5.3 Ferromagnetism

Origins of Ferromagnetism

Ferromagnetism is the phenomenon where certain materials develop spontaneous magnetization, meaning they become magnetic even without an applied external field. This behavior originates from quantum mechanical interactions between electron spins that favor parallel alignment of magnetic moments. The result is materials with strong, permanent magnetic properties that underpin technologies from electric motors to data storage.

This section covers the exchange interaction that drives ferromagnetic ordering, the nature of spontaneous magnetization, and the Curie temperature that marks the boundary between ferromagnetic and paramagnetic behavior.

Exchange Interaction

The exchange interaction is the quantum mechanical mechanism responsible for ferromagnetic ordering. It arises because of the Pauli exclusion principle: electrons with parallel spins must occupy different spatial states, which changes the Coulomb energy between them. In ferromagnetic materials, this energy difference favors parallel spin alignment.

The interaction is described by the Heisenberg Hamiltonian:

$H = -J \sum_{\langle i,j \rangle} \mathbf{S}_i \cdot \mathbf{S}_j$

where the sum runs over nearest-neighbor pairs. The exchange integral $J$ determines the type and strength of magnetic ordering:

• Positive $J$: favors parallel spin alignment (ferromagnetism)
• Negative $J$: favors antiparallel alignment (antiferromagnetism)

The magnitude of $J$ depends on the overlap of electron wavefunctions between neighboring atoms, which is why only certain elements and crystal structures support ferromagnetism.

Spontaneous Magnetization

Below the Curie temperature, a ferromagnetic material develops a net magnetization without any external field applied. This happens because the exchange interaction aligns magnetic moments within regions called domains.

• The maximum achievable magnetization is the saturation magnetization $M_s$, reached when all moments are fully aligned.
• $M_s$ depends on the material's crystal structure and electronic configuration. For example, iron has $M_s \approx 1.7 \times 10^6$ A/m at room temperature.
• Spontaneous magnetization can be measured using techniques like vibrating sample magnetometry (VSM).

Note that a bulk ferromagnet may show zero net magnetization if its domains cancel each other out, even though each domain is individually magnetized.

Curie Temperature

The Curie temperature $T_C$ is the critical temperature above which thermal energy overcomes the exchange interaction, destroying long-range ferromagnetic order. Above $T_C$, the material becomes paramagnetic.

• $T_C$ is set by the strength of exchange interactions in the material.
• It varies widely: iron has $T_C = 1043$ K, cobalt 1388 K, nickel 627 K, and gadolinium only 293 K (near room temperature).
• $T_C$ can be measured experimentally using thermomagnetic analysis, where magnetization is tracked as a function of temperature.

The transition at $T_C$ is typically second-order (continuous), meaning the magnetization drops smoothly to zero rather than jumping discontinuously.

Magnetic Domains

A ferromagnetic material doesn't usually behave like a single giant magnet. Instead, it breaks up into magnetic domains, regions where all the magnetic moments point in the same direction. Different domains point in different directions, which can reduce or cancel the net magnetization of the bulk sample.

Domains exist because they lower the total energy of the system. A single uniformly magnetized sample would have large stray fields outside it, costing significant magnetostatic energy. By forming domains with opposing magnetizations, the material reduces those stray fields.

Domain Wall Structure

The boundary between two adjacent domains is called a domain wall. Within the wall, the magnetization gradually rotates from one domain's direction to the other's.

Two main types exist:

• Bloch walls: the magnetization rotates out of the plane of the wall. These are common in bulk materials.
• Néel walls: the magnetization rotates within the plane of the wall. These tend to appear in thin films where out-of-plane rotation is energetically costly.

Domain wall width is set by the competition between exchange energy (which prefers gradual rotation, making walls wider) and magnetocrystalline anisotropy (which prefers magnetization along specific axes, making walls narrower). Typical widths range from 10 to 100 nm. Domain wall motion is one of the primary mechanisms by which magnetization reversal occurs.

