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Semi-magnetic topological insulators have received wide attention because of their unique electrical properties, including the emergent half-quantized linear Hall effect. However, nonlinear Hall effects in these materials have not been studied. In this work, the nonlinear Hall effect in semi-magnetic topological insulators is investigated, and its dependence on the orientation of the magnetic moment in the magnetic layer is explored. By using both analytical method and numerical method, it is demonstrated that the nonlinear Hall conductance is more sensitive to the horizontal component of the magnetic moment than 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, it is shown that the nonlinear Hall effect is governed by the Berry dipole moment, whose magnitude and direction vary with the tilt of the magnetic moment, thereby offering a unique signature of its orientation. The potential for using both linear and nonlinear Hall effects to map the direction of the magnetic moment in semi-magnetic topological insulators is highlighted in this work. Besides, the measurement of the nonlinear Hall effect can be directly implemented using existing experimental setups, without the need for additional modifications. The findings offer 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
Figure 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 $, the magnetic moment is tuned along the z direction; (b) in the presence of an in-plane magnetic field, the magnetic moment will tilt along the direction of the magnetic field; (c) the direction of the magnetic moment in spherical coordinates, where ϕ is the polar angle and θ is the azimuth angle.
图 2 (a), (b)$ \theta=0 $时, 霍尔电导$ \sigma_{xy} $和贝里偶极矩$ D_y $关于费米面$ E_{\mathrm{F}} $的依赖关系, 不同颜色的曲线对应不同的极角ϕ. (c), (d)霍尔电导$ \sigma_{xy} $和贝里偶极矩$ D_y $关于费米面$ E_{\mathrm{F}} $和极角ϕ的相图, 不同颜色分别表示霍尔电导和贝里偶极矩的强度
Figure 2. (a) The Hall conductance $ \sigma_{xy} $ and (b) Berry dipole moment $ D_y $ as functions of the Fermi surface $ E_{\mathrm{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_{\mathrm{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 $时随θ角的依赖关系; (b)—(d) 贝里偶极矩$ D_{x} $, $ D_{y} $和霍尔电导$ \sigma_{xy} $随着ϕ和θ角的依赖关系. 不同颜色分别表示$ D_x $, $ D_y $, 和$ \sigma_{xy} $的强度. 该图中固定费米面为$ E_{\mathrm{F}}=-0.5 $
Figure 3. (a) Numerically calculated Berry dipole moment $ D_{x/y} $ as functions of θ with$ \phi=0.2\pi $; (b)–(d) $ 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_{\mathrm{F}}=-0.5 $ in the numerical calculations.
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[1] Qi X L, Zhang S C 2011 Rev. Mod. Phys. 83 1057
Google Scholar
[2] Hasan M Z, Kane C L 2010 Rev. Mod. Phys. 82 3045
Google Scholar
[3] Bansil A, Lin H, Das T 2016 Rev. Mod. Phys. 88 021004
Google Scholar
[4] Bernevig B A, Felser C, Beidenkopf H 2022 Nature 603 41
Google Scholar
[5] Tokura Y, Yasuda K, Tsukazaki A 2019 Nat. Rev. Phys. 1 126
Google Scholar
[6] Liu J, Hesjedal T 2021 Adv. Mater. 35 2102427
Google Scholar
[7] Chang C Z, Zhang J, Feng X, Shen J, Zhang Z, Guo M, Li K, Ou Y, Wei P, Wang L L, Ji Z Q, Feng Y, Ji S, Chen X, Jia J, Dai X, Fang Z, Zhang S C, He K, Wang Y, Lu L, Ma X C, Xue Q K 2013 Science 340 167
Google Scholar
