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Localization and Kosterlitz-Thouless Transition in Disordered Graphene

Yan-Yang Zhang,1 Jiangping Hu,1 B.A. Bernevig,2 X.R. Wang,3 X.C Xie,4, 5 and W.M Liu4

1 Department of Physics, Purdue University, West Lafayette, Indiana 47907, USA2 Princeton Center for Theoretical Science, Jadwin Hall, Princeton University, Princeton, NJ 08544

3 Physics Department, The Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong SAR, China.4 Institute of Physics, Chinese Academy of Sciences, Beijing 100080, China.

5Department of Physics, Oklahoma State University, Stillwater, Oklahoma 74078, USA

(Dated: October 28, 2018)

We investigate disordered graphene with strong long-range impurities. Contrary to the commonbelief that delocalization should persist in such a system against any disorder, as the system is ex-pected to be equivalent to a disordered two-dimensional Dirac Fermionic system, we find that statesnear the Dirac points are localized for sufficiently strong disorder and the transition between thelocalized and delocalized states is of Kosterlitz-Thouless type. Our results show that the transitionoriginates from bounding and unbounding of local current vortices.

PACS numbers: 71.30.+h, 72.10.-d, 72.15.Rn, 73.20.Fz

It is well-known that the electronic spectrum ofgraphene can be approximately described by relativis-tic Dirac Femions[1, 2]. This is due to the linear dis-persion relation at low energies near two valleys associ-ated to two inequivalent points K and K′ at the cor-ner of the Brillouin zone[3]. The relativistic dispersiongives rise to several remarkable phenomena. Unlike non-relativistic Schrodinger fermions in two dimensions [9],Dirac fermions cannot be trapped by a barrier due tothe Klein paradox, a property of relativistic quantummechanics[4]. Theories based on the two-dimensional(2D) single flavor Dirac Hamiltonian also predict thatDirac fermions cannot be localized by disorder[5, 6, 7, 8].

The great majority of experimental and theoreticalstudies of graphene [2, 10] has focused on the effect of therelativistic electronic dispersion on different phenomenasuch as Landau level structure or quantum Hall ferro-magnetism. However, the validity of single flavor Diracfermion picture for disordered graphene is only approxi-mate and relies on two premisses: (1) The spatial rangeof the impurities is long enough to avoid inter-valleyscattering[5] (for short-range impurities, strong inter-valley scattering can lead to localization[5, 11, 12]); (2)Even in a disordered graphene with long-range impuritiesthat completely suppress inter-valley scattering, a singlevalley Dirac Hamiltonian is only valid when weak impu-rities are considered. Since the approximated linear rela-tivistic dispersion is valid near the Dirac valley points, Kand K′, the approximation cannot be carried out in a re-gion with a strong impurity where the deviation from theDirac point is large enough so that higher order correc-tions to the energy spectrum become relevant[5]. There-fore, the application of single valley Dirac Hamiltonianto disordered graphene is limited to weak long-range im-purities. Indeed, localized states in disordered graphenenear Dirac points have been observed experimentally [13]and numerically[14, 15]. All the above beg the physicalquestion: how does graphene behave in the presence of

strong long-range impurities?In this letter we investigate several novel phenomena

induced by disorder with strong long-range impurities ingraphene. We calculate the scaling properties of dis-ordered graphene in the framework of a tight-bindingmodel and finite-size scaling. Instead of delocalizationwe find that, in the presence of strong long-range impu-rities, states near the Dirac points are localized. Local-ization arises from enhanced backscattering due to thedeviation from linear dispersion in the strong impurityregime. We show that there is a metal-insulator tran-sition (MIT) as a function of the disorder strength andchemical potential. On the delocalized (metallic) side,the conductance is independent of the system size, whichis a characteristic of the Kosterlitz-Thouless (K-T) [16]type transition in conventional 2D systems with randommagnetic field [18, 19] or correlated disorder [20]. We ver-ify the Kosterlitz-Thouless transition nature of the MITby explicitly identifying the bounding and unboundingvortex-anti-vortex local currents in the system.The π electrons in graphene are described by the tight

binding Hamiltonian (TBH)

