8.2 Ferromagnetism and antiferromagnetism

Magnetic dipole moments

Magnetic dipole moments describe the strength and orientation of a tiny magnetic source, analogous to a small bar magnet with a north and south pole. They arise from the motion of charged particles: electrons orbiting a nucleus and the intrinsic spin of electrons and nuclei.

Origin of magnetic moments

Magnetic moments in atoms come from two sources:

• Orbital magnetic moments result from electrons moving around the nucleus. This orbital motion acts like a current loop, generating a magnetic field.
• Spin magnetic moments arise from the intrinsic angular momentum of electrons, a purely quantum mechanical property with no classical analogue.

The total magnetic moment of an atom is the vector sum of all orbital and spin contributions from its electrons.

Magnetic moment of atoms

An atom's net magnetic moment depends on its electronic configuration and how the orbital and spin angular momenta couple together. Atoms with completely filled electron shells have zero net moment because all contributions cancel. Only atoms with partially filled shells can carry a net magnetic moment.

Hund's rules determine the ground-state arrangement of electron spins and orbital angular momenta by minimizing total energy. They predict:

1. Maximize total spin $S$ (fill orbitals with parallel spins first).
2. Maximize total orbital angular momentum $L$ consistent with rule 1.
3. Total angular momentum $J = |L - S|$ for less-than-half-filled shells, $J = L + S$ for more-than-half-filled shells.

The Bohr magneton ($\mu_B$) is the natural unit for atomic magnetic moments:

$\mu_B = \frac{e\hbar}{2m_e} \approx 9.274 \times 10^{-24} \text{ J/T}$

Magnetic moment of electrons

Electrons carry an intrinsic spin magnetic moment of approximately one Bohr magneton. In an external magnetic field, the spin can orient either "up" or "down" (two quantum states, $m_s = \pm \frac{1}{2}$).

The interaction between spin magnetic moments on neighboring atoms is what ultimately drives the collective magnetic ordering seen in ferromagnets and antiferromagnets.

Exchange interaction

The exchange interaction is the quantum mechanical mechanism responsible for aligning (or anti-aligning) magnetic moments in solids. It originates from the interplay of two effects: the Pauli exclusion principle (no two electrons can share the same quantum state) and electrostatic (Coulomb) repulsion between electrons.

The exchange energy between two neighboring spins can be written as:

$E_{ex} = -2J \, \mathbf{S}_1 \cdot \mathbf{S}_2$

where $J$ is the exchange constant. When $J > 0$, parallel alignment is favored (ferromagnetism). When $J < 0$, antiparallel alignment is favored (antiferromagnetism).

Direct exchange

Direct exchange occurs when the electron wave functions on neighboring atoms overlap directly. It is a short-range interaction, effective only when atoms are close enough for significant orbital overlap. Direct exchange is most relevant in materials with tightly bound electrons, such as the 3d transition metals (Fe, Co, Ni).

Indirect exchange (RKKY interaction)

Indirect exchange, often called the RKKY interaction (Ruderman-Kittel-Kasuya-Yosida), occurs in metals where localized magnetic moments interact through conduction electrons. A localized moment polarizes the surrounding conduction electrons, and that polarization is "felt" by neighboring moments.

A distinctive feature of RKKY coupling is that it oscillates between ferromagnetic and antiferromagnetic as a function of the distance between moments. This oscillatory behavior is critical in rare earth metals and multilayer magnetic structures.

Superexchange

Superexchange is an indirect exchange mechanism found in ionic solids, especially transition metal oxides. Here, two magnetic ions (e.g., $\text{Mn}^{2+}$) interact through a non-magnetic ion (typically $\text{O}^{2-}$) sitting between them.

The sign and strength of superexchange depend on the bond angle between the magnetic and oxygen ions. According to the Goodenough-Kanamori rules, a 180° bond angle typically produces strong antiferromagnetic coupling, while a 90° angle tends to give weaker ferromagnetic coupling. Classic examples of superexchange-driven antiferromagnets include MnO and NiO.

Ferromagnetic ordering

In ferromagnetic ordering, the magnetic moments of atoms align parallel to each other, producing a large net magnetization even without an external field. This happens because the exchange constant $J$ is positive, making parallel alignment the lowest-energy configuration. Ferromagnetic order exists only below a critical temperature called the Curie temperature ($T_C$).

Spontaneous magnetization

Spontaneous magnetization is the net magnetic moment that appears in a ferromagnet without any applied field. Below $T_C$, thermal energy is too weak to disrupt the exchange-driven parallel alignment.

