---
title: "Antiferromagnetism, Ferrimagnetism, and Domains"
module: Magnetism in Solids
moduleNumber: 9
lessonNumber: 3
order: 903
summary: >
  A negative exchange coupling orders neighboring spins antiparallel. Two-sublattice
  molecular-field theory gives a Néel temperature marked by a cusp in the
  susceptibility, and unequal sublattices leave a net moment — ferrimagnetism, the
  magnetism of the ferrites. A ferromagnet breaks into domains to reduce its
  magnetostatic energy, separated by Bloch walls whose width is set by the
  competition between exchange and magnetocrystalline anisotropy, and the
  irreversible motion of those walls produces the hysteresis loop.
topics: [Magnetism in Solids]
draft: false
sources:
  - book: Kittel
    ref: "Ch. 12 — Ferromagnetism and Antiferromagnetism"
  - book: Ashcroft & Mermin
    ref: "Ch. 33 — Magnetic Ordering"
  - book: Hook & Hall
    ref: "Ch. 8 — Magnetic Order"
---

When the exchange integral between neighbors is negative, the Heisenberg energy
$-2J\vec S_i\cdot\vec S_j$ 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 $\vec M_A = -\vec M_B$ and the total magnetization vanishes. The
molecular-field method of the [previous lesson](/condensed-matter/magnetism/exchange-and-ferromagnetism)
now needs one field per sublattice. Each sublattice feels a field from its
neighbors on the other,

$$
\vec B_A = -\lambda\mu_0 \vec M_B, \qquad \vec B_B = -\lambda\mu_0 \vec M_A,
$$

with $\lambda > 0$ 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
$\vec M_A = -\vec M_B \neq 0$: staggered order with no net moment.

$$
% caption: Antiferromagnet (left): two sublattices of equal and opposite moment
% cancel. Ferrimagnet (right): the sublattice moments are unequal and leave a net
% magnetization.
\begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  % antiferromagnet: up on A sites, down on B sites, equal length
  \begin{scope}
    \draw[acc, very thick, ->] (0,-0.35) -- (0,0.35);
    \draw[acc, very thick, ->] (2,-0.35) -- (2,0.35);
    \draw[acc, very thick, ->] (1,0.65) -- (1,1.35);
    \draw[black, very thick, ->] (1,0.35) -- (1,-0.35);
    \draw[black, very thick, ->] (0,1.35) -- (0,0.65);
    \draw[black, very thick, ->] (2,1.35) -- (2,0.65);
    \node[black, anchor=north] at (1,-0.6) {antiferromagnet: cancels};
  \end{scope}
  % ferrimagnet: long up on A sites, short down on B sites
  \begin{scope}[xshift=5.4cm]
    \draw[acc, very thick, ->] (0,-0.45) -- (0,0.45);
    \draw[acc, very thick, ->] (2,-0.45) -- (2,0.45);
    \draw[acc, very thick, ->] (1,0.55) -- (1,1.45);
    \draw[black, very thick, ->] (1,0.22) -- (1,-0.22);
    \draw[black, very thick, ->] (0,1.22) -- (0,0.78);
    \draw[black, very thick, ->] (2,1.22) -- (2,0.78);
    \node[black, anchor=north] at (1,-0.6) {ferrimagnet: net moment};
  \end{scope}
\end{tikzpicture}
$$

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

$$
\chi = \frac{C}{T + \theta}, \qquad \theta = T_N
$$

in the simplest case of only inter-sublattice coupling. The crucial sign change is
in the denominator: the intercept of $1/\chi$ versus $T$ falls at the negative
temperature $-\theta$, the mirror image of the ferromagnet's positive intercept.
The ratio $\theta/T_N$, 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 $\chi_\parallel \to 0$ as $T \to 0$. With the field
perpendicular, both sublattices cant slightly toward it against a restoring
exchange torque, giving a temperature-independent $\chi_\perp$. A powder averages
the two. The result is a **cusp** in the susceptibility at $T_N$: rising on the
paramagnetic side as $1/(T+\theta)$, peaking at $T_N$, and falling below it.

