Review of the tight-binding method applicable to the properties of moiré superlattices

TL;DR

This review surveys tight-binding TB methods for moiré superlattices, emphasizing atomistic detail, lattice relaxation, and large unit cells that challenge first-principles approaches. It covers graphene-based, TMD-based, and hBN-based moiré Hamiltonians, detailing Slater–Koster hoppings, spin–orbit coupling, interlayer couplings, and mean-field many-body extensions such as Hartree, Hartree–Fock, and Hubbard-U terms. The article also surveys computational strategies including exact diagonalization, linear-scaling random-state methods (KPM, TBPM), and machine-learning–driven TB parameterization, along with software packages that facilitate moiré TB modeling. Two illustrative examples—the graphene moiré dodecagonal quasicrystal and Rydberg moiré excitons in WSe$_2$/TBG—demonstrate the power of TB to capture both spectral features and real-space localization in large-scale systems. The work highlights how TB serves as a bridge to low-energy continuum models and guides future directions in parameterization, relaxation, and data-driven moiré materials discovery, with potential impact on quantum technologies and optoelectronics.

Abstract

Moiré superlattices have emerged as a versatile platform for exploring a wide range of ex- otic quantum phenomena. Unlike angstrom-scale materials, the moiré length-scale system contains a large number of atoms, and its electronic structure is significantly modulated by the lattice relaxation. These features pose a huge theoretical challenge. Among the available theoretical approaches, tight-binding (TB) methods are widely employed to predict the electronic, transport, and optical properties of systems such as twisted graphene, twisted transition-metal dichalcogenides (TMDs), and related moiré materials. In this review, we pro- vide a comprehensive overview of atomistic TB Hamiltonians and the numerical techniques commonly used to model graphene-based, TMD-based and hBN-based moiré superlattices. We also discuss the connection between atomistic TB descriptions and effective low-energy continuum models. Two examples of different moiré materials and geometries are provided to emphasize the advantages of the TB methods. This review is intended to serve as a theoretical and practical guide for those seeking to apply TB methods to the study of various properties of moiré superlattices.

Figures (14)

• Figure 1: (a) The atomic structure of TBG with $\theta=3.15^{\circ}$. The moiré unit cell is illustrated with a black parallelogram. (b) Band structure of TBG with $\theta=5.08^{\circ}$ obtained by performing TB (solid line) and ab initio (dot) calculations. In the TB calculation, the hopping parameters are $t_0 = 2.7 eV$ and $t_1 = 0.48 eV$. (c) Fermi velocity ratio $V_{bi}/V_{mono}$ of TBG versus angle $\theta$. Red dot for the ab initio calculations and black cross for the TB calculations. The velocity close to 0 at angle $\theta = 1.08^\circ$ with integer pair $(30,31)$. (d) Distribution of one eigenstate at K point with energy $E=0$, in the unit cell of TBG with $\theta = 1.08^\circ$. Black small dots are the positions of all atoms, red dots are atoms where 80% of the states are localized. Inset shows the local density of states (DOS) of the AA stacking (solid red line) and the total DOS (dashed black line). Adapted with permission from trambly2010localization. Copyright (2010) American Chemical Society.
• Figure 2: (a) The band structure of TBG with $\theta=1.08^\circ$ by taking long-range Hartree corrections into account at electron filling number $\nu=0$ (left side) and $\nu=-1$ (right side). The gray lines are the band structure without Hartree corrections. The dashed line is the Fermi level at each filling. Adapted with permission from rademaker2019charge. Copyright (2019) by the American Physical Society. (b) The first valence (upper) and conduction (lower) flat bands obtained by including the Hartree-Fock interaction at filling number $\nu=0$ for TBG with $\theta=1.16^\circ$. The dashed line is the Fermi level. Adapted with permission from gonzalez2020time. Copyright (2020) by the American Physical Society.
• Figure 3: (a) Band structure of TBG with $\theta=1.5^\circ$ obtained from a scaled (solid lines) and unscaled (dashed lines) TB models. Adapted with permission from gonzalez2017electrically. Copyright (2017) by the American Physical Society. (b) The calculated flat bands and spin $z$ magnetization of TBG with angle $\theta = 0.8^\circ$ by considering the effect of local mean-field interactions. At angle $\theta = 0.8^\circ$, the second bands from both conduction and valence bands became flat. The interaction strength is $U=2t$, where $t$ the nearest-neighbor hopping within one layer, and the electron filling number is $\nu = -6$, corresponding to half-filling of the second band. The red (blue) color indicates a positive (negative) expectation value $\langle S_z \rangle=M_z$ of the spin operator. The calculation was performed by using a rescaling method. Adapted with permission from wolf2019electrically. Copyright (2019) by the American Physical Society.
• Figure 4: Tight-binding band structure of twisted homobilayer MoS$_2$ at $\theta = 3.15^\circ$. (a) Bands obtained from a TB model from Ref. zhan2020tunabilitykuang2022flat. Adapted with permission from kuang2022flat. Copyright (2022) by the American Physical Society. (b) Bands calculated from the TB model from Ref. vitale2021flat. Adapted under the terms of the CC BY license from Ref. vitale2021flat. Copyright (2021) IOP Publishing.
• Figure 5: Tight-binding band structure of twisted heterobilayer TMDs for (a) twisted bilayer WSe$_2$/MoS$_2$ and (b) MoSe$_2$/WS$_2$ heterostructure at twist angle $\theta = 4.5^\circ$vitale2021flat. Adapted under the terms of the CC BY license from Ref. vitale2021flat. Copyright (2021) IOP Publishing.