I-V forward current characteristics of Si Schottky diodes are measured in the 80-291 K temperature range. The apparent decrease of the electron barrier height at the metal-semiconductor (M-S) interface, q(phi B.I-V), and the increase of the ideality factor, n, with the temperature decrease (with q(phi B.I-V) and n calculated by the thermionic theory of Schottky diode current) are connected with the appearance of an insulating part of the space charge layer of the Schottky diode near to the M-S interface when the diode temperature is decreased. Electron tunnelling from occupied deep levels to unoccupied ones in this insulating part of the space charge layer is considered as an electron transport mechanism in these M-S Schottky diodes at temperatures near to 77 K. Electron tunnelling via deep levels at epsilon(t) eV below the Si conduction band and variable-range hopping when the deep levels have constant energy distribution in the Si energy gap are considered separately. After the forward pulse voltage, current transients are observed in these Si Schottky diodes. The amplitude of these current transients increases and their time-constant remains unchanged when the Schottky diode temperature decreases. These current transients are also connected with electron tunnelling in the insulating part of the diode space charge layer near to the M-S interface.