Coulomb drag at zero temperature
aa r X i v : . [ c ond - m a t . m e s - h a ll ] O c t Coulomb drag at zero temperature
Alex Levchenko and Alex Kamenev
Department of Physics, University of Minnesota, Minneapolis, MN 55455, USA (Dated: October 8, 2007)We show that the Coulomb drag effect exhibits saturation at small temperatures, when calculatedto the third order in the interlayer interactions. The zero-temperature transresistance is of the or-der h/ ( e g ), where g is the dimensionless sheet conductance. The effect is therefore the strongestin low mobility samples. This behavior should be contrasted with the conventional ( second or-der) prediction that the transresistance scales as a certain power of temperature and is (almost)mobility-independent. The result demonstrates that the zero-temperature drag is not an unambigu-ous signature of a strongly-coupled state in double-layer systems. PACS numbers: 73.23.-b, 73.50.-h, 73.61.-r
Coulomb drag effect has proven to be a sensitive probeof electron-electron ( e-e ) interactions. The phenomenonis usually observed [1, 2, 3, 4, 5, 6] in double-layer sys-tems, where electrons interact through the long-rangedCoulomb forces. A current, passing through one of thelayers (the active layer), induces a voltage across the sec-ond (passive) layer. The ratio between the two, the socalled drag transresistance ρ D , carries a valuable infor-mation about the state of electrons in each of the layers,as well as the nature of their mutual interactions.The transresistance for weakly interacting electronswas calculated [7, 8, 9, 10, 11] in the second order inthe screened interlayer interaction and found to be givenby ρ D ( T ) = 0 . he (cid:18) TE F (cid:19) κd ) ( k F d ) , (1)where d is the separation between the layers, E F and k F are the Fermi energy and momentum correspondinglyand κ is the inverse Thomas-Fermi screening radius. Thisresult is in a reasonable agreement with a number of ex-periments [1, 2, 3, 4]. Its main feature is the quadratic temperature dependence, which may be traced back tothe phase volume accessible for the interlayer e-e scat-tering. The second order effect requires the electron-holeasymmetry, i.e. the difference in velocity between elec-trons and holes on the opposite sides of the Fermi surface.Such an asymmetry scales as E − F for each of the two lay-ers, giving rise to the factor E − F in Eq. (1). The latterserves as the dimensional scale, which normalizes the T dependence.On the other hand, the systems with strong interlayercorrelations are predicted to exhibit a nonzero drag tran-sresistance ( ∝ h/e ) even at zero temperature . The exam-ples include 1D charge density waves at exact commensu-rability [12], as well as Quantum Hall bilayer structuresat the total filling factor ν = 1 [13]. In the latter systemthe effect was likely observed experimentally in Ref. [14].This raises a question if ρ D (0) may serve as an unambigu-ous indicator of a strongly-correlated state in a systemat hand. I.e. whether the drag transresistance undergoes a quantum phase transition between the weakly-coupledstate, where it is strictly zero, and a strongly-coupledphase, where it is finite.In this Letter we give a strong argument against sucha scenario. We show that ρ D (0) = 0 already in weaklyinteracting bilayer systems. To this end we evaluate thetransresistance in the third order in the (screened) in-terlayer interactions and find a constant temperature-independent contribution ρ D ( T ) = 0 . he g κd ) ; T < h/τ , (2)where g = 25 . k Ω /ρ (cid:3) is the dimensionless conductance(here ρ (cid:3) is the resistance of the single layer) and τ is theelastic scattering time. Drag effect, saturating at smalltemperatures, is therefore not an automatic indicator ofa strongly correlated state.There are general reasons to expect that the third or-der effect may be qualitatively different from the secondorder one, Eq. (1). Indeed, the third order transresis-tance does not rely on the electron-hole asymmetry. Thisis because the corresponding linear-response diagramsinvolve four -leg vertices (see below) which do not van-ish within linearized dispersion relation approximation(i.e. in the electron-hole symmetric case). Therefore,the result is expected to be independent on the Fermienergy, E F . Since we are interested in the lowest tem-peratures, it is natural