Planar Hall effect in topological materials

Ba Jia-Yan; Chen Fu-Yang; Duan Hou-Jian; Deng Ming-Xun; Wang Rui-Qiang

doi:10.7498/aps.72.20230905

Abstract

The planar Hall effect (PHE) is one of the hot topics in the field of condensed matter physics. In recent years, the PHE has received great attention especially in topological materials such as topological insulators and topological semimetals, and great progress has been made. Unlike the scenario in ordinary Hall effect, the transverse current, magnetic field, and electric field in the PHE can appear in the same plane and cannot be explained by the Lorentz force, which largely depends on the anisotropy of the magnetoresistivity. With the development of nonlinear effect in topological material, the PHE has been extended to a nonlinear regime, which has also been extensively studied experimentally. To explain the linear and nonlinear PHEs observed experimentally, various microscopic mechanisms have been proposed theoretically. In this paper, the research progress of the linear and nonlinear PHEs of topological materials is introduced theoretically and experimentally, and various extrinsic and intrinsic mechanisms leading to the linear and nonlinear PHEs are analyzed in depth. The physical mechanisms of the linear PHE mainly include the tilt of Dirac cone, magnon scattering, chiral anomaly (or chiral-anomaly-like), shift effect, and Berry curvature, whereas ones of the nonlinear PHE mainly include the nonlinear Drude term, shift effect, Berry curvature dipole, magnon scattering, chiral anomaly, and Berry-connection polarizability. In addition, the relevant problems to be solved and the future development directions are also proposed.

Keywords:

Authors and contacts

Guangdong Basic Research Center of Excellence for Structure and Fundamental Interactions of Matter, Guangdong Provincial Key Laboratory of Quantum Engineering and Quantum Materials, School of Physics, South China Normal University, Guangzhou 510006, China
Guangdong-Hong Kong Joint Laboratory of Quantum Matter, Frontier Research Institute for Physics, South China Normal University, Guangzhou 510006, China

Corresponding author: Wang Rui-Qiang, wangruiqiang@m.scnu.edu.cn

Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 11874016, 12274146, 12174121, 12104167) and the Natural Science Foundation of Guangdong Province (Grant Nos. 2021A1515010369).

