Semiconductor Transport: The Einstein Relations
From the theorem of equipartition of energy, which states that molecules in thermal equilibrium have the same average energy associated with each independent degree of freedom of their motion. In each of the three dimensions we have,
The expression for mobility is
Rearranging the equation (2) for the velocity, equation (3) for the mean free collision time, and substituting into equation (1) noting that the length is
A similar argument can be shown to give the Einstein relation for holes in the valence band.