Deep vs shallow nature of oxygen vacancies and consequent<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>n</mml:mi></mml:math>-type carrier concentrations in transparent conducting oxides

The source of $n$-type conductivity in undoped transparent conducting oxides has been a topic of debate for several decades. The point defect of most interest in this respect is the oxygen vacancy, but there are many conflicting reports on the shallow versus deep nature of its related electronic states. Here, using a hybrid quantum mechanical/molecular mechanical embedded cluster approach, we have computed formation and ionization energies of oxygen vacancies in three representative transparent conducting oxides: ${\mathrm{In}}_{2}{\mathrm{O}}_{3},{\mathrm{SnO}}_{2}$, and ZnO. We find that, in all three systems, oxygen vacancies form well-localized, compact donors. We demonstrate, however, that such compactness does not preclude the possibility of these states being shallow in nature, by considering the energetic balance between the vacancy binding electrons that are in localized orbitals or in effective-mass-like diffuse orbitals. Our results show that, thermodynamically, oxygen vacancies in bulk ${\mathrm{In}}_{2}{\mathrm{O}}_{3}$ introduce states above the conduction band minimum that contribute significantly to the observed conductivity properties of undoped samples. For ZnO and ${\mathrm{SnO}}_{2}$, the states are deep, and our calculated ionization energies agree well with thermochemical and optical experiments. Our computed equilibrium defect and carrier concentrations, however, demonstrate that these deep states may nevertheless lead to significant intrinsic $n$-type conductivity under reducing conditions at elevated temperatures. Our study indicates the importance of oxygen vacancies in relation to intrinsic carrier concentrations not only in ${\mathrm{In}}_{2}{\mathrm{O}}_{3}$, but also in ${\mathrm{SnO}}_{2}$ and ZnO.