ,ω,q)ensuresthatthe expression 0 Aswewillshowbelow,thenon-linearsusceptibilityΓde- under the square root in Eq. (14) is always positive (the caysexponentiallyforq max(T,µ). Therefore,inview minus signs in θ (ǫ,ω,q) reflect the contribution of both 0 ≫ of Eq. (3) the limit d 0 is equivalent to the condition electrons and holes and appear after summing over all → µ v/d (the case µ >v/d is discussed in Sec. IVA). branches of the Dirac spectrum in graphene) i ≪ 4 θ (ǫ,ω,q)=θ(vq 2ǫ+ω )[θ( ω vq) θ(ω vq)]+θ(vq ω )[θ(2ǫ+ω vq) θ( vq 2ǫ ω)]. (15) 0 −| | − − − − −| | − − − − − Approximating21 the impurity scatteringtime by a constantτ (the effect ofenergy-dependentτ(ǫ) is discussedbelow in Sec. IVB), we can express the non-linear susceptibility in terms of dimensionless variables ω vq W = , Q= , 2T 2T and find the following form eτ µ Γ(ω,q)= 2 q g W,Q; , (16a) − π T (cid:16) (cid:17) 1 z√1 z2 W2 1 dz − I (z;W,Q;x), W >Q Q2 − z2 W2/Q2 2 | |  q 0 − g(W,Q;x)= 1 W2R∞dz z√z2−1 I(z;W,Q;x), W <Q (16b) − − Q2 z2 W2/Q2 | |  q R1 − zQ+W +x zQ+W x zQ W x zQ W +x I(z;W,Q;x)=tanh tanh − +tanh − − tanh − . (16c) 2 − 2 2 − 2 From Eq. (16c) it is clear that at the Dirac point the 2. Single-layer conductivity in graphene non-linear susceptibility vanishes: I(x=0)=0 Γ(µ=0)=0. (17) In order to find the drag coefficient ρD one needs to ⇒ knowthesingle-layerconductivityσ . Underourassump- i Thus there is no drag atthe Dirac point (physically,due tions the single-layer conductivity is completely deter- to electron-hole symmetry). mined by the weak impurity scattering and can be written in the form µ C. Perturbative results for the Coulomb drag in σ =e2Tτh , (19) 0 0 T graphene (cid:16) (cid:17) where 1. Drag conductivity in graphene ∞ 2 z 2 x, x 1, Using Eqs.(10)and(16)we canfindthe dragconduc- h0(x)= π−Z∞ dzcosh2|z|+ x2 = π (2ln2, x≪≫1. tivity in graphene (7). As usual for an isotropic system (cid:0) (cid:1) (20) in the absence of external magnetic fields σαβ =δαβσ . D D Now, in the limit α 0 and d 0 we find → → 3. The drag coefficient µ µ σ =α2e2T2τ2f 1, 2 , (18a) D 0 T T (cid:16) (cid:17) Finally, using Eqs. (1), (18a), and (19) we find that where the dimensionless function f (x ,x ) is defined as 0 1 2 the drag coefficient can indeed be expressed in the form ∞ ∞ (6),withthedimensionlessfunctionr (x ,x )definedas 4 dW 0 1 2 f (x ,x )= QdQ g(x )g(x ). (18b) 0 1 2 π2 sinh2W 1 2 f0(x1,x2) Z0 Z0 r0(x1,x2)= h (x )h (x ) (21) 0 1 0 2 where we have suppressedthe arguments of the function In the case of two identical layers this function depends g(W,Q;x) for brevity. on one variable only In general the function f (x ,x ) has to be computed 0 1 2 numerically. Below, we evaluate this function analyti- f (x,x) 0 r (x)= , cally in the physically interesting limiting cases of large 0 h2(x) 0 x(the lowtemperatureregime)andsmallx(the vicinity of the Dirac point). and is shown in Fig. 2. 5 1.8 r0(x, x) The numerical coefficient N1 can now be determined by (8 2/3) ln(x /2) / x 2 usingtheapproximation(22)inEqs.(16b)and(16b)and 1.6 1.41x 2 then evaluating the integral in Eq. (18b). The result is 1.4 1.1. (24) -2 e 1.2 N1 ≈ 2 1.0 Therefore the drag conductivity in the vicinity of the /D 0.8 Diracpointisindependentoftemperatureandispropor- tional to the square of the chemical potential 0.6 0.4 σD(µ T) α2e2µ2τ2 1. (25) ≪ ≈ N 0.2 In this case the drag conductivity is independent of T. 0.0 Thesingle-layerconductivitycanbe foundbyexpand- 0 5 10 15 20 ing the function h (x). Thus in the vicinity of the Dirac x = / T 0 point the conductivity of a single graphene sheet due to impurity scattering [see Eq. (5)] is14 FIG. 2: [Color online] The dimensionless drag coefficient 4ln2 r0(x,x) (two identical