Consistent derivation of impurity resistivity from the force-balance equation.

Abstract

The adiabatic impurity resistivity calculation based upon the force-balance equation is studied with use of the method of closed-time-path Green's functions. In this, the effects due to both the noncommutability of the center-of-mass fluctuations at different times and the exact noncanonical commutation relations between the coordinates and momenta of relative electrons are considered. We show that the leading higher-order resistivity terms generated by the noncommutability of center-of-mass fluctuations fully cancel among themselves and make no contribution to the impurity resistivity. As a result the center-of-mass fluctuations can be regarded as classical variables. When canonical commutation relations are approximated for relative electrons, the resulting adiabatic resistivity vanishes. We also demonstrate that the standard adiabatic resistivity formula can be obtained only if the exact noncanonical commutation relations are employed for relative electrons. In the presence of a momentum-conserving inelastic scattering time due to a heat bath and/or direct electron-electron interactions, we confirm that the impurity resistivity (isothermal) is devoid of the divergences of Van Hove's ``${\ensuremath{\lambda}}^{2}$t''-series expansion. This result is in agreement with a phenomenological study based on the Boltzmann equation.