转角石墨烯体系的拓扑特性和轨道磁性

刘健鹏; 戴希

doi:10.7498/aps.69.20200506

摘要

本文介绍了转角双层石墨烯和多层石墨烯中的电子结构、拓扑性质以及轨道磁性. 在转角双层石墨烯中, 由于两层石墨烯之间的相对旋转会形成具有长周期的摩尔条纹. 由转角产生的摩尔势场会在摩尔超元胞中产生方向相反的赝磁场, 与两层的石墨烯中的狄拉克电子耦合, 从而产生赝朗道能级. 而魔角石墨烯中的每个谷和自旋自由度的两条平带就等价于两个具有相反陈数的零赝朗道能级. 这样的赝朗道能级表示可以很自然地解释一系列“魔角”的来源, 也对理解魔角双层石墨烯中观测到的关联绝缘态和量子反常霍尔效应具有重要意义. 本文进一步讨论了转角多层石墨烯, 并发现转角多层石墨烯体系中普遍存在具有非平庸拓扑性质的平带. 这些拓扑平带通常具有非零的谷陈数, 并且在一定近似下可以由一个普适的规律描述. 本文还讨论了转角石墨烯体系中的拓扑平带所具有的轨道磁性. 如果时间反演对称性自发破缺, 转角石墨烯体系会处于一个谷极化的基态. 这样的谷极化基态是一个在摩尔尺度上的轨道磁性态, 在摩尔超胞中具有纳米尺度的环状电流分布. 之前的理论提出在转角双层石墨烯体系中观测到的关联绝缘态的本质就是一种净磁矩为零的“摩尔轨道反铁磁态”. 当体系的$C_{2z}$对称性被氮化硼衬底破坏时, 转角石墨烯中的谷极化基态则变成了一种“摩尔轨道铁磁态”, 它不仅具有(量子)反常霍尔效应, 也具有新奇的磁光效应和非线性光学响应.

关键词:

Abstract

We review and discuss the electronic structures, topological properties and orbital magnetism in twisted bilayer (TBG) and multilayer graphene systems. Moiré pattern is formed in twisted bilayer graphene due to the mutual twist of the two graphene layers. The moiré potential induced by the twist can generate opposite pseudo magnetic fields in the Moiré supercell, which are coupled with the Dirac fermions and generate two sets of pseudo Landau levels with opposite Chern numbers $\pm1$. The two flat bands for each valley each spin of TBG are equivalent to the two zeroth pseudo Landau levels with opposite Chern numbers and opposite sublattice polarizations. Such a pseudo-Landau-level representation has significant implications on the quantum anomalous Hall states observed at integer fillings of the flat bands in TBG at the magic angle. The origin of the magic angle can also be naturally explained by using the pseudo-Landau-level picture. We further discuss twisted multilayer graphene systems, and show that topological flat bands generally exist in the twisted multilayer graphene systems. These topological flat bands have nonzero valley Chern numbers, which can be described by a succinct formula under certain approxmations. These topological flat bands in twisted bilayer and multilayer graphene systems are associated with orbital magnetism. A valley polarized state in the twist graphene system is an orbital magnetic state with nontrivial current-loop pattern in the moiré supercell. The experimentally observed correlated insulating states at $\pm 1/2$ fillings and at charge neutrality point of magic-angle TBG can be valley polarized states, which are associated with compensating current loops and induce staggered orbital magnetizations on the moiré length scale. If $C_{2z}$ symmetry is broken due to the alignment of hexagonal boron nitride substrate, then a valley-polarized ground state would be a moiré orbital ferromagnetic state, which exhibits not only (quantum) anomalous Hall effect, but also novel magneto-optical and nonlinear optical responses.