layers). The solid line shows the re- σ0 e2Tτ. (26) sult of numerical evaluation usingEqs. (21), (20), (18b), and ≈ π (16). Thereddashedlineshowstheasymptoticbehavior(27) Therefore the drag coefficient is inthevicinityoftheDiracpoint. Thegreenshort-dashedline shows theasymptotic behavior (33) at small temperatures. ~ π2 µ2 ~ µ2 ρ (µ T) α2 N1 =1.41α2 . (27) D ≪ ≈ e216ln22T2 e2T2 The expressions (6) and (21) give the perturbative re- ThisresultisrepresentedinFig.2bythedashedredline. sult for the drag coefficient in graphene-based bi-layer systems in terms of the dimensionless functions h (x) 0 and f (x ,x ). The applicability of this “universal” re- B. Low temperature limit 0 1 2 sultanditsexperimentalrelevanceisdiscussedinSec.IV. In the opposite limit II. ASYMPTOTIC BEHAVIOR OF THE DRAG µ T BETWEEN TWO IDENTICAL LAYERS ≫ we notice that the function I is given by Eq. (22) with 2 Considernowthecasewherethetwolayersinthedrag interchanged W and x and can be written as experiment are identical, i.e. are kept at the same tem- 4W ∂ sinhx perature and chemical potential. This case has not been I(x 1) . (28) ≫ ≈ Q ∂zcoshzQ+coshx yet realized experimentally in graphene-based systems, but has a long history of theoretical research2,3. Then the integral over z in Eq. (16b) is dominated by z 1sothatthealgebraicfunctionofz intheintegrand ≫ may be approximated by unity. The corresponding con- A. Vicinity of the Dirac point tribution (16b) to the non-linear susceptibility may now be approximated by the expression23 We start with situation where the graphene sheets are tuned close to the Dirac point, i.e. W W2 sinhx g(x, W <Q)=4 1 . (29) µ T. | | Qs − Q2 coshQ+coshx ≪ GiventhatattheDiracpointthenonlinearsusceptibility Now the momentum integral in Eq. (18b) is logarithmic vanishes [see Eq. (17)], we can expand it to the lowest andis dominated by largevalues of momentumQ W. ≫ order in the small parameter µ/T. Expanding first the Then the function (18b) takes the form expression (16c) we find ∞ ∞ 64 dWW2 dQ sinh2x I(x 0) 4x sinhW sinhzQ . (22) f0(x,x)= π2 sinh2W Q (coshQ+coshx)2. → ≈− (coshzQ+coshW)2 Z0 WZ (30) ThiscorrespondstothequadraticexpansionofEq.(18b) TheratioofthehyperbolicfunctionsinEq.(30)issimilar to the step function: it’s equal to unity for Q x and f (x,x) x2. (23) vanishesatlargervaluesofmomentumQ x. T≪herefore 0 1 ≈N ≫ 6 xeffectivelyactsastheuppercut-offandthemomentum The dimensionless function f (x) is characterizedby the 1 integral can be approximated by a logarithm two limits. If the second layer is also close to the Dirac point, then ∞ dQ sinh2x x ln . f (x 1) x, Q (coshQ+coshx)2 ≈ W 1 ≪ ≈N1 Z W whichyieldsastraightforwardgeneralizationofEq.(23): Therefore, in the low temperature limit the leading con- f (x ,x ) x x . If, on the other hand the chem- 0 1 2 1 1 2 ≈ N tribution to the drag conductivity is logarithmic in the ical potential of the second layer is large compared to chemical potential and quadratic in temperature T, then the drag conductivity (18b) can be calculated by using the linear approximation(22) for g(x ) and the 32 µ 1 σD(T µ) α2e2T2τ2ln . (31) low temperature approximation (29) for g(x2). Due to ≪ ≈ 3 T the exponential decay of Eq. (22) at Q W, the mo- ≫ The single-layer conductivity at low temperatures is mentum integral is dominated by Q 1 and thus the ∼ determinedbythechemicalpotential14 andthusisgiven combination of the hyperbolic functions in Eq. (29) can by the Drude formula be replaced by unity. Thus in this limit the drag conductivity in independent of the chemicalpotential of the σ0 (2/π)e2µτ. (32) second layer and ≈ Consequently the drag coefficient is similar to Eq. (2). f (x 1) , 3.26. (34) 1 2 2 ≫ ≈N N ≈ ~ 8π2T2 