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The Hall effect refers to the generation of a voltage in a direction perpendicular to the applied current. Since its discovery in 1879, the Hall effect family has become a huge group, and its in-depth study is an important topic in the field of condensed matter physics. The newly discovered nonlinear Hall effect is a new member of Hall effects. Unlike most of previous Hall effects, the nonlinear Hall effect does not need to break the time-reversal symmetry of the system but requires the spatial inversion asymmetry. Since 2015, the nonlinear Hall effect has been predicted and observed in several kinds of materials with a nonuniform distribution of the Berry curvature of energy bands. Experimentally, when a longitudinal alternating current (AC) electric field is applied, a transverse Hall voltage will be generated, with its amplitude proportional to the square of the driving current. Such a nonlinear Hall signal contains two components: one is an AC transverse voltage oscillating at twice the frequency of the driving current, and the other is a direct current (DC) signal converted from the injected current. Although the history of the nonlinear Hall effect is only a few years, its broad application prospects in fields of wireless communication, energy harvesting, and infrared detectors have been widely recognized. The main reason is that the frequency doubling and rectification of electrical signals via some nonlinear Hall effects are achieved by an inherent quantum property of the material - the Berry curvature dipole moment, and therefore do not have the thermal voltage thresholds and/or the transition time characteristic of semiconductor junctions/diodes. Unfortunately, the existence of the Berry curvature dipole moment has more stringent requirements for the lattice symmetry breaking of the system apart from the spatial inversion breaking, and the materials available are largely limited. This greatly reduces the chance to optimize the signal of the nonlinear Hall effect and limits the application and development of the nonlinear Hall effect. The rapid development of van der Waals stacking technology in recent years provides a brand new way to design, tailor and control the symmetry of lattice, and to prepare artificial moiré crystals with certain physical properties. Recently, both theoretical results and experimental studies on graphene superlattices and transition metal dichalcogenide superlattices have shown that artificial moiré superlattice materials can have larger Berry curvature dipole moments than those in natural non-moiré crystals, which has obvious advantages in generating and manipulating the nonlinear Hall effect. On the other hand, abundant strong correlation effects have been observed in two-dimensional superlattices. The study of the nonlinear Hall effect in two-dimensional moiré superlattices can not only give people a new understanding of the momentum space distribution of Berry curvatures, contributing to the realization of more stable topological transport, correlation insulating states and superfluidity states, but also expand the functional space of moiré superlattice materials which are promising for the design of new electronic and optoelectronic devices. This review paper firstly introduces the birth and development of the nonlinear Hall effect and discusses two mechanisms of the nonlinear Hall effect: the Berry curvature dipole moment and the disorder. Subsequently, this paper summaries some properties of two-dimensional moiré superlattices which are essential in realizing the nonlinear Hall effect: considerable Berry curvatures, symmetry breaking effects, strong correlation effects and tunable band structures. Next, this paper reviews theoretical and experimental progress of nonlinear Hall effects in graphene and transition metal dichalcogenides superlattices. Finally, the future research directions and potential applications of the nonlinear Hall effect based on moiré superlattice materials are prospected.