Keywords:

作者及机构信息

刘健鹏^1,2,
戴希²

1.
上海科技大学物质科学与技术学院, 上海　201210

2.
香港科技大学物理系, 香港

通信作者: 刘健鹏, liujp@shanghaitech.edu.cn ; 戴希, daix@ust.hk

基金项目: 香港研究资助局 (批准号: GRF16300918) 资助的课题

Authors and contacts

Liu Jian-Peng^1,2,
Dai Xi²

1.
School of Physical Science and Technology, Shanghai Tech University, Shanghai 201210, China

2.
Department of Physics, Hong Kong University of Science and Technology, Hong Kong, China

Corresponding author: Liu Jian-Peng, liujp@shanghaitech.edu.cn ; Dai Xi, daix@ust.hk

Funds: Project supported by the Hong Kong Research Grants Council Project, China (Grant No. GRF16300918

文章全文

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施引文献

图 1 (a)转角双层石墨烯的摩尔条纹示意图, 插图展示两层石墨烯在不同区域层间距离的褶皱起伏; (b) 转角双层-双层石墨烯体系的示意图; (c)转角石墨烯体系的摩尔布里渊区示意图

Fig. 1. (a) Schematic illustration of the moiré pattern in twisted bilayer graphene, the inserted shows the wrinkles of the graphene for different layer distances; (b) schematic illustration of twisted double bilayer graphene system; (c) moiré Brillouin zone of twisted graphene systems.

下载: 全尺寸图片幻灯片

图 2 (a)转角双层石墨烯在魔角时的能带; (b)转角双层石墨烯中K谷两条平带的威尔逊圈 (以$2{\text{π}}$为单位)

Fig. 2. (a) Band structures of twisted bilayer graphene at the magic angle; (b)Wilson loops of the two flat bands from the K valley of twisted bilayer graphene (in $2{\text{π}}$).

下载: 全尺寸图片幻灯片

图 3 (a) 转角双层-双层石墨烯在$1.24 ^{\circ}$时的能带; (b) AB-AB堆垛的转角双层-双层石墨烯在$1.24 ^{\circ}$时K谷第一个导带陈数随转角$\theta$和垂直偏压$U_{\rm{d}}$的变化

Fig. 3. (a) Band structures of twisted double bilayer graphene at $\theta=1.24^{\circ}$; (b) the Chern number of the first conduction band for the K valley of AB-AB stacked twist-ed double bilayer graphene vs. the twist angle $\theta$ and vertical potential drop $U_{\rm{d}}$.

下载: 全尺寸图片幻灯片

图 4 魔角双层石墨烯平带对应的实空间电流密度分布　(a) K谷, $\varDelta_{\rm M}=0$; (b) $K'$谷, $\varDelta_{\rm M}=0$; (c) K谷, $\varDelta_{\rm M}=15\, {\rm{meV}}$; (d) $K'$谷, $\varDelta_{\rm M}=15\, {\rm{meV}}$, 图中黑色箭头代表电流方向, 颜色编码表示电流诱导的磁场强度, 单位为T

Fig. 4. Real-space current-density distribution contributed by the flat bands of magic-angle twisted bilayer graphene: (a) K valley, $\varDelta_{\rm M}=0$; (b) $K'$ valley, $\varDelta_{\rm M}=0$; (c) K valley, $\varDelta_{\rm M}=15\, {\rm{meV}}$; (d) $K'$ valley, $\varDelta_{\rm M}=15\, {\rm{meV}}$. The black arrows indicate the dire-ctions of the current density, and the color coding indicates the magnetic field induced by the current in unites of Tesla.

下载: 全尺寸图片幻灯片

图 5 (a)转角双层石墨烯在魔角时K谷平带贡献的轨道磁矩($M_ {\rm {orb}}$)随化学势 ($\varepsilon_{\rm{F}}$)的变化; (b)转角双层-双层石墨烯在$\theta=1.24^ {\circ}$时K谷平带贡献的轨道磁矩随化学势的变化

Fig. 5. (a) The orbital magnetization contributed by the flat bands of the K valley of twisted bilayer graphene at the magic angle; (b) the orbital magnetization contributed by the flat bands of K valley for twisted double bilayer graphene at $\theta=1.24^{\circ}$.

下载: 全尺寸图片幻灯片

表 1 由赝朗道能级图像推算出的魔角

Table 1. Magic angles derived from the pseudo Landau-level picture.

n 1 2 3 4 5

文献[4] 1.05° 0.50° 0.35° 0.24° 0.20°
(7)式 1.05° 0.52° 0.35° 0.26° 0.21°

下载: 导出CSV

map

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转角石墨烯体系的拓扑特性和轨道磁性