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References

[1]	Hall E H 1879 Am. J. Math. 2 287 Google Scholar
[2]	Nagaosa N, Sinova J, Onoda S, MacDonald A H, Ong N P 2010 Rev. Mod. Phys. 82 1539 Google Scholar
[3]	Hall E H 1880 Philos. Mag. 10 301 Google Scholar
[4]	Karplus R, Luttinger J M 1954 Phys. Rev. 95 1154 Google Scholar
[5]	Klitzing K V, Dorda G, Pepper M 1980 Phys. Rev. Lett. 45 494 Google Scholar
[6]	König M, Wiedmann S, Brüne C, Roth A, Buhmann H, Molenkamp L W, Qi X L, Zhang S C 2007 Science 318 766 Google Scholar
[7]	Tsui D C, Stormer H L, Gossard A C 1982 Phys. Rev. Lett. 48 1559 Google Scholar
[8]	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
[9]	Fu L, Kane C L 2008 Phys. Rev. Lett. 100 096407 Google Scholar
[10]	Wang C M, Sun H P, Lu H Z, Xie X C 2017 Phys. Rev. Lett. 119 136806 Google Scholar
[11]	Zhang C, Zhang Y, Yuan X, Lu S, Zhang J, Narayan A, Liu Y, Zhang H, Ni Z, Liu R, Choi E S, Suslov A, Sanvito S, Pi L, Lu H Z, Potter A C, Xiu F 2019 Nature 565 331 Google Scholar
[12]	Tang H X, Kawakami R K, Awschalom D D, Roukes M L 2003 Phys. Rev. Lett. 90 107201 Google Scholar
[13]	Seemann K M, Freimuth F, Zhang H, Blügel S, Mokrousov Y, Bürgler D E, Schneider C M 2011 Phys. Rev. Lett. 107 086603 Google Scholar
[14]	Bowen M, Friedland K J, Herfort J, Schönherr H P, Ploog K H 2005 Phys. Rev. B 71 172401 Google Scholar
[15]	He P, Zhang S S L, Zhu D, Shi S, Heinonen O G, Vignale G, Yang H 2019 Phys. Rev. Lett. 123 016801 Google Scholar
[16]	Weng Z J, Ba J Y, Ke Y H, Duan H J, Deng M X, Wang R Q 2022 Phys. Rev. B 106 195134 Google Scholar
[17]	Wei Y W, Feng J, Weng H 2023 Phys. Rev. B 107 075131 Google Scholar
[18]	Huang Y X, Feng X, Wang H, Xiao C, Yang S A 2023 Phys. Rev. Lett. 130 126303 Google Scholar
[19]	Berry M V 1984 Proc. R. Soc. London A 392 45 Google Scholar
[20]	Pugh E M, Rostoker N 1953 Rev. Mod. Phys. 25 151 Google Scholar
[21]	Berger L 1970 Phys. Rev. B 2 4559 Google Scholar
[22]	Miyasato T, Abe N, Fujii T, Asamitsu A, Onoda S, Onose Y, Nagaosa N, Tokura Y 2007 Phys. Rev. Lett. 99 086602 Google Scholar
[23]	Deng M X, Luo W, Wang R Q, Sheng L, Xing D Y 2017 Phys. Rev. B 96 155141 Google Scholar
[24]	Jungwirth T, Sinova J, Mašek J, Kučera J, MacDonald A H 2006 Rev. Mod. Phys. 78 809 Google Scholar
[25]	Xiao D, Chang M C, Niu Q 2010 Rev. Mod. Phys. 82 1959 Google Scholar
[26]	Ge Z, Lim W L, Shen S, Zhou Y Y, Liu X, Furdyna J K, Dobrowolska M 2007 Phys. Rev. B 75 014407 Google Scholar
[27]	Yin G, Yu J X, Liu Y, Lake R K, Zang J, Wang K L 2019 Phys. Rev. Lett. 122 106602 Google Scholar
[28]	Wadehra N, Tomar R, Varma R M, Gopal R K, Singh Y, Dattagupta S, Chakraverty S 2020 Nat. Commun. 11 874 Google Scholar
[29]	Joshua A, Ruhman J, Pecker S, Altman E, Ilani S 2013 Proc. Natl. Acad. Sci. USA 110 9633 Google Scholar
[30]	Taskin A A, Legg H F, Yang F, Sasaki S, Kanai Y, Matsumoto K, Rosch A, Ando Y 2017 Nat. Commun. 8 1340 Google Scholar
[31]	Nandy S, Sharma G, Taraphder A, Tewari S 2017 Phys. Rev. Lett. 119 176804 Google Scholar
[32]	Deng M X, Duan H J, Luo W, Deng W Y, Wang R Q, Sheng L 2019 Phys. Rev. B 99 165146 Google Scholar
[33]	Deng M X, Qi G Y, Ma R, Shen R, Wang R Q, Sheng L, Xing D Y 2019 Phys. Rev. Lett. 122 036601 Google Scholar
[34]	Deng M X, Ba J Y, Ma R, Luo W, Wang R Q, Sheng L, Xing D Y 2020 Phys. Rev. Research 2 033346 Google Scholar
[35]	Li X S, Wang C, Deng M X, Duan H J, Fu P H, Wang R Q, Sheng L, Xing D Y 2019 Phys. Rev. Lett. 123 206601 Google Scholar
[36]	Chen J N, Yang Y Y, Zhou Y L, Wu Y J, Duan H J, Deng M X, Wang R Q 2022 Phys. Rev. B 105 085124 Google Scholar