to focus on the diffusive regime,where T ≪ h/τ . In this regime there are no any otherrelevant energy, which may provide a scale for a temper-ature dependence. Hence, the temperature-independentresult, Eq. (2), is not entirely unexpected. Moreover,the four-leg vertex, mentioned above, is known to play acentral role in the low-temperature transport of diffusivemetals. It is exactly this object that gives rise to singu-lar Altshuler-Aronov (AA) corrections to the intra layerconductance [15].Coming from another perspective, it is certainly un-usual to find a temperature-independent result for thequantity which relies on the e-e scattering rate. Indeed,the latter is proportional to the available phase volumearound the Fermi surface, which scales as T . However,in addition to the occupation numbers the scattering rateis proportional to a certain matrix element (the overlapintegral of six wave functions for the third order process,considered here). In diffusive systems such matrix ele-ments are known to be singularly enhanced in the limitwhere all involved states are close in energy [16]. It isexactly this enhancement that leads to singular e-e in-teraction effects in the low-temperature diffusive limit[17]. In the case of the third order transconductance in2D the smallness of the phase volume is exactly com-pensated by the divergence of the corresponding matrixelements. This yields the temperature independent tran-sresistance, Eq. (2). The diffusive enhancement of thematrix elements is less pronounced in cleaner systems.Hence, in the clean limit, g → ∞ , the zero temperaturedrag, Eq. (2), disappears.There are two limitations on the applicability of Eq. (2)at the very low temperatures. (i) Once the tempera-ture length L T = p hD/T , where D is the diffusionconstant, reaches the sample size L , the growth of thematrix elements is saturated. As a result, ρ D ∝ T at T < E Th , here E Th = hD/L is the Thouless energy.(ii) If the sample size is very big, one may enter theregime of disorder and/or interaction induced localiza-tion. The relevant temperature scale is that where AAcorrection σ (cid:3) = e /h [ g − π − ln( h/T τ )] [15] is significant,i.e. T ∼ ( h/τ ) e − πg . At smaller temperatures the diffu-sive approximation breaks down and our result, Eq. (2),is not applicable.A natural question is why in experiments of e.g.Refs. [1, 2, 3] the low-temperature saturation of ρ D ( T )was not observed. In order to answer, one may estimatethe saturation temperature T ∗ by equating Eqs. (1) and(2). This way, one finds T ∗ ≈ E F ( k F d ) g − / . Employ-ing the parameters of e.g. Ref. [2]: E F ≈ K , g ≈ k F d ≈
4, one finds T ∗ ≈ .
25K and the residual re-sistance ρ D ≈ . m Ω as it follows from the Eq.(2). Atthe same time the lowest temperature reported in Ref. [2] T ≈ .
5K and the corresponding drag ρ D ≈ . m Ω arejust above the expected saturation. The similar situa-tion is true regarding most of the other reports of theCoulomb drag Refs. [1, 3, 4, 5].It is rather likely, though, that the saturation observedby Lilly et. al
Ref. [18] in ν = 1 Quantum Hall bilayersystem in the composite fermion regime is a manifesta-tion of Eq. (2). Indeed, it was shown [19] that the dif-fusive corrections in the composite fermion regime arerather similar to those in zero magnetic field. Virtu-ally the only difference is a substantial downward renor-malization of the composite fermion conductance g cf , ascompared to the zero field one, g . Estimating g cf ≈ E cfF ≈
5K for the samples of Ref. [18], one finds T ∗ ≈ .
15K and ρ D (0) ≈
2Ω in a good agreement withRef. [18]. To verify Eq. (2) more experiments in zeromagnetic field with smaller g or/and smaller tempera- / 2 ( ) cl A r / 2 ’ ( ’) q A r / 2 ( ) cl A r / 2 ’ ( ’) q A r / 2/ 2’ r r / 2 ( ) cl A r / 2 ’ ( ’) q A r / 2 ( ) cl A r / 2 ’ ( ’) q A r / 2/ 2’ r r
