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When 2D materials with different lattice constants or lattice rotation angles are stacked together, a periodic moiré pattern will appear. Such moiré superlattice introduces a new two dimensional periodic potential, which can greatly change the physical properties of the original systems. Recent experimental studies of moiré superlattices formed by graphene on graphene and graphene on hexagonal boron nitride have revealed very rich strong correlation effects and topological effects due to novel states in superlattice minibands. It has been shown that flat bands in graphene-based moiré superlattice systems can host both topological states and strongly correlated states, which can be controlled by an external electric field. In bilayer graphene, ABC stacked trilayer graphene and twisted bilayer-bilayer graphene, the number of valence and conduction bands near the Dirac point and even the band topology and bandwidth can be changed by varying the stacking angle between graphene layers or the applied bias voltage. Moreover, the competition between kinetic energy and coulomb interaction depends on the bandwidth and the external electric field, and at the so-called magic angle mott insulator states and superconductivity were observed. Twisted bilayer-bilayer graphene has also been predicted to show similar intriguing properties, including electrically tunable strongly correlated insulators, superconductivity and many rich topological states. In graphene-based moiré systems, the combination of topological states and strong correlations is expected to lead to a broad range of novel phenomena that are not achievable in other material systems. Therefore, graphene moiré systems is likely to bring substantial progress to the study of topological materials. In this paper, we review theoretical and experimental investigations of the topological properties of graphene moiré superlattices, including topological domain wall states in bilayer graphene and topological effects in twisted bilayer graphene, ABC trilayer graphene and twisted double bilayer graphene. The origins of topological properties of these systems are discussed as well as topological phenomena observed in various experiments. Finally, recent near-field optical studies of the band structure and novel topological properties of graphene moiré superlattices are discussed.
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Keywords:
- graphene /
- moiré superlattice /
- topology /
- near-field optics
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图 1 (a) 石墨烯、单层过渡金属硫族化合物(TMDs)等材料的二维蜂窝晶格; (b)当单层石墨烯与h-BN基底产生相互作用, 空间反演对称性就会被破坏, 单层TMDs不具有空间反演对称性结构, 在双层石墨烯和双层TMDs中反演对称性可以通过施加z方向的电场打开或关闭; (c) 反演对称性破缺的狄拉克体系在能谷处打开能隙, 箭头表示能谷光学跃迁, 圆形箭头表示不同的圆偏振光[28]
Figure 1. (a) 2D hexagonal lattice, representing graphene, monolayer transition metal dichalcogenides (TMDs), etc; (b) In monolayer graphene, inversion symmetry is broken when monolayer graphene interacts with h-BN substrate. The monolayer TMDs have structures that lack inversion symmetry. Inversion symmetry in bilayer graphene and TMDs can be switched on/off by an electric field applied in the z-direction; (c) An energy gap is opened in Dirac systems with broken inversion symmetry. The arrows indicate interband transitions at different valleys, and the circular arrows represent different circularly polarized light[28].
图 2 (a) 双层石墨烯中剪切型畴壁的示意图与BA、鞍点(Saddle point, SP)和AB堆叠的能带结构. 红色和粉色的箭头表示束缚在畴壁上的手性拓扑模; (b), (c) 在正(负)层间偏压
${V_{\rm{i}}}$ 作用下K谷畴壁的能带结构[61]Figure 2. (a) Schematic representation of a shear domain wall in bilayer graphene and the band structure of BA, Saddle point (SP), and AB stacking. Red and magenta wavy arrows represent chiral topological modes bound to the domain wall; (b), (c) Band structure of the wall under a positive (negative) interlayer bias
${V_{\rm{i}}}$ for the K valley[61].
图 3 (a)转角双层石墨烯莫尔超晶格示意图[10]; (b)小布里渊区示意图,
${K_{\rm{s}}}$ ,${K'_{\rm{s}}}$ 和${\varGamma _{\rm{s}}}$ 代表小布里渊区中的点[10]; (c)与h-BN对齐的转角双层石墨烯中小能带处的能带结构,${\theta _{\rm{M}}} = 1.20^\circ $ [8]Figure 3. (a) The Moiré superlattice as seen in twisted bilayer graphene[10]; (b) schematic representation of the mini Brillouin zone.
${K_{\rm{s}}}$ ,${K'_{\rm{s}}}$ and${\varGamma _{\rm{s}}}$ denote points in the mini Brillouin zone[10]; (c) band structure for valley + of the twisted bilayer graphene aligned with h-BN in the mini Brillouin zone${\theta _{\rm{M}}} = 1.20^\circ $ [8].
