When you have an energy barrier in the contact, there are three possible mechanisms for the electrical transport: thermionic emission, hoping or tunneling. Depending on the system, but at sufficient high temperature and low bias voltage thermionic emission should dominate. In general, as you increase the bias voltage or lower the temperature you have a lower contribution of the thermionic emission current. A rule of thumb that people follow is that the TE model can be fitted to just the low voltage region in the semilog plot.
@ Jorge, but as vibhor pointed above, in the lower voltages space-charge recombination is taking place and if I fit at lower voltages, I will be getting wrong contact parameters.
You are right, but you can't go too high in voltage range because hoping and tunneling can overcome the thermionic emission electrons. It is a compromise.