Multivalency of Sn in Cu$_{2}$ZnSnS$_{4}$

The highly efficient ternary chalcopyrites such as 2(CuInSe$_{2})$ = Cu$_{2}$In$_{2}$Se$_{4}$ can be replaced as absorber materials in photovoltaic cells by Cu$_{2}$ZnSnS$_{4}$ (band gap $\sim $ 1.5 eV) in order to avoid the costly element In. Yet, the question remains whether a Fermi-level pinning defect can form spontaneously in these quaternary materials just as the In$_{Cu}$ intrinsic \textit{DX} centers in CuInSe$_{2}$ [1]. Here we study theoretically the deep gap levels introduced by the Sn$_{Cu}$ and Sn$_{Zn}$ defects in Cu$_{2}$ZnSnS$_{4}$. We find that these originate from the multi-valency of Sn, which can change into a +II oxidation state instead of the normal +IV state. Such a transition can even occur for Sn on its native site. Thus, we compare to the respective defect behavior in the kesterite Cu$_{2}$ZnGeSe$_{4}$, which has a similar band gap, but a less pronounced multi-valency of the respective IV-valent element Ge. \\[4pt] [1] S. Lany and A. Zunger, Phys. Rev. Lett. 100, 016401 (2008).