In-Plane Magnetization-Induced Quantum Anomalous Hall Effect

The quantum Hall effect can only be induced by an out-of-plane magnetic field for two-dimensional electron gases, and similarly, the quantum anomalous Hall effect has also usually been considered for systems with only out-of-plane magnetization. In the present work, we predict that the quantum anomalous Hall effect can be induced by in-plane magnetization that is not accompanied by any out-of-plane magnetic field. Two realistic two-dimensional systems, ${\mathrm{Bi}}_2{\mathrm{Te}}_3$ thin film with magnetic doping and HgMnTe quantum wells with shear strains, are presented and the general condition for the in-plane magnetization-induced quantum anomalous Hall effect is discussed based on the symmetry analysis. Nonetheless, an experimental setup is proposed to confirm this effect, the observation of which will pave the way to search for the quantum anomalous Hall effect in a wider range of materials.

Figure 1

In-plane magnetization-induced QAH effect in a ${\mathrm{Bi}}_2{\mathrm{Te}}_3$ film. (a) Top view of ${\mathrm{Bi}}_2{\mathrm{Te}}_3$. The blue circles (red rhombuses) represent the Te atoms and the yellow stars represent the Bi atoms. The notation Te(1), Bi(2), Te(3) show the first three atomic layers away from the top surface. The three black lines ($M_1$, $M_2$, and $M_3$) indicate the three reflection planes. (b) The band gap as a function of $g_x$ and $g_y$. The $g_x-g_y$ plane is divided in six insulating regimes separated by the three metal lines ($L_{1,2,3}$) where the band gap is closed. The $+(-)$ in each area indicates the sign of the Hall conductance. Here the parameters $g_x$ and $g_y$ are rescaled with the energy unit ${\epsilon }^{*}=v/\lambda =0.23\mathrm{eV}$ [19]. (c) The band gap in the $g_y-g_z$ plane. The red solid line indicates the band gap closing. The white dashed line corresponds to the white line in panel (b). (d) The Hall conductance is plotted versus the chemical potential for different magnitudes of the in-plane magnetization. The blue, black, and red solid lines correspond to the $y$ direction magnetization $g_y=-0.5{\epsilon }^{*}$, $-0.3{\epsilon }^{*}$, $-0.1{\epsilon }^{*}$, respectively, while the blue, black, and red dashed lines correspond to $g_y=0.5{\epsilon }^{*}$, $0.3{\epsilon }^{*}$, $0.1{\epsilon }^{*}$.

Figure 2

Experimental setup. (a) The proposed experimental configuration to confirm the in-plane magnetization-induced QAH effect. The arrow indicates the direction of magnetization. Three planes denote three reflection planes of the ${\mathrm{Bi}}_2{\mathrm{Te}}_3$ crystal. (b) The Hall conductance is plotted versus the angle of the in-plane magnetization at different Fermi energies. The red crosses, and black dashed and blue solid lines correspond to $E_f=0$, $-0.002{\epsilon }^{*}$, and $0.02{\epsilon }^{*}$, respectively. The Hall conductance oscillates between $\pm e^2/h$ with a $2\pi /3$ period whether the chemical potential is in the electron or hole band. Here we take $g_y=0.1{\epsilon }^{*}$ [21].

Figure 3

In-plane magnetization induced QAH effect in HgMnTe quantum wells. (a) The band gap as a function of $g_{2z}$ and ${ϵ}_{xz}$ with a finite $g_{1x}=8\mathrm{meV}$. (b) The band gap as a function of ${ϵ}_{xz}$ and ${ϵ}_{yz}$ with $g_{1x}=4\mathrm{meV}$, $g_{2x}=3\mathrm{meV}$. (c) Schematic plot of the experimental setup. The angle between the strain vector $\mathit{ϵ}$ (green arrow) and the magnetization vector $\mathit{m}$ (red arrow) is denoted as $\phi$. The green plane denotes the reflection plane preserved by the strain vector $\mathit{ϵ}$ while the red plane denotes the reflection plane preserved by the magnetization vector $\mathit{m}$. (d) The band gap as a function of $g_{1x}$ and ${ϵ}_{xz}$ with $g_{2z}=0$. The other parameters are taken as $M=-3\mathrm{meV}$, $B=0.85\mathrm{eV}·{\mathrm{nm}}^2$, $D=0.67\mathrm{eV}·{\mathrm{nm}}^2$, $A=0.38\mathrm{eV}·\mathrm{nm}$, $g_{1(2)y}=g_{1z}=0$, and ${ϵ}_{yz}=0$.