[37]	Sulaev A, Zeng M, Shen S Q, Cho S K, Zhu W G, Feng Y P, Eremeev S V, Kawazoe Y, Shen L, Wang L 2015 Nano Lett. 15 2061 Google Scholar
[38]	Liu Q, Fei F, Chen B, Bo X, Wei B, Zhang S, Zhang M, Xie F, Naveed M, Wan X, Song F, Wang B 2019 Phys. Rev. B 99 155119 Google Scholar
[39]	Wang Y, Lee P A, Silevitch D M, Gomez F, Cooper S E, Ren Y, Yan J Q, Mandrus D, Rosenbaum T F, Feng Y 2020 Nat. Commun. 11 216 Google Scholar
[40]	Wu B, Pan X C, Wu W, Fei F, Chen B, Liu Q, Bu H, Cao L, Song F, Wang B 2018 Appl. Phys. Lett. 113 011902 Google Scholar
[41]	Zheng S H, Duan H J, Wang J K, Li J Y, Deng M X, Wang R Q 2020 Phys. Rev. B 101 041408 Google Scholar
[42]	Shan W Y, Lu H Z, Shen S Q 2010 New J. Phys. 12 043048 Google Scholar
[43]	Rakhmilevich D, Wang F, Zhao W, Chan M H W, Moodera J S, Liu C, Chang C Z 2018 Phys. Rev. B 98 094404 Google Scholar
[44]	Onose Y, Ideue T, Katsura H, Shiomi Y, Nagaosa N, Tokura Y 2010 Science 329 297 Google Scholar
[45]	Fu L 2009 Phys. Rev. Lett. 103 266801 Google Scholar
[46]	Nomura M, Souma S, Takayama A, Sato T, Takahashi T, Eto K, Segawa K, Ando Y 2014 Phys. Rev. B 89 045134 Google Scholar
[47]	Armitage N P, Mele E J, Vishwanath A 2018 Rev. Mod. Phys. 90 015001 Google Scholar
[48]	Hasan M Z, Xu S Y, Belopolski I, Huang S M 2017 Annu. Rev. Condens. Matter Phys. 8 289 Google Scholar
[49]	Yan B, Felser C 2017 Annu. Rev. Condens. Matter Phys. 8 337 Google Scholar
[50]	Das K, Agarwal A 2019 Phys. Rev. B 99 085405 Google Scholar
[51]	Ma D, Jiang H, Liu H, Xie X C 2019 Phys. Rev. B 99 115121 Google Scholar
[52]	Wang Z Y, Cheng X C, Wang B Z, Zhang J Y, Lu Y H, Yi C R, Niu S, Deng Y, Liu X J, Chen S, Pan J W 2021 Science 372 271 Google Scholar
[53]	Yuan X, Zhang C, Zhang Y, Yan Z, Lyu T, Zhang M, Li Z, Song C, Zhao M, Leng P, Ozerov M, Chen X, Wang N, Shi Y, Yan H, Xiu F 2020 Nat. Commun. 11 1259 Google Scholar
[54]	Puphal P, Pomjakushin V, Kanazawa N, Ukleev V, Gawryluk D J, Ma J, Naamneh M, Plumb N C, Keller L, Cubitt R, Pomjakushina E, White J S 2020 Phys. Rev. Lett. 124 017202 Google Scholar
[55]	Nielsen H B, Ninomiya M 1983 Phys. Lett. B 130 389 Google Scholar
[56]	Huang S M, Xu S Y, Belopolski I, Lee C C, Chang G, Wang B, Alidoust N, Bian G, Neupane M, Zhang C, Jia S, Bansil A, Lin H, Hasan M Z 2015 Nat. Commun. 6 7373 Google Scholar
[57]	Weng H, Fang C, Fang Z, Bernevig B A, Dai X 2015 Phys. Rev. X 5 011029 Google Scholar
[58]	Xu S Y, Belopolski I, Alidoust N, Neupane M, Bian G, Zhang C, Sankar R, Chang G, Yuan Z, Lee C C, Huang S M, Zheng H, Ma J, Sanchez D S, Wang B, Bansil A, Chou F, Shibayev P P, Lin H, Jia S, Hasan M Z 2015 Science 349 613 Google Scholar
[59]	Lv B Q, Xu N, Weng H M, Ma J Z, Richard P, Huang X C, Zhao L X, Chen G F, Matt C E, Bisti F, Strocov V N, Mesot J, Fang Z, Dai X, Qian T, Shi M, Ding H 2015 Nat. Phys. 11 724 Google Scholar
[60]	Xiong J, Kushwaha S K, Liang T, Krizan J W, Hirschberger M, Wang W, Cava R J, Ong N P 2015 Science 350 413 Google Scholar
[61]	Li C Z, Wang L X, Liu H, Wang J, Liao Z M, Yu D P 2015 Nat. Commun. 6 10137 Google Scholar
[62]	Lv Y Y, Li X, Zhang B B, Deng W Y, Yao S H, Chen Y B, Zhou J, Zhang S T, Lu M H, Zhang L, Tian M, Sheng L, Chen Y F 2017 Phys. Rev. Lett. 118 096603 Google Scholar
[63]	Goswami P, Tewari S 2013 Phys. Rev. B 88 245107 Google Scholar
[64]	Rylands C, Parhizkar A, Burkov A A, Galitski V 2021 Phys. Rev. Lett. 126 185303 Google Scholar
[65]	Burkov A 2015 Science 350 378 Google Scholar