As temperature increases toward $T_C$, thermal fluctuations progressively disorder the spins, and the spontaneous magnetization decreases continuously. At $T_C$ it drops to zero. The temperature dependence near $T_C$ follows a power law described by mean-field theory (or more accurately by critical exponents from renormalization group theory).

Curie temperature

The Curie temperature ($T_C$) marks the phase transition between the ferromagnetic and paramagnetic states. Above $T_C$, thermal energy overcomes the exchange interaction, and the moments orient randomly.

$T_C$ is an intrinsic property that scales with the strength of the exchange interaction. Some representative values:

Above $T_C$, the magnetic susceptibility follows the Curie-Weiss law: $\chi = \frac{C}{T - T_C}$, where $C$ is the Curie constant.

Magnetic domains

A ferromagnet below $T_C$ doesn't necessarily appear magnetized on a macroscopic scale. Instead, it breaks into magnetic domains, regions where all moments point in the same direction, but different domains point in different directions.

Why do domains form? A single uniformly magnetized sample would have a large magnetostatic energy due to stray fields outside the material. Splitting into domains with opposing magnetizations reduces this energy. The trade-off is the energy cost of creating domain walls.

Domain size and shape depend on the material's composition, crystal structure, sample geometry, and any applied field.

Domain walls

Domain walls are the transition regions between adjacent domains with different magnetization directions. The magnetization rotates gradually across the wall rather than flipping abruptly.

The wall width is set by a competition:

• Exchange energy favors a wide wall (gradual rotation keeps neighboring spins nearly parallel).
• Magnetocrystalline anisotropy energy favors a narrow wall (spins away from the easy axis cost energy).

The typical domain wall width $\delta$ scales as $\delta \sim \sqrt{A/K}$, where $A$ is the exchange stiffness and $K$ is the anisotropy constant.

Two main types of domain walls exist:

• Bloch walls: the magnetization rotates out of the plane of the wall. Common in bulk materials.
• Néel walls: the magnetization rotates within the plane of the wall. Common in thin films where Bloch walls would create large surface charges.

Antiferromagnetic ordering

In antiferromagnetic ordering, neighboring magnetic moments align antiparallel to each other. The exchange constant $J$ is negative, so antiparallel alignment minimizes the energy. The opposing moments cancel, giving zero net magnetization. This order persists below the Néel temperature ($T_N$).

Néel temperature

The Néel temperature ($T_N$) is the critical temperature below which long-range antiferromagnetic order develops. Above $T_N$, the material becomes paramagnetic.

Named after Louis Néel (Nobel Prize, 1970), the susceptibility of an antiferromagnet above $T_N$ follows: $\chi = \frac{C}{T + \theta}$, where $\theta$ is a positive constant related to the strength of the antiferromagnetic exchange. Note the plus sign, which distinguishes this from the Curie-Weiss law for ferromagnets.

Sublattice magnetization

The crystal structure of an antiferromagnet can be decomposed into two (or more) interpenetrating sublattices. Within each sublattice, all moments point the same way, but the sublattices have equal and opposite magnetizations, so they cancel exactly.

The sublattice magnetization decreases with increasing temperature and vanishes at $T_N$, similar to how spontaneous magnetization behaves in a ferromagnet.

Types of antiferromagnets

• Simple (collinear) antiferromagnets have a bipartite lattice with equal and opposite moments on two sublattices. MnO is the textbook example, with a Néel temperature of 118 K.
• Frustrated antiferromagnets arise when the lattice geometry (e.g., triangular or pyrochlore) prevents all exchange interactions from being satisfied simultaneously. This geometric frustration can lead to exotic ground states like spin liquids or spin ices.
• Canted antiferromagnets (also called weak ferromagnets) have moments that are almost antiparallel but tilted slightly, producing a small net moment. Hematite ($\alpha$-$\text{Fe}_2\text{O}_3$) is a well-known example. The canting is often caused by the Dzyaloshinskii-Moriya interaction, an antisymmetric exchange term arising from spin-orbit coupling.

Spin-flop transition

When a strong magnetic field is applied along the easy axis of an antiferromagnet, a spin-flop transition can occur. At a critical field strength, the antiparallel moments suddenly reorient into a canted configuration, tilting toward the field direction. This produces a jump in magnetization.

The transition results from competition between two energies:

• The antiferromagnetic exchange, which wants moments antiparallel.
• The Zeeman energy ($-\mathbf{m} \cdot \mathbf{B}$), which wants moments aligned with the field.

