Multiscale simulations are used to study the structural relaxation in twisted graphene bilayers and the associated electron diffraction patterns. The initial twist forms an incommensurate moiré pattern that relaxes to a commensurate microstructure comprised of a repeating pattern of alternating low-energy AB and BA domains surrounding a high-energy AA domain. The simulations show that the relaxation mechanism involves a localized rotation and shrinking of the AA domains that scales in two regimes with the imposed twist. For small twisting angles, the localized rotation tends to a constant; for large twist, the rotation scales linearly with it. This behavior is tied to the inverse scaling of the moiré pattern size with twist angle and is explained theoretically using a linear elasticity model. The results are validated experimentally through a simulated electron diffraction analysis of the relaxed structures. A complex electron diffraction pattern involving the appearance of weak satellite peaks is predicted for the small twist regime. This new diffraction pattern is explained using an analytical model in which the relaxation kinematics are described as an exponentially-decaying (Gaussian) rotation field centered on the AA domains. Both the angle-dependent scaling and diffraction patterns are in quantitative agreement with experimental observations. A Matlab program for extracting the Gaussian model parameters accompanies this paper.
Bibliographical noteFunding Information:
We thank Philip Kim, Hyobin Yoo, Rebecca Engelke for useful discussion on FFT and the experimental observations, Mitchell Luskin, Paul Cazeaux on mathematical theory of bilayer structure and diffraction, and Efthimios Kaxiras for helpful comments. We acknowledge the support of the Army Research Office ( W911NF-14-1-0247 ) under the MURI program.
© 2017 Elsevier Ltd
- Electron diffraction
- Moiré pattern
- Multiscale simulation
- Twisted graphene bilayers