FIG. 1: The four-leg vertex, central to the third-order drageffect, as well as to the intralayer AA correction. Externalwavy lines represent fluctuating vector potentials; the ladderis the diffusion propagator D ( r − r ′ , ω ). tures are needed.The four-leg vertex, which is a building block for di-agrams of the third order drag transconductance is de-picted in Fig. 1. It describes an induced non-linear in-teraction of electromagnetic fields through excitation ofelectron-hole pairs in a given layer. The vertex is non-local because of the diffusive propagation of the electron-hole excitations within the layer. The latter is encodedin the propagator D α ( q, ω ) = 1 ν α D α q − iω , (3)where ν α is the density of states of the layer α = 1 , D α is its diffusion coefficient. Notice that the dimension-less conductance is expressed as g α = ν α D α .We work with the Keldysh technique [20, 21, 22]. In itsframework the fluctuating electromagnetic potentials ac-quire an additional index: classical ( cl ) or quantum ( q ),which stay for symmetric and antisymmetric combina-tions of the fields propagating forward and backward intime, correspondingly. The proper indices are indicatedin Fig. 1. The fact that the four-leg vertex of this verystructure is unique in the leading order in 1 /g α may berigorously proven within Keldysh non-linear sigma-model[21]. In fact, it is exactly this vertex which gives rise tothe singular AA correction [15]. The latter is obtainedby pairing one classical and one quantum electromagneticpotentials, while the two remaining ones represent an ex-ternal (classical) electric field along with induced current[21].It is convenient to work in a gauge, where the Coulombinteractions are mediated by the longitudinal vector po-tentials, rather than the scalar potentials. An advantageof using such a gauge is that both internal and externalpotentials, as well as the current sources are all expressedthrough the same type of field. This makes the structureof the vertex, Fig. 1, particularly symmetric. Moreover,the gauge may be chosen in a way that the propagatorof the longitudinal vector potentials V αβ = 2 i h A clα A qβ i automatically includes the vertex renormalization by thedisorder [21] V αβ ( q, ω ) = q V Rαβ ( q, ω )( D α q − iω )( D β q − iω ) , (4)where V Rαβ ( q, ω ) is the 2 × V R = ˆ V + ˆ V ˆΠ ˆ V R ,whereˆ V = 2 πe q (cid:18) e − qd e − qd (cid:19) , ˆΠ = ν D q D q − iω ν D q D q − iω ! . (5)Note that the polarization operator ˆΠ( q, ω ) has no off-diagonal elements, reflecting the absence of tunneling be-tween the layers.We are now on the position to evaluate the third orderdrag transconductance. The corresponding diagrams areconstructed from the two vertices of Fig. 1: one for eachof the layers, Fig. (2). Remarkably, there are only twoways to connect them, using the propagators (4) (recallthat h A qα A qβ i = 0, [22]). The analytic expression for thesum of the two diagrams of Fig. 2 is given by σ D = 32 e T g g ∞ Z d ω dΩ4 π F F X q,Q Im h D ( q, ω ) D ( q, ω ) V ( q, ω ) V (cid:16) q − Q, ω − Ω (cid:17) V (cid:16) q Q, ω (cid:17) i . (6)The two functions F ( ω, Ω) and F ( ω, Ω) originate fromthe integration over the fast electronic energy ε , Fig. 1,in the active and passive layers correspondingly. In thedc limit they are given by F ( ω, Ω) =
T ∂∂
Ω [ B (Ω + ω/ − B (Ω − ω/ , (7a) F ( ω, Ω) = 2 − B (Ω + ω/ − B (Ω − ω/
2) + B ( ω ) , (7b) B ( ω ) = ωT coth (cid:16) ω T (cid:17) . (7c)To make the farther calculations more compact, we re-strict ourselves to the identical layers. We shall first con-sider the experimentally most relevant case of the long-ranged coupling, where κd ≫
1. Here κ = 2 πe ν is theThomas-Fermi inverse screening radius. In this limit theeffective interlayer interaction potential, Eqs. (4), (5), ac-quires a simple form V ( q, ω ) = 1 g κdDq − iω . (8) q,q/2+Q, /2+q/2-Q, /2- R (q, ) R (q, ) R (q, ) R (q, )q,q/2-Q, /2-q/2+Q, /2+q,q/2+Q, /2+q/2-Q, /2- R (q, ) R (q, ) R (q, ) R (q, )q,q/2-Q, /2-q/2+Q, /2+ FIG. 2: Two diagrams for the drag transconductance σ D inthe third order in the interlayer interactions, V ( q, ω ), de-noted by wavy lines. The intralayer diffusion propagators D α ( q, ω ), Eq. (3), are denoted by ladders. Next, we substitute Eqs. (3), (7) and (8) into Eq. (6) andperform the energy and momentum integrations. The in-spection of the integrals shows that both energies ω andΩ are of the order of the temperature ω ∼ Ω ∼ T (in com-pliance with the phase volume considerations) [23]. Onthe other hand, the characteristic value of the transferredmomenta is q ∼ Q ∼ p T / ( Dκd ) ≪ p T /D , cf. Eq. (8).Therefore one may disregard Dq as compared to iω inthe expressions for D α ( q, ω ), Eq. (3), approximating theproduct D D in Eq. (6) by − ω − . This factor representsthe diffusive enhancement of the matrix elements, men-tioned in the introduction. Such