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Keywords:
- nonlinear Hall effect /
- moiré superlattice /
- two-dimensional materials /
- Berry curvature dipole
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图 2 二维莫尔超晶格的特性 (a) 2.6°转角MoS2的局部应变图[94]; (b)系统中平均应力大小与旋转角度的关系[94]; (c)无应力的情况下, 转角WSe2第一莫尔价带的贝里曲率分布, 白色六边形对应莫尔布里渊区[30]; (d)沿zigzag方向引入0.6%应力后的贝里曲率分布[30], 不均匀的分布可以诱导贝里曲率偶极矩的产生; (e) 1.08°转角双层石墨烯的能带结构, 蓝线标注的两个能带十分平坦[36]; (f)不同转角下能带带宽W(蓝色粗线)与库仑相互作用能U(细彩色线)比较[36]; (g)转角过渡金属硫族化合物中可调控的电子特性, 蓝色区域为费米液体(T2)态, 红色区域为奇异金属(T-linear)态, 灰色区域为绝缘态[39]
Figure 2. Characteristics of two dimensional moiré superlattices: (a) Calculated patterns of local strain for twisted MoS2 at twist angle 2.6°[94]; (b) the average (local) strain in the system as a function of twist angle[94]; (c) the Berry curvature of the top moiré valence band of twisted WSe2 without strain. The white hexagon indicates the moiré Brillouin zone[30]; (d) the Berry curvature distribution after introducing a strain strength of 0.6% along zigzag direction[30], the unbalanced distribution results in finite Berry curvature dipole; (e) the band energy E of magic angle twisted bilayer graphene calculated using an ab initio tight-binding method, the bands shown in blue are ultra-flat[36]; (f) comparison between the bandwidth W (thick blue line) and the on-site Coulomb interaction energy U (thin coloured lines for different values of κ) for different twist angles θ[36]; (g) a summary of tunable electronic properties in twisted transition metal dichalcogenides verified by different temperature dependences, blue showed Fermi liquid (T2) behaviour, red showed strange metal (T-linear) behaviour, grey showed insulating behaviour[39].
图 3 石墨烯超晶格中非线性霍尔效应的理论研究 (a) 1.05°魔角石墨烯在0.3%应力下的能带结构(红线和黑线), 无应力时的能带结构为绿色虚线[29]; (b)第一莫尔导带的贝里曲率分布[29]; (c), (d)贝里曲率偶极矩x (c)和y (d)方向的分量在莫尔布里渊区内的分布, 插图为贝里曲率和偶极矩集中区域的放大图[29]; (e)贝里曲率偶极矩对旋转角度和掺杂的依赖性, 转角接近第一个魔角时, 带边出现约200 Å的巨大贝里曲率偶极矩[22]; (f)贝里曲率偶极矩对于费米面和应力的依赖性[22]; (g)拓扑相变过程中能带边缘附近贝里曲率分布的变化, 贝里曲率偶极矩在相变过程中发生符号反转[27]
Figure 3. Theoretical studies of the nonlinear Hall effect in graphene superlattices: (a) Band structure of twisted bilayer graphene with a twist angle θ = 1.05° and uniaxial strain of 0.3% (red and black lines), the band structure for the unstrained twisted bilayer graphene is shown as green dotted lines[29]; (b) Berry curvature of the bottom moiré conduction band[29]; (c), (d) distribution of the x (c) and y (d) component of the Berry curvature dipole, insets in each panel are the enlarged regions where the corresponding Berry curvature and dipole are concentrated[29]; (e) twist angle and doping dependence of the Berry curvature dipole, a giant Berry curvature dipole of order ~200 Å appears near all the band edges for a twist angle near the first magic angle[22]; (f) evolution of the Berry curvature dipole with respect to both the Fermi energy EF and the strain ε[22]; (g) the change of the Berry curvature distribution near the band edges during the topological phase transition, Berry curvature dipole undergoes a sign reversal across the phase transition[27].
