Two-dimensional Dirac fermions are used to discuss quasiparticles in graphene
in the presence of impurity scattering. Transport properties are completely
dominated by diffusion. This may explain why recent experiments did not find
weak localization in graphene. The diffusion coefficient of the quasiparticles
decreases strongly with increasing strength of disorder. Using the Kubo
formalism, however, we find a robust minimal conductivity that is independent
of disorder. This is a consequence of the fact that the change of the diffusion
coefficient is fully compensated by a change of the number of delocalized
quasiparticle states.Comment: 4 pages, 1 figur

K. Ziegler

O. Madelung

English

arXiv

Ziegler, Klaus

Online-Publikationserver Augsburg

Robust Transport Properties in Graphene

Ziegler, Klaus G.

OPUS Augsburg

Robust Transport Properties in GrapheneK. Ziegler*Institut für Physik, Universität Augsburg, D-86135 Augsburg, Germany(Received 24 April 2006; published 26 December 2006)Two-dimensional Dirac fermions are used to discuss quasiparticles in graphene in the presence ofimpurity scattering. Transport properties are completely dominated by diffusion. This may explain whyrecent experiments did not find weak localization in graphene. The diffusion coefficient of the quasi-particles decreases strongly with increasing strength of disorder. Using the Kubo formalism, however, wefind a robust minimal conductivity that is independent of disorder. This is a consequence of the fact thatthe change of the diffusion coefficient is fully compensated by a change of the number of delocalizedquasiparticle states.DOI: 10.1103/PhysRevLett.97.266802 PACS numbers: 73.20.Jc, 71.55.Ak, 72.10.Bg, 81.05.UwRecent experimental studies of single graphite layers(graphene) have revealed interesting transport properties[1–3]. A quantum Hall effect was found in the presence ofan external magnetic field with Hall plateaus xy 2n 12e2=h (n  0; 1; . . . ). This result can be ex-plained by the band structure of graphene which has twonodes (or valleys) due to the hexagonal lattice and a lineardispersion around each of these nodes [4]. If the Fermienergy is near these nodes the quasiparticles can be de-scribed as fourfold degenerated (two valleys and two spinsorientations) 2D Dirac fermions. Another interesting ob-servation is the existence of a minimal conductivity minxxwhich occurs if the Fermi energy is exactly at the nodes.This quantity shows a remarkable stability with minxx 3 . . . 5 e2=h, even if the mobility of the studied sampleschanges by almost a factor 6 from   0:15 m2=V s to  0:85 m2=V s [1]. The varying mobilities in differentsamples indicate a varying density of impurities. Thetransport mechanism must be related to diffusion of elec-trons and holes, caused by impurity scattering. The lattercan formally be described by a random potential. Then, interms of the Dirac fermions, the impurities can create gapfluctuations, i.e., a random Dirac mass, because a gap caneasily be opened at the nodes of the band structure, forinstance, by a staggered potential [5].Existing theories of 2D Dirac fermions predicted thevalue xx  e2=h for a single Dirac node [6–10], (i.e.,minxx  4e2=h) such that there is a quantitative discrep-ancy by a factor 1= in comparison with the experimentalobservation, as discussed in Ref. [1]. In recent papersseveral authors applied the Landauer formula, instead ofthe Kubo formula used in previous studies, to determinethe minimal conductivity also as minxx  4e2=h for arectangular system with aspect ratio W=L 1 [11,12].Nomura and MacDonald argued that minxx could be en-hanced by Coulomb scattering, leading to minxx  4e2=h[13]. Several possibilities for the value of minxx , usingdifferent approaches to the linear response were also dis-cussed in Ref. [14].The effect of quantum interference due to impurityscattering was studied in a recent experiment [3]. It wasfound that there is no weak localization in a single gra-phene sheet. Multilayer graphite films, on the other hand,exhibit clearly weak localization. These observations in-dicate that graphene has special transport