H =∑

i

Vic†i ci + t

∑

〈i,j〉(c†i cj +H.c), (1)

where c†i (ci) creates (annihilates) an electron on site iwith coordinate ri, t (∼2.7eV) is the hopping integral be-tween the nearest neighbor carbon atoms with distancea/

√3 (a ∼ 2.46A is the lattice constant), and Vi is the

potential energy. In the presence of disorder, Vi is thesum of contributions from NI impurities randomly cen-tered at {rm} among N sites Vi =

∑NI

m=1 Um exp(−|ri −rm|2/(2ξ)), where Um is randomly distributed within(−W/2,W/2) in units of t. Different random configu-rations of graphene samples with same size, ξ, W andni ≡ NI/N constitute an ensemble with definite disorderstrength. This model has been widely used in investigat-ing the transport properties in graphene[21, 22, 23].

2

At zero temperature, the two terminal dimensionlessconductance gL of the sample between perfect leads atFermi energy EF can be written in terms of Landauer-Buttiker formula [24]:

gL(EF ) = 2Tr(tt†), (2)

where t is the transmission matrix and the factor 2 ac-counts for spin degeneracy. Equation (2) can be numer-ically evaluated by recursive Green’s function method[25] for systems with rather large size. For the pur-pose of scaling, the contact effect should be subtractedfrom gL to yield the “intrinsic conductance”, g, definedas 1/g = 1/gL − 1/(2NC), where NC is the number ofpropagating channels at Fermi energy EF and 1/(2NC)is the contact resistance[26]. The conductance g thenreceives contributions solely from the bulk and thus hasthe same scaling property as if it were obtained by thetransfer matrix method [27]. The scaling function[9, 27]

β =d〈ln g〉d lnL

, (3)

〈. . .〉 being the average over random ensemble, is used todetermine the localization properties; β < 0 and β > 0correspond to the insulator and the metal, respectively.We plot the size dependence of 〈ln g〉 with ξ = 1.73a,

nI = 1%, for different EF and W in Fig. 1. The sam-ples are set to be square shaped with length L. Peri-odic boundary conditions in the transverse direction areadopted to exclude the edge states of the zigzag edges[22].The potential range ξ here is chosen long enough to avoidobvious inter-valley scattering [5, 22], and the scalingξ/Lx ∼ 0 is irrelevant. When EF < Ec = 0.1t (Fig.1 (a)) or W > Wc = 2t (Fig. 1 (b)), 〈g〉 is monotonicallydecreasing with increasing L, which means the wavefunc-tions are localized. Otherwise, when EF > Ec = 0.1t(Fig. 1 (a)) orW < Wc = 2t (Fig. 1 (b)), 〈ln g〉 curves fordifferent sizes merge, suggesting a delocalized state withfinite conductance in the thermodynamic limit. However,they are not real metals with β > 0. All the states withW ∈ (0,Wc) are within the metal-insulator transition(MIT) region with β = 0. Even in the cases of extremelyweak disorder with W = 0.25t and W = 0.1t (see theinset of Fig. 1 (b)), except for a vanishing even-odd likefluctuation, 〈ln g〉(L) doesn’t seem to show a tendencyto be increasing nor decreasing. In Fig. 2, the universalβ(ln g) is plotted from the same data in Fig. 1, showing acritical conductance ln gc ∼ 1 separating the delocalizedstates with β = 0 and localized states with β < 0. Thisphenomenon corresponds to a disorder-driven Kosterlitz-Thouless (K-T) type transition that has been observedin many disordered 2D systems[17, 18, 19, 20]. As canbe seen from Fig. 1 (a) and the phase diagram (inset ofFig. 2), states in the low energy region are more easilylocalized.The existence of localized states near the Dirac point

is in contrast to the belief that Dirac fermions are robust

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FIG. 1: (Color online) The scaling of conductance for longrange disorder (ξ = 1.73a, nI = 1%): (a)〈ln g〉 as func-tions of the Fermi energy EF with fixed disorder strengthW = 2t(note: the bandwidth is 6t)); (b) 〈ln g〉 as functions ofdisorder strength W with fixed Fermi energy EF = 0.1t. Theinsets are the same data plotted as functions of size L. Each〈ln g〉 is an average over 100 ∼ 400 random realizations.

