Relationship between resistance, localization length, and inelastic-scattering length

An integral equation for the average resistance of a wire sample of length L, R(L), is obtained in terms of the probability density function for inelastic scattering and the average quantum resistance due to elastic scattering within the sample. This equation yields the average sample resistance as a function of the sample length L, the electron localization length ξ, and the inelastic-scattering length scrl(T), which depends on temperature. For the metallic regime, L>ξ>scrl(T), and the insulating regime, L>scrl(T)>ξ, analytic expressions for the average resistivity ρ(T) in terms of ξ and scrl(T) are obtained. Our approach allows a unified treatment of both regimes. The relationship of the present results to those of Abrahams et al. and Thouless on the conductance of thin wires is discussed.