Electronic and crystal structure of Cu 2 − x S : Full-potential electronic structure calculations

Electronic structure calculations are presented for ${\mathrm{Cu}}_{2\ensuremath{-}x}\mathrm{S}$ using the full-potential linearized muffin-tin orbital method. In the simple cubic antifluorite structure, ${\mathrm{Cu}}_{2}\mathrm{S}$ is found to be semimetallic both in the local density approximation (LDA) and using the quasiparticle self-consistent $GW$ $(\mathrm{QS}GW)$ method. This is because the $\mathrm{Cu}\phantom{\rule{0.2em}{0ex}}d$ bands comprising the valence band maximum are degenerate at the $\ensuremath{\Gamma}$ point and the fact that the $\mathrm{Cu}\phantom{\rule{0.2em}{0ex}}s$ band, which can be considered to be the lowest conduction band, lies slightly below it at $\ensuremath{\Gamma}$. Small deviations from the ideal antifluorite positions for the Cu atoms, however, open a small gap between the $\mathrm{Cu}\phantom{\rule{0.2em}{0ex}}d$ valence and $\mathrm{Cu}\phantom{\rule{0.2em}{0ex}}s$-like conduction bands because the symmetry breaking allows the $\mathrm{Cu}\phantom{\rule{0.2em}{0ex}}s$ and $\mathrm{Cu}\phantom{\rule{0.2em}{0ex}}d$ bands to hybridize. Supercell models are constructed for cubic and hexagonal chalcocite ${\mathrm{Cu}}_{2}\mathrm{S}$ as well as cubic digenite ${\mathrm{Cu}}_{1.8}\mathrm{S}$ by means of a weighted random number structure generating algorithm. This approach generates models with Wyckoff site occupancies adjusted to those obtained from experimental x-ray diffraction results and with the constraint that atoms should stay within reasonable distance from each other. The band structures of these models as well as of the low-chalcocite monoclinic structure $({\mathrm{Cu}}_{96}{\mathrm{S}}_{48})$ are calculated in LDA with an additional $\mathrm{Cu}\phantom{\rule{0.2em}{0ex}}s$ shift obtained from the $\mathrm{QS}GW$-LDA difference for the antifluorite structure. Even with this correction, smaller band gaps of about $0.4--0.6\phantom{\rule{0.3em}{0ex}}\mathrm{eV}$ (increasing from cubic to hexagonal to monoclinic) than experimentally observed $(1.1--1.2\phantom{\rule{0.3em}{0ex}}\mathrm{eV})$ are obtained for the ${\mathrm{Cu}}_{2}\mathrm{S}$ composition. Decreasing the Cu content of ${\mathrm{Cu}}_{2\ensuremath{-}x}\mathrm{S}$ in the range $0.06lxl0.2$ is found to essentially dope the $p$-type material by placing the Fermi level $0.2--0.3\phantom{\rule{0.3em}{0ex}}\mathrm{eV}$ below the valence band maximum but also to increase the gap between highest partially filled valence band and lowest conduction bands to about $0.7--1.0\phantom{\rule{0.3em}{0ex}}\mathrm{eV}$. This results from a reduced $\mathrm{Cu}\phantom{\rule{0.2em}{0ex}}d$-band width. Thus, the optical band gap or onset of optical absorption increases in part but not exclusively due to the Moss-Burstein effect. The total energies of the structures are found to increase from monoclinic to hexagonal to cubic to antifluorite. This is consistent with the fact that the simple antifluorite structure is not observed and that the systems change from monoclinic to hexagonal to cubic with increasing temperature, under the assumption that the electronic energy of the system dominates the free energy. We find that ${\mathrm{Cu}}_{2}\mathrm{S}$ is unstable toward the formation of Cu vacancies even in thermodynamic equilibrium with bulk Cu metal. The experimental data on the band gaps and optical absorption are discussed. We find no evidence for indirect band gaps in the hexagonal materials and argue that the experimental results are consistent with this in spite of previous reports to the contrary. The presence of a second onset of absorption located about $0.5\phantom{\rule{0.3em}{0ex}}\mathrm{eV}$ higher than the minimum band gap observed in experiment is explained by a rise in conduction band density of states at this energy in our calculations. The calculated increase in gap with decreasing Cu concentration is in agreement with experimental observations.