Introduction

The study of magnetization processes in magnetic materials has been in the last fifty years the focus of considerable research for its application to magnetic recording technology. In fact, the design of nowadays widespread magnetic storage devices, such as the hard-disks which are within computers on our desktops, requires the knowledge of the ``microscopic'' phenomena occurring within magnetic media. In this respect, it is known that some materials, referred to as ferromagnetic materials, present spontaneous magnetization at room temperature, which is the result of ``spontaneous'' alignment of the elementary magnetic moments that constitute the medium. Roughly speaking, from a phenomenological point of view, one has a medium whose magnetization state can be changed by means of appropriate external magnetic fields. The magnetic recording technology exploits the magnetization of ferromagnetic media to store information. The first example of magnetic storage device was the magnetic core memory prototype, realized by IBM in 1952, and used in the IBM 405 Alphabetical Accounting Machine. The working principle of magnetic core memories is very simple. One can think about several cores placed at the nodal positions of an array-type structure made with horizontal and vertical wired lines, as sketched in Fig. 1. Each core is basically a bistable unit, capable of storing one bit (binary digit), which is the smallest piece of binary-coded information (can be let's say ``0'' or ``1''). In Figure 1, on the right, it is illustrated the writing mechanism of the IBM 2361 Core Storage Module. Basically, the target magnetic core can be ``switched'', from 0 to 1 or viceversa, by addressing it with the horizontal and vertical current lines which pass through the core. The currents flowing in the addressing wires generate a magnetic field that can change the magnetic state of the core. Nevertheless, the magnetic field produced by the single current line is designed to be not sufficient to switch a core. Therefore, the only core that switches is the only one traversed by two currents, namely the one addressed by the horizontal and the vertical current lines. It turns out that a collection of magnetic cores can store a sequence of bits, namely can record a piece of information. After magnetic core memories, magnetic tapes (or, equivalently, floppy disks) have been used, but the most widespread magnetic storage device is certainly the hard-disk. In this respect, it is evident from the photography in Fig. 1 that the first prototypes of magnetic storage devices had dimensions in the order of meters. The progress made by research activity performed worldwide in this subject has led to exponential decay of magnetic device dimensions. In fact, modern recording technology deals with magnetic media whose characteristic dimensions are in the order of microns and submultiples. It is sufficient to mention that commercial hard disks are capable of storing more that 100 Gbit (gigabit $\sim10^9$ bits) per square inch!

Figure 1:(left) The first magnetic core memory, from the IBM 405 Alphabetical Accounting Machine. The photo shows the single drive lines through the cores in the long direction and fifty turns in the short direction. The cores are 150 mm inside diameter, 240 mm outside, 45 mm high. This experimental system was tested successfully in April 1952. (right) Writing mechanism of magnetic cores memory.

Recently the possibility to realize magnetic random access memories (from now on MRAMs), similar in principle to magnetic core memories, has been investigated, but, at the moment no commercial realization of MRAMs is present on the market. However, both hard disks and MRAMs rely on flat pieces of magnetic materials having the shape of thin-films. Typically, the information, coded as bit sequences, is connected to the magnetic orientation of these films, which have dimensions in the order of microns and submultiples.

Figure 2: Simple representation of Read/Write magnetic recording device present in hard disks realizations.

Let us now consider the simple scheme of principle of hard disk, depicted in Fig. 2. The recording medium is a flat magnetic material that is thin-film shaped. The read and write heads are separate in modern realizations, since they use different mechanisms. In fact, as far as the writing process is concerned, one can see that the writing head is constituted by a couple of polar expansions made of soft materials, excited by the current flowing in the writing coil. The fringing field generated by the polar expansions is capable to change the magnetization state of the recording medium. Generally the recording medium is made with magnetic materials that have privileged magnetization directions. This means that the recording medium tends to be naturally magnetized either in one direction (let's say `1' direction) or in the opposite (`0' direction). In this sense, pieces of the material can behave like bistable elements. The bit-coded information can be therefore stored by magnetizing pieces of the recording medium along directions 0 or 1. The size of the magnetized bit is a critical design parameter for hard disks. In addition, for the actual data rates, magnetization dynamics cannot be neglected in the writing process. The reading mechanism currently relies on a magnetic sensor, called spin valve, which exploits the giant magneto-resistive (GMR) effect. Basically, the spin valve is constituted by a multi-layers structure. Typically two layers are made with ferromagnetic material. One is called free layer since its magnetization can change freely. The other layer, called pinned layer, has fixed magnetization. If suitable electric current passes through the multi-layers, significant changes in the measured electric resistance can be observed depending on the mutual orientation of the magnetization in the free and pinned layer. Let us see how this can be applied to read data magnetically stored on the recording medium. Basically, the spin valve is placed in the read head almost in contact with the recording medium [1]. Then, when the head moves over the recording medium, the magnetization orientation in the free layer is influenced by the magnetic field produced by magnetized bits on the recording medium. More specifically, when magnetization in the free layer and magnetization in the pinned layer are parallel, the electrical resistance has the lowest value. Conversely, the antiparallel configuration of magnetization in the free layer and pinned layer yields the highest value of the resistance. Thus, by observing the variation in time of the electrical resistance (that is, the variation of the read current passing trough the multilayers) of the GMR head, the bit sequence stored on the recording medium can be recognized.

Figure 3: Typical array structure for magnetic random access memories (MRAMs).

