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We use density functional theory based first-principles method to investigatethe bandstructure and phase stability in the laterally grown hexagonalC$_x$(BN)$_{1-x}$, two-dimensional Graphene and $h$-BN hybrid nanomaterials,which were synthesized by experimental groups recently (Liu $et al$, NatureNanotech, 8, 119 (2013)). Our detail electronic structure calculations on suchmaterials, with both armchair and zigzag interfaces between the Graphene and $h$-BN domains, indicate that the band-gap decreases non-monotonically with theconcentration of Carbon. The calculated bandstructure shows the onset of Diraccone like features near the band-gap at high Carbon concentration ($x \sim0.8$). From the calculated energy of formation, the phase stability ofC$_x$(BN)$_{1-x}$ was studied using a regular solution model and the system wasfound to be in the ordered phase below a few thousand Kelvin. Furthermore,using the Boltzmann transport theory we calculate the electrical resistivityfrom 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

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