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The discovery of GMR proved that the resistance of a ferromagnetic conductor can depend considerably on its magnetisation configuration. This means that there is an interaction between the conduction electrons and the magnetisation, which can lead to changes in the electric conductivity. We may argue that — following the third Newton’s law — if the magnetisation can affect the flow of an electric current, there should be also an effect in the opposite direction: the flow of an electric current may affect the magnetisation dynamics. Studies of the interaction between a spin polarised current and the magnetisation of a ferromagnetic conductor were carried out in the seventies by the pioneering works of Berger [14], who already predicted the possibility for a current to move a domain wall. Only in 1996, however, the spin transfer torque between the itinerant electrons and the magnetisation was quantitatively taken into

account in two independent works by Slonczewski [3] and Berger [9], and the Landau- Lifshitz-Gilbert equation was extended by adding the so-called spin transfer torque, the torque exerted by the electric spin polarised current on the magnetisation. These works predicted, on the one hand, the possibility for steady magnetisation precession driven by a constant electric current and, on the other, the possibility for current driven switching of the magnetisation. Both phenomena are relevant for applications such as microwave generation and magnetic random access memories (MRAM) and greatly stimulated the research on spintronics in the last decade. The research field is today very active. Here we mention two research areas which are particularly relevant for the studies presented in this thesis:

• The research area focusing on multilayered films and nanopillars similar to the one of Fig. 3.2, often called spin-valves. This is the system considered by Slonczewski in the aforementioned paper [3]. The difference with respect to the GMR setup lies in the higher current density regime: if in the GMR effect the applied electric current is weak and is used just to probe the magnetisation of the free ferromag- netic layer, in the case considered by Slonczewski, the current is stronger and is used to actively control the dynamics of the magnetisation in the free layer. The theoretical description of Slonczewski has been experimentally verified, showing that the spin transfer can indeed induce switching [15, 16] and magnetisation precession [10, 17].

• The research area studying systems made by a single homogeneous ferromagnetic material, such as ferromagnetic nanowires or films, where a spin polarised electric current interacts with the magnetisation patterns developed inside the sample, such as domain walls or vortices. This is a quite recent area of research and has received considerable attention both from theoretical and experimental studies. It has been experimentally shown that a current flowing through a ferromagnetic nanowire can induce the movement of the domain walls which are developed in- side it (see Fig. 3.3). Such studies [18, 19, 20] have also been explained with

theoretical models [21]. These models are often1 based on the theory by Zhang

and Li [23], where the Landau-Lifshitz equation is extended by including addi- tional torques, which capture the interaction between an electric current and a locally inhomogeneous magnetisation.

1

Actually, several models have been proposed. Initially it was assumed that the magnetic moment of the conduction electron adiabatically follows the local magnetic moment [22]. Later a non-adiabatic correction was added [23, 24].

Figure 3.3: Magnetic transmission X-ray microscopy (MTXM) showing a domain wall inside a nanowire. A current pulse (j ∼ 1012_A/m2_{) can be used to move it. Repetitive measurements reveal} the stochastic nature of the current induced domain wall motion (reproduction from [20]).

The main problem in both the two research areas is that the current density required

in order to obtain significant effects is often too high (between 1010 and 1012A/m2),

causing excessive Joule heating and thus the meltdown or deterioration of the sample. There is then a high interest in finding systems where the spin transfer torque effects are maximised and require lower current density. In this thesis we investigate exchange spring systems in the form of multilayer films, a case which lies between the case of spin-valve and the case of homogeneous ferromagnetic nanowire. Indeed, exchange spring systems are multilayer systems which still can develop artificial domain walls with shape and size which can be controlled, first, during manufacturing (by selecting a suitable geometry) and, later, by applying an appropriate magnetic field. This is an extremely important feature, since the size and shape of a domain wall have a critical role in determining its interaction with the applied electric current [18]. Moreover the recent experimental discovery of significant GMR in exchange spring multilayers [25], suggests that spin-transfer-torque may play a role in these systems.

The numerical spin-transfer studies that we present in this thesis are all based on the Zhang-Li model, the same model [23] which has been successfully employed in the theoretical understanding of current-driven domain wall motion in ferromagnetic nanowires. The applicability of this model to systems made by different materials, such as exchange spring systems, needs to be discussed carefully, since the different spin transport properties of the layers may lead to effects which are not taken into account by the model. We will return on this point later in the thesis. In the next section we enter into the details of the derivation of the Zhang-Li model, exploring

closely the physics of spin transport in ferromagnets with inhomogeneous magnetisation configuration.