µ ρ (µ T) α2 ln . (33) Now, the single-layer conductivity in the first layer is D ≫ ≈ e2 3 µ2 T given by Eq. (26), while in the second layer we should considerbothlimits. Ifµ isalsosmallthentheresulting This is to be expected, since atlow temperatures T µ 2 ≪ drag coefficient is a trivial generalization of Eq. (27): the phase-space argument yielding the T2 dependence is justifiedandtheelectron-holeasymmetrydeterminesthe ~ µ µ dependence on the chemical potential. The logarithmic ρD(µ1,µ2 ≪T)≈1.41α2e2 T122. (35) factor is of course beyond such qualitative estimates. This result (33) is represented in Fig. 2 by the dashed Intheoppositelimitthedragconductivityisindependent greenlineandisshowntogetherwiththeresultofthedi- of the properties of the second layer, since the integrals rectnumericalcalculation(shownbythe solidblue line). inEq.(18b)aredominatedbytheregionwherebothfre- Notethatthedragconductivity(31)wascalculatedwith quency and momentum are of order T. The single-layer logarithmic accuracy. conductivity in the second layer however is determined by µ [see Eq. (32)] and thus 2 III. ASYMPTOTIC BEHAVIOR OF THE DRAG ρ (µ T µ ) α2π2N2 ~ µ1 =5.8α2 ~ µ1. (36) BETWEEN INEQUIVALENT LAYERS D 1 ≪ ≪ 2 ≈ 8ln2 e2µ e2µ 2 2 Consider now the more realistic9 situation where the B. One layer with high carrier density two layersare characterizedby different chemical potentials. We will still assume that the temperatures of the two layers are the same. Secondly,ifthechemicalpotentialinoneofthelayersis Note that in this Section the subscripts of the chemi- muchlargerthantemperature(withoutlossofgenerality cal potentials µ and µ do not indicate the passive and we may also assume that the chemical potential of the 1 2 active layers, but rather simply distinguish between two other layer is also smaller) layers with different carrier density. µ T, µ >µ , 2 2 1 ≫ then arguing along the lines of the previous subsection A. One layer near the Dirac point we find that the drag conductivity is independent of the largest chemical potential Firstly, suppose that one of the two layers is characterized by a small chemical potential σ =α2e2T2τ2f (µ /T). D 2 1 µ1 ≪T. Again, we characterize the dimensionless function f2(x) bythetwolimits. Thesituationwhenthesecondlayeris Then in Eq. (18b) the function g(x ) may be expanded 1 nearitsDiracpointwasalreadydiscussedintheprevious using Eq. (22): subsection. Therefore, similar to Eq. (34) µ σD =α2e2µ1Tτ2f1 T2 . f2(x 1) 2x. ≪ ≈N (cid:16) (cid:17) 7 The drag coefficient in this case is given by Eq. (36). (i) (iia) (iii) (iv) In the case where µ > µ T the calculation is 2 1 ≫ T 2ln 1 similar to that presented in Sec. IIB. However now the N combinationofthehyperbolicfunctions[comingfromthe v approximation(29)]inthemomentumintegralsimilarto 2 T 2ln 2 N d Eq.(30)comprisestwostepfunctionsandtheintegration e 2 T 2ln is cut off by the sfm2(axller c1h)emic3a2llpnoµte1n,tial. Hence / () D T T4 2 v4d 4 4 ≫ ≈ 3 T thedragconductivityisindependentofthelargerchemi- 2 calpotentialµ . Thedragcoefficientinthelimitoflarge 2 µ (but smaller than µ ) is a generalizationof Eq. (33) 2 1 ρD(µ2 >µ1 ≫T)≈α2e~28π32µTµ2 lnµT1. (37) T TN dv Nvd 1 2 IV. BEYOND THE LOWEST ORDER FIG. 3: [Color online] The sketch of the drag conductivity PERTURBATION THEORY (in the units of α2e2τ2) as a function of the chemical potential illustrating the results of Section IVA1 in the case T ≪Nαv/d. The blue line corresponds to the quadratic de- A. Finite inter-layer spacing and static screening pendence (25) in the vicinity of the Dirac point. The four regionsatlargechemicalpotentialscorrespondtothefourre- Let us now discuss what happens for non-zero inter- gions discussed in Section