Magnetocrystalline Anisotropy

Intuition

A spin in vacuum is rotationally invariant: it can point any direction with no energy cost. But the same spin in a crystal feels the lattice through its electronic orbitals — and orbitals know all about the crystal axes. Spin-orbit coupling is the bridge: it ties the spin to the orbital, the orbital to the lattice, and therefore the spin to the crystal axes.

The energetic consequence is that the magnetization of a ferromagnetic crystal does not point with equal ease in every direction. Some directions are easy — the magnetization settles there spontaneously — and some are hard — pushing $\vec{M}$ there costs measurable energy.

Without anisotropy there would be no hysteresis, no domain walls, and no magnetic memory at all. $\vec{M}$ could rotate freely to zero in any soft direction and the material would forget every bit you wrote on it. Anisotropy is the reason a refrigerator magnet stays magnetized.

Formal definition

The magnetocrystalline anisotropy energy is the part of the free energy that depends on the orientation of $\vec{M}$ relative to the crystal axes. For a single-domain particle of volume $V$ it can always be expanded in the direction cosines $({\alpha }_1,{\alpha }_2,{\alpha }_3)$ of $\vec{M}$ along the crystal axes.

Two cases dominate practice:

Uniaxial (Co hcp, all single-axis crystals):

$$E_{\text{ani}}\approx K_{u1}V{\sin }^2\theta +K_{u2}V{\sin }^4\theta +\dots ,$$

where $\theta$ is the angle between $\vec{M}$ and the easy axis and $K_{u1}$ is the uniaxial anisotropy constant in J/m${}^3$.

Cubic (Fe bcc, Ni fcc):

$$E_{\text{ani}}\approx K_{c1}V({\alpha }_1^2{\alpha }_2^2+{\alpha }_2^2{\alpha }_3^2+{\alpha }_3^2{\alpha }_1^2)+\dots$$

The sign and magnitude of $K_{c1}$ pick out which crystal directions are easy.

Key results

1. Origin: spin-orbit coupling

The exchange interaction by itself is isotropic: it cares only about relative spin orientation, not absolute direction in space. What ties the spins to the lattice is the spin-orbit Hamiltonian ${\mathcal{H}}_{\text{SO}}\propto \vec{L}\cdot \vec{S}$. Through the orbital, $\vec{S}$ inherits the symmetry of the local electrostatic environment, and the anisotropy energy is what survives after averaging over the partially filled d- or f-shell.

Two consequences follow immediately:

• 4f magnets (rare earths) have enormous anisotropy — the f orbitals barely hybridize with the lattice so spin-orbit is unscreened. SmCo${}_5$ and NdFeB inherit anisotropies of $10^7$ J/m${}^3$.
• 3d magnets (Fe, Co, Ni) have modest anisotropy — strong hybridization with neighbours quenches most of the orbital angular momentum, leaving residual anisotropies of $10^4$ – $10^5$ J/m${}^3$.

2. Uniaxial anisotropy and the bit

For $K_{u1}>0$ the energy is minimal at $\theta =0$ and $\theta =\pi$ — two equivalent ends of the easy axis. The energy surface looks like two polar caps; rotating $\vec{M}$ into the equator costs $K_{u1}V$.

These two stable states are the two minima of one classical bit. Every uniaxial ferromagnetic grain — in a hard disk, in an MRAM cell, in a permanent magnet — is fundamentally a Stoner–Wohlfarth bit sitting in such a double-well potential. See Stoner–Wohlfarth for the dynamics.

For $K_{u1}<0$ the easy plane replaces the easy axis: $\vec{M}$ prefers any direction perpendicular to the unique axis, but no specific direction within that plane — the so-called easy-plane case.

3. Cubic anisotropy in Fe and Ni

In a cubic crystal the sign of $K_{c1}$ picks one of two distinguished sets of directions:

• $K_{c1}>0$ (Fe): easy axes are the $⟨100⟩$ edges of the cube. There are six equivalent easy directions, so a single cubic crystal can host six possible orientations of $\vec{M}$.
• $K_{c1}<0$ (Ni): easy axes are the $⟨111⟩$ body diagonals, with eight equivalent orientations.

Iron and nickel both have several equivalent easy axes, which makes their magnetic memory richer (and their domain structure more complicated) than that of a uniaxial cobalt crystal.

4. Orders of magnitude

| Material | $K_u$ or $|K_{c1}|$ (kJ/m${}^3$) | Hardness class | Typical use | | -------- | ------------------------------- | -------------- | ----------- | | Permalloy (Ni${}_{80}$Fe${}_{20}$) | $<1$ | very soft | sensors, HF transformers | | Soft Fe | $\sim 50$ | soft | transformer cores | | Co (hcp) | $\sim 500$ | moderate | thin-film magnets | | SmCo${}_5$ / NdFeB | $10^4$ – $10^5$ | very hard | permanent magnets |

The same number $K_u$ that fixes the coercivity $H_K=2K_u/({\mu }_0{M}_s)$ also fixes the energy barrier $K_{u}V$ to thermal switching — that is why it is the central material parameter of magnetism engineering.

5. Beyond magnetocrystalline anisotropy

Anisotropy in a real sample comes from several sources, only one of which is magnetocrystalline:

• Shape anisotropy — stray-field energy of an elongated grain prefers $\vec{M}$ along the long axis. Effective anisotropy $\frac{1}{2}{\mu }_0{M}_s^2(N_{\perp }-N_{\parallel })$.
• Surface / interface anisotropy — broken symmetry at a film surface generates a perpendicular term $K_s/t$. Crucial in MRAM free layers.
• Strain-induced (magnetoelastic) — strain $\epsilon$ feeds into the energy through the magnetostriction constants $\lambda$.

The micromagnetic energy functional lumps all of these into a single effective anisotropy term.

Summary

Without anisotropy a ferromagnet has no memory. Magnetocrystalline anisotropy is the spin-orbit-mediated rule that ties spins to the crystal lattice: it picks easy axes, sets the height of the double-well bit, and through $H_K=2K_u/({\mu }_0{M}_s)$ caps the coercivity of any single-domain particle.

The contrast between soft ($K_u\to 0$) and hard ($K_u\gtrsim 10^5$ kJ/m${}^3$) materials is the contrast between a ferromagnet that forgets and one that remembers — and it is set almost entirely by spin-orbit physics on the atomic scale.

Connections

• micromagnetic-energy — anisotropy as one of four competing energies
• stoner-wohlfarth — the minimal hysteretic model built from $K_u$
• hysteresis — what anisotropy buys you macroscopically
• magnetic-domains — domain walls cost $\sqrt{A{K}_u}$ per area
• stoner-model — band-magnetism counterpart of the same 3d electrons

References

• A. Hubert & R. Schäfer, Magnetic Domains (Springer, 1998), Ch. 3.
• R. Skomski, Simple Models of Magnetism (Oxford, 2008), Ch. 3.
• J. M. D. Coey, Magnetism and Magnetic Materials (Cambridge, 2010), Ch. 7.