1 Introduction

Magnetism has a quantum origin and magnetization processes depend on the atomic magnetic moments, the exchange interaction, the spin-orbit coupling, etc. For this reason the "ab-initio" calculations can be useful to obtain the intrinsic magnetic parameters. But from the computational point of view these calculations are very expensive, they require long time and limit the system size to several hundred atoms. Therefore an "ab-initio" study of a macroscopic system is intractable. Additionally, the complete account of temperature fluctuations and dynamics on "ab-initio" level remains a future challenge. In this situation, multiscale modeling can provide a description on all scales using results from one scale as input parameters to the model of the next scale. Thus the "ab-initio" calculations (on a system with a small size) provide the magnetic parameters which will be included in the atomistic simulations. In these atomistic simulations we can take into account the temperature effects and obtain the temperature dependence of the magnetic parameters. The thermodynamic evaluation of macroscopic parameters such as temperature-dependent magnetization $ Ms(T)$, anisotropy $ K(T)$ [126] and exchange stiffness A(T) [139] can be used later as an input to large-scale micromagnetic simulations, based, for example on open source programs such as Magpar [140] or OOMMF [141]. This "hierarchichal" multiscale scheme was proposed in Ref. [121]. A scheme of this model is represented in Fig. 6.1.
Figure 6.1: Scheme showing a hierarchical multiscale model .

In this chapter we will describe the properties of magnetic thin films in the framework of a generalised Heisenberg model for localized magnetic moments with additional magnetostatic interactions and uniaxial anisotropy.

$\displaystyle \mathcal{H}=-\frac{1}{2}\sum_{i,j}\sum_{\alpha,\beta}S_{i}^{\alph...
...}^{\beta}+ \sum_{i}\mathrm{d(\mathbf{S_{i}})}+H_{MAG}  ;  \alpha,\beta=x,y,z,$ (78)

Here $ J_{ij}^{\alpha, \beta}$ are the elements of the exchange tensor $ \mathcal{J}_{ij}$ between the spins located at sites i and j, $ \alpha$ and $ \beta$ represent the coordinates $ x,y,z$. $ \mathrm{d(\mathbf{S_{i}})}$ is the anisotropy matrix, and $ \mathbf{S_{i}}$ is the spin vector at site i, ( $ \mid\mathbf{S_{i}}\mid=1$). First we determine the parameters appearing in the model from the first principle calculations and then study the thermodynamical macroscopic properties by means of the constrained Monte Carlo method [126], presented in chapter 4.

The self -consistent "ab-initio" calculations are performed in terms of the fully relativistic screened Khon-Korringa-Rostocker (SKKR) method [86]. Within this method , spin-polarization and relativistic effects (in particular, spin-orbit coupling) are treated on equal theoretical footing by solving the Kohn-Sham-Dirac equation. We employed the technique to map the electronic structure calculations into the Heisenberg model following L. Udvardi et. al [142]. The anisotropy energies have been extracted from the difference of the layer resolved band energies for the magnetization perpendicular and parallel to the surface. Since the spin-orbit coupling plays a key role in the magnetic anisotropy energy, the calculations were done by a fully relativistic spin polarized code, which was supplied by Dr. L. Szunyogh of the Budapest University of Technology and Economics, and henceforth we will call it the Budapest-Vienna Code.

Rocio Yanes