Anharmonic effects on thermodynamic properties of a graphene monolayer

We extend the unsymmetrized self-consistent-field method (USF) for anharmonic crystals to layered non-Bravais crystals to investigate structural, dynamical and thermodynamic properties of a free-standing graphene monolayer. In this theory, the main anharmonicity of the crystal lattice has been included and the quantum corrections are taken into account in an ℏ-expansion for the one-particle density matrix. The obtained result for the thermal expansion coefficient (TEC) of graphene shows a strong temperature dependence and agrees with experimental results by Bao et al. (Nat. Nanotechnol., 4 (2009) 562). The obtained value of TEC at room temperature (300 K) is $-6.4\times10^{-6}\ \text{K}^{-1}$ and it becomes positive for $T>T_{\alpha}=358\ \text{K}$ . We find that quantum effects are significant for $T