[66]	Ali M N, Xiong J, Flynn S, Tao J, Gibson Q D, Schoop L M, Liang T, Haldolaarachchige N, Hirschberger M, Ong N P, Cava R J 2014 Nature 514 205 Google Scholar
[67]	Shekhar C, Nayak A K, Sun Y, Schmidt M, Nicklas M, Leermakers I, Zeitler U, Skourski Y, Wosnitza J, Liu Z, Chen Y, Schnelle W, Borrmann H, Grin Y, Felser C, Yan B 2015 Nat. Phys. 11 645 Google Scholar
[68]	Chen L, Chang K 2020 Phys. Rev. Lett. 125 047402 Google Scholar
[69]	Ong N P, Liang S 2021 Nat. Rev. Phys. 3 394 Google Scholar
[70]	Lei Y H, Zhou Y L, Duan H J, Deng M X, Lu Z E, Wang R Q 2021 Phys. Rev. B 104 L121117 Google Scholar
[71]	Ghosh S, Sinha D, Nandy S, Taraphder A 2020 Phys. Rev. B 102 121105 Google Scholar
[72]	Sipe J E, Shkrebtii A I 2000 Phys. Rev. B 61 5337 Google Scholar
[73]	Sodemann I, Fu L 2015 Phys. Rev. Lett. 115 216806 Google Scholar
[74]	Kang K, Li T, Sohn E, Shan J, Mak K F 2019 Nat. Mater. 18 324 Google Scholar
[75]	Xu S Y, Ma Q, Shen H, Fatemi V, Wu S, Chang T R, Chang G, Valdivia A M M, Chan C K, Gibson Q D, Zhou J, Liu Z, Watanabe K, Taniguchi T, Lin H, Cava R J, Fu L, Gedik N, Jarillo-Herrero P 2018 Nat. Phys. 14 900 Google Scholar
[76]	Du Z Z, Wang C M, Lu H Z, Xie X C 2018 Phys. Rev. Lett. 121 266601 Google Scholar
[77]	Du Z Z, Lu H Z, Xie X C 2021 Nat. Rev. Phys. 3 744 Google Scholar
[78]	He P, Zhang S S L, Zhu D, Liu Y, Wang Y, Yu J, Vignale G, Yang H 2018 Nat. Phys. 14 495 Google Scholar
[79]	Rao W, Zhou Y L, Wu Y j, Duan H J, Deng M X, Wang R Q 2021 Phys. Rev. B 103 155415 Google Scholar
[80]	Dyrda A, Barnaś J, Fert A 2020 Phys. Rev. Lett. 124 046802 Google Scholar
[81]	Zarezad A N, Barnaś J, Qaiumzadeh A, Dyrda A 2023 Phys. Status Solidi RRL 2200483 Google Scholar
[82]	Yasuda K, Tsukazaki A, Yoshimi R, Takahashi K S, Kawasaki M, Tokura Y 2016 Phys. Rev. Lett. 117 127202 Google Scholar
[83]	Yasuda K, Tsukazaki A, Yoshimi R, Kondou K, Takahashi K S, Otani Y, Kawasaki M, Tokura Y 2017 Phys. Rev. Lett. 119 137204 Google Scholar
[84]	Zyuzin A A, Hook M D, Burkov A A 2011 Phys. Rev. B 83 245428 Google Scholar
[85]	Battilomo R, Scopigno N, Ortix C 2021 Phys. Rev. Research 3 L012006 Google Scholar
[86]	Kheirabadi N, Langari A 2022 Phys. Rev. B 106 245143 Google Scholar
[87]	Nandy S, Sodemann I 2019 Phys. Rev. B 100 195117 Google Scholar
[88]	Gao Y, Yang S A, Niu Q 2014 Phys. Rev. Lett. 112 166601 Google Scholar
[89]	Wang C, Gao Y, Xiao D 2021 Phys. Rev. Lett. 127 277201 Google Scholar
[90]	Liu H, Zhao J, Huang Y X, Wu W, Sheng X L, Xiao C, Yang S A 2021 Phys. Rev. Lett. 127 277202 Google Scholar
[91]	Liu H, Zhao J, Huang Y X, Feng X, Xiao C, Wu W, Lai S, Gao W B, Yang S A 2022 Phys. Rev. B 105 045118 Google Scholar
[92]	Kumar N, Guin S N, Felser C, Shekhar C 2018 Phys. Rev. B 98 041103 Google Scholar
[93]	Singha R, Roy S, Pariari A, Satpati B, Mandal P 2018 Phys. Rev. B 98 081103 Google Scholar
[94]	Huang D, Nakamura H, Takagi H 2021 Phys. Rev. Research 3 013268 Google Scholar
[95]	Wu M, Tu D, Nie Y, Miao S, Gao W, Han Y, Zhu X, Zhou J, Ning W, Tian M 2022 Nano Lett. 22 73 Google Scholar
[96]	Kozuka Y, Isogami S, Masuda K, Miura Y, Das S, Fujioka J, Ohkubo T, Kasai S 2021 Phys. Rev. Lett. 126 236801 Google Scholar
[97]	Lao B, Liu P, Zheng X, Lu Z, et al. 2022 Phys. Rev. B 106 L220409 Google Scholar
[98]	Li L, Wu Y, Liu X, Liu J, Ruan H, Zhi Z, Zhang Y, Huang P, Ji Y, Tang C, Yang Y, Che R, Kou X 2023 Adv. Mater. 35 2207322 Google Scholar
[99]	Wang Y, Mambakkam S V, Huang Y X, Wang Y, Ji Y, Xiao C, Yang S A, Law S A, Xiao J Q 2022 Phys. Rev. B 106 155408 Google Scholar
[100]	Li L, Cao J, Cui C, Yu Z M, Yao Y 2023 Phys. Rev. B 108 085120 Google Scholar

