Novel electrical properties of moiré graphene systems

Zhang Shi-Hao; Xie Bo; Peng Ran; Liu Xiao-Qian; Lü Xin; Liu Jian-Peng

doi:10.7498/aps.72.20230120

Abstract

In this review, we discuss the electronic structures, topological properties, correlated states, nonlinear optical responses, as well as phonon and electron-phonon coupling effects of moiré graphene superlattices. First, we illustrate that topologically non-trivial flat bands and moiré orbital magnetism are ubiquitous in various twisted graphene systems. In particular, the topological flat bands of magic-angle twisted bilayer graphene can be explained from a zeroth pseudo-Landau-level picture, which can naturally explain the experimentally observed quantum anomalous Hall effect and some of the other correlated states. These topologically nontrivial flat bands may lead to nearly quantized piezoelectric response, which can be used to directly probe the valley Chern numbers in these moiré graphene systems. A simple and general chiral decomposition rule is reviewed and discussed, which can be used to predict the low-energy band dispersions of generic twisted multilayer graphene system and alternating twisted multilayer graphene system. This review further discusses nontrivial interaction effects of magic-angle TBG such as the correlated insulator states, density wave states, cascade transitions, and nematic states, and proposes nonlinear optical measurement as an experimental probe to distinguish the different “featureless” correlated states. The phonon properties and electron-phonon coupling effects are also briefly reviewed. The novel physics emerging from band-aligned graphene-insulator heterostructres is also discussed in this review. In the end, we make a summary and an outlook about the novel physical properties of moiré superlattices based on two-dimensional materials.

Keywords:

Authors and contacts

1.
School of Physical Science and Technology, ShanghaiTech University, Shanghai 201210, China
2.
Laboratory for Topological Physics, ShanghaiTech University, Shanghai 201210, China

Corresponding author: Liu Jian-Peng, liujp@shanghaitech.edu.cn

Funds: Project supported by the National Natural Science Foundation of China (Grant No. 12174257), the National Key R&D Program of China (Grant No. 2020YFA0309601), and the Start-up Grant of ShanghaiTech University, China

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图 1 (a)转角双层石墨烯和(b)转角双层-双层石墨烯体系的莫尔晶格结构的图示. (c)转角石墨烯体系的布里渊区, 其中红色和蓝色六角分别代表底部多层石墨烯和顶部多层石墨烯的布里渊区, 而小黑色六角代表着转角石墨烯超晶格的莫尔布里渊区. (d)魔角双层石墨烯的能带. (e)魔角双层石墨烯平带赝朗道能级表示的示意图^[37,45]. (f), (g)魔角双层石墨烯K谷平带贡献的实空间电流密度分布^[37], 子晶格势能分别为(f)${\varDelta} _{\rm{M}}$ = 0 meV和(g)${\varDelta} _{\rm{M}}$ = 15 meV

Figure 1. Schematic illustration of the moiré lattice structures of (a) twisted bilayer graphene and (b) twisted double bilayer graphene. (c) Brillouin zone of the twisted graphene systems: the blue and red hexagons represent the atomic Brillouin zones of the bottom and top layers, and the small black hexagon denotes the mini-Brillouin zone of the moiré superlattices. (d) Energy bands of magic-angle twisted bilayer graphene. (e) Schematic illustration of the pseudo-Landau-level representation of the flat bands in magic-angle twisted bilayer graphene (TBG)^[37,45]. The real-space current density distribution contributed by flat bands in the K valley of magic-angle TBG with the staggered sublattice potential ${\varDelta} _{\rm{M}}$ = 0 meV in (f), and ${\varDelta} _{\rm{M}}$ = 15 meV in (g)^[37]

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图 2 (a)交错转角多层石墨烯体系的晶格结构示意图; (b)手性近似下A-ABA-A体系的能带^[56], 存在两套平带, 其中实线为K能谷的能带, 虚线为$K'$能谷的能带

Figure 2. (a) Schematic illustration of the lattice structure of alternating twist multilayer graphene; (b) band structures of A-ABA-A system which include two sets of flat bands^[56]. The solid and dashed blue lines denote the bands from the K and $K'$ valleys, respectively

