# Spin Wave Theory for Thin Films

We have introduced in Chapters 3 and 4 basic methods to study general bulk properties of spin waves excited in systems of interacting spins. We have taken advantage of the periodic crystalline structure to use the Fourier transforms and the periodic boundaiy conditions which allowed us to simplify calculations via the sum rules and the crystal symmetry.

When the invariance by translation is broken because of the presence of impurities, defects or surfaces, the calculation becomes more complicated. Often we have to modify methods established for the bulk and to introduce new techniques. Physically, the loss of the spatial periodicity causes a change in bulk properties of materials. The physics of disorder has been and still is a major research domain for more than 40 years. We can mention some well-studied disordered systems such as spin glasses, amorphous compounds and doped semiconductors. There is another domain which has been intensively developed in the past decades: the physics of nanomaterials. Nanomaterials such as ultrafine particles, ultrathin films and nanoribbons do not have periodic structures due to their nanometric dimension. These tiny objects have been used in many industrial applications which rapidly change our

*Physics of Magnetic Thin Films: Theory and Simulation *Hung T. Diep

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In this chapter, we present a few standard methods to study a family of simple magnetic systems where the loss of translational invariance is caused by the existence of surfaces or interfaces such as semi-infinite crystals, thin films, or multilayers. More complicated systems such as systems of non-collinear spin configurations are presented in the following chapters.

Surface effects are very important because they modify drastically properties of a bulk material when the so-called aspect ratio, i.e., the ratio of the number of surface atoms to the total number of atoms, becomes important as it is the case in small particles or in films of a few atomic layers. Surface physics has been intensively developed during the last 30 years. Among the main reasons for that rapid and successful development we can mention the interest in understanding the physics of low-dimensional systems and an immense potential of industrial applications of thin films [36, 72, 374]. In particular, theoretically it has been shown that systems of continuous spins (XY and Heisenberg) in two dimensions (2D) with a short-range interaction cannot have a long-range order at finite temperatures [231]. In the case of thin films, it has been shown that low-lying localized spin waves can be found at the film surface [281] and effects of these localized modes on the surface magnetization at finite temperatures (T) and on the critical temperature have been investigated by the Green's function technique [75, 78]. Experimentally, objects of nanometric size such as ultrathin films and nanoparticles have also been intensively studied because of numerous and important applications in industry. An example is the so-called giant magnetoresistance used in data storage devices, magnetic sensors, etc. [18, 22,134,345]. Recently, much interest has been attracted towards practical problems such as spin transport, spin valves and spin-transfer torques, due to numerous applications in spintronics. A family of topological magnetic structures called "skyrmions" are currently under intensive investigations. They have already important applications in various domains (see Chapters 14 and 15).

## Surface Effects

All crystals terminate in space by a surface. It can have various geometries and situations. The simplest one is a clean surface which is a perfect crystalline atomic plane such as (100) and (111) planes. It can be spherical as in the case of spherical aggregates. In general, the surface can include impurities, dislocations, vacancies, islands, steps, ... A surface can also chemisorb or physisorb alien atoms. Chemisorbed atoms form a strong chemical binding with surface atoms while physisorbed atoms are physically bound to surface atoms by weak potentials such as the long-range van der Waals interaction. They are sometimes called "adatoms.” It is obvious that the more the surface is disordered, the more it is difficult to study. Spectacular effects are often observed with well controlled and well characterized surfaces. Today, sophisticated techniques allow to create a surface with desired characteristics.

At the surface, atoms do not have the same environment as those inside the crystal. Due to the lack of neighbors and various neighboring defects and geometries, electronic states of surface atoms are modified in one way or in another, giving rise to changes in their effective interactions with neighboring atoms. The density of states shows often surface states which modify the filling of electronic bands, the position of the Fermi level and the magnetic moment of surface atoms. In addition, the surface anisotropy and surface exchange interaction can be very different from those of the interior atoms. We mention here some remarkable observations:

- (i) For a thin film, the dipolar interaction favors an in-plane spin configuration. However, when the film thickness becomes very small a perpendicular anisotropy comes into play to favor a spin configuration perpendicular to the film surface at low temperatures [36]. In addition, the sign and amplitude of the surface exchange may also change with decreasing thickness. These modifications can cause interesting surface behaviors such as magnetic ordering reconstructions near the surface and localized surface spin waves. Also, the competition between the dipolar interaction and the perpendicular surface anisotropy can give rise to the so-called "re-orientation transition” which turns the perpendicular configuration into the in-plane one at a finite temperature.
- (ii) Perturbations in the cohesive interaction which binds surface atoms can yield a modification of the lattice constant at the surface (contraction or dilatation) and even a reconstruction of surface geometry.
- (iii) Perturbations in electronic energy bands can give rise to anomalies in electronic (charge and/or spin) transport observed in multilayers and near the surface.

It is not the purpose of this chapter to give experimental data on problems raised above. There exist a great number of handbooks and reviews [35, 36, 72, 73, 374] for that purpose. In this chapter, we present some fundamental aspects which are well understood at present. Our purpose is to provide a theoretical framework to understand microscopic mechanisms which lead to macroscopic surface effects such as low surface magnetization, low transition temperature, surface phase transition and surface spin- configuration instability. We will concentrate our attention to simple methods which allow us to study properties of magnetic semiinfinite crystals and thin films.

## Surface Effects in Magnetism

We consider a ferromagnetic thin film of *N _{T}* atomic layers. The surface is denoted by index

*n*= 1 and the last layer by

*n*=

*N*We suppose the Oz axis is perpendicular to the film surface.

_{T}.### Surface Magnons

We have studied the bulk spin waves in Chapter 3. The amplitude of a bulk spin wave mode does not vary in space. In general, near magnetic perturbation sources such as magnetic impurities and surfaces, spin waves can be spatially localized. Such modes are called "surface spin wave modes" or "surface-localized modes." The amplitude of a surface-localized mode decays when it propagates

Figure 8.1 A surface spin wave mode (top) and a bulk mode (bottom).

from the surface into the bulk. We show in Fig. 8.1 a surface mode and a bulk mode, for comparison.

We express the amplitude of a surface mode by t/„(k) = Ae'^{k r}" where *n* denotes the index of the layer and r„ = Гц + z is the position of a lattice site of the n-th layer at the *z* position *z = **na *on the Oz axis, *a* being the distance between two successive layers in the *z* direction. In such a notation, a surface mode corresponds to a complex wave vector **к **= *ki* + *ik _{2}* where k

_{2}is non-zero. Its amplitude t/„(k) а

*е~*therefore, diminishes while propagating into the crystal interior (increasing n). We will show some examples in the following.

^{кгпа},### Reconstruction of Surface Magnetic Ordering

As we said above, surface exchange interaction and surface anisotropy may suffer from modifications not only in their magnitudes but also in their signs. These modifications may result in a rearrangement of the magnetic ordering at the surface to minimize the system energy. In the case of Heisenberg spin model, we can have a non-collinear spin configuration in the vicinity of the surface as will be seen below.

### Surface Phase Transition

Basic methods leading to fundamental properties of phase transitions in the bulk have been presented in Chapter 5. Phase transition is a collective phenomenon which takes place when the system changes its symmetiy. At the transition, the system spins become strongly correlated at a macroscopic scale. In a system with a finite size such as fine particles, the theoretical definition of the phase transition is not rigorously obeyed. For example, the correlation length in a second-order phase transition cannot go to infinity in a finite system. Nevertheless, anomalies in physical quantities can be observed in small systems. Finite-size scaling relations have been established [10, 95, 380] to allow us to obtain properties of the phase transition such as universality class at the infinite-size limit. In thin films, the infinite dimension of the film planes makes transitions possible. The characteristics of the phase transition in films depend on the surface conditions: If the film thickness is important then the influence of surface parameters is small since the aspect ratio is small. However, for ultrathin films, surface parameters become dominant making the surface phase transition very different from the bulk one.

There exist a large number of books on the surface phase transition. The reader is referred to, for example, reviews given in references [35, 54, 72, 87] for more details. One of the remarkable results is the existence of surface critical exponents and surface scaling laws which are different from the bulk ones [54].