Quantum Hall Plateau Transition in Graphene with Spatially Correlated Random Hopping

Abstract

We investigate how the criticality of the quantum Hall plateau transition in disordered graphene differs from those in the ordinary quantum Hall systems, based on the honeycomb lattice with ripples modeled as random hoppings. The criticality of the graphene-specific n = 0 Landau level is found to change dramatically to an anomalous, almost exact fixed point as soon as we make the random hopping spatially correlated over a few bond lengths. We attribute this to the preserved chiral symmetry and suppressed scattering between K and K' points in the Brillouin zone. The results suggest that a fixed point for random Dirac fermions with chiral symmetry can be realized in free-standing, clean graphene with ripples.PACS numbers: 73.43.-f, 72.10.-d, 71.23.-k After the seminal observation of the anomalous quantum Hall effect (QHE) in graphene, [1,2,3] fascination expands with the graphene QHE. One crucial question that is not fully explored is: what exactly is the role of the chiral symmetry in the problem? This has to do with a most significant feature of double Dirac cones (at K and K' in the Brillouin zone) in graphene. Although a single Dirac cone would already imply a characteristic Landau level structure with the zero-energy level, if we really want to look at the effect of disorder on the graphene Landau levels, we have to go back to the honeycomb lattice for which the chiral (A-B sub-lattice) symmetry [4,5] and the associated valley (K and K') degrees of freedom enter as an essential ingredient. The effect of disorder should then be sensitive to the nature of disorder, i.e., bond disorder or potential disorder, which determines the presence or otherwise of the chiral symmetry [6,7,8], and whether the disorder is short-ranged or long-ranged, which controls the scattering between K and K' points.For Dirac fermions, effects of random gauge fields induced by ripples in the two-dimensional plane have been discussed [9,10,11], and the stability of zero modes has been argued in terms of the index theorem and the chiral symmetry [1,4,5,12]. More recently, the plateau-toplateau transition for random Dirac fermions has been discussed, where the particle-hole symmetry is shown to make the zero-energy Landau level robust [13]. As for the criticality, however, the result [13] shows nothing special about the n = 0 Landau level, but this is obtained for a model of the Dirac fermions for which the randomness is introduced as a scalar random potential, so the chiral symmetry is degraded.On the other hand, the actual randomness in graphene, even when atomically clean, is known to have ripples, i.e, long-ranged corrugation of the graphene plane [9]. In fact, while a monolayer graphene naively contradicts with the well-known theorem that two-dimensional crystals should be thermodynamically unstable, one explanation attributes the stability to the ripples [14]. In this sense, we can take the disorder coming from ripples in graphene as an intrinsic disorder. Since the ripples consist of random bending of the honeycomb lattice, its ma...

Correlated random hopping disorder in graphene at high magnetic fields: Landau level broadening and localization properties

“…In particular, the inclusion of a finite correlation length on the hopping (off-diagonal) disorder model was recently reported to induce an anomalously sharp n = 0 LL compared to higher levels. 25,26 It was found that the width of the zeroth LL (Γ 0 ) shrinks to zero as soon as the hopping correlation length λ exceeds the lattice parameter a, in line with analytical studies of the effects of long-range chiral disorder.…”

“…[21][22][23][24][25][26][27][28][29] The broadening of the LLs differs among disorder models. In particular, the inclusion of a finite correlation length on the hopping (off-diagonal) disorder model was recently reported to induce an anomalously sharp n = 0 LL compared to higher levels.…”

“…25,26 We present a systematic study of the shape of the lowest LLs and their localization properties as a function of the hopping disorder correlation length, as well as of other relevant parameters of the system, such as disorder amplitude and magnetic field. We find that Γ 0 decays exponentially with the correlation length, never fully vanishing for any finite λ.…”

Narrow depression in the density of states at the Dirac point in disordered graphene

“…14 Yet, which type of disorder is encountered in real samples is still completely unclear or only partly known in some special cases. This lack of knowledge is particularly unfortunate as the definite type of disorder entirely determines the physical properties, 15,16,17,18,19,20,21 e.g., leading to complete Anderson localization in the case of short range electrostatic scattering potentials via chiral symmetry breaking and scattering between valleys. 22,23 For disordered systems, even the single particle density of states (DOS) near the Dirac point remains still under debate.…”

Chiral symmetry and fermion doubling in the zero-mode Landau levels of massless Dirac fermions with disorder

“…Specifically, the effect on the n = 0 Landau level, which is essential to the anomalous Hall effect of massless Dirac fermions, is examined from the viewpoint of the fermion doubling and the symmetry of the system. In our previous work 12 , we have shown that the n = 0 Landau level of the honeycomb lattice (graphene) becomes anomalously sharp even in the presence of disorder, if the disorder respects the chiral symmetry and is spatially correlated over a distance exceeding a few lattice constants. Conversely, we have also pointed out that when the disorder is spatially uncorrelated, the n = 0 Landau level is broadened just like the other Landau levels, even if the chiral symmetry is respected by the disorder.…”