Voltage-Driven Magnetization Switching via Dirac Magnetic Anisotropy and Spin-Orbit Torque in Topological-Insulator-Based Magnetic Heterostructures

Electric field control of magnetization dynamics is fundamentally and technologically important for future spintronic devices. Here, based on electric field control of both magnetic anisotropy and spin-orbit torque, two distinct methods are presented for switching the magnetization in topological-insulator– (TI) magnetic-TI hybrid systems. The magnetic anisotropy energy in magnetic TIs is formulated analytically as a function of the Fermi level, and it is confirmed that the out-of-plane magnetization is always favored for the partially occupied surface band. Also proposed is a transistorlike device with the functionality of a nonvolatile magnetic memory that uses voltage-driven writing and the (quantum) anomalous Hall effect for readout. For magnetization reversal, by using parameters of $\mathrm{Cr}$-doped $({\mathrm{Bi}}_{1-x}{\mathrm{Sb}}_x{)}_2{\mathrm{Te}}_3$, we find the estimated source-drain current density and gate voltage are on the order of ${10}^4$–${10}^5{\mathrm{A}/\mathrm{cm}}^2$ and 0.1 V, respectively, below 20 K and the writing requires no external magnetic field. Also discussed is the possibility of magnetization switching by the proposed method in TI–ferromagnetic-insulator bilayers with the magnetic proximity effect.

Figure 1

Current-induced nonequilibrium spin polarization $\mathrm{\Delta }\mathit{\mu }/d$ scaled by thickness $d$ of a ferromagnet as a function of $E_F$ for different values of $\mathrm{\Delta }$ (with $m_z=1$): (a) $x$ component $\mathrm{\Delta }{\mu }_x/d$; (b) $y$ component $\mathrm{\Delta }{\mu }_y/d$; (c) $\mathrm{\Delta }{\mu }_x/d$ and (d) $\mathrm{\Delta }{\mu }_y/d$ at $E_F=95$ meV (corresponding carrier density approximately ${10}^{12}{\mathrm{cm}}^{-2}$) as a function of $m_z$ for different values of $\mathrm{\Delta }$. In these graphs, we use $v_F=4.0\times {10}^5$ m/s, $d=10$ nm, and $E_x=0.1\mathrm{V}/\mu \mathrm{m}$. The details of the calculations are given in the text.

Figure 2

(a) Massless (dashed line) and massive (solid line) surface state dispersions at $k_y=0$ in which $E_F$ denotes the Fermi level measured from the Dirac point ($E_{\mathbf{k}s}=0$) of the original massless surface bands, $2\mathrm{\Delta }$ is the surface band gap due to an exchange interaction, and $2{\mathrm{\Delta }}_c$ is the bulk band gap. $k_F$ and $k_c$ correspond to the Fermi wave vector and the cutoff wave vector, respectively. (b) Scaled magnetic anisotropy energy $K_u/d$ as a function of $E_F$ for different values of ${\mathrm{\Delta }}_c$ and $\mathrm{\Delta }$. Red and blue lines are for ${\mathrm{\Delta }}_c=150$ meV and ${\mathrm{\Delta }}_c=100$ meV, respectively. In this plot, we use $v_F=4.0\times {10}^5$ m/s and $d=10$ nm.

Figure 3

Geometry (side view) of the FET-like device comprising a magnetic-TI film (with thickness $d$) sandwiched between a nonmagnetic TI and a dielectric attached to a top electric gate $V_G$ in which the Dirac electron system should appear on the top surface of the magnetic TI [17]. $d_D$ is the thickness of the dielectric and $l_x$ is the length of the conduction channel. $V_S$ is the voltage difference between the source electrode and the drain electrode. Current flows on the $x$-$y$ plane depicted by a yellow line that corresponds to the TI surface state. The arrows denote the initial (red) and final (blue) magnetization directions in the magnetization reversal.

Figure 4

(a) Time evolution of each component of magnetization with a pulsed source voltage ($V_S$). The numerical calculation is performed with static $V_G=0.32$ V, $V_S=1.5$ V, and $t_S=1$ ns. (b),(c) Corresponding magnetization-switching trajectories during the duration (b) from 1 to 2 and (c) from 3 to 4 in the upper panel in (a). The vertical arrows denote the initial (red) and final (blue) magnetization directions in the magnetization reversal. The green arrow on the surface of the magnetic TI (black cube) indicates the direction of the current-induced spin polarization: $\hat{\mathit{\mu }}=\mathit{\mu }/|\mathit{\mu }|$. The SOT is active from 1 (3) to the black star.

Figure 5

(a) Time evolution of each component of magnetization with a pulsed gate voltage $V_G$. The numerical calculation is performed with $V_G=0.38$ V, $V_S=0.3$ V, and $t_G=6.7$ ns. (b),(c) Corresponding magnetization-switching trajectories during the duration (b) from 1 to 2 and (c) from 2 to 3 in the upper panel in (a). The red, blue, and green arrows and the black star have the same meanings as in Fig. 4.

Figure 6

(a) Final-state diagram of $m_z$ at $V_S=1.5$ V as a function of pulse duration $t_S$ and gate voltage $V_G$. (b) Final-state diagram of $m_z$ at $V_G=0.39$ V as a function of pulse duration $t_G$ and source-drain voltage $V_S$.

Figure 7

(a) Top view of device proposed in Fig. 3 (here the gate and dielectric layers are hidden), where $l_x$ and $l_y$ are the lengths of the source-drain and Hall directions, respectively. (b) Source-drain current density $J_S$ of proposed device versus $V_S$ for different values of $V_G$. (c) Corresponding Hall voltage $|V_H|$ versus $V_S$ for different values of $V_G$. (d) Transfer characteristics obtained for different values of $V_S$, indicating the FET operation (on-off) of the proposed device. The insets show the band pictures corresponding to the on-off states. In these plots, we assume $|m_z|=1$, $d=7$ nm, and $l_x=l_y=10\mu \mathrm{m}$.

Figure 8

(a) $R_{\mathrm{bulk}/\mathrm{surface}}=n_b{d}_{\mathrm{TI}}/n_s$ as a function of $T$ for different magnetic-TI thicknesses $d_{\mathrm{TI}}$. (b) Final-state diagram at $T=20$ K as a function of pulse duration $t_G$ with $V_G=0.39$ V and source-drain voltage $V_S$. The influence of the thermal fluctuation emerges in a longer $t_G$ region with larger $V_S$.