A schematic figure of a spin-valve pillar. The cross section of the free layer is roughly . is the magnetization vector in the free layer and is the dynamical quantity of interest. is the direction of polarization of the spin current.
Regions of chaos in the space for an applied alternating magnetic field with an amplitude of and a frequency of : (a) without anisotropy field, , and (b) with anisotropy field of strength along the direction. The dark regions indicate values for which the dynamics is chaotic, i.e., regions where the largest Lyapunov exponent is positive. In (a) chaos is rarely noticed for lower values of . The other parameters are , , and . The points are plotted at intervals of 5 Oe along both axes, and hence the figure offers only limited resolution in the dark (chaotic) regions.
Period doubling route to chaos as is varied. The figure is a plot of the minimum values of over several periods for the given parameter values (a) without anisotropy and (b) with anisotropy of along the direction. The applied dc field is . All the other parameters remain the same as in Fig. 2. The corresponding Lyapunov spectrum is shown as an inset.
Regions of chaos (dark stem) and periodicity (light wings) in the parameter space of dc and frequency (a) without anisotropy and (b) with anisotropy along the direction. The left over regions show multiply periodic behavior. All other parameters remain the same as in Fig. 3. The power spectrum at the two dark points in (a) (255,25) and (280,25) are shown in Figs. 5(a) and 5(b), respectively.
The power spectrum distribution corresponding to periodic, (inset), and chaotic, , scenarios in Fig. 4(a). The first peak in the inset is seen at . The anisotropy is taken zero, and all other parameters are the same as in Fig. 4(a).
The power spectrum distribution in the limit at certain values of (indicated on each spectrum) where periodic behavior is noted. Multiply periodic behavior is noticed for other values of in the range shown. The current magnitudes vary linearly and decrease with the frequency of oscillation (inset). , while all other parameters are the same as in Fig. 3.
Regions of multiply periodic dynamics for the system with the dc applied field fixed at and nonzero anisotropy. All the other parameters remain the same as in Fig. 4(b). Synchronization is noted in the unshaded regions, while chaotic dynamics is not noticed in the parameter range shown in the figure. Islands of multiply periodic behavior appear between regions of periodic behavior for low frequencies. For higher frequencies, the dynamics is exclusively multiply periodic.
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