Conclusions and Outlook

In this thesis some categories of magnetization phenomena, connected with high-end technological applications have been investigated. The followed approach has its foundations in micromagnetics, which is capable of properly describing magnetic phenomena on sub-micron scale. A first step has been done by studying magnetization dynamics under the hypothesis that the particles were uniformly magnetized. In this framework, it has been recalled that magnetization dynamical processes of technological interest, like damping switching and precessional switching, can be treated with analytical approaches present in literature, which provide critical design parameters like critical fields, as well as switching time and, for precessional switching, the time tolerance allowed in order to have successful switching. A slightly different context in the framework of uniform mode theory, has been touched, regarding some aspects of the LLG dynamics under circularly polarized fields, which arises in typical ferromagnetic resonance experiments. In particular, a special perturbative technique, based on mathematical background in the framework of dynamical systems theory, has been developed for the study of quasiperiodic solutions of LLG equation under circularly polarized field. In this respect, the problem of finding quasiperiodic solutions has been turned into the determination of limit cycles of a suitable modification of LLG equation, obtained by introducing the appropriate rotating reference frame. Basically, the result is that the study of the existence, the number, the stability of limit cycles of the dissipative LLG dynamics can be performed, at first order, on the conservative dynamics. The analytical results have been confirmed by numerical simulations that indeed show the accuracy of the predictions. In addition, this technique, which permits also to study the (global) bifurcations of the limit cycles, has been applied to the study of LLG dynamics driven by spin-transfer torque. This subject is currently the focus of considerable research for its applications to current-driven switching of MRAMs cells. In particular, by using the above perturbative technique, and on the other hand the analytical treatment of the conservative dynamics, it is possible to predict analytically the critical values of the electric current and magnetic fields that rule complicated behaviors, like the onset of self-oscillations and the current-driven switching, observed in recent experiments on spin-injection. Moreover, with this technique characteristics of the self-oscillations like frequency and amplitude can be analytically computed from the knowledge of the conservative dynamics. As soon as the hypothesis of uniformly magnetized particles has been abandoned, the problem of the numerical computation of the magnetostatic field, which has been recognized to be the bottleneck of micromagnetic computations, has been analyzed. The two most used methods, respectively for finite differences and finite elements discretizations, have been described. Afterwards it has been shown with micromagnetic simulations of damping and precessional switching, that the former is an intrinsically non-uniform process, whereas the latter can be reasonably considered quasi-uniform also for dimensions of hundreds of nanometers (half micron) and moderately low anisotropy. In this respect, by computing the switching time with the uniform mode analysis, reliable switching can be obtained. It has been demonstrated that the accuracy of the prediction increases for increasing anisotropy of the material. Moreover, for moderately soft materials, with $K_1$ in the order of $10^4\div10^5$ J/m$^3$, a tolerance of $\pm 25\%$ on the pulse amplitude is allowed. The predicted time window agree with micromagnetic simulations with a precision of few picoseconds. Then the fast switching of tilted granular media has been analyzed by means of a uniform mode approximation. The medium has been considered as a collection of noninteracting grains with dispersion of easy axes and initial conditions. In this framework, the necessary condition for the switching of the whole granular medium, i.e. the applied field amplitude range which makes fast switching possible, has been derived by analyzing the single grain dynamics, first conservative and then dissipative. A set of micromagnetic simulations have confirmed the predictions made with the uniform mode analysis. Thus, the main result of this study is the fact that the uniform mode theory can be applied to study processes of technological interest, and in some cases it provides critical design parameters. Beside the above analysis, the problem of geometrical integration of LLG equation has been addressed. In fact, the mostly used numerical time-stepping techniques do not preserve the fundamental properties of LLG dynamics, namely magnetization magnitude conservation, Lyapunov structure for constant in time applied field and hamiltonian structure in the conservative case. There is an interesting example in literature of how the missed fulfillment of magnitude conservation leads to inaccurate computation of magnetostatic field. Moreover, the quantitative discordance arising in the solution of micromagnetic standard problems suggests that the particular choice of numerical methods may affect the results of the computations. Therefore, we are convinced that a numerical model has to qualitatively reproduce the properties of the continuous model as best as possible, but, nevertheless it must have a feasible computational cost. In this respect, the proposed implicit mid-point rule technique has revealed very efficient from both points of view. First of all, it can be applied to any spatial discretization, like finite differences and finite elements, which preserves the formal structure of the effective field. Then, we have shown that the mid-point discretized LLG equation exactly fulfills magnetization magnitude conservation regardless of the time step. In addition, in case of constant applied field, the discrete dynamics has itself a Lyapunov structure regardless of the time step, and in the case of conservative dynamics ($\alpha=0$) the discretized free energy is preserved regardless of the time step and the hamiltonian structure is preserved up to third order in the time step. The implicit nature of the mid-point time-stepping leads to the solution of a nonlinear system of equations at each time step. Therefore, special and reasonably fast quasi-Newton iterative procedure has been developed to solve this system. Since the solution is approximate depending on the tolerance of the quasi-Newton procedure, the precision in the fulfillment of the LLG dynamics properties has been checked a posteriori. In particular, for finite differences spatial discretization, we have solved the micromagnetic standard problem no. 4. As far as the accuracy tests are concerned, the magnitude has been preserved within machine precision for each cell and the Lyapunov structure is preserved with a relative error in the order of $10^{-8}$. Conservative simulations of the same problem show that the free energy is preserved with relative error also in the order of $10^{-8}$. As far as computational effort is concerned, the use of quasi-Newton technique which approximates the full jacobian matrix of the nonlinear system of equations as a sparse matrix, allows the use of fast iterative methods (GMRES) for the inversion of the linear systems arising in the single quasi-Newton iteration. The moderately low measured simulation times, together with the fact that the time step can be chosen much larger than explicit methods due to mid-point rule unconditional stability, make this method a good candidate for accurate micromagnetic simulations. Future work in this framework could be made by developing a finite element code with mid-point rule time-stepping, which would permit to treat magnetic bodies with in principle arbitrary shape. Moreover the computational cost could be lowered by implementing suitable preconditioning for the GMRES method. In addition, the inclusion of the spin-transfer torque term in the code would permit to investigate non-uniform spin-injection phenomena in multi-layers structures. Finally, the inclusion of thermal effects in magnetization dynamics model and, consequently in micromagnetic simulations, would be a considerable improvement of the investigation. This direction will be pursued in future activities.