Origins of the $p$-type nature and cation deficiency in ${\mathrm{Cu}}_2\mathrm{O}$ and related materials

While most of crystalline wide gap oxides are both stoichiometric and insulating, a handful of them including ZnO and ${\mathrm{In}}_2{\mathrm{O}}_3$ are naturally anion-deficient and electron conductors. Even fewer of the oxides are naturally cation-deficient and hole conductors, the arch-type of which is ${\mathrm{Cu}}_2\mathrm{O}$. Based on first principles calculation of equilibrium nonstoichiometry and defect stability, we explain why the ${\mathrm{Cu}}^{\left(\mathrm{I}\right)}\left(d^{10}\right)$ oxide-based materials are both $p$-type and naturally cation-deficient, and why cation vacancies lead to delocalized, conductive states, whereas in other oxides (e.g., ZnO and MgO), they lead to localized, nonconductive states.

Figure 1

(Color online) The cuprite structure of ${\mathrm{Cu}}_2\mathrm{O}$ and possible interstitial positions. Cu and O atoms are given as blue and red spheres, respectively. The cuprite structure consists of two interpenetrating cristobalite lattices that are distinguished by using two different shades for Cu atoms in each of the lattices. Interstitial atoms are labeled “c” and “d” (yellow spheres) according to their Wyckoff positions (symmetries $D_{3d}$ and $D_{2d}$, respectively).

Figure 2

(Color online) (a) The local density of states of the ideal Cu vacancy $V_{\mathrm{Cu}}$ in ${\mathrm{Cu}}_2\mathrm{O}$ and (b) of $V_{\mathrm{Zn}}$ in ZnO, indicating the symmetric ($a_g$ and $a_1$, red) and nonsymmetric ($a_u$ and $t_2$, blue) representations according to the $D_{3d}$ and the (local) $T_d$ symmetries in the ${\mathrm{Cu}}_2\mathrm{O}$ and ZnO lattices, respectively. (c) Schematic band diagram of bulk ${\mathrm{Cu}}_2\mathrm{O}$ and (d) ZnO, constructed from the interaction of the symmetry-adapted cation-vacancy dangling bond levels with the atomic orbitals of the free metal ions.

Figure 3

Formation energies for intrinsic defects in ${\mathrm{Cu}}_2\mathrm{O}$ in (a) Cu-rich/O-poor conditions and (b) Cu-poor/O-rich conditions. Only lowest energy charge states are shown; ionization levels are indicated with a dot.

Figure 4

Equilibrium Fermi level and defect concentrations calculated for [(a) and (c)] Cu-rich/O-poor conditions and [(b) and (d)] Cu-poor/O-rich conditions. The hole concentration and Fermi level are given both in equilibrium at growth temperature $T_g$ and at room temperature $\left(298\mathrm{K}\right)$.