The incommensurate stacking of multilayered two-dimensional materials is a challenging problem from a theoretical perspective and an intriguing avenue for manipulating their physical properties. Here we present a multiscale 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.
Bibliographical noteFunding Information:
We thank Stephen Carr, Shiang Fang, Steven Torrisi, and Emine Kucukbenli, Philip Kim, Liang Fu, Ke Wang for insightful discussions. This work was supported by the STC Center for Integrated Quantum Materials, NSF Grant No. DMR-1231319, ARO MURI Award No. W911NF-14-0247, NSF DMREF Award No. 1922165, and NSF Award No. DMS-1819220. Calculations were performed on the Odyssey cluster supported by the FAS Division of Science, Research Computing Group at Harvard University.