Effects of site asymmetry and valley mixing on Hofstadter-type spectra of bilayer graphene in a square-scatter array potential

TL;DR

This paper investigates how site asymmetry and valley mixing influence Hofstadter spectra in bilayer graphene under a magnetic field and periodic scatter array, revealing complex band structure deformations and intervalley scattering effects.

Contribution

It introduces a detailed theoretical analysis of valley mixing and site-energy asymmetry effects on Hofstadter spectra in bilayer graphene using a Bloch-wave expansion and effective Hamiltonian.

Findings

Valley mixing couples Landau subbands from different valleys.

Perpendicular electric field deforms or destroys Hofstadter band structures.

Intervalley scattering and interference are significant in the presence of a scatter array.

Abstract

Under a magnetic field perpendicular to an monolayer graphene, the existence of a two-dimensional periodic scatter array can not only mix Landau levels of the same valley for displaying split electron-hole Hofstadter-type energy spectra, but also couple two sets of Landau subbands from different valleys in a bilayer graphene. Such a valley mixing effect with a strong scattering strength has been found observable and studied thoroughly in this paper by using a Bloch-wave expansion approach and a projected $2\times 2$ effective Hamiltonian including interlayer effective mass, interlayer coupling and asymmetrical on-site energies due to a vertically-applied electric field. For bilayer graphene, we find two important characteristics, i.e., mixing and interference of intervalley scatterings in the presence of a scatter array, as well as a perpendicular-field induced site-energy asymmetry which deforms severely or even destroy completely the Hofstadter-type band structures due to the dependence of Bloch-wave expansion coefficients on the applied electric field.