## 3 Results and Discussion  
In this section, the main results of this work applied to an outwardly Neel skyrmion hosted in a Co dot of radius \(R_{\rm d}\) = 200 nm and \(t\) = 0.8 nm having the core magnetization \(m_{z}\) (\(r\) = 0) = \(-\)1 (\(S\) = \(-\)1) and \(\mathbf{H}_{\rm ext}\) applied along \(+\)\(z\) are discussed. The magnetic parameters are the ones used in the analytical and numerical calculations in [13,14]. In particular, the following magnetic parameters at \(T\) = 0 K were used: saturation magnetization \(M_{\rm s}\) = 600 KA/m, exchange stiffness constant \(A\) = 20 pJ/m, IDMI parameter \(D\) = 3.0 mJ/m\({}^{2}\), \(K_{\rm u}\) = 0.6 MJ/m\({}^{3}\) (for example, Co). The magnetic parameters \(A\), \(D\) and \(K_{\rm u}\) at non-zero temperature are scaled from their zero temperature values, by using the scaling laws \(A(m)\) = \(A\) (\(T\) = 0 K) \(m\) (\(T\))\({}^{3/2}\), \(D\) (\(T\) = 0 K) \(m\)(\(T\))\({}^{3/2}\), and \(K_{\rm u}\) (\(T\) = \(K_{\rm u}\) (\(T\) = 0 K) \(m\)(\(T\))\({}^{3.6}\)[15]. In Table 1 are summarized the values of \(a\) and of \(D_{\rm 0 sky}\) used in the numerical calculations [13,14]. Table 1 does not contain the values of \(a\) and \(D_{\rm 0 sky}\) at \(T\) = 0 K since the shown calculations are for \(T\) \(\geq\) 50 K.  
In Figure 2, the 2D skyrmion diameters distribution calculated via Equation (2) (solid black lines) is compared to the 3D distribution (dashed red lines) [13] obtained by means of Equation (1) for three different temperatures, \(T\) = 100 K, 200 K and 300 K, respectively and, at fixed \(T\), for three values of the external bias field, \(\mu_{0}\)\(H_{\rm ext}\) = 0 mT, 25 mT and 50 mT, respectively. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{\(T\) (K)} & \(a\) (\(\times\)10-5 /m\({}^{2}\)) & \(D_{\rm 0 sky}\) (mm) & \(a\) (\(\times\)10-5 /m\({}^{2}\)) & \(D_{\rm 0 sky}\) (mm) & \(a\) (\(\times\)10-5 /m\({}^{2}\)) & \(D_{\rm 0 sky}\) (mm) \\  & \(\mu_{0}H\) = 0 mT & \(\mu_{0}H\) = 0 mT & \(\mu_{0}H\) = 25 mT & \(\mu_{0}H\) = 50 mT & \(\mu_{0}H\) = 50 mT & \(\mu_{0}H\) = 50 mT \\ \hline
50 & 6.42 & 30.83 & 11.15 & 24.42 & 15.03 & 21.22 \\
100 & 5.15 & 34.43 & 9.67 & 26.83 & 13.85 & 22.42 \\
150 & 3.86 & 39.64 & 8.46 & 29.23 & 12.57 & 24.02 \\
200 & 2.69 & 46.85 & 7.33 & 32.03 & 11.28 & 26.03 \\
250 & 1.62 & 58.46 & 6.33 & 35.24 & 10.04 & 28.43 \\
300 & 0.71 & 81.28 & 5.32 & 39.64 & 9.02 & 30.83 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Calculated \(a\) and \(D_{\rm 0 sky}\) used in the numerical calculations [13,14]. One notes that the 2D and 3D distributions are almost superimposed at low temperature (\(T=100\) K) for any amplitudes of the external bias field, while at higher temperatures (\(T=200\) K, 300 K) there is a shift of the 3D distribution towards higher values together with a broadening of the distribution itself marked by an increase of the full width at half maximum (FWHM). The shift and the broadening of the 3D distributions are attenuated with increasing the amplitude of the external bias field. Therefore, \(H_{\text{ext}}\) partially masks the effect due to the increase of the dimensionality on the skyrmion diameters distribution. The displacement of the maximum passing from the 2D to the 3D skyrmions distribution shown in Figure 2 is highlighted in Figure 3 displaying the most probable skyrmion diameters of the 2D and 3D distributions. The calculation of the most probable diameters for the 2D (3D) case was performed according to Equation (4) (Equation (3)). The general trend is a monotonic increase of the most probable diameter vs. \(T\) according to both approaches with a strong deviation from linearity for \(T>200\) K and especially in the absence of an external field. However, while for \(T<150\) K the most probable diameters of the 2D and 3D distributions are almost superimposed, at higher temperatures \(D_{\text{sky}}^{\text{mp}\,\text{3D}}\) values are larger than \(D_{\text{sky}}^{\text{mp}\,\text{2D}}\) ones especially in the absence of an external bias field.