This chapter is in part taken from Abert (2013) .
Model | Description | Length Scale |
---|---|---|
Atomic level theory | Quantum mechanical ab initio calculations | \(< 1\,\text{nm}\) |
Micromagnetic theory | Continuous description of the magnetization | \(1-1000\,\text{nm}\) |
Domain theory | Description of domain structure | \(1-1000\,\mu\text{m}\) |
Phase theory | Description of ensembles of domains | \(> 0.1\,\text{mm}\) |
Ferromagnetic materials from the viewpoint of theoretical physics are most accurately described by the theory of quantum mechanics. In this theory the ferromagnet is described by a \(N\) -body problem whose complexity grows exponentially with the number of involved bodies \(N\) Thouless (1961) . Thus analytical calculations in this framework are restricted to very small systems.
Different models have been proposed in order to approximately describe ferromagnetic materials on a macroscopic scale. Depending on the simplifications introduced by a particular model it is able to describe the system accurately only under certain assumptions and on a certain length scale. Table 1 gives an overview of established models for the description of ferromagnets on different length scales.
For the description of ferromagnetism on the micron scale the theory of micromagnetism has proved to be a reliable tool. In contrast to domain theory it is able to resolve the inner structure of domain walls. On the other hand the micromagnetic equations can be solved numerically for relatively large system compared to atomistic approaches.
This chapter is organized as follows. In Sec. 1 the assumptions and simplifications of the micromagnetic model are discussed. Section 2 provides an overview over the different energy contributions of a ferromagnetic body. In Sec. 3 the Landau-Lifshitz-Gilbert equation is introduced, which describes the magnetization dynamics in micromagnetism. Section 4 and 5 discuss extensions, limits and solutions of the micromagnetic equations.
Magnetic matter basically consists of magnetic dipoles called elementary magnets. These dipoles can be identified with spins and orbital angular momentum of charges on the atomic level. Macroscopic magnetic properties are consequently derived from the interaction and superposition of these dipoles.
In the special case of ferromagnetic materials, electrons with overlapping wave functions favor a parallel spin alignment due to the so-called exchange interaction, see Sec. 2.1. The alignment of elementary magnets \(\boldsymbol{m}_i\) at places \(\boldsymbol{r}_i\) can thus be assumed to be locally almost parallel
\[\boldsymbol{m}_i \approx \boldsymbol{m}_j \quad \text{for} \quad |\boldsymbol{r}_i - \boldsymbol{r}_j| < \lambda \label{eq:neighbors_equal}\]
where \(\lambda\) is a measure for the range of the exchange interaction, called the exchange length. Further a homogeneous density of elementary magnets is assumed. Taking into account these assumptions the discrete distribution of magnetic moments \(\boldsymbol{m}_i\) can be well approximated by a continuous vector density \({\boldsymbol{M}_{}(\boldsymbol{r})}\) such that
\[\int_\Omega {\boldsymbol{M}_{}(\boldsymbol{r})} \,\text{d}\boldsymbol{r}\approx \sum_i \mathbb{1}_\Omega (\boldsymbol{r}_i) \boldsymbol{m}_i \label{eq:discrete_to_continuum}\]
holds approximately for volumes \(\Omega\) of the size \(\lambda^3\) and bigger. It is important to understand that the existence and the range of the exchange interaction is crucial for a good approximation in Eqn. \(\text{\ref{eq:discrete_to_continuum}}\) . The vector field \(\boldsymbol{M}\) is called magnetization. Due to the homogeneous density of elementary magnets it has a constant norm
\[{\boldsymbol{M}_{}(\boldsymbol{r})} = M_{\text{s}} \cdot {\boldsymbol{m}_{}(\boldsymbol{r})} \quad \text{with} \quad | {\boldsymbol{m}_{}(\boldsymbol{r})} | = 1. \label{eq:normalized_magnetization}\]
where \(M_{\text{s}}\) is called the saturation magnetization. In the following the unit vector field \(\boldsymbol{m}\) which should not be confused with the moments \(\boldsymbol{m}_i\) will often be used instead of \(\boldsymbol{M}\) .
The continuous magnetization field is a common parameter in classical electrodynamics Jackson (1999) . Micromagnetism extends the classical field theory by non classical effects such as the exchange interaction. These effects are expressed in the framework of a continuum theory, see Sec. 2. Moreover, micromagnetism describes the dynamics of the magnetization field by the Landau-Lifshitz-Gilbert equation, see Sec. 3. Due to this combination of classical field theory and quantum mechanics, micromagnetism is often referred to as semi-classical continuum theory.
The total energy of a ferromagnet with respect to the magnetization is influenced by a multitude of physical effects. While some of these effects have a classical description, like the demagnetization energy and the Zeeman energy, others have a quantum mechanical origin and have to be adapted to the continuum theory of micromagnetism, e.g. the exchange energy and the anisotropy energy. By finding local minima of the energy functional, stable magnetization configurations can be obtained. However, the total energy does also play an important role for the dynamics of a ferromagnet as will be seen in Sec. 3.
The characteristic feature of ferromagnetic materials is the existence of spontaneous magnetization. From classical electrodynamics it is known that neighboring spins energetically favor an antiparallel alignment Jackson (1999) . Consequently macroscopic magnetic bodies are expected to avoid a uniform magnetization, which in fact is the case for paramagnetic and diamagnetic materials.
However, the elementary magnets in ferromagnetic materials are subject to the so-called exchange interaction. This quantummechanical effect leads to an energetically favored parallel alignment of neighboring spins and thus to macroscopic uniform magnetization configurations.
The exchange energy is derived from the Coulomb energy of two indistinguishable particles with overlapping wave functions. A two-particle system of fermions features an antisymmetric overall wave function. For a singlet spin configuration this leads to a symmetric orbital wavefunction, whereas a triplet spin configuration leads to an antisymmetric orbital wave function. The expectation value of the two-particle distance is larger for antisymmetric orbital wave functions. Hence the triplet spin configuration leads to a lowered Coulomb energy and is thus energetically favored. In the classical picture the triplet configuration corresponds to a parallel alignment of spins. A detailed discussion of the exchange interaction can be found in any textbook on quantummechanics, e.g. Griffiths (1994) .
