One of the most important tasks in the development of high-performance spintronic devices is the preparation of two-dimensional (2D) magnetic layers with long-range exchange interactions. ${\mathrm{MN}}_4$ embedded graphene $\left({\mathrm{MN}}_4\text{−}\mathrm{G}\right)$ layers, with $\mathrm{M}$ being transition metal elements, are experimentally accessible 2D layers which exhibit interesting magnetic properties. In this paper, by employing the spin-polarized density functional theory (SP-DFT), we study ${\mathrm{MN}}_4\text{−}\mathrm{G}$ layers with a special focus on the behavior of the indirect $\mathrm{M}\text{−}\mathrm{M}$ exchange interactions and demonstrate that the ${\mathrm{MN}}_4\text{−}\mathrm{G}$ layers with $\mathrm{M}=\mathrm{Fe}$, Mn, and Co, are 2D anisotropic magnetic layers with Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction. By examining the electronic configurations of the $\mathrm{M}$ atoms for various $\mathrm{M}\text{−}\mathrm{M}$ spacers, we demonstrate that the RKKY interaction in such layers are tunable and exhibit an unusual, prolonged decay ($r^{-n}$, with $0.5\le n\le 2)$. In addition, we isnvestigate the influence of the ${\mathrm{CuN}}_4$ moiety in the graphene host, and show that, in contrary to the other ${\mathrm{MN}}_4\text{−}\mathrm{G}$ layers, the 2D ${\mathrm{CuN}}_4\text{−}\mathrm{G}$ layer behaves as decoupled one-dimensional spin chains, regardless of the spacer lengths.

• Received 2 May 2022
• Accepted 18 July 2022

DOI:https://doi.org/10.1103/PhysRevB.106.064407