We study localization properties of the Dirac-like electronic states in monolayers of graphite. In the framework of a general disorder model, we discuss the conditions under which such standard localization effects as the logarithmic temperature-dependent conductivity correction appear to be strongly suppressed, as compared to the case of a two-dimensional electron gas with parabolic dispersion, in agreement with recent experimental observations

Khveshchenko, D.V.

Carolina Digital Repository

Electron Localization Properties in GrapheneD. V. KhveshchenkoDepartment of Physics and Astronomy, University of North Carolina, Chapel Hill, North Carolina 27599, USA(Received 6 March 2006; published 19 July 2006)We study localization properties of the Dirac-like electronic states in monolayers of graphite. In theframework of a general disorder model, we discuss the conditions under which such standard localizationeffects as the logarithmic temperature-dependent conductivity correction appear to be strongly sup-pressed, as compared to the case of a two-dimensional electron gas with parabolic dispersion, inagreement with recent experimental observations.DOI: 10.1103/PhysRevLett.97.036802 PACS numbers: 73.20.Fz, 72.15.Rn, 73.20.JcAfter several decades of predominantly application-driven studies, graphene has finally been recognized as aunique example of the system of two-dimensional fermi-ons with linear dispersion and pseudorelativistic kinemati-cal properties. The recent advances in microfabrication ofgraphitic samples that are only a few carbon layers thick[1] have made it possible to test the early theoreticalpredictions of the anomalous properties of this system [2].The most striking experimental observation up-to-datewas that of an unusual (half-integer) quantization of theHall conductivity [1], which is characteristic of the Diracnature of the quasiparticle excitations in graphene [3]. Theother properties manifesting this peculiar single-particlekinematics have been revealed by magnetotransport mea-surements [1], including theBpdependence of the ener-gies of (nonequidistant) Landau levels and the intrinsic shift of the phase of the Shubnikov–de Haas oscillations[4].A further insight into the properties of electron states inrealistic (disordered) graphene can be gained from thosesingle-particle phenomena that involve quantum interfer-ence between electronic waves scattered off of multipleimpurities. Conceivably, these properties can be affectedby the Dirac kinematics, too, and therefore they might beexpected to differ from their counterparts in the conven-tional two-dimensional electron gas (2DEG) with para-bolic electron dispersion.For one, contrary to the situation in the conventional2DEG, the available experimental data do not manifest anypronounced weak-localization effects even at the lowestaccessible temperatures where one would expect the gra-phene samples to be well in the diffusive regime [5].Motivated by these findings, in the present work we setout to study electron localization (or a lack thereof) ingraphene.The electronic band structure of graphene is character-ized by the presence of a pair of inequivalent nodal pointsat the wave vectors ~K1;2  2=9a3p; 3, where a isthe lattice spacing. At these two and the four other points inthe Brillouin zone obtainable from ~K1;2 with a shift by oneof the reciprocal lattice vectors ~Q1;2  2=3a3p;1,the valence and conduction bands touch upon each other asa pair of opposing cones with the opening angle given bythe Fermi velocity vF.In the leading approximation, quasiparticle excitationsin the vicinity of the nodal points (hereafter referred to asvalleys) can be described by the Dirac Hamiltonian [2] Ĥ  vF1̂  ̂xkx  ̂z  ̂yky; (1)where ̂i is the triplet of the Pauli matrices acting in thespace of spinors   A;B composed of the values ofthe electron wave function on the A and B sublattices of thehexagonal lattice of graphene, whereas the triplet ̂i acts inthe valley subspace. In the absence of magnetic field, theHamiltonian (1) remains a unity matrix in the physical spinsubspace.In the presence of disorder, the quasiparticles experienceboth intra- and intervalley scattering. Upon averaging overdisorder, the most general form of the elastic four-fermionvertex function is represented by the expression Ŵ  g41 ̂z21 ̂z21	 ̂z21	 ̂z2 g21 ̂z21	 ̂z21	 ̂z21 ̂z2 g1̂  ̂	  ̂	  ̂ g3̂  ̂  ̂	  ̂	; (2)where we used the customary notations for the processes offorward scattering