Dirac fermion quantization on graphene edges: Isospin-orbit coupling, zero modes and spontaneous valley polarization

TL;DR

This paper investigates the electronic boundary states of graphene with complex edges, revealing a valley-dependent zero mode and spontaneous valley polarization linked to isospin-orbit coupling, with implications for local conductivity.

Contribution

It demonstrates the existence of a zero-energy valley-specific bound state and spontaneous valley polarization due to isospin-orbit coupling at graphene edges.

Findings

Presence of a zero-energy valley-specific bound state.

Spontaneous valley polarization breaking Kramers' symmetry.

Relation between valley polarization and local electric Hall conductivity.

Abstract

The paper addresses boundary electronic properties of graphene with a complex edge structure of the armchair/zigzag/armchair type. It is shown that the finite zigzag region supports edge bound states with discrete equidistant spectrum obtained from the Green's function of the continuum Dirac equation. The energy levels exhibit the coupling between the valley degree of freedom and the orbital quantum number, analogous to a spin-orbit interaction. The characteristic feature of the spectrum is the presence of a zero mode, the bound state of vanishing energy. It resides only in one of the graphene valleys, breaking spontaneously Kramers' symmetry of the edge states. This implies the spontaneous valley polarization characterized by the valley isospin $\pm 1/2$. The polarization is manifested by a zero-magnetic field anomaly in the local tunneling density of states, and is directly related to the local electric Hall conductivity.