Thermodynamics of quantum crystalline membranes

We investigate the thermodynamic properties and the lattice stability of two-dimensional crystalline membranes, such as graphene and related compounds, in the low-temperature quantum regime T→0. A key role is played by the anharmonic coupling between in-plane and out-of-plane lattice modes that, in the quantum limit, has very different consequences from those in the classical regime. The role of retardation, namely of frequency dependence, in the effective anharmonic interactions turns out to be crucial in the quantum regime. We identify a crossover temperature, T*, between the classical and quantum regimes, which is ∼70-90 K for graphene. Below T*, the heat capacity and thermal expansion coefficient decrease as power laws with decreasing temperature, tending to zero for T→0 as required by the third law of thermodynamics.We thank SURFsara (www.surfsara.nl) for the support Grant No. (MP-282-13) in using the Lisa Compute Cluster. B.A. acknowledges support from Fundação para a Ciência e a Tecnologia (Portugal), through Grant No. SFRH/BD/78987/2011. R.R. acknowledges financial support from the Juan de la Cierva Programe (MEC, Spain). E.C. acknowledges support from the European project FP7-PEOPLE-2013-CIG “LSIE_2D” and Italian National MIUR Prin project 20105ZZTSE. R.R. and F.G. gratefully acknowledge the financial support from MINECO, Spain, through Grant No. FIS2011-23713. M.I.K and A.F. acknowledge funding from the European Union Seventh Framework Programme under Grant Agreement No. 604391 Graphene Flagship