Magnetic spectrum of trigonally warped bilayer graphene - semiclassical analysis, zero modes, and topological winding numbers

TL;DR

This paper analyzes the energy spectrum of bilayer graphene with stacking mismatches, revealing how topological invariants influence Landau levels and degeneracies, and employing semiclassical methods for spectrum description.

Contribution

It introduces a topological and semiclassical framework to understand the fine structure and merging of Dirac points in bilayer graphene with stacking defects.

Findings

Four Dirac points move and merge depending on stacking mismatch

Landau-level degeneracies are governed by topological winding numbers

Semiclassical description effectively captures the spectrum across parameter ranges

Abstract

We investigate the fine structure in the energy spectrum of bilayer graphene in the presence of various stacking defaults, such as a translational or rotational mismatch. This fine structure consists of four Dirac points that move away from their original positions as a consequence of the mismatch and eventually merge in various manners. The different types of merging are described in terms of topological invariants (winding numbers) that determine the Landau-level spectrum in the presence of a magnetic field as well as the degeneracy of the levels. The Landau-level spectrum is, within a wide parameter range, well described by a semiclassical treatment that makes use of topological winding numbers. However, the latter need to be redefined at zero energy in the high-magnetic-field limit as well as in the vicinity of saddle points in the zero-field dispersion relation.