Thermal expansion in 2D honeycomb structures: Role of transverse phonon modes

Graphene and its derivatives including hexagonal BN are notorious for their large negative thermal expansion over a wide range of temperature which is quite unusual. We attempt to analyze this unusual behavior on the basis of character of the phonon modes. The linear thermal expansion coefficients (LTEC) of two-dimensional honeycomb structured pure graphene, h-BN and B/N doped graphene are studied using density functional perturbation theory (DFPT) under quasi harmonic approximation. The dynamical matrix and the phonon frequencies were calculated using VASP code in interface with phonopy code. The approach is first applied to pure graphene to calculate thermal expansion. The results agree with earlier calculations using similar approach. Thereafter we have studied the effect of B/N doping on LTEC and also compared it with LTEC of h-BN sheet. The LTEC of graphene is negative in the whole temperature range under study (0-1000K) with a room temperature (RT) value of -3.51X10-6K-1. The value of LTEC at room temperature becomes more negative with B/N doping in graphene as well as for h-BN sheet. In order to get an insight into the cause of negative thermal expansion, we have computed the contribution of individual phonon modes of vibration. We notice that it is principally the ZA (transverse acoustic) mode which is responsible for negative thermal expansion. It has been concluded that transverse mode in 2D hexagonal lattices have an important role to play in many of the thermo dynamical properties of 2D structures.