Cited By

图 1 (a)$ {\rm{Bi}}_{2-x}{\rm{Sb}}_{x}{\rm{Te}}_{3} $的薄膜双门霍尔棒控器件简图; (b)自旋极化杂质背散射以及背散射锁定的示意图; (c)磁场下PHE和纵向磁电阻振幅随着角度的变化; (d)磁场下的PHE振幅(左轴)以及有效总载流子密度(右轴)关于门电压的依赖性^[30]

Figure 1. (a) A sketch of the $ {\rm{Bi}}_{2-x}{\rm{Sb}}_{x}{\rm{Te}}_{3} $ dual-gate Hall-bar device and the measurement configuration; (b) schematic diagram of spin-polarized impurity backscattering and backscattering locking; (c) PHE and longitudinal magnetoresistance variation with angle at magnetic field; (d) dependence of PHE amplitude (left axis) and effective total carrier density (right axis) about gate voltage under magnetic field. Image cited from Ref.[30]

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图 2 (a)和(b)展现了包含动量平方项的狄拉克锥在平面磁场下发生倾斜; (c)和(d) 展示了加入面内磁场后造成的背散射锁定的解除, 从而导致各向异性的纵向磁电阻率^[41]

Figure 2. (a) and (b) exhibit the Dirac cone containing the momentum squared term tilted under a planar magnetic field; in (c) and (d) the unlocking of the backscattering caused by the addition of an in-plane magnetic field is shown, resulting in an anisotropic longitudinal magnetoresistivity. Image cited from Ref.[41]

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图 3 (a)平面霍尔电阻(planar Hall resistivity, PHR)和(b)各向异性磁电阻(anisotropic magnetoresistivity, AMR)随着磁场与电场的角度$ \theta_{\rm{B}} $振荡; (c) 在不同化学势下, PHR和AMR的振幅与杂质势U的关系; (d) PHR和AMR的振幅与化学势μ的关系(其中$ U=50 $)^[41]

Figure 3. (a) Planar Hall resistivity (PHR) and (b) anisotropic magnetoresistivity (AMR) oscillate with the angle $ \theta_{\rm{B}} $ of the magnetic and electric fields; (c) amplitudes of PHR and AMR versus impurity potential U at different chemical potentials; (d) amplitudes of PHR and AMR versus chemical potential μ where U = 50. Image cited from Ref.[41]

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图 4 平面霍尔电阻(planar Hall resistivity, PHR)随化学势的演化图　(a)不同杂质势取值对双峰的影响; (b)不同磁场强度对双峰的影响. 插入图展示了PHR振幅随磁场强度B的变化, 当B较小时呈二次方关系, 较大时呈线性关系^[41]

Figure 4. Evolution of the double-peak structure in planar Hall resistivity (PHR) amplitude $ \Delta \rho _{xy} $ with (a) Effect of different U values on double-peaks, and (b) effect of different magnetic field strength on double-peaks. The inset of (b) shows the $ \Delta \rho _{xy} $ as a function of the magnetic field strength B. Image cited from Ref.[41]

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图 5 (a)铁磁绝缘体/拓扑绝缘体双层异质结构装置以及所选坐标系示意图, 其中$ \theta_{{\rm{B}}} $为外磁场与x 轴的夹角. 这里假设铁磁体被完全磁化(S与B 平行). (b)各向同性费米表面上的磁振子散射示意图^[16]

