  Three new novel phases of carbon nitride (CN) bilayer, which are named as
\alpha-C$_{2}$N$_{2}$, \beta-C$_{2}$N$_{2}$ and \gamma-C$_{4}$N$_{4}$,
respectively, have been predicted in this paper. All of them are consisted of
two CN sheets connected by C-C covalent bonds. The phonon dispersions reveal
that all these phases are dynamically stable, since no imaginary frequency is
found for them. Transition path way between \alpha-C$_{2}$N$_{2}$ and
\beta-C$_{2}$N$_{2}$ is investigated, which involves bond-breaking and
bond-reforming between C and N. This conversion is difficult, since the
activation energy barrier is found to be 1.90 eV per unit cell, high enough to
prevent the transformation at room temperature. Electronic structures
calculations show that they are all semiconductors with indirect band gap of
3.76 / 5.22 eV, 4.23 / 5.75 eV and 2.06 / 3.53 eV by PBE / HSE calculation,
respectively. The \beta-C$_{2}$N$_{2}$ has the widest band gap among the three
phases. From our results, the three new two-dimensional materials have
potential applications in the electronics, semiconductors, optics and
spintronics.