properties.It will be shown in the following that (1) a calculation,based on linear response theory (Kubo formula), gives aconductivity of minxx  e2=h for the pure system,(2) weak scattering leads to a linear Boltzmann conductiv-ity similar to what was observed experimentally, and(3) there are no weak (anti-)localization corrections dueto a spontaneously broken supersymmetry which createsdiffusive fermions.Starting from the Kubo formula [15], the conductivitytensor  of a system with Hamiltonian H at inversetemperature   1=kBT and for frequency ! reads ei@lim!0Z 01ei!tTreH; r	eiHtjeiHtdt: (1)The current operator is given by the Hamiltonian as thecommutator j  ieH; r	. For noninteracting fermionswith single-particle energy eigenstates jki (i.e., Hjki kjki) a lengthy but straightforward calculation yields   ie2@Xk1;k2hk1jH; r	jk2ihk2jrjk1ifk2  fk1k1  k2 ! i(2)with Fermi function f  1=1 e. The identityhk2jH; r	jk1i  k2  k1hk2jrjk1i and the Dirac deltafunction k    lim	!0	=k  2  	2	 allowus to express the conductivity in Eq. (2) as a double integralwith respect to two energies , 0:PRL 97, 266802 (2006) P H Y S I C A L R E V I E W L E T T E R Sweek ending31 DECEMBER 20060031-9007=06=97(26)=266802(4) 266802-1 © 2006 The American Physical Society   ie2@ZZTrfH; r	H  0H; r	H  g1 0 ! if0  f 0dd0: (3)An alternative expression is obtained by writing the Diracdelta functions in terms of the Green’s function Gz H  z1. Using the identity GzH; r	Gz0  rGz0 GzrGzrz z0Gz0;the diagonal elements of the conductivity tensor read   ie2@182lim	1;	2!0ZZ Xr;r0r  r02Xs1;s21s1s20   is1	1  s2	2	 TrnGrr00  is1	1Gr0r is2	2	f0  f 0 ! idd0; (4)where Trn is the trace related to n additional degrees offreedom (e.g., n  2 for the spinor index of 2D Diracfermions).In the special case of graphene the Hamiltonian reads insublattice representation [4,5] H  1h1  2h2; (5)where j (j  1, 2, 3) are Pauli matrices. In Fourierrepresentation with wave vector ~k  k1; k2 the coeffi-cients of the Pauli matrices in a pure system are h1  tX3j1cos ~aj  ~k; h2  tX3j1sin ~aj  ~k;with the lattice vectors of the honeycomb lattice ~a13p=2;1=2, ~a2  0;1, and ~a33p=2;1=2. H canbe diagonalized as Hdiagek;ek with ekh21h22q.The current operator transforms under Fourier transforma-tion as ieH; r	 ! e@H=@k.There are six points at ~k  4=33p; 0, (2=33p,2=3), and (2=33p, 2=3) where ek vanishes,corresponding with the two nodes. The commutators arein the diagonal representation of H H; r	12H; r	21 1e2kh2@h1@k h1@h2@k2; (6)whose value is 9=4 at all nodes. The 2D k integration ateach node can be expressed by an integration with respectto h1 and h2 as d2k  Jdh1dh2, where the Jacobian is J 4=9 at all nodes. Therefore, after the angular integrationaround the nodes the integration is given by J2=3ekdek  8=27ekdek0  ek  :This can be inserted in the conductivity of Eq. (3) and afterthe summation over all nodes the conductivity  readsat low temperatures (1)  ie2h1227Z 0H;r	12H;r	212ek! iH;r	21H;r	122ek! idek:Inserting the commutators from Eq. (6) yields eventually Re 22 e2hZ 02ek !dek 2e2h: (7)Another factor of 2 comes from the spin-1=2 degeneracy ofthe quasiparticles. Thus our calculation gives for the mini-mal conductivity minxx  e2=h.In a pure graphene sheet there is only ballistic transport.Consequently, the diffusion coefficient D is infinite. On theother hand, if the Fermi energy is exactly at the nodes, therelated density of states  vanishes. From this point ofview, the conductivity, expressed by the Einstein relationas minxx / D, depends very sensitively on the limits of themodel parameters (e.g., the dc limit !! 0). A moreinstructive situation is a system with randomly distributedscatterers that may lead to diffusion (i.e., D<1) or evento Anderson localization (i.e.,D  0) [16–19]. A source ofdisorder in the tight-binding Hamiltonian H of Eq. (5) is arandomly fluctuating nearest-neighbor hopping rate. For aqualitative discussion of random scattering, theHamiltonian is approximated by the 2D Dirac Hamiltonian HD  i1r1  i2r2 m3:A randomly fluctuating gap is introduced by a randomDirac mass m with Gaussian distribution of zero meanand variance g. The transformation property of the tight-binding Hamiltonian HT  2H2 (8)is also obeyed by the random Dirac Hamiltonian HD afterrotating 1 ! 