Here we will use the classical Heisenberg Hamiltonian for two neighboring spins as starting point to derive a micromagnetic expression for the exchange energy
\[E_{i,j} = - J \, \boldsymbol{S}_i \cdot \boldsymbol{S}_j \label{eq:energetics_heisenberg}\]
where \(J\) is the so-called exchange integral and \(\boldsymbol{S}_i\) and \(\boldsymbol{S}_j\) are two neighboring classical spins. With the magnitude of the spins \(S = |\boldsymbol{S}_i| = |\boldsymbol{S}_j|\) und the unit vectors \(\boldsymbol{n}_i = \boldsymbol{S}_i / S\) and \(\boldsymbol{n}_j = \boldsymbol{S}_j / S\) this can be written as
\begin{align} E_{i,j} &= - J \, S^2 \, \boldsymbol{n}_i \cdot \boldsymbol{n}_j \label{eq:exchange_discrete} \\ &= - J \, S^2 \, [ 1 - \frac{1}{2} (\boldsymbol{n}_i - \boldsymbol{n}_j)^2 ]\end{align}
The exchange energy of a magnetic body is calculated by summing up
\[E = \sum_{i,j} - J_{ij} \, S^2 \, [ 1 - \frac{1}{2} (\boldsymbol{n}_i - \boldsymbol{n}_j)^2 ]\]
where \(J_{ij}\) is the exchange integral for spins \(i\) and \(j\) . That means \(J_{ij} \neq 0\) only for exchange coupled, usually neighboring, spins. In the theory of micromagnetism this expression has to be adapted to the continuous magnetization field \(\boldsymbol{m}\) . The scalar product in Eqn. \(\text{\ref{eq:exchange_discrete}}\) is analogously given by
\[{\boldsymbol{m}_{}(\boldsymbol{r})} \cdot \boldsymbol{m}(\boldsymbol{r} + \Delta \boldsymbol{r}) = 1 - \frac{1}{2} [{\boldsymbol{m}_{}(\boldsymbol{r})} - \boldsymbol{m}(\boldsymbol{r} + \Delta \boldsymbol{r})]^2 \label{eq:exchange_continuous_1}\]
where \(\Delta \boldsymbol{r}\) is chosen as distance vector between two exchange coupled magnetic moments. Expanding \(\boldsymbol{m}(\boldsymbol{r} + \Delta \boldsymbol{r})\) in \(\Delta \boldsymbol{r}\) up to first order and inserting into Eqn. \(\text{\ref{eq:exchange_continuous_1}}\) yields
\[{\boldsymbol{m}_{}(\boldsymbol{r})} \cdot \boldsymbol{m}(\boldsymbol{r} + \Delta \boldsymbol{r}) \approx 1 - \frac{1}{2} \sum_i (\Delta \boldsymbol{r} \cdot \boldsymbol{\nabla}m_i)^2. \label{eq:exchange_continuous_2}\]
The energy for a magnetic body is obtained by summing up contributions from different distance vectors \(\Delta \boldsymbol{r}_i\) depending on the crystal structure and integration over the magnetic body
\[E = \int_\Omega \sum_i A_i \, {\boldsymbol{m}_{}(\boldsymbol{r})} \cdot \boldsymbol{m}(\boldsymbol{r} + \Delta \boldsymbol{r}_i) \,\text{d}\boldsymbol{r}\]
where \(A_i\) are called exchange constants and include the exchange integral and the magnitude of the spins involved. Inserting Eqn. \(\text{\ref{eq:exchange_continuous_2}}\) yields the general expression
\[E = C + \int_\Omega \sum_{i,j,k} A_{jk} \frac{\partial m_i}{\partial x_j} \frac{\partial m_i}{\partial x_k} \,\text{d}\boldsymbol{r}\label{eq:exchange_continuous_3}\]
for the exchange energy. The constant \(C\) results from the integration of the constant term of Eqn. \(\text{\ref{eq:exchange_continuous_2}}\) and can be omitted without changing the physics of the system. Here \(A_{jk}\) is a matrix of exchange constants. By rotation of the coordinate system this matrix can be diagonalized Döring (1966) which yields
\[E = \int_\Omega \sum_{i,j} A_{j} \left( \frac{\partial m_i}{\partial x_j} \right)^2 \,\text{d}\boldsymbol{r}. \label{eq:exchange_continuous_4}\]
In the case of cubic and isotropic materials the exchange constant does not depend on the spatial dimension und thus Eqn. \(\text{\ref{eq:exchange_continuous_4}}\) further reduces to
\[E = A \int_\Omega \sum_{i} (\boldsymbol{\nabla}m_i)^2 \,\text{d}\boldsymbol{r}= A \int_\Omega (\boldsymbol{\nabla}\boldsymbol{m})^2 \,\text{d}\boldsymbol{r}. \label{eq:exchange_continuous_final}\]
This expression turns out to accurately describe most materials and is usually used in micromagnetics. The exchange constant \(A\) is determined experimentally.
Note that this result was derived from the classical Heisenberg model that assumes localized spins. In metallic ferromagnets the spins are not localized and the Heisenberg model does not apply. However, Eqn. \(\text{\ref{eq:exchange_continuous_final}}\) still describes the exchange interaction phenomenologically up to first order Hubert and Schäfer (1998) .
The demagnetization energy, also called magnetostatic energy or stray-field energy, is the energy of the magnetization in the magnetic field created by the magnetization itself. This means that this energy contribution accounts for the dipole–dipole interaction of the elementary magnets.
