  Single-atom-thick two-dimensional materials such as graphene usually have a
hexagonal lattice while the square-planar lattice is uncommon in the family of
two-dimensional materials. Here, we demonstrate that single-atom-thick
transition metal nitride CrN$_4$ monolayer is a stable free-standing layer with
a square-planar network.
  The stability of square-planar geometry is ascribed to the combination of N=N
double bond, Cr-N coordination bond, and $\pi$-d conjugation, in which the
double $\pi$-d conjugation is rarely reported in previous studies.
  This mechanism is entirely different from that of the reported
two-dimensional materials, leading to lower formation energy and more robust
stability compared to the synthesized g-C$_3$N$_4$ monolayer.
  On the other hand, CrN$_4$ layer has a ferromagnetic ground state, in which
the ferromagnetic coupling between two Cr atoms is mediated by electrons of the
half-filled large $\pi$ orbitals from $\pi$-d conjugation.
  The high-temperature ferromagnetism in CrN$_4$ monolayer is confirmed by
solving the Heisenberg model with Monte Carlo method.
