Electronic Structure, Phase Stability and Resistivity of Hybrid Hexagonal C$_x$(BN)$_{1-x}$ Two-dimensional Nanomaterial: A First-principles Study

We use density functional theory based first-principles method to investigate the bandstructure and phase stability in the laterally grown hexagonal C$_x$(BN)$_{1-x}$, two-dimensional Graphene and $h$-BN hybrid nanomaterials, which were synthesized by experimental groups recently (Liu $et al$, Nature Nanotech, 8, 119 (2013)). Our detail electronic structure calculations on such materials, with both armchair and zigzag interfaces between the Graphene and $h$-BN domains, indicate that the band-gap decreases non-monotonically with the concentration of Carbon. The calculated bandstructure shows the onset of Dirac cone like features near the band-gap at high Carbon concentration ($x \sim 0.8$). From the calculated energy of formation, the phase stability of C$_x$(BN)$_{1-x}$ was studied using a regular solution model and the system was found to be in the ordered phase below a few thousand Kelvin. Furthermore, using the Boltzmann transport theory we calculate the electrical resistivity from the bandstrcture of C$_x$(BN)$_{1-x}$ at different temperature ($T$), which shows a linear behaviour when plotted in the logarithmic scale against $T^{-1}$, as observed experimentally