$$
% caption: Magnetic susceptibility of an antiferromagnet peaks at the Néel
% temperature. Above it the Curie–Weiss form rises toward T_N; below it the
% ordered sublattices resist alignment and chi falls.
\begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (6.4,0) node[below right] {temperature};
  \draw[->, black] (0,0) -- (0,3.6) node[left] {susceptibility};
  \draw[black, dashed] (2.8,0) -- (2.8,2.9);
  \node[black, anchor=north] at (2.8,-0.05) {$T_N$};
  % below T_N: rising from a lower value to the peak
  \draw[acc, very thick, domain=0.2:2.8, samples=60, variable=\x]
    plot ({\x},{1.4 + 1.5*(\x/2.8)*(\x/2.8)});
  % above T_N: Curie-Weiss decay from the peak
  \draw[acc, very thick, domain=2.8:6.2, samples=90, variable=\x]
    plot ({\x},{2.9*2.8/\x*0.99 + 0.02});
  \fill[black] (2.8,2.9) circle (2pt);
  \node[black, anchor=south] at (2.8,2.95) {cusp};
\end{tikzpicture}
$$

## 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 $\text{Fe}_3\text{O}_4$, the
lodestone of the ancient world, is the prototype: its $\text{Fe}^{3+}$ ions occupy
both tetrahedral and octahedral sites in antiparallel arrangement and cancel,
while the $\text{Fe}^{2+}$ 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.

$$
% caption: Flux-closure domains: dividing the crystal into oppositely magnetized
% regions with triangular closure domains routes the flux entirely inside the
% sample and removes the external field energy.
\begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[black, thick] (0,0) rectangle (3.2,3.2);
  % diagonal closure lines
  \draw[black] (0,0) -- (1.6,1.6) -- (0,3.2);
  \draw[black] (3.2,0) -- (1.6,1.6) -- (3.2,3.2);
  % up domain (left), down domain (right)
  \draw[acc, very thick, ->] (0.75,0.7) -- (0.75,2.5);
  \draw[acc, very thick, ->] (2.45,2.5) -- (2.45,0.7);
  % closure domains: horizontal
  \draw[acc, very thick, ->] (0.9,2.75) -- (2.3,2.75);
  \draw[acc, very thick, ->] (2.3,0.45) -- (0.9,0.45);
  \node[black, anchor=north] at (1.6,-0.15) {closed inside the crystal};
\end{tikzpicture}
$$

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

$$
\delta \sim \sqrt{\frac{A}{K}}, \qquad \sigma \sim \sqrt{A K}.
$$

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

$$
% caption: A Bloch wall: the magnetization rotates gradually across many atomic
% planes. A wider wall lowers exchange energy but raises anisotropy energy; the
% width delta balances the two.
\begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[black] (0,-1.1) -- (0,1.1);
  \draw[black] (6.0,-1.1) -- (6.0,1.1);
  \node[black, anchor=north] at (0,-1.1) {domain};
  \node[black, anchor=north] at (6.0,-1.1) {domain};
  % rotating spins: angle sweeps 0 to 180 across the wall
  \foreach \i in {0,1,2,3,4,5,6,7,8,9,10}{
    \pgfmathsetmacro{\ang}{180 - \i*18}
    \draw[acc, very thick, ->] (\i*0.6,0) -- ++({0.5*cos(\ang)},{0.5*sin(\ang)});
  }
  \draw[black, <->] (1.2,-0.8) -- (4.8,-0.8) node[midway, below] {wall width};
\end{tikzpicture}
$$

## 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 $\text{SmCo}_5$ and
  $\text{Nd}_2\text{Fe}_{14}\text{B}$ — have large $H_c$ and wide loops. Their
  strong anisotropy pins the walls, so they retain magnetization against
  demagnetizing fields. These are permanent magnets.

$$
% caption: Ferromagnetic hysteresis loop. Starting demagnetized, M rises to
% saturation; on reversing H it retains a remanence M_r at zero field and needs a
% coercive field H_c to vanish. The loop area is the energy lost per cycle.
\begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (-3.2,0) -- (3.2,0) node[below right] {applied $H$};
  \draw[->, black] (0,-2.6) -- (0,2.6) node[left] {$M$};
  % upper branch (increasing then reverse)
  \draw[acc, very thick]
    (-2.9,-1.9) .. controls (-1.2,-1.85) and (-1.0,1.6) .. (0,1.85)
    .. controls (1.4,2.05) and (2.6,2.05) .. (2.9,2.05);
  % lower branch
  \draw[acc, very thick]
    (2.9,1.9) .. controls (1.2,1.85) and (1.0,-1.6) .. (0,-1.85)
    .. controls (-1.4,-2.05) and (-2.6,-2.05) .. (-2.9,-2.05);
  % markers
  \fill[black] (0,1.85) circle (1.6pt);
  \node[black, anchor=west] at (0.1,1.5) {$M_r$};
  \fill[black] (-1.0,0) circle (1.6pt);
  \node[black, anchor=south east] at (-1.0,0.05) {$H_c$};
\end{tikzpicture}
$$

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](/condensed-matter/magnetism/spin-waves-and-magnons) of the final
lesson.