Local density of states in disordered graphene

TL;DR

This paper investigates how disorder affects the density of states in two lattice models of graphene, revealing divergent behavior in one model and bounded behavior in the other, despite similar continuum limits.

Contribution

It provides a comparative analysis of DOS behavior in honeycomb and square lattice models under disorder, highlighting differences not apparent from their continuum descriptions.

Findings

Upper bound for DOS on SQL at Dirac point confirmed numerically

No upper bound for DOS on HCL with random vector potential, possibly diverging

Models exhibit different behaviors despite similar continuum limits

Abstract

We study two lattice models, the honeycomb lattice (HCL) and a special square lattice (SQL), both reducing to the Dirac equation in the continuum limit. In the presence of disorder (gaussian potential disorder and random vector potential), we investigate the behaviour of the density of states (DOS) numerically and analytically. While an upper bound can be derived for the DOS on the SQL at the Dirac point, which is also confirmed by numerical calculations, no such upper limit exists for the HCL in the presence of random vector potential. A careful investigation of the lowest eigenvalues indeed indicate, that the DOS can possibly be divergent at the Dirac point on the HCL. In spite of sharing a common continuum limit, these lattice models exhibit different behaviour.