## 3 Results and Discussion  
The in-plane lattice constant of 7.32 A was taken from the experimental lattice constant of Laves phase of GdFe\({}_{2}\), with lattice mismatches of 3.6% (Rh)-14.2% (Os), as depicted in Figure 3a. From the total energy calculation, it was confirmed that the hollow site is the most energetically favorable to stack the TM layer (see Figure 3). The atoms of GdFe\({}_{2}\) and TM capping layer were fully relaxed by atomic force calculations. Figure 2: (a) Total magnetocrystalline anisotropy (MCA) energy and (b) effective Dzyaloshinskii–Moriya interaction (DMI) of GdFe\({}_{2}\) with TM capping layer. After structural optimization, the interface distances between the TM capping layer and the GdFe\({}_{2}\) is presented in Figure 3b. As the atomic number becomes larger in the \(4d\) and \(5d\) TMs, the interlayer distances increase monotonically. Induced spin moments of the TMs for TM/GdFe\({}_{2}\) are presented in Figure 3c. The Rh and Ir capping layers, which are the Co-group elements, are found to have the largest moments of 0.98 and 0.80 \(\mu_{B}\). For all of the TM/GdFe\({}_{2}\), the direction of magnetization is favored to perpendicularly orientate to the film plane. Interestingly, the MCA energy and DMI of GdFe\({}_{2}\) films capped by \(5d\) TMs are significantly larger than those of GdFe\({}_{2}\) with \(4d\) TMs. In particular, the Ir-capped GdFe\({}_{2}\) film exhibits the largest MCA energy of 14.1 meV and effective DMI of 1.6 meV. We attribute the substantial enhancement of MCA energy and DMI in GdFe\({}_{2}\) with the \(5d\) capping layer to the strong SOC of the \(5d\) orbitals because the SOC is proportional to the fourth power of the atomic number. Since the \(4d\) also exhibit similar trend with \(5d\), Rh has the largest magnetic moments and MCA energy among other \(4d\) TMs. This is related to the band-filling effect and orbital hybridization. The calculated energy dispersion E(q) of spin spirals is presented in Figure 1 along the high-symmetry direction, \(\bar{\Gamma}\)-\(\bar{\text{K}}\) for GdFe\({}_{2}\) capped by Rh and Ir which exhibit the largest magnetic moment, MCA energy, and effective DMI among the \(4d\) and \(5d\) TM elements, respectively. In the results without SOC, a minimum point of the energy dispersion is observed at the \(\bar{\Gamma}\) point, and it degenerates for right-(\(q>0\)) and left-rotating (\(q<0\)) spirals. For both Rh- and Ir-capped films, it is confirmed that the out-of-plane direction is an easy magnetization axis due to SOC (see Figure 2a). As a result of imperfect inversion symmetry at the interface, the SOC for spin spirals derives DMI in system [33; 34]. Therefore, DMI leads to non-collinear spin structures with the magnetic moments on  
Figure 3: (a) Side view and top view of GeFe\({}_{2}\) film capped by a transition-metal (TM) monolayer. Blue, gray, and red balls represent Gd, Fe, and TM atoms, respectively. TM atoms are on the hollow site of GeFe\({}_{2}\); (b) Interface distances between the TM capping layer and GeFe\({}_{2}\) after structural optimization; (c) Magnetic moments of TM atoms, induced by GeFe\({}_{2}\). an oblique angle. In case of the inclusion of the DMI, the E(q) has the lowest value for a homogeneous cycloidal flat spin spiral state with a particular rotational sense [35]. As presented in Figure 1, an energy minimum of \(-\)0.50 meV/atom and \(-\)0.35 meV/atom compared to the ground magnetic state appears for a right-rotating spin spiral for GdFe\({}_{2}\) films with Rh and Ir capping, respectively. A skyrmion can be considered to be an intermediate state between spin spiral state and ferromagnetic state in a magnetic material because it rises from the competition between the exchange interaction that is responsible for the ferromagnetic state and the anisotropic exchange that generates spin spiral behavior.