Figure 5. (a) Schematic diagram of the ferromagnetic insulator/topological insulator bilayer heterostructure device and the chosen coordinate system, where $ \theta_{{\rm{B}}} $ is the angle between the external magnetic field and the x-axis. We assume that the ferromagnet is fully magnetized, which leads to $ {\boldsymbol{S}} // {\boldsymbol{B}} $. (b) Schematic of the magnon scattering for isotropic Fermi surface. Image cited from Ref.[16]

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图 6 (a)在磁场B存在下, 左手性和右手性费米子填充的能谱; (b) 存在额外平行于磁场B 的电场E时左手性和右手性费米子的能谱填充图^[65]

Figure 6. (a) Energy spectra of left-handed and right-handed fermions in the presence of a magnetic field B; (b) energy spectra of left-handed and right-handed fermions in the presence of an electric field E additionally parallel to the magnetic field B. Image cited from Ref.[65]

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图 7 (a)平面霍尔效应测量装置示意图; (b)平面霍尔电导率振幅随磁场强度B 的变化(其中插入图为纵向电导率); (c), (d)$ B=5\;{\rm{T}} $ 时纵向磁电导率和平面霍尔电导率随角度$ \theta $ 的变化^[31]

Figure 7. (a) Schematic diagram of the planar Hall effect measurement device; (b) amplitude of planar Hall conductivity as a function of magnetic field (the inset is the longitudinal conductivity); (c), (d) variation of longitudinal magnetoconductivity and planar Hall conductivity with angle $ \theta $ for $ B=5\;{\rm{T}} $. Image cited from Ref.[31]

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图 8 塞曼场不存在[(a), (c)]和存在[(b), (d)]时的Weyl锥和费米子填充, 第一行和第二行分别为$ {\boldsymbol{E}=0} $和$ {\boldsymbol{E}\neq 0} $. 我们注意两个锥之间的手征化学势可以分别由(c)手征反常, 或是(d)倾斜效应产生^[70]

Figure 8. Weyl cone and fermion filling in the absence [(a), (c)] and presence [(b), (d)] of the Seeman field with $ {\boldsymbol{E}=0} $ and $ {\boldsymbol{E}\neq 0} $ in the first and second rows, respectively. We note that the chiral chemical potential between the two Weyl cones can be generated by (c) the chiral anomaly, or (d) the tilt effect, respectively. Image cited from Ref.[70]

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图 9 非线性平面霍尔效应和非线性磁电阻的测量示意图^[15]

Figure 9. Schematic illustration of the simultaneous measurements of nonlinear planar Hall effect and nonlinear magnetoresistance. Image cited from Ref.[15]

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图 10 (a)电场作用于三维拓扑绝缘体上, 横向产生电场E的二阶非线性自旋电流$ Q_{y}^{x} $; (b)外加磁场$ {\boldsymbol{B}} {/ /} {\boldsymbol{E}} $, 横向非线性自旋电流部分转换为电荷电流$ J_{y}({\boldsymbol{E}}^{2}) $, 产生非线性霍尔效应^[15]

Figure 10. (a) When an electric field E is applied to three dimensional (3D) topological insulators, a transverse nonlinear spin current $ Q_{y}^{x} $ at the second order of E is generated; (b) when an external magnetic field $ {\boldsymbol{B}} {/ /} {\boldsymbol{E}} $, the transverse nonlinear spin current is partially converted into a charge current $ J_{y}({\boldsymbol{E}}^{2}) $, giving rise to the nonlinear Hall effect. Image cited from Ref.[15]

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图 11 (a)平面磁场作用下的位移效应Dirac锥的变化示意图; (b)位移效应的自旋阀结构示意图; (c)磁场B相关的净自旋极化$ |{\boldsymbol{S}}| $^[79]

Figure 11. (a) Schematic illustration of Dirac cones of top and bottom surfaces in topological insulator thin films with shift effect induced by the in-plane magnetic field B; (b) schematic pictures of spin valve structure with shift effect; (c) the dependence of net spin polarization $ |{\boldsymbol{S}}| $ on the field strength B. Image cited from Ref.[79]

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图 12 (a) CBST/BST/InP样品的二次谐波横向电压$ V_{y}^{2\omega} $和二次谐波纵向电压$ V_{x}^{2\omega} $在$ xy $平面上的角依赖性; (b)$ {\boldsymbol{J}} {/ /} {\boldsymbol{M}} $时的$ V_{y}^{2\omega} $的起源说明^[83]

Figure 12. (a) Angular dependence of second harmonic transverse (Hall) voltage $ V_{y}^{2\omega} $ and second harmonic longitudinal voltage $ V_{x}^{2\omega} $ in $ xy $ plane for the CBST/BST/InP sample; (b) illustration of the origin of $ V_{y}^{2\omega} $ under $ {\boldsymbol{J}} {/ /} {\boldsymbol{M}} $ configuration. Image cited from Ref.[83]

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表 1 拓扑材料中的线性和非线性平面霍尔效应的主要物理机制

Table 1. Main physics mechanisms of linear and nonlinear planar Hall effect in topological materials

平面霍尔效应	物理机制	内外禀
线性	狄拉克锥倾斜	外禀
	磁振子散射
	(类)手征反常
	位移效应
	贝里曲率	内禀
非线性	非线性Drude项	外禀
	位移效应
	贝里偶极子
	磁振子散射
	手征反常
	贝里联络极化	内禀

