7.6 Ferromagnetism

Ferromagnetic materials

Ferromagnetism is the mechanism behind the strongest magnetic effects you'll encounter in everyday materials. Unlike paramagnetism (where moments align weakly with an applied field) or diamagnetism (where moments oppose it weakly), ferromagnetic materials develop large, persistent magnetizations. They can be magnetized by an external field and retain that magnetization after the field is removed. This property makes them essential in motors, generators, transformers, and data storage.

The key to ferromagnetism is the exchange interaction, a quantum mechanical effect that causes neighboring electron spins to align parallel to each other. This alignment creates macroscopic magnetization far stronger than what thermal statistics alone would predict.

Iron, cobalt, nickel

Iron (Fe), cobalt (Co), and nickel (Ni) are the three classic ferromagnetic elements. Their ferromagnetism originates from unpaired electrons in their 3d orbitals, which carry a net magnetic moment. The exchange interaction between these 3d electrons is strong enough to maintain spin alignment up to high temperatures.

Each element loses its ferromagnetism above its Curie temperature:

• Iron: $T_c = 1043 \text{ K}$
• Cobalt: $T_c = 1388 \text{ K}$
• Nickel: $T_c = 627 \text{ K}$

Cobalt's high Curie temperature makes it useful in applications that must operate at elevated temperatures.

Rare earth elements

Rare earth elements like neodymium (Nd), samarium (Sm), and dysprosium (Dy) derive their magnetic properties from unpaired electrons in their 4f orbitals. The 4f electrons are more localized than 3d electrons, which gives rare earths very large magnetic moments per atom but weaker direct exchange coupling between neighboring atoms.

In practice, rare earths are combined with transition metals to get the best of both: the large moment of the rare earth and the strong exchange coupling of the 3d metal. The result is high-performance permanent magnets like neodymium-iron-boron (NdFeB), which has the highest energy product of any commercial permanent magnet, and samarium-cobalt (SmCo), which offers excellent high-temperature stability.

Alloys and compounds

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

• Permalloy (Ni-Fe, ~80% Ni): very high permeability and near-zero magnetostriction, used in magnetic shielding and sensor cores
• Alnico (Al-Ni-Co): high coercivity and good temperature stability, used in older permanent magnets
• Ferrites (e.g., magnetite, $\text{Fe}_3\text{O}_4$): ceramic magnetic materials with high electrical resistivity, which suppresses eddy currents. This makes ferrites ideal for transformer and inductor cores at high frequencies.

Note that ferrites are technically ferrimagnetic (their sublattice moments are antiparallel but unequal), not truly ferromagnetic. However, they're often discussed alongside ferromagnets because their macroscopic behavior is similar.

Magnetic domains

A uniformly magnetized sample would have enormous magnetostatic energy due to the demagnetizing field it creates. To minimize total energy, ferromagnetic materials break up into magnetic domains, regions where all the moments point in the same direction. Adjacent domains point in different directions, so the external stray field is reduced.

The domain structure represents an energy balance between several competing terms: exchange energy (favoring uniform magnetization), magnetostatic energy (favoring flux closure), anisotropy energy (favoring alignment along easy axes), and domain wall energy (penalizing the creation of new walls).

Domain walls

Domain walls are the thin transition regions between adjacent domains where the magnetization rotates from one direction to another. The two main types are:

• Bloch walls: the magnetization rotates out of the plane of the wall. These are typical in bulk materials.
• Néel walls: the magnetization rotates within the plane of the wall. These are common in thin films where the Bloch configuration would create too much surface magnetic charge.

The wall width $\delta$ is set by the competition between exchange energy (which favors a wide, gradual rotation) and anisotropy energy (which favors a narrow, abrupt transition). A standard estimate is:

$\delta \sim \pi \sqrt{\frac{A}{K}}$

where $A$ is the exchange stiffness and $K$ is the anisotropy constant. Typical wall widths in iron are on the order of tens of nanometers.

