Scaling and interaction-assisted transport in graphene with one-dimensional defects
aa r X i v : . [ c ond - m a t . m e s - h a ll ] M a r Scaling and interaction-assisted transport in graphene with one-dimensional defects
M. Kindermann School of Physics, Georgia Institute of Technology, Atlanta, Georgia 30332, USA (Dated: March 2010)We analyze the scattering from one-dimensional defects in intrinsic graphene. The Coulombrepulsion between electrons is found to be able to induce singularities of such scattering at zerotemperature as in one-dimensional conductors. In striking contrast to electrons in one space dimen-sion, however, repulsive interactions here can enhance transport. We present explicit calculationsfor the scattering from vector potentials that appear when strips of the material are under strain.There the predicted effects are exponentially large for strong scatterers.
PACS numbers: 72.80.Vp, 71.45.Gm, 72.10.Fk, 73.63.-b
The Coulomb repulsion between electrons can haveprofound consequences for the scattering of electrons inconductors. In one-dimensional (1D) conductors it dra-matically suppresses the conductance through impurities,with a singularity at temperature T = 0. This is one ofthe hallmarks of the Luttinger liquid state of interactingelectrons in 1D [1]. A particularly intuitive understand-ing of this effect is due to Matveev, Yue, and Glazman[2, 3]. In their approach interactions cause extra elec-tron scattering from Friedel oscillations, which inhibitstransport. Friedel oscillations are density modulationsthat form when electron waves incident on an impurityinterfere with backscattered waves. In 1D at T = 0 theFriedel oscillation around an impurity is not integrableover space and the induced scattering amplitude divergeslogarithmically in Born approximation.Friedel oscillations in two dimensions (2D) at nonzerochemical potential can be integrated and have muchweaker effects for both, point-like [4–6] and 1D de-fects [7, 8]. Interactions play a more important rolein graphene [9–11] at zero chemical potential (“intrin-sic graphene”), which forms a so-called marginal Fermiliquid [12]. There is, for instance, experimental evidence[13] of a singular renormalization of the Fermi velocity[12, 14]. Nevertheless, also in intrinsic graphene scatter-ing from point-like impurities of both, scalar and vectorpotential character, does not receive any singular correc-tions from the Coulomb interaction [15, 16] [23].In this Letter we show that the Coulomb interactionin intrinsic graphene can have a singular impact on scat-terers that are extended in one space dimension. A qual-itative analysis of the Friedel oscillations at such scat-terers reveals the reason: the Friedel oscillation due toan electron with wavevector k ′ scattering with ampli-tude r from a 1D defect fills a strip of width ∼ /k ′ .The corresponding exchange potential has support in thesame strip, implying a factor ∼ r/k ′ in the induced Bornscattering amplitude δr ex . A second factor 1 /k ′ comesfrom the Coulomb interaction. For energy-independent r the exchange interaction with electrons in all filled statesthus results in δr ex ∝ r R filled d k ′ /k ′ . Since in intrinsicgraphene at T = 0 the entire valence band is filled, δr ex is logarithmically divergent. This suggests that when theCoulomb interaction does not cause a Hartree potential itcan have the same drastic effects for 1D scatterers in 2Dintrinsic graphene as for point defects in 1D conductors.In contrast to the familiar situation in 1D [1], how-ever, we demonstrate that the Coulomb interaction ingraphene can enhance electron transport. This intriguingeffect is due to bound states that straddle defects. Theexchange interaction of transport electrons with electronsin such bound states opens an additional transport pathacross the defect, which increases the electric current. FIG. 1: A vector potential A y in a narrow strip (dark blue)forms a scattering barrier in a sheet of otherwise ballisticgraphene (light blue with contacts in green). A y may be in-duced by strain or, alternatively, by a pair of current-carryingwires (brown). We exemplify the above effects with the scattering