Micromagnetics

The exchange energy is a phenomenological continuum description of the quantum-mechanical exchange interaction. It is written as:

${\displaystyle E_{\text{exch}}=A\int _{V}\left((\nabla m_{x})^{2}+(\nabla m_{y})^{2}+(\nabla m_{z})^{2}\right)\mathrm {d} V}$

where A is the exchange constant; m_x, m_y and m_z are the components of m; and the integral is performed over the volume of the sample.

The exchange energy tends to favor configurations where the magnetization varies only slowly across the sample. This energy is minimized when the magnetization is perfectly uniform.

Magnetic anisotropy arises due to a combination of crystal structure and spin-orbit interaction. It can be generally written as:

${\displaystyle E_{\text{anis}}=\int _{V}F_{\text{anis}}(\mathbf {m} )\mathrm {d} V}$

where F_anis, the anisotropy energy density, is a function of the orientation of the magnetization. Minimum-energy directions for F_anis are called easy axes.

Time-reversal symmetry ensures that F_anis is an even function of m. The simplest such function is

${\displaystyle F_{\text{anis}}(\mathbf {m} )=-Km_{z}^{2}}$.

where K is called the anisotropy constant. In this approximation, called uniaxial anisotropy, the easy axis is the z direction.

The anisotropy energy favors magnetic configurations where the magnetization is everywhere aligned along an easy axis.

The Zeeman energy is the interaction energy between the magnetization and any externally applied field. It's written as:

${\displaystyle E_{\text{Z}}=-\mu _{0}\int _{V}\mathbf {M} \cdot \mathbf {H} _{\text{a}}\mathrm {d} V}$

where H_a is the applied field and µ₀ is the vacuum permeability.

The Zeeman energy favors alignment of the magnetization parallel to the applied field.

Example of micromagnetic configuration. Compared to a uniform state, the flux closure structure lowers the energy of the demagnetizing field, at the expense of some exchange energy.

The demagnetizing field is the magnetic field created by the magnetic sample upon itself. The associated energy is:

${\displaystyle E_{\text{demag}}=-{\frac {\mu _{0}}{2}}\int _{V}\mathbf {M} \cdot \mathbf {H} _{\text{d}}\mathrm {d} V}$

where H_d is the demagnetizing field. This field depends on the magnetic configuration itself, and it can be found by solving:

${\displaystyle \nabla \cdot \mathbf {H} _{\text{d}}=-\nabla \cdot \mathbf {M} }$

${\displaystyle \nabla \times \mathbf {H} _{\text{d}}=0}$

where −∇·M is sometimes called magnetic charge density. The solution of these equations (c.f. magnetostatics) is:

${\displaystyle \mathbf {H} _{\text{d}}=-{\frac {1}{4\pi }}\int _{V}\nabla \cdot \mathbf {M} {\frac {\mathbf {r} }{r^{3}}}\mathrm {d} V}$

where r is the vector going from the current integration point to the point where H_d is being calculated.

It is worth noting that the magnetic charge density can be infinite at the edges of the sample, due to M changing discontinuously from a finite value inside to zero outside of the sample. This is usually dealt with by using suitable boundary conditions on the edge of the sample.

The energy of the demagnetizing field favors magnetic configurations that minimize magnetic charges. In particular, on the edges of the sample, the magnetization tends to run parallel to the surface. In most cases it is not possible to minimize this energy term at the same time as the others. The static equilibrium then is a compromise that minimizes the total magnetic energy, although it may not minimize individually any particular term.

The magnetoelastic energy describes the energy storage due to elastic lattice distortions. It may be neglected if magnetoelastic coupled effects are neglected. There exists a preferred local distortion of the crystalline solid associated with the magnetization director m, . For a simple model, one can assume this strain to be isochoric and fully isotropic in the lateral direction, yielding the deviatoric ansatz

$${\displaystyle \mathbf {\varepsilon } _{0}(\mathbf {m} )={\frac {3}{2}}E\,\left[\mathbf {m} \otimes \mathbf {m} -{\frac {1}{3}}\mathbf {1} \right]}$$

where the material parameter E > 0 is the magnetostrictive constant. Clearly, E is the strain induced by the magnetization in the direction m. With this ansatz at hand, we consider the elastic energy density to be a function of the elastic, stress-producing strains ${\displaystyle \mathbf {\varepsilon } _{e}:=\mathbf {\varepsilon } -\mathbf {\varepsilon } _{0}}$. A quadratic form for the magnetoelastic energy is

