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A theoretical investigation of electronic transport in metal contacts to polycrystalline silicon thin films is presented. Calculations based upon the reported values of grain-boundary potentials indicate that the thermionic emission theory may be applied to the majority-carrier transport only for low bias voltages. At larger bias voltages, one needs to take account of the voltage lost to the space-charge regions adjacent to the grain boundaries, and a transition from electrode-limited to bulk-limited majority-carrier transport results. We further demonstrate that the injection of minority carriers can dominate the dark current for a range of grain size and interface state densities at the grain boundaries. Under these conditions, the current obeys an dependence reminiscent of space-charge recombination, although the origin of the current in this case is minority-carrier diffusion current, with recombination only at grain boundaries in the neutral region. This is a special case of the "high-injection" regime observed in single crystals, but in the present situation it is found even for low bias voltages as a consequence of the band bending at the grain boundaries, which makes the material nearly intrinsic at these points. Finally, we show that the effective minority-carrier diffusion length for the injected carriers under dark conditions itself increases with bias voltage approximately as , in striking contrast to previous treatments.