^{1,a)}and M. Lakshmanan

^{2,b)}

### Abstract

We study the bifurcation and chaos scenario of the macromagnetization vector in a homogeneous nanoscale-ferromagnetic thin film of the type used in spin-valve pillars. The underlying dynamics is described by a generalized Landau–Lifshitz–Gilbert (LLG) equation. The LLG equation has an especially appealing form under a complex stereographic projection, wherein the qualitative equivalence of an applied field and a spin-current induced torque is transparent. Recently, chaotic behavior of such a spin vector has been identified by Li *et al.* [Li *et al.*Phys. Rev. B74, 054417 (2006)] using a spin-polarized current passing through the pillar of constant polarization direction and periodically varying magnitude, owing to the spin-transfer torque effect. In this paper, we show that the same dynamical behavior can be achieved using a periodically varying applied magnetic field in the presence of a constant dc magnetic field and constant spin current, which is technically much more feasible, and demonstrate numerically the chaotic dynamics in the system for an infinitely thin film. Further, it is noted that in the presence of a nonzero crystal anisotropy field, chaotic dynamics occurs at much lower magnitudes of the spin current and dc applied field.

Bolstered by the importance of giant magnetoresistance (GMR), a sequence of experimental and theoretical developments in the past few years on current induced switching of magnetization in nanoscale ferromagnets has thrown open several prospects in next generation magnetic memory devices. The direct role of spin-polarized current, as against the traditional applied field, in controlling spin dynamics has brought in the possibility of new types of current-controlled memory devices and microwave resonators. The system under consideration is primarily a nanoscale spin-valve pillar structure, with one

*free*ferromagnetic layer and another

*pinned*layer separated by a nonferromagnetic conducting layer. The behavior of the dynamical quantity of interest, the magnetization field in the free layer, is well modeled by an extended Landau–Lifshitz equation with Gilbert damping, which is a fascinating nonlinear dynamical system. The free layer is usually assumed to be of single magnetic domain. Owing to the highly nonlinear nature of the Landau–Lifshitz–Gilbert (LLG) equation it is imperative to study the chaotic dynamical regime of the magnetization field. Indeed, several recent experiments have exclusively focused on chaos aspect of the system. In this paper, we have shown that a small applied periodically varying (ac)magnetic field, in the presence of a constant spin current and a steady applied magnetic field, can induce parametric regimes, displaying a broad variety of dynamics and period doubling route to chaos. A numerical study of the effects of a nonzero anisotropy field reveals chaotic dynamics at much lower magnitudes of the spin current and applied dc field. This could be an important factor to consider in microwave resonator applications of spin-valve pillars.

S.M. wishes to thank DST, India, for funding through the FASTTRACK scheme. The work of M.L. forms part of a DST-IRHPA research project and is supported by a DST-Ramanna fellowship.

I. INTRODUCTION

II. THE EXTENDED LANDAU–LIFSHITZ EQUATION AND COMPLEX REPRESENTATION

III. CHAOTIC DYNAMICS

A. Regions of chaos in the presence of an applied alternating field

1. Case A: No anisotropy

2. Case B: Nonzero anisotropy

B. Periodic, multiply periodic, and chaotic dynamics

IV. DISCUSSION AND CONCLUSION

### Key Topics

- Magnetic fields
- 38.0
- Chaos
- 25.0
- Chaotic dynamics
- 20.0
- Alternating current power transmission
- 19.0
- Magnetic anisotropy
- 19.0

## Figures

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.

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.

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.

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.

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 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.

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.

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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