  Ab-initio density functional perturbation theory (DFPT) has been employed to
study thermodynamical properties of pure and doped graphene sheet and the
results have been compared with available theoretical and experimental data.
The concentration of B and N has been varied upto 50% of the carbon atoms in
graphene. Phonon frequencies are essential ingredients entering into such a
calculation, which have been computed by using the dynamical matrix provided by
VASP software in combination with phonopy code in the harmonic approximation.
This easily provides us the Helmholtz free energy and leads us to numerical
estimates of various thermodynamical properties. The results for specific heat
are in good agreement with various theoretical and experimental studies
obtained earlier for pure graphene. Interesting new results have been reported
for B and N substituted structure. It has been observed that specific heat
decreases with the increase in concentration of doping while the entropy
increases. Further, large doping concentrations result in unstable sheets
resulting in imaginary frequencies in the transverse directions. The
instability needs to be compensated by external strains and that has been
assumed while carrying out Brilluoin summations. These results will be useful
for calculation of thermal conductivity of doped graphene and thus feasibility
of using these for device applications. Our preliminary results for thermal
expansion have indicated negative thermal expansion behavior of pure graphene
at low temperatures which needs investigation on doped graphene as well.
