Anomalous Hall effect in magnetic topological insulators: Semiclassical framework

The anomalous Hall effect (AHE) is studied on the surface of a 3D magnetic topological insulator. By applying a modified semiclassical framework, all three contributions to the AHE, the Berry curvature effect, the side-jump effect and the skew scattering effects are systematically treated, and analytical expressions for the conductivities are obtained in terms of the Fermi level, the spatial orientation of the surface magnetization and the concentration of magnetic and nonmagnetic impurities. We demonstrate that the AHE can change sign by altering the orientation of the surface magnetization, the concentration of the impurities and also the position of the Fermi level, in agreement with recent experimental observations. We show how each contribution to the AHE, or even the whole AHE, can be turned off by properly adjusting the given parameters. For example, one can turn off the anomalous hall conductivity in a system with in-plane magnetization by pushing the system into the fully metallic regime.

Figure 1

Schematical overview of the system under study: a magnetic impurity on the surface of a TI, with its magnetization in the $yz$ plane, tilted over an angle $\theta$. An electron with initial wave vector ${\mathit{k}}_e$ approaches the impurity and elastically scatters off the impurity with ${\varphi }_{{\mathit{k}}^{\prime }}$ as the polar angle for ${{\mathit{k}}^{\prime }}_e$.

Figure 2

${\mathbf{v}}_{\mathit{k}}^{m.sj}$ is shown as a function of ${\varphi }_{\mathit{k}}$ and $\theta$, with ${\vartheta }_{\mathit{k}}^m=1$. The background color shows $cos\phi$, with $\phi$ the angle between ${\mathbf{v}}_{\mathit{k}}^{m.sj}$ and ${\mathbf{v}}_{0k}$.

Figure 3

${\sigma }_{xy}^{m.sj}$ and ${\sigma }_{yx}^{m.sj}$ are plotted in terms of $\theta$ for some different values of $m$ in (a), and in terms of $m$ for some different values of $\theta$ in panel (b), respectively. In the inset of (b), ${\mathrm{\Lambda }}_{{\mathit{k}}_F}$ is plotted against $m$. As it is clear from the main window of (b), ${\sigma }_{ij}^{m.sj}$ is inherited its nonmonotonic behavior from ${\mathrm{\Lambda }}_{{\mathit{k}}_F}$.

Figure 4

Effective mean free path of electrons during different scattering events are plotted against $\theta$ for $m=1.5$ and $\mathbf{E}=E\widehat{x}$. ${\tilde{\lambda }}_{1,1}^{m.ad,s}$, ${\tilde{\lambda }}_{1,1}^{m.s,c}$, ${\tilde{\lambda }}_{1,1}^{m.a1,s}$, and ${\tilde{\lambda }}_{1,1}^{m.a2,s}$ are the effective mean free paths of electrons during a magnetic side-jump scattering, a magnetic longitudinal scattering, a conventional magnetic skew scattering and an intrinsic magnetic skew scattering, respectively.

Figure 5

${\sigma }_{ij}^{\text{tot}.m.sj}$ is plotted in terms of $m$ for some values of $\theta$ in the main panel, and in terms of $\theta$ for some different values of $m$ in the inset.

Figure 6

${\sigma }_{xy}^{\text{tot}.sj}$ is plotted in terms of $\theta$ and $m$.

Figure 7

${\sigma }_{xy}^{m.sk2}$ is plotted in terms of $\theta$ for some different values of $m$ in the main window of (a), and against $m$ for some different values of $\theta$ in the inset of this panel. ${\sigma }_{yx}^{m.sk2}$ is plotted in (b) for the same choice of parameters $\theta$ and $m$.

Figure 8

${\sigma }_{xy}^{\mathrm{tot}.sk2}$ and ${\sigma }_{yx}^{\mathrm{tot}.sk2}$ as functions of $\theta$ and $m$, are plotted in (a) and (b), respectively.

Figure 9

${\sigma }_{xy}^{\mathrm{int}.\mathrm{AHE}}[\theta ,m]$ and ${\sigma }_{yx}^{\mathrm{int}.\mathrm{AHE}}[\theta ,m]$ are plotted in (a) and (b), respectively.

Figure 10

${\sigma }_{ij}^{\mathrm{int}.\mathrm{AHE}}$ is plotted as function of $m$ for two values of $\theta =0$ and $\frac{\pi }{2}$.

Figure 11

${\sigma }_{yx}^{m.sk1}$ is plotted in terms of $\theta$ and $m$ in (a) and (b), respectively.

Figure 12

${\sigma }_{yx}^{\mathrm{tot}.sk1}\left[{\eta }^{nm}\right]$ is plotted in terms of $\theta$ and $m$ for $\nu =0.1$ and $\nu =100$ in (a) and (b), respectively.

Figure 13

${\sigma }_{xy}^{\mathrm{AHE}}$ and ${\sigma }_{yx}^{\mathrm{AHE}}$ are plotted respectively in the first and second columns, in terms of $\theta$ and $m$, for some values of $n_{\mathrm{im}}$. First, second, and third rows correspond to $n_{\mathrm{im}}=0.4$, 0.7 and 1, respectively.

Figure 14

${\sigma }_{xy}^{\mathrm{AHE}}$ is plotted in terms of $n_{\mathrm{im}}$ and $m$ for some values of $\theta$ and $\nu$. The blue part corresponds to negative values of ${\sigma }_{xy}^{\mathrm{AHE}}$ and the red part corresponds to positive values. The first, second, and third rows correspond to $\nu =10$, 0.1, and 0, respectively. Also, first, second and third column correspond to $\theta =\frac{\pi }{6}$, $\frac{\pi }{3}$, and $0.44\pi$, respectively.

Figure 15

${\sigma }_{xy}^{\mathrm{AHE}}$ and ${\sigma }_{yx}^{\mathrm{AHE}}$ are plotted in the first and the second columns, respectively. In the first row, these conductivities are plotted against $m$ for $n_{\mathrm{im}}=n_{\mathrm{inm}}=1$ and $\nu =1$, and in the second row, against $n_{\mathrm{im}}$ for $m=3$ and $\nu =1$. The red and blue curves correspond to the cases $\theta =0$ and $\frac{\pi }{2}$, respectively. It is clear that, in contrast to the case $\theta =\frac{\pi }{2}$, sign changes occur with respect to $m$ or $n_{\mathrm{im}}$ for $\theta =0$.

Figure 16

All contributions to the AHE (solid lines and the total conductivity (dashed line), plotted against $\theta$. In (a), $m=1.6$, $n_{\mathrm{im}}=n_{\mathrm{inm}}=1$, ${\eta }_1^m={\eta }_1^{nm}=0.8$ and in (b), $m=2$, $n_{\mathrm{im}}=n_{\mathrm{inm}}=1$, ${\eta }_1^m={\eta }_1^{nm}=1$.