Domain alignment

In a demagnetized state, domains are oriented so that their magnetizations roughly cancel, giving zero net magnetization. When you apply an external field $\mathbf{H}$, two things happen in sequence:

1. Domain wall motion (at low fields): walls shift so that domains aligned with $\mathbf{H}$ grow at the expense of unfavorably oriented domains.
2. Domain rotation (at higher fields): the magnetization within remaining domains rotates toward $\mathbf{H}$.

At saturation, the entire sample is effectively a single domain aligned with the field.

Domain size vs. material dimensions

In bulk materials, domain sizes are typically on the order of micrometers. As the physical dimensions of the material shrink toward this scale, the domain structure changes dramatically:

• In thin films, domains may adopt stripe or maze patterns, and Néel walls replace Bloch walls.
• In nanoparticles below a critical size (roughly 10–100 nm depending on the material), it becomes energetically unfavorable to form a domain wall at all. The particle becomes a single-domain particle.
• Below an even smaller size, thermal fluctuations can randomly flip the single-domain moment, leading to superparamagnetism: the particle behaves paramagnetically despite having ferromagnetic internal order.

Spontaneous magnetization

Spontaneous magnetization is the net magnetization that exists within each domain even without an applied field. It arises because the exchange interaction aligns neighboring spins below the Curie temperature. This is what distinguishes ferromagnets from paramagnets, where moments only align in response to an external field.

Curie temperature

The Curie temperature $T_c$ marks the phase transition between the ferromagnetic and paramagnetic states. Below $T_c$, exchange interactions dominate thermal fluctuations and spontaneous magnetization exists. Above $T_c$, thermal energy wins and the material becomes paramagnetic.

Just above $T_c$, the magnetic susceptibility follows the Curie-Weiss law:

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

where $C$ is the Curie constant. The susceptibility diverges as $T \to T_c^+$, reflecting the onset of long-range magnetic order.

Temperature dependence

Spontaneous magnetization $M_s(T)$ is maximum at $T = 0$ and decreases monotonically as temperature rises, dropping to zero at $T_c$. Near $T = 0$, the decrease follows Bloch's $T^{3/2}$ law due to spin-wave excitations:

$M_s(T) = M_s(0)\left[1 - \left(\frac{T}{T_c}\right)^{3/2}\right]$

(This is an approximation valid at low temperatures.) The full temperature dependence can be described by the Brillouin function $B_J(x)$, which accounts for the quantum mechanical nature of the angular momentum. In the mean-field (Weiss) model, $M_s(T)$ is found self-consistently from:

$M_s = N g \mu_B J \, B_J\!\left(\frac{g \mu_B J \, B_{\text{eff}}}{k_B T}\right)$

where $B_{\text{eff}}$ includes the Weiss molecular field $\lambda M_s$.

Exchange interaction

The exchange interaction is fundamentally quantum mechanical. It arises from the overlap of electron wavefunctions combined with the Pauli exclusion principle. The Heisenberg exchange Hamiltonian between two neighboring spins is:

$\mathcal{H} = -2J_{\text{ex}} \, \mathbf{S}_i \cdot \mathbf{S}_j$

where $J_{\text{ex}}$ is the exchange integral. When $J_{\text{ex}} > 0$, parallel alignment is favored (ferromagnetism). When $J_{\text{ex}} < 0$, antiparallel alignment is favored (antiferromagnetism).

The magnitude of $J_{\text{ex}}$ determines both the spontaneous magnetization and the Curie temperature. The Bethe-Slater curve relates $J_{\text{ex}}$ to the ratio of interatomic distance to the 3d orbital radius, explaining why Fe, Co, and Ni are ferromagnetic while Mn and Cr (with smaller ratios) are antiferromagnetic.

Magnetic hysteresis

When you cycle a ferromagnetic material through an applied field, the magnetization doesn't retrace the same path. This path-dependence is hysteresis, and it occurs because domain wall motion and rotation involve irreversible processes: walls get pinned on defects and grain boundaries, and energy is dissipated as they jump free.