between fermions from the same (g4)and different (g2) valleys, as well as those of ‘‘backward’’scattering (g1) with the momentum transfer close to ~K1 	~K2. Besides, we have included the possibility of ‘‘um-klapp’’ scattering (g3) where the total momentum of twofermions changes by 2 ~K1 	 ~K2  ~K2 	 ~K1  ~Q. A jus-tification for this (not immediately obvious, consideringthat the total momentum changes by only a fraction of thereciprocal lattice vector) extension of the disorder modelwill be discussed later.In the framework of the self-consistent Born approxi-mation (SCBA), the effect of disorder on the quasiparticlePRL 97, 036802 (2006) P H Y S I C A L R E V I E W L E T T E R Sweek ending21 JULY 20060031-9007=06=97(3)=036802(4) 036802-1 © 2006 The American Physical Societyspectrum is described by the self-energy obeying the equation ̂ R!; ~k Z d ~q22TrŴĜR!; ~k ~q; (3)where the retarded Green function is given by the expression Ĝ R!; ~k !R1̂  1̂ vF1̂  ̂xkx  ̂z  ̂yky!R2 	 v2Fk2 (4)which includes a chemical potential > 0, thus allowingone to account for variable electron density.Notably, the solution to Eq. (3) turns out to be diagonalin the valley and sublattice subspaces, and its valuêR0; 0  i1̂  1̂ at !  ~k  0 determines the inverseelastic lifetime  F2g4  g1; (5)where F  	Rd ~k=43 Im TrĜ; ~k is the density ofstates (DOS) at the Fermi energy (per spin).The bulk of the data of Ref. [1] pertains to the metallicregime of relatively high dopings (> ), as indicated bythe measured mobilities 104–6 cm2=V s at electron den-sities ne  1011–13 cm	2, which correspond to 10–100 K and  100–1000 K. In this regime, the DOSvalue F  kF=2vF controlled by a finite radius of theFermi surface kF  =vF  4ne1=2 is only weaklyaffected by disorder.Under these conditions, the double-pole Green function(4) given by a four-by-four matrix can be well approxi-mated by a pair of two-by-two single-pole matrices repre-senting quasiparticle excitations near the two Dirac points Ĝ 1;2!; ~k 1̂ ̂x cosk  ̂y sink2!	 k  i; (6)where k  vFk	 and k  tan	1ky=kx.Also, in the high-density regime, the presence of a largeparameter = facilitates a systematic account of quantuminterference corrections to the SCBA results. The leadingquantum correction to the zeroth-order (Drude) conductiv-ity is given by the standard single-Cooperon (fan-shaped)diagram. From the technical standpoint, however, a calcu-lation of the corresponding correction appears to be ratherinvolved due to the matrix structure of the Green function(6) and the vertex (2).The Drude conductivity itself is given by the standardexpression 0  e2=h=, where the renormaliza-tion factor  accounts for a ladder series of vertex correc-tions associated with one of the two current operators~J  vFcosk; sink inserted into the fermion loop inthe diagrammatic representation of the Kubo formula.Being given by a nonsingular (angular momentum m 1) diffusion mode in the expansion over the angular har-monics eimk ,  is a function of the parameters gi.Expanding the full expression for the Cooperon in thesame basis as that used in Eq. (2), we obtain equations forthe corresponding amplitudes Ci, each of which is a matrixin the direct product of two sublattice subspaces Ĉ1  g11̂ g2Ĥ12Ĉ1  g1Ĥ21Ĉ2;Ĉ2  g21̂ g2Ĥ12Ĉ2  g1Ĥ21Ĉ1;Ĉ3  g31̂ g4Ĥ11Ĉ3  g3Ĥ22Ĉ4;Ĉ4  g41̂ g4Ĥ11Ĉ4  g3Ĥ22Ĉ3:(7)Here Ĥij!; ~q Rd ~k22 ĜRi 	!=2; k q=2 ĜAj !=2	 	; q=2	 k stands for the convolution of apair of Green functions. Computing it for !, vFq ,we obtain Ĥ11;12!; q  Ĥ22;21!; q F41i!2	v2Fq21621̂  1̂	121i!2	v2Fq21623cos2k  sin2k4̂x ̂x 121i!2	v2Fq2162cos2k  3sin2k4̂y  ̂y: (8)It can be readily seen that Eq. (7) splits into two pairs which only couple Ĉ1;2 and Ĉ3;4, respectively. Their solutions read Ĉ 1;2!; q 2c1;2F1̂  1̂	 ̂x  ̂x  ̂y  ̂y  ̂z  ̂zg4 	 g2=g2  g1  v2Fq2=162 	 i!=2;Ĉ3;4!; q 2c3;4F1̂  1̂	 ̂x  ̂x 	 ̂y  ̂y 	 ̂z  ̂zg1 	 g3=g3  g4  v2Fq2=162 	 i!=2;(9)where c1g21g1g4g4	g22g1g1g2; c2g2g4g1g2	g22g21g4	g22g1g1g2; c3g3g1g3g4g1g3g3g4; c4g1g4g23g1g3g3g4:(10)PRL 97, 036802 (2006) P H Y S I C A L R E V I E W L E T T E R Sweek ending21 JULY 2006036802-2The quantum conductivity correction (including spin) in-volves the Ĉ1 and Ĉ4 components of the Cooperon T  	e2hFv2F163Z d ~q22TrĈ10; q  Ĉ40; q(11)whose contributions turn out to be negative and positive,respectively.At low temperatures, Eq. (11) can be cast in the form T 2e2hlnmax=; jg4 	 g2j=g1  g2c1max=; jg1 	 g3j=g4  g3c4(12)where we introduced an inelastic phase relaxation rateT (of either Coulombic or phonon origin, whicheveris dominant) providing a finite cutoff in the momentumintegration.The analysis of Eq. (12) reveals that, in the absence of afine-tuning between the amplitudes of backward scatteringand umklapp processes (g1  g3), the Ĉ4 Cooperon ac-quires a gap jg1 	 g3j=g4  g3. On the other hand,the Ĉ1 mode remains gapless, provided that all the forwardscattering processes are controlled by the same amplitude(i.e., g2  g4), which implies the onset of rather conven-tional weak localization at T ! 