图 4 石墨烯超晶格中非线性霍尔效应的实验研究 (a)非线性霍尔电压(左轴)和纵向电导率的平方(右轴)对转角双层+双层石墨烯中垂直电场的依赖性, 能带的谷陈数在两个区域中发生变化(区域 I 具有浅绿色背景, 区域 II 具有浅蓝色背景)[20]; (b)贝里曲率偶极矩在两各区域内符号相反[20]; (c)石墨烯/氮化硼超晶格中的非线性电导率与迁移率的立方成线性关系, 表明观测到的非线性霍尔效应由无序引起[24]; (d)杂质、声子和混合偏斜散射对非线性霍尔信号的贡献对温度依赖性[23]; (e)非线性霍尔效应的相图, 虚线表示贝里曲率偶极矩和散射对非线性霍尔信号的贡献相等的位置, 红色(蓝色)区域代表偶极矩(散射)主导的区域[26]; (f)转角单层+单层石墨烯中的二次谐波霍尔电压符号随掺杂和垂直电场改变[26]
Figure 4. Experimental observations of the nonlinear Hall effect in graphene superlattices: (a) Dependence of normalized nonlinear Hall voltage (left axis) and square of longitudinal conductivity (right axis) on displacement field in twisted double bilayer graphene, the valley Chern numbers of the bands change across the two regimes (regime I with the light-green background and regime II with the light-blue background)[20]; (b) extracted Berry curvature dipole shows opposite sign for the two regimes[20]; (c) the nonlinear conductivity scales linearly with the cube of mobility in graphene/BN superlattices, indicating disorder-induced nonlinear Hall effect[24]; (d) temperature dependence of the distribution of the contributions to the nonlinear Hall signal from impurity, mixed, and phonon skew scatterings[23]; (e) phase diagram of the nonlinear Hall effect, the dashed line represents the position where the Berry curvature dipole and the scattering contribute equally to the nonlinear Hall signal, the red (blue) area represents the dipole (scattering) dominated area[26]; (f) the second harmonic Hall voltage in twisted bilayer graphene change signs when tuning the filling and displacement field[26].
图 5 过渡金属硫族化合物超晶格中非线性霍尔效应的研究 (a)转角WSe2中在半填充附近观察到巨大的非线性霍尔信号[30]; (b)非相互作用框架中理论计算的偶极矩(下图)可用于理解远离半填充位置(上图)的实验数据[30]; (c)半填充时观察到的巨大信号(黑点)可用有效质量发散模型(红线)解释[30]; (d) Dx (x分量贝里曲率偶极矩)对Vz (垂直电场)和nh (空穴填充数)依赖性, Dx在相变点附近显著增大[31]; (e)转角双层WTe2的能带结构(θ = 29.4°)[32]; (f)转角双层WTe2 (θ=29.4°)和普通双层WTe2的贝里曲率偶极矩随温度的变化[32]; (g)非线性霍尔信号在莫尔势存在(P < 0)和不存在时(P > 0)随填充因子的变化. 仅当 P ＜ 0 时, 才能在关联绝缘状态下观察到非线性霍尔信号[33]
Figure 5. Studies of the nonlinear Hall effect in transition metal dichalcogenides superlattices: (a) Second harmonic Hall voltage versus filling in twisted WSe2, a sharp peak is observed near the half-filling[30]; (b) theoretical calculated dipole in non-interacting picture (bottom panel) can be used to understand the experimental data away from half-filling (top panel)[30]; (c) theoretical fitting using the effective mass divergence formula (red line) can be used to understand the observed giant signal at half filling (black dots)[30]; (d) Vz (out of plane displacement field) and nh (number of holes per unit cell) dependence of Dx (x component Berry dipole), Dx is strongly enhanced near the phase transition point[31]; (e) the band structure of twisted bilayer WTe2 (θ = 29.4°)[32]; (f) the temperature dependence of Berry curvature dipole for twisted bilayer WTe2 (θ = 29.4°) and prefect bilayer WTe2[32]; (g) filling-factor-dependent nonlinear Hall signal for with (P < 0) and without (P > 0) moiré potential, respectively. Nonlinear anomalous Hall resets are observed at the correlated insulating states only for P < 0[33].
图 6 非线性霍尔效应的潜在研究和应用 (a)利用化学气相沉积制备具有大应力结构的莫尔超晶格阵列[157]; (b)利用非线性霍尔效应探测莫尔超晶格的相变[158]; (c)利用二维超晶格器件进行高灵敏度应力探测[159]; (d) 基于二维莫尔超晶格中非线性霍尔效应的太赫兹探测[19]
Figure 6. Potential research area and application of the nonlinear Hall effect: (a) Fabrication of moiré superlattice arrays with large stress structures by chemical vapor deposition[157]; (b) probing the phase transiton of moiré superlattice using nonlinear Hall effect[158]; (c) high-sensitivity strain detection using two-dimensional superlattice devices[159]; (d) terahertz detection based on nonlinear Hall effect in moiré superlattice[19].
