Multilayered Atomic Relaxation in van der Waals Heterostructures

When two-dimensional van der Waals materials are stacked to build heterostructures, moiré patterns emerge from twisted interfaces or from a mismatch in the lattice constant of individual layers. Relaxation of the atomic positions is a direct, generic consequence of the moiré pattern, with many implications for the physical properties. Moiré-driven atomic relaxation may be naively thought to be restricted to the interfacial layers and thus irrelevant for multilayered heterostructures. However, we provide experimental evidence for the importance of the three-dimensional nature of the relaxation in two types of van der Waals heterostructures: First, in multilayer graphene twisted on graphite at small twist angles ($\theta \approx 0.14°$), we observe propagation of relaxation domains even beyond 18 graphene layers. Second, we show how for multilayer ${\mathrm{PdTe}}_2$ on ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ the moiré lattice constant depends on the number of ${\mathrm{PdTe}}_2$ layers. Motivated by the experimental findings, we develop a continuum approach to model multilayered relaxation processes based on the generalized stacking fault energy functional given by ab initio simulations. Leveraging the continuum property of the approach enables us to access large-scale regimes and achieve agreement with our experimental data for both systems. Furthermore, it is well known that the electronic structure of graphene sensitively depends on local lattice deformations. Therefore, we study the impact of multilayered relaxation on the local density of states of the twisted graphitic system. We identify measurable implications for the system, experimentally accessible by scanning tunneling microscopy. Our multilayered relaxation approach is not restricted to the discussed systems and can be used to uncover the impact of an interfacial defect on various layered systems of interest.