Thin-Film Magnetization Dynamics on the Surface of a Topological Insulator

We theoretically study the magnetization dynamics of a thin ferromagnetic film exchange coupled with a surface of a strong three-dimensional topological insulator. We focus on the role of electronic zero modes imprinted by domain walls (DWs) or other topological textures in the magnetic film. Thermodynamically reciprocal hydrodynamic equations of motion are derived for the DW responding to electronic spin torques, on the one hand, and fictitious electromotive forces in the electronic chiral mode fomented by the DW, on the other. An experimental realization illustrating this physics is proposed based on a ferromagnetic strip, which cuts the topological insulator surface into two gapless regions. In the presence of a ferromagnetic DW, a chiral mode transverse to the magnetic strip acts as a dissipative interconnect, which is itself a dynamic object that controls (and, inversely, responds to) the magnetization dynamics.

Figure 1

Schematic of a DW in a ferromagnetic strip with an out-of-plane easy ($z$) axis anisotropy, deposited on the surface of a TI. The DW (of width ${\lambda }_{\mathrm{dw}}$) is parametrized, according to Eq. (2), by two soft dynamic coordinates: its position $x_{\mathrm{dw}}(t)$ and azimuthal angle ${\varphi }_{\mathrm{dw}}(t)$. At the DW position, $x_{\mathrm{dw}}$, the magnetization $\mathbf{m}$ lies fully in the $xy$ plane (forming angle ${\varphi }_{\mathrm{dw}}$ with the $x$ axis). A chiral electron mode (of width $\xi \ll {\lambda }_{\mathrm{dw}}$) formed in the TI under the DW carries transport current $I_{\mathrm{dw}}$ at its exit point, which is governed by the voltage $V_y$ applied to the TI surface at its entrance and the fictitious electromotive force generated by the DW dynamics along its length. An Onsager-reciprocal spin torque affects DW dynamics in the presence of $I_{\mathrm{dw}}$.

Figure 2

The Onsager reciprocity relates voltage-induced DW dynamics (via spin torques) in the top panel [Eq. (14)] to the magnetization-dynamics-generated current (via fictitious electromotive force) in the bottom panel [Eq. (16)]. Note that the DWs in the bottom panel are mapped back onto their time-reversed parents in the top panel by a $\pi$ rotation in the $xy$ plane. This means that $\dot{Q}$ pumped by ${\dot{\varphi }}_{\mathrm{dw}}$ for the right chiral mode is the same in both panels. The left DW is treated as pinned (and thus magnetically inert) in our treatment. However, when the electron-electron interactions are taken into account, electrostatic charge imbalance produced by fictitious forces near one DW could induce currents also along the other DW, making such a double-DW system generally coupled.