Extended Kohler${}^{,}$s Rule of Magnetoresistance

Abstract

A notable phenomenon in topological semimetals is the violation of Kohler${}^{,}$s rule, which dictates that the magnetoresistance $MR$ obeys a scaling behavior of $MR=f(H/\ddot{I}{}_0$), where $MR=[\ddot{I}{}_H-\ddot{I}{}_0]/\ddot{I}{}_0$ and $H$ is the magnetic field, with $\ddot{I}{}_H$ and $\ddot{I}{}_0$ 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 $MR=f[H/(n_T\ddot{I}{}_0)]$, 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 bandstructure. 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 BaFe${}_2$(As${}_{1-x}$P${}_x$)${}_2$ to settle a long-standing debate on the scaling behavior of the normal-state magnetoresistance of a superconductor, namely, $MR$ ~ $ta{n}^2\hat{I}{¸}_H$, where $\hat{I}{¸}_H$ 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.