The demagnetization energy can be derived from classical electromagnetics. Maxwell’s equations for electrostatics, assuming a vanishing current \(\boldsymbol{J}\) , are given by
\begin{align} \boldsymbol{\nabla}\cdot \boldsymbol{B} &= 0 \label{eq:demag_zero_div} \\ \boldsymbol{\nabla}\times \boldsymbol{H} &= 0 \label{eq:demag_zero_curl}\end{align}
where the magnetic flux \(\boldsymbol{B}\) is connected to the magnetic field \(\boldsymbol{H}\) via the magnetization \(\boldsymbol{M}\)
\[\boldsymbol{B} = \mu_0 (\boldsymbol{H} + \boldsymbol{M}).\]
Equation \(\text{\ref{eq:demag_zero_curl}}\) is equivalent to the magnetic field \(\boldsymbol{H}\) being the gradient of a scalar potential \(u\) . Hence the demagnetization-field is the solution to the system
\begin{align} \Delta u &= \boldsymbol{\nabla}\cdot \boldsymbol{M} \label{eq:demag_poisson} \\ \boldsymbol{H} &= - \boldsymbol{\nabla}u. \label{eq:demag_scalar_potential}\end{align}
The boundary condition to this system is given in an asymptotical fashion by
\[{u_{}(\boldsymbol{r})} = \mathcal{O}(1 / |\boldsymbol{r}|) \text{ for } |\boldsymbol{r}| \rightarrow \infty. \label{eq:demag_open_boundary}\]
This condition states that the potential drops to zero at infinity. It is often referred to as open boundary condition. Equation \(\text{\ref{eq:demag_poisson}}\) is Poisson’s equation and can be solved with the well known Green’s function of the Laplacian, which naturally satisfies the boundary condition \(\text{\ref{eq:demag_open_boundary}}\) , see Jackson (1999)
\[{u_{}(\boldsymbol{r})} = - \frac{1}{4 \pi} \int \frac{\boldsymbol{\nabla}' \cdot {\boldsymbol{M}_{}(\boldsymbol{r}')}}{|\boldsymbol{r} - \boldsymbol{r}'|} \,\text{d}\boldsymbol{r}' \label{eq:demag_whole_space}\]
where the integration is carried out over the whole space. In the case of an ideal magnetic body the magnetization is only defined on a finite region \(\Omega\)
\begin{align} |{\boldsymbol{M}_{}(\boldsymbol{r})}| = \left\{ \begin{array}{cl} M_{\text{s}} & \text{if } \boldsymbol{r} \in \Omega \\ 0 & \text{else} \end{array} \right.\end{align}
which leads to a discontinuity of \(\boldsymbol{M}\) on the boundary \(\partial \Omega\) .
Figure 1: Sketch of the limiting procedure used for the derivation of the boundary term in the integral solution of the scalar potential. The discontinuity of the magnetization across the sample boundary \(\partial \Omega\) is smoothed out in a region \(\Omega_{\text{d}}\) surrounding the sample.
In this case the solution of Eqn. \(\text{\ref{eq:demag_whole_space}}\) can be obtained by a limiting process. Consider a finite region \(\Omega_{\text{d}}\) that surrounds the magnetic body \(\Omega\) . Further we assume a smooth decay of the magnetization within \(\Omega_{\text{d}}\) , see Fig. 1. With the magnetization and its divergence vanishing in the outside region \(\mathbb{R}^3 \backslash \Omega \cup \Omega_{\text{d}}\) Eqn. \(\text{\ref{eq:demag_whole_space}}\) can be written as
\[{u_{}(\boldsymbol{r})} = - \frac{1}{4 \pi} \left[ \int_{\Omega} \frac{\boldsymbol{\nabla}' \cdot {\boldsymbol{M}_{}(\boldsymbol{r}')}}{|\boldsymbol{r} - \boldsymbol{r}'|} \,\text{d}\boldsymbol{r}' + \int_{\Omega_{\text{d}}} \frac{\boldsymbol{\nabla}' \cdot {\boldsymbol{M}_{}(\boldsymbol{r}')}}{|\boldsymbol{r} - \boldsymbol{r}'|} \,\text{d}\boldsymbol{r}' \right]. \label{eq:demag_split}\]
Applying Green’s theorem to the integral over \(\Omega_{\text{d}}\) yields
\[\int_{\Omega_{\text{d}}} \frac{\boldsymbol{\nabla}' \cdot {\boldsymbol{M}_{}(\boldsymbol{r}')}}{|\boldsymbol{r} - \boldsymbol{r}'|} \,\text{d}\boldsymbol{r}' = \int_{\partial \Omega_{\text{d}}} \frac{{\boldsymbol{M}_{}(\boldsymbol{r}')} \cdot \boldsymbol{n}}{|\boldsymbol{r} - \boldsymbol{r}'|} \,\text{d}\boldsymbol{s}' - \int_{\Omega_{\text{d}}} {\boldsymbol{M}_{}(\boldsymbol{r}')} \cdot \boldsymbol{\nabla}' \frac{1}{|\boldsymbol{r} - \boldsymbol{r}'|} \,\text{d}\boldsymbol{r}' \label{eq:demag_green}\]
where \(\,\text{d}\boldsymbol{s}'\) is the area measure to \(\boldsymbol{r}'\) and \(\boldsymbol{n}\) is the unit outward normal. In the limit of a rapidly decaying magnetization \(\boldsymbol{M}\) the region \(\Omega_{\text{d}}\) can become infinitely small without changing the result of Eqn. \(\text{\ref{eq:demag_green}}\) . In this case the right-hand side of Eqn. \(\text{\ref{eq:demag_green}}\) reduces to the boundary integral since the volume integral has a finite integrand and is carried out over an infinitely small volume. The boundary \(\partial \Omega_{\text{d}}\) consists of an inner and an outer boundary. The integral over the outer boundary vanishes, because of a vanishing magnetization \(\boldsymbol{M}\) . The inner boundary coincides with the boundary of the magnetic body \(\partial \Omega\) except for the orientation. Thus the boundary integral can be replaced by an integral over \(\partial \Omega\) with opposite sign. Inserting into Eqn. \(\text{\ref{eq:demag_split}}\) results in
\[{u_{}(\boldsymbol{r})} = - \frac{1}{4 \pi} \left[ \int_\Omega \frac{\boldsymbol{\nabla}' \cdot {\boldsymbol{M}_{}(\boldsymbol{r}')}}{|\boldsymbol{r} - \boldsymbol{r}'|} \,\text{d}\boldsymbol{r}' - \int_{\partial \Omega} \frac{{\boldsymbol{M}_{}(\boldsymbol{r}')} \cdot \boldsymbol{n}}{|\boldsymbol{r} - \boldsymbol{r}'|} \,\text{d}\boldsymbol{s}' \right]. \label{eq:demag_jackson}\]
The expressions \(\rho = - \boldsymbol{\nabla}\boldsymbol{M}\) and \(\sigma = \boldsymbol{M} \cdot \boldsymbol{n}\) are often referred to as magnetic volume charges and magnetic surfaces charges respectivly. An alternative expression to Eqn. \(\text{\ref{eq:demag_jackson}}\) can be obtained by applying Green’s theorem
\[{u_{}(\boldsymbol{r})} = \frac{1}{4 \pi} \int_\Omega {\boldsymbol{M}_{}(\boldsymbol{r}')} \cdot \boldsymbol{\nabla}' \frac{1}{|\boldsymbol{r} - \boldsymbol{r}'|} \,\text{d}\boldsymbol{r}'. \label{eq:demag_potential_convolution}\]
The demagnetization field \(\boldsymbol{H}_{\text{demag}}\) is calculated as negative gradient of the potential \(u\) with respect to \(\boldsymbol{r}\)
\[{\boldsymbol{H}_{\text{demag}}(\boldsymbol{r})} = - \boldsymbol{\nabla}{u_{}(\boldsymbol{r})} = \int_\Omega \boldsymbol{\widetilde{N}}(\boldsymbol{r} - \boldsymbol{r}') {\boldsymbol{M}_{}(\boldsymbol{r}')} \,\text{d}\boldsymbol{r}' \label{eq:demag_field}\]
with the so-called demagnetization tensor \(\boldsymbol{\widetilde{N}}\) given by
\[\boldsymbol{\widetilde{N}}(\boldsymbol{r} - \boldsymbol{r}') = - \frac{1}{4 \pi} \boldsymbol{\nabla}\boldsymbol{\nabla}' \frac{1}{|\boldsymbol{r} - \boldsymbol{r}'|}.\]
According to classical electrodynamics the energy is given by
\begin{align} E &= - \frac{\mu_0}{2} \int_\Omega \boldsymbol{M} \cdot \boldsymbol{H}_{\text{demag}} \,\text{d}\boldsymbol{r}\label{eq:demag_energy}\\ &= - \frac{\mu_0}{2} \int \!\!\!\! \int_\Omega {\boldsymbol{M}_{}(\boldsymbol{r})} \boldsymbol{\widetilde{N}}(\boldsymbol{r} - \boldsymbol{r}') {\boldsymbol{M}_{}(\boldsymbol{r}')} \,\text{d}\boldsymbol{r}\,\text{d}\boldsymbol{r}'\end{align}
where the factor \(1/2\) accounts for the fact that the field is generated by the magnetization itself. Due to the integration over \(\boldsymbol{r}\) and \(\boldsymbol{r}'\) every dipole–dipole interaction between \({\boldsymbol{M}_{}(\boldsymbol{r})}\) and \({\boldsymbol{M}_{}(\boldsymbol{r}')}\) contributes twice to the result, which is corrected by this prefactor.
