Negative intercept of the apparent zero-temperature extrapolated linear-in- T metallic resistivity

We consider the well-known phonon-scattering-induced high-temperature linear-in-T metallic resistivity, showing that a naive extrapolation of the effective linearity from high temperatures to $T=0$ leads to an apparent zero-temperature negative resistivity. The precise magnitude of this extrapolated $T=0$ negative resistivity depends on the temperature regime from where the extrapolation is carried out, and approaches the correct physical result of zero resistivity at $T=0$ only if the extrapolation starts from TTD, where TD is the Debye temperature. We establish a theoretical relationship between the negative intercept and the slope of the linear-in-T resistivity as a function of the temperature T from where the extrapolation is carried out. Experimental implications of our finding are discussed for the much-discussed Planckian behavior of the transport scattering rate.