$${\displaystyle E_{\text{m-e}}={\frac {1}{2}}[\mathbf {\varepsilon } -\mathbf {\varepsilon } _{0}(\mathbf {m} )]:\mathbb {C}$$ :[\mathbf {\varepsilon } -\mathbf {\varepsilon } _{0}(\mathbf {m} )]}

where ${\displaystyle \mathbb {C}$ :=\lambda \mathbf {1} \otimes \mathbf {1} +2\mu \mathbb {I} } is the fourth-order elasticity tensor. Here the elastic response is assumed to be isotropic (based on the two Lamé constants λ and μ). Taking into account the constant length of m, we obtain the invariant-based representation

$${\displaystyle E_{\text{m-e}}={\frac {\lambda }{2}}{\mbox{tr}}^{2}[\mathbf {\varepsilon } ]+\mu \,{\mbox{tr}}[\mathbf {\varepsilon } ^{2}]-3\mu E{\big \{}{\mbox{tr}}[\mathbf {\varepsilon } (\mathbf {m} \otimes \mathbf {m} )]-{\frac {1}{3}}{\mbox{tr}}[\mathbf {\varepsilon } ]{\big \}}.}$$

This energy term contributes to magnetostriction.

The effective field is the local field felt by the magnetization. It can be described informally as the derivative of the magnetic energy density with respect to the orientation of the magnetization, as in:

${\displaystyle \mathbf {H} _{\mathrm {eff} }=-{\frac {1}{\mu _{0}M_{s}}}{\frac {\mathrm {d} ^{2}E}{\mathrm {d} \mathbf {m} \mathrm {d} V}}}$

where dE/dV is the energy density. In variational terms, a change dm of the magnetization and the associated change dE of the magnetic energy are related by:

${\displaystyle \mathrm {d} E=-\mu _{0}M_{s}\int _{V}(\mathrm {d} \mathbf {m} )\cdot \mathbf {H} _{\text{eff}}\,\mathrm {d} V}$

Since m is a unit vector, dm is always perpendicular to m. Then the above definition leaves unspecified the component of H_eff that is parallel to m. This is usually not a problem, as this component has no effect on the magnetization dynamics.

From the expression of the different contributions to the magnetic energy, the effective field can be found to be:

${\displaystyle \mathbf {H} _{\mathrm {eff} }={\frac {2A}{\mu _{0}M_{s}}}\nabla ^{2}\mathbf {m} -{\frac {1}{\mu _{0}M_{s}}}{\frac {\partial F_{\text{anis}}}{\partial \mathbf {m} }}+\mathbf {H} _{\text{a}}+\mathbf {H} _{\text{d}}}$

The terms of the Landau-Lifshitz-Gilbert equation: precession (red) and damping (blue). The trajectory of the magnetization (dotted spiral) is drawn under the simplifying assumption that the effective field H_eff is constant.

This is the equation of motion of the magnetization. It describes a Larmor precession of the magnetization around the effective field, with an additional damping term arising from the coupling of the magnetic system to the environment. The equation can be written in the so-called Gilbert form (or implicit form) as:

${\displaystyle {\frac {\partial \mathbf {m} }{\partial t}}=-|\gamma |\mathbf {m} \times \mathbf {H} _{\mathrm {eff} }+\alpha \mathbf {m} \times {\frac {\partial \mathbf {m} }{\partial t}}}$

where γ is the electron gyromagnetic ratio and α the Gilbert damping constant.

It can be shown that this is mathematically equivalent to the following Landau-Lifshitz (or explicit) form:[1]

${\displaystyle {\frac {\partial \mathbf {m} }{\partial t}}=-{\frac {|\gamma |}{1+\alpha ^{2}}}\mathbf {m} \times \mathbf {H} _{\mathrm {eff} }-{\frac {\alpha |\gamma |}{1+\alpha ^{2}}}\mathbf {m} \times (\mathbf {m} \times \mathbf {H} _{\text{eff}}),}$

Where ${\displaystyle \alpha }$ is the Gilbert Damping constant, characterizing how quickly the damping term takes away energy from the system (${\displaystyle \alpha }$ = 0, no damping, permanent precession).