Topological charge pumping effect by the magnetization dynamics on the surface of three-dimensional topological insulators

We discuss a current dynamics on the surface of a three-dimensional topological insulator induced by magnetization precession of an attached ferromagnet. It is found that the magnetization dynamics generates a direct charge current when the precession axis is within the surface plane. This rectification effect is due to a quantum anomaly and is topologically protected. The robustness of the rectification effect against first-varying exchange field and impurities is confirmed by explicit calculation.

Figure 1

A ferromagnet (FM) is attached on the surface of the TI. The direction of the magnetization $\widehat{\mathbf{M}}$ of the FM is externally controlled. The current is pumped on the surface of the TI by the rotation of the magnetization. The precession axis is assumed in the $y$-$z$ plane.

Figure 2

The current on the surface of the TI induced by the rotating exchange field for $\phi =\pi /2,\pi /2+0.2$ and $r_0=1/\sqrt{2}$. The straight line denotes the full-wave rectification given by Eqs. (10). The dashed line represents the current when the rotation axis of the exchange field deviates from the $x$-$y$ plane. The abrupt change occurs at $\widehat{M_z}=0$ and is smoothed by the high-frequency mode in the short time period $\delta t\sim \hslash /\Delta$.

Figure 3

The current for $\phi =\pi /4,\pi /4+0.05,0$ and $r_0=1/\sqrt{2}$. For $\phi =\pi /4$, the $z$ component of the rotating exchange field $\widehat{M_z}$ touches zero. For $\phi >\pi /4$ the abrupt change of the current appears when $\widehat{M_z}=0$.

Figure 4

The diagrams that contribute the current in the first order in $A$ and $m_2$.