Gap opening in single-layer graphene in the presence of periodic scalar and vector potentials within the continuum model

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. With the perturbation theory, we derive an analytical expression (gap equation) for gap opening at the Dirac point. Furthermore, the bandgap from the gap equation agrees well with the exact result, when the applied potentials are weak.