Finite difference method for transport properties of massless Dirac fermions

Abstract

We adapt a finite difference method of solution of the two-dimensional massless Dirac equation, developed in the context of lattice gauge theory, to the calculation of electrical conduction in a graphene sheet or on the surface of a topological insulator. The discretized Dirac equation retains a single Dirac point (no ``fermion doubling''), avoids intervalley scattering as well as trigonal warping, and preserves the single-valley time-reversal symmetry $(=\text{symplectic}\text{ }\text{symmetry})$ at all length scales and energies---at the expense of a nonlocal finite difference approximation of the differential operator. We demonstrate the symplectic symmetry by calculating the scaling of the conductivity with sample size, obtaining the logarithmic increase due to antilocalization. We also calculate the sample-to-sample conductance fluctuations as well as the shot-noise power and compare with analytical predictions.