Origins of transparent conductivity in SnO$_{2}$

SnO$_{2}$ belongs to a small class of materials that can conduct electricity while remaining transparent to visible light. Along with ITO it is widely used for contacts in flat-panel displays, solar cells, light emitting diodes and other optoelectronic applications. The origin of the observed unintentional $n$-type conductivity in SnO$_{2}$ is generally attributed to oxygen vacancies, $V_{\rm O}$. Using density functional calculations along with the LDA+$U$ method we show that $V_{\rm O}$ is not a shallow donor, but has a deep $\epsilon$(2+/0) level at $\sim$2.0 eV below the conduction band. The Sn interstitial is a shallow donor; however, its formation energy is very high making it very unlikely to be incorporated. Instead of point defects we propose that impurities, and in particular hydrogen, are responsible for the observed conductivity. We find that the hydrogen interstitial H$_{\rm i}$ has low formation energy and acts as a shallow donor in SnO$_{2}$. However, its migration barriers are small and therefore it is not stable at elevated temperatures. {\it Substitutional} hydrogen (H$_{\rm O}$), also acts as a shallow donor, and is more stable. Unlike H$_{\rm i}$ the formation energy of H$_{\rm O}$ depends on the abundance of oxygen and hence explains the experimentally observed dependence of conductivity on oxygen partial pressure in SnO$_{2}$.