Hysteresis loop

The hysteresis loop (also called the B-H or M-H loop) is traced by:

1. Starting from a demagnetized state and increasing $H$ until the material saturates at $M_s$.
2. Decreasing $H$ back to zero. The magnetization doesn't return to zero but instead remains at the remanence $M_r$.
3. Reversing $H$ until the magnetization reaches zero. The field required to do this is the coercivity $H_c$.
4. Continuing to saturate in the opposite direction, then cycling back.

The area enclosed by the loop equals the energy dissipated per cycle per unit volume. This is why hysteresis losses matter in AC applications like transformers.

Saturation magnetization

Saturation magnetization $M_s$ is the magnetization when all magnetic moments in the material are aligned. At this point, increasing the field further produces no additional magnetization. $M_s$ is an intrinsic property determined by the number of magnetic atoms per unit volume and the moment per atom.

For iron at room temperature, $M_s \approx 1.7 \times 10^6 \text{ A/m}$ (or equivalently, $\mu_0 M_s \approx 2.15 \text{ T}$).

Remanent magnetization

Remanent magnetization $M_r$ (or remanence) is what remains after you remove the applied field from a saturated sample. It reflects how much of the domain alignment persists without external drive. The ratio $M_r / M_s$ (the squareness ratio) indicates how "square" the hysteresis loop is. A ratio near 1 means the material holds most of its saturation magnetization, which is desirable for permanent magnets.

Coercivity

Coercivity $H_c$ is the reverse field needed to bring the magnetization to zero from saturation. It measures how resistant the material is to demagnetization.

• Soft magnetic materials have low $H_c$ (e.g., permalloy: $H_c \sim 1 \text{ A/m}$). They magnetize and demagnetize easily, making them ideal for transformer cores and magnetic shielding.
• Hard magnetic materials have high $H_c$ (e.g., NdFeB: $H_c \sim 10^6 \text{ A/m}$). They resist demagnetization, making them suitable for permanent magnets and magnetic recording media.

Magnetic anisotropy

Magnetic anisotropy means the energy of magnetization depends on which direction the magnetization points relative to the material's structure. This creates easy axes (low-energy directions) and hard axes (high-energy directions). Anisotropy is what gives permanent magnets their ability to maintain a preferred magnetization direction.

Magnetocrystalline anisotropy

This anisotropy comes from spin-orbit coupling: the electron spins interact with the orbital motion, which in turn is coupled to the crystal lattice. The result is that certain crystallographic directions are energetically preferred.

For a uniaxial crystal (e.g., hexagonal cobalt), the anisotropy energy density to lowest order is:

$E_a = K_1 \sin^2\theta$

where $\theta$ is the angle between the magnetization and the easy axis, and $K_1$ is the first anisotropy constant. For cubic crystals (e.g., iron, nickel), the expression involves direction cosines:

$E_a = K_1(\alpha_1^2 \alpha_2^2 + \alpha_2^2 \alpha_3^2 + \alpha_3^2 \alpha_1^2)$

Iron has $K_1 > 0$, making $\langle 100 \rangle$ the easy axis. Nickel has $K_1 < 0$, making $\langle 111 \rangle$ the easy axis.

Shape anisotropy

A magnetized sample creates a demagnetizing field $\mathbf{H}_d = -N \mathbf{M}$, where $N$ is the demagnetizing factor (a tensor that depends on geometry). For a non-spherical sample, $N$ is smaller along the long axis, so the demagnetizing energy is lower when the magnetization points along the long axis. This makes the long axis an easy axis.

For a prolate ellipsoid, the shape anisotropy energy density is:

$E_{\text{shape}} = \frac{1}{2}\mu_0 (N_a - N_c) M_s^2 \sin^2\theta$

where $N_a$ and $N_c$ are the demagnetizing factors along the short and long axes, respectively. Shape anisotropy dominates in elongated nanoparticles and thin film elements.

