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Placing a two-dimensional lattice such as graphene on another with a small rotation θ (the “twist angle”) gives rise to periodic moiré patterns (see figure below) for a superlattice with a lattice constant much larger (≈ 1/θ) than the single layer lattice constant. Many quantum phenomena of scientific and technological interest emerge naturally at the moiré length scales of layered 2D materials which makes those materials an exciting platform to explore quantum materials and prototype quantum devices. For example, correlated electronic phases such as superconductivity (Jarillo-Herrero, et al. 2018) have been recently observed in twisted bilayer graphene (tBLG). Such transformational results have opened up a new era in the investigation and exploitation of quantum phenomena.
Although each layer has a periodic structure when isolated, multi-layer 2D systems are generally no longer periodic when the individual layers have incommensurate lattice constants or are rotated on another, and the existing theory and computational methods based on periodicity are no longer applicable. To overcome this challenge, Luskin and his collaborators have proposed novel theoretical foundations, analysis, and computational methods for the mechanics, electronic structure, and transport properties of incommensurate 2D heterostructures.
Local Configuration (Disregistry) in Moiré Domains
The figure above shows that the local configuration (disregistry, shift, stacking) of the blue lattice uniformly samples the unit cell of the red lattice on the moiré domain. Our local configuration space approach gives a unified theoretical and computational framework for the mechanics, electronic structure, transport, and diffraction of incommensurate moiré heterostructures.
Electronic Density of States for Twisted Bilayer Graphene
The electronic density of states can be exactly formulated for lattices by integrating the Bloch waves (modulated plane wave eigenfunctions) over the Brillouin zone (fundamental domain of the reciprocal lattice), but twisted bilayer graphene and other moiré materials do not have a periodic structure.
We have given an exact formulation for the electronic density of states of incommensurate structures by integrating the local density of states over the space of local configurations. This gives an an efficient computational method for small twist angles by sampling the local configuration and computing the local density of states with a kernel polynomial method.
We have formulated more efficient local configurations methods in momentum space by exploiting the concentration of low energy states near high symmetry points of the Brillouin zone.
Structural Relaxation in the Moiré Domain
Unrelaxed (left) and Relaxed (right)
The energy landscape with respect to the possible bilayer stackings is not uniform in incommensurate twisted multilayer structures, so the multilayer structures mechanically relax to enhance the regions of lowest stacking energy (AB and BA stacking) within each moiré cell while confining the regions of highest stacking energy to domain walls separating the moiré cells. We have presented a new conceptual approach to modeling and computing the mechanical relaxation pattern of general weakly coupled incommensurate deformable multilayers by parametrizing the relaxation pattern by local configuration space rather than real space.
Electronic Band Structure for Relaxed TBG
The electronic band structure describes many fundamental material properties such as electrical conductivity. The theoretical discovery of flat bands (zero group velocity, so the interaction energy is important) in TBG near the "magic angle" of 1.05° by Bistritzer and MacDonald in 2011 (see computed band structure above) led to their prediction of strongly correlated phases that was later experimentally confirmed by Jarillo-Herrero, et al. in 2018.
The accurate prediction of electronic properties requires that the band structure be computed for the actual relaxed structure. At 3.0° (top), the relaxation gives a small moiré band gap opening up near the first band crossing at ± 350 eV. After relaxations near the magic angle at .1° (middle), the band is slightly less flat and the moiré band gaps near ± 40 eV are larger.
We have developed an accurate computational model for the band structure of relaxed TBG guided by rigorous error estimates. In the above figure, electronic band structure is displayed along high-symmetry lines of the moiré Brillouin zone at a single monolayer K valley for three twist angles, 3.0° (top), 1.1° (middle), and 0.3° (bottom). The first column shows the band structure for unrelaxed TBG, while the second shows that of relaxed TBG. The momentum axes are labeled in terms of the high-symmetry points of the reciprocal lattice of the moiré supercell, not the graphene monolayer cells.
Electron Transport in Incommensurate Heterostructures
The optical conductivity and other transport properties have been formulated and computed for periodic systems by the Kubo--Greenwood conductivity in terms of a double integral over the Brillouin zone of the conductivity function with respect to a current-current correlation measure. We have given an exact formula for the large body limit of incommensurate multilayered systems by formulating the Kubo--Greenwood conductivity as a double integral over the space of local configurations in both real space and momentum space. The computation of the Kubo conductivity in incommensurate systems is significantly more challenging than the computation of the electron density of states. We have demonsrated that the incommensurate structure can be exploited by decomposing the current-current correlation measure into local contributions and deducing an approximation scheme which is exponentially convergent in terms of domain size.
S. Etter, D. Massatt, M. Luskin, and C. Ortner. Modeling and computation of Kubo conductivity for 2D incommensurate bilayers. Multiscale Model. Simul., 18:1525–1564, 2020.
D. Massatt, S. Carr, and M. Luskin. Efficient computation of Kubo conductivity for incommensurate 2D heterostructures. Eur. Phys. J. B, 93, 2020.
A. Watson, D. Margetis, and M. Luskin. Mathematical aspects of the Kubo formula for electrical conductivity with dissipation. Jpn. J. Indust. Appl. Math., 40:1765–1795, 2023.
