  Phase stability and properties of two-dimensional phosphorus carbide,
P$_x$C$_{1-x}$, are investigated using the first-principles method in
combination with cluster expansion and Monte Carlo simulation. Monolayer
P$_x$C$_{1-x}$ is found to be a phase separating system which indicates
difficulty in fabricating monolayer P$_x$C$_{1-x}$ or crystalline
P$_x$C$_{1-x}$ thin films. Nevertheless, a bottom-up design approach is used to
determine the stable structures of P$_x$C$_{1-x}$ of various compositions which
turn out to be superlattices consisting of alternating carbon and phosphorus
nanoribbons along the armchair direction. Results of first-principles
calculations indicate that once these structures are produced, they are
mechanically and thermodynamically stable. All the ordered structures are
predicted to be semiconductors, with band gap ranging from 0.2 to 1.2 eV. In
addition, the monolayer P$_x$C$_{1-x}$ are predicted to have high carrier
mobility, and high optical absorption in the ultraviolet region which shows a
red-shift as the P:C ratio increases. These properties make 2D P$_x$C$_{1-x}$
promising materials for applications in electronics and optoelectronics.