Stress anisotropy

Mechanical stress modifies the magnetic anisotropy through magnetoelastic coupling. The stress anisotropy energy density is:

$E_\sigma = -\frac{3}{2}\lambda_s \sigma \cos^2\theta$

where $\lambda_s$ is the saturation magnetostriction coefficient, $\sigma$ is the applied stress, and $\theta$ is the angle between the magnetization and the stress direction. If $\lambda_s > 0$ (as in iron), tensile stress creates an easy axis along the stress direction. If $\lambda_s < 0$ (as in nickel), tensile stress creates a hard axis along the stress direction.

Magnetization processes

The path from a demagnetized state to saturation involves distinct physical mechanisms that dominate at different field strengths. Understanding these processes explains the shape of the magnetization curve.

Domain wall motion

At low applied fields, magnetization changes primarily through domain wall displacement. Walls move so that domains with magnetization components along $\mathbf{H}$ expand. This process has two regimes:

1. Reversible wall bowing: at very low fields, walls bend elastically around pinning sites and return to their original position if the field is removed.
2. Irreversible wall jumps (Barkhausen jumps): at higher fields, walls break free from pinning sites and jump to new positions. These jumps are irreversible and are the main source of hysteresis at moderate fields. You can actually hear them as crackling noise if you wrap a coil around a sample and amplify the induced voltage.

Domain rotation

Once wall motion has largely completed and unfavorable domains have been consumed, further magnetization increase requires rotating the magnetization of the remaining domains away from their easy axes toward the applied field direction. This costs anisotropy energy and requires stronger fields.

At very high fields, the magnetization is nearly saturated and only a small component remains to be rotated. In this regime, the approach to saturation follows a characteristic $1/H^2$ law for cubic crystals.

Magnetization curves

The initial (virgin) magnetization curve starts from the demagnetized state and shows three characteristic regions:

1. Low-field region: reversible wall motion produces a gentle, roughly linear increase. The slope here defines the initial permeability $\mu_i$.
2. Intermediate-field region: irreversible wall motion causes a steep rise in magnetization. This is where most of the magnetization change occurs.
3. High-field region: domain rotation dominates, and the curve bends over as it approaches $M_s$.

The differential susceptibility $\chi_{\text{diff}} = dM/dH$ peaks in the intermediate region and drops toward zero near saturation.

Applications of ferromagnetism

The unique combination of high magnetization, controllable domain structure, and tunable hysteresis properties makes ferromagnetic materials indispensable across electrical engineering and technology.

Permanent magnets

Permanent magnets require high $M_r$, high $H_c$, and a large maximum energy product $(BH)_{\max}$, which quantifies the maximum magnetostatic energy the magnet can supply externally.

• NdFeB: highest $(BH)_{\max}$ (~400 kJ/m³), but limited to temperatures below ~150°C without special grades
• SmCo: lower energy product than NdFeB but excellent thermal stability up to ~300°C
• Ferrite magnets: much cheaper, lower performance, widely used in consumer products
• Alnico: good temperature stability but low coercivity, easily demagnetized by external fields

Electromagnets

An electromagnet uses current through a coil to generate a controllable magnetic field. Placing a ferromagnetic core inside the coil multiplies the field by the relative permeability $\mu_r$ of the core material. Soft magnetic materials with high permeability and low coercivity are chosen so the core magnetizes and demagnetizes easily with minimal energy loss.

The field inside a solenoid with a ferromagnetic core is approximately:

$B = \mu_r \mu_0 n I$

where $n$ is the turns per unit length and $I$ is the current.

Data storage devices

In hard disk drives (HDDs), data is stored as the magnetization direction of tiny regions in a thin ferromagnetic film (the recording medium). Each region represents a binary bit. The read/write head uses a localized magnetic field to flip the magnetization of individual bits.

Modern HDDs use perpendicular magnetic recording, where the magnetization points out of the film plane, allowing higher bit densities than the older longitudinal recording. Materials with high anisotropy (like CoPtCr alloys) are needed to keep the bits thermally stable at small sizes, countering the superparamagnetic limit.