IVA1. If T ≫ Nαv/d, then the layer spacing d. For simplicity, we will consider the case region(iia)shouldbereplacedby(ii),thelogarithmicdepen- of two identical layers. Generalization to the case of in- denceT2ln(1/α)shouldbereplacedbyT2ln[v/(Td)],andthe equivalent layers is achieved along the lines of Sec. III. limits v/d and T/(Nα) should be exchanged. considered the effect of the unscreened inter-layer inter- 1. Vanishing interaction strength action. Consider now the role of the static screening. If both d and α are small then we may approximate the Inthelimitα 0theinter-layerspacingappearsonly → inter-layer interaction (9) by the expression in the exponential factor in the unscreened inter-layer interaction (8). Therefore, the momentum integration e−qd aatcqsumiraelsl maofimrmenutappiseruncucht-aonffgeqd.< v/d, but the behavior D1R2 =− q +2ΠR. (38) 2πe2 As we have seen in Sec. II, the behavior of the drag coefficientinthevicinityoftheDiracpointisdetermined At low temperatures the leading contribution to the byfrequenciesandmomentaoftheorderofT (withinour static polarization operator is perturbationtheorythesingle-layerconductivitiesinthe 2k two layers are independent of each other and d). Under ΠR(ω =0)= F. (39) the assumption (3) these momenta are small compared πv to v/d: taking into account the exponential in Eq. (8) Thus the interaction propagator can be written as yields a small correction to the numerical coefficient in Eqs. (27) and (35): 1.41 1.41+4.06Td/v. 2παv2 → R = e−qd. Intheoppositelimitoflowtemperaturewehavefound D12 −vq+2Nαµ the behavior of the drag conductivity that is determined by large momenta. In the result (33) the upper limit Here N = 4 is due to spin and valley degeneracy. The of the logarithm is given by the chemical potential [due unscreenedinteractionisagoodapproximationaslongas to the step-like behavior of the non-linear susceptibility the inversescreening length is small comparedto typical (29)], while the lower limit is given by temperature as momenta, i.e. as long as Nαµ T. In the opposite ≪ the typical value of frequency in Eq. (30). regime the lower limit of the logarithm in Eq. (31) will Clearly, if finite d is taken into account then the loga- be determined by the screening length. rithmicbehaviorinEq.(33)maychange: theupperlimit Combining the above arguments we conclude that in- ofthe logarithmwillnowdepend onthe relativevalueof creasingthechemicalpotentialthefollowingfourregimes the chemical potential and inverse inter-layer spacing. may be gradually achieved. At the same time the lower limit of the logarithm also (i)Nαµ T µ v/d. Thisregimewasconsidered ≪ ≪ ≪ depends on our approximations. Indeed, so far we have in Sec. II leading to Eqs. (31) and (33). 8 (ii) Nαµ T v/d µ. If the chemical poten- need to use the full expression (9) for the interaction ≪ ≪ ≪ tial is increased beyond the inverse inter-layer spacing, propagator; (ii) the four regimes specified in the previ- then the momentum integration in Eq. (30) is cut off by oussubsectionmaynotexist,since itmighthappenthat v/d instead of µ. The logarithmic behavior of the drag T/(Nα) T < v/(Nαd) v/d. In this case there are ≪ ≪ conductivity Eq. (31) will be modified and σ no longer only two distinctive regimes (for µ T): (a) µ v/d, D ≫ ≪ depends on the chemical potential and (b) µ v/d. The latter regime is usually identified ≫ with the Fermi-liquid result3,9,20,24,25, i.e. Eq. (43). v σ α2e2T2τ2ln . (40) In this subsection we derive the approximate expres- D ∼ Td sion for the drag conductivity for large values of the (iia) T Nαµ µ v/d. Depending on the actual chemical potential µ T, which accounts for the pos- valuesofT≪,d,and≪αiti≪spossiblethatNαµexceedstem- sibility