Stable spatially discrete envelope-function model for graphene-like band structures

TL;DR

This paper introduces a stable first-order difference scheme for modeling graphene's band structure, effectively removing spurious states and preserving physical accuracy in numerical simulations.

Contribution

It develops a first-order discretization method for the Dirac equation in graphene, eliminating fermion doubling and ensuring accurate, monotonic dispersion relations near the origin.

Findings

First-order scheme removes spurious states in graphene models

Dispersion relations are strictly monotonic and linear near zero momentum

Method maintains physical observables and is applicable to current-carrying systems

Abstract

The spurious states found in numerical implementations of envelope function models for semiconductor heterostructures and nanostructures have been shown to be readily removed by employing a first-order difference scheme. This approach is applied to the band structure of graphene. The massless Dirac equation is identical to a simple two-band model with zero energy gap. A first-order discretization of this equation produces strictly monotonic dispersion relations with the desired linear dependence on k near the origin, thus removing the "fermion doubling" anomaly associated with formulation on a computational mesh. The first-order formulation produces an ambiguity in the form of the Hamiltonian; both forms produce identical results for physical observeables, and are related by both unitary transformations and time reversal. Other details needed to evaluate the properties of current-carrying systems, including current density expressions, open boundary conditions and the treatment of heterojunctions, are also developed.