At even higher fields, a second transition (the spin-flip transition) can force all moments fully parallel to the field.

Ferromagnetic materials

Ferromagnetic materials exhibit spontaneous magnetization and respond strongly to external fields. They have high magnetic susceptibility and are the basis for most magnetic technology.

Iron, nickel, and cobalt

Iron (Fe), nickel (Ni), and cobalt (Co) are the three elemental ferromagnets at room temperature. Their ferromagnetism originates from partially filled 3d electron shells and direct exchange between 3d electrons.

• Fe: $T_C = 1043$ K, body-centered cubic structure, highest saturation magnetization of the three (~2.2 $\mu_B$/atom)
• Co: $T_C = 1388$ K, hexagonal close-packed structure, strongest magnetocrystalline anisotropy among the three
• Ni: $T_C = 627$ K, face-centered cubic structure, lowest moment (~0.6 $\mu_B$/atom)

Rare earth metals

Several rare earth elements are ferromagnetic, including gadolinium (Gd), dysprosium (Dy), and terbium (Tb). Their magnetism comes from highly localized 4f electrons, which carry large orbital and spin moments.

Rare earth ferromagnets typically have strong magnetocrystalline anisotropy, making them essential ingredients in high-performance permanent magnets (e.g., $\text{Nd}_2\text{Fe}_{14}\text{B}$, $\text{SmCo}_5$) and magnetostrictive devices (e.g., Terfenol-D).

Ferromagnetic alloys

Combining ferromagnetic elements with other metals can tailor magnetic properties for specific applications:

• Permalloy ($\sim$80% Ni, 20% Fe): very high magnetic permeability, low coercivity. Used in magnetic shielding and transformer cores.
• Alnico (Al-Ni-Co-Fe): high remanence and moderate coercivity. Used in permanent magnets before rare earth magnets became dominant.
• SmCo$_5$ and Nd$_2$Fe$_{14}$B: rare earth permanent magnet alloys with extremely high coercivity and energy product. Used in motors, generators, and hard disk drives.

Ferrites and garnets

Ferrites and garnets are ceramic (oxide-based) magnetic materials. Technically, most ferrites are ferrimagnetic rather than ferromagnetic: they have two sublattices with unequal, antiparallel moments, giving a net magnetization.

• Ferrites have a spinel structure ($\text{AB}_2\text{O}_4$). Examples: magnetite ($\text{Fe}_3\text{O}_4$), Mn-Zn ferrite, Ni-Zn ferrite.
• Garnets have the structure $\text{A}_3\text{B}_5\text{O}_{12}$. The most important is yttrium iron garnet (YIG, $\text{Y}_3\text{Fe}_5\text{O}_{12}$), used in microwave and magneto-optical devices.

Their key advantage is high electrical resistivity, which suppresses eddy currents and makes them ideal for high-frequency applications where metallic ferromagnets would suffer large losses.

Antiferromagnetic materials

Antiferromagnetic materials have zero net magnetization due to the cancellation of opposing sublattice moments. While they don't produce strong macroscopic fields, they are increasingly important for spintronics applications because their dynamics are ultrafast and they are insensitive to external magnetic perturbations.

Transition metal oxides

Transition metal oxides are the most common antiferromagnets. The antiferromagnetic order arises from superexchange through oxygen ions.

These materials were historically important for confirming the theory of antiferromagnetism through neutron diffraction experiments.

Rare earth compounds

Some rare earth compounds exhibit antiferromagnetic ordering driven by indirect exchange (RKKY) between rare earth ions. Note that EuO is actually ferromagnetic ($T_C \approx 69$ K), not antiferromagnetic. A genuine rare earth antiferromagnet is EuTe ($T_N \approx 9.6$ K).

The magnetic behavior of these compounds is strongly influenced by crystal field effects and the large orbital angular momentum of 4f electrons.

Perovskites

Perovskites have the general formula $\text{ABX}_3$, where A and B are cations and X is usually oxygen. Several perovskites are antiferromagnetic:

• LaMnO$_3$: antiferromagnetic with $T_N \approx 140$ K. The parent compound of the colossal magnetoresistance manganites.
• YCrO$_3$: antiferromagnetic with weak ferromagnetism due to spin canting.

Antiferromagnetic perovskites are actively studied for multiferroic applications, where magnetic and electric order coexist and couple to each other (e.g., $\text{BiFeO}_3$).

