Enhancing magnetic coupling in MN₄-graphene via strain engineering

MN₄-embedded graphene (MN₄-G) layers, with transition metal elements M, are experimentally accessible two-dimensional (2D) materials and show great potential for stable nanoscale magnetization. In these materials, the exchange couplings between magnetic atoms are predominantly governed by Ruderman-Kittel-Kasuya-Yosida (RKKY) coupling, exhibiting an unusual prolonged decay of r^-, where r is the M-M separation distance, and 0.5 ≤ n ≤ 2. In this paper, we explore the effects of induced strain on the electronic and magnetic properties of MN₄-G layers through ab initio density functional theory. We employ a specific method to apply strain by positioning atoms from one layer within the equilibrium structure of another layer, thereby inducing strain in the form of either tension or compression. The induced strain results in an approximate ±0.4% variation in the unit-cell area of the MN₄-G lattice. Our findings reveal that while the exchange coupling mechanism remains unaffected, the strength, amplitude, and decay rate of the RKKY coupling are significantly influenced by the induced strain. Notably, the CoN₄-G layer exhibits a remarkable increase in the strength and oscillation amplitude of the RKKY coupling, along with a reduced decay rate. Additionally, the electronic and magnetic properties of the CuN₄-G layers remain unchanged under induced strain.