The electronic structure of crystalline materials reflects the influence of underlying discrete symmetries of the atomic lattice. When these symmetries are broken, associated degeneracies in the spectrum of states often split, with important consequences for potential electronic and spintronic applications. The mathematical language of group theory can be used to capture the effect of these symmetries and provide insight into the origins of spectrum features obtained from essentially empirical ab initio numerical calculations such as density functional theory. A salient example comes from the unusual electronic structure of atomically thin two-dimensional "four-six-ene" semiconductors such as tin sulfide, germanium telluride, etc. (related to group-V phosphorene but formed from group IV and VI). In this case, group theoretic methods provide a straightforward framework for understanding the consequences of inversion symmetry breaking due to inequivalent sublattice atomic identity. In particular, the quantum states at the edge of the fundamental band gap – relevant to all transport, optoelectronic, and spintronic properties – are shown to directly inherit their character from nearby points of high symmetry in the reciprocal lattice, where the form of allowable energetic interactions is constrained.

Abstract: Band-edge states in the indirect-gap group-IV metal monochalcogenide monolayers (four-six-enes such as SnS, GeTe, etc.) inherit the properties of nearby reciprocal space points of high symmetry at the Brillouin zone edge. We employ group theory and the method of invariants to capture these essential symmetries in effective Hamiltonians including spin-orbit coupling, and use perturbation theory to shed light on the nature of the band-edge states. In particular, we show how the structure of derived wave functions leads to specific dominant momentum and spin scattering mechanisms for both valence holes and conduction electrons, we analyze the direct optical transitions across the band gap, and expose the interactions responsible for subtle features of the local dispersion relations.