Quasiparticle self-consistent GW energy band calculations for Ge3N4 phases

Quasiparticle self-consistent $GW$ (G Green's function, W screened Coulomb interaction) band structure calculations are presented for the $\ensuremath{\alpha}, \ensuremath{\beta}$, and $\ensuremath{\gamma}$ phases of ${\mathrm{Ge}}_{3}{\mathrm{N}}_{4}$. The $GW$ corrections to the gap are substantial and the gaps are found to be 3.85, 3.86, and 3.56 eV for $\ensuremath{\alpha}, \ensuremath{\beta}, \ensuremath{\gamma}$ respectively, at the experimental lattice constants. Only the $\ensuremath{\beta}$ phase has a direct gap. The $\ensuremath{\alpha}$ and $\ensuremath{\beta}$ phases, which are closely related to each other in crystal structure, are consequently also found to have rather similar band structures. Nonetheless, some significant differences are found near the valence band maximum, which are related to the different interlayer interactions in the $c$ direction. In the cubic spinel phase, the conduction band minima of $s$ and $p$ character are found close to each other. Effective masses are calculated for the relevant valence and conduction band edges and for the cubic case analyzed in terms of the Kohn-Luttinger effective Hamiltonian. The optical response functions from direct interband transitions are calculated.