Flat bands and related novel quantum states in two-dimensional systems

Zhang Ruo-Han; Ren Hui-Ying; He Lin

doi:10.7498/aps.71.20220225

Abstract

In flat bands of two-dimensional materials, the mass of charge carriers increases dramatically and the Coulomb energy of the charge carriers can be much larger than the quenched kinetic energy. When the flat band is partially filled, electron-electron interactions can drive electrons to form exotic correlated phases, such as quantum Hall ferromagnetism, fractional quantum Hall effect, superconductivity, and quantum anomalous Hall effect. Therefore, flat bands in two-dimensional materials have attracted much attention very recently. In the past few years, the strongly correlated phenomena in flat bands have become a hot topic in community of condensed matter physics. There are several different methods, such as using a perpendicular magnetic field, introducing strained structures, and introducing a twist angle, to realize the flat bands in two-dimensional materials. In this review article, we summarize the methods to realize flat bands in two-dimensional systems and introduce the related novel electronic states when the flat band is partially filled.

Keywords:

Authors and contacts

Center for Advanced Quantum Studies, Department of Physics, Beijing Normal University, Beijing 100875, China

Corresponding author: He Lin, helin@bnu.edu.cn

Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 12141401, 11974050).

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References

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图 1 (a)电子关联示意图^[4]; (b) Hubbard近似下的金属-绝缘体相变示意图^[5]

Figure 1. (a) Schematic diagram of electronic correlation^[4]; (b) schematic diagram of metal-insulator phase transition under Hubbard approximation^[5].

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图 2 平带的实现方法　(a) 理论模拟的石墨烯在磁场中的能级(上图)及连续极限下B = 12 T时电子态密度随杂质浓度的变化(下图, 虚线表示朗道能级的位置)^[16]; (b) 从0—6 T不同磁场下的朗道能级光谱^[17] (曲线偏移以保证清晰度; 隧道设定点, ${V}_{\mathrm{B}}=350\;\mathrm{m}\mathrm{V}, I=400\;\mathrm{p}\mathrm{A}$); (c) ABC堆垛三层石墨烯的晶体结构紧密结合图(左图)和低能电子能带结构图(右图)^[56]; (d) 在Kagome晶格中诱导平带的破坏性量子干涉示意图^[68]

Figure 2. Method of introducing flat bands: (a) Calculated energy levels of a graphene in a magnetic field (figure above) and the electronic density of states as a function of impurity concentration in the continuum limit for B = 12 T (figure bellow, the dashed line indicates the position of the Landau level)^[16]; (b) Landau level spectra for various applied magnetic fields from 0–6 T (The curves are offset for clarity; tunneling set point, ${V}_{\mathrm{B}}=350\;\mathrm{m}\mathrm{V}, I=400\;\mathrm{p}\mathrm{A}$)^[17]; (c) tight-binding diagrams (figure left) and the predicted band structure (figure right) of ABC trilayer graphene^[56]; (d) schematic diagram of destructive quantum interference inducing a flat band in the Kagome lattice^[68].

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图 3 二维材料量子霍尔铁磁态研究进展　(a) 外加磁场分别为9 T (○), 25 T (□), 30 T (◇), 37 T (△), 42 T (▽), 45 T (☆)时单层石墨烯霍尔电导$ {\sigma }_{xy} $随背栅$ {V}_{\mathrm{g}} $的变化关系, 除了9 T的实验温度是30 mK外, 其他数据的实验温度均为1.4 K (左上插图为25 T时单层石墨烯的纵向电阻$ {R}_{xx} $和霍尔电阻$ {R}_{xy} $; 右下插图为在30 mK下B = 9.0 T (○), 11.5 T (五边形), 17.5 T (六边形)时狄拉克点附近的$ {\sigma }_{xy} $数据)^[14]; (b) 在$ n=-1 $朗道能级能量处(上图, –141 meV)和n = 0朗道能级未填充的劈裂峰之一能量处(下图, 7.1 meV)探测的3 nm×3 nm STS成像图 (黑色六边形为石墨烯晶格示意图; 插图为依据基态电子态在实空间的分布情况, 石墨烯朗道能级半填充时体系可能的相: 倾斜的反铁磁相和PSP相)^[90]; (c) 电荷中性点处的双端电阻随磁场和温度的变化关系(虚线显示了F相的量化螺旋边缘输运的近似极限; 插图为零朗道能级对称破缺态的边缘色散的示意图, 表现为在边缘打开一个能隙)^[93]

Figure 3. Progress in the study of quantum Hall ferromagnetic state of two-dimensional materials: (a)$ {\sigma }_{xy} $ as a function of ${V}_{\rm g}$ at different magnetic fields of 9 T (○), 25 T (□), 30 T (◇), 37 T (△), 42 T (▽), 45 T (☆), except the experimental temperature of 9 T is 30 mK, the experimental temperature of other data is 1.4 K (Left upper inset, R_xx and R_xy for the same device measured at B = 25 T; right inset, detailed $ {\sigma }_{xy} $ near the Dirac point for B = 9 T (circle), 11.5 T (pentagon), and 17.5 T (hexagon) at 30 mK)^[14]; (b) 3 nm×3 nm STS map taken at –141 meV corresponding to the energy of n = –1 Landau level (figure above) and at the energy of 7.1 meV, which corresponds to the one of the empty peak in the n = 0 Landau level (figure bellow) ( The black hexagon is a schematic diagram of the graphene lattice; the image inset shows the two possible phases of the system with the graphene Landau level half-filling which are based on the distribution of the ground-state electronic states in the real space: canted antiferromagnetic phase and PSP phase)^[90]; (c) two-terminal resistance at the charge neutral point versus magnetic field and temperature for a different contact configuration (The dashed line shows the approximate limit of quantified helical edge transport for the F-phase; the image inset shows the schematic of the edge dispersion of the zeroth Landau level broken symmetry states showing the opening of a gap at the edge)^[93].

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表 1 平带的实现方法及相关新奇量子物态

Table 1. Implementation methods of flat bands and their corresponding resulting novel quantum states.

平带实现方法		新奇量子物态
平带实现方法		量子霍尔铁磁态	分数量子霍尔态	关联绝缘态	超导态	量子反常霍尔效应	Wigner 晶体
外加强磁场		[14]	[100, 101]	—	—	—	[149, 150]
构筑应变结构		—	—	—	—	—	—
引入转角	魔角双层石墨烯	—	—	[172]	[41]	[123, 126]	—
	魔角三层石墨烯	—	—	[44]	[44, 173]	—	—
	双层-双层转角石墨烯	—	—	[45]	—	—	—
	转角过渡金属硫化物	—	—	[53]	—	—	[163, 165]
ABC堆叠三层石墨烯		[61]	—	[105]	[105]	[127]
Kagome结构		—	—	—	—	—	—

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