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图 3 (a)两层都与hBN对齐的魔角双层石墨烯y方向电极化关于应变分量$u_{xx}$的变化^[81]; (b)由价带平带(红色三角)和导带平带(蓝色圆点)贡献的压电响应$\gamma_{yxx}$关于转角θ的变化^[81], 其中实心点代表连续模型的结果, 空心点代表紧束缚模型的结果, 而水平虚线代表理想的量子化值

Figure 3. (a) Plots of polarization along y direction vs. strain $u_{xx}$ for hBN-aligned magic-angle TBG^[81]; (b) twist-angle (θ) dependence of $\gamma_{yxx}$ contributed by the valence (red triangles) and conduction (blue circles) flat bands, where the solid ones are the results from continuum model and the hollow ones are the results from tight-binding model^[81]. The horizontal-dashed lines in panel (b) mark the ideal quantized values

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图 4 A-AB堆垛转角单层-双层石墨烯的(a)压电响应和(c)谷陈数, 以及AB-BA堆垛双层-双层转角石墨烯的(b)压电响应和(d)谷陈数^[81]. 这里$U_{\rm{d}}$为位移电场. 在图(a)和(b)中, $\gamma_{yxx}$的值显示为$|4\gamma_{yxx}^0 |=1127\;\rm{pC/m}$的倍数. 空白格子代表过于接近拓扑相变而无法准确计算压电响应和陈数的区域

Figure 4. (a), (b) Piezoelectric tensor and (c), (d) valley Chern numbers of all flat bands in (a), (c) twisted monolayer-bilayer graphene and (b), (d) twisted double bilayer systems^[81]. $U_{\rm{d}}$ is displacement field here. In panels (a) and (b), the values of $\gamma_{yxx}$ are shown in the units of $|4\gamma_{yxx}^0 |=1127\;\rm{pC/m}$. The blank patches indicate points, which are too close to gap closures

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图 5 (a)魔角双层石墨烯的非相互作用能带; (b)包含高能子空间库仑势能效应的平带色散^[82]; (c)考虑库仑屏蔽效应后介电常数随波矢的变化^[82]; (d)上图为ν = 3填充处的相互作用单粒子能谱, 下图为ν = –3填充处的相互作用单粒子能谱^[82]; (e) $\nu=1$时密度波态的实空间电荷分布^[82]; (f)不同自发对称性破缺态下的非线性光学响应^[82]

Figure 5. (a) Non-interacting energy bands of magic-angle twisted bilayer graphene; (b) flat-band dispersions including remote-band Hartree-Fock potentials^[82]; (c) wave vector dependence of the effective dielectric constant^[82]; (d) single-particle excitation spectra at ν = 3 filling (upper panel) and at ν = –3 filling (lower panel)^[82]; (e) real-space distributions of charge density at ν = 1 filling^[82]; (f) nonlinear optical response of different symmetry-breaking states^[82]

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图 6 (a)转角多层石墨烯体系在半填充时自旋极化和谷极化态的竞争关系. “SP”和“VP”分别代表自旋极化态和谷极化态; (b)转角双层-双层石墨烯半填充处的实空间电荷分布^[83]; (c)—(e)转角双层-单层石墨烯体系在不同Hubbard参数和磁场下衍生出的不同陈数的绝缘态, 用红线标记^[83]

Figure 6. (a) Competition between spin polarized and valley polarized states in twisted multilayer graphene system at half filling. “SP” and “VP” stand for “spin polarized” and “valley polarized” respectively; (b) charge density distribution in real space at half filling for twisted double bilayer graphene^[83]; (c)–(e) calculated gapped states with different Chern numbers remarked by red lines under magnetic fields in the twisted bilayer-monolayer graphene system^[83]

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图 7 (a)魔角双层石墨烯的声子态密度, 蓝色线条为$0—2.4$THz的低频声子的声子态密度^[85]; (b)魔角双层石墨烯在Γ点的声子软模, 莫尔原胞用黑色六边形标注出^[85]; (c)魔角双层石墨烯在K点的声子极化, 其中$\sqrt{3}\times\sqrt{3}$超胞用黑色虚线标注^[85]; (d)魔角双层石墨烯在M点的声子模, 双倍莫尔超胞用黑色虚线标注^[85]