Transformers and inductors

Transformer and inductor cores use soft ferromagnetic materials to provide a high-permeability, low-reluctance path for magnetic flux. The key performance requirements are:

• High permeability to maximize inductance and flux linkage
• Low coercivity to minimize hysteresis losses per cycle
• High resistivity (or laminated construction) to minimize eddy current losses

At power frequencies (50/60 Hz), silicon steel (Fe-Si) laminations are standard. At higher frequencies (kHz to MHz), ferrite cores or amorphous/nanocrystalline alloys are used because their high resistivity suppresses eddy currents.

Ferromagnetic resonance

Ferromagnetic resonance (FMR) occurs when an oscillating (RF) magnetic field drives the magnetization of a ferromagnetic sample into precession around a static applied field, and the driving frequency matches the natural precession frequency. At resonance, the sample absorbs maximum RF energy.

FMR is both a characterization tool (for measuring anisotropy fields, damping constants, and g-factors) and the basis for microwave devices like YIG filters and circulators.

Resonance frequency

The magnetization precesses around the effective field at the Larmor frequency, modified by the sample geometry and anisotropy. For a thin film magnetized in-plane, the Kittel formula gives:

$f_r = \frac{\gamma}{2\pi} \sqrt{H(H + M_s)}$

For a general case including an anisotropy field $H_A$:

$f_r = \frac{\gamma}{2\pi} \sqrt{(H + H_A)(H + H_A + M_s)}$

Here $\gamma$ is the gyromagnetic ratio ($\gamma = g \mu_B / \hbar \approx 1.76 \times 10^{11} \text{ rad/(s·T)}$ for $g \approx 2$). Typical FMR frequencies for common ferromagnets in laboratory fields are in the GHz range.

Damping mechanisms

The precession doesn't continue indefinitely. Damping causes the magnetization to spiral toward equilibrium. The Landau-Lifshitz-Gilbert (LLG) equation describes this:

$\frac{d\mathbf{M}}{dt} = -\gamma \mathbf{M} \times \mathbf{H}_{\text{eff}} + \frac{\alpha}{M_s} \mathbf{M} \times \frac{d\mathbf{M}}{dt}$

where $\alpha$ is the dimensionless Gilbert damping parameter. Typical values range from $\alpha \sim 0.001$ (low-damping materials like YIG) to $\alpha \sim 0.01$ (transition metals like permalloy).

Damping sources include:

• Intrinsic (Gilbert) damping: spin-orbit coupling transfers energy from the precessing magnetization to the lattice (phonons)
• Extrinsic damping: two-magnon scattering (from surface roughness or defects), eddy currents, and spin pumping (transfer of spin angular momentum into adjacent non-magnetic layers)

Spin waves

Spin waves (magnons) are the collective excitations of the spin system in a ferromagnet. Think of them as propagating disturbances in the magnetization orientation, analogous to phonons in a crystal lattice.

The dispersion relation for exchange-dominated spin waves (short wavelengths) is:

$\hbar \omega = 2J_{\text{ex}} S a^2 k^2 + g \mu_B B_0$

where $a$ is the lattice constant, $k$ is the wavevector, and $B_0$ is the applied field. At long wavelengths, dipolar interactions modify the dispersion, and the waves are called magnetostatic waves (forward volume, backward volume, or surface modes depending on geometry).

Spin waves are quantized as magnons and follow Bose-Einstein statistics. Their thermal excitation is responsible for the Bloch $T^{3/2}$ law for the decrease of $M_s(T)$ at low temperatures.

Magnetostriction

Magnetostriction is the coupling between magnetization and mechanical strain in a ferromagnet. When you magnetize the material, it physically changes shape; when you stress it, its magnetization changes. Both effects arise from the same magnetoelastic coupling energy.