Domain Formation Energetics

Domain structure is determined by minimizing the total energy, which has several competing contributions:

• Magnetostatic energy: the energy of stray fields outside the material. Forming multiple domains reduces this.
• Exchange energy: favors uniform magnetization within each domain. Creating domain walls costs exchange energy.
• Magnetocrystalline anisotropy energy: favors magnetization along certain crystal axes, influencing which domain orientations are preferred.
• Magnetoelastic energy: arises from coupling between magnetization and mechanical strain.

The equilibrium domain size balances these terms. Smaller domains reduce magnetostatic energy but require more domain walls (costing exchange and anisotropy energy). Micromagnetic simulation tools like OOMMF and MuMax3 are used to model these competing effects numerically.

Single-Domain Particles

When a magnetic particle is small enough, forming a domain wall costs more energy than tolerating the stray field of a uniformly magnetized particle. Below this critical single-domain size (typically 10-100 nm depending on the material), the particle remains uniformly magnetized.

• Single-domain particles have no domain walls, so their magnetization reverses by coherent rotation rather than wall motion.
• Above the blocking temperature, thermal fluctuations can randomly flip the particle's magnetization, producing superparamagnetic behavior: the particle has a large magnetic moment but no stable remanence.
• Applications include ferrofluids, biomedical imaging contrast agents, and magnetic hyperthermia therapy.
• Characterization tools include SQUID magnetometry and Mössbauer spectroscopy.

Magnetization Process

When you apply an external magnetic field to a ferromagnetic material, the magnetization changes through two main mechanisms: domain wall motion (walls shift so that favorably oriented domains grow) and magnetization rotation (moments within domains rotate toward the field direction). The relative importance of each depends on the field strength and the material's microstructure.

Hysteresis Loop

The hysteresis loop is a plot of magnetization $M$ versus applied field $H$ as the field is cycled. It's the most important characterization tool for ferromagnetic materials.

Key features of the loop:

• Saturation magnetization $M_s$: the maximum magnetization when all moments are aligned with the field.
• Remanence $M_r$: the magnetization remaining when the field is reduced to zero.
• Coercivity $H_c$: the reverse field needed to bring the magnetization back to zero.
• Loop area: proportional to the energy dissipated per cycle (hysteresis loss).

At low fields, magnetization increases mainly through reversible domain wall motion. At intermediate fields, irreversible wall motion and domain nucleation dominate. Near saturation, the remaining misaligned moments rotate into the field direction.

Coercivity vs. Remanence

These two quantities together determine whether a material is "hard" or "soft" magnetically:

• Coercivity $H_c$: how resistant the material is to demagnetization. High $H_c$ means the material holds its magnetization stubbornly.
• Remanence $M_r$: how much magnetization the material retains after the field is removed.

Both properties depend on material composition, grain size, defect density, and processing history.

Magnetic Anisotropy

Magnetic anisotropy is the tendency of magnetization to prefer certain directions over others. It directly affects coercivity, domain structure, and the shape of the hysteresis loop.

The main types are:

• Magnetocrystalline anisotropy: intrinsic preference for magnetization along specific crystal axes (e.g., the $\langle 100 \rangle$ directions in iron). Originates from spin-orbit coupling.
• Shape anisotropy: arises from the geometry of the sample. An elongated sample is easier to magnetize along its long axis due to demagnetizing field effects.
• Stress (magnetoelastic) anisotropy: mechanical strain alters the preferred magnetization direction through magnetostriction.

Anisotropy can be engineered for specific applications. For example, perpendicular magnetic anisotropy in thin films is critical for high-density magnetic recording. Measurement techniques include torque magnetometry and ferromagnetic resonance (FMR).

Ferromagnetic Materials

Only a relatively small number of elements are ferromagnetic on their own, but combining elements into alloys and compounds opens up a vast range of tunable magnetic properties.

Transition Metals

Iron, cobalt, and nickel are the three ferromagnetic elements you encounter most often. Their ferromagnetism comes from partially filled 3d electron shells, which produce uncompensated magnetic moments.

