Emergent Atomic Environments in Twisted Bilayer Graphene and Their Use in the Prediction of the Vibrational Properties

While Bernal stacked bilayer graphene bears two distinct atom types in its lattice, there exists no analytical framework addressing the number of atomic environments that emerge in twisted bilayer graphene superlattices. Here, we computationally analyze 140 different twisted bilayer superlattices using descriptor functions to study the emergent local environments. Our study reveals that the number of atoms with unique local environments depend on the superlattice size linearly manifesting itself on two distinct lines in accordance with the respective space groups. We then introduce the use of local environments in the investigation of vibrational properties. Local phonon density of states of the atoms with unique local environments are calculated by molecular dynamics simulations and are used to train a machine learning model. This model is used to predict the phonon density of states of twisted bilayer structures. Performance of the trained model is cross validated and discussed thoroughly via different selection of training and test sets. It is shown that the model proves effective in predicting the vibrational properties of any given twisted bilayer structure. Since the generic method presented reaches far beyond twisted bilayer graphene the possible applications ranging from non-periodic structures to strain induced moir\'e superlattices are also discussed.