Linear-in-T resistivity in dilute metals: A Fermi liquid perspective

We consider a short-range deformation potential scattering model of electron-acoustic phonon interaction to calculate the resistivity of an ideal metal as a function of temperature (T) and electron density (n). We consider both 3D metals and 2D metals, and focus on the dilute limit, i.e., low effective metallic carrier density of the system. The main findings are: (1) a phonon scattering induced linear-in-T resistivity could persist to arbitrarily low T in the dilute limit independent of the Debye temperature ($T_D$) although eventually the low-T resistivity turns over to the expected Bloch-Gruneisen (BG) behavior with $T^5$ ($T^4$) dependence, in 3D (2D) respectively; (2) because of low values of n, the phonon-induced resistivity could be very high in the system; (3) the resistivity shows an intrinsic saturation effect at very high temperatures (for $T>T_D$), and in fact, decreases with increasing T above a high crossover temperature with this crossover being dependent on both $T_D$ and n in a non-universal manner. We also provide high-T linear-in-T resistivity results for 2D and 3D Dirac materials. Our work brings out the universal features of phonon-induced transport in dilute metals, and we comment on possible implications of our results for strange metals, emphasizing that the mere observation of a linear-in-T metallic resistivity at low temperatures or a very high metallic resistivity at high temperatures is not necessarily a reason to invoke an underlying quantum critical strange metal behavior. We discuss the temperature variation of the effective transport scattering rate showing that the scattering rate could be below or above $k_BT$, and in particular, purely coincidentally, the calculated scattering rate happens to be $k_BT$ in normal metals with no implications for the so-called Planckian behavior.