Exact continuum theory of anti-Klein tunneling in bilayer graphene

TL;DR

This paper analytically derives exact conditions for anti-Klein tunneling in bilayer graphene, accounting for trigonal warping and crystallographic orientation, revealing valley-dependent effects and potential experimental applications.

Contribution

It provides the first exact analytical conditions for anti-Klein tunneling in bilayer graphene including trigonal warping effects, extending previous approximate models.

Findings

Anti-Klein tunneling occurs at oblique and normal incidence depending on orientation.

Valley dependence of anti-Klein tunneling is demonstrated.

Differences between 4-component and 2-component models are highlighted.

Abstract

Exact conditions for anti-Klein transmission zeros are found analytically with a 4-component continuum approach which includes trigonal warping. Anti-Klein tunneling occurs at oblique incidence on steps and barriers with soft and hard walls as well as in the known case of normal incidence on a hard step. The necessary energy and angle of incidence depend on the crystallographic orientation of the step or barrier. At normal incidence on an armchair step in unbiased bilayer graphene, anti-Klein tunneling occurs because both the continuum and the tight binding Hamiltonians are invariant under layer and site interchange. At oblique incidence, anti-Klein tunneling is valley-dependent even in the absence of trigonal warping. An experimental arrangement that functions both as a detector of anti-Klein tunneling and a valley polarizer is suggested. There are cases where anti-Klein tunneling occurs in the 4-component theory but not in the 2-component approximation.