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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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