Tuning magnetic anisotropy in Fe$_{5}$GeTe$_{2}$ monolayer through doping and strain

Controlling magnetic anisotropy energy (MAE) in two-dimensional (2D) ferromagnetic materials is crucial for designing novel spintronic devices. Using first-principles calculations, we systematically investigate the magnetic properties of monolayer Fe$_5$GeTe$_2$ (F5GT) under two scenarios: (I) Co and Ni doping, and (II) compressive and tensile strains. Our results show that the F5GT monolayer exhibits a weak in-plane MAE, which can be significantly enhanced by Co doping. Additionally, a $1\%$ compressive strain switches the magnetic easy axis from in-plane to out-of-plane, while $4\%$ compressive strain can further enhance the out-of-plane MAE. Spin-orbit coupling (SOC) matrix analysis reveals that the enhancement of in-plane MAE in Co-doped F5GT (Co-F5GT) arises from changes in $\left\langle {\it p}_{x} \left\vert L_z\right\vert {\it p}_{y} \right\rangle$ of Te and $\left\langle {\it d}_{xy} \left\vert L_z\right\vert {\it d}_{x^2+y^2} \right\rangle$ of Fe(2) and Fe(3). The effect of compressive strain is primarily attributed to a substantial increase in the positive contribution from $\left\langle {\it d}_{xy} \left\vert L_z\right\vert {\it d}_{x^2+y^2} \right\rangle$ of Fe(1).