## 2 Analytical Model
### Skyrmion Energy  
It has been considered an outwardly directed radial Neel skyrmion with a magnetization pointing along \(-z\) at the core center (skyrmion number \(S=-1\)) stabilized in an ultrathin magnetic cylindrical dot of radius \(R_{\rm d}\) in a cylindrical reference frame (\(\rho,\varphi,z\)) with \(\rho\) the radial coordinate, \(\varphi\) the azimuthal coordinate and \(z\) the coordinate perpendicular to the dot plane. Neel skyrmion's texture, expressing the orientation of the magnetization, is \({\bf m}(x,y)\ =\ \sin\ \theta(\rho)\hat{\rho}+\cos\ \theta(\rho)\hat{z}\) with \({\bf m}={\bf M}/M_{\rm s}\), \({\bf m}=(m_{\rho},m_{z})\) and \(\theta\) the polar angle. In Figure 1, a pictorial representation of the ferromagnetic cylindrical dot (e.g., Co) hosting the Neel magnetic skyrmion is shown together with the cylindrical reference frame. The skyrmion has a cylindrical symmetry with respect to the out-of-plane direction (\(z\)-axis). The cylindrical dot is assumed to be in contact with a thin layer of heavy metal (e.g., Pt) providing a consistent IDMI interaction (not shown). Figure 1: A sketch of the cylindrical dot hosting the Néel skyrmion with \(R_{\rm d}\) the dot radius and \(t\) the dot thickness. A cylindrical reference frame (\(\rho,\varphi,z\)) is also shown. The skyrmion energy \(E\) was numerically calculated as a volume integral of the skyrmion energy density \(\varepsilon_{\text{tot}}\), namely \(E\Big{(}r_{\text{sky}}\Big{)}\ =\ \int\varepsilon_{\text{tot}}\Big{(}r,r_{\text{sky}} \Big{)}dV\ =\ \int\limits_{-t/2}^{t/2}dz\int\limits_{0}^{2\pi}d\varphi\int\limits_{0}^{R_{ d}}\varepsilon_{\text{tot}}\Big{(}r,r_{\text{sky}}\Big{)}\rho d\rho\). Here, \(t\) is the dot thickness, \(r_{\text{sky}}=R_{\text{sky}}/l_{\text{exch}}\) is the dimensionless skyrmion radius with \(R_{\text{sky}}\) the skyrmion radius and \(l_{\text{exch}}=\sqrt{2A}/(\mu_{0}M_{\text{s}}{}^{2})\) the exchange length being \(\mu_{0}\) the vacuum permeability. For the skyrmion energy calculation it has been used as the ansatz distribution \(\Theta_{0}(r)\ =\ 2\text{arctan}[\frac{\beta}{e}e^{-\xi r}]\) expressing the magnetization at equilibrium in excellent agreement with the magnetization distribution of micromagnetic simulations [15]. Here, \(B=r_{\text{sky}}\exp\ [\xi\ r_{\text{sky}}]\) being \(\xi=\sqrt{Q-1}\ (Q=2K_{\text{u}}/(\mu_{0}M_{\text{s}}{}^{2})\) is the quality factor with \(K_{\text{u}}\) the uniaxial anisotropy constant), \(r_{\text{sky}}=R_{\text{sky}}/l_{\text{exch}}\) is the dimensionless skyrmion radius and \(r=\rho/l_{\text{exch}}\) is the dimensionless radial coordinate. The skyrmion energy density \(\varepsilon=\varepsilon_{\text{exch}}+\varepsilon_{\text{IDMI}}+\varepsilon_{ \text{ani}}+\varepsilon_{\text{ext}}\) contains all the relevant contributions. In particular, \(\varepsilon_{\text{exch}}\ =\ A(\
abla\mathbf{m})^{2}\) is the exchange energy density with \(A\) the material exchange stiffness constant, \(\varepsilon_{\text{IDMI}}\ =\ D\ [m_{2}(\
abla\cdot\mathbf{m})-(\mathbf{m} \cdot\
abla)m_{2}]\) is IDMI energy density with \(D\) the DMI parameter, \(\varepsilon_{\text{ani}}\ =\ K_{\text{u}}\Big{(}1-m_{2}^{2}\Big{)}+\frac{1}{2}\mu_{0}M_{ \text{s}}^{2}m_{2}^{2}\) is the anisotropy energy density including the perpendicular uniaxial anisotropy energy density and the magnetostatic energy density in the ultrathin approximation, and \(\varepsilon_{\text{ext}}=-M_{\text{s}}\ B_{\text{ext}}\ m_{2}\) is the Zeeman energy density with \(B_{\text{ext}}=\mu_{0}H_{\text{ext}}\) the amplitude of the external bias field parallel to the \(z\)-axis. The computation of the 2D configurational entropy is based on the calculation of the skyrmion energy. Details about the numerical calculation of the skyrmion energy can be found in [13; 14]. As done for the 3D model, the main hypothesis allowing a simplification of the calculation of the configurational entropy is the harmonic approximation.