Micromagnetics

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Micromagnetics deals with the interactions between magnetic moments on sub-micrometre length scales. These are governed by several competing energy terms. Dipolar energy is the energy which causes magnets to align north to south pole. Exchange energy will attempt to make the magnetic moments in the immediately surrounding space lie parallel to one another (if the material is ferromagnetic) or antiparallel to one another (if antiferromagnetic). Anisotropic energy is low when the magnetic moments are aligned along a particular crystal direction. The Zeeman energy is at its lowest when magnetic moments lie parallel to an external magnetic field.

Since the most efficient magnetic alignment (also known as a configuration) is the one in which the energy is lowest, the sum of these four energy terms will attempt to become as small as possible at the expense of the others, yielding complex physical interactions.

The competition of these interactions under different conditions is responsible for the overall behavior of a magnetic system.

History

Micromagnetics as a field (i.e. that which deals specifically with the behaviour of (ferro)magnetic materials at sub-micrometer length scales) was introduced in 1963 when William Fuller Brown, Jr. published a paper on antiparallel domain wall structures. Until comparatively recently computation micromagnetics has been prohibitively expensive in terms of computational power, but smaller problems are now solveable on a modern desktop PC.

Landau-Lifshitz-Gilbert equation

Generally, a form[1] of the Landau-Lifshitz-Gilbert equation:

$\frac{\partial\textbf M}{\partial t} = - |\gamma| \textbf{M} \times \textbf{H}_{\mathrm{eff}} +\alpha\,\frac{\textbf M}{M_S}\times\frac{\partial \textbf{M}} {\partial t}\qquad (1)$

is used to solve time-dependent micromagnetic problems, where $\textbf{M}$ is the magnetic moment per unit volume, $\textbf{H}_{\mathrm{eff}}$ is the effective magnetic field, $\alpha$ is the Gilbert phenomenological damping parameter and $\gamma$ is the electron gyromagnetic ratio. Furthermore, $M_S$ is the magnitude of the magnetization vector $\textbf M\,.$

Equation (1) can be shown to be equivalent to the more complicated form

$\frac{\partial\textbf M}{\partial t} = - \frac{|\gamma|}{1+\alpha^2} \textbf{M} \times \textbf{H}_{\mathrm{eff}} -\frac{|\gamma |\alpha}{1+\alpha^2}\,\frac{\textbf M}{M_S}\times{\textbf{M}}\times{\textbf H_{\textrm{eff}}}\qquad (1')\,$

Originally, in 1935, Landau and Lifshitz used this expression without the denominator $(1+\alpha^2)$, which arose from Gilbert's modification in 1955.

Landau-Lifshitz equation

If in (1) we put the Gilbert damping parameter $\alpha =0$, then we get the famous, damping-free, Landau-Lifshitz equation (LLE)

${\partial\textbf{M}\over \partial t} = - |\gamma| \textbf{M} \times \textbf{H}_{\mathrm{eff}}.\qquad (2)$

The effective field

An essential merit of the micromagnetic theory concerns the answer on the question, how the effective field $\textbf H_{\mathrm{eff}}$ depends on the relevant interactions, namely, (i), on the exchange interaction; (ii), on the so-called anisotropy interaction; (iii), on the magnetic dipole-dipole interaction; and, (iv), on the external field (the so-called Zeeman field).

The answer is somewhat involved: let the energies corresponding to (i) and (ii) be given by

$E_\mathrm{exchange} =A\int_V\,\mathrm dV\,\sum_{j=1}^3\,\frac{\partial \textbf m}{\partial x_j}\cdot\frac{\partial \textbf m}{\partial x_j}\,_{|\textbf r}$

and

$E_\mathrm{anisotropy}\, =\int_V\,\mathrm dV\,F_\mathrm{anisotropy}(\textbf m)\,_{|\textbf r}\,.$

Here we use the decomposition $\textbf M=M_s\,\textbf m$ of the magnetization vector into its magnitude MS and the direction vector $\textbf m\,,$ while A is the so-called exchange constant. V is the magnetic volume.

Then we have: $\textbf H_{\mathrm{eff}}(\textbf r)=\frac{2A}{M_s}\,\nabla^2 \textbf m\,_{|\textbf r} -\frac{\partial F_\mathrm{anisotropy}\,(\textbf m)}{M_s\,\partial\textbf m}\,_{|\textbf r}+\int_V\,\mathrm dV'\,M_S\,\frac{\textbf m(\textbf r')\cdot (\textbf r -\textbf r')}{|\textbf r -\textbf r'|^3}+\textbf H_\mathrm{external}.$[2][3][4]

Here the third term on the r.h.s. is the internal field produced at the position $\textbf r$ by the dipole-dipole interaction, whereas the fourth term is the external field, also called the Zeeman field. Usually the first and the third term play the dominating role, usually a competing one, in this complicated sum. In particular: due to the third term the effective field is a nonlocal function of the magnetization, i.e. although the Landau-Lifshitz-Gilbert equation looks relatively harmless, one is actually dealing with a complicated nonlinear set of integro-differential equations.

Applications

Apart from conventional magnetic domains and domain-walls, the theory also treats the statics and dynamics of topological line and point configurations, e.g. magnetic vortex and antivortex states;[5] or even 3d-Bloch points,[6][7] where, for example, the magnetization leads radially into all directions from the origin, or into topologically equivalent configurations. Thus in space, and also in time, nano- (and even pico-)scales are used.

The corresponding topological quantum numbers[7] are thought to be used as information carriers, to apply the most recent, and already studied, propositions in information technology.

Magnetism

Footnotes and references

1. ^ There are different (equivalent) forms of the Landau-Lifshitz-Gilbert equation.
2. ^ Here the minus sign at the second place on the r.h.s. is obvious: the magnetization chooses that direction which is lowest in energy.
3. ^ We use the cgs system of units. In the SI system, in the third term on the r.h.s. an additional factor $\frac{\mu_0}{4\pi\mu_0}$ appears.
4. ^ Note that certain transformations of $\textbf H_{\mathrm{ eff}}(\textbf r)$ are always allowed, e.g. one can add any modification parallel to $\textbf m(\textbf r)\,,$ since this does not change $\textbf m\times\textbf H_{\textrm{eff}}.$
5. ^ S. Komineas, N. Papanicolaou: Dynamics of vortex-antivortex pairs in ferromagnets, in: arXiv:0712.3684v1, (2007)
6. ^ A. Thiaville et al., Micromagnetic study of Bloch-point-mediated vortex core reversal, in: Phys. Rev. B, vol. 67 (9), 094410 (2003), doi:10.1103/PhysRevB.67.094410
7. ^ a b W. Döring, Point singularities in micromagnetism, J. Appl. Phys. 39, 1006 (1968), [1]