Tuning the magnetic interactions in van der Waals Fe3GeTe2 heterostructures: A comparative study of ab initio methods

We investigate the impact of mechanical strain, stacking order, and external electric fields on the magnetic interactions of a two-dimensional (2D) van der Waals heterostructure in which a 2D ferromagnetic metallic ${\mathrm{Fe}}_{3}{\mathrm{GeTe}}_{2}$ monolayer is deposited on germanene. Three distinct computational approaches based on ab initio methods are used, (i) the Green's function method, (ii) the generalized Bloch theorem, and (iii) the supercell approach, and a careful comparison is given. First, the shell-resolved exchange constants are calculated for the three Fe atoms within the unit cell of the freestanding ${\mathrm{Fe}}_{3}{\mathrm{GeTe}}_{2}$ monolayer. We find that the results obtained with approaches (i) and (ii) are in good qualitative agreement and also in good qualitative agreement with previously reported values. An electric field of $\mathcal{E}=\ifmmode\pm\else\textpm\fi{}0.5$ V/\AA{} applied perpendicular to the ${\mathrm{Fe}}_{3}{\mathrm{GeTe}}_{2}$/germanene heterostructure leads to significant changes in the exchange constants. We show that the Dzyaloshinskii-Moriya interaction (DMI) in ${\mathrm{Fe}}_{3}{\mathrm{GeTe}}_{2}$/germanene is mainly dominated by the nearest neighbors, resulting in a good quantitative agreement between approaches (i) and (ii). Furthermore, we demonstrate that the DMI is highly tunable by strain, stacking, and electric field, leading to a large DMI comparable to that of ferromagnetic/heavy metal interfaces, which have been recognized as prototypical multilayer systems to host isolated skyrmions. The geometrical change and hybridization effect explain the origin of the high tunability of the DMI at the interface. The electric-field-driven DMI obtained by approach (iii) is in qualitative agreement with the more accurate ab initio method used in approach (ii). However, the field effect on the DMI is overestimated by approach (iii) by about 50%. This discrepancy is attributed to the different implementations of the electric field and basis sets used in the ab initio methods applied in approaches (ii) and (iii). The magnetocrystalline anisotropy energy (MAE) can also be drastically changed by the application of compressive or tensile strain in the ${\mathrm{Fe}}_{3}{\mathrm{GeTe}}_{2}$/germanene heterostructure. The application of an electric field, in contrast, leads only to relatively small changes in the MAE for electric fields of up to 1 V/\AA{}.