Antiferromagnetism, Ferrimagnetism, and Domains

When the exchange integral between neighbors is negative, the Heisenberg energy $- 2 J S_{i} \cdot S_{j}$ is lowered by antiparallel alignment. The moments still order, but the ordered state carries little or no net magnetization. This lesson covers the two-sublattice mean-field theory of that order, the ferrimagnets whose sublattices do not cancel, and the domain structure and hysteresis that make a piece of iron a useful magnet.

Two-sublattice antiferromagnetism

Divide a lattice with antiferromagnetic coupling into two interpenetrating sublattices $A$ and $B$ such that every $A$ site has only $B$ neighbors and vice versa. In the ground state all $A$ moments point one way and all $B$ moments the opposite way, so $M_{A} = - M_{B}$ and the total magnetization vanishes. The molecular-field method of the previous lesson now needs one field per sublattice. Each sublattice feels a field from its neighbors on the other,

B_{A} = - λ μ_{0} M_{B}, B_{B} = - λ μ_{0} M_{A},

with $λ > 0$ for antiferromagnetic coupling. Each sublattice magnetization follows the Brillouin function of its own field, giving a coupled pair of self-consistent equations. Below a critical temperature they have a solution with $M_{A} = - M_{B} \neq = 0$ : staggered order with no net moment.

Antiferromagnet (left): two sublattices of equal and opposite moment cancel. Ferrimagnet (right): the sublattice moments are unequal and leave a net magnetization.

The ordering temperature is the Néel temperature $T_{N}$ . Above it the material is paramagnetic, and the same linear-response calculation that gave the Curie–Weiss law now gives

χ = \frac{C}{T + θ}, θ = T_{N}

in the simplest case of only inter-sublattice coupling. The crucial sign change is in the denominator: the intercept of $1/ χ$ versus $T$ falls at the negative temperature $- θ$ , the mirror image of the ferromagnet's positive intercept. The ratio $θ / T_{N}$ , ideally one, measures the neglected same-sublattice coupling.

Below $T_{N}$ the susceptibility is anisotropic. With the field along the sublattice (easy) axis, the two sublattices are held rigidly antiparallel and a small field barely tilts them, so $χ_{∥} \to 0$ as $T \to 0$ . With the field perpendicular, both sublattices cant slightly toward it against a restoring exchange torque, giving a temperature-independent $χ_{⊥}$ . A powder averages the two. The result is a cusp in the susceptibility at $T_{N}$ : rising on the paramagnetic side as $1/ (T + θ)$ , peaking at $T_{N}$ , and falling below it.

Magnetic susceptibility of an antiferromagnet peaks at the Néel temperature. Above it the Curie–Weiss form rises toward T_N; below it the ordered sublattices resist alignment and chi falls.

Ferrimagnetism and the ferrites

If the two sublattices carry unequal moments — different ions, or different numbers of ions per cell — their antiparallel arrangement leaves a net moment. This is ferrimagnetism, and materials showing it are ferrites, typically oxides with the spinel or garnet structure. Magnetite $Fe_{3} O_{4}$ , the lodestone of the ancient world, is the prototype: its $Fe^{3 +}$ ions occupy both tetrahedral and octahedral sites in antiparallel arrangement and cancel, while the $Fe^{2 +}$ ions on the octahedral sites are left uncompensated and supply the moment.

A ferrimagnet looks like a ferromagnet from outside — a net spontaneous magnetization vanishing at an ordering temperature, a hysteresis loop — but its internal order is antiferromagnetic. Because the two sublattice magnetizations can have different temperature dependences, a ferrimagnet can have a compensation point below its ordering temperature where the sublattices exactly cancel and the net moment passes through zero before reappearing. The ferrites are electrically insulating, which suppresses the eddy currents that plague metallic magnets at high frequency, and this makes them the material of microwave components and transformer cores.

Domains and the magnetostatic energy

A single crystal of iron below $T_{c}$ is magnetized to saturation within any small region, yet a bulk piece can have zero net moment. The exchange energy is minimized by uniform magnetization, but a uniformly magnetized body creates a strong external field, and the energy stored in that field — the magnetostatic or demagnetizing energy — is large. The solid lowers its total energy by breaking into domains, regions of uniform magnetization pointing in different directions so that the external flux nearly cancels. A four-domain flux-closure pattern confines the flux entirely within the crystal and eliminates the external field.

Flux-closure domains: dividing the crystal into oppositely magnetized regions with triangular closure domains routes the flux entirely inside the sample and removes the external field energy.

The domains cannot be arbitrarily small, because each boundary between them costs energy. That boundary is the Bloch wall, a layer in which the magnetization rotates gradually from one domain orientation to the other. Two energies compete to set its width. Exchange favors a wide wall: neighboring spins want to be nearly parallel, so the total turning of the moment should be spread over many atomic planes. Magnetocrystalline anisotropy favors a narrow wall: the crystal has easy axes along which the magnetization prefers to lie, and any spin pointing between them costs anisotropy energy, so the wall should pass through the hard directions as quickly as possible.

Balancing an exchange stiffness $A$ (energy per unit length, of order $J S^{2} / a$ ) against an anisotropy energy density $K$ gives the wall width and energy per unit area

δ \sim \frac{A}{K}, σ \sim A K .

For iron $δ$ is tens of nanometers, hundreds of atomic planes. A particle smaller than $δ$ cannot support a wall and stays single-domain, a fact that governs magnetic recording media and rock magnetism.

A Bloch wall: the magnetization rotates gradually across many atomic planes. A wider wall lowers exchange energy but raises anisotropy energy; the width delta balances the two.

Hysteresis and the magnetization curve

Magnetizing a demagnetized ferromagnet drives its domain structure through a sequence of irreversible changes. At small applied field the domains already aligned with the field grow at the expense of those opposed, their Bloch walls sweeping through the crystal. Walls snag on defects and impurities and then break free suddenly (the Barkhausen jumps), so wall motion is irreversible and dissipative. At larger fields the remaining domains rotate their magnetization into the field direction against the anisotropy, and the sample reaches saturation.

On reducing the field the magnetization does not retrace its path: at zero field a remanent magnetization $M_{r}$ remains, and a reverse coercive field $H_{c}$ is needed to bring it back to zero. The loop of $M$ against $H$ is the hysteresis loop, and the energy dissipated per cycle equals its enclosed area. Two regimes of material follow:

Soft magnets — silicon steel, permalloy, the ferrites — have small $H_{c}$ and narrow loops. Their walls move easily, so they magnetize and demagnetize with little loss. These are transformer and motor cores.
Hard magnets — alnico, the rare-earth compounds $SmCo_{5}$ and $Nd_{2} Fe_{14} B$ — have large $H_{c}$ and wide loops. Their strong anisotropy pins the walls, so they retain magnetization against demagnetizing fields. These are permanent magnets.

Ferromagnetic hysteresis loop. Starting demagnetized, M rises to saturation; on reversing H it retains a remanence M_r at zero field and needs a coercive field H_c to vanish. The loop area is the energy lost per cycle.

The domain structure and its hysteresis are the low-energy, long-wavelength face of magnetic order. The lowest-energy excitations of the ordered state itself — the collective spin precessions that reduce the magnetization at low temperature — are the spin waves of the final lesson.