Dissipationless Anomalous Hall Current in the Ferromagnetic Spinel CuCr$_2$Se$_{4-x}$Br$_x$

Abstract

In a ferromagnet, an applied electric field $\bf E$ invariably produces an anomalous Hall current ${\bf J}_H$ that flows perpendicular to the plane defined by $\bf E$ and $\bf M$ (the magnetization). For decades, the question whether ${\bf J}_H$ is dissipationless (independent of the scattering rate), has been keenly debated without experimental resolution. In the ferromagnetic spinel CuCr$_2$Se$_{4-x}$Br$_x$, the resistivity $\rho$ (at low temperature) may be increased 1000 fold by varying $x$(Br), without degrading the $\bf M$. We show that ${\bf J}_H/E$ (normalized per carrier, at 5 K) remains unchanged throughout. In addition to resolving the controversy experimentally, our finding has strong bearing on the generation and study of spin-Hall currents in bulk samples.