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Native point defects in CuIn$_{1-x}$Ga$_x$Se$_{2}$: hybrid density functional calculations predict origin of p- and n-type conductivity

We have performed a first-principles study of the p- and n-type conductivity in CuIn$_{1-x}$Ga$_x$Se$_{2}$ due to native point defects, based on the HSE06 hybrid functional. Band alignment shows that the band gap becomes larger with $x$ due to the increasing conduction band minimum, rendering it hard to establish n-type conductivity in CuGaSe$_{2}$. From the defect formation energies, we find that In/Ga$_{\mathrm{Cu}}$ is a shallow donor, while V$_{\mathrm{Cu}}$, V$_{\mathrm{In}/\mathrm{Ga}}$ and Cu$_{\mathrm{In}/\mathrm{Ga}}$ act as shallow acceptors. Using total charge neutrality of ionized defects and intrinsic charge carriers to determine the Fermi level, we show that under In-rich growth conditions In$_{\mathrm{Cu}}$ causes strongly n-type conductivity in CuInSe$_{2}$. Under In-poor growth conditions the conductivity type in CuInSe$_{2}$ alters to p-type and compensation of the acceptors by In$_{\mathrm{Cu}}$ reduces, as observed in photoluminescence experiments. In CuGaSe$_{2}$, the native acceptors pin the Fermi level far away from the conduction band minimum, thus inhibiting n-type conductivity. On the other hand, CuGaSe$_{2}$ shows strong p-type conductivity under a wide range of Ga-poor growth conditions. Maximal p-type conductivity in CuIn$_{1-x}$Ga$_x$Se$_{2}$ is reached under In/Ga-poor growth conditions, in agreement with charge concentration measurements on samples with In/Ga-poor stoichiometry, and is primarily due to the dominant acceptor Cu$_{\mathrm{In}/\mathrm{Ga}}$.