against localization, especially in the presence of longrange impurities that can effectively prohibit the inter-valley scattering. In order to gain insight in the natureof the localization transition, we now turn back to thedispersion structure of realistic graphene. In the absenceof disorder (Vi ≡ 0), the upper (+) and the lower (−)bands touch at two Dirac points K = (2π

3a, 2π

3√3a) and

K′ = (2π3a,− 2π

3√3a). When |E| ≤ t, the dispersion consists

of two valleys centered at Dirac points. Near each Diracpoint, e.g. K, the energy bands can be expanded as[5]

E±(q) = ±3ta

2|q| ±

√3ta2

8sin (3α(q))|q|2 +O(q3), (4)

where q ≡ k − K is the momentum measured from K

and α(q) ∈ [0, 2π) is the angle of vector q. The firstterm in the r.h.s of (4) corresponds to the Dirac Hamil-tonian, but non-linear terms will be prominent when q

(or E) is increased. Even when |E| < t, the second termin (4) can lead to non-trivial consequences. First, thequadratic dependence on momentum |q|2 gives rise tonon-vanishing backscattering probability and thus a ten-dency to localization. Second, the angular dependencefactor sin (3α(q)) (“trigonal warping”) breaks the perfectsymmetry of the cone-like valley. The pseudo-time rever-

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FIG. 2: (Color online) The scaling function β = d〈ln g〉d lnL

ob-tained from the data in Fig. 1. The inset is the schematicphase diagram for ξ = 1.73a, nI = 1%.

sal symmetry [8, 22] restricted to each valley is destroyed.When |E| > t, the linear approximation and double-valley structure collapses completely. Although graphenecannot be experimentally doped to a bulk Fermi en-ergy far away from the neutral point (Dirac points) (e.g.,EF ∼ t), the local potential of impurities might still behigh enough to create non-Dirac scatterings. The ob-served localization originates from the non-Dirac behav-ior due to higher order corrections to the dispersion.To confirm this, let us consider the simplest case of a

single long range impurity in the center of a graphenesheet. If inter-valley interaction is effectively prohib-ited and the regime of Klein tunneling[4] holds, thereshould be no bound states, no matter how high the po-tential barrier is. After diagonalizing the Hamiltonianfor a graphene sheet with N sites, the spatial extensionof eigenstate |ψn〉 =

∑N

i=1 anic†i |0〉 with eigen-energy En

can be characterized by the participation ratio

Rn = (

N∑

i=1

a2ni)2/(N

N∑

i=1

a4ni), (5)

which is a measure of the portion of the space where theamplitude of wavefunction differs markedly from zero.For an extended state, R has a finite value (typicallyclose to 1/3 in the presence of disorder), whereas for alocalized state R approaches zero proportional to (1/N)[28]. The results for ξ = 1.73a with different potentialheight V ≥ 0 is plotted in Fig. 3. For small V (Fig. 3(a) and (b)), where the electronic behaviors inside andoutside the barrier are Dirac-like (Fig. 3 (e) and (f)),there are no bound states. When V is increased, boundstates with small R begin to appear in the negative en-ergy region near the Dirac point, as seen in Fig. 3 (c).For positive injected energy (orange arrow with solid linein Fig. 3 (g)), the electron is not far from K both in-side and outside the barrier, so the regime of Klein tun-neling is still valid and the electron cannot be trapped.

