Among the various effects predicted by this stability analysis, two of them are of particular physical interest: current-controlled nonlinear ferromagnetic resonance and phase-locking between current-induced magnetization oscillations and rf field. This analysis reveals the remarkable result that the rf input powers capable of inducing spin-wave instabilities are bounded from below as well as from above. Therefore, two P-modes only will exist for any arbitrary choice of the control parameters. Second, the study of the precessional magnetization dynamics is also of importance in its own right, because this study lays the foundation for the analysis of the precessional switching of magnetization which is extensively discussed in Chapter 6. Chapter 2 deals with the discussion of the origin of the Landau—Lifshitz equation.
This fact will be instrumental in the development of the averaging technique for the magnetization dynamics discussed in Chapter 5. The phase portrait for this case and the corresponding energy graph are shown in Fig. However, the complications due to the nonautonomous nature of the dynamics can be circumvented by taking advantage of the rotational symmetry of the problem. In the averaging technique, the first-order differential equation for the magnetic free energy is derived through the averaging of certain terms over precession cycles. Mayergoyz , Springer-Verlag, 1991, pp.
The formal similarity of Eq. Comparison with the numerical results in Fig. The book is concerned with large magnetization motions when the nonlinear nature of the Landau-Lifshitz equation is strongly pronounced. It turns out that the most general and natural way to account for this constraint is to use the Helmholtz decomposition for vector fields defined on the unit sphere. Switching events are treated as bifurcations and the dynamic analog and generalization of the Stoner—Wohlfarth astroid is introduced. The analytical expressions for these periods as functions of energy are given in terms of the complete elliptical integrals. The classification of phase portraits of the autonomous dynamics in the rotating frame is introduced and the detailed analysis of bifurcations i.
Because of rapid advances in the medical sciences, in particular, independent verification of diagnoses and drug dosages should be made British Library Cataloguing in Publication Data Mayergoyz, I. The chapter is concluded with the discussion of the Poincaré—Melnikov theory, which is conceptually similar to the averaging technique. These forms clearly reveal that these equations have two distinct fast and slow time scales associated with precession and damping, respectively. Finally, the comprehensive qualitative and quantitative understanding of nonlinear magnetization dynamics is of interest in its own right, because it may reveal new physics and, in this way, it may eventually lead to new technological applications. In the typical case when the product of the damping coefficient and radio frequency is quite small, the approximate formula of P. It is also worth remarking that no contributions to Heff from eddy currents are included. This is because the high-energy regions of the phase portraits are very fine mixtures of two basins of attraction, and the smaller the damping constant α, the more intricate and finer the entanglement of the two basins of attraction in the high-energy regions.
It turns out that the most general and natural way to account for this constraint is to use the Helmholtz decomposition for vector fields defined on the unit sphere. To identify the precessional trajectories over which this balancing occurs, the appropriate Melnikov function is introduced and the analytical expressions for this Melnikov function are derived in terms of elliptical integrals for various central regions of the phase portrait of precessional magnetization dynamics. Numerical Techniques for Magnetization Dynamics Analysis Publisher Summary 11. Finally, the spatially uniform magnetization dynamics deserves special attention because it is the simplest albeit nontrivial case of nonlinear magnetization dynamics. This argument justifies the solution of Eq. However, the spin-polarized current injection does affect the stability of the critical points. This time averaging can be carried out analytically by using the formulas derived in Chapter 4 for the precessional dynamics.
He served as a consultant for many years for the Research and Development Center of General Electric Company and has been selected as a visiting research fellow of this center. Traditionally, this study has been driven by ferromagnetic resonance problems. This leads to the reduction of the number of state variables to two. However, this linearization approach is rather limited in scope and has little relevance to magnetic data storage technology, where the magnetic writing process results in large magnetization motions. In terms of spherical coordinates, the two terms in the right-hand side of Eq.
The approach presented below is simpler because it takes explicit advantage of the rotational symmetry of the problem and clearly separates fast and slow time scales of magnetization dynamics. In this respect, the existence of P-modes, studied in the previous chapter, suggests that the analysis of spin-wave effects can be carried out at least for systems with uniaxial symmetry for fields of arbitrary amplitude and frequency. The quantity hI depends on ω only. We shall discuss the first-order instability by considering the associated behavior of the one-period-map characteristic multipliers introduced in the previous section. In these situations, the critical points of magnetization dynamics are distinct from micromagnetic equilibrium states and the only constraint which remains is the confinement of the magnetization dynamics to the unit sphere.
Thus, these equilibria can be classified as energy maxima, energy minima and energy saddles. Circles labeled A through E are points where the field crosses bifurcation lines and magnetization switching may occur see Fig. Chapter 6 is concerned with the analytical study of precessional switching of magnetization in thin films. This slow relaxational dynamics is completely decoupled from the fast precessional dynamics of mx and my. The analytical conservative solutions solid, dashed and dotted lines are the same as in Fig. Furthermore, chaos is precluded, because the phase space of autonomous magnetization dynamics is two-dimensional. These magnetostatic modes naturally appear when exchange forces can be neglected and magnetostatic boundary conditions are dominant.
A spontaneous symmetry breaking occurs in the magnetization response, with the appearance of out-of-plane self-oscillations characterized by a nonzero average value of mz. Thus, no explicit time dependence is present anymore in the right-hand side of Eq. Arrows indicate foldover jumps occurring under the largest current. The shape of the trajectory is determined only by the damping constant α and by the initial orientation of the magnetization. This general result is of little value for strongly nonlinear randomly perturbed magnetization dynamics. Switching events are treated as bifurcations and the dynamic analog and generalization of the Stoner—Wohlfarth astroid is introduced.