Figure 7. (a) Phonon density of states (DOS) of magic-angle TBG (MATBG)^[85], where the blue line shows the low-frequency DOS from 0 to 2.4 THz; (b) soft phonon modes in the MATBG at the Γ point, where the black hexagon marks the moiré primitive cell^[85]; (c) phonon polarizations at K point in the MATBG, in which the $\sqrt{3}\times\sqrt{3}$ moiré supercell are marked with dashed black lines^[85]; (d) phonon modes at M point in the MATBG, in which the double moiré supercell are marked with dashed black lines^[85]

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图 8 (a)八极矩声子模式被冻结后在魔角双层石墨烯电中性点打开能隙^[85] (b)能带间隙随声子振幅线性增加^[85]. (c)魔角双层石墨烯中八极矩声子的电声耦合强度随费米能级的变化^[85]; (d)四极矩声子被冻结后产生四极矩的电荷序^[85]

Figure 8. (a) Flat bands of magic-angle TBG with the octupolar-type phonon modes under frozen mode approximation^[85]; (b) increasing bandgap as a function of average displacement amplitudes^[85]; (c) strength of electron-phonon coupling verse Fermi level in the magic-angle TBG^[85]; (d) charge order with the quadrupolar-type phonon modes under frozen mode approximation^[85]

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图 9 (a)单层石墨烯与绝缘衬底上的长程电荷序耦合; (b)电子转移从石墨烯转移到衬底中形成的长程电荷序产生超晶格势场反作用于石墨烯的电子上, 进而在电子间相互作用的驱动下, 在狄拉克点打开能隙并大大增强在其附近的费米速度^[86]

Figure 9. (a) Coupling between graphene and long-range charge order within insulating substrate; (b) long-range charge order in the substrate with the charge transfer from graphene to substrate, can provide superlattice potential on the graphene’s electron and open a gap at the Dirac point driven bt electron interaction. And the Fermi velocity near the Dirac point is enhanced^[86]

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表 1 手性分解规则的例子. 在${{K}}_{\mu}({{K}}^{\prime}_{\mu})$点有m条色散为$E({\boldsymbol{k}})\sim{\boldsymbol{k}}^{n}$的能带, 记为$(m, n)$

Table 1. Typical cases for generic partition rules. $(m, n)$ represents that there are m energy bands with $E({\boldsymbol{k}})\sim{\boldsymbol{k}}^{n}$ dispersion at the ${{K}}_{\mu}({{K}}^{\prime}_{\mu})$ point.

手性分解	平带数量	K点能带	${{K}}^{\prime}$点能带
A-AB+A	2	(1, 1)	0
A-AB+AB	2	(1, 2)	0
A-A-A	2	(1, 1)	0
A-A-AB+AC	2	(1, 1), (1, 2)	0
AB-A-BA	2	(1, 1)	0
A-AB+A-A	4	0	0
A-ABC-A	2	/	/
A-AB+ABC-A	4	0	0

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表 2 具有非零的非线性光学响应的对称破缺相

Table 2. Three types of ordered states with non-vanishing nonlinear optical responses

序参量	对称性允许的非线性光导率分量
${\boldsymbol{\tau}}_z$	$\sigma_{xx}^{x} = -\sigma_{xy}^{y} = -\sigma_{yx}^{y} = -\sigma^{x}_{yy}$
$({\boldsymbol{\tau}}_z{\boldsymbol{\sigma}}_x, {\boldsymbol{\sigma}}_y)$	$\begin{array}{c} \sigma^{x}_{xx, x} = \sigma_{xy, x}^{y}+\sigma_{yx, x}^{y}+\sigma_{yy, x}^{x},\quad \sigma^{y}_{yy, y} = \sigma_{xx, y}^{y}+\sigma_{xy, y}^{x}+\sigma_{yx, y}^{x}, \\ \sigma^{x}_{xx, x} = -\sigma^{y}_{yy, y},\quad \sigma_{xy, x}^{y} = -\sigma_{yx, y}^{x},\quad \sigma_{yx, x}^{y} = -\sigma_{xy, y}^{x},\quad \sigma_{yy, x}^{x} = -\sigma_{xx, y}^{y}\end{array}$
${\boldsymbol{\sigma}}_z$	$\sigma_{xx}^{x} = -\sigma_{xy}^{y} = -\sigma_{yx}^{y} = -\sigma^{x}_{yy},~~\sigma_{xx}^{y} = \sigma_{xy}^{x} = \sigma_{yx}^{x} = -\sigma^{y}_{yy}$

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Vol. 72, Issue 6 - 2023-03-20

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