表 1 非线性霍尔效应两种机制的比较
Table 1. Comparison between dipole- and disorder-induced nonlinear Hall effect.
机制 对称性要求*(二维体系) 信号方向 标度关系 贝里曲率偶极矩导致 C1, C1v 仅在霍尔方向 $ {V}_{\perp }^{2 w}/{\left({V}_{/ /}^{w}\right)}^{2}\propto {\sigma }_{xx}^{0} $ 无序导致 边跳作用 C1, C1v, C3, C3h, C3v, D3h, D3 各个方向都有 $ {V}_{\perp }^{2 w}/{\left({V}_{/ /}^{w}\right)}^{2}\propto {\sigma }_{xx}^{2} $ 斜散射 注: *表示此处旋转轴为 z 轴, 镜面 v 代表 yz 或 xz 平面, 镜面 h 代表 xy 平面. 
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表 2 不同材料系统中非线性霍尔效应实验总结. 最大V 2w为实验中测得的最大二倍频输出, V w, I w为此时输入电压、电流, 非线性霍尔效应强度可由$V^{2w}/(V^w)^2$或$V^{2w}/(I^w)^2$反映
Table 2. Nonlinear Hall effect observed in different systems. V 2w is the observed highest second harmonic signal, V w and I w are the input voltage and current at this time, respectively. The strength of the nonlinear Hall effect can be determined by $V^{2w}/(V^w)^2$ or $V^{2w}/(I^w)^2$.
体系 维度 主导机制 温度/K 最大
V 2w/VVw/V Iw/A $V^{2w}/(V^w)^2 $$ /\rm V^{-1}$ $V^{2w}/(I^w)^2 $$ /\rm (V{\cdot}A^{-2})$ 双层WTe2[12] 2 贝里曲率偶极矩 10—100 $ 2 \times {10}^{-4} $ $ {1 \times 10}^{-2} $ $ {1 \times 10}^{-6} $ 2 $ {2 \times 10}^{8} $ 多层WTe2[13] 2 贝里曲率偶极矩& 斜散射 1.8—100 $ 2.5 \times {10}^{-5} $ $ 7 \times {10}^{-4} $ — 51 — 多层WTe2[86] 2 贝里曲率偶极矩 80 $ 9 \times {10}^{-6} $ — $ 8 \times {10}^{-6} $ $ 5 \times {10}^{-2} $ $ {1.4 \times 10}^{5} $ 双层MoTe2[87] 2 贝里曲率偶极矩& 斜散射 1.6—100 $ 1.3 \times {10}^{-4} $ — $ 9.7 \times {10}^{-5} $ $ 2 \times {10}^{-3} $ $ {1.4 \times 10}^{4} $ Bi2Se3[79] 2 斜散射 2—200 $ 1.5 \times {10}^{-5} $ — $ 1.5 \times {10}^{-3} $ — 6.7 LaAlO3/SrTiO3
异质结[93]2 贝里曲率偶极矩 1.5 $ 1.2 \times {10}^{-4} $ — $ 2 \times {10}^{-4} $ — $ {3 \times 10}^{3} $ WTe2 (面内
直流电场中)[82]2 贝里曲率偶极矩 5—286 $ 8 \times {10}^{-6} $ — $ 5 \times {10}^{-5} $ — $ {3.2 \times 10}^{3} $ 有应力的WSe2[84] 2 贝里曲率偶极矩 50—140 $ 1.2 \times {10}^{-5} $ — $ 4.5 \times {10}^{-6} $ — $ {5.9 \times 10}^{5} $ 波纹状graphene[85] 2 贝里曲率偶极矩 1.5—15 $ 1.2 \times {10}^{-6} $ — $ 1.2 \times {10}^{-7} $ — $ {8.3 \times 10}^{7} $ Twisted double bilayer graphene[20] 2 贝里曲率偶极矩 1.5—25 $ 4 \times {10}^{-5} $ $ 1.3 \times {10}^{-4} $ $ 8 \times {10}^{-8} $ $ 2 \times {10}^{3} $ $ {6.3 \times 10}^{9} $ Graphene/BN