Spinels

Spinels have the formula $\text{AB}_2\text{X}_4$. In these structures, magnetic cations occupy tetrahedral (A) and octahedral (B) sites, and superexchange pathways connect them through oxygen.

Examples of antiferromagnetic spinels include chromite ($\text{FeCr}_2\text{O}_4$) and hercynite ($\text{FeAl}_2\text{O}_4$). Their magnetic properties can be tuned by substituting different cations or by controlling the degree of cation inversion (how cations distribute between the two site types).

Magnetic anisotropy

Magnetic anisotropy means a material's magnetic properties depend on direction. It determines which directions are "easy" (energetically favorable) or "hard" for the magnetization to point. Anisotropy is central to understanding domain structure, hysteresis, and the performance of permanent magnets.

Magnetocrystalline anisotropy

This is an intrinsic property arising from spin-orbit coupling: the interaction between an electron's spin and its orbital motion, which "locks" the magnetization to preferred crystallographic directions.

• In uniaxial materials (e.g., hexagonal Co, $\text{BaFe}_{12}\text{O}_{19}$), there is a single easy axis. The anisotropy energy density is: $E_a = K_1 \sin^2\theta + K_2 \sin^4\theta + \ldots$ where $\theta$ is the angle between the magnetization and the easy axis.
• In cubic materials (e.g., Fe, Ni), the anisotropy energy depends on the direction cosines of the magnetization relative to the crystal axes.

Materials with large $K_1$ values (like $\text{SmCo}_5$ with $K_1 \approx 1.7 \times 10^7$ J/m$^3$) are excellent candidates for permanent magnets.

Shape anisotropy

Shape anisotropy comes from the demagnetizing field that a magnetized sample creates. The demagnetizing field opposes the magnetization and depends on the sample geometry.

• In a long rod, the demagnetizing factor is small along the long axis, making it the easy axis.
• In a thin film, the easy axis lies in the plane of the film.

Shape anisotropy is exploited in magnetic recording media (elongated grains) and in patterned nanostructures for spintronics.

Stress anisotropy

Also called magnetoelastic anisotropy, this arises when mechanical stress modifies the preferred magnetization direction through magnetostriction (the coupling between magnetic order and lattice strain).

Tensile or compressive stress can rotate the easy axis, depending on the sign of the magnetostriction coefficient. This effect is used in magnetostrictive sensors and actuators, where applied stress controls the magnetic state or vice versa.

Anisotropy constants

Anisotropy constants ($K_1, K_2, \ldots$) are material-specific parameters that quantify the strength of magnetic anisotropy. $K_1$ represents the dominant contribution and sets the energy scale for switching the magnetization direction.

These constants are temperature-dependent (generally decreasing toward zero at $T_C$) and can be measured experimentally via torque magnetometry, ferromagnetic resonance, or singular point detection. Their values directly determine properties like coercivity and domain wall width.

Magnetic hysteresis

When you cycle a ferromagnet through an applied magnetic field, the magnetization doesn't retrace the same path. This path-dependent behavior is magnetic hysteresis, and it arises because domain wall motion is irreversible: walls get pinned by crystal defects, grain boundaries, and impurities, and energy is dissipated as they jump between pinning sites.

Hysteresis loop

The hysteresis loop plots magnetization $M$ versus applied field $H$. Starting from a demagnetized state and increasing $H$, the magnetization rises along the initial magnetization curve until it saturates at $M_s$. Decreasing $H$ back through zero and to negative values traces out the characteristic loop.

Key quantities read from the loop:

• Saturation magnetization ($M_s$): the maximum magnetization when all moments are aligned.
• Remanent magnetization ($M_r$): the magnetization remaining when $H = 0$ after saturation.
• Coercive field ($H_c$): the reverse field needed to bring $M$ back to zero.

The area enclosed by the loop equals the energy dissipated per cycle, which matters for applications involving AC fields.

Coercivity and remanence

Coercivity ($H_c$) measures how resistant a material is to demagnetization. It depends on microstructure: defects, grain size, and inclusions that pin domain walls all increase $H_c$.

Remanence ($M_r$) is the "memory" of the material after the field is removed. A high $M_r/M_s$ ratio (called the squareness ratio) means the material retains most of its magnetization.

The maximum energy product $(BH)_{\text{max}}$ is the figure of merit for permanent magnets, representing the maximum energy stored in the magnetic field outside the magnet. Modern Nd-Fe-B magnets achieve $(BH)_{\text{max}} \approx 400$ kJ/m$^3$.