Lattice relaxation and substrate effects of graphene moiré superlattice

Zhan Zhen; Zhang Ya-Lei; Yuan Sheng-Jun

doi:10.7498/aps.71.20220872

Abstract

When two two-dimensional (2D) materials with different lattice constants or with different rotation angles are superimposed, a moiré superlattice can be constructed. The electronic properties of the superlattice are strongly dependent on the stacking configuration, twist angle and substrate. For instance, theoretically, when the rotation angle of twisted bilayer graphene is reduced to a set of specific values, the so-called magic angles, flat bands appear near the charge neutrality, and the electron-electron interaction is significantly enhanced. The Mott insulator and unconventional superconductivity are detected in the twisted bilayer graphene with a twist angle around 1.1°. For a moiré pattern with a large enough periodicity, lattice relaxation caused by an interplay between van der Waals force and the in-plane elasticity force comes into being. The atomic relaxation forces atoms to deviate from their equilibrium positions, and thus making the system reconstructed. This review mainly focuses on the effects of the lattice relaxation and substrates on the electronic properties of the graphene superlattices. From both theoretical and experimental point of view, the lattice relaxation effects on the atomic structure and electronic properties of graphene-based superlattices, for example, the twisted bilayer graphene, twisted trilayer graphene, graphene-hexagonal boron nitride superlattice and twisted bilayer graphene-boron nitride superlattice are discussed. Finally, a summary and perspective of the investigation of the 2D material superlattice are presented.

Keywords:

Authors and contacts

1.
School of Physics and Technology, Wuhan University, Wuhan 430072, China
2.
Wuhan Institute of Quantum Technology, Wuhan 430206, China

Corresponding author: Yuan Sheng-Jun, s.yuan@whu.edu.cn

Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 12174291, 12047543) and the National Key R&D Program of China (Grant No. 2018YFA0305800).

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Cited By

图 2 (a) $ \theta =0.48^{\circ} $的TBG的STM形貌图 (100 nm $ \times $ 100 nm, 超晶格格矢$ L1\approx L2\approx L3\approx 29.6 $ nm, STM图在V = 100 mV和I_t = 1.0 nA下采集); (b) 超晶格结构中AA和AB区域的STS, 两条实(虚)线表示探测不同AA (AB)位置的STS, 证明实验数据的可重复性; (c) TBG非弛豫结构(上)和弛豫结构(下)中AA和AB区域的LDOS; (d) 形变势能V的分布; (e) 赝磁场B的分布; (f) $ \theta =0.48^{\circ} $的TBG理论计算模型图; (g) 路径M-N-P上原子在平面内(|Δd|)和z方向上(|Δz|)的位移; (h), (i) 原子分别在平面内(h)和z方向上(i)位移的实空间分布图^[19]

Figure 2. (a) STM topography image (100 nm $ \times $ 100 nm) for TBG with $ \theta =0.48^{\circ} $ (The three moiré wavelengths are $ L1\approx L2\approx L3\approx 29.6 $ nm, sample bias $ V=100 $ mV, tunneling current $ {I}_{\mathrm{t}}=1.0 $ nA); (b) logarithmic dI/dV spectra measured at AA and AB regions (The two solid/dashed lines were taken at different AA/AB regions to show reproducibility, and curves are vertically shifted for clarity); (c) calculated LDOS in the AA and AB regions for deformed (upper) and rigidly twisted (bottom) cases; (d) calculated local potential V; (e) calculated pseudo-magnetic fields B; (f) schematic model of the moiré pattern of TBG with $ \theta =0.48^{\circ} $; (g) absolute magnitude of different in-plane atomic displacements and out-of-plane displacements for the deformed system along the path M-N-P; (h), (i) maps of the absolute magnitude of the in-plane atomic displacement |Δd| (h) and out-of-plane displacement |Δz| (i) in deformed systems^[19].

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图 1 $ \theta =6.01^{\circ} $时TBG结构示意图(黑色实线框表示莫尔超晶格的原胞; 圆圈表示4种高对称堆垛结构, 分别为AA堆垛(红色)、AB堆垛(蓝色)、DW堆垛(紫色)和BA堆垛(绿色))

Figure 1. Schematics of the atomic configuration of TBG with$ \theta =6.01^{\circ} $ (The moiré supercell is outlined in black line. High-symmetry stacking regions of AA, AB, DW and BA are marked by the red, blue, purple and green circles, respectively).