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[1]	Hall E H 1879 Am. J. Math. 2 287 Google Scholar
[2]	Nagaosa N, Sinova J, Onoda S, MacDonald A H, Ong N P 2010 Rev. Mod. Phys. 82 1539 Google Scholar
[3]	Hall E H 1880 Philos. Mag. 10 301 Google Scholar
[4]	Karplus R, Luttinger J M 1954 Phys. Rev. 95 1154 Google Scholar
[5]	Klitzing K V, Dorda G, Pepper M 1980 Phys. Rev. Lett. 45 494 Google Scholar
[6]	König M, Wiedmann S, Brüne C, Roth A, Buhmann H, Molenkamp L W, Qi X L, Zhang S C 2007 Science 318 766 Google Scholar
[7]	Tsui D C, Stormer H L, Gossard A C 1982 Phys. Rev. Lett. 48 1559 Google Scholar
[8]	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
[9]	Fu L, Kane C L 2008 Phys. Rev. Lett. 100 096407 Google Scholar
[10]	Wang C M, Sun H P, Lu H Z, Xie X C 2017 Phys. Rev. Lett. 119 136806 Google Scholar
[11]	Zhang C, Zhang Y, Yuan X, Lu S, Zhang J, Narayan A, Liu Y, Zhang H, Ni Z, Liu R, Choi E S, Suslov A, Sanvito S, Pi L, Lu H Z, Potter A C, Xiu F 2019 Nature 565 331 Google Scholar
[12]	Tang H X, Kawakami R K, Awschalom D D, Roukes M L 2003 Phys. Rev. Lett. 90 107201 Google Scholar
[13]	Seemann K M, Freimuth F, Zhang H, Blügel S, Mokrousov Y, Bürgler D E, Schneider C M 2011 Phys. Rev. Lett. 107 086603 Google Scholar
[14]	Bowen M, Friedland K J, Herfort J, Schönherr H P, Ploog K H 2005 Phys. Rev. B 71 172401 Google Scholar
[15]	He P, Zhang S S L, Zhu D, Shi S, Heinonen O G, Vignale G, Yang H 2019 Phys. Rev. Lett. 123 016801 Google Scholar
[16]	Weng Z J, Ba J Y, Ke Y H, Duan H J, Deng M X, Wang R Q 2022 Phys. Rev. B 106 195134 Google Scholar
[17]	Wei Y W, Feng J, Weng H 2023 Phys. Rev. B 107 075131 Google Scholar
[18]	Huang Y X, Feng X, Wang H, Xiao C, Yang S A 2023 Phys. Rev. Lett. 130 126303 Google Scholar
[19]	Berry M V 1984 Proc. R. Soc. London A 392 45 Google Scholar
[20]	Pugh E M, Rostoker N 1953 Rev. Mod. Phys. 25 151 Google Scholar
[21]	Berger L 1970 Phys. Rev. B 2 4559 Google Scholar
[22]	Miyasato T, Abe N, Fujii T, Asamitsu A, Onoda S, Onose Y, Nagaosa N, Tokura Y 2007 Phys. Rev. Lett. 99 086602 Google Scholar
[23]	Deng M X, Luo W, Wang R Q, Sheng L, Xing D Y 2017 Phys. Rev. B 96 155141 Google Scholar
[24]	Jungwirth T, Sinova J, Mašek J, Kučera J, MacDonald A H 2006 Rev. Mod. Phys. 78 809 Google Scholar
[25]	Xiao D, Chang M C, Niu Q 2010 Rev. Mod. Phys. 82 1959 Google Scholar
[26]	Ge Z, Lim W L, Shen S, Zhou Y Y, Liu X, Furdyna J K, Dobrowolska M 2007 Phys. Rev. B 75 014407 Google Scholar
[27]	Yin G, Yu J X, Liu Y, Lake R K, Zang J, Wang K L 2019 Phys. Rev. Lett. 122 106602 Google Scholar
[28]	Wadehra N, Tomar R, Varma R M, Gopal R K, Singh Y, Dattagupta S, Chakraverty S 2020 Nat. Commun. 11 874 Google Scholar
[29]	Joshua A, Ruhman J, Pecker S, Altman E, Ilani S 2013 Proc. Natl. Acad. Sci. USA 110 9633 Google Scholar
[30]	Taskin A A, Legg H F, Yang F, Sasaki S, Kanai Y, Matsumoto K, Rosch A, Ando Y 2017 Nat. Commun. 8 1340 Google Scholar
[31]	Nandy S, Sharma G, Taraphder A, Tewari S 2017 Phys. Rev. Lett. 119 176804 Google Scholar
[32]	Deng M X, Duan H J, Luo W, Deng W Y, Wang R Q, Sheng L 2019 Phys. Rev. B 99 165146 Google Scholar
[33]	Deng M X, Qi G Y, Ma R, Shen R, Wang R Q, Sheng L, Xing D Y 2019 Phys. Rev. Lett. 122 036601 Google Scholar
[34]	Deng M X, Ba J Y, Ma R, Luo W, Wang R Q, Sheng L, Xing D Y 2020 Phys. Rev. Research 2 033346 Google Scholar
[35]	Li X S, Wang C, Deng M X, Duan H J, Fu P H, Wang R Q, Sheng L, Xing D Y 2019 Phys. Rev. Lett. 123 206601 Google Scholar
[36]	Chen J N, Yang Y Y, Zhou Y L, Wu Y J, Duan H J, Deng M X, Wang R Q 2022 Phys. Rev. B 105 085124 Google Scholar
[37]	Sulaev A, Zeng M, Shen S Q, Cho S K, Zhu W G, Feng Y P, Eremeev S V, Kawazoe Y, Shen L, Wang L 2015 Nano Lett. 15 2061 Google Scholar
[38]	Liu Q, Fei F, Chen B, Bo X, Wei B, Zhang S, Zhang M, Xie F, Naveed M, Wan X, Song F, Wang B 2019 Phys. Rev. B 99 155119 Google Scholar
[39]	Wang Y, Lee P A, Silevitch D M, Gomez F, Cooper S E, Ren Y, Yan J Q, Mandrus D, Rosenbaum T F, Feng Y 2020 Nat. Commun. 11 216 Google Scholar
[40]	Wu B, Pan X C, Wu W, Fei F, Chen B, Liu Q, Bu H, Cao L, Song F, Wang B 2018 Appl. Phys. Lett. 113 011902 Google Scholar
[41]	Zheng S H, Duan H J, Wang J K, Li J Y, Deng M X, Wang R Q 2020 Phys. Rev. B 101 041408 Google Scholar
[42]	Shan W Y, Lu H Z, Shen S Q 2010 New J. Phys. 12 043048 Google Scholar
[43]	Rakhmilevich D, Wang F, Zhao W, Chan M H W, Moodera J S, Liu C, Chang C Z 2018 Phys. Rev. B 98 094404 Google Scholar
[44]	Onose Y, Ideue T, Katsura H, Shiomi Y, Nagaosa N, Tokura Y 2010 Science 329 297 Google Scholar
[45]	Fu L 2009 Phys. Rev. Lett. 103 266801 Google Scholar
[46]	Nomura M, Souma S, Takayama A, Sato T, Takahashi T, Eto K, Segawa K, Ando Y 2014 Phys. Rev. B 89 045134 Google Scholar
[47]	Armitage N P, Mele E J, Vishwanath A 2018 Rev. Mod. Phys. 90 015001 Google Scholar
[48]	Hasan M Z, Xu S Y, Belopolski I, Huang S M 2017 Annu. Rev. Condens. Matter Phys. 8 289 Google Scholar
[49]	Yan B, Felser C 2017 Annu. Rev. Condens. Matter Phys. 8 337 Google Scholar
[50]	Das K, Agarwal A 2019 Phys. Rev. B 99 085405 Google Scholar
[51]	Ma D, Jiang H, Liu H, Xie X C 2019 Phys. Rev. B 99 115121 Google Scholar
[52]	Wang Z Y, Cheng X C, Wang B Z, Zhang J Y, Lu Y H, Yi C R, Niu S, Deng Y, Liu X J, Chen S, Pan J W 2021 Science 372 271 Google Scholar