Depending on the crystal structure of a ferromagnetic material, it energetically favors the alignment of the magnetization parallel to certain axes. This energy contribution results from spin-orbit interactions Hubert and Schäfer (1998) and is referred to as anisotropy energy. The energetically favored axes are called easy axes. These axes are undirected which means that a local minimum of the energy at \(\boldsymbol{m}_{\text{min}}\) implies a local minimum at \(-\boldsymbol{m}_{\text{min}}\) with
\[E(\boldsymbol{m}_{\text{min}}) = E(-\boldsymbol{m}_{\text{min}}) \label{eq:aniso_symmetry}\]
Depending on the lattice structure, a material may have one or more of these easy axes. In the simplest case a material has a single easy axis. This uniaxial anisotropy energy is given by
\[E = - \int_\Omega [ K_{\text{u1}} ( \boldsymbol{m} \cdot \boldsymbol{e}_{\text{u}} )^2 + K_{\text{u2}} ( \boldsymbol{m} \cdot \boldsymbol{e}_{\text{u}} )^4 ] \,\text{d}\boldsymbol{r}\label{eq:aniso_uniaxial}\]
where \(\boldsymbol{e}_{\text{u}}\) is a unit vector pointing in the direction of the easy axis and \(K_{\text{u1}}\) and \(K_{\text{u2}}\) are called anisotropy constants. This phenomenological expression is the result of a Taylor expansion up to fourth order. Only even powers are considered in order to fullfill the symmetry condition \(\text{\ref{eq:aniso_symmetry}}\) . Uniaxial anisotropy occurs in materials with a hexagonal or tetragonal crystal structure, e.g. cobalt.
Materials with a cube-symmetric lattice structure naturally feature three easy axes \(\boldsymbol{e}_i\) which are pairwise orthogonal
\[\boldsymbol{e}_i \cdot \boldsymbol{e}_j = \delta_{ij}.\]
The energy of such a cubic anisotropy is the result of an expansion in the magnetization components along the easy axes
\[E = \int_\Omega [ K_{\text{c1}} (m_1^2 m_2^2 + m_2^2 m_3^2 + m_3^2 m_1^2) + K_{\text{c2}} m_1^2 m_2^2 m_3^2 ] \,\text{d}\boldsymbol{r}\label{eq:aniso_cubic}\]
where \(m_i = \boldsymbol{e}_i\cdot\boldsymbol{m}\) is the magnetization component in direction of the anisotropy axis \(\boldsymbol{e}_i\) . Again terms which violate the symmetry condition \(\text{\ref{eq:aniso_symmetry}}\) are neglected. Moreover, only contributions which are constant under permutation of magnetization components \(m_i\) are considered in order to comply with the cubic symmetry. Cubic anisotropy occurs in materials such as iron which has a body-centered cubic structure or nickel which has a face-centered cubic structure.
Although the expressions for the anisotropy energy in Eqn. \(\text{\ref{eq:aniso_uniaxial}}\) and \(\text{\ref{eq:aniso_cubic}}\) have a pure phenomenological origin, they are able to describe anisotropy effects with a high accuracy. In practical applications the energy expressions are often reduced to the lowest order term.
The Zeeman energy of a ferromagnetic body is the energy of the magnetization in an external field \(\boldsymbol{H}_{\text{zeeman}}\) given by
\[E = - \mu_0 \int_\Omega \boldsymbol{M} \cdot \boldsymbol{H}_{\text{zeeman}} \,\text{d}\boldsymbol{r}.\]
The central equation in micromagnetism for the description of magnetization dynamics is the Landau-Lifshitz-Gilbert equation, which was originally proposed in Landau and Lifshitz (1935) . In this work from 1935 by Landau and Lifshitz the motion of the magnetization is described by a precessional term and a damping term. While the precessional term is physically derived in this work, the damping term is purely phenomenological. In 1955 Gilbert derived an equivalent expression for the Landau-Lifshitz-Gilbert equation from a Lagrangian formulation, where the damping is treated more strictly, see Gilbert (1955) and Gilbert (2004) .
In the following a classical Lagrangian method as well as a quantum mechanical approach are presented briefly to derive the Landau-Lifshitz-Gilbert equation. Both approaches give insight to different aspects of the micromagnetic model and are thus of interest for the application of this theory.
