^{1,a)}, P. P. Horley

^{1}, V. K. Dugaev

^{2}, J. Barnaś

^{3}and W. Dobrowolski

^{4}

### Abstract

Magnetization dynamics of a single-domain ferromagnet is studied theoretically using the methods developed for self-organization phenomena. Time evolution of the magnetization is described by the Landau-Lifshitz-Gilbert equation with the spin-transfer torque included. The equilibrium and stationary states are investigated as a function of spin current and external magnetic field. The presented bifurcation diagram allows the margins of a neutral stability mode of the equilibrium and stationary states to be determined. An envelope equation for the magnetization switching is derived. The switching time between different states is found to be comparable to the half-width of the time derivative of the system energy, which allows the energy flow due to spin current and the magnetization switching velocity to be related. Dynamics of the phase states in external magnetic field and in the presence of spin current is analyzed using different methods of numerical analysis.

We thank V. R. Vieira for numerous discussions. This work is partly supported by STCU Grant No. 3098 in Ukraine, the FCT Grant POCI/FIS/58746/2004 (Portugal), and the Polish State Committee for Scientific Research under Grant Nos. 4 T11F 014 24 and 2 P03B 053 25.

I. INTRODUCTION

II. THEORETICAL MODEL AND METHOD

A. Equilibrium solutions

B. Stationary solutions

III. RESULTS OF NUMERICAL CALCULATIONS AND DISCUSSION

A. Analysis of equilibrium and stationary states

B. Phase evolution of the system

C. Magnetization switching time

IV. SUMMARY

### Key Topics

- Magnetic fields
- 40.0
- Magnetization dynamics
- 13.0
- Self organized systems
- 9.0
- Magnetic anisotropy
- 8.0
- Spintronic devices
- 7.0

## Figures

Schematic of the system considered in this article.

Schematic of the system considered in this article.

Equilibrium states of the system as a function of magnetic field (here and in the following, d.u. means dimensionless units).

Equilibrium states of the system as a function of magnetic field (here and in the following, d.u. means dimensionless units).

Magnetic field dependence of the magnetic moment components for the stationary states calculated for . The curves labeled with 1, 2, and 3 correspond, respectively, to , , and for the state ; whereas the curves 4, 5, and 6 represent , , and for the state . (The data corresponding to the curves 3 and 5 have been multiplied by a factor of 40.)

Magnetic field dependence of the magnetic moment components for the stationary states calculated for . The curves labeled with 1, 2, and 3 correspond, respectively, to , , and for the state ; whereas the curves 4, 5, and 6 represent , , and for the state . (The data corresponding to the curves 3 and 5 have been multiplied by a factor of 40.)

Magnetic field dependence of the solutions of the characteristic equation: real (curves 1 and 3) and imaginary (curves 2 and 4) parts of , real parts of (curves 5 and 6). Stationary states : curves 1–6, equilibrium states : curves . The curves 1,3, and are multiplied by a factor of 100.

Magnetic field dependence of the solutions of the characteristic equation: real (curves 1 and 3) and imaginary (curves 2 and 4) parts of , real parts of (curves 5 and 6). Stationary states : curves 1–6, equilibrium states : curves . The curves 1,3, and are multiplied by a factor of 100.

Bifurcation diagram of neutral stability for stationary (, solid curves) and equilibrium (, dashed curves) states for and different values of : 1 and correspond to ; 2 and correspond to ; 3 and correspond to .

Bifurcation diagram of neutral stability for stationary (, solid curves) and equilibrium (, dashed curves) states for and different values of : 1 and correspond to ; 2 and correspond to ; 3 and correspond to .

Dependence of on external magnetic field for the stationary state at different observation times, calculated for .

Dependence of on external magnetic field for the stationary state at different observation times, calculated for .

Dependence of on external magnetic field for the equilibrium state at different observation times.

Dependence of on external magnetic field for the equilibrium state at different observation times.

Phase portraits and main system characteristics vs applied magnetic field: (a) density plot, (b) power spectral density , (c) evolution of trajectory tracing curve, and (d) Hausdorff dimension and maximum Lyapunov exponent .

Phase portraits and main system characteristics vs applied magnetic field: (a) density plot, (b) power spectral density , (c) evolution of trajectory tracing curve, and (d) Hausdorff dimension and maximum Lyapunov exponent .

The system evolution with the applied magnetic field : density plots for different values of the spin current.

The system evolution with the applied magnetic field : density plots for different values of the spin current.

Time evolution of (curves 1, 2, 3) and (curves ), calculated for and different applied magnetic fields: for ; for ; for . The thin oscillating curves correspond to calculated numerically.

Time evolution of (curves 1, 2, 3) and (curves ), calculated for and different applied magnetic fields: for ; for ; for . The thin oscillating curves correspond to calculated numerically.

Magnetization component (curves 1–4) and (curves ) as a function of magnetic field for and different spin currents: the curves are for ; for ; for ; and for .

Magnetization component (curves 1–4) and (curves ) as a function of magnetic field for and different spin currents: the curves are for ; for ; for ; and for .

Dependence of the switching time on for different spin currents: The curve 1 is for , 2 for , 3 for . Behavior of the parameters and with spin current is given in the inset.

Dependence of the switching time on for different spin currents: The curve 1 is for , 2 for , 3 for . Behavior of the parameters and with spin current is given in the inset.

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