[53]	Yuan X, Zhang C, Zhang Y, Yan Z, Lyu T, Zhang M, Li Z, Song C, Zhao M, Leng P, Ozerov M, Chen X, Wang N, Shi Y, Yan H, Xiu F 2020 Nat. Commun. 11 1259 Google Scholar
[54]	Puphal P, Pomjakushin V, Kanazawa N, Ukleev V, Gawryluk D J, Ma J, Naamneh M, Plumb N C, Keller L, Cubitt R, Pomjakushina E, White J S 2020 Phys. Rev. Lett. 124 017202 Google Scholar
[55]	Nielsen H B, Ninomiya M 1983 Phys. Lett. B 130 389 Google Scholar
[56]	Huang S M, Xu S Y, Belopolski I, Lee C C, Chang G, Wang B, Alidoust N, Bian G, Neupane M, Zhang C, Jia S, Bansil A, Lin H, Hasan M Z 2015 Nat. Commun. 6 7373 Google Scholar
[57]	Weng H, Fang C, Fang Z, Bernevig B A, Dai X 2015 Phys. Rev. X 5 011029 Google Scholar
[58]	Xu S Y, Belopolski I, Alidoust N, Neupane M, Bian G, Zhang C, Sankar R, Chang G, Yuan Z, Lee C C, Huang S M, Zheng H, Ma J, Sanchez D S, Wang B, Bansil A, Chou F, Shibayev P P, Lin H, Jia S, Hasan M Z 2015 Science 349 613 Google Scholar
[59]	Lv B Q, Xu N, Weng H M, Ma J Z, Richard P, Huang X C, Zhao L X, Chen G F, Matt C E, Bisti F, Strocov V N, Mesot J, Fang Z, Dai X, Qian T, Shi M, Ding H 2015 Nat. Phys. 11 724 Google Scholar
[60]	Xiong J, Kushwaha S K, Liang T, Krizan J W, Hirschberger M, Wang W, Cava R J, Ong N P 2015 Science 350 413 Google Scholar
[61]	Li C Z, Wang L X, Liu H, Wang J, Liao Z M, Yu D P 2015 Nat. Commun. 6 10137 Google Scholar
[62]	Lv Y Y, Li X, Zhang B B, Deng W Y, Yao S H, Chen Y B, Zhou J, Zhang S T, Lu M H, Zhang L, Tian M, Sheng L, Chen Y F 2017 Phys. Rev. Lett. 118 096603 Google Scholar
[63]	Goswami P, Tewari S 2013 Phys. Rev. B 88 245107 Google Scholar
[64]	Rylands C, Parhizkar A, Burkov A A, Galitski V 2021 Phys. Rev. Lett. 126 185303 Google Scholar
[65]	Burkov A 2015 Science 350 378 Google Scholar
[66]	Ali M N, Xiong J, Flynn S, Tao J, Gibson Q D, Schoop L M, Liang T, Haldolaarachchige N, Hirschberger M, Ong N P, Cava R J 2014 Nature 514 205 Google Scholar
[67]	Shekhar C, Nayak A K, Sun Y, Schmidt M, Nicklas M, Leermakers I, Zeitler U, Skourski Y, Wosnitza J, Liu Z, Chen Y, Schnelle W, Borrmann H, Grin Y, Felser C, Yan B 2015 Nat. Phys. 11 645 Google Scholar
[68]	Chen L, Chang K 2020 Phys. Rev. Lett. 125 047402 Google Scholar
[69]	Ong N P, Liang S 2021 Nat. Rev. Phys. 3 394 Google Scholar
[70]	Lei Y H, Zhou Y L, Duan H J, Deng M X, Lu Z E, Wang R Q 2021 Phys. Rev. B 104 L121117 Google Scholar
[71]	Ghosh S, Sinha D, Nandy S, Taraphder A 2020 Phys. Rev. B 102 121105 Google Scholar
[72]	Sipe J E, Shkrebtii A I 2000 Phys. Rev. B 61 5337 Google Scholar
[73]	Sodemann I, Fu L 2015 Phys. Rev. Lett. 115 216806 Google Scholar
[74]	Kang K, Li T, Sohn E, Shan J, Mak K F 2019 Nat. Mater. 18 324 Google Scholar
[75]	Xu S Y, Ma Q, Shen H, Fatemi V, Wu S, Chang T R, Chang G, Valdivia A M M, Chan C K, Gibson Q D, Zhou J, Liu Z, Watanabe K, Taniguchi T, Lin H, Cava R J, Fu L, Gedik N, Jarillo-Herrero P 2018 Nat. Phys. 14 900 Google Scholar
[76]	Du Z Z, Wang C M, Lu H Z, Xie X C 2018 Phys. Rev. Lett. 121 266601 Google Scholar
[77]	Du Z Z, Lu H Z, Xie X C 2021 Nat. Rev. Phys. 3 744 Google Scholar
[78]	He P, Zhang S S L, Zhu D, Liu Y, Wang Y, Yu J, Vignale G, Yang H 2018 Nat. Phys. 14 495 Google Scholar
[79]	Rao W, Zhou Y L, Wu Y j, Duan H J, Deng M X, Wang R Q 2021 Phys. Rev. B 103 155415 Google Scholar
[80]	Dyrda A, Barnaś J, Fert A 2020 Phys. Rev. Lett. 124 046802 Google Scholar
[81]	Zarezad A N, Barnaś J, Qaiumzadeh A, Dyrda A 2023 Phys. Status Solidi RRL 2200483 Google Scholar
[82]	Yasuda K, Tsukazaki A, Yoshimi R, Takahashi K S, Kawasaki M, Tokura Y 2016 Phys. Rev. Lett. 117 127202 Google Scholar
[83]	Yasuda K, Tsukazaki A, Yoshimi R, Kondou K, Takahashi K S, Otani Y, Kawasaki M, Tokura Y 2017 Phys. Rev. Lett. 119 137204 Google Scholar
[84]	Zyuzin A A, Hook M D, Burkov A A 2011 Phys. Rev. B 83 245428 Google Scholar
[85]	Battilomo R, Scopigno N, Ortix C 2021 Phys. Rev. Research 3 L012006 Google Scholar
[86]	Kheirabadi N, Langari A 2022 Phys. Rev. B 106 245143 Google Scholar
[87]	Nandy S, Sodemann I 2019 Phys. Rev. B 100 195117 Google Scholar
[88]	Gao Y, Yang S A, Niu Q 2014 Phys. Rev. Lett. 112 166601 Google Scholar
[89]	Wang C, Gao Y, Xiao D 2021 Phys. Rev. Lett. 127 277201 Google Scholar
[90]	Liu H, Zhao J, Huang Y X, Wu W, Sheng X L, Xiao C, Yang S A 2021 Phys. Rev. Lett. 127 277202 Google Scholar
[91]	Liu H, Zhao J, Huang Y X, Feng X, Xiao C, Wu W, Lai S, Gao W B, Yang S A 2022 Phys. Rev. B 105 045118 Google Scholar
[92]	Kumar N, Guin S N, Felser C, Shekhar C 2018 Phys. Rev. B 98 041103 Google Scholar
[93]	Singha R, Roy S, Pariari A, Satpati B, Mandal P 2018 Phys. Rev. B 98 081103 Google Scholar
[94]	Huang D, Nakamura H, Takagi H 2021 Phys. Rev. Research 3 013268 Google Scholar
[95]	Wu M, Tu D, Nie Y, Miao S, Gao W, Han Y, Zhu X, Zhou J, Ning W, Tian M 2022 Nano Lett. 22 73 Google Scholar
[96]	Kozuka Y, Isogami S, Masuda K, Miura Y, Das S, Fujioka J, Ohkubo T, Kasai S 2021 Phys. Rev. Lett. 126 236801 Google Scholar
[97]	Lao B, Liu P, Zheng X, Lu Z, et al. 2022 Phys. Rev. B 106 L220409 Google Scholar
[98]	Li L, Wu Y, Liu X, Liu J, Ruan H, Zhi Z, Zhang Y, Huang P, Ji Y, Tang C, Yang Y, Che R, Kou X 2023 Adv. Mater. 35 2207322 Google Scholar
[99]	Wang Y, Mambakkam S V, Huang Y X, Wang Y, Ji Y, Xiao C, Yang S A, Law S A, Xiao J Q 2022 Phys. Rev. B 106 155408 Google Scholar
[100]	Li L, Cao J, Cui C, Yu Z M, Yao Y 2023 Phys. Rev. B 108 085120 Google Scholar

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