Flexural mode of graphene on a substrate

Out of plane vibrations are suppressed in graphene layers placed on a substrate. These modes, in suspended samples, are relevant for the understanding of properties such as the resistivity, the thermal expansion coefficient, and other. We present a general framework for the study of the properties of out of plane modes in graphene on different substrates. We use the model to estimate the substrate induced changes in the thermal expansion coefficient, or in the temperature dependence of the resistivity.