Low-temperature T$^{2}$ resistivity in the underdoped pseudogap phase versus T-linear resistivity in the overdoped strange-metal phase of cuprate superconductors

The transport experiments demonstrate a dramatic switch from the low-temperature linear in temperature (T-linear) resistivity in the overdoped strange-metal phase of cuprate superconductors to the low-temperature quadratic in temperature (T-quadratic) resistivity in the underdoped pseudogap phase, however, a consensus on the origin of this unusual switch is still lacking. Here the resistivity in the underdoped pseudogap phase of cuprate superconductors is investigated using the Boltzmann transport equation. The resistivity originates from the electron umklapp scattering mediated by the spin excitation, however, the dominant contribution mainly comes from the antinodal umklapp scattering. In particular, a low temperature $T_{\rm scale}$ scales with $Δ^{2}_{p}$ in the underdoped regime due to the opening of a momentum dependent spin pseudogap, where $Δ_{p}$ is the minimal umklapp vector at the antinode. Moreover, this $T_{\rm scale}$ decreases with the increase of doping in the underdoped regime, and then is reduced to a very low temperature in the overdoped regime. In the underdoped regime, the resistivity is T-quadratic at the low temperatures below $T_{\rm scale}$, where the strength of the T-quadratic resistivity weakens as the doping is raised. However, in the overdoped regime, the resistivity is T-linear at the low temperatures above $T_{\rm scale}$. The results in this paper together with the recent study on the resistivity in the overdoped regime therefore show that the electron umklapp scattering from a spin excitation responsible for the low-temperature T-linear resistivity in the overdoped regime naturally produces the low-temperature T-quadratic resistivity in the underdoped regime resulting from the opening of a momentum dependent spin pseudogap.