Electronic properties of graphene hexagonal boron nitride moir\'{e} superlattice

TL;DR

This paper develops a continuum model to analyze the electronic structures and quantum Hall effects of graphene on hBN moiré superlattices, revealing valley splitting and band gap features influenced by stacking and magnetic fields.

Contribution

It introduces a new effective continuum model derived from a microscopic Hamiltonian to study electronic properties of graphene-hBN moiré superlattices with various rotation angles.

Findings

Band gap opens at superlattice Brillouin zone corners.

Valley splitting is more pronounced in bilayer graphene on hBN.

Fractal spectrum evolves with magnetic field.

Abstract

We theoretically investigate the electronic structures of moir\'{e} superlattices arising in monolayer / bilayer graphene stacked on hexagonal boron nitride (hBN) in presence and absence of magnetic field. We develop an effective continuum model from a microscopic tight-binding lattice Hamiltonian, and calculate the electronic structures of graphene-hBN systems with different rotation angles. Using the effective model, we explain the characteristic band properties such as the gap opening at the corners of the superlattice Brillouin zone (mini-Dirac point). We also investigate the energy spectrum and quantum Hall effect of graphene-hBN systems in uniform magnetic field and demonstrate the evolution of the fractal spectrum as a function of the magnetic field. The spectrum generally splits in the valley degrees of freedom ($K$ and $K'$) due to the lack of the inversion symmetry, and the valley splitting is more significant in bilayer graphene on hBN than in monolayer graphene on hBN because of the stronger inversion-symmetry breaking in bilayer.