of Nα > 1 (p≫ossible for small α 1, but large ≪ peraturewhilethechemicalpotentialisstillsmallerthan N 1; here we still consider identical layers). In this ≫ the inverse inter-layer spacing µ v/d. Then instead of regime the single-layer conductivity (32) is large (since the previous regime we find ≪ µτ Tτ 1) and therefore we can still limit our con- ≫ ≫ siderationtothediagraminFig.1. Moreover,forµ T σ α2e2T2τ2ln 1 . (41) we can somewhat relax the condition (5) on the inte≫rac- D ∼ Nα tionstrengthandrequiretheelectron-electronscattering time to be larger than the impurity scattering time (iii) T Nαµ v/d µ. Increasing the chemi- ≪ ≪ ≪ calpotentialfurther leads to the regime where the static T2 µ screeningcannolongerbeneglected. Nowthelowerinte- τee τ τ−1 α2 α2Tτ . ≫ ⇒ ≫ µ ⇒ ≪ T grationlimitinEq.(30)iseffectivelygivenbytheinverse screening length rather than the frequency. The upper Now we can follow the usual steps leading to the Fermi- limitisstilldeterminedbytheinter-layerspacing. There- liquid result (43). We consider only the static screening forethe dragconductivity againdepends logarithmically by approximating the polarization operator by Eq. (39) on the chemical potential24 (see Appendix C for details). We further assume23 that the dominant contribution to the drag conductivity v σ α2e2T2τ2ln , (42) comes from the region vq > ω. Finally, we assume that D ∼ Nαµd inthatregionthe resultofthe momentumintegralisde- butnowthis isadecreasingfunction,indicatingtheexis- termined by the upper integration limit and is therefore independent of ω. This allows us to evaluate the fre- tence of the absolute maximum of the drag conductivity quency integral and represent the drag conductivity in as a function of the chemical potential. terms of the single integral over momenta. Under these (iv) T v/d Nαµ µ. Finally, if the chem- ≪ ≪ ≪ assumptions we find similarly to Eqs. (30) ical potential is so large that the screening length becomes smaller than the inter-layer spacing the momen- µ Td tum integral in Eq. (30) is no longer logarithmic. In σ =α2e2T2τ2f ;α; , (44a) D 0 this regimewe recoverthe standardFermi-liquidresult3. (cid:18)T v (cid:19) Note,thatinthisregimethestep-likecombinationofthe hyperbolic function in the non-linear susceptibility (29) ∞ 32 dQQ3e−4λQ is completely ineffective and may be replaced by unity. f (x;α;λ) Giventhatinthisregimetheintegrationisdominatedby 0 ≈ 3 [(Q+α˜(x))2 α˜(x)2e−4λQ]2 Z1 − momenta large compared to temperature the non-linear susceptibilitymaybefurtherlinearizedinfrequency. The sinh2x , resulting expression ×(coshQ+coshx)2 σ = ζ(3) e2τ2T2 , κ =4αk , (43) where D 4 (k d)2(κd)2 F F N α˜(x)= αx. (44b) differsfromthatofRef.3onlybythefactorreflectingval- 2 leydegeneracyingraphene. Suchexpressionforthedrag Here we have approximated the function g(x, W < Q) conductivity was previously obtained in Ref. 20. The | | byEq.(29),butneglectedthefrequencyunderthesquare results of this subsection are illustrated in Fig 3. root. In the limit µ T the combination of the hy- ≫ perbolic functions in g(x, W < Q) has the form of the | | stepfunction,whicheffectivelycutsofftheintegrationat 2. Intermediate interaction strength and µ≫T Q x. Thus we arrive at the approximate expression ∼ The above results rely on the smallness of the inter- 32 x dQQ3e−4λQ action strength α. However, if Nα > 1, then (i) the f (x;α;λ) . (44c) 0 ≈ 3 [(Q+α˜(x))2 α˜(x)2e−4λQ]2 approximation (38) might not be justified and we would Z1 − 9 The results of the previous subsection can now be re- 0.8 = 0.01, Td /v = 0.1 covered for α 1 by neglecting the terms proportional ≪ direct numerical evaluation to α˜2 in the denominator. In contrast, the Fermi-liquid