## 3 Results
### Eigenmode of Isolated Elliptical Nanodots
First of all, one notices that the number of modes that appear in the range of frequencies up to 30 GHz rapidly increases with the lateral dimensions. For instance, for the 200 \(\times\) 100 nm\({}^{2}\) dots one finds more than ten eigenmodes, i.e., about twice the number of modes observed in the smallest dots as well as in the 100 nm wide wires studied in the previous paragraph. Moreover, in absence of DMI the lowest mode (0,0) tends to be much more localized at the dot edges if compared to the case of the smallest dots, so that it loses its \"fundamental\" character, being replaced by the (2,0) mode. In fact, the latter is characterized by a maximum amplitude in the central region of the dot and presents the largest average dynamical magnetization, as seen in the calculated spectra plotted in the left-hand side of Figures 4 and 5. However, also for the larger dots when the value of \(D\) is lifted from zero, the frequencies of all the modes decrease and the odd (0,1) mode becomes visible. The latter actually becomes the most intense peak in the spectrum, overcoming the (2,0) mode, as \(D\) approaches the value of 2 mJ/m\({}^{2}\), as shown in the spectra on the right-hand side of Figures 4 and 5. Finally, let us extend the analysis of the frequency evolution of the eigenfrequencies to large values of \(D\), in order to cross check also in the case of the elliptical dots the softening associated to the transition to a chiral ground state. Figure 6 presents the calculated frequencies for \(D\) ranging up to 5 mJ/m\({}^{2}\) for the elliptical dots with different lateral dimensions. One finds a behavior that is similar to what observed for the wires, although in this case the mode softening occurs at relatively larger values of \(D\). The slope of curve corresponding to the first odd mode (1,0) is clearly steeper than that of the corresponding curves of the low-frequency (0,0) and (0,2) modes. Therefore, mode-crossing is observed around \(D\) = 3 mJ/m\({}^{2}\) and then the frequency of the soft mode approaches zero for a value of \(D\) just above 4 mJ/m\({}^{2}\). Actually, this specific value of \(D\) is about 4.3 mJ/m\({}^{2}\) in the smallest dots and somehow smaller for the larger dots, reflecting the different effective fields inside the dots related to their lateral dimensions. For larger values of \(D\), the ground state undergoes a transition to a chiral non-uniform state and the dynamics becomes very complex, so the spectrum is completely changed. Figure 4: Central panel: dependence of the frequencies of the spin wave eigenmodes on the DMI constant \(D\) for the elliptical dot of lateral dimensions 200 \(\times\) 100 nm\({}^{2}\). On the left- and right-hand sides, the intensity spectra corresponding to \(D\) = 0 and 2 mJ/m\({}^{2}\) are reported, respectively, together with the color insets that represent the spatial profile of the dynamical magnetization, expressed as the product of the modulus of the dynamical magnetization by the sign of its phase. The animations with the movies of the dynamics of all the modes for several values of \(D\) are available in the Supplemental Material. To conclude, let us notice and interesting aspect connected with the eigenmodes intensity at relatively large values of \(D\), which is easily seen in Figure 6. As \(D\) overcomes the values of about 2 mJ/m\({}^{2}\), there is a clear evolution of the modes intensity. In particular, one sees that the character of \"fundamental mode\", i.e., the largest intensity, is not anymore related to the (0,0) or (2,0) modes at lower frequencies. Rather, one finds that modes with relatively high-order assume a considerable intensity, while the soft mode at low frequency becomes less intense. For instance, for \(D\) between 3.5 and 4 mJ/m\({}^{2}\) the spectrum is dominated by modes between 15 and 20 GHz.