Magnetic order: ferro-, antiferro- and ferrimagnets and the exchange that makes them

Spins lock into order when a bridge atom makes neighbours agree

Intuition

Picture a row of people on a bus who must each sit next to a stranger. Left alone, they face every which way. If something makes each person want to face the same way as, or the opposite way to, the person next to them, the whole row falls into a pattern. In a crystal, that something is magnetic exchange.

Magnetic atoms in a periodic lattice can have long-range magnetic order. In a ferromagnet (FM) all spins point the same way. In an antiferromagnet (AFM) they alternate. Most uses of solid magnets rely on ordered states, not paramagnetic or diamagnetic ones. Even for a simple cubic cell with equal parallel spins there are several arrangements.

Order comes from magnetic exchange, an electrostatic effect among the outer-shell electrons of a magnetic atom and its neighbours. The energy of the crystal falls when like charges (electrons) move further apart, which is electron delocalisation. Delocalisation is subject to selection rules on spin direction. That is why spin order appears. The lecture names two types: superexchange and double exchange.

Compare

Superexchange

Indirect, between non-neighbouring magnetic atoms, through a non-magnetic atom such as O. Partly filled $e_g$ orbitals overlap a full oxygen p orbital. By the Pauli principle, the spins in the shared orbital pair up as opposites. Favours antiferromagnetic order.

Double exchange

An unpaired electron hops between magnetic ions of different oxidation states (such as Mn³⁺ and Mn⁴⁺). The hop is easy only if the spin does not flip, so spins align (Hund’s rule). Favours ferromagnetic order. In mixed-valence crystals it gives conduction.

Exam detail

The learning outcomes ask you to explain the different exchange interactions that lead to bulk magnetism, and to identify a material as diamagnetic, paramagnetic, ferromagnetic or antiferromagnetic from magnetometry. Know which exchange gives which order, and the shape of $\chi$, $1/\chi$ and $M$ for each.

Ferromagnets keep a magnetisation below Tc, and a loop shows how they respond

A ferromagnet has a net spontaneous magnetisation, $M_{sat}$, below the Curie temperature, $T_C$. The net moment appears on cooling in a second-order phase transition. Above $T_C$ the susceptibility follows the Curie-Weiss law, a modified version of Curie law.

$\chi = \frac{C}{T - \theta}$

$C$ is the Curie constant and $\theta$ the Weiss constant. For ferromagnets $\theta > 0$ and $\theta = T_C$. On the slide, $\chi^{-1}$ is linear above $T_C$ and reaches zero at $T_C$ on the extrapolated line.

Below $T_C$, a magnetised ferromagnet can be reoriented by an applied field. The shape of the hysteresis loop sets its use. A slim loop is easily reoriented and good for transformer cores. A square loop with a large coercive field $H_c$ is hard to reorient and good for data storage. The loop is labelled with saturation magnetisation, remnant magnetisation (on the $M$ axis at $H = 0$) and coercive field $H_c$ (on the $H$ axis). Because of the domain structure there is often no net magnetisation until a field has been applied at least once. This is the magnetic version of the ferroelectric loop in the last topic.

Antiferromagnets cancel, ferrimagnets do not quite

In an antiferromagnet the moments point in opposite directions at zero field and cancel, so there is no net magnetisation. At high temperature it also follows $\chi = C/(T - \theta)$, with $\theta < 0$ and $\theta = -T_N$, where $T_N$ is the Néel temperature. The susceptibility has a cusp at $T_N$. In plain words, $\chi^{-1} = (T - \theta)/C$ reaches zero at $T = \theta$, so for an antiferromagnet the straight line of $\chi^{-1}$ against $T$ extrapolates to a negative temperature, $-T_N$. For a ferromagnet it extrapolates to a positive one, $T_C$. The lecture states the sign of $\theta$ (slides 22 and 26); the extrapolation is that formula rearranged. The lecture says magnetism increases more rapidly above $T_N$ because thermal energy helps align the spins. The slide marks this sketch as a worksheet question.

Try it: worked example

Find chi for a material that follows the Curie-Weiss law with C = 1 (arbitrary units), theta = -30 K at T = 300 K.

Show answer and steps

Answerchi = 1/330 (about 3.0 x 10^-3 in these units). The negative theta marks an antiferromagnet.

1. Use $\chi = C/(T - \theta)$.
2. Put in $C = 1$, $T = 300$ and $\theta = -30$.
3. $T - \theta = 300 - (-30) = 330$.
4. $\chi = 1/330$.
5. $\theta < 0$, so it is an antiferromagnet with $T_N = 30$ K. This is a practice case: the lecture gives no numeric example.

Ferrimagnets have opposing moments like antiferromagnets, but the antiparallel moments do not cancel, so there is a spontaneous (zero-field) magnetisation. The example is magnetite, Fe₃O₄, “lodestone”. It is a ferrite with an inverse spinel structure, in which Fe³⁺O₄ tetrahedra sit among sheets of edge-sharing (M²⁺,Fe³⁺)O₆ octahedra. It conducts well because of Fe²⁺ ⇌ Fe³⁺ + e⁻. The tetrahedral and octahedral sites carry opposite but unequal moments, leaving a net moment in the direction of the stronger one. There are twice as many octahedra as tetrahedra.

Optional aside

The lecture recalls site filling from Lecture 5: spinel is (A²⁺)tet (B³⁺)₂ oct O₄ and inverse spinel is (B³⁺)tet (A²⁺B³⁺)oct O₄, where tet and oct label tetrahedral and octahedral sites. A closing slide, marked beyond examinable, says delocalised electrons show temperature-independent (Pauli) paramagnetism, a constant positive susceptibility from a small spin imbalance near the Fermi energy.

Pitfall

Ferrimagnet and antiferromagnet both have opposing moments. Only the antiferromagnet cancels completely. If there is a net zero-field magnetisation with opposing moments, it is a ferrimagnet.

Recall

Which sign of theta goes with which order?

Positive theta, equal to T_C, is a ferromagnet. Negative theta, equal to minus T_N, is an antiferromagnet.