0.Conversely, making the C1 mode gapful and invertingthe sign of the conductivity correction would only bepossible under the condition jg1 	 g3j  jg2 	 g4j, whichis unlikely to be satisfied for any realistic impurity poten-tial that yields equal amplitudes of the processes of intra-and intervalley forward scattering.Somewhat more generic would be an apparent antiloc-alizing behavior that can set in at intermediate tempera-tures, namely, at g1=g2 < T< , provided thatg1  g2  g4 and g3  0 (c1  1=2, c4  1). At stilllower temperatures the overall sign of Eq. (12) wouldrevert to negative, thereby giving rise to a nonmonotonictemperature dependence of the lnT term in theconductivity.The above conclusions pertain to the general disordermodel (2). However, in the situation where disorder isrealized as a random distribution of impurities with aconcentration ni and a (short-range) potential uq, thereare only two independent parameters, g2  g4 niu201 Fu02;g1  g3 niu2 ~K1 	 ~K21 Fu ~K1 	 ~K22;(13)which represent the T̂ matrix computed for an arbitrarystrength of disorder, the customary Born and unitarity(where the scattering phase approaches =2) limits corre-sponding to u! 0 and 1, respectively. In the case of agenuine long-range (unscreened) impurity potential, astrong momentum dependence of uq would make theexplicit expressions for gi more involved, though.Provided that the relations (13) between the parametersgi hold, one obtains c1  c4  1=2, and the logarithmicterm in Eq. (12) identically vanishes due to an exactcancellation between the contributions of the localizing(Ĉ1) and antilocalizing (Ĉ4) Cooperon modes, therebyresulting in the absence of the leading lnT correction tothe conductivity.It has to be stressed, however, that the sought-aftersuppression of (anti)localization would only be possibledue to an opening of the umklapp channel. In turn, thelatter requires an emergence of a crystal superstructurewith the wave vector 2 ~K1 	 ~K2  2=3 ~Q1  ~Q2 orequivalent.While an isolated sheet of weakly interacting graphenewould apparently lack such a superstructure, it is conceiv-able that the latter could emerge, if a commensurate sub-strate were used during the process of microfabrication [6].Alternatively, a commensurate corrugation could occurdue to the Coulomb correlations that can induce a spatiallyperiodic pattern of the electron density itself [7]. Thepossibility of a spontaneous formation of such chargedensity wave states has long been discussed in the generalcontext of degenerate semimetals and, specifically, in gra-phene [8].Next, we contrast our calculations against the argumentspresented in Ref. [9] where the possibility of an antiloca-lizationlike behavior in graphene was originally pointedout. The authors of this work asserted that the equations forthe Cooperon mode involve a disorder-induced vertex Wk;	k;p;	p  nih ~pjuj ~kih	 ~pjuj 	 ~ki niu20eik	p1 cosk 	p (14)that makes the full Cooperon amplitude negative for ~k 	 ~p where eik	p  	1, which fact was claimed to beinstrumental for the onset of antilocalizing behavior atg1  g3  0, although no further technical details wereprovided.However, the above argument appears to be invalid,since the actual vertex that ought to be used in the con-struction of the Cooperon is the particle-hole (exchange)amplitude Wk;p;p;k  nih ~pjuj ~kih ~kjuj ~pi, which is given byEq. (14) without the phase factor in question and whichremains non-negative for all ~k and ~p.The true origin of a possible antilocalizing behavior atintermediate temperatures can be traced back to the nega-tive signature of the expansion of the Cooperon mode Ĉ4 inthe product basis ̂a  ̂b, as opposed to that of Ĉ1 [seeEq. (9)]. Interpreting the sublattice index as a fictitious spinone-half ( ~S1;2  ~1;2=2), one can associate the Ĉ4 compo-nent with the singlet mode Ĉ4 / 1̂  1̂	12 ~S1  ~S22],which is known to be a common source of antilocalizationin 2DEG with a spin-orbit coupling