Band Symmetries in Two-Dimensional Materials

The symmetries of the energy bands are of fundamental importance for understanding many properties of a material. Here we develop a general scheme to determine the irreducible representations of Bloch functions for a given wave vector. Using a tight-binding picture and exploiting the fact that the atomic orbitals are localized in the vicinity of the atomic sites, we demonstrate that this problem can be factorized into one characterizing the atomic orbitals times one characterizing the crystal-periodic plane waves. Each of these subproblems permits a universal classification, independent of the details of a particular crystal structure. We apply this general scheme to two-dimensional materials including transition metal dichalcogenides (such as MoS₂, WS₂, MoSe₂, and WSe₂) and few-layer graphene. We demonstrate that the irreducible representations characterizing the energy bands are not always uniquely determined by the symmetry of a crystal structure. However, we also show that this ambiguity does not affect observable physics such as selection rules or the effective Hamiltonians for Bloch states that can be derived by means of the theory of invariants.