With an appropriate Lagrange functional the Landau-Lifshitz-Gilbert equation can be obtained by the Lagrangian formalism. Since the magnetization \(\boldsymbol{M}\) is assumed to be normalized, see Eqn. \(\text{\ref{eq:normalized_magnetization}}\) , its motion may locally be well described by the rotation of a rigid body. The angular velocity of a rigid body in terms of the Euler angles reads
\[\boldsymbol{\Omega} = \begin{pmatrix} \dot{\phi} \sin(\theta) \sin(\psi) + \dot{\theta} \cos(\psi) \\ \dot{\phi} \sin(\theta) \cos(\psi) + \dot{\theta} \sin(\psi) \\ \dot{\phi} \cos(\theta) + \dot{\psi} \end{pmatrix} \label{eq:llg_euler}\]
where the shorthand notation \(\dot{f} = \partial_t f\) is used. Note that the angular velocity \(\boldsymbol{\Omega}\) as well as the angles \(\theta\) , \(\phi\) and \(\psi\) have to depend on the location \(\boldsymbol{r}\) in order to describe the magnetization field \({\boldsymbol{m}_{}(\boldsymbol{r})}\) . Without loss of generality the rotation axes can be chosen such that the \(r_3\) -axis points into the direction of the magnetization in every point and thus
\[\boldsymbol{m} = \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix}.\]
In the rigid-body picture the magnetization is described by a rotationally symmetric stick. Since the angle \(\psi\) describes the rotation of the stick around its symmetry axis it may be set to \(\psi = 0\) without loss of generality Landau and Lifshitz (1969) . Thus Eqn. \(\text{\ref{eq:llg_euler}}\) reduces to
\[\boldsymbol{\Omega} = \begin{pmatrix} \dot{\theta} \\ \dot{\phi} \sin(\theta) \\ \dot{\phi} \cos(\theta) + \dot{\psi} \end{pmatrix}.\]
The Lagrangian of a dynamical system in general is given by
\[\mathcal{L} = T(\boldsymbol{q}, \dot{\boldsymbol{q}}) - V(\boldsymbol{q})\]
where \(T\) is the kinetic energy and \(V\) is the potential energy of the system and \(\boldsymbol{q}\) and \(\dot{\boldsymbol{q}}\) are the generalized coordinates and their time derivatives respectively. In micromagnetics the potential energy is given by the energy contributions discussed in Sec. 2. The kinetic energy however has no classical explanation since the magnetization does not have inertia in the classical sense. In Gilbert (1955) Gilbert proposed the Lagrangian
\[\mathcal{L} = - \frac{M_{\text{s}}}{\gamma} \dot{\phi} \cos(\theta) - U(\theta, \phi). \label{eq:llg_lagrangian}\]
As shown by Wegrowe et al. in Wegrowe and Ciornei (2012) this expression is equivalent to
\begin{align} \mathcal{L} &= \frac{1}{2} I [\dot{\phi} \cos(\theta) + \dot{\psi}]^2 - U(\theta, \phi) \\ &= \frac{1}{2} I \Omega_3^2 - U(\theta, \phi) \label{eq:llg_lagrangian_wegrowe}\end{align}
which describes a classical rotation energy with \(I\) being the moment of inertia. However, in this energy only the rotation around the symmetry axis of the magnetization is considered. From a classical point of view it is not clear why \(\Omega_1\) and \(\Omega_2\) may be neglected. In Wegrowe and Ciornei (2012) this choice of \(\mathcal{L}\) is discussed in detail.
In the following we will stick with the Lagrangian \(\text{\ref{eq:llg_lagrangian}}\) as proposed by Gilbert. According to the Lagrangian formulation, the equation of motion in terms of the generalized coordinates \(\boldsymbol{q} = (\theta, \phi)\) is given by
\[\frac{\text{d}}{\text{d}t} \frac{\delta \mathcal{L}}{\delta \dot{q_i}} - \frac{\delta \mathcal{L}}{\delta q_i} + \frac{\delta D}{\delta \dot{q_i}} = 0 \label{eq:llg_lagrange_formulation}\]
where \(\delta / \delta f\) denotes a functional derivative and \(D\) is an additional dissipative Rayleigh function that accounts for energy losses of the system. Inserting Eqn. \(\text{\ref{eq:llg_lagrangian}}\) into Eqn. \(\text{\ref{eq:llg_lagrange_formulation}}\) yields
\begin{align} \dot{\phi} \sin(\theta) &= \frac{\gamma}{M_{\text{s}}} \left[ \frac{\delta U}{\delta \theta} + \frac{\delta D}{\delta \dot{\theta}} \right] \label{eq:llg_lagrange_result_1} \\ \dot{\theta} &= - \frac{\gamma}{M_{\text{s}} \sin(\theta)} \left[ \frac{\delta U}{\delta \phi} + \frac{\delta D}{\delta \dot{\phi}} \right]. \label{eq:llg_lagrange_result_2}\end{align}
These equations of motion describe the dynamics of the magnetization by the Euler angles \({\phi_{}(\boldsymbol{r})}\) and \({\theta_{}(\boldsymbol{r})}\) . In order to obtain a description in cartesian coordinates, the time derivative of the magnetization in cartesian coordinates is considered
\[\partial_t \boldsymbol{m} = \boldsymbol{\Omega} \times \boldsymbol{m} = \begin{pmatrix} \dot{\phi} \sin(\theta) \\ - \dot{\theta} \\ 0 \end{pmatrix}. \label{eq:llg_rotating_frame}\]
Furthermore, since the magnetization is a unit vector field \(|\boldsymbol{m}| = 1\) , variations of the Euler angles \(\phi\) and \(\theta\) are given by
\begin{align} \delta m_1 &= \sin(\theta) \delta \phi \\ \delta m_2 &= \delta \theta.\end{align}
Hence Eqn. \(\text{\ref{eq:llg_lagrange_result_1}}\) and \(\text{\ref{eq:llg_lagrange_result_2}}\) can be written as
\begin{align} \partial_t m_1 &= \frac{\gamma}{M_{\text{s}}} \left[ \frac{\delta U}{\delta m_2} + \frac{\delta D}{\delta \dot{m_2}} \right] \\ \partial_t m_2 &= - \frac{\gamma}{M_{\text{s}}} \left[ \frac{\delta U}{\delta m_1} + \frac{\delta D}{\delta \dot{m_1}} \right]\end{align}
or equivalently
\[\partial_t \boldsymbol{m} = - \frac{\gamma}{M_{\text{s}}} \boldsymbol{m} \times \left( \frac{\delta U}{\delta \boldsymbol{m}} + \frac{\delta D}{\delta \dot{\boldsymbol{m}}} \right).\]
Choosing the reasonable dissipative function \(D = \alpha^\ast (\partial_t \boldsymbol{m})^2\) with \(\alpha^\ast \geq 0\) results in the Landau-Lifshitz-Gilbert equation
\[\partial_t \boldsymbol{m} = - \gamma (\boldsymbol{m} \times \boldsymbol{H}_{\text{eff}}) + \alpha (\boldsymbol{m} \times \partial_t \boldsymbol{m}) \label{eq:llg_implicit}\]
where \(\boldsymbol{H}_{\text{eff}}\) is the so-called effective field given by
\[\boldsymbol{H}_{\text{eff}} = - \frac{1}{\mu_0 M_{\text{s}}} \frac{\delta U}{\delta \boldsymbol{m}}. \label{eq:llg_effective_field}\]
and \(\alpha \geq 0\) is a damping constant.
