Impurity Resistivity under Thermalized Condition
The standard impurity resistivity based upon the force-balance equation is derived with use of the method of closed-time-path Green’s functions. In this the effects from both the noncommutability of the center of mass fluctuations at different times and the exact noncannonical commutation relations between the coordinates and momentum of relative electrons are considered. In the presence of a short momentum-conserving inelastic scattering time due to electron-electron interaction, fast thermalization among charge carriers can be achieved. Under this condition, the (thermalized) impurity resistivity will have different form than the standard impurity resistivity and it is practically given by the lowest order electron-impurity term in the force balance equation. We also demonstrate that this conclusion is consistent with results based upon the Boltzmann equation in a relaxation time approximation.