General Principles of Micromagnetics

Classical (zero-temperature) micromagnetics describe magnetic materials in a nanometric lengthscale where each discretization unit is represented by a large magnetic moment that could be described in a semiclassical approximation, i.e., no quantum effects are taken into account. The principle of micromagnetics is due to Brown [107] who derived the micromagnetic equations for the equilibrium properties. The method consists in discretizing the magnetic film in finite differences [108-110] or finite elements [111-114], writing the energy contribution of each magnetic discretization unit, and minimizing the total energy of the system. No temperature effects or dynamics are included at this stage. The total energy normally comprises the contributions of Zeeman energy (external field):

Jy where M(r) is the magnetization of the specimen and V is its volume; the energy of anisotropy in the uniaxial case has the expression:

Jy where K is the anisotropy constant, e(r) is a unit vector that defines its direction, and m (r) is the unit magnetization vector; the energy of exchange:

Jy where A is the exchange constant, and the magnetostatic energy:

The magnetostatic energy, which has the nature of very long-range decaying interactions, requires the most effort of the computation. It should be found by solving the Poisson equation of motion (SGSM units are used):

Hmagn ^Umagn>

AUinside _ 4^VM

AU^outside_0

magn

where Umagn is the magnetostatic potential inside and outside the magnetic body, with the boundary conditions:

j outside tj inside _ tj o

Umagn Umagn

dn dn and n defines the unit vector, normal to the surface of the magnetic sample. At the present time, two main methods have become widely used for its calculation: the Fast Fourier transform [115, 116] and FEM-BEM (finite element-boundary element method), which directly solves the Poisson equation [111-114]. For the energy minimization, the conjugate gradient method [113, 114] or the relaxation method [108, 109] are very useful. Another way to minimize the energy is the integration of the equation of motion of the magnetization (the Landau-Lifshitz-Gilbert equation) with a large damping constant [117]. The advantage of this method is that it provides the dynamical information for the magnetization motion together with the static information.

For simulation of a nanostructured material, it is important to generate numerically the geometrical microstructure (crystalline grain structure). Although at the beginning, many calculations used regular hexagonal arrays [118, 119], lately, it has been widely accepted that the Voronoi discretization [111, 112, 120, 121] naturally and realistically describes the nanostructure with the desired log-normal distribution. Figure 13 represents an example of such a structure generated for micromagnetic simulations. Another way to introduce the nanostructure into the calculation is the use of closely packed spherical particles [122, 123] with log-normal distribution.

The role of the magnetization-discretization length has been discussed intensively in the literature. Depending on the desired size of calculations, more or less details in the magnetization distribution are required. As a consequence, the micromagnetic calculations have been performed in systems by using different discretization scales, ranging from one discretization element equal to one grain [119, 124], passing through the grains with special discretization at the intergrain boundaries [111], and down to atomistic models [125]. However, it must be noted that there exist two

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