In Sec. 3.1 it is shown that the Landau-Lifshitz-Gilbert equation can be obtained by application of the classical Lagrange formalism. However, this method is based on a particular choice of the Lagrangian, which is not completely justifiable by classical theory.
In this section a quantum mechanical approach is discussed to derive the Landau-Lifshitz-Gilbert equation. As described in Sec. 1 the continuous magnetization field approximates a discrete distribution of spins. In quantum mechanics the components of the spin are described by the spin operators \(\hat{S}_i\) . In the Heisenberg picture the time development of the spin operators are given by
\[\frac{\text{d}\hat{S}_j}{\text{d}t} = \frac{1}{i \hbar} \left[ \hat{S}_j, \hat{H} \right]. \label{eq:llg_qm_heisenberg}\]
The contributions to the Hamiltonian in the context of micromagnetics will depend on the spin. Thus it is reasonable to expand the Hamiltonian in the spin operators
\begin{align} \left[ \hat{S}_j, \hat{H} \right] &= \sum_k \frac{\partial \hat{H}}{\partial \hat{S}_k} \left[ \hat{S}_k, \hat{S}_j \right] + \mathcal{O}(\hbar^2) \notag \\ &= i \hbar \sum_{k,l} \frac{\partial \hat{H}}{\partial \hat{S}_k} \epsilon_{jkl} \hat{S}_l + \mathcal{O}(\hbar^2).\end{align}
Inserting into Eqn. \(\text{\ref{eq:llg_qm_heisenberg}}\) and using the vector notation \(\hat{\boldsymbol{S}} = (\hat{S}_1, \hat{S}_2, \hat{S}_3)\) yields
\[\frac{\text{d}\hat{\boldsymbol{S}}}{\text{d}t} = - \hat{\boldsymbol{S}} \times \frac{\partial \hat{H}}{\partial \hat{\boldsymbol{S}}} + \mathcal{O}(\hbar).\]
In the classical limit the spin operators may be replaced by the continuous magnetization vector \(\boldsymbol{M} = M_{\text{s}} \boldsymbol{m}\) . Further the second term can be neglected since \(\hbar \rightarrow 0\) . Identifying \(\partial \hat{H} / \partial \hat{\boldsymbol{S}}\) with the effective field \(\boldsymbol{H}_{\text{eff}}\) in Eqn. \(\text{\ref{eq:llg_effective_field}}\) leads to
\[\partial_t \boldsymbol{m} = - \gamma' (\boldsymbol{m} \times \boldsymbol{H}_{\text{eff}}).\]
This equation describes the precession of the magnetization around the effective field without any loss of energy. According to the original work of Landau and Lifshitz Landau and Lifshitz (1935) a phenomenological damping term is added in order to account for these losses. This damping term is constructed such that it is perpendicular to the precessional term. Further it should conserve the magnetization norm leading to
\[\partial_t \boldsymbol{m} = - \gamma' (\boldsymbol{m} \times \boldsymbol{H}_{\text{eff}}) - \alpha' \boldsymbol{m} \times (\boldsymbol{m} \times \boldsymbol{H}_{\text{eff}}). \label{eq:llg_explicit}\]
This is the explicit form of the Landau-Lifshitz-Gilbert equation. It can be shown that this form is equivalent to the implicit form in Eqn. \(\text{\ref{eq:llg_implicit}}\) . This is done by inserting Eqn. \(\text{\ref{eq:llg_implicit}}\) for \(\partial_t \boldsymbol{m}\) on the right-hand side of Eqn. \(\text{\ref{eq:llg_implicit}}\) and using the vector identity \(\boldsymbol{a} \times (\boldsymbol{b} \times \boldsymbol{c}) = (\boldsymbol{a} \cdot \boldsymbol{c}) \boldsymbol{b} - (\boldsymbol{a} \cdot \boldsymbol{b}) \boldsymbol{c}\) . The coefficients of the different versions of the Landau-Lifshitz-Gilbert equation satisfy the relations
\begin{align} \gamma' &= \gamma / (1 + \alpha^2) \\ \alpha' &= \alpha \gamma / (1 + \alpha^2).\end{align}
A full quantum mechanical description of a spin subject to exchange interaction, anisotropy and Zeeman field is given in Bode et al. (2012) where the Landau-Lifshitz-Gilbert equation is also obtained in a limit case.
Figure 2: Time evolution of a single magnetic moment as described by the Landau-Lifshitz-Gilbert equation. The motion can be divided into a precessional and a damping part. (a) Precessional motion around the effective field. (b) Damped motion. The magnetization relaxes towards the effective field. (c) Resulting motion including precession and damping.
The Landau-Lifshitz-Gilbert equation describes the motion of the magnetization in an effective field defined by Eqn. \(\text{\ref{eq:llg_effective_field}}\) . This motion can be described as the sum of a precessional term and a damping term. Figure 2 illustrates these two terms for the explicit formulation of the Landau-Lifshitz-Gilbert equation \(\text{\ref{eq:llg_explicit}}\) .
In the case of the Zeeman energy and the demagnetization energy, the effective field is an actual field in the classical sense. The Zeeman field is directly given by \(\boldsymbol{H}_{\text{zeeman}}\) , the demagnetization field is computed via Eqn. \(\text{\ref{eq:demag_field}}\) .