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FIG. 3: (Color online) Left column ((a)→(d)): The partici-pation ratio R as functions of energy E for a graphene withN = 70 × 40, in the presence of a single impurity at thecenter with ξ = 1.73a (long range) with different potentialheight V ≥ 0. Right column ((e)→(h)): Schematic diagramsof scattering process corresponding to their left counterparts.The electron is injected from the left, scattered by the barrierat the center and eventually transmitted to the right (thickorange arrows). The dispersion configurations in these threeregions around K are plotted, where the red lines mark theideal Dirac dispersion E±(k) = ± 3ta

2|k-K| and the olive part

is that for graphene calculated from TBH. Discrepancies be-tween them at high energy can be clearly seen.

On the other hand, for negative energy (orange arrowwith dashed line in Fig. 3 (g)), the electron sees a non-Dirac barrier (pointed by the yellow arrow). This causesstrong back-scattering and localization around the impu-rity. When V is increased further (Fig. 3 (h)), even elec-trons in the positive Dirac region will encounter strongback-scattering in the barrier and will be localized (Fig. 3(d)). For negative V (not shown here), all the results aresimilar, except that the localized states now first appearin the positive region near Dirac point. In conclusion,the localized states originate from back-scattering at thebarrier due to its deviation from Dirac behavior at theFermi level. The states near the Dirac point with lowdensity of states will be more sensitive to backscatteringand will be localized first, as in the case of conventionaldisordered systems[25, 26, 27].Why is the MIT in disordered graphene of K-T type?

The K-T transition is a typical topological transitionwhich has been understood as unbounding of vortex-anti-vortex pairs [16]. For instance, in the high temperaturephase 2D XY model, a plasma of unbounded vortices andanti-vortices of local spins gives rise to an exponential de-

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FIG. 4: (Color online) Typical configurations of local currentsin (red arrows) and potential Vn (color contour) on two sidesof K-T type MIT with N = 56× 32 sites, ξ = 1.73a, nI = 1%and EF = 0.1t. (a): W = 1.1t (delocalized); (b): W = 2.9t(localized). The size of arrows is proportional to the logarithmof current value. Carbon hexagons with topological chargen 6= 0 are marked explicitly with blue numbers. Both plots arein the same random realization of impurities, with differentpotential height therefore effectively different W .

cay of spin correlation function; in the low temperaturephase, vortices and anti-vortices are bound to each other,leading to a power law correlation function. This can beclearly seen in the present problem if the local currentsare identified with local spins in XY model.

The bond current vector il→m(EF ) per unit energypointing along the bond between sites l and m can becalculated using Green’s functions[24, 29, 30]. It ismore convenient to investigate the “current flow vector”il =

∑m il→m defined on site l, where the vectorial sum-

mation is taken over the nearest neighbors of site l [29].The current flow il is a vector with angle θl ∈ [0, 2π). Thetopological charge n of local currents on a closed pathcan now be defined as usual: n = 1

2π

∮∇θ · dℓ. In Fig.

4, typical distributions of local currents on both sides ofMIT are plotted. As expected from the K-T picture, inthe delocalized phase (Fig. 4 (a)), vortices (n > 0) and

anti-vortices (n < 0) are closely bounded, correspondingto the “low temperature” phase of 2D XY model withquasi-long range correlations. In the localized phase (Fig.4 (b)), there are a large number of current vortices andanti-vortices. Many of them are unbounded, correspond-ing to the “high temperature” phase of 2D XY modelwithout long range correlations. This offers an explicitpicture of the microscopic origin of the disorder drivenK-T transition in graphene.

In conclusion, we find a Kosterlitz-Thouless typemetal-to-insulator transition as a function of disorderstrength or Fermi energy in disordered graphene withstrong long-range impurities. We explicitly demonstratethe KT nature of transition by showing the boundingand unbounding of local current vortexes. One uniquefeature about the K-T transition is a scaling of expo-nential form near the transition point. The conductanceg ∝ exp(− α√

W−Wc

) where α is a constant. Recently,

the MIT near the neutral point of graphene has beenobserved in graphene nanoribbons[13] which are quasi-one-dimensional systems. Our results can be tested inexperiments with large nanoribbon radius.

We thank D.-X. Yao, W.-F. Tsai and C. Fang for usefuldiscussions. YYZ and JPH were supported by the NSFunder grant No. PHY-0603759.

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