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图 3 (a)—(c) $ \theta =1.05^{\circ}\mathrm{ }\left(\mathrm{a}\right) $, $ \theta =0.53^{\circ}\mathrm{ }\left(\mathrm{b}\right) $和$ \theta =0.35^{\circ}\mathrm{ }\left(\mathrm{c}\right) $的面内应变张量$ \mathit{u}\left(\mathit{r}\right) $(白色箭头表示原子在平面内的位移矢量; 彩色条表示原子在面内的旋转角度$\Delta \theta =\nabla \times {\boldsymbol u }$, 正值表示顺时针旋转; 莫尔结构的晶胞由黑色边框标记; 3个高对称区域分别是AA, AB和DW); (d)—(f) $ \theta =1.05^{\circ} $ (d), $ \theta =0.53^{\circ} $ (e)和$ \theta =0.35^{\circ} $ (f)的层间距(ILS)在实空间的分布

Figure 3. (a)–(c) In-plane strain $ \mathit{u}\left(\mathit{r}\right) $ in twisted bilayer graphene with (a) $ \theta =1.05^{\circ} $, (b) $ \theta =0.53^{\circ} $ and (c) $ \theta =0.35^{\circ} $ (The in-plane displacements are visualized with white arrows; the color data denotes the local value of the in-plane twist of the atoms with respect to their original position ($\Delta \theta =\nabla \times {\boldsymbol u})$, and the positive values indicate counterclockwise rotation. The moiré supercell is outlined in black, and the high-symmetry stacking regions of AA, AB and DW are illustrated); (d)–(f) the interlayer spacing of TBG with (d) $ \theta =1.05^{\circ} $, (e) $ \theta =0.53^{\circ} $ and (f) $ \theta =0.35^{\circ} $.

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图 4 (a), (d), (g) 三类魔角的DOS分布; (b), (e), (h) 三类魔角非弛豫体系中不同能量下LDOS在实空间的分布; (c), (f), (i) 三类魔角弛豫体系中不同能量下LDOS在实空间的分布. 其中(a)—(c)$ \theta =1.05^{\circ} $; (d)—(f) $ \theta =0.53^{\circ} $; (g)—(i) $ \theta =0.35^{\circ} $

Figure 4. (a), (d), (g) DOS distributions of three types of magic angles; (b), (e), (h) LDOS distributions in real space at different energies in non-relaxation systems of three types of magic angle non-relaxation systems; (c), (f), (i) LDOS distributions in real space at different energies in relaxation systems of three types of magic angle. The rotation angle θ is 1.05° (a)–(c), 0.53° (d)–(f), 0.35° (g)–(i).

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图 5 (a) tTLG-AÃA-6.01体系的侧视图(上)和俯视图(下); (b), (c) 分别为tTLG-AÃA-1.35体系和tTLG-ÃAB-1.05体系中原子在z方向的位移Δz; (d)—(f) tTLG-AÃA-1.35体系中DOS (d)和不同能量点LDOS (e), (f)在实空间的分布; (g)—(i) tTLG-ÃAB-1.05体系中DOS (g)和不同能量点LDOS (h), (i)在实空间的分布^[64]

Figure 5. (a) Side (upper) and top (lower) views of tTLG-AÃA-6.01; (b), (c) the displacement Δz of atoms in the z direction for tTLG-AÃA-1.35 and tTLG-ÃAB-1.05, respectively; (d)–(f) distribution of DOS (d) and LDOS at different energy points in real space (e), (f) of tTLG-AÃA-1.35. (g)–(i) distribution of DOS (g) and LDOS at different energy points in real space (h), (i) of tTLG-ÃAB-1.05^[64].

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图 6 (a), (c) tTLG-AÃA-1.89体系分别在常压和4 GPa高压条件下原子在z方向的位移Δz; (b), (e) 体系在常压条件下的能带、DOS (b)和范霍夫奇点处LDOS在实空间的分布(e); (d), (f) 体系在4 GPa高压条件下的能带、DOS (d)和范霍夫奇点处LDOS在实空间的分布(f)^[72]

Figure 6. (a), (c) Out-of-plane displacement Δz of relaxed tTLG-AÃA-1.89 without and with 4 GPa vertical pressure, respectively; (b), (e) the band structure, DOS (b) and LDOS mappings of van Hove singularities (e) of tTLG-AÃA-1.89 under ambient pressure; (d), (f) the band structure, DOS (d) and LDOS mappings of van Hove singularities (f) of tTLG-AÃA-1.89 with 4 GPa pressure^[72].

