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 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 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 , QQ, (46a) the function (46b)Qdzo2eQs(2zn2+oQtW2va−2n−iWsh222z)122Wfor2Q, Q=>|W|W|: |, dz − I(z)K(z), W

Coulomb drag in graphene: perturbation theory

Similar Coulomb drag in graphene: perturbation theory