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

We present analytical relations for the dark recombination current of a $pn^+$ 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 $V_{\rm oc}$, so that our dark $J(V)$ relations determine $V_{\rm oc}$ for a given short circuit current $J_{\rm sc}$. We finally explicitly show how $V_{\rm oc}$ depends on the grain boundary defect state configurations.