Prediction of single-atom-thick transition metal nitride ${\mathrm{CrN}}_4$ with a square-planar network and high-temperature ferromagnetism

Single-atom-thick two-dimensional (2D) materials such as graphene usually have a hexagonal lattice while the square-planar lattice is uncommon in the family of 2D materials. Here, we demonstrate that single-atom-thick transition metal nitride ${\mathrm{CrN}}_4$ 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 2D materials, leading to lower formation energy and more robust stability than the synthesized $g-{\mathrm{C}}_3{\mathrm{N}}_4$ monolayer. On the other hand, the ${\mathrm{CrN}}_4$ layer has a ferromagnetic (FM) ground state, in which the FM 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 ${\mathrm{CrN}}_4$ monolayer is confirmed by solving the Heisenberg model with the Monte Carlo method.

Figure 1

Atomic structure of square ${\mathrm{CrN}}_4$ monolayer with all atoms in a plane. The unit cell is marked with a dashed line square.

Figure 2

(a) Phonon spectra of ${\mathrm{CrN}}_4$ monolayer. (b) The evolution of total potential energy with the time at 1000 K for 10 ps, where the insets are the top and side views of the final configuration of ${\mathrm{CrN}}_4$ monolayer after 10 ps molecular dynamics simulation.

Figure 3

The convex hull diagram of Cr–N and Be–N systems.

Figure 4

Partial density of states of Cr $3d$ and N $2p$ suborbitals in ${\mathrm{CrN}}_4$ sheet. The insets are the diagrams of the relevant orbital shape.

Figure 5

Band structure of ${\mathrm{CrN}}_4$ monolayer in ferromagnetic phase with different functional methods: (a) PBE, (b) GGA $+ U$, (c) SCAN, (d) HSE. The Fermi energy is set to zero.

Figure 6

(a) Top view of atomic structure of ${\mathrm{CrN}}_4$. $J_1$ and $J_2$ are the nearest and next-nearest neighbored exchange interaction. [(b)–(d)] The sketches of FM, coll-AFM, and neel-AFM.

Figure 7

The susceptibility $\chi$ and average magnetic moment $M$ as functions of temperature for the Heisenberg model on a square lattice, and the parameters $J_1, J_2$, and $A$ of the Heisenberg model derived from the various functional calculations: (a) PBE, (b) GGA $+ U$, (c) SCAN, and (d) HSE.

Figure 8

(a)Atomic structure of ${\mathrm{CrN}}_4$ monolayer with the exchange coupling $J_1, J_2$, and $J_3$. (b) Ferromagnetic order (FM). (c) Antiferromagnetic order I (AFMI). (d) Antiferromagnetic order II (AFMII). (e) Antiferromagnetic order III (AFMIII).

Figure 9

Sequence numbers of structures predicted by calypso. The red symbols represent the typical structures, and the blue symbols mean the repetitive one to the structure represented by the left red symbol.

Figure 10

The typical structures predicted by calypso, the energies of which are displayed in Fig. 9 with red symbols.