Existence of Dirac cones in Brillouin zone of diperiodic atomic crystals according to group theory

TL;DR

This paper uses group theory to identify conditions under which Dirac cones can exist in the Brillouin zone of non-magnetic, weak spin-orbit coupling, diperiodic atomic crystals, revealing symmetry constraints on their electronic band structures.

Contribution

It systematically analyzes all points in the Brillouin zone of 80 diperiodic groups to establish symmetry-based criteria for Dirac cone existence in such materials.

Findings

Dirac cones can exist at certain Brillouin zone points under specific symmetry conditions.

Complete linear dispersion is forbidden for certain irreducible representations.

A tight-binding model illustrates the theoretical results.

Abstract

We have considered non-magnetic materials with weak spin-orbit coupling, that are periodic in two non-collinear directions, and finite in third, orthogonal direction. In some cases, combined time-reversal and crystal symmetry of such systems, allows the existence of Dirac cones at certain points in the reciprocal space. We have investigated in a systematic way, all points of Brillouin zone of all 80 diperiodic groups and have found sufficient conditions for the existence of s=1/2 Dirac fermions, with symmetry-provided band touching at the vertex of the Dirac cones. Conversely, complete linear dispersion is forbidden for orbital wave-functions belonging to two-dimensional irreducible representations (irreps) of little groups that do not satisfy certain group-theoretical conditions given in this paper. Our results are illustrated by a tight-binding example.