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We investigated theoretically the specific heat of graphene, bilayer grapheneand twisted bilayer graphene taking into account the exact phonon dispersionand density of states for each polarization branch. It is shown that contraryto a conventional believe the dispersion of the out-of-plane acoustic phonons -referred to as ZA phonons - deviates strongly from a parabolic law startingfrom the frequencies as low as ~100 1/cm. This leads to the frequency-dependentZA phonon density of states and the breakdown of the linear dependence of thespecific heat on temperature T. We established that ZA phonons determine thespecific heat for T<200 K while contributions from both in-plane andout-of-plane acoustic phonons are dominant for 200 K < T < 500 K. In thehigh-temperature limit, T>1000 K, the optical and acoustic phonons contributeapproximately equally to the specific heat. The Debye temperature for grapheneand twisted bilayer graphene was calculated to be around ~1861 - 1864 K. Ourresults suggest that the thermodynamic properties of materials such as bilayergraphene can be controlled at the atomic scale by rotation of the sp2-carbonplanes.

Phonon Engineering of the Specific Heat of Twisted Bilayer Graphene: The Role of the Out-of-Plane Phonon Modes