static screening Eq. (44e) result (43) can be obtained by (i) assuming α˜ 1 and 0.6 ≫ keeping only the terms proportionalto α˜2 in the denom- 2 -e inator, (ii) assuming x 1/4λ and thus replacing the 2 ≫ ulopwpeerriinntteeggrraattiioonnlliimmiittbbyyiznefirnoi.tyM,aonredo(vieiir),rfeoprlaµcingvt/hde ) / 0.4 ≫ ( theintegrationlimitcanbeextendedsuchthattheresult D becomes a function of one single parameter 0.2 f (x;α;λ) f˜(4λα˜), (44d) 0 0 ≈ 0.0 where 0 4 8 12 16 ∞ 32 dZZ3e−Z x = / T f˜(y)= . (44e) 0 3 [(Z +y)2 y2e−Z]2 Z0 − FIG. 4: [Color online] Results of the numerical evaluation of The function (44e) describes the crossover between the thedragcoefficientforα=0.01andTd/v=0.1. Thesquares regimes (iii) and (iv) of the previous subsection (see representthecalculation ofEq.(7)withtheonlyapproxima- Fig.3). Thiscanbeseenbyevaluatingtheintegralinthe tionthatthepolarization operatorinthescreenedinter-layer interaction (9) was evaluatedin theabsence ofdisorder. The two limits (here γ 0.577216 is the Euler’s constant) 0 ≈ red line corresponds to the same calculation, but replacing the polarization operator by Eq. (39), i.e. taking into ac- 32 1 11 f˜(y 1) ln γ , (45a) count only static screening. Theblue linecorresponds tothe 0 0 ≪ ≈ 3 y − − 6 approximate expression (44e),valid for µ≫v/d. (cid:18) (cid:19) 64ζ(3) 10 = 0.3, Td /v = 0.2 f˜(y 1) . (45b) 0 ≫ ≈ y4 1 direct numerical evaluation static screening 0.1 Fermi-liquid asymptotic It turns out that by numerical reasons this crossover Eq. (44e) spans a large interval of values of the chemical poten- 2T 0.01 2 1E-3 tial such that the Fermi-liquid result (43) is practically 2 e unattainableingraphene-baseddragmeasurements9. We 2 1E-4 illuFsitrrsatteofthailsl pwoeinetvainluFatiges.th4e-w5h.ole expression (7). In / ) 11EE--65 ( order to do that we need to evaluate the full non-linear D 1E-7 susceptibility (16). Inadditionwe needto determine the 1E-8 polarization operator in graphene at finite temperature and chemical potential. In the ballistic regime we ne- 1E-9 glect the effect of disorder on the polarization operator. 1 10 100 1000 This is the only approximation in this calculation, see x = / T Appendix C for details. The role of the disorder is discussed in the following subsections. The corresponding data is shown in Figs. 4 - 5 by blue squares. Further re- FIG. 5: [Color online] Results of the numerical evaluation of sultsofthenumericalevaluationofthedragconductivity the drag conductivity for α = 0.3 and Td/v = 0.2 shown (7) are shown in Appendix D. in the log-log scale. The straight green line represents the Then we can simplify calculations by using the static Fermi-liquid result (43). The solid blue line corresponds to approximation(39)forthepolarizationoperator. There- theapproximate expression (44e),valid for µ≫v/d. sult of this calculation is shown in Figs. 4 - 5 by the red line. For weak interaction α = 0.01 and relatively small inter-layer spacing (such that Td/v = 0.1) the results Finally, we can evaluate the approximate expression of the static screening approximation are indistinguish- (44e). This expression was derived assuming µ v/d. ≫ able from the “full” calculation as can be seen in Fig. 4. For the parameter values used in Figs. 4 - 5 the ex- At the same time, for “intermediate” values9,25 α=0.3, pression (44e) fits the exact calculation for µ/T > 10. Td/v = 0.2, the static screening approximation “works” However, the Fermi-liquid asymptotic (represented by well for small and large chemical potentials, while some- the straight green line in Fig. 5) is not reached until what overestimating the overallpeak height. µ/T > 200. Clearly this is very far from the param- 10 eter regime relevant to the experimental observation of In the simple case τ(ǫ) = const, the function K(z) theCoulombdragingraphene9,25. Itisthereforenotsur- simplifies to prising that the Fermi-liquid-like approximationsoveres- timate the observed values of the drag9. The data in 1 W2/Q2 K(z)= 2z − , (47) Fig.5showthatthe asymptotic results,suchasEq.