However, the exchange field as well as the anisotropy field have to be calculated via the variational derivative in Eqn. \(\text{\ref{eq:llg_effective_field}}\) . The exchange energy density is given by the integrand of Eqn. \(\text{\ref{eq:exchange_continuous_final}}\) . Applying the chain rule of variational calculus yields
\begin{align} {\boldsymbol{H}_{\text{ex}}(\boldsymbol{r})} &= - \frac{A}{\mu_0 M_{\text{s}}} \frac{\delta }{\delta \boldsymbol{m}} (\boldsymbol{\nabla}\boldsymbol{m})^2 \\ &= \frac{2 A}{\mu_0 M_{\text{s}}} \Delta \boldsymbol{m} \label{eq:effective_field_exchange}\end{align}
Analogously the effective fields for uniaxial and cubic anisotropy are obtained by computing the variational derivative of Eqn. \(\text{\ref{eq:aniso_uniaxial}}\) and \(\text{\ref{eq:aniso_cubic}}\) respectively
\begin{align} {\boldsymbol{H}_{\text{u}}(\boldsymbol{r})} &= \frac{2 K_{\text{u1}}}{\mu_0 M_{\text{s}}} \boldsymbol{e}_{\text{u}} (\boldsymbol{e}_{\text{u}} \cdot \boldsymbol{m}) + \frac{4 K_{\text{u2}}}{\mu_0 M_{\text{s}}} \boldsymbol{e}_{\text{u}} (\boldsymbol{e}_{\text{u}} \cdot \boldsymbol{m})^3 \\ {\boldsymbol{H}_{\text{c}}(\boldsymbol{r})} &= - \frac{2 K_{\text{c1}}}{\mu_0 M_{\text{s}}} \begin{pmatrix} m_1 m_2^2 + m_1 m_3^2 \\ m_2 m_3^2 + m_2 m_1^2 \\ m_3 m_1^2 + m_3 m_2^2 \end{pmatrix} - \frac{2 K_{\text{c2}}}{\mu_0 M_{\text{s}}} \begin{pmatrix} m_1 m_2^2 m_3^2 \\ m_1^2 m_2 m_3^2 \\ m_1^2 m_2^2 m_3 \end{pmatrix}.\end{align}
Figure 3: A so-called Gaussian pillbox \(\Omega_{\text{p}}\) on the outer boundary of the sample \(\Omega\). The discontinuity of the magnetization \(\boldsymbol{m}\) across the sample boundary \(\partial \Omega\) is smoothed out within the pillbox, as sketched on the right-hand side. Properties of the discontinuous system are obtained by considering the limit \(d \rightarrow 0\).
Due to the first derivate in time the Landau-Lifshitz-Gilbert equation is an initial value problem. In order to find a solution \(\boldsymbol{m}(\boldsymbol{r}, t)\) for \(t > t_0\) the initial magnetization \(\boldsymbol{m}(\boldsymbol{r}, t_0)\) has to be known.
Depending on the effective field, additional boundary conditions are required in order to find a unique solution of the Landau-Lifshitz-Gilbert equation. In fact, even with given boundary conditions the uniqueness of solutions to the Landau-Lifshitz-Gilbert equation could only be shown for special cases, see Carbou and Fabrie (2001a; Carbou and Fabrie 2001b; Alouges and Soyeur 1992) . From the effective field contribution discussed in Sec. 3.3, the exchange field is the only one that adds the need of boundary conditions due to its second order in space.
If this so-called exchange boundary condition is applied on the boundary of an ideal magnetic body, i.e. a boundary where the magnetization rapidly drops to zero, it is uniquely defined. Consider the Landau-Lifshitz-Gilbert equation without damping and with the exchange field as effective field.
\[\partial_t \boldsymbol{m} = - \gamma \boldsymbol{m} \times \Delta \boldsymbol{m}. \label{eq:llg_boundary_llg}\]
Since at the boundary of an ideal magnetic body the Laplacian of the magnetization \(\Delta \boldsymbol{m}\) is not defined, a limiting procedure is used to obtain the boundary condition. Like in Sec. 2.2 the magnetization is considered to decay continuously in a finite interval \(d\) . Equation \(\text{\ref{eq:llg_boundary_llg}}\) is integrated over a so-called Gaussian pillbox as sketched in Fig. 3. Application of Green’s theorem yields
\begin{align} \int_{\Omega_{\text{p}}} \partial_t m_i \,\text{d}\boldsymbol{r}&= \int_{\Omega_{\text{p}}} \epsilon_{ijk} m_j \Delta m_k \,\text{d}\boldsymbol{r}\label{eq:llg_boundary_green_1} \\ &= \int_{\partial \Omega_{\text{p}}} \epsilon_{ijk} m_j \frac{\partial m_k}{\partial \boldsymbol{n}} \,\text{d}\boldsymbol{r}- \int_{\Omega_{\text{p}}} \epsilon_{ijk} \boldsymbol{\nabla}m_j \cdot \boldsymbol{\nabla}m_k \,\text{d}\boldsymbol{r}. \label{eq:llg_boundary_green_2}\end{align}
The second integral on the right-hand side of Eqn. \(\text{\ref{eq:llg_boundary_green_2}}\) vanishes due to the skew-symmetric Levi-Civita tensor \(\epsilon\) . The boundary of the pillbox \(\partial \Omega_{\text{p}}\) coincides with the boundary of the magnetic body \(\Omega\) on one side except for the orientation. The other side of the pillbox is outside the magnetic body where \(|\boldsymbol{m}| = 0\) . Hence Eqn. \(\text{\ref{eq:llg_boundary_green_2}}\) reads
\[\int_{\Omega_{\text{p}}} \partial_t m \,\text{d}\boldsymbol{r}= - \int_{\partial_\Omega} \boldsymbol{m} \times \frac{\partial \boldsymbol{m}}{\partial \boldsymbol{n}} \,\text{d}\boldsymbol{r}\]
In the limit \(d \rightarrow 0\) the volume integral on the left-hand-side of Eqn. \(\text{\ref{eq:llg_boundary_green_1}}\) vanishes. Since the faces of the pill box can be chosen arbitrarily the boundary condition
\[\boldsymbol{m} \times \frac{\partial \boldsymbol{m}}{\partial \boldsymbol{n}} = 0\]
must hold in every boundary point. Further since \(|\boldsymbol{m}|=1\) the normal derivative of the magnetization is perpendicular to the magnetization in every point \(\partial \boldsymbol{m} / \partial \boldsymbol{n} \perp \boldsymbol{m}\) and thus
\[\frac{\partial \boldsymbol{m}}{\partial \boldsymbol{n}} = 0. \label{eq:exchange_boundary_condition}\]
This is the so-called exchange boundary condition, which is the right choice if the boundary of the computational domain \(\Omega\) coincides with the boundary of an ideal magnet as shown above. Equation \(\text{\ref{eq:exchange_boundary_condition}}\) was originally derived by Rado and Weertman in Rado and Weertman (1959) . Before, it was shown by Brown that the same boundary condition has to hold in energetic equilibrium Brown Jr. (1963) . Depending on further contributions to the effective field, such as surface anisotropy, this boundary condition changes accordingly, see Hubert and Schäfer (1998) .
