We present temperature-dependent measurements of I-V curves in the dark and under illumination in order to elucidate the dominant transport mechanisms in amorphous silicon-crystalline silicon (a-Si:H/c-Si) heterojunction solar cells. ZnO:Al/(p)a-Si:H/(n)c-Si/(n+)a-Si:H cells are compared with inversely doped structures and the impact of thin undoped a-Si:H buffer layers on charge carrier transport is explored. The solar cell I-V curves are analyzed employing a generalized two-diode model which allows fitting of the experimental data for a broad range of samples. The results obtained from the fitting are discussed using prevalent transport models under consideration of auxiliary data from constant-final-state-yield photoelectron spectroscopy, surface photovoltage, and minority carrier lifetime measurements. Thus, an in-depth understanding of the device characteristics is developed in terms of the electronic properties of the interfaces and thin films forming the heterojunction. It is shown that dark I-V curve fit parameters can unequivocally be linked to the open circuit voltage under illumination which opens a way to a simplified device assessment.