  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.