As mentioned in Sec. 1 the magnetization \(\boldsymbol{m}\) is assumed to be normalized everywhere. This feature is preserved by the Landau-Lifshitz-Gilbert equation. Consider the time derivative of the squared magnetization
\[\partial_t |\boldsymbol{m}|^2 = \partial_t (\boldsymbol{m} \cdot \boldsymbol{m}) = 2 \partial_t \boldsymbol{m} \cdot \boldsymbol{m}.\]
Inserting Eqn. \(\text{\ref{eq:llg_implicit}}\) immediately yields \(\partial_t |\boldsymbol{m}|^2 = 0\) and thus also \(\partial_t |\boldsymbol{m}| = 0\) .
If the energy functional of a magnetic system does not explicitly depend on the time, i.e. if the external field is constant in time, the time derivative of the energy density may be written as
\begin{align} \partial_t U &= \frac{\delta U}{\delta \boldsymbol{m}} \cdot \partial_t \boldsymbol{m} \\ &= - \mu_0 M_{\text{s}} \boldsymbol{H}_{\text{eff}} \cdot \partial_t \boldsymbol{m}.\end{align}
Replacing \(\partial_t \boldsymbol{m}\) with Eqn. \(\text{\ref{eq:llg_explicit}}\) and spatial integration yields
\begin{align} \partial_t E &= \mu_0 M_{\text{s}} \int_\Omega \boldsymbol{H}_{\text{eff}} [ \gamma' \boldsymbol{m} \times \boldsymbol{H}_{\text{eff}} + \alpha' \cdot \boldsymbol{m} \times (\boldsymbol{m} \times \boldsymbol{H}_{\text{eff}}) ] \,\text{d}\boldsymbol{r}\\ &= - \mu_0 M_{\text{s}} \alpha' \int_\Omega | \boldsymbol{m} \times \boldsymbol{H} |^2 \,\text{d}\boldsymbol{r}\label{eq:llg_energy_loss}\\ &\leq 0.\end{align}
This means that the energy of a magnetic system is always a non increasing function in time. The Landau-Lifshitz-Gilbert equation is said to have Lyapunov structure d’Aquino, Serpico, and Miano (2005; Cimr á k 2007) .
In the special case of no damping \(\alpha = 0\) the right-hand side of Eqn. \(\text{\ref{eq:llg_energy_loss}}\) vanishes
\[\partial_t E = 0.\]
In this case the energy of the system ist preserved and the Landau-Lifshitz-Gilbert equation has Hamiltonian structure.
As discussed in Sec. 1 the micromagnetic model introduces a set of simplifications to the quantum mechanical description of magnetism. While justified for a broad range of applications, these simplifications may be inappropriate in some cases. The successfull application of micromagnetism to physical problems requires a solid understanding of the origin of this model. It is crucial to consider its underlying assumptions and simplifications in order to predict the validity of simulation results for specific physical problems. Without claiming to be complete, this section discusses a number of limits and extension to the basic micromagnetic model as introduced in the preceding sections.
The central assumption in micromagnetics is the homogeneous saturation magnetization \(M_{\text{s}}\) . This assumption is justified by the fact that ferromagnetic materials are subject to exchange coupling which leads to locally almost perfectly aligned magnetic moments.
However, certain magnetic processes involve the creation of magnetic singularities called Bloch points. At these Bloch points the magnetization changes rapidly in space, which is inconsistent with the basic assumption of a homogeneous \(M_{\text{s}}\) in micromagnetics. Despite this fact it was shown in Thiaville et al. (2003) that micromagnetic simulations involving the creation of Bloch points are able to describe the corresponding processes in accordance with experiments, although the energy density at the Bloch point is underestimated.
Another example for the violation of the micromagnetic assumption of a locally homogeneous magnetization is given by thermal effects. Thermal effects are most naturally reflected by local perturbation of magnetic moments. Perturbation of a single magnetic moment, however, obviously breaks the homogeneity of the magnetization. In the framework of classical micromagnetics a possible approach for consideration of finite temperature is the reduction of the saturation magnetization according to a mean-field approximation, see Kittel and McEuen (1996) . Another approach is to add a fluctuating field to the effective field, which converts the Landau-Lifshitz-Gilbert equation into a stochastic differential equation, see Lyberatos, Berkov, and Chantrell (1993) . Both of these techniques do not account for local changes in the saturation magnetization and as a result both methods fail to describe the magnetization dynamics correctly when approaching the Curie temperature. This deficiency is overcome by the Landau-Lifshitz-Bloch equation, which extends the Landau-Lifshitz-Gilbert equation not only by a fluctuating field, but also by a term that allows the change of the magnetization modulus Garanin (1997) .
Other successfull extensions to the micromagnetic model include the description of spin polarized currents and its interactions with the magnetization configuration Zhang and Li (2004) . A famous application for this interaction is the magnetic racetrack memory proposed by Parkin et al. in Parkin, Hayashi, and Thomas (2008) .
The micromagnetic model was designed to describe the energetics and dynamics in a homogeneous material. Thus the description of interfaces between different materials gives rise to another limit of the model. The magnetic coupling at these interfaces is often not entirely clear and can be subject to quantum mechanical effects that are not covered by the micromagnetic model. These effects include the Ruderman-Kittel-Kasuya-Yosida interaction (RKKY) Parkin and Mauri (1991) and the Dzyaloshinskii-Moriya Interaction (DMI) Sergienko and Dagotto (2006) . Work has been done on the integration of these effects into micromagnetic theory and computations, see Heide, Bihlmayer, and Bl ü gel (2008; Lopez-Diaz et al. 2012; Thiaville et al. 2012) . A common simplification for the description of thin multilayer materials however is the assumption of a bulk material with efficient material constants. These efficient material constants are determined by experiments.
The Landau-Lifshitz-Gilbert equation with effective field contributions as introduced in the preceding sections is a non-linear partial differential equation in space and time. Apart from some simplified edge cases the arising system of equations cannot be solved analytically. This work is dedicated to the numerical solution of the micromagnetic equations. In the following chapters the discrete solution of the different subproblems is discussed in detail. An overview over existing methods is given and novel methods are introduced. Finally the open-source finite-element micromagnetic simulation code magnum.fe is presented.
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