  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. MN4 embedded graphene (MN4-G) layers,
with 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 MN4-G
layers with a special focus on the behavior of the indirect M-M exchange
interactions, and demonstrate that the MN4-Gs with M = Fe, Mn and Co, are 2D
anisotropic magnetic layers with Ruderman-Kittel-Kasuya-Yosida (RKKY)
interaction. By examining the electronic configurations of the M atoms for
various M-M spacers, we demonstrate that the RKKY interaction in such layers
are tunable and exhibiting an unusual prolonged decay (r to the power of minus
n, n between 0.5 and 2). In addition, we investigate the influence of the CuN4
moiety in the graphene host, and show that, in contrary to the other MN4-Gs,
the 2D CuN4-G layer behaves as decoupled one-dimensional spin chains,
regardless of the spacer lengths.
