  Analytic expressions are presented for the dark current-voltage relation
$J(V)$ of a $pn^+$ junction with positively charged columnar grain boundaries
with high defect density. These expressions apply to non-depleted grains with
sufficiently high bulk hole mobilities. The accuracy of the formulas is
verified by direct comparison to numerical simulations. Numerical simulations
further show that the dark $J(V)$ can be used to determine the open circuit
potential $V_{\rm oc}$ of an illuminated junction for a given short-circuit
current density $J_{\rm sc}$ . A precise relation between the grain boundary
properties and $V_{\rm oc}$ is provided, advancing the understanding of the
influence of grain boundaries on the efficiency of thin film polycrystalline
photovoltaics like CdTe and $\mathrm{Cu(In,Ga)Se_2}$.
