## 2 Methods  
We used DFT as implemented in the Quantum Espresso [26] and Fleur code [27] to investigate the electronic and magnetic properties of GdFe\({}_{2}\)/TMs film. For the TMs capping layers, we have considered Ru, Rh, Pd, and Ag in 4\(d\) and Os, Ir, Pt and Au in 5\(d\). For the exchange-correlation potential we adapted the generalized gradient approximation (GGA). The wave functions were expanded by a plane-wave basis set with an optimized cutoff energy of 350 Ry, and the Brillouin zone was sampled via a \(12\times 12\times 1\)\(k\)-point mesh. Different mesh values from 36 to 256 were tested to ensure the precision of our calculations, with the convergence criterion being 0.1 \(\upmu\)eV. The convergence with respect to cutoff was also carefully checked.  
Total energy E(q) is calculated along the paths of \(\bar{\Gamma}\)-\(\bar{K}\) and \(\bar{\Gamma}\)-\(\bar{M}\) which have the highest symmetry among other directions in the two-dimensional Brillouin zone (2D BZ). E(q) with and without SOC [28] are separately displayed in Figure 1. In the 2D BZ, we characterize spin spiral phase using the wave vector \(\mathbf{q}\) with a constant angle of \(\phi\), where \(\phi\) is defined as \(\mathbf{q}\cdot\mathbf{R}\).  
Figure 1: Energy dispersion E(q) of homogeneous cycloidal flat spin spirals in high-symmetry direction \(\bar{\Gamma}\)-\(\bar{\mathrm{K}}\) for (**a**) GdFe\({}_{2}\)/Rh and (**b**) GdFe\({}_{2}\)/Rh films. Filled and empty symbols represent E(q) with and without SOC, respectively. The energy is given relative to the magnetic ground state. The dispersion is fitted to the Heisenberg model (dotted line) and includes the DMI and MCA (solid line).  
In order to examine the magnetically characteristic of GdFe\({}_{2}\) films with TM capping layers, we adopt the atomistic spin model given by References [29; 30; 31]:  
\[H=-\sum\limits_{ij}{{{J}_{ij}}({{m}_{i}}\cdot{{m}_{j}})}-\sum\limits_{ij}{{{D}_{ ij}}({{m}_{i}}\times{{m}_{j}})}+\sum\limits_{i}{{{K}({{m}_{i}^{\pi}})}^{2}}-\sum \limits_{i}{{{\mu}_{s}}({{B}}\cdot{{m}_{i}})}. \tag{1}\]  
By using Equation (1), we can describe the magnetic interactions between two neighbor Fe atoms with spins of \(\textbf{M}_{i}\) and \(\textbf{M}_{j}\) at sites \(\textbf{R}_{i}\) and \(\textbf{R}_{j}\), respectively. Here, \({{m}_{i}}\) is defined as \(\textbf{M}_{i}/{{\mu}_{s}}\). Both energy dispersion curves (with and without SOC) are calculated and fitted to extract the parameters for the exchange interactions (\({{J}_{ij}}\)) and the DMI (\({{D}_{ij}}\)).  
We then compute the magnetic state by solving the Landau-Lifshitz-Gilbert (LLG) equation,  
\[\frac{{d\textbf{S}_{i}}}{{dt}}=-\gamma^{\prime}{\textbf{S}_{i}}\times({ \textbf{B}_{i}^{\text{eff}}}+{\textbf{B}_{i}^{\text{th}}})-\gamma^{\prime} \alpha{\textbf{S}_{i}}\times[{\textbf{S}_{i}}\times({\textbf{B}_{i}^{\text{ eff}}}+{\textbf{B}_{i}^{\text{th}}})]. \tag{2}\]  
Here \(\alpha\) denotes the Gilbert damping parameter. When \(\gamma\) is the gyromagnetic ratio, \(\gamma^{\prime}\) represents \(\dfrac{\gamma}{1+\alpha^{2}}\). \({\textbf{B}_{i}^{\text{eff}}}\) is the effective magnetic field at site \(i\), and \({\textbf{B}_{i}^{\text{th}}}\) is the thermal noise. The LLG simulations were done with mumax3 [32]. For the present systems we use material parameters obtained from DFT: \(K\) = 2-14 meV and \(D\) = 0.2-1.6 meV (see Figure 2). To verify the numerical stability of the simulations, calculations with different cell sizes were performed. Finally, the thin films are discretized in a 400 \(\times\) 400 \(\times\) 2 mesh with periodic boundary conditions in in-plane directions.  
The MCA energy was calculated using the force theorem and defined as the total energy difference between the magnetization perpendicular to the [100]-plane and parallel to the [100]-plane. Therefore, MCA energy \(E_{\text{MCA}}\) = E\({}_{[100]}-E_{[001]}\), where \(E_{[100]}\) and \(E_{[001]}\) are the total energies with the magnetization aligned along the [100] and [001] of the magnetic anisotropy, respectively.