1.Department of Electronic and Computing Engineering, The Hong Kong University of Science and Technology, Hong Kong 999077, China 2.Department of Physics, The Hong Kong University of Science and Technology, Hong Kong 999077, China
Fund Project:Project supported by the HK University Grant Committee (Grant Nos. ECS26200520, N_HKUST626/18, 26302118, 16305019), the HKUST ECE Start-up Fund, and the HKUST Postdoctoral Fellowship Matching Fund (Grant No. NA389)
Received Date:15 December 2020
Accepted Date:02 April 2021
Available Online:07 June 2021
Published Online:20 June 2021
Abstract:The magnetic response in a two-dimensional material has received increasing attention in recent years. The magnetic effects and related quantum transport originate from Berry curvature, which is associated with crystal symmetry and many quantum effects including electrons’ orbital magnetism, spin-orbit coupling, and magnetoelectricity. The importance of studying the magnetic response in the two-dimensional material lies in two aspects. First, the magnetic response of two-dimensional material provides a platform to investigate the coupling between the above-mentioned intrinsic quantum effects and their couplings. Second, it possesses the potential applications in energy-efficient quantum and spintronic devices. Here, we review the experimental research progress made in recent years. In particular, we focus on the research progress of the valley Hall and magnetoelectric effect, quantum non-linear Hall effect, anomalous Hall, and quantum anomalous Hall effect in two-dimensional materials such as graphene, transition-metal chalcogenides, and twisted bilayer graphene. For each session, we first introduce these phenomena and their underlying physics by using crystal symmetries and band structures. Then, we summarize the experimental results and identify unsolved problems. At last, we provide an outlook in this emerging research direction. Keywords:two-dimensional material/ orbital magnetism/ quantum effects/ Berry curvature
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3.量子非线性霍尔效应当纵向载流子垂直通过磁场或垂直于磁性材料的磁矩方向运动时, 会获得横向于磁场或者磁矩方向的速度, 在材料的两端产生电压, 这一现象被称为霍尔效应或反常霍尔效应[49]. 如图2(a)所示, 无论是霍尔效应还是反常霍尔效应, 霍尔电压与电流成正比, 故也称为线性霍尔效应. 线性霍尔效应要求系统不具备时间反演对称性. 因此, 无法在不施加磁场的情况下, 在非磁性材料中观察到线性霍尔效应[28,49,50]. 与线性霍尔效应不同, 量子非线性霍尔效应却允许时间反演对称性的存在. 2015年, Sodemann和Fu[47]首次提出了量子非线性霍尔效应. 他们认为由贝里曲率产生的布洛赫电子的反常速度可引起二阶霍尔电压. 图2(a)比较了线性霍尔效应和非线性霍尔效应. 对于非线性霍尔效应, 向材料输入交流电流时, 可在材料横向两端获得二阶霍尔电压, 并且其大小与电流强度的二次方成正比. Sodemann和Fu[47]预言二阶霍尔电压与贝里曲率偶极矩成正比. 通过分析晶体的点群结构, 可以获得允许非零贝里曲率偶极矩存在的对称性. 在三维块状材料中, 非零贝里曲率偶极矩可以在18种点群的晶体结构中存在, 这18种点群被称作回旋点群(gyrotropic point group)[51]. 在二维材料中, 非零的贝里曲率偶极矩只能存在于具有C1, C1V, C2 (需要旋转对称轴在二维材料面内) 和C2V的晶体结构中. 这些晶体结构对称性都非常低, 至多有一条镜面对称轴或二重旋转对称轴存在于二维晶面内. 依据材料的贝里曲率和晶体结构, 他们提出了在以下三类材料中可以发现量子非线性霍尔效应: 拓扑绝缘体、二维过渡金属硫族化合物以及三维外尔半金属. 相较于体电子态, 拓扑绝缘体的表面态存在富集的贝里曲率并且具有较低的晶体对称性. 单层过渡金属硫族化合物拥有强自旋轨道耦合, 并缺乏对称中心, 可产生可观的贝里曲率. 另外, 不具有对称中心的外尔半金属由于外尔点的存在, 也允许非零的贝里曲率偶极矩存在. 2018年, You等[52]通过ab initio方法计算了不同晶体结构的过渡金属硫族化合物的贝里曲率. 他们预测Td相的单层过渡金属硫族化合物可具有非零的贝里曲率耦合极矩. 而在1H和1T' 相的过渡金属硫族化合物中, 可通过施加面内应力和垂直电场破坏面内旋转对称性和空间反演对称性实现非零的贝里曲率耦合极矩. Zhang等[53]报道了类似的研究结果. 他们通过计算, 预测Td相的二碲化钨和1T' 的二碲化钼中存在量子非线性霍尔效应. 2019年Shi和Song[54]通过计算预测可以通过电场调控1T' 相过渡金属硫族化合物的贝里曲率耦合极矩. 以上计算显示, 十分有希望在过渡金属硫族化合物, 特别是Td相的二碲化钨中, 观察到量子非线性霍尔效应. 图 2 碲化钨中量子非线性霍尔效应示意图 (a)线性和非线性霍尔电压随电流的变化[47]; (b)碲化钨在不同方向上的晶体结构示意图; (c)纵向电压和非线性霍尔电压与电流施加方向的关系[27]; (d)非线性霍尔电压与材料电导率的关系. 插图表示了非线性霍尔效应的两种来源: 贝里曲率和电子偏散射输运[27] Figure2. Illustration of the quantum nonlinear Hall effect: (a) Dependence of linear and non-linear Hall voltage on applied currents[47]; (b) crystal structure of WTe2; (c) angular dependence of longitudinal voltage and non-linear Hall voltage[27]; (d) relationship between nonlinear Hall voltage and conductance. The inset shows two origins of nonlinear Hall voltage: Intrinsic Berry curvature and skew scattering[27].