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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.
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Keywords:
- planar Hall effect /
- nonlinear effect /
- topological materials
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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]Fig. 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]
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]
Fig. 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]
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]Fig. 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]
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]
Fig. 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]
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]Fig. 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]
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]
Fig. 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]
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]Fig. 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]
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]Fig. 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]
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]Fig. 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]
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]Fig. 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]
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]Fig. 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]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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