• Iron: $T_C = 1043$ K, moment $\approx 2.2 \, \mu_B$/atom
• Cobalt: $T_C = 1388$ K, moment $\approx 1.7 \, \mu_B$/atom
• Nickel: $T_C = 627$ K, moment $\approx 0.6 \, \mu_B$/atom

These metals form the basis for most technologically important magnetic alloys. Their electronic structure is studied using techniques like X-ray magnetic circular dichroism (XMCD), which can separate orbital and spin contributions to the magnetic moment.

Rare Earth Elements

Elements like gadolinium, terbium, and dysprosium are ferromagnetic due to partially filled 4f electron shells. Compared to the 3d transition metals, rare earths have:

• Much stronger magnetocrystalline anisotropy (because 4f electrons have large orbital angular momentum)
• Higher atomic magnetic moments
• Lower Curie temperatures (gadolinium: 293 K, meaning it's only ferromagnetic near or below room temperature)

The real power of rare earths comes from combining them with transition metals. NdFeB (neodymium-iron-boron) magnets, for instance, combine the high anisotropy of Nd with the high magnetization of Fe to produce the strongest permanent magnets available, with energy products $(BH)_{max}$ exceeding 400 kJ/m$^3$.

Alloys and Compounds

By alloying multiple elements, you can tailor magnetic properties for specific applications:

• Permalloy ($\text{Ni}_{80}\text{Fe}_{20}$): extremely low coercivity and high permeability, ideal for magnetic shielding and sensor elements.
• Alnico (Al-Ni-Co alloys): high remanence and good temperature stability, used in older permanent magnet designs.
• Yttrium iron garnet (YIG): a ferrimagnetic insulator with extremely low magnetic damping, important for microwave devices.
• Heusler alloys: some exhibit half-metallic ferromagnetism, where only one spin channel conducts. This makes them promising for spintronic devices that require high spin polarization.

Characterization of these materials often relies on neutron diffraction (to determine magnetic structure) and Mössbauer spectroscopy (to probe local magnetic environments).

Quantum Mechanical Description

Classical electromagnetism cannot explain ferromagnetism. The Bohr-van Leeuwen theorem proves that classical statistical mechanics predicts zero net magnetization for any system in thermal equilibrium. Ferromagnetism is fundamentally quantum mechanical, arising from exchange interactions between electron spins.

Heisenberg Model

The Heisenberg model treats magnetic atoms as localized spins interacting with their neighbors:

$H = -\sum_{\langle i,j \rangle} J_{ij} \mathbf{S}_i \cdot \mathbf{S}_j$

Here $J_{ij}$ is the exchange integral between spins at sites $i$ and $j$, and $\mathbf{S}_i$ is the spin operator at site $i$. The model captures the essential physics of ferromagnetism: when $J_{ij} > 0$, the ground state has all spins aligned.

Extensions of the basic model include:

• Anisotropy terms that break rotational symmetry
• Zeeman coupling to an external field $\mathbf{B}$
• Next-nearest-neighbor interactions for more realistic descriptions

The Heisenberg model is exactly solvable in 1D (no long-range order at finite temperature by the Mermin-Wagner theorem for isotropic interactions) but requires approximations in 2D and 3D.

Mean Field Theory

Mean field theory (also called molecular field theory, introduced by Weiss) simplifies the many-body problem by replacing the effect of all neighboring spins on a given spin with a single average field.

The steps are:

1. Assume each spin experiences an effective field $\mathbf{B}_{eff} = \mathbf{B}_{ext} + \lambda \mathbf{M}$, where $\lambda$ is the molecular field constant.
2. Calculate the thermal average of the magnetization using the Brillouin function.
3. Solve self-consistently for $M(T)$.

This approach correctly predicts:

• Spontaneous magnetization below $T_C$
• The Curie-Weiss law for susceptibility above $T_C$
• A second-order phase transition at $T_C$

However, mean field theory overestimates $T_C$ (because it neglects fluctuations) and gets the critical exponents wrong. It predicts $\beta = 1/2$ for the magnetization exponent, while experiments on 3D ferromagnets give $\beta \approx 0.36$.

Spin Waves

At low temperatures, the excitations of a ferromagnet are not individual spin flips but collective, wave-like disturbances called spin waves. The quantized spin wave excitation is called a magnon.