图 4 (a) ABC TLG/h-BN的莫尔超晶格示意图, 为了图像清晰, 只显示了顶部h-BN和底部石墨烯最上层的原子[13]; (b) ABC堆叠三层石墨烯/h-BN体系示意图, 垂直电场使顶部和底部石墨烯层之间的电子能量差为
$\Delta V$ [12]; (c), (d)分别为没有和有垂直电场时的小布里渊区处的能带图; (d)垂直电场在顶部和底部石墨烯层之间产生20 mev的电位差, 导致了一个带宽减小的孤立的空穴型小能带, 增强了强关联作用, 从而生成了可调节的Mott绝缘体态[13]Figure 4. (a) Schematic of ABC TLG/h-BN Moiré superlattice. Only atoms of the top h-BN layer and the bottom graphene layer are shown for clarity[13]; (b) illustration of the ABC stacked trilayer graphene/h-BN system. A vertical electric field introduces an energy difference
$\varDelta V$ for electrons between the top and the bottom graphene layer[12]; (c), (d) energy dispersion of the two electron and hole minibands without and with a vertical electrical field, respectively. The vertical electrical field in (d) generates a potential difference of 20 meV between the top and bottom graphene layers, leading to an isolated hole minibands with strongly suppressed bandwidth. The reduced electronic bandwidth relative to the Coulomb interaction enhances the electron correlation, and leads to the tunable Mott insulator states[13].
图 5 ABC TLG/h-BN, 纵向电阻率图和不同磁场下的霍尔电阻率图 (a) T = 1.5 K时以
${V_{\rm{t}}}$ 和${V_{\rm{b}}}$ 为函数的纵向电阻率图, 箭头分别表示掺杂(n)和电位移场(D)的方向. 理论预言D < 0时空穴小能带为拓扑非平庸态(即陈数C ≠ 0), D > 0时为拓扑平庸态(C = 0); (b)在1/4填充和D = –0.5 V/nm时不同的温度下的霍尔电阻${\rho _{yx}}$ , 显示出清晰的反常霍尔效应(AH)的信号并伴随着很强的磁滞回线. 在温度T = 0.06 K时, 横向电阻$\rho _{yx}^{{\rm{AH}}} = 8\;{\rm{k}}\Omega $ 和矫顽场${H_{\rm{c}}} = 30\;{\rm{mT}}$ . 插图: 矫顽场和AH信号与温度的函数[14]Figure 5. ABC TLG/h-BN, color plot of the longitudinal resistivity and Magnetic field dependent
${\rho _{yx}}$ : (a) Longitudinal resistivity as a function of${V_{\rm{t}}}$ and${V_{\rm{b}}}$ at T = 1.5 K. The arrows show the direction of changing doping (n) and displacement field (D), respectively. It was predicted theoretically that the hole miniband is topological (Chern number C ≠ 0) for D < 0 and trivial (C = 0) for D > 0; (b) magnetic field dependent${\rho _{yx}}$ at 1/4 filling and D = –0.5 V/nm at different temperatures. The Hall resistivity displays a clear AH signal with strong ferromagnetic hysteresis. At the base temperature of T = 0.06 K, the AH signal can be as high as$\rho _{yx}^{{\rm{AH}}} = 8\;{\rm{k}}\Omega $ and the coercive field is${H_{\rm{c}}} = 30\;{\rm{mT}}$ . Inset: Extracted coercive field${H_{\rm{c}}}$ and AH signal$\rho _{yx}^{{\rm{AH}}}$ as a function of temperature[14]
图 6 (a) 红外s-SNOM实验技术示意图. AB、BA和AA表示双层石墨烯堆积方式的周期性改变; (b) (左)显示转角双层石墨烯中由畴壁晶格形成的纳米光子晶体. 这种反差是由于畴壁的局部光学导电性增强造成的. (右)转角双层石墨烯样品的暗场TEM图像; (c), (d) 分别为
${\lambda _{\rm{p}}}$ = 135 nm和282 nm时获得的红外s-SNOM图像[65]Figure 6. (a) Schematic of the IR s-SNOM experimental technique. AB, BA, and AA label periodically occurring stacking types of graphene layers; (b) (Left) isualizing the nano-light photonic crystal formed by the domain wall in twisted bilayer graphene. The contrast is due to enhanced local optical conductivity at domain wall. (Right) Dark-field TEM image of a twisted bilayer graphene sample; (c), (d) IR s-SNOM images obtained for
${\lambda _{\rm{p}}}$ = 135 nm and 282 nm, respectively[65].
图 7 (a) 通过唯象模型得到的石墨烯/h-BN电子能带结构的三维表示; (b)
${E_{\rm{F}}}$ 小于${E_\mu }$ ~170 meV时的光学跃迁; (c)${E_{\rm{F}}}$ 大于${E_\mu }$ 时莫尔势导致多个额外的光学跃迁通道, 这些跃迁提高电导率, 同时对等离激元波长产生了一个额外的带间跃迁贡献[80]Figure 7. (a) 3D representation of the electronic band structure of graphene/h-BN obtained from the phenomenological model; (b) Optical transitions at 170 mev, for a magnitude of the
${E_{\rm{F}}}$ smaller than${E_{\rm{\mu }}}$ ; (c) For a magnitude of${E_{\rm{F}}}$ larger than${E_\mu }$ one finds multiple additional channels for optical transitions, all initiated by the moirépotential. These transitions enhance the conductivity and also yield an interband contribution to the plasmonic wavelength in addition to intraband contribution[80].
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