Scattering universality classes of side jump in the anomalous Hall effect

The anomalous Hall conductivity has an important extrinsic contribution known as the side jump contribution, which is independent of both scattering strength and disorder density. Nevertheless, we discover that side jump has strong dependence on the spin structure of the scattering potential. We propose three universality classes of scattering for the side jump contribution, having the characters of being spin independent, spin conserving, and spin flip, respectively. For each individual class, the side jump contribution takes a different unique value. When two or more classes of scattering are present, the value of side jump is no longer fixed but varies as a function of their relative disorder strength. As system control parameters such as temperature change, due to the competition between different classes of disorder scattering, the side jump Hall conductivity could flow from one class dominated limit to another class dominated limit. Our result indicates that magnon scattering plays a role distinct from normal impurity scattering and phonon scattering in the anomalous Hall effect because they belong to different scattering classes.

Figure 1

(a)–(d) are the four conductivity diagrams of side jump that correspond to the distribution function correction. $+$ and $-$ represent the upper and lower bands, respectively. $\Upsilon$ stands for the renormalized velocity vertex which is dressed by a ladder-like diagram as shown in (e).

Figure 2

Diagrams corresponding to the part of side jump from fourth-order scattering process (intrinsic skew scattering).

Figure 3

Two diagrams corresponding to skew scattering contribution.

Figure 4

${\sigma }_{\mathit{xy}}^0$ plotted as a function of the Fermi energy ${\varepsilon }_F$ for each of the three universality classes. ${\sigma }_{\mathit{xy}}^0$ is measured in units of $e^2/(4\pi )$, and ${\varepsilon }_F$ is measured in units of $\Delta$ which is half of the band gap.

Figure 5

${\sigma }_{\mathit{xy}}^0$ plotted as a function of the Fermi energy ${\varepsilon }_F$ for fixed values of $\zeta$. ${\sigma }_{\mathit{xy}}^0$ is measured in units of $e^2/(4\pi )$, and ${\varepsilon }_F$ is measured in units of $\Delta$.

Figure 6

${\sigma }_{\mathit{xy}}^0$ versus the ratio of scattering rates $\zeta$ for fixed values of Fermi energy ${\varepsilon }_F$. ${\sigma }_{\mathit{xy}}^0$ is measured in units of $e^2/(4\pi )$. The plot shows the crossover from class A dominated regime to class C dominated regime as $\zeta$ increases.