Magnetic anisotropy

Magnetic anisotropy is the directional dependence of a material's magnetic properties. The magnetic moment of magnetically anisotropic materials will tend to align with an easy axis, which is an energetically favorable direction of spontaneous magnetization. The two opposite directions along an easy axis are usually equivalent, and the actual direction of magnetization can be along either of them (see spontaneous symmetry breaking).

In contrast, a magnetically isotropic material has no preferential direction for its magnetic moment unless there is an applied magnetic field.

Magnetic anisotropy is a prerequisite for hysteresis in ferromagnets: without it, a ferromagnet is superparamagnetic.^[1]

Sources

There are several sources of magnetic anisotropy:^[2]

Magnetocrystalline anisotropy: The atomic structure of a crystal introduces preferential directions for the magnetization.
Shape anisotropy: When a particle is not perfectly spherical, the demagnetizing field will not be equal for all directions, creating one or more easy axes.
Magnetoelastic anisotropy: Tension may alter magnetic behaviour, leading to magnetic anisotropy.
Exchange anisotropy: Occurs when antiferromagnetic and ferromagnetic materials interact.^[3]

Anisotropy energy of a single-domain magnet

Suppose that a ferromagnet is single-domain in the strictest sense: the magnetization is uniform and rotates in unison. If the magnetic moment is $\scriptstyle {\boldsymbol {\mu }}$ and the volume of the particle is $\scriptstyle V$ , the magnetization is $\scriptstyle {\mathbf {M}}={\boldsymbol {\mu }}/V=M_{s}\left(\alpha ,\beta ,\gamma \right)$ , where $\scriptstyle M_{s}$ is the saturation magnetization and $\scriptstyle \alpha ,\beta ,\gamma$ are direction cosines (components of a unit vector) so $\scriptstyle \alpha ^{2}+\beta ^{2}+\gamma ^{2}=1$ . The energy associated with magnetic anisotropy can depend on the direction cosines in various ways, the most common of which are discussed below.

Uniaxial

A magnetic particle with uniaxial anisotropy has one easy axis. If the easy axis is in the $z$ direction, the anisotropy energy can be expressed as one of the forms:

E=KV\left(1-\gamma ^{2}\right)=KV\sin ^{2}\theta ,

where $\scriptstyle V$ is the volume, $\scriptstyle K$ the anisotropy constant, and $\scriptstyle \theta$ the angle between the easy axis and the particle's magnetization. When shape anisotropy is explicitly considered, the symbol $\scriptstyle {\mathcal {N}}$ is often used to indicate the anisotropy constant, instead of $\scriptstyle K$ . In the widely used Stoner–Wohlfarth model, the anisotropy is uniaxial.

Triaxial

A magnetic particle with triaxial anisotropy still has a single easy axis, but it also has a hard axis (direction of maximum energy) and an intermediate axis (direction associated with a saddle point in the energy). The coordinates can be chosen so the energy has the form

\displaystyle E=K_{a}V\alpha ^{2}+K_{b}V\beta ^{2}.

If $\scriptstyle K_{a}>K_{b}>0,$ the easy axis is the $z$ direction, the intermediate axis is the $y$ direction and the hard axis is the $x$ direction.^[4]

Cubic

A magnetic particle with cubic anisotropy has three or four easy axes, depending on the anisotropy parameters. The energy has the form

E=KV\left(\alpha ^{2}\beta ^{2}+\beta ^{2}\gamma ^{2}+\gamma ^{2}\alpha ^{2}\right).

If $\scriptstyle K>0,$ the easy axes are the $x,y,$ and $z$ axes. If $\scriptstyle K<0,$ there are four easy axes characterized by $x=\pm y=\pm z$ .

Notes

References

Aharoni, Amikam (1996). Introduction to the Theory of Ferromagnetism. Clarendon Press. ISBN 0-19-851791-2.
Donahue, Michael J.; Porter, Donald G. (2002). "Analysis of switching in uniformly magnetized bodies". IEEE Transactions on Magnetics. 38 (5): 2468–2470. Bibcode:2002ITM....38.2468D. doi:10.1109/TMAG.2002.803616.
McCaig, Malcolm (1977). Permanent magnets in theory and practice. Pentech press. ISBN 0-7273-1604-4.
Meiklejohn, W.H.; Bean, C.P. (1957-02-03). "New Magnetic Anisotropy". Physical Review. 105 (3): 904–913. Bibcode:1957PhRv..105..904M. doi:10.1103/PhysRev.105.904.
Tyablikov, S. V. (1995). Methods in the Quantum Theory of Magnetism (Translated to English) (1st ed.). Springer. ISBN 0-306-30263-2.

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[1] Aharoni 1996

[2] McCaig 1977

[3] Meiklejohn & Bean 1957

[4] Donahue & Porter 2002