超晶格[24]2 斜散射 1.6—120 $ 1.3 \times {10}^{-4} $ $ 9 \times {10}^{-3} $ $ 5 \times {10}^{-6} $ 1.6 $ {5.2 \times 10}^{6} $ Twisted bilayer graphene[23] 2 斜散射 1.7—80 $ 1 \times {10}^{-3} $ $ 6 \times {10}^{-3} $ $ 1 \times {10}^{-6} $ $ 27 $ $ {1 \times 10}^{9} $ Twisted double bilayer graphene[21] 2 贝里曲率偶极矩& 斜散射 1.7—20 $ 2 \times {10}^{-3} $ — $ 1 \times {10}^{-6} $ — $ {2 \times 10}^{9} $ Twisted bilayer graphene[26] 2 贝里曲率偶极矩 1.5—80 $ 2.3 \times {10}^{-6} $ $ 6.8 \times {10}^{-4} $ $ 1 \times {10}^{-7} $ $ 5 \times {10}^{2} $ $ {2.3 \times 10}^{8} $ Twisted WSe2[30] 2 贝里曲率偶极矩 1.5-30 $ 2 \times {10}^{-2} $ $ 4 \times {10}^{-3} $ $ 5 \times {10}^{-11} $ $ 1.2 \times {10}^{3} $ $ {8 \times 10}^{18} $ WTe2/WSe2
超晶格[33]2 贝里曲率偶极矩 30—100 $ 1.5 \times {10}^{-3} $ — $ 1 \times {10}^{-6} $ — $ {1.5 \times 10}^{9} $ 多层MoTe2[71] 3 无序散射 2—40 $ 4 \times {10}^{-7} $ $ 2 \times {10}^{-3} $ — $ {10}^{-1} $ — WTe2块材[88] 3 贝里曲率偶极矩或无序散射 1.4—4.2 $ 1.8 \times {10}^{-6} $ — $ 4 \times {10}^{-3} $ — $ {1.1 \times 10}^{-1} $ Cd3As2[88] 3 贝里曲率偶极矩或无序散射 1.4-4.2 $ 7.5 \times {10}^{-7} $ — $ 3.5 \times {10}^{-3} $ — $ {6.1 \times 10}^{-2} $ NbP (Pt电极)[89] 3 无序散射 300—350 $ 9 \times {10}^{-5} $ — $ 5 \times {10}^{-5} $ — $ {3.6 \times 10}^{4} $ TaIrTe4[72] 3 贝里曲率偶极矩& 无序散射 2—300 $ 1 \times {10}^{-4} $ — $ 6 \times {10}^{-4} $ — $ {2.8 \times 10}^{2} $ Ce3Bi4Pd3[90] 3 贝里曲率偶极矩 0.4—4 $ 8 \times {10}^{-7} $ — $ 1 \times {10}^{-2} $ — $ {8 \times 10}^{-3} $ (Pb1–xSnx)1–yInyTe[91] 3 贝里曲率偶极矩 3—40 $ 4 \times {10}^{-8} $ — $ 6 \times {10}^{-5} $ — 11 Pb1–xSnxTe[74] 3 贝里曲率偶极矩 5—300 $ 1 \times {10}^{-3} $ — $ 3 \times {10}^{-5} $ — $ {1.1 \times 10}^{6} $ ZrTe5[75] 3 贝里曲率偶极矩 2—100 $ 1 \times {10}^{-5} $ $ 1.1 \times {10}^{-2} $ — $ 8 \times {10}^{-2} $ — BaMnSb2[76] 3 贝里曲率偶极矩 100—400 $ 4 \times {10}^{-4} $ — $ 2 \times {10}^{-4} $ — $ {1 \times 10}^{4} $ α-(BEDT-TTF)2I3[92] 3 贝里曲率偶极矩 4.2—40 $ 1.3 \times {10}^{-6} $ — $ 1 \times {10}^{-3} $ — 1.3
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