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图 7 (a), (b) 石墨烯/氮化硼之间转角$ \theta =0^{\circ} $的体系在晶格弛豫后的紧束缚模型参数(λ = 13.8 nm; 从左到右分别为在位能$ {V}_{D} $和碳原子最近邻的跃迁振幅t1, t2, t3, 彩色条的单位是t = 2.7 eV); (c) 非弛豫和弛豫结构石墨烯的DOS分布; (d) 不同$ \theta $下的DOS (箭头表示超晶格狄拉克点随转角的减小而向高能部分移动; 当$ \theta =1.85^{\circ} $, 超晶格狄拉克点消失; 相应的莫尔长度分别为 λ = 13.8, 11.9, 6.7 nm); (e) 转角$ \theta =0^{\circ} $时不同能量下的准本征态在实空间的分布 (左侧和右侧分别是A子晶格和B子晶格的准本征态; 对于接近超晶格狄拉克点的能量, 可以形成一个清晰的莫尔条纹^[41])

Figure 7. (a), (b) Modified tight-binding parameters for a relaxed sample of graphene on hBN with θ = 0° (λ = 13.8 nm; from left to right, the on-site potential $ {V}_{D} $ and the hopping parameters t1, t2, and t3. The color bars are in units of t = 2.7 eV); (c) DOS distributions of unrelaxed and relaxed graphene; (d) DOS for different angles θ (As indicated by the arrows, superlattice Dirac point moves towards the high-energy part with the decreasing of the turning angle; the superlattice Dirac point disappears at $ \theta =1.85^{\circ} $. The corresponding moiré lengths λ = 13.8, 11.9, 6.7 nm, respectively); (e) amplitude of the quasi eigenstates for different energies in real space for θ = 0° (The left-hand panels show sublattice A and the right-hand panels show sublattice B. For energies closer to the extra Dirac cones, a clear moiré pattern can be distinguished)^[41].

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图 8 (a) TBG/hBN的结构示意图; (b) TBG/hBN的俯视图和高对称堆垛结构; (c) 不同$ {\theta }_{\mathrm{b}\mathrm{o}\mathrm{t}} $ 体系的面内形变u(r)和面内转角$\Delta \theta =\nabla \times {\boldsymbol u}$, 白色箭头是原子的面内位移; (d)—(f) 不同$ {\theta }_{\mathrm{b}\mathrm{o}\mathrm{t}} $体系的能带图和DOS (彩色条表示每个谷$ \langle{{\widehat{V}}_{z}}\rangle $的带, 如果属于K谷状态(红色), 则$\langle{{\widehat{V}}_{z}}\rangle \approx 1$, 如果属于K'谷状态(蓝色), 则$\langle{{\widehat{V}}_{z}}\rangle \approx -1$); (g), (h) 晶格重构引起的形变势能$ {V}_{D} $与赝磁场${\boldsymbol B}=\nabla \times {\boldsymbol A}$ (红色箭头代表矢势A(r), TBG的转角固定为$ 1.05^{\circ} $^[102])

Figure 8. (a) Schematic of the atomic configuration of TBG/hBN; (b) top view and high-symmetry stacking regions of the atomic configuration of TBG/hBN; (c) in-plane strain u(r) and in-plane rotation angle $\Delta \theta =\nabla \times {\boldsymbol u}$ with varying $ {\theta }_{\mathrm{b}\mathrm{o}\mathrm{t}} $ (The in-plane displacements are visualized with white arrows); (d)–(f) band structure and DOS of TBG/hBN with different $ {\theta }_{\mathrm{b}\mathrm{o}\mathrm{t}} $ (The color bar denotes the band for each valley $ \langle{{\widehat{V}}_{z}}\rangle $ with $ \langle{{\widehat{V}}_{z}}\rangle\approx 1 $ if a state belongs to valley K and $ \langle{{\widehat{V}}_{z}}\rangle\approx -1 $ if a state belongs to valley K'); (g) the deformation potential $ {V}_{D} $ and (h) pseudo-magnetic field ${\boldsymbol B}=\nabla \times {\boldsymbol A}$ induced by lattice relaxations in the TBG/hBN with $ {\theta }_{\mathrm{b}\mathrm{o}\mathrm{t}}=0.53^{\circ} $ (The vector field A(r) is visualized with red arrows in (h) TBG is fixed to $ {\theta }_{\mathrm{t}\mathrm{b}\mathrm{g}}=1.05^{\circ} $ in all cases)^[102].

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图 9 (a) 三明治结构hBN/TBG/hBN的形貌图和高对称堆垛结构; (b)—(d) 不同$ {\theta }_{\mathrm{t}\mathrm{o}\mathrm{p}} $和$ {\theta }_{\mathrm{b}\mathrm{o}\mathrm{t}} $组成体系的(从左到右)能带图、面内转角、形变势能和赝磁场 (TBG的转角固定为$ {\theta }_{\mathrm{t}\mathrm{b}\mathrm{g}}=1.05^{\circ} $)^[102]

Figure 9. (a) Schematic structure of the hBN/TBG/hBN system and the different high-symmetry stackings in the superlattice; (b)–(d) panels from left to right display the band structure, in-plane twist of the atoms with respect to their original position, scalar potential and pseudo-magnetic field of the systems with different $ {\theta }_{\mathrm{t}\mathrm{o}\mathrm{p}} $ and $ {\theta }_{\mathrm{b}\mathrm{o}\mathrm{t}} $ (TBG is fixed to $ 1.05^{\circ} $^[102]).

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