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We have performed a first-principles study of the p- and n-type conductivityin CuIn$_{1-x}$Ga$_x$Se$_{2}$ due to native point defects, based on the HSE06hybrid functional. Band alignment shows that the band gap becomes larger with$x$ due to the increasing conduction band minimum, rendering it hard toestablish n-type conductivity in CuGaSe$_{2}$. From the defect formationenergies, we find that In/Ga$_{\mathrm{Cu}}$ is a shallow donor, whileV$_{\mathrm{Cu}}$, V$_{\mathrm{In}/\mathrm{Ga}}$ andCu$_{\mathrm{In}/\mathrm{Ga}}$ act as shallow acceptors. Using total chargeneutrality of ionized defects and intrinsic charge carriers to determine theFermi level, we show that under In-rich growth conditions In$_{\mathrm{Cu}}$causes strongly n-type conductivity in CuInSe$_{2}$. Under In-poor growthconditions the conductivity type in CuInSe$_{2}$ alters to p-type andcompensation of the acceptors by In$_{\mathrm{Cu}}$ reduces, as observed inphotoluminescence experiments. In CuGaSe$_{2}$, the native acceptors pin theFermi level far away from the conduction band minimum, thus inhibiting n-typeconductivity. On the other hand, CuGaSe$_{2}$ shows strong p-type conductivityunder a wide range of Ga-poor growth conditions. Maximal p-type conductivity inCuIn$_{1-x}$Ga$_x$Se$_{2}$ is reached under In/Ga-poor growth conditions, inagreement with charge concentration measurements on samples with In/Ga-poorstoichiometry, and is primarily due to the dominant acceptorCu$_{\mathrm{In}/\mathrm{Ga}}$.

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