Discrete-to-continuum modelling of weakly interacting incommensurate two-dimensional lattices

We analyze a description of twisted graphene bilayers, that incorporates the deformation of the layers using state of the art interlayer atomic potentials, and a modification of the hopping parameters between layers in the light of the classic Slonczewski-Weiss-McClure parametrisation. We obtain narrow bands in all cases, but that their nature can be rather different. We will show how to describe the results by equivalent continuum models. Even though such models can be constructed, their complexity can vary, requiring many coupling parameters to be included, and the full in-layer dispersion must be taken into account. The combination of all these effects will have a large impact on the wave functions of the flat bands, and that modifications in details of the underlying models can lead to significant changes. A robust conclusion is that the natural strength of the interlayer couplings is higher than usually assumed, leading to shifts in the definition of the magic angles. The structure at the edges of the narrow bands, at the Γ point of the Brillouin Zone is also strongly dependent on parametrization. As a result, the existence, and size, of band gaps between the flat bands and the neighboring ones are changed. Hence, the definition of Wannier functions, and descriptions based on local interactions are strongly dependent on the description of the model at the atomic scale. * Francisco. Guinea@imdea.org † Niels.Walet@manchester.ac.uk; https://www.research.manchester.ac.uk/portal/niels.walet.html AA AB FIG. 1. Examples of a graphene bilayer in (approximate) AA and AB alignment. arXiv:1903.00364v2 [cond-mat.str-el]

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“…This problem is also discussed in Ref. [74] using an analytic description of domain wall formation, but for rectangular domains.…”

Section: Classical Atomistic Simulationsmentioning

confidence: 99%

Continuum models for twisted bilayer graphene: Effect of lattice deformation and hopping parameters

2019

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“…The study of bilayer graphene has grown recently due to the discovery of superconductivity in low-temperature bilayer graphene with a small relative twist [3]. These twisted bilayer graphene systems form a large scale moiré pattern [12,13,23] that is incommensurate [11,14,18,21], or aperiodic, which has motivated the development of methods to overcome the theoretical and computational challenge posed by the lack of periodicity. Most current physics investigations overcome the lack of periodicity by utilizing the lowenergy approximation of Bistritzer and MacDonald [1,9] that restricts interlayer scattering to nearest neighbor reciprocal superlattice vectors.…”

Section: Introductionmentioning

confidence: 99%

Electronic Observables for Relaxed Bilayer 2D Heterostructures in Momentum Space

Massatt¹,

Carr²,

Luskin³

2021

Preprint

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We generalize the transformations and duality found in incommensurate 2D systems between real space, configuration space, momentum space, and reciprocal space to study electronic observables of incommensurate bilayers in the tight-binding framework using a wide class of applicable Hamiltonians. We then apply this generalization to obtain the effects of mechanical relaxation on nearly aligned materials in momentum space, which produce in-plane incommensurate scattering. The relaxation scattering is long-ranged in this case, which likewise changes the momentum space numerical scheme convergence rate. We study this convergence theoretically, and perform a numerical study on twisted bilayer graphene at small angles with mechanical relaxation.

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“…Crystal relaxation in vdW heterostructures has been studied in the continuum limit by energy minimization [12][13][14][15][16][17][18] . In twisted bilayer graphene, for instance, relaxation enlarges the AB/BA stacking regions (the equilibrium stacking) and forms a thin domain line in between adjacent AB/BA stacking regions, while the AA stacking regions (high energy stacking order) shrink to localized spots.…”

Section: Introductionmentioning

confidence: 99%

Moiré of Moiré: Modeling Mechanical Relaxation in Incommensurate Trilayer van der Waals Heterostructures

Zhu,

Cazeaux,

Luskin

et al. 2019

Preprint

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The incommensurate stacking of multi-layered two-dimensional materials is a challenging problem from a theoretical perspective and an intriguing avenue for manipulating their physical properties.Here we present a multi-scale model to obtain the mechanical relaxation pattern of twisted trilayer van der Waals (vdW) heterostructures with two independent twist angles, a generally incommensurate system without a supercell description. We adopt the configuration space as a natural description of such incommensurate layered materials, based on the local environment of atomic positions, bypassing the need for commensurate approximations. To obtain the relaxation pattern, we perform energy minimization with respect to the relaxation displacement vectors. We use a continuum model in combination with the Generalized Stacking Fault energy to describe the interlayer coupling, obtained from first-principles calculations based on Density Functional Theory. We show that the relaxation patterns of twisted trilayer graphene and WSe2 are "moiré of moiré", as a result of the incommensurate coupling two bilayer moiré patterns. We also show that, in contrast to the symmetry-preserving in-plane relaxation in twisted bilayers, trilayer relaxation can break the two-fold rotational symmetry about the xy-plane when the two twist angles are equal.

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Discrete-to-continuum modelling of weakly interacting incommensurate two-dimensional lattices

Cited by 8 publications

References 41 publications

Continuum models for twisted bilayer graphene: Effect of lattice deformation and hopping parameters

Continuum models for twisted bilayer graphene: Effect of lattice deformation and hopping parameters

Electronic Observables for Relaxed Bilayer 2D Heterostructures in Momentum Space

Moiré of Moiré: Modeling Mechanical Relaxation in Incommensurate Trilayer van der Waals Heterostructures

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