2 and 2 ! 1. This property is crucialfor the formation of a diffusion mode in two dimensions[10]. Models which violate this property, e.g., by an addi-tional term proportional to a 2 2 unit matrix, may lead tolocalization of states near the Fermi energy [18,19]. Inter-valley scattering is ignored by the approximation H  HDsuch that we have only independent Dirac cones. The effectof the random mass can be studied by applying a perturba-tion theory, using a partial summation of an infinite seriesof most relevant contributions. On the level of the averagedsingle-particle Green’s function this leads to a self-energyterm 	 in the Green’s function: Gi  Gi i	.This is formally equivalent to a mean-field approximationof a supersymmetric functional-integral approach, wherethe random Dirac mass is replaced by a random super-matrix [10]. Then 	 is obtained as a solution ofPRL 97, 266802 (2006) P H Y S I C A L R E V I E W L E T T E R Sweek ending31 DECEMBER 2006266802-2 	  g	 Z	 2  k2	1kdk=: (9)The Green’s function can be expressed again by a pertur-bation expansion, now in terms of fluctuations around themean-field approximation Gi. This expansion can beinserted into the conductivity (3). The leading term fori  EF (Fermi energy) is the Boltzmann conductivity Re e2 	2@Zfek  EF2  	2	2 ek  EF2  	2	2gekdek: (10)Here  is an approximation of the two commutators inEq. (3) and  is the density of states of pure Dirac fermi-ons: E  0jEj, where 0 depends on the cutoff  ofthe spectrum of H. This is a classical result for the con-ductivity of Dirac fermions that was already anticipated byFradkin [6] and discussed in the context of d-wave super-conductors by Lee [7] and for graphene by Peres et al. [8].The conductivity at EF  0 is minxx  2e2 0=h and doesnot depend on 	. Thus minxx is independent of the strengthof impurity scattering g. Sufficiently away from EF  0the conductivity becomes linear, as shown in Fig. 1. Thisbehavior agrees well with the experimentally observedlinear conductivity [1].The next question is whether or not quantum interfer-ence effects are important. The corresponding correctionsin the conductivity are given by the next order terms of theexpanded Green’s function. This includes a logarithmicterm (Cooperons) due to a massless mode of the fluctua-tions around the mean-field approximation. Previous stud-ies found that the corrections give an antilocalization effectfor conventional scatterers, i.e., an increase of the conduc-tivity due to quantum interference effects [20]. Additionalterms in the Hamiltonian of second order in the momentumcan suppress weak antilocalization [21].A crucial question, implied by the antilocalization effectand the absence of weak-localized corrections in experi-ments [3], is whether or not the states in graphene arelocalized. The conductivity in Eq. (3) does not directlyaddress localization, in contrast to the alternative expres-sion in Eq. (4): The long-distance behavior of the Green’sfunctions is directly related to the behavior of the quantumstates. For this purpose we return to Eq. (4), consider theminimal conductivity (i.e., EF  0), and take the limits! 0 and ! 1 (	1, 	2 ! 0 are implicit) to obtain minxx  e2@!8Z !=2!=2Xr;r0r  r02Xs1;s21s1s2Tr2Grr0!=2 is1	1Gr0r!=2 is2	2	d: (11)Assuming that the integrand is finite for 0<! 1 it canbe pulled out of the integral for   0. The major contri-bution comes from s1  s2, where the poles of the Green’sfunction are on different sides of the real axis. A finitefactor 0 takes care of a correction in comparison with theexact value of the integral minxx e2@02!2Xr;r0r  r02Tr2Grr0!=2 i	1Gr0r!=2 i	1	: (12)For localized states the sum, averaged with respect torandomness, is finite due to the exponential decay for jrr0j  1. In the dc limit !! 