bar-rier shown in Fig. 1, formed by a vector potential in anarrow strip of an otherwise ballistic sheet of intrinsicgraphene. Such vector potentials are induced by strain[17, 18] or by nearby electric currents, see Fig. 1. Bysymmetry vector potentials in graphene do not produce aHartree potential. As suggested by the above argument,the Coulomb interaction therefore induces extra scatter-ing whose amplitude diverges at low T , when the elec-tron wavelength is much longer than the width a of thebarrier and scattering is energy-independent. The effectturns out to be exponentially enhanced for strong barri-ers, where it strongly impacts transport even at moder-ate T . In addition, the scatterer of Fig. 1 hosts boundstates that enhance transport through electron exchangeas motivated above. As a consequence, the barrier ofFig. 1 becomes entirely transparent at T = 0, in strikingcontrast to impurities in interacting 1D conductors thatcompletely block the electric current at T = 0. Model:
In the setup of Fig. 1 a vector potential A = (0 , A y ) ; A y = ˜ χea h Θ (cid:16) x + a (cid:17) − Θ (cid:16) x − a (cid:17)i (1)[Θ( x ) is the step function] is applied to a sheet of clean,intrinsic graphene that we first assume to be infinite, L →∞ . The noninteracting Hamiltonian in valley γ thus is H γ = v σ γ · ( p − e A ) . (2)Here, σ γ = ( σ x , γσ y ) is a vector of Pauli matrices, p isthe electron momentum, v the electron velocity, and − e the electron charge. The above model has parity sym-metry, P H P = H , where P ψ ( x, y ) = σ y ψ ( − x, y ), andparticle-hole (PH) symmetry, σ z Hσ z = − H . In intrinsicgraphene both symmetries are also respected by electron-electron interactions [24]. Without restriction we assume˜ χ >
0. Transport through the barrier at ˜ χ < x -axis. The above modelhas been analyzed in Ref. [17] on a noninteracting level,where A suppresses linear transport exponentially in ˜ χ if kT ≪ ˜ χv/a (we set ¯ h = 1). Below we study the effects ofthe Coulomb repulsion between electrons on this trans-port problem for short barriers (Λ a ≫
1, but ln Λ a > ∼ V C ( q ) =2 π ˜ r s v/ | q | at interaction parameter ˜ r s and wavevector q [25]. In the weakly interacting limit ˜ r s ≪ V HF dueto Friedel oscillations at the barrier. In addition, A hereinduces bound states with an extra contribution to V HF .The potential V HF is strongly constrained by the sym-metries of the setup. Most importantly, a Hartree po-tential is precluded by PH-symmetry. Also, the ex-change potential V ex is spin- and valley-diagonal. Trans-port in both valleys is identical since they are relatedas H − = U † v H U v , where U v = σ x commutes with thecurrent σ x through the barrier [26]. We thus set γ = 1.The compound scatterer composed of A and V ex isconveniently described by the transfer matrix of elec-trons at zero energy M = T ( ∞ , −∞ ) | ε =0 [6]. The P -and PH-symmetries of our model imply σ y M σ y = M − and σ z M σ z = M , respectively. Together with currentconservation σ x M † σ x = M − they constrain M to takethe form M = exp( χσ z ). Low energy transport in thesetup of Fig. 1 with weak many-body interactions is thuscharacterized by a single parameter χ . This reduces ouranalysis to a calculation of χ . At ˜ r s = 0 we have χ = ˜ χ . First order in ˜ r s : To begin with, we calculate the in-teraction correction to ˜ χ at first order in ˜ r s . Computingthe correction to M due to scattering from the exchangepotential created by the noninteracting electron states inBorn approximation we find in the limit a ≪ v/kTχ = ˜ χ − ˜ r s sinh ˜ χF ( ˜ χ ) h ln vkT a − ˜ χ + O (1) i (3)with the positive function F ( χ ) = coth χ (cosh 3 χ − χ ) arcsin(tanh χ ) + 2 cosh 2 χ π − cosh χ sinh χ. (4)The first order result Eq. (3) manifests itself ina non-analytic temperature-dependence of the conduc-tance through the barrier, which takes the form G = 4 e h kT W ln 2 πv (cid:16) χ tanh χ ln tanh χ (cid:17) . (5)In G , the logarithmic temperature dependence of the in-teraction correction to χ competes with the factor T dueto the linear density of states. The predicted non-analyticscale-dependence of χ is thus more easily observed in thenormalized conductance G r = G/ ( G | ˜ χ =0 ) [27].Due to its logarithmic T -dependence the first ordercorrection to ˜ χ may grow large even at ˜ r s ≪
