  The widely used crystal structures for both heptazine-based and
triazine-based two-dimensional (2D) graphitic carbon nitride (g-C$_3$N$_4$) are
the flat P-6m2 configurations. However, the experimentally synthesized 2D
g-C$_3$N$_4$ possess thickness ranging in 0.2-0.5 nm, indicating that the
theoretically used flat P-6m2 configurations are not the correct ground states.
In this work, we propose three new corrugated structures P321, P3m1 and Pca21
with energies of 66 (86), 77 (87) and 78 (89) meV/atom lower than that of the
corresponding heptazine-based (triazine-based) g-C$_3$N$_4$ in flat P-6m2
configuration, respectively. These corrugated structures have very similar
periodic patterns to the flat P-6m2 ones and they are difficult to be
distinguished from each other according to their top-views. The optimized
thicknesses of the three corrugated structures ranging in 1.347-3.142 {\AA} are
in good agreement with the experimental results. The first-principles results
show that these corrugated structural candidates are also semiconductors with
band gaps slightly larger than those of the correspondingly flat P-6m2 ones.
Furthermore, they possess also suitable band edge positions for
sun-light-driven water-splitting at both $pH=0$ and $pH=7$ environments. Our
results show that these three new structures are more promising candidates for
the experimentally synthesized g-C$_3$N$_4$.
