Jing Xu, Fei Han, Ting-Ting Wang, Laxman R. Thoutam, Samuel E. Pate, Mingda Li, Xufeng Zhang, Yong-Lei Wang, Roxanna Fotovat, Ulrich Welp, Xiuquan Zhou, Wai-Kwong Kwok, Duck Young Chung, Mercouri G. Kanatzidis, Zhi-Li Xiao

A notable phenomenon in topological semimetals is the violation of Kohler’s rule, which dictates that the magnetoresistance MR obeys a scaling behavior of $\mathrm{MR}=f(H/{\rho}_{0})$, where $\mathrm{MR}=[\rho (H)-{\rho}_{0}]/{\rho}_{0}$ and $H$ is the magnetic field, with $\rho (H)$ and ${\rho}_{0}$ being the resistivity at $H$ and zero field, respectively. Here, we report a violation originating from thermally induced change in the carrier density. We find that the magnetoresistance of the Weyl semimetal TaP follows an extended Kohler’s rule $\mathrm{MR}=f[H/({n}_{T}{\rho}_{0})]$, with ${n}_{T}$ describing the temperature dependence of the carrier density. We show that ${n}_{T}$ is associated with the Fermi level and the dispersion relation of the semimetal, providing a new way to reveal information on the electronic band structure. We offer a fundamental understanding of the violation and validity of Kohler’s rule in terms of different temperature responses of ${n}_{T}$. We apply our extended Kohler’s rule to ${\mathrm{BaFe}}_{2}{({\mathrm{As}}_{1-x}{\mathrm{P}}_{x})}_{2}$ to settle a long-standing debate on the scaling behavior of the normal-state magnetoresistance of a superconductor, namely, $\mathrm{MR}\sim {\mathrm{tan}}^{2}{\theta}_{H}$, where ${\theta}_{H}$ is the Hall angle. We further validate the extended Kohler’s rule and demonstrate its generality in a semiconductor, InSb, where the temperature-dependent carrier density can be reliably determined both theoretically and experimentally.