Multi-layered atomic relaxation in van der Waals heterostructures

TL;DR

This paper investigates the three-dimensional atomic relaxation in van der Waals heterostructures, combining experimental evidence with a continuum model to understand how relaxation propagates through multiple layers and affects electronic properties.

Contribution

It introduces a continuum modeling approach for multi-layered relaxation in van der Waals heterostructures, validated by experimental data on graphene and PdTe2 systems.

Findings

Relaxation domains propagate beyond 18 graphene layers.

Moiré lattice constant varies with the number of PdTe2 layers.

Multi-layer relaxation impacts local electronic density of states.

Abstract

When two-dimensional van der Waals materials are stacked to build heterostructures, moir\'e patterns emerge from twisted interfaces or from mismatch in lattice constant of individual layers. Relaxation of the atomic positions is a direct, generic consequence of the moir\'e pattern, with many implications for the physical properties. Moir\'e driven atomic relaxation may be naively thought to be restricted to the interfacial layers and thus irrelevant for multi-layered 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 multi-layer graphene twisted on graphite at small twist angles ($\theta\approx0.14^\circ$) we observe propagation of relaxation domains even beyond 18 graphene layers. Second, we show how for multi-layer PdTe$_2$ on Bi$_2$Se$_3$ the moir\'e lattice constant depends on the number of PdTe$_2$ layers. Motivated by the experimental findings, we developed a continuum approach to model multi-layered 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 multi-layered 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 multi-layered 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.