We fabricate and characterize graphene/Si heterojunctions in different configurations, by extensively studying the voltage and the temperature behaviour of device parameters and optical response. In particular, we demonstrate tunable Schottky barrier height and record photo-responsivity in a new-concept device made of a single-layer CVD graphene transferred onto a matrix of nanotips patterned on n-type Si wafer. The original layout, where nano-sized graphene/Si heterojunctions alternate to graphene areas exposed to the electric field of the Si substrate, which acts both as diode cathode and transistor gate, results in a two-terminal barristor with single-bias control of the Schottky barrier. The nanotip patterning favors light absorption, and the enhancement of the electric field at the tip apex improves photo-charge separation and enables internal gain by impact ionization. These features render the device a photodetector with responsivity comparable or superior to commercial photodiodes. We prove that the multi-junction approach does not add extra-inhomogeneity to the Schottky barrier height distribution. We also introduce a modified phenomenological diode equation, which well describes the experimental I-V characteristics of a graphene/Si diode both in forward and reverse bias.