of the Rashba and/orDresselhaus kind.Before concluding, a few more comments are in order.Considering the possibility of the electron density tuning inPRL 97, 036802 (2006) P H Y S I C A L R E V I E W L E T T E R Sweek ending21 JULY 2006036802-3the experimental setups of Ref. [1], it would also be ofinterest to extend the analysis of localization effects to thelow-density regime where a finite DOS is dominated bydisorder. A similar situation has been previously studied inthe context of a normal quasiparticle transport in dirtyd-wave superconductors, where   0 and the spectrumpossesses an exact particle-hole symmetry [10]. By anal-ogy with that work, for <  one readily finds that theself-consistent DOS and bare conductivity of graphene aregiven by the expressions F  =2 lnvF=a and0  4e2=h, respectively.Unlike in the metallic (high-doping) regime, however,the relative smallness of the quantum conductivity correc-tions can only be controlled by such a parameter as the(inverse) number of valleys, whose actual value is Nv  2.Moreover, even for an artificially large Nv the calculationof T=0  1=Nv ln=T involves additionaldiagrams containing one conventional [made out of oneretarded (R) and one advanced (A) Green functions, or‘‘RA’’] as well as one anomalous (‘‘RR’’ or ‘‘AA’’)Cooperon mode, the latter having a small gap [11,12].Although in the case of dirty d-wave superconductorsthese new contributions have been argued to reverse thesign of the overall correction to the spin and thermalconductivities [13], we do not find such a behavior ingraphene even in the particle-hole symmetrical limit  0, thanks to the formal differences between the Dirac-likedescriptions of the two systems. Thus, the aforementionedcaveat notwithstanding, our results suggest that the con-ductivity correction in graphene might retain its signthroughout the entire range of electron densities.Also, while being the most general model of a randomscalar potential due to short-range (screened) impurities,the vertex (2) obviously misses out on those types ofdisorder that can be best represented by either a randommass of the Dirac fermions or random vector potential. Theformer might be relevant in the situations where a spatiallyinhomogeneous charge or spin density wave-type orderingemerges [7,8], whereas the latter describes topologicallattice defects (dislocations, disclinations, and cracks)that can be thought of as sources of a random magneticflux (RMF). The corresponding random vector potentialappears to long-range correlated, h ~A~q ~A	 ~qi  42nd=q2,where nd is the density of defects.In an abstract setting, the RMF problem for Dirac fer-mions has been studied in Ref. [14]. Applying the results ofthat work, we predict that the RMF-induced elastic scat-tering gives rise to an additional quasiparticle width d minv2Fnd=; vFn1=2d  [obtaining this result requires one todevelop the SCBA for a gauge-invariant counterpart ofEq. (4), since the non-gauge-invariant self-energy (3) di-verges with the system size L as d /lnLp].It also follows from the conclusions drawn in Ref. [14]that, in the absence of umklapp, the RMF localizationscenario is likely to belong to the so-called ‘‘C’’ universal-ity class [11] which controls the limit T ! 0. However, atintermediate temperatures the localization behavior is ex-pected to be governed by a crossover from C to yet another(‘‘A’’) class [11], due to the predominantly small-anglenature of the RMF scattering.In summary, we carried out a comprehensive analysis ofquantum interference effects in disordered graphene andidentified the conditions under which the quantum con-ductivity correction becomes positive, negative, and zero,respectively. The results of this work provide a furtherinsight into the quantum properties of this novel pseudor-elativistic two-dimensional electron system.This research was supported by the NSF under GrantNo. DMR-0349881. The author acknowledges valuablecommunications with E. Abrahams, A. Geim, V.Gusynin, and H. Suzuura.[1] K. S. Novoselov et al., Science 306, 666 (2004); Nature(London) 438, 197 (2005); Y. Zhang et al., ibid. 438, 201(2005); Appl. Phys. Lett. 86, 073104 (2005); Phys. Rev.Lett. 94, 176803 (2005).[2] G. Semenoff, Phys. Rev. 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Electron localization properties in graphene

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Electron localization properties in graphene

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