• Spin waves are analogous to phonons in a crystal lattice, but for the spin system rather than the atomic positions.
• For a simple ferromagnet, the dispersion relation at long wavelengths is $\omega \propto k^2$ (quadratic), in contrast to the linear dispersion of acoustic phonons.
• Spin waves reduce the magnetization from its zero-temperature saturation value. This leads to Bloch's $T^{3/2}$ law: $M(T) = M(0)[1 - (T/T_0)^{3/2}]$ at low temperatures.
• Spin waves are measured experimentally using inelastic neutron scattering and Brillouin light scattering.

Temperature Dependence

Temperature is one of the most important variables controlling ferromagnetic behavior. As temperature increases, thermal fluctuations progressively disrupt spin alignment until long-range order is destroyed at the Curie temperature.

Curie-Weiss Law

Above the Curie temperature, the magnetic susceptibility follows the Curie-Weiss law:

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

where $C$ is the Curie constant (proportional to the square of the effective moment) and $T_C$ is the Curie temperature.

• The susceptibility diverges as $T \to T_C^+$, signaling the onset of ferromagnetic order.
• Plotting $1/\chi$ vs. $T$ should give a straight line with an x-intercept at $T_C$. This is a standard method for estimating $T_C$ from experimental data.
• Deviations from linearity near $T_C$ indicate that mean field theory is breaking down and critical fluctuations are important.
• A positive intercept ($T_C > 0$) indicates ferromagnetic interactions; a negative intercept suggests antiferromagnetic interactions (Curie-Weiss law with $\theta < 0$).

Critical Exponents

Near the phase transition at $T_C$, physical quantities follow power-law behavior characterized by critical exponents:

• $\beta$: magnetization near $T_C$, where $M \propto (T_C - T)^\beta$
• $\gamma$: susceptibility above $T_C$, where $\chi \propto (T - T_C)^{-\gamma}$
• $\delta$: critical isotherm at $T = T_C$, where $M \propto H^{1/\delta}$

These exponents are universal: they depend only on the dimensionality of the system and the symmetry of the order parameter, not on microscopic details. For the 3D Heisenberg model, $\beta \approx 0.36$, $\gamma \approx 1.39$, and $\delta \approx 4.8$. Mean field theory predicts $\beta = 0.5$, $\gamma = 1$, $\delta = 3$.

Critical exponents are measured using high-precision magnetometry and neutron scattering near $T_C$.

Magnetic Phase Transitions

The ferromagnetic-to-paramagnetic transition at $T_C$ is the most studied magnetic phase transition.

• It is typically second-order (continuous): the magnetization goes smoothly to zero, and there is no latent heat, but the specific heat shows a divergence or cusp.
• Some materials exhibit first-order magnetic transitions, where the magnetization drops discontinuously. This can happen when the transition is coupled to a structural change (e.g., in MnAs or some Heusler alloys).
• External factors like applied field, hydrostatic pressure, and chemical substitution can shift $T_C$ or even change the order of the transition.
• The magnetocaloric effect (temperature change upon adiabatic magnetization/demagnetization) is largest near $T_C$ and is exploited in magnetic refrigeration research.

Magnetic Ordering

Ferromagnetism is just one type of long-range magnetic order. The sign and range of exchange interactions, combined with the crystal structure, determine which ordering pattern a material adopts.

Ferromagnetic vs. Antiferromagnetic

Ferrimagnetic Materials

Ferrimagnets have two or more magnetic sublattices with antiparallel alignment, but the moments on the sublattices are unequal. This gives a net spontaneous magnetization, similar to ferromagnets, but with a different microscopic origin.

• Magnetite ($\text{Fe}_3\text{O}_4$) is the classic example: Fe$^{3+}$ ions on tetrahedral sites and Fe$^{2+}$/Fe$^{3+}$ on octahedral sites have antiparallel but unequal moments.
• Ferrimagnetic oxides (ferrites) typically have high electrical resistivity, making them useful at high frequencies where eddy current losses would be problematic in metallic ferromagnets.
• YIG has exceptionally low magnetic damping, making it important for microwave and magnonic applications.