0 this would lead to a vanish-ing minxx .In order to evaluate the expression in Eq. (12) it isconvenient to return to the functional integral ofRef. [10]. It was found that the underlying supersymmetryof the integral is spontaneously broken by the mean-fieldsolution of Eq. (9). The main consequence of this effect is adiffusive fermionic mode, similar to the Goldstone mode insystems with a rotational symmetry, that can be formallydescribed by a complex Grassmann field r. This allowsus to write hTr2Grr0iGr0ri	i 4	2g2Zrr0eS00D 	;(13)where the action depends on the solution 	 of Eq. (9): S00 4	2g	 Z g4	 k2kkd2k: (14)The latter contains the diffusion coefficient D g4	 (15)  0 10 20 30 40 50 60 70-1 -0.5  0  0.5  1Conductivity [e^2/h]Fermi EnergyFIG. 1. Conductivity of graphene calculated in mean-fieldapproximation [from Eq. (10)].PRL 97, 266802 (2006) P H Y S I C A L R E V I E W L E T T E R Sweek ending31 DECEMBER 2006266802-3that depends strongly on the variance g of the distributionof random scatterers.The minimal conductivity is obtained from Eq. (13) forsmall   i!=2, together with Eq. (12), as minxx  0e2h: (16)This agrees with the result of the mean-field approximationin Eq. (10), except for the (undetermined) prefactors. Thevariation of 0 with g for 0  g  1 can be neglected.Comparing it with the result in Eq. (7) we conclude that therenormalization factor is 0  2. Thus, in contrast to thediffusion coefficient D the minimal conductivity minxx doesnot depend on g. This implies the absence of correctionsdue to quantum interference on characteristic length scales,in agreement with recent experimental observations [3].The fact that minxx is so robust with respect to impurityscattering can be understood in terms of the Einsteinrelation minxx / D, where the conductivity is separatedinto the diffusion coefficient D and the averaged density ofstates  at the Fermi energy EF  0. The latter is calcu-lated from a functional integral similar to Eq. (13) as  	=g [10]. Then with D of Eq. (15) the minimal conduc-tivity reads minxx / D /		 		 i!=2:For small  the mean-field Eq. (9) gives 	  e=g, whichimplies for D and  D  ge=g=4;   e=g=g: (17)Thus, moving away from the ballistic limit g  0, theconductivity should fall rapidly with increasing randompotential fluctuations due to a decreasing D in the Einsteinrelation. On the other hand, the density of states  in-creases correspondingly so that in minxx the influence ofrandom scattering is compensated. This is in agreementwith the direct evaluation of minxx in Eq. (16). The resultsfor  andD in Eq. (17) describe a nonperturbative effect ofdisorder which is not visible within an expansion in powersof g.Strong potential scattering by charged impurities in thesubstrate, for instance, can lead to a destruction of themassless fermion mode used in Eq. (14). This can causelocalization and a vanishing minxx , at least at very lowtemperatures. The localized regime cannot be treatedwithin a conventional field theory but would require eithera numerical finite-size scaling approach [22] or a strong-disorder expansion.In conclusion, we have studied the conductivity of gra-phene, using a model of 2D Dirac fermions. In the case of apure system the ballistic transport leads to a minimalconductivity minxx  e2=h. In the case of impurity scat-tering we found pure diffusion for any strength of Gaussiandistributed scatterers, where the diffusion coefficient de-pends strongly on the distribution. The minimal conduc-tivity minxx , on the other hand, does not depend on thestrength of impurity scattering because the change of thediffusion coefficient is completely compensated by achange of the density of diffusive states.This research was supported in part by the Sonder-forschungsbereich 484.*Electronic address: Klaus.Ziegler@Physik.Uni-Augsburg.de[1] K. S. Novoselov et al., Nature (London) 438, 197 (2005).[2] Y. 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B 48, 4204 (1993).[19] V. M. Pereira et al., Phys. Rev. Lett. 96, 036801 (2006).[20] H. Suzuura and T. Ando, Phys. Rev. Lett. 89, 266603(2002).[21] E. McCann et al., Phys. Rev. Lett. 97, 146805 (2006).[22] A. MacKinnon and B. Kramer, Phys. Rev. Lett. 47, 1546(1981).PRL 97, 266802 (2006) P H Y S I C A L R E V I E W L E T T E R Sweek ending31 DECEMBER 2006266802-4

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