It is possible to say something also about MRAMs prototypes. The magnetic random access memories follow a working principle very similar to the older magnetic core memories. In fact, they present the same cell array structure as their predecessors, but each cell is constituted by a magnetic multi-layers structure rather than a magnetic core (see Fig. 3). The reading mechanism is based on GMR effect, whereas the writing process is conceptually analogous to the one seen for magnetic core memories. Thus, an MRAM cell can be switched by addressing it with the current lines (bit lines in Fig. 3). The switching is realized by means of the magnetic field pulse produced by the sum of horizontal and vertical current. This magnetic field pulse can be thought as applied in the film plane at 45$^\circ$ off the direction of the magnetization. In this situation, the magnetic torque, whose strength depends on the angle between field and magnetization, permits the switching of the cell. This behavior is simple in principle, but it is very hard to realize in practice on a nanometric scale. In fact, the array structure must be designed such that the magnetic field produced by only one current line cannot switch the cells. Conversely, the field produced by two currents must be such that it switches only the target cell. Recently, to circumvent the problems of switching MRAMs cells with magnetic field, the possibility of using spin-polarized currents, injected directly in the magnetic free layer with the purpose to switch its magnetization, has been investigated. In particular, this possibility has been first predicted by the theory developed by J. Slonczewski in 1996 (see Ref. [44]) and then observed experimentally [45,46,48]. The interaction between spin-polarized currents and the magnetization of the free layer is permitted by suitable quantum effects. From a ``macroscopic'' point of view, these effects produce a torque acting on the magnetization of the free layer. The resulting dynamics may indeed exhibit very complicated behaviors.

Figure 4: Magnetic Recording Disk Technology: Practical Challenges in Delivering the Areal Density Performance [2].

The above situations are only few examples of technological problems which require to be investigated by means of theoretical models. Now, referring to hard disk technology, at the present time the main challenges and issues can be summarized as follows:

1. Higher areal density.
2. Improved thermal stability of magnetized bits.
3. Increasing read/write speed in recording devices ($<1$ ns)

The first two points are strongly connected, since the smaller is the size of the bit, the stronger are the thermal fluctuations which tend to destabilize the configuration of the ``magnetized bit''. For this reasons, as far as the bit size decreases, it has been recognized that the use of perpendicular media, constituted of grains in which the bit is magnetized in the direction normal to the film plane, leads to better thermal stability. In fact, by looking at Fig. 4, the future perspectives in hard disk design show that the use of perpendicular media, patterned media and heat-assisted magnetic recording technology will possibly yield [2] areal densities towards 1 Terabit/in$^2$ by the year 2011. Thus, being the spatial scale of magnetic media in the order of, more or less, hundred nanometers, magnetic phenomena has to be analyzed by theoretical models with appropriate resolution. This is the case of micromagnetics, which is a continuum theory that stands between quantum theory and macroscopic theories like mathematical hysteresis models (Preisach, etc.). Moreover, as far as the read/write speed increases (frequencies in the order of GHz and more), dynamic effects cannot be neglected. Therefore, as a result, the design of modern ultra-fast magnetic recording devices cannot be done out of the framework of magnetization dynamics. This is the motivation for the research activity that will be illustrated in the following chapters.

In chapter 1 the micromagnetic model and the Landau-Lifshitz-Gilbert (LLG) equation will be introduced to describe magnetization phenomena in ferromagnetic bodies. First, an approach in terms of the free energy associated with the magnetic body will be presented to derive the static equilibrium conditions for magnetization vector field. Then, the dynamic effects due to the gyromagnetic precession will be introduced. Both Landau-Lifshitz and Landau-Lifshitz-Gilbert equation will be presented. Phenomenological Gilbert damping will be analyzed in terms of Rayleigh dissipation function. In chapter 2 the study of magnetization dynamics in uniformly magnetized particles will be addressed. In particular, first the static Stoner-Wohlfarth model and then magnetization switching processes will be analyzed. In addition, novel analytical techniques to study magnetization dynamics under circularly polarized external fields and magnetization dynamics driven by spin-polarized currents will be introduced and deeply discussed. In this respect, it will be shown how some behaviors indeed observed in experiments, can be explained in terms of bifurcations of fixed points and limit cycles of the LLG dynamical system. As a further step, in chapter 3, the assumption of magnetization spatial uniformity will be removed and the problem of studying thin-films reversal processes of technological interest will be addressed. In this respect, as preliminary step, the issue of the computation of magnetostatic fields, which is still the bottleneck of micromagnetic simulations, will be illustrated together with the mostly used methods at this time. Then, a comparison of damping and precessional switching processes in thin-films will be performed, showing that fast precessional switching can be considered spatially quasi-uniform and, therefore, its crucial aspects can be analyzed by means of uniform mode theory discussed in chapter 2. Finally, a uniform mode analysis will be applied to the fast switching of granular tilted media which represents one of the most promising solutions for high density magnetic storage in future hard disks. In chapter 4, the problem of the geometrical numerical integration of LLG equation will be considered. In particular, mid-point rule time-stepping will be applied to the LLG equation. In fact, it will be shown that the fundamental properties of magnetization dynamics, embedded in the continuous model, are reproduced by the mid-point discretized LLG equation regardless of the time step. In addition, since the resulting numerical scheme is implicit, special and reasonably fast quasi-Newton technique will be developed to solve the nonlinear system of equations arising at each time step. The proposed mid-point technique will be validated on the micromagnetic standard problem no. 4 which concerns with thin-films reversal processes. Finally, discussion on numerical results and computational cost will be performed. In the end, some conclusions about the results obtained and the possible future work will be drawn.