Anomalous Hall Heat Current and Nernst Effect in the ${\mathrm{C}\mathrm{u}\mathrm{C}\mathrm{r}}_2{\mathrm{S}\mathrm{e}}_{4-x}{\mathrm{B}\mathrm{r}}_x$ Ferromagnet

In a ferromagnet, an anomalous Hall heat current, given by the off-diagonal Peltier term ${\alpha }_{xy}$, accompanies the anomalous Hall current. By combining Nernst, thermopower, and Hall experiments, we have measured how ${\alpha }_{xy}$ varies with hole density and lifetime $\tau$ in ${\mathrm{C}\mathrm{u}\mathrm{C}\mathrm{r}}_2{\mathrm{S}\mathrm{e}}_{4-x}{\mathrm{B}\mathrm{r}}_x$. At low temperatures $T$, we find that ${\alpha }_{xy}$ is independent of $\tau$, consistent with anomalous-velocity theories. Its magnitude is fixed by a microscopic geometric area $\mathcal{A}\sim 34{\AA }^2$. Our results are incompatible with some models of the Nernst effect in ferromagnets.