spatial scales separationimplies that the four-leg vertices, Fig. 1, are effectivelyspatially local, while the three interlayer interaction linesare long-ranged.Rescaling energies by T and momenta by p T / ( Dκd ),one may reduce the expression (6) for the transconduc-tance to σ D = ( e /h ) g − ( κd ) − × [dimensionless inte-gral]. The latter integral does not contain any parame-ters and is free from divergences in all directions. It isthus simply a number that may be evaluated numerically[24]. In the limit σ D ≪ ( e /h ) g α the transresistance isrelated to σ D by ρ D = σ D h / ( e g g ), resulting finallyin Eq. (2).To emphasize the fact that the scale separation, dis-cussed above, is not crucial for having the low tem-perature saturation, we briefly consider the case of theshort-ranged interlayer interactions, V R ( q, ω ) = V . Thelatter may be relevant, if interactions are screened bye.g. metallic back gate. One employs then Eqs. (3), (4)and rescales the energies by the temperature, while themomenta by p T /D . This way the transconductance,Eq. (6), once again reduces to the dimensionless and pa-rameter free integral. The latter is convergent in all di-rections and may be readily evaluated, resulting in ρ D = 0 . he g ( νV ) . (9)Notice, that the effect is expected to have the negativesign for the short-ranged attractive interactions. This ob-servation may have relevance for oppositely doped doublelayer structures.The low temperature saturation of the Coulomb dragwas discussed previously in Refs. [25] and [26]. Bothof them considered essentially different and somewhatmore exotic mechanisms. The zero temperature satura-tion suggested in Ref. [25] relies on the assumption thatthe electrons in both layers are scattered by exactly thesame disorder potential. Ref. [26] focuses on the stronglycoupled regime, where the pairing order parameter is sup-pressed by disorder.To conclude, we studied the Coulomb drag phe-nomenon in weakly interacting bilayer systems. Wefound that effect saturates at small temperatures, whencalculated to the third order in the interlayer interac-tions. The saturation of drag relies on the presence ofdisorder and scales inversely with mobility. It does notrequire, though, any correlations of the disorder poten-tial in the two layers. The effect was possibly observedin Ref. [18], although more experiments in lower mobilitysamples and zero magnetic field are highly desirable.We are grateful to D. Bagrets, L. Glazman, I. Gornyi,F. von Oppen, A. Savchenko, B. Shklovskii, A. Sternfor stimulating discussions. This work was supported byNSF Grant No. DMR 0405212. A.K. is also supportedby the A. P. Sloan foundation. [1] P.M. Solomon, P.J. Price, D.J. Frank, and D.C. LaTulipe, Phys. Rev. Lett. , 2508 (1989).[2] T.J. Gramila, J.P. Eisenstein, A.H. MacDonald,L.N. Pfeiffer, and K.W. West, Phys. Rev. Lett. , 1196 (1992).[4] M. Kellogg, J.P. Eisenstein, L.N. Pfeiffer, andK.W. West, Solid State Commun. , 515 (2002).[5] A.S. Price, A.K. Savchenko, B.N. Narozhny, G. Allison,D.A. Ritchie, Science , 99 (2007).[6] P. Pillarisetty, H. Noh, D.C. Tsui, E.P. De Poortere,E. Tutuc, and M. Shayegan, Phys. Rev. Lett. , 016805(2002). [7] B. Laikhtman, P.M. Solomon, Phys. Rev. B , 9921(1990).[8] A.-P. Jauho and H. Smith, Phys. Rev. B , 4420 (1993).[9] L. Zheng and A.H. MacDonald, Phys. Rev. B , 8203(1993).[10] A. Kamenev and Y. Oreg, Phys. Rev. B , 7516 (1995).[11] K. Flensberg, B.Y.-K. Hu, and A.-P. Jauho, Phys. Rev.B , 14761 (1995).[12] Y. V. Nazarov and D. V. Averin, Phys. Rev. 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Kamenev, in Nanophysics: Coherence and Transport ,edited by H.Bouchiat et. al. page 177, (Elsevier 2005).[23] This behavior should be contrasted with AA effect [15,17], where the frequency integral comes from the range T . ω . h/τ .[24] The evaluation of the integrals over momenta is substan-tially simplified by the local nature of the vertex (in thelimit κd ≫ Z ∞ d r r Im[ K ( rµ ) K ( rµ + ) K ( rµ − )] = M ( x, y ) , where K ( rµ ) is the modified Bessel function, which isthe 2D Fourier transform of the interaction potential (8).Here µ = √− i x and µ ± = p − i( ± y + x/ p Dκd/ (2 T ) and x = ω/T ; y = Ω /T .Finally the number of interest is given by − π − ∞ Z Z d x d y F ( x, y ) F ( x, y ) x − M ( x, y ) ≈ . , 152 (1999).[26] F. Zhou and Y. B. Kim, Phys. Rev. B59