Joule effect

The Joule effect (or direct magnetostriction) is the change in dimensions of a ferromagnet upon magnetization. The saturation magnetostriction $\lambda_s$ is defined as the fractional change in length when the material goes from demagnetized to saturated:

$\lambda_s = \frac{\Delta L}{L}\bigg|_{\text{saturation}}$

Typical values:

• Iron: $\lambda_s \approx -7 \times 10^{-6}$ (contracts along the field direction)
• Nickel: $\lambda_s \approx -33 \times 10^{-6}$
• Terfenol-D: $\lambda_s \approx 1000 \times 10^{-6}$ (giant magnetostriction)

Note that the Joule effect is not linear in $H$. The strain depends on the square of the magnetization direction cosines, making it an even function of the field (the material elongates or contracts regardless of field sign). The relationship $\lambda = \Delta L / (L \cdot H)$ sometimes seen is a rough linearization valid only over a limited field range.

Villari effect

The Villari effect (inverse magnetostriction) is the converse: applying mechanical stress changes the magnetization. This is the basis for magnetoelastic sensors. If you apply tensile stress $\sigma$ to a material with $\lambda_s > 0$, the easy axis shifts toward the stress direction, increasing the magnetization component along that axis.

Applications include:

• Torque sensors on rotating shafts
• Force and pressure sensors
• Non-contact stress measurement in structural health monitoring

Magnetostrictive materials

For actuator and sensor applications, you want materials with large $|\lambda_s|$:

• Terfenol-D ($\text{Tb}_{0.3}\text{Dy}_{0.7}\text{Fe}_2$): $\lambda_s \sim 1000$ ppm, the workhorse giant magnetostrictive material. Used in sonar transducers and precision actuators.
• Galfenol (Fe-Ga): $\lambda_s \sim 200$ ppm, mechanically robust and machinable, unlike the brittle Terfenol-D.
• Nickel: $\lambda_s \sim -33$ ppm, historically important and still used in some ultrasonic transducers.

Magnetic force microscopy

Magnetic force microscopy (MFM) is a variant of atomic force microscopy (AFM) that maps the magnetic stray field above a sample surface. It provides nanoscale images of domain structures, domain walls, and other magnetic features without requiring special sample preparation.

Principles of operation

MFM typically operates in a two-pass (lift mode) technique:

1. First pass: the tip scans in tapping mode to record the surface topography.
2. Second pass: the tip retraces the topography at a set lift height (typically 20–100 nm above the surface), detecting only the long-range magnetic force.

The magnetic tip (usually coated with a CoCr alloy) interacts with the stray field gradients from the sample. The force gradient $\partial F_z / \partial z$ shifts the cantilever's resonance frequency, and this shift is recorded to produce the MFM image. Attractive interactions (tip and sample moments aligned) appear as one contrast, while repulsive interactions appear as the opposite.

Resolution and sensitivity

Spatial resolution in MFM is primarily limited by the tip geometry and the tip-sample separation. Sharper tips and smaller lift heights give better resolution but also increase sensitivity to topographic artifacts.

• Typical lateral resolution: 20–50 nm (down to ~10 nm with specialized tips)
• Vertical sensitivity: sub-nanometer deflections detectable
• The technique measures field gradients, not absolute field values, so quantitative interpretation requires modeling

Recent advances include using magnetic nanoparticle tips and nitrogen-vacancy (NV) center magnetometry for improved resolution and quantitative field mapping.

Imaging magnetic domains

MFM is widely used to visualize:

• Domain patterns in thin films and patterned nanostructures
• Domain wall positions and types (Bloch vs. Néel)
• Magnetic vortex cores in micron-scale disks
• Bit patterns in magnetic recording media
• Skyrmion lattices in materials with Dzyaloshinskii-Moriya interaction

The magnetic contrast arises because adjacent domains with different magnetization orientations produce different stray field patterns above the surface. Domain walls appear as lines of contrast change. MFM images complement other techniques like Kerr microscopy (which is optical and has lower resolution but faster acquisition) and Lorentz TEM (which images domains in transmission with very high resolution but requires thin, electron-transparent samples).