2.1 Multiscale character of magnetism

Several length scales of different nature participate nontrivially in the magnetic phenomena. A good example is the magnetization reversal in a magnetic material that contains different kinds of defects as interfaces, voids, twins, etc. The reversal starts by nucleation near a defect, which is few nanometers and even angstroms in size. Posteriorly, a domain wall is formed that has a width of tens or hundreds of nanometers. The reversal creates a full spectrum wave length magnon distribution that transfers the energy across the sample [Suhl 98]. Additionally, the microstructure (which includes the defect profile) and the extrinsic parameters (e.g sample shape) determine the behavior of the material and the maximum domain size. The microstructure has a length scale from few nanometers to micrometers. The interactions have also different range in magnetism: from the long range dipolar interaction to the rapidly decaying exchange interaction.

Since the atomic magnetism has a quantum origin, we need quantum tools to describe it. Even assuming permanent magnetic moments, classical physics is not able to predict ferromagnetism [Hernando 01, Page 35]. Ab-initio calculations can be used to calculate the intrinsic parameters as magnetic moment, exchange energies, lattice sizes, etc. together with the energy reduction due to defect presence. The most widely used formalism is Density Functional Theory (DFT) [Dreizler 90] and its different approximations. However, it is extremely expensive in computational requirements and largest computationally affordable systems in our days consist of hundred of atoms. Simulation sizes comparable to domain wall width are not reachable for a large range of magnetic materials.

On the other hand micromagnetics, a continuum approximation, can be used in a range of systems in which the main length is comparable to the domain wall width as in artificially nanostructured elements (e.g. magnetic dots). Quite extended regions with the size of up to one hundred times the domain wall width can be simulated with this technique. In order to be more realistic the micromagnetic simulations must go hand in hand with the growing processes and characterization, since they need as input parameters the structural material parameters, such as the distribution of grain sizes, the presence of different chemical phases, the texture, etc. The tendency in the last years is to perform simulation with the greatest level of complexity of the media. However, other necessary parameters, such as the intergranular exchange and local values of magnetization, can not be obtained from standard magnetometric techniques and only ab-initio calculations could provide them, although in idealized situations.

In nowadays the trend is multiscale modeling, with the aim of using the ab-initio parameters in large scale calculations. Both in the European Union and in the United States mulsticale modeling programs have been regarded as an scientific priority for the next years [Department of Energy USA ,Cordis Europa ]. The notion of multiscale modeling stems from the necessity to treat inherently multiscale problems, as the magnetic systems. A nice example of such a modeling is the study of domain wall pinning by antiphase boundary in CoPt [Antropov 03]. Different schemes of multiscale model can be implemented depending on the simulation purposes. Fig. 2.1 shows a hierarchical model. In this model the small scale simulation(ab-initio) provides data for the final inclusion in the largest scale simulation. Additional finite temperature atomistic simulations are performed to obtain temperature dependent parameters. Other implementations are possible (see Section 2.2.4), in which two regions at different discretization levels are simulated at the same time to account for a defect on an atomistic scale.

The main problem of multiscale modeling is to link together different types of models. For example, the link between ab-initio calculations and the atomistic model is not obvious when magnetic moments are not localized or the exchange interactions are not of the Heisenberg type. The link between atomistic and micromagnetic model presents difficulties because the high-frequency part of the spinwave spectrum is truncated in micromagnetics. A related model is coarse grained modeling [Dobrovitski 00] that includes an intermediate region that is neither atomistic nor micromagnetic. The aim of the region is the correct propagation of spin waves that otherwise would be reflected by the interface.

Furthermore, different time scales appear in magnetism. From the femto seconds ultrafast magnetic dynamics [Beaurepaire 96] a vast range of magnetic processes appear, each of them with a different time scale. The magnetization precession occurs at picoseconds. The processes corresponding to the relaxation of energy into the lattice vibrations have time scales of $10^{-10}-10^{-11} s$. If a domain wall is propagating driven by current in a pinning media, its velocity can be as slow as $0.3 m/s$ [Kläui 05]. Finally, larger time scales corresponding to slow relaxation are interesting from the point of view of magnetic recording media stability and magnetic viscosity experiments.

Figure 2.1: Diagram showing a hierarchical multiscale model and its relation with experiments.