Possible \textit{n}-type carrier producers in In$_2$O$_3$(ZnO)$_k$ homologous compounds

In$_2$O$_3$(ZnO)$_k$ (${\rm k = integers}$) homologous compounds are promising intrinsic \textit{n}-type transparent conducting semiconductors.\footnote{T. Moriga, et.al., J. Am. Ceram. Soc. \textbf{81}, 1310 (1998).} To find out the carrier producers, we investigated the energetics and thermodynamic properties of \textit{n}-type defects and their complexes in In$_2$O$_3$(ZnO)$_k$, with the k=3 phase as prototype, using the first-princiles density functional method. We calculated the defect formation energies and defect transition energy levels of oxygen vacancies (V$_{\rm O}$), substitutional indium on zinc sites (In$_{\rm Zn}$), zinc and indium interstitials (Zn$_i$ and In$_i$) on different atomic sites, and also some V$_O$--In$_{\rm Zn}$ and V$_{\rm O}$--Zn$_i$ defect complexes. We find, under the experimental growth condition of O-poor and $T=1300$$^{\circ}$C, that V$_{\rm O}$, In$_{\rm Zn}$, and V$_{\rm O}$-In$_{\rm Zn}$ complexes have much lower formation energies than the others, among which V$_{\rm O}$ will stay in the neutral charged state and the latter two are the most possible \textit{n}-type carrier producers. The V$_{\rm O}$-In$_{\rm Zn}$ complex tends to form between V$_{\rm O}$ and In$_{\rm Zn}$ in the same atomic layer; thus its distribution should be affected by the site-preference of V$_{\rm O}$.