Theoretical evidence of the Berry-phase mechanism in anomalous Hall transport: First-principles studies of $\mathrm{Cu}{\mathrm{Cr}}_2{\mathrm{Se}}_{4-x}{\mathrm{Br}}_x$

To justify the origin of the anomalous Hall effect (AHE), it is highly desirable to have the system parameters tuned continuously. By quantitative calculations, we show that the doping-dependent sign reversal in $\mathrm{Cu}{\mathrm{Cr}}_2{\mathrm{Se}}_{4-x}{\mathrm{Br}}_x$, observed but not understood, is nothing but direct evidence for the Berry-phase mechanism of the AHE. Systematic calculations explain well the experiment data for the whole doping range where the impurity scattering rate is changed by several orders with Br substitution. Further sign change is also predicted, which may be tested by future experiments.

Figure 1

The calculated total and projected densities of states of $\mathrm{Cu}{\mathrm{Cr}}_2{\mathrm{Se}}_4$. The Fermi level is located at zero energy.

Figure 2

(Color online) The anomalous Hall conductivity ${\sigma }_{xy}$ as a function of doping $x$ in $\mathrm{Cu}{\mathrm{Cr}}_2{\mathrm{Se}}_{4-x}{\mathrm{Br}}_x$. For the theoretical results, the open circles are ${\sigma }_{xy}$ for the clean limit $(\delta =0)$ and the triangles are ${\sigma }_{xy}$ with finite $\delta$ in the Kubo formula, where the doping-dependent $\delta$ is determined from the ab initio–calculated plasma frequency (shown in the lower panel) and the experimental longitudinal resistivity (see the text part for details). The square crosses are experimental results from Ref. 12.

Figure 3

The calculated band structure of $\mathrm{Cu}{\mathrm{Cr}}_2{\mathrm{Se}}_{4-x}{\mathrm{Br}}_x$ (upper panel) and the sum of Berry curvature over the occupied bands ${\Omega }^z\left(\mathit{k}\right)={\sum }_n{f}_{nk}{\Omega }_n^z\left(\mathit{k}\right)$ for two characteristic Fermi level positions corresponding to doping $x=0.21$ and 0.37, respectively. The lower panels show the ${\Omega }^z\left(\mathit{k}\right)$ in a special plane of the BZ for different doping $x$.