Magic angles in the chiral model of twisted bilayer graphene are parameters for which the chiral version of the Bistritzer–MacDonald Hamiltonian exhibits a flat band at energy zero. We compute the sums over powers of (complex) magic angles and use that to show that the set of magic angles is infinite. We also provide a new proof of the existence of the first real magic angle, showing also that the corresponding flat band has minimal multiplicity for the simplest possible choice of potentials satisfying all symmetries. These results indicate (though do not prove) a hidden integrability of the chiral model.
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Modeling of Electronic Dynamics in Twisted Bilayer Graphene
We consider the problem of numerically computing the quantum dynamics of an
electron in twisted bilayer graphene. The challenge is that atomic-scale models of the dynamics are
aperiodic for generic twist angles because of the incommensurability of the layers. The Bistritzer--
MacDonald PDE model, which is periodic with respect to the bilayer's moir\'e pattern, has recently
been shown to rigorously describe these dynamics in a parameter regime. In this work, we first prove
that the dynamics of the tight-binding model of incommensurate twisted bilayer graphene can be
approximated by computations on finite domains. The main ingredient of this proof is a speed of
propagation estimate proved using Combes--Thomas estimates. We then provide extensive numerical
computations, which clarify the range of validity of the Bistritzer--MacDonald model.
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- Award ID(s):
- 1922165
- PAR ID:
- 10561572
- Publisher / Repository:
- SIAM
- Date Published:
- Journal Name:
- SIAM Journal on Applied Mathematics
- Volume:
- 84
- Issue:
- 3
- ISSN:
- 0036-1399
- Page Range / eLocation ID:
- 1011 to 1038
- Subject(s) / Keyword(s):
- partial differential equations, electronic properties of materials, 2D materials, moir\'e materials, numerical analysis, twisted bilayer graphene
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Abstract -
Momentum space transformations for incommensurate two-dimensional electronic structure calculations are fundamental for reducing computational cost and for representing the data in a more physically motivating format, as exemplified in the Bistritzer--MacDonald model [Proc. Natl. Acad. Sci. USA, 108 (2011), pp. 12233--12237]. However, these transformations can be difficult to implement in more complex systems such as when mechanical relaxation patterns are present. In this work, we aim for two objectives. First, we strive to simplify the understanding and implementation of this transformation by rigorously writing the transformations between the four relevant spaces, which we denote real space, configuration space, momentum space, and reciprocal space. This provides a straightforward algorithm for writing the complex momentum space model from the original real space model. Second, we implement this for twisted bilayer graphene with mechanical relaxation effects included. We also analyze the convergence rates of the approximations and show the tight-binding coupling range increases for smaller relative twists between layers, demonstrating that the 3-nearest neighbor coupling of the Bistritzer--MacDonald model is insufficient when mechanical relaxation is included for very small angles. We quantify this and verify with numerical simulation.more » « less
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Anisotropic hopping in a toy Hofstadter model was recently invoked to explain a rich and surprising Landau spectrum measured in twisted bilayer graphene away from the magic angle. Suspecting that such anisotropy could arise from unintended uniaxial strain, we extend the Bistritzer–MacDonald model to include uniaxial heterostrain and present a detailed analysis of its impact on band structure and magnetotransport. We find that such strain strongly influences band structure, shifting the three otherwise-degenerate van Hove points to different energies. Coupled to a Boltzmann magnetotransport calculation, this reproduces previously unexplained nonsaturating
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