Helical Spin Structures

In some materials, competing exchange interactions (e.g., ferromagnetic nearest-neighbor and antiferromagnetic next-nearest-neighbor) stabilize helical or spiral magnetic structures, where the spin direction rotates progressively from one atomic layer to the next.

• Examples include MnSi and FeGe, where the Dzyaloshinskii-Moriya interaction (an antisymmetric exchange arising from spin-orbit coupling in non-centrosymmetric crystals) stabilizes helical order.
• Under applied magnetic fields or at certain temperatures, helical magnets can host magnetic skyrmions: topologically protected, nanoscale spin textures with potential applications in spintronic memory and logic.
• Polarized neutron scattering is the primary tool for determining helical magnetic structures.

Applications of Ferromagnetism

Ferromagnetic materials are among the most technologically important classes of materials. Their applications span energy conversion, information storage, and emerging quantum technologies.

Permanent Magnets

Permanent magnets retain their magnetization without an external field and are used wherever a static magnetic field is needed: electric motors, generators, loudspeakers, magnetic bearings, and MRI machines.

The figure of merit for a permanent magnet is the maximum energy product $(BH)_{max}$, which measures the maximum energy density the magnet can supply to an external circuit.

• Ferrite magnets (e.g., $\text{BaFe}_{12}\text{O}_{19}$): inexpensive, moderate performance, widely used.
• Alnico: good temperature stability, but lower coercivity than modern alternatives.
• NdFeB: highest $(BH)_{max}$ of any commercial magnet (up to ~450 kJ/m$^3$), but performance degrades above ~150°C.
• SmCo: excellent high-temperature performance, but expensive due to cobalt and samarium costs.

A major research direction is developing rare-earth-free permanent magnets to reduce dependence on critical raw materials.

Magnetic Storage Devices

Digital data storage relies heavily on ferromagnetic thin films:

• Hard disk drives (HDDs): data is stored as magnetized regions in a thin CoPtCr-based alloy film. Read heads use the giant magnetoresistance (GMR) or tunneling magnetoresistance (TMR) effect to detect the stray fields from recorded bits.
• Magnetic tape: uses iron oxide or metal particle coatings for archival and backup storage.
• Current research into heat-assisted magnetic recording (HAMR) uses a laser to locally heat the recording medium, temporarily reducing its coercivity so that smaller, more thermally stable grains can be written. This pushes areal densities beyond 1 Tb/in$^2$.

Spintronics

Spintronics exploits the electron's spin degree of freedom in addition to its charge. Ferromagnetic materials are central to this field because they provide spin-polarized currents.

• Magnetic tunnel junctions (MTJs): two ferromagnetic layers separated by a thin insulating barrier. The resistance depends on the relative orientation of the two magnetizations (TMR effect). MTJs are the core element of MRAM (magnetic random-access memory).
• Spin-transfer torque (STT): a spin-polarized current can exert a torque on a ferromagnetic layer, switching its magnetization without an external field. This enables current-controlled writing in STT-MRAM.
• Magnonics: an emerging subfield that uses spin waves (magnons) rather than charge currents to carry and process information, potentially offering lower power dissipation.

Experimental Techniques

Characterizing ferromagnetic materials requires probing their magnetization, domain structure, and spin dynamics across multiple length and time scales.

Magnetometry Methods

Magnetometry measures bulk magnetic properties like magnetization, susceptibility, and anisotropy.

• Vibrating sample magnetometer (VSM): the sample oscillates inside pickup coils, inducing a voltage proportional to the magnetic moment. Fast and versatile.
• SQUID magnetometer: uses a superconducting quantum interference device to detect extremely small magnetic signals (sensitivity ~$10^{-8}$ emu). The gold standard for measuring weak moments.
• Alternating gradient magnetometer (AGM): applies an alternating field gradient to the sample and measures the resulting force. Offers high sensitivity for thin films and small samples.
• Torque magnetometry: measures the torque exerted on a sample in a uniform