1. At thecorresponding low temperatures, when ˜ r s ln( v/kT a ) > ∼
1, the above perturbative calculation looses validity, as inthe 1D case [2, 3]. The origin of the logarithmic diver-gence here is, however, different. In 1D the divergencestems from a Friedel oscillation density ∝ /x at T = 0.In the present 2D case the T = 0 Friedel oscillation den-sity is ∝ /x and integrable. But the non-locality of V ex gives rise to the same logarithmic divergencies as in 1D. RG-analysis:
We extend our analysis into the regime˜ r s ≪
1, but ˜ r s ln( v/kT a ) > ∼
1, by a renormalization group(RG) calculation that re-sums the perturbation series inthe “leading logarithm” approximation [2, 3]. To this endwe successively integrate out shells of wavevectors [ k ′ , bk ′ ]with b > k ′ ≪ /a . Foreach shell we compute the exchange potential V ex k ′ ,b due tothe corresponding states. Thereafter we renormalize thetransfer matrix for low energy electrons at wavevector k by V ex k ′ ,b . The resulting χ characterizes the wavefunctionsin the subsequent shell [ k, bk ]. The logarithmic scale-dependence of δχ , Eq. (3), allows us to assume k ≪ k ′ inthis process. Introducing l = − ln k ′ a we thus find that dχdl = − r s sinh χF ( χ ) , (6)where χ = ˜ χ at l = 0. Here, r s is a scale-dependentinteraction parameter that is renormalized as dr s /dl = − r / r s = ˜ r s / (1 + ˜ r s l/ l -dependent r s is solved best by introducing y = ln(1 + ˜ r s l/ dχ/dy = − χF ( χ ).We start the detailed analysis of Eq. (6) assuming ˜ χ ≫
1. As long as χ ≫ χ →∞ F ( χ ) = 2 / π to find dχdy = − π e χ . (7)Eqs. (6) and (7) make two remarkable predictions:First, the minus signs in both equations imply that theCoulomb interaction reduces the barrier strength. It thus enhances transport through the studied scatterer, in ev-ident contrast with the familiar situation in 1D [1–3].Second, that enhancement is exponentially large in χ such that for sufficiently large ˜ χ the Coulomb interac-tion strongly impacts electron transport even at r s l ≪ | x | < a/
2, but leak into the region | x | > a/
2. Electronsin bound states have the same probability for being tothe left of the barrier as to the right, guaranteed by P -symmetry [28]. The bound states therefore straddle thebarrier and the exchange interaction with electrons inthose bound states opens an additional path for electronsto traverse the barrier. This explains the minus signs inEqs. (6) and (7). The resulting enhancement of the bar-rier transmission t is of order r s l and can easily exceed theexponentially small noninteracting t ∝ exp( − ˜ χ ), whichimplies the exponential renormalization of χ , Eq. (7).This exponential renormalization limits our RG anal-ysis to ˜ r s ≪ π exp( − ˜ χ ). Also, the bound state ener-gies ε b = ± vk y sech χ are exponentially small at χ ≫ χ is thus cut-off at κ b = kT cosh χ | k ′ = κ b /v and it requires exponentially lowtemperatures kT ≪ exp( − ˜ χ ) v/a . Eq. (7) is solved by χ = − ln (cid:18) e − ˜ χ + 43 π y (cid:19) , (8)where y is evaluated at κ b , y = ln[1 − (˜ r s /
4) ln κ b a ]. Theself-consistent solution of Eq. (8) with this χ -dependent y determines the normalized conductance, which for χ ≫ G r = 8 exp( − χ ) /
3. In analogy withthe Kondo problem we define the temperature scale T ∗ where the transmission across the barrier becomes of or-der unity, G r ≃
1. At ˜ r s ≪ π exp( − ˜ χ ) we find kT ∗ = va e − π/ ˜ r s . (9)At wavevectors k ′ < κ b the bound states do not renor-malize χ anymore, but the propagating states continue todo so and dχ/dy = 3 / κ b ≫ k ′ ≫ kT /v . This now decreases the conductance, by a factor (1+˜ r s χ | k ′ = κ b / − if χ | k ′ = κ b ≫
1. In Fig. 2 we plot the normalized resis-tance G − for various parameter values. Some of themare within the regime ˜ r s ≪ π exp( − ˜ χ ), where Eq. (8) isrigorously justified, others at ˜ r s < ∼ π exp( − ˜ χ ), where Eq.(8) may still be qualitatively correct. At ˜ r s > π exp( − ˜ χ )Eq. (8) does not have a unique solution anymore. Clearly,the predicted effect is large even for moderate ˜ χ > ∼ κ b a > ∼
