After the carrier drift velocity at the semiconductor/metal interface is considered, current transport in Schottky diodes under a forward electric field is physically modelled. This model reveals that the ideality factor can be physically originated from the drift velocity and the drift velocity can also reduce the effective Schottky barrier height. This proposed model predicts that both the ideality factor and the Schottky barrier height depend on temperature, voltage and doping density, which agree well with the experimental results reported in the literature. The proposed diode current model also predicts a linear dependent relation between the reciprocal of the ideality factor and the effective Schottky barrier height, which is validated by experimental results. Such a model is useful to better understand the thermionic emission current physically in semiconductor/metal contact. It is also useful to characterise the material properties by using the ideality factor.