Continuum Models for 2D Moiré Materials
We have rigorously derived continuum models from discrete tight-binding models of 2D moiré materials. Our first work rigorously derived the celebrated Bistritzer-MacDonald for twisted bilayer graphene and proved that the model is accurate in the magic angle regime. Our current work if focused on rigorously deriving higher-order models including relaxation for more general 2D moiré materials.
Interacting TBG with Systematic Modeling of Structural Relaxation
Energy of converged state per moiré site, energy difference to ground state. α interpolates between HF interacting chiral limit and HF relaxed interacttng BM model. Interpolation is initialized at QH, VH, VP, KIVC, and TIVC ground states of the HF interacting chiral model α=0.
Energy of converged state per moiré site, energy difference to ground state. α interpolates between HF interacting chiral limit and HF relaxed interacttng BM model. Interpolation is initialized at QH, VH, VP, KIVC, and TIVC ground states of the HF interacting chiral model α=0.
C2zT symmetry order parameter during interpolation between interacting chiral model (α=0) and relaxed interacting BM model (α=1).
We introduced a novel reduced-order model of TBG’s many-body electronic properties which systematically accounts for the effects of structural relaxation. We do this by first modeling relaxation by coupling linear elasticity to a stacking penalty energy computed from density functional theory (DFT). We then incorporate the effects of relaxation into a single-particle electronic tight-binding model. We simplify this tight-binding model by focusing on wave-packets spectrally concentrated at the monolayer Dirac points in order to derive a moiré-scale continuum PDE model which we refer to as the relaxed BM model. We finally add a Coulomb electron–electron interaction and project both parts of the many-body Hamiltonian down to the relaxed BM model’s flat bands. We compute Hartree–Fock minimizers for the new model and predict symmetry-breaking phase transitions when the atomic structure is very close to the structure predicted by our relaxation model.
Twisted Trilayer Graphene
Twisted trilayer honeycomb lattice in real space with θ12 = 5.3° and θ23=7.7° . Magnified insert on left.
Twisted trilayer honeycomb lattice in real space with θ12 = 5.3° and θ23=7.7° . Magnified insert on left.
Real space relaxation pattern for twisted trilayer graphene with θ12=1.73°, θ23 = 2.24° .
Real space relaxation pattern for twisted trilayer graphene with θ12=1.73°, θ23 = 2.24° .
Moiré of moiré of lengths λmn as a function of θ12 for θ23 = 2.8°. Each color corresponds to a different set of harmonics (m; n).
Moiré of moiré of lengths λmn as a function of θ12 for θ23 = 2.8°. Each color corresponds to a different set of harmonics (m; n).
Electronic band structures and DOS at θ12 =1.4°, and θ23 = 2.8°..
Electronic band structures and DOS at θ12 =1.4°, and θ23 = 2.8°..
We introduced twisted trilayer graphene (tTLG) with two independent twist angles as an ideal system for the precise tuning of the electronic interlayer coupling to maximize the effect of correlated behaviors as observed experimentally in our collaboration with the Ke Wang group. The addition of a third layer introduces a new degree of freedom, the second twist angle, allowing for the further tuning of electron correlations. In twisted trilayer graphene (tTLG) with two consecutive twist angles, θ12 and θ23, the beating of two bilayer moiré patterns leads to higher-order patterns (moiré of moiré). The length scale of these is orders of magnitude larger than the bilayer moiré. Unlike in twisted bilayer graphene (tBLG) where only the lowest-order moiré pattern dominates in the continuum limit, the dominant harmonic is twist-angle dependent in tTLG. The issue with applying the real space continuum model to obtain the relaxation pattern in trilayer systems is the lack of periodicity. While bilayer systems always have a moiré supercell, trilayer systems are generally incommensurate and thus lack a periodic supercell even in the continuum limit. Therefore, a more general description beyond the supercell approximation is required. Here, we introduce the configuration space to describe the local environment of every position in the continuum and parametrize the trilayer system in a four dimensional configuration space.
Z. Zhu, S. Carr, D. Massatt, M. Luskin, and E. Kaxiras. Twisted trilayer graphene: A precisely tunable platform for correlated electrons. Phys. Rev. Lett., 125:11604 (6 pp), 2020.
Z. Zhu, P. Cazeaux, M. Luskin, and E. Kaxiras. Modeling mechanical relaxation in incommensurate trilayer van der Waals heterostructures. Phys. Rev. B, page 224107 (14 pp), 2020.
K.-T. Tsai, X. Zhang, Z. Zhu, Y. Luo, S. Carr, M. Luskin, E. Kaxiras, and K. Wang. Correlated superconducting and insulating states in twisted trilayer graphene moiré of moiré superlattices. Phys. Rev. Lett., 127:166802 (7pp)(18pp supplementary material), 2021.
W. Ren, K. Davydov, Z. Zhu, J. Ma, K. Watanabe, T. Taniguchi, E. Kaxiras, M. Luskin, and K. Wang. Tunable inter-moiré physics in consecutively-twisted trilayer graphene. Phys. Rev. B (Editors’ Suggestion, 110:115404, 2024.
K. Davydov, Z. Zhu, N. Friedman, E. Gramowski, Y. Li, J. Tavakley, K. Watanabe, T. Taniguchi, M. Luskin, E. Kaxiras, and K. Wang. Tunable atomically enhanced moiré Berry curvatures in twisted triple bilayer graphene. Tunable atomically enhanced moiré Berry curvatures in twisted triple bilayer graphene. Phys. Rev. B, 111:L161120, 2025.
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