[8] Mogi M, Kawamura M, Yoshimi R, Tsukazaki A, Kozuka Y, Shirakawa N, Takahashi K S, Kawasaki M, Tokura Y 2017 Nat. Mater. 16 516
Google Scholar
[9] Mogi M, Okamura Y, Kawamura M, Yoshimi R, Yasuda K, Tsukazaki A, Takahashi K S, Morimoto T, Nagaosa N, Kawasaki M, Takahashi Y, Tokura Y 2022 Nat. Phys. 18 390
Google Scholar
[10] Chang C Z, Liu C X, MacDonald A H 2023 Rev. Mod. Phys. 95 011002
Google Scholar
[11] Qi X L, Hughes T L, Zhang S C 2008 Phys. Rev. B 78 195424
Google Scholar
[12] Essin A M, Moore J E, Vanderbilt D 2009 Phys. Rev. Lett. 102 146805
Google Scholar
[13] Varnava N, Vanderbilt D 2018 Phys. Rev. B 98 245117
Google Scholar
[14] Chen R, Li S, Sun H P, Liu Q, Zhao Y, Lu H Z, Xie X C 2021 Phys. Rev. B 103 L241409
Google Scholar
[15] Liu C, Wang Y, Li H, Wu Y, Li Y, Li J, He K, Xu Y, Zhang J, Wang Y 2020 Nat. Mater. 19 522
Google Scholar
[16] Mogi M, Kawamura M, Tsukazaki A, Yoshimi R, Takahashi K S, Kawasaki M, Tokura Y 2017 Sci. Adv. 3 eaao1669
Google Scholar
[17] Xiao D, Jiang J, Shin J H, Wang W, Wang F, Zhao Y F, Liu C, Wu W, Chan M H, Samarth N, Chang C Z 2018 Phys. Rev. Lett. 120 056801
Google Scholar
[18] Lian Z, Wang Y, Wang Y, Dong W H, Feng Y, Dong Z, Ma M, Yang S, Xu L, Li Y, Fu B, Li Y, Jiang W, Xu Y, Liu C, Zhang J, Wang Y 2025 Nature 641 70
Google Scholar
[19] Zhu K, Cheng Y, Liao M, Chong S K, Zhang D, He K, Wang K L, Chang K, Deng P 2024 Nano Lett. 24 2181
Google Scholar
[20] Du Z Z, Lu H Z, Xie X C 2021 Nat. Rev. Phys. 3 744
Google Scholar
[21] Suárez-Rodríguez M, Juan F, Souza I, Gobbi M, Casanova F, Hueso L 2025 Nat. Mater. 24 1005
Google Scholar
[22] Du L, Huang Z, Zhang J, Ye F, Dai Q, Deng H, Zhang G, Sun Z 2024 Nat. Mater. 23 1179
Google Scholar
[23] Wu F, Xu Q, Wang Q, Chu Y, Li L, Tang J, Liu J, Tian J, Ji Y, Liu L, Yuan Y, Huang Z, Zhao J, Zan X, Watanabe K, Taniguchi T, Shi D, Gu G, Xu Y, Xian L, Yang W, Du L, Zhang G 2023 Phys. Rev. Lett. 131 256201
Google Scholar
[24] Gao Y, Yang S A, Niu Q 2014 Phys. Rev. Lett. 112 166601
Google Scholar
[25] Sodemann I, Fu L 2015 Phys. Rev. Lett. 115 216806
Google Scholar
[26] Shvetsov O O, Esin V D, Timonina A V, Kolesnikov N N, Deviatov E V 2019 JETP Lett. 109 715
Google Scholar
[27] Dzsaber S, Yan X, Taupin M, Eguchi G, Prokofiev A, Shiroka T, Blaha P, Rubel O, Grefe S E, Lai H H, Si Q, Paschen S 2021 Proc. Natl. Acad. Sci. U.S.A. 118 e2013386118
Google Scholar
[28] Qin M S, Zhu P E, Ye X G, Xu W Z, Song Z H, Liang J, Liu K, Liao Z M 2021 Chin. Phys. Lett. 38 017301
Google Scholar
[29] Huang M, Wu Z, Hu J, Cai X, Li E, An L, Feng X, Ye Z, Lin N, Law K T, Wang N 2022 Natl. Sci. Rev. 10 nwac232
Google Scholar
[30] Ho S C, Chang C H, Hsieh Y C, Lo S T, Huang B, Vu T H Y, Ortix C, Chen T M 2021 Nat. Electron. 4 116
Google Scholar
[31] Shen S Q 2017 Topological Insulators (Singapore: Springer) pp207–229
[32] Wang J, Lian B, Qi X L, Zhang S C 2015 Phys. Rev. B 92 081107(R
Google Scholar
[33] Morimoto T, Furusaki A, Nagaosa N 2015 Phys. Rev. B 92 085113
Google Scholar
[34] Xiao D, Chang M C, Niu Q 2010 Rev. Mod. Phys. 82 1959
Google Scholar
[35] Li R H, Heinonen O, Burkov A, Zhang S 2021 Phys. Rev. B 103 045105
Google Scholar
[36] Nandy S, Zeng C, Tewari S 2021 Phys. Rev. B 104 205124
Google Scholar
[37] Wang Y, Zhang Z, Zhu Z G, Su G 2024 Phys. Rev. B 109 085419
Google Scholar
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