Gap opening of single-layer graphene under the continuum model

TL;DR

This paper theoretically investigates how periodic scalar and vector potentials can induce a bandgap at the Dirac point in single-layer graphene, providing analytical and numerical insights into the conditions for gap opening.

Contribution

It derives an analytical gap equation under the continuum model showing how combined potentials can open a bandgap in graphene.

Findings

A gap equation at the Dirac point is derived analytically.

The bandgap from the equation matches numerical results for weak potentials.

Coupling of scalar and vector potentials can generate a mass term at the Dirac point.

Abstract

Gap opening at the Dirac point of the single-layer graphene with periodic scalar and vector potentials has been theoretically investigated under the continuum model. The symmetry analysis indicates that the two-fold degeneracy at the Dirac point can be lifted when the potentials break both the chiral symmetry and the time-reversal symmetry. A gap equation at the Dirac point is obtained analytically with perturbation theory. It is shown that a mass term at the Dirac point would be generated by coupling of vector and scalar potentials. This gap equation could be considered as a criterion for gap opening at the Dirac point, which is confirmed by the numerical calculation. Furthermore, the bandgap from the gap equation agrees well with the exact result, when the applied potentials are weak.