1. Its observation therefore does not requireaccess to large temperature intervals. D l500010 000 G r - r s ‡ Π ã -Χ r s ‡ Π ã -Χ r s ‡ Π ã -Χ FIG. 2: Normalized resistance G − at ˜ χ = 5 and kT In reality the unscreened and ballistic seg-ments of graphene around the barrier in Fig. 1 have a fi-nite length L (typically limited by contacts or by the elas-tic mean free path). Our predictions then require that theentire Friedel oscillation fits into those segments, imply-ing an additional cut-off κ L = coth ˜ χ/L for the RG-flowand necessitating L ≫ a . Due to the strong renormal-ization at χ ≫ 1, however, the observation of the effectsshown in Fig. 2 does not require excessively long samples.While inelastic processes are negligible in the aboveanalysis of the renormalization of χ at r s ≪ kT G r ) − ≫ r L/v .The cut-off κ L further limits the regime where our pre-dictions can be observed directly in the conductance to r G r ≪ v/kT L ≪ tanh ˜ χ . At r G r > ∼ v/kT L the obser-vation of G − requires experimental differentiation fromthe inelastic background resistance, e.g. by varying ˜ χ .The vector potential A is induced by strain in the x -direction if that is along the “armchair” direction of thegraphene lattice [17]. Strain, however, also generates ascalar, ”deformation” potential V < ∼ ˜ χv/a [17]. Thisbreaks PH-symmetry, a Hartree-potential appears, andour predictions require V ≪ max { v coth ˜ χ/L, kT } /r s .Also the bound states are strongly affected by V andconsequently Eq. (7) holds only if V ≪ ˜ χ exp( − ˜ χ ) v/a , atight constraint when ˜ χ ≫ 1. We remark that Eq. (8)(strictly valid at ˜ χ ≫ 1, but possibly qualitatively cor-rect also at ˜ χ ≃ 1) predicts strong effects like those ofFig. 2 down to ˜ χ ≃ 1, when this extra constraint dis-appears. To avoid a nonzero V , A can be generatedalternatively by a pair of wires at x = 0, but differentdistances d , d ≃ a from the graphene plane (see Fig.1). Opposite currents I and − I through such wires cre-ate a vector potential that falls off as 1 /x at x ≫ a andthat has integral ˜ χ = R dx A y ( x ) = µ I ( d − d ). In ourlimit r s ≪ 1, but l ≫ Conclusions: We have shown that the Coulomb repul-sion between electrons can have profound and unconven-tional effects on transport in intrinsic graphene. Generalconsiderations predict singular transport across energy-independent 1D defects without Hartree potential. In ad-dition we have shown that interactions in graphene canenhance transport through static scatterers, in surprisingcontrast with the familiar result for 1D conductors. Wehave presented detailed calculations for vector potentialscatterers that are induced by strain. There the aboveeffects have exponentially large experimental signatures.The author thanks Yu. V. Nazarov for discussion andthe KITP at UCSB for hospitality, where this work wassupported in part by the NSF, Grant No. PHY05-51164. [1] C. L. Kane and M. P. A. 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Guinea, New J.Phys. , 318 (2006).[23] Staggered potentials do have singular interaction correc-tions [15, 16], but they require atomically sharp disorder.[24] In the case of PH symmetry this is seen easiest in secondquantized form, when H is invariant under the transfor-mation c → σ z c † , where c is the electron annihilationoperator. PH-symmetry requires vanishing chemical po-tential and interaction with the ion charge density.[25] Screening by the electrons in the graphene sheet itself isnegligible at the wavevectors k ′ ≫ kT /v that make thedominant contributions to the analyzed effects [21, 22].[26] In case A is due to strain, ˜ χ has opposite sign in the twovalleys [17]. Because of the invariance of transport under˜ χ → − ˜ χ , however, also this does not affect our results.[27] G r is measured best when A is not due to strain, butelectric currents that can easily be switched on and off.[28] The bound states are non-degenerate (apart from thespin and valley degeneracies).[29] This equation has corrections of order χk ′ a . It is strictlyvalid for k ′ a ≪ ˜ χ (e.g. when gates around x = 0 screenshort wavelength Friedel oscillations), or at 1 / ˜ χ y ≪ ∝ χχ