**Abbreviations**

The following abbreviations are used in this manuscript:


### **Appendix A. Energy Minimization**

Because in this particular study we are not interested in the dynamic evolution of the magnetization but only in static, converged magnetic structures, the integration of the LLG equation in the code fulfils the practical role of guiding the system along a path of energy-minimization in an iterative way. Since the dynamics of the magnetization during the transition from the initial to the converged state is irrelevant for this work, we are free to choose a conveniently large damping parameter *α* = 0.5 in order to accelerate the energy minimization. Furthermore, we remove the precession term by setting *γ* = 0 in Equation (5), thereby effectively using a a direct energy minimization scheme instead of following the path of the magnetization dynamics described by the LLG equation. The numerical integration is done with a Dormand-Prince algorithm [52], and the effective field values are refreshed several times during each time step. The choice of a large value of the damping allows us to use time steps of up to 1 ps, which is about ten times larger than the step size that we would usually employ in dynamic simulations with low damping. With these parameters, and owing to the GPU acceleration of our code, it takes only a short time (between several minutes and a few hours) to simulate the magnetic structures discussed in this work. To calculate the skyrmion states, we either start from a random configuration or saturate the magnetization along the positive *z* direction and subsequently let the system relax in the presence of an external magnetic field aligned along the negative *z*direction. The *z* axis is oriented parallel to the surface normal, as shown in Figure 1.
