Quantum Hall states in graphene with correlated hopping disorder

Ripples are believed to be one of the important sources of long-range disorder in graphene. Observed both in suspended, as well as in graphene deposited on substrates, smooth ripples can be modeled in the tight-binding Hamiltonian by locally changing the hopping term. We investigate the density of states and the participation ration (PR) of a graphene single-layer sheet with correlated hopping disorder in the quantum Hall regime. We find that for hopping correlation lengths $\lambda$ larger than the lattice parameter, the width of the $n$th Landau Level (LL) increases with $n$. The $n=0$ LL splits into two peaks, but as $\lambda$ increases their widths are dramatically reduced. We observe that this width reduction becomes particularly pronounced when $\lambda$ is of the order or higher than the magnetic length. The analysis of the PR suggests that, with increasing $\lambda$, the localization length decreases for the states from the $n>0$ LLs, while it increases for the $n=0$ Landau level.