Unveiling The Evolution of Charge Density Wave Order in Twisted Bilayers of NbSe2—A Machine Learning Approach

Niobium diselenide has attracted significant attention over the past decade due to the coexistence of superconductivity and a charge density wave (CDW), which have been experimentally observed down to the monolayer limit. Their coexistence, and evolution with varying numbers of layers and different twist angles are central topics in twistronics, and would benefit from accurate atomistic simulations. A fundamental question that remains largely unexplored in the literature is whether CDWs persist within moiré structures and how they are altered compared to the pristine monolayer. Traditional first-principles methods encounter limitations in addressing such questions due to the computational resources required to model the long-wavelength moiré pattern. For instance, investigating a 1-degree twist angle would necessitate approximately 10,000 atoms, rendering such simulations impractical. This study adopts a practical approach by leveraging ab-initio data to develop accurate machine learning interatomic potentials thanks to the ALLEGRO architecture, an open-source code for building highly scalable and accurate equivariant deep learning interatomic potentials. We explore the formation and evolution of CDW order in monolayers and twisted bilayers. Our results are validated against density functional theory calculations, encompassing structural relaxation and phonon dispersions, with minimal errors observed in energy and forces. We find that the CDWs persist in a mosaic-like pattern in the moiré lattice. Expanding our approach to multilayers and considering variables like doping, strain, substrates, and proximity effects promises to refine CDW behaviors. These advancements could lead to new technological applications and scientific insights in this field.