The steady orbit radius u 0(J) allows finding the STNO generation frequency , which increases approximately linearly with J increasing up to the second critical click here current value J c2 when the steady oscillation state becomes unstable (see Figure 2). The instability is related with the vortex core polarity reversal reaching a core critical velocity or the vortex core expelling from the dot increasing the current density J [12, 16]. We simulated selleck inhibitor the values of J c2 = 2.7, 5.0, and 10.2 MA/cm2 for the dot thickness L = 5, 7, and 10 nm, respectively. The calculated STNO frequency is 15 to 20% higher

than the simulated one due to overestimation of within TVA for β =0.1. The calculated nonlinear frequency part is very close to the simulated one, except the vicinity of J c2, APO866 price where the analytical model fails. Figure 2 The vortex steady-state oscillation frequency vs. current. The nanodot thickness L is 5 nm (1), 7 nm (2), and 10 nm (3), and radius is R = 100 nm. The frequency is shown within the current range of the stable vortex steady-state orbit, J c1 < J < J c2. Solid black lines are calculations by Equation 5; red squares mark the simulated points. Inset: the nonlinear vortex frequency coefficient vs. the dot thickness for R = 100 nm and J = 0 accounting all energy contributions (1) and only magnetostatic contribution (2). Our comparison of the calculated dependences u 0(J) and ω G (J) with simulations is principally different from

the comparison conducted in a paper [19],

where the authors compared Equations 5 and 7 with their simulations fitting the model-dependent nonlinear coefficients N and λ from the same simulations. One can compare Figures 1 and 2 with the results by Grimaldi et al. [20], buy Regorafenib where the authors had no success in explaining their experimental dependences u 0(J) and ω G (J) by a reasonable model. The realistic theoretical nonlinear frequency parameter N for Py dots with L = 5 nm and R = 250 nm should be larger than 0.11 that the authors of [21] used. N = 0.25 can be calculated from pure magnetostatic energy in the limit β → 0 (inset of Figure 2). Accounting all the energy contributions in Equation 4 yields N = 0.36, which is closer to the fitted experimental value N = 0.50. The system (6) can be solved analytically in nonlinear case. Its solution describing transient vortex dynamics is (8) where u(0) is the initial vortex core displacement and is the inverse relaxation time for J > J c1 (order of 100 ns). at t → ∞ and J = J c1. If J < J c1, the orbit radius u(t, J) decreases exponentially to 0 with the relaxation time . The divergence of the relaxation times τ ± at J = J c1 allows considering a breaking symmetry second-order phase transition from the equilibrium value u 0 = 0 to finite defined by Equation 7. Equations 7 and 8 represent mean-field approximation to the problem and are valid not too very close to the value of J = J c1, where thermal fluctuations are important [13, 21].