Engineering of the thermodynamic properties of bilayer graphene by atomic plane rotations: the role of the out-of-plane phonons.

We investigated theoretically the specific heat of graphene, bilayer graphene and twisted bilayer graphene taking into account the exact phonon dispersion and density of states for each polarization branch. It is shown that contrary to a conventional belief the dispersion of the out-of-plane acoustic phonons - referred to as ZA phonons - deviates strongly from a parabolic law starting from the frequencies as low as ∼100 cm(-1). This leads to the frequency-dependent ZA phonon density of states and the breakdown of the linear dependence of the specific heat on temperature T. We established that ZA phonons determine the specific heat for T ≤ 200 K while contributions from both in-plane and out-of-plane acoustic phonons are dominant for 200 K ≤ T ≤ 500 K. In the high-temperature limit, T > 1000 K, the optical and acoustic phonons contribute approximately equally to the specific heat. The Debye temperature for graphene and twisted bilayer graphene was calculated to be around ∼1861-1864 K. Our results suggest that the thermodynamic properties of materials such as bilayer graphene can be controlled at the atomic scale by rotation of the sp(2)-carbon planes.