Rigorous foundations of the effective mass approximation: analyticity and symmetry of band extrema

Abstract

The effective mass approximation is widely used across models of carrier transport, optical response, and excitons in semiconductors and insulators, but its validity hinges on the assumption that the band dispersion $E_n(\mathbf{k})$ at the relevant extremum is analytic. We prove that analyticity holds at any non-degenerate extremum for Kohn-Sham density functional theory (DFT) with local or hybrid exchange-correlation functionals, for Hartree-Fock, and for band-edge $G_0{W}_0$ quasiparticle energies in gapped systems. Band non-analyticity (or warping) in these settings is therefore intrinsically tied to degeneracy. We then use group theory to determine the symmetry-allowed form of the effective mass tensor for each of the 32 crystallographic point groups, providing a stringent consistency check on first-principles calculations. As a representative application, we show that the electron and hole effective masses at the $K$ point of monolayer MoS${}_2$ must be strictly isotropic at the DFT and $G_0{W}_0$ levels.