Mechanism of Corrugated Graphene Moiré Superstructures on Transition-Metal Surfaces

A graphene layer on a transition-metal (TM) surface can be either corrugated or flat, depending on the type of the substrate and its rotation angle with respect to the substrate. It was broadly observed that the degree of corrugation generally decreases with the increase of rotation angle or the decrease of Moiré pattern size. In contrast to a flat graphene on a TM surface, a corrugated graphene layer has an increased binding energy to the substrate and a concomitant elastic energy. Here, we developed a theoretical model about the competition between the binding energy increase and the elastic energy of corrugated graphene layers on TM surfaces in which all the parameters can be calculated by density functional theory (DFT) calculations. The agreement between the theoretical model and the experimental observations of graphene on various TM surfaces, for example, Ru(0001), Rh(111), Pt(111), and Ir(111), substantiated the applicability of this model for graphene on other TM surfaces. Moreover, the morphology of a graphene layer on an arbitrary TM surface can be theoretically predicted through simple DFT calculations based on the model. Our work thus provides a theoretical framework for the intelligent design of graphene/TM superstructures with the desired structure.