Effect of inelastic scattering on impurity resistivity

The standard impurity resistivity formula ρ=m${\lambda }^2$/${\mathrm{Ne}}^2$〈τ〉 in the absence of phonons and electron-electron interactions is clearly understood to be derived under adiabatic conditions. Considering coupling to a heat bath of phonons, a charged carrier can interact with other carriers via the electron-phonon interaction and thus acquires a momentum-conserving inelastic scattering time ${\tau }_{\mathrm{in}}$. The attendant phonon-mediated electron-electron interaction promotes a tendency toward thermalization of the drifted system. On the basis of the force-balance-equation approach, we demonstrate that under isothermal conditions the impurity resistivity is devoid of the divergencies of van Hove’s ‘‘${\lambda }^2$t’’ series expansion. Electron-electron interactions will yield similar results. In the limit of ${\tau }_{\mathrm{in}\ll \mathrm{\tau }}$ (the impurity scattering time), the impurity resistivity reduces to the expression given by the lowest-order term in the force balance equation, ρ=(m${\lambda }^2$/${\mathrm{Ne}}^2$)〈1/τ〉. We also show that this conclusion is consistent with results based upon the Boltzmann equation in a relaxation-time approximation.