Phonon Limited Resistivity to O(T2)

Abstract

The theory of phonon-limited resistivity (ρ) of metals as developed by Baym and others has been extended to include the anharmonic effects using Green’s function technique. The higher order correlation functions involving 3 and 4 operators have been obtained from the corresponding Green’s functions. The physical significance of the contributions from these correlation functions to ρ is pointed out. The contribution to ρ from the third order correlation functions can be identified with the interference term and the fourth order correlation functions can be identified with the correction to ρ from the Debye Waller factor and the first term of the multiphonon series. The derived expressions are valid for all temperatures. In the high temperature limit the Debye Waller factor and multiphonon contributions are found to be of O(T2). All other terms of O(T2) come from the cubic, quartic shifts, phonon widths and the interference term. Thus the formula for ρ in the high temperature limit is found to be ρ = AT + BT2. Numerical estimates for potassium and sodium of the Debye-Waller factor and multiphonon contributions to ρ indicate that there is a strong cancellation between them for potassium (correction to ρ is of the order of 1.3% at T ≃ Tm), but for sodium this correction is as large as 2–5% in the temperature range 160°K to 370°K.