3. Non-uniform Magnetization Dynamics in thin-films reversal processes

When one wants to remove the simplifying hypothesis that the body is uniformly magnetized, some issues arise which considerably complicate the solution of Landau-Lifshitz-Gilbert (LLG) equation. First of all, LLG equation becomes an integro-partial differential equation in the unknown vector field $\textbf{{m}}(\textbf{r},t)$. Moreover, the LLG equation is nonlinear and this implies that, in general, it is not possible to find exact analytical solutions. Therefore, the most general method to solve LLG equation lies on appropriate numerical techniques. In this respect, usually a semi-discretization approach is adopted. First, a spatial discretization of the equation is performed, by using finite differences or finite elements methods [52]. As a result, a discretized version of the micromagnetic free energy and a corresponding system of ordinary differential equations are derived. Finally, this system of ordinary differential equations is numerically solved with appropriate time-stepping schemes. In this framework, many problems arise. One is the fact that micromagnetics, although is applicable in principle to magnetic bodies within a broad spatial scale (form few nm to many $\mu$m), cannot be practically used for dimensions exceeding $1 \mu$m. In fact, for sub-micron particles, numerical simulations reasonably agree with experimental results, whereas for increasing dimensions of the bodies the agreement with experimental observations is only qualitative. Amikam Aharoni pointed out few years ago [51] that the reasons of such a `failure' can be found in bad understanding of theoretical results, like nucleation theory [5], as well as bad approximations and rough discretization in energy computations. More specifically, regarding the latter point, he emphasized that the correctness of the results strongly depends from an accurate computation of magnetostatic field. In fact, once that magnetization becomes space-dependent and for arbitrary body shape, the analytical expression (2.19) does not hold anymore. For these reasons, few years ago, some researchers at NIST proposed a set of micromagnetic standard problems [79], with simple rectangular geometries and material properties, to test micromagnetic numerical codes. In this respect, the standard problem no. 1, involving a $2 \times 1 \times 0.02 \mu$m permalloy thin-film, represents the most appropriate example of the above issues. In fact, the results provided by different groups showed qualitative agreement between the computed hysteresis loops, but the quantitative evaluations of coercive fields were different even for two orders of magnitude! After that, much more attention has been paid to the correct formulation of numerical models. It has been recognized that the bottleneck of micromagnetic simulations is always the fast and accurate evaluation of the magnetostatic demagnetizing field. In the following sections we will present mostly used methods for demagnetizing field computation. Afterwards, we will perform a comparison between damping and precessional switching processes for rectangular thin-film geometry and we will show that damping switching is intrinsically a non-uniform process, involving domain nucleation and wall motion, whereas precessional switching can be reasonably considered a quasi-uniform process also for body dimensions of hundreds of nanometers. Finally, in the framework of quasi-uniform magnetization dynamics, we will analyze the fast switching, below Stoner-Wohlfarth field, of tilted media for magnetic recording.