  Using first-principles density functional theory calculations, we investigate
a family of stable two-dimensional crystals with chemical formula $A_2B_2$,
where $A$ and $B$ belong to groups IV and V, respectively ($A$ = C, Si, Ge, Sn,
Pb; $B$ = N, P, As, Sb, Bi). Two structural symmetries of hexagonal lattices
$P\bar{6}m2$ and $P\bar{3}m1$ are shown to be dynamically stable, named as
$\alpha$- and $\beta$-phases correspondingly. Both phases have similar cohesive
energies, and the $\alpha$-phase is found to be energetically favorable for
structures except CP, CAs, CSb and CBi, for which the $\beta$-phase is favored.
The effects of spin-orbit coupling and Hartree-Fock corrections to
exchange-correlation are included to elucidate the electronic structures. All
structures are semiconductors except CBi and PbN, which have metallic
character. SiBi, GeBi and SnBi have direct band gaps, whereas the remaining
semiconductor structures have indirect band gaps. All structures have quartic
dispersion in their valence bands, some of which make the valence band maximum
and resemble a Mexican hat shape. SnAs and PbAs have purely quartic valence
band edges, i.e. $E{\sim}{-}\alpha k^4$, a property reported for the first
time. The predicted materials are candidates for a variety of applications.
Owing to their wide band gaps, CP, SiN, SiP, SiAs, GeN, GeP can find their
applications in optoelectronics. The relative band positions qualify a number
of the structures as suitable for water splitting, where CN and SiAs are
favorable at all pH values. Structures with quartic band edges are expected to
be efficient for thermoelectric applications.
