Charged grain boundaries and carrier recombination in polycrystalline thin film solar cells

We present analytical relations for the dark recombination current of a $p{n}^{+}$ junction with positively charged columnar grain boundaries in the high defect density regime. We consider two defect state configurations relevant for positively charged grain boundaries: a single donor state and a continuum of both acceptors and donors. Compared to a continuum of acceptor+donor states, or to the previously studied single acceptor+donor state, the grain boundary recombination of a single donor state is suppressed by orders of magnitude. We show numerically that superposition holds near the open-circuit voltage $V_{\text{oc}}$ , so that our dark $J\left(V\right)$ relations determine $V_{\text{oc}}$ for a given short circuit current $J_{sc}$ . We finally explicitly show how $V_{oc}$ depends on the these two grain boundary defect state configurations.