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Semi-magnetic topological insulators have garnered significant interest for their unique electronic properties, including the emergent half-quantized linear Hall effect. However, nonlinear Hall effects in these materials remain unexplored. This work systematically investigates the nonlinear Hall effect in semi-magnetic topological insulators and explores its dependence on the orientation of the magnetic moment in the magnetic layers. Using both analytical and numerical methods, we demonstrate that the nonlinear Hall conductance is more sensitive to the horizontal components of the magnetic moment compared to the linear Hall conductance, which predominantly depends on the vertical component of the magnetic moment. Our results reveal that the nonlinear Hall conductance can serve as a sensitive probe to detect changes in the orientation of the magnetic moment in experiments. Specifically, we show that the nonlinear Hall effect is governed by the Berry dipole moment, whose magnitude and direction vary with the tilt of the magnetic moment, offering a unique signature of its orientation. This work highlights the potential for using both linear and nonlinear Hall effects to map the direction of the magnetic moment in semi-magnetic topological insulators. Besides, the measurement of the nonlinear Hall effect can be directly implemented using existing experimental setups, without the need for additional modifications. The findings provide insights into the quantum transport behavior of the semi-magnetic topological insulator and pave the way for new experimental techniques to manipulate and probe their magnetic properties.
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图 1 半磁性拓扑绝缘体的磁矩方向可以通过施加外磁场进行调控 (a) 在垂直磁场$ B_z $的作用下, 磁矩朝向z方向. (b) 加入面内磁场后, 磁矩会沿着磁场方向发生倾斜. (c) 球坐标下的磁矩方向表示, 其中ϕ为极角, θ为方位角
Fig. 1. In the semi-magnetic topological insulator, the direction of the magnetic moment can be controlled by applying an external magnetic field. (a) In the presence of a vertical magnetic field $ B_z $, the magnetic moment is tuned along the z direction. (b) In the presence of an in-plane magnetic field, the magnetic moment will tilt along the direction of the magnetic field. (c) The direction of the magnetic moment in spherical coordinates, where ϕ is the polar angle and θ is the azimuth angle.
图 2 $ \theta=0 $时, (a)霍尔电导$ \sigma_{xy} $和(b)贝里偶极矩$ D_y $关于费米面$ E_F $的依赖关系, 其中不同颜色的曲线对应于不同的极角ϕ. (c)霍尔电导$ \sigma_{xy} $和(d)贝里偶极矩$ D_y $关于费米面$ E_F $和极角ϕ的相图, 其中不同颜色分别描述了霍尔电导和贝里偶极矩的强度.
Fig. 2. (a) The Hall conductance $ \sigma_{xy} $ and (b) Berry dipole moment $ D_y $ as functions of the Fermi surface $ E_F $ when $ \theta=0 $, where different colors represent different polar angles ϕ. The phase diagrams of (c) the Hall conductance $ \sigma_{xy} $ and (d) the Berry dipole moment $ D_y $ as functions of the Fermi surface $ E_F $ and polar angle ϕ, where different colors describe the strength of Hall conductance and Berry dipole moment, respectively.
图 3 (a) 贝里偶极矩$ D_{x/y} $在$ \phi=0.2\pi $时随θ角的依赖关系. (b), (c)和(d)分别描述了贝里偶极矩$ D_{x} $, $ D_{y} $和霍尔电导$ \sigma_{xy} $随着ϕ和θ角的依赖关系, 其中不同颜色分别描述了$ D_x $, $ D_y $, 和$ \sigma_{xy} $的强度. 在该图中, 我们固定费米面为$ E_F=-0.5 $
Fig. 3. (a) The numerically calculated Berry dipole moment $ D_{x/y} $ as functions of θ with$ \phi=0.2\pi $. (b), (c) and (d) show $ D_{x} $, $ D_{y} $, and $ \sigma_{xy} $ as functions of θ and ϕ, respectively. The colors describe the strength of $ D_x $, $ D_y $, and $ \sigma_{xy} $. We fix the Fermi surface as $ E_F=-0.5 $ in the numerical calculations.
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