Incommensurate stacking provides an intriguing avenue for manipulating the physical properties of layered two-dimensional materials, but is a challenging problem from a theoretical perspective. Here, we present a multi-scale model to obtain the mechanical relaxation pattern and electronic structure of twisted trilayer van der Waals heterostructures with two independent twist angles. This serves as a prototype system of a generally incommensurate system without a supercell description.
To study mechanical properties, we adopt configuration space as a natural description of incommensurate layers, which describes the local environment of each atomic position. We minimize the total energy, parameterized using Density Functional Theory calculations, to obtain the relaxation pattern. For the electronic properties, we focus on twisted trilayer graphene. We adopt a k.p effective theory derived from a low-energy expansion around the Dirac point. Our results suggest that the twisted trilayer systems are interesting from theoretical and mathematical points of view, as well as a promising platform to study correlated physics.