Emergence of symmetric and asymmetric Dirac points under periodic electric and vector potentials in ABA-trilayer graphene superlattice

Abstract

We have theoretically investigated the impact of both periodic electric and vector potentials on the low energy spectrum of ABA-trilayer graphene superlattice. Finite energy Dirac points appear in the energy spectrum due to the application of the vector potential. These extra Dirac points are symmetric about k_y = 0 plane for equal barrier and well widths. For different barriers and well widths, one Dirac point shifts away and the second Dirac point shifts towards the k_y = 0 plane. The extra Dirac points are not only generated from the original Dirac point but also emerge from the valleys developed in the energy spectrum. The application of both electric and vector potentials with identical barrier and well widths breaks the symmetry of the spectrum about the Fermi level. When the electric and vector potentials are asymmetric with all three layers having the same electric potentials, the energy spectrum becomes asymmetric about the Fermi level, and this asymmetric behavior of both potentials annihilates the original Dirac point from the spectrum. When all the layers have different electric potentials and both electric and vector potentials are asymmetric, the spectrum becomes asymmetric again, but this time the asymmetry of the spectrum occurs across the k_y = 0 plane.