(43) − z2 W2/Q2 − may sometimes be achieved only at the extreme values of parameters. In order to describe the effect in the in- and we recover Eqs. (16b) and (16b). termediate (or realistic) parameter regime, one needs to evaluate Eq. (7) with only minimal approximations. 2. Vicinity of the Dirac point B. Energy-dependent scattering time Taking into accountenergydependence ofthe scattering time τ(ǫ) modifies the non-linear susceptibility and The drag coefficient (21) and the asymptotic results thus changes the Coulomb drag. In the limit µ T the ≪ of Sec. II and Sec. III, as well as the numerical data region Q W becomes important. As can be seen from ∼ of Sec. IVA, were obtained assuming a constant im- Eqs. (16b) and (16b), if τ(ǫ) = const, then precisely at purity scattering time τ. In graphene, the scattering Q = W the non-linear susceptibility vanishes. Other- | | time strongly depends on the type of the impurities and wise, Eqs. (46) may contain a divergence. usually depends on energy26–30. In the context of the Consider for example Coulomb scatterers. Then26,27 Coulomb drag in graphene a similar issue (namely, the τ(ǫ)=τ2 ǫ. (48) momentum-dependent scattering time) was investigated 0| | in Ref. 25 in the framework of the kinetic equation. In this case the function (47) depends on the relation between W and Q | | 1. Non-linear susceptibility Q, Q>W, K(z) 2 (49) Consider now the effect of the energy (or momen- →− (z W , Q<W, | | tum) dependence of the scattering time. Within our assumption (5) we can still consider the ballistic Green’s and as a result the integral (18b) contains a logarithmic functions (A10), only now instead of being the overall divergence at Q= W since now | | factor, the scattering time is a part of the integrand g(x )g(x ) W2 Q2 −1. (50) in the non-linear susceptibility (12). Due to the δ- 1 2 ∝| − | function form of the Green’s functions we can focus on Sameresultholds forthe caseofstrongshort-rangedim- theenergy-dependentτ regardlessofthemicroscopicim- purities that also yield the linear dependence τ ǫ (up purity model. ∼| | tologarithmicrenormalizationwhichisinessentialinthe Repeating the steps leading to Eq. (16) with the present context)28. energy-dependent τ(ǫ) we find that the functions (16b) Similarly, in the case of weak short-rangeddisorder29 and (16b) become (see Appendices A and B for details) τ−1(ǫ)=γ ǫ, (51) 1 √1 z2 | | g(x)=Z0∞dz2q√zWQ2−22 −11 I(z)K(z), |W|>Q, (46a) the function (46b)Qdzo2eQs(2zn2+oQtW2va−2n−iWsh222z)122Wfor2Q, Q=>|W|W|: |, dz − I(z)K(z), W <Q, K(z) 2 where Z1 2q1− WQ22 | | Therefore→th−e nonz-l|iWne|azr2(sQzu22sQc+e2pW−tib2Wi−lit2y2)2Qco2n,taQin<st|hWe|s.ame zW Qτ(T[zQ W]) K(z)= − − (46b) logarithmic divergence (50). zQ W τ(T) − Finally,eveninthecaseoflogarithmicrenormalization zW +Qτ( T[zQ+W]) ofthescatteringtime28,30thefunction(47)doesnotvan- − , ish for Q = W leading to Eq. (50). We conclude, that −zQ+W τ(T) | | Eq.(50) is the generic behavior,while the vanishing (for so that the explicit factor of the scattering time in Q= W ) non-linear susceptibility (16) is the artifact of | | Eq. (16a) should be understood as τ(T). This choice the approximationτ(ǫ)=const. of the prefactor is not essential and is dictated by the The divergence indicates that in the region W Q ∼ discussion of the case µ T below. the δ-function approximation for the Green’s functions ≪