  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.
