  Graphene and its derivatives distinguish themselves for their large negative
thermal expansion even at temperatures as high as 1000K. The linear thermal
expansion coefficients (LTEC) of two-dimensional honeycomb structured pure
graphene and B/N doped graphene are analyzed using ab initio density functional
perturbation theory (DFPT) employed in VASP software under quasiharmonic
approximation. One of the essential ingredients required is the phonon
frequencies for a set of points in the Brillouin zone and their volume
dependence. These were obtained from the dynamical matrix which was calculated
using VASP code in interface with phonopy code. In particular, the transverse
acoustic modes (ZA) behave drastically differently as compared to planer modes
and so also their volume dependence. Using this approach firstly thermal
expansion for pure graphene is calculated. The results agree with earlier
calculations using similar approach. Thereafter we have studied the effect of
boron and nitrogen doping on LTEC. The LTEC of graphene is found to be negative
in the entire range of temperature under study (0-1000K) and its value at room
temperature (RT) is around -3.26X10-6K-1. The value of LTEC at RT becomes more
negative with B/N doping in graphene. 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 modes which
are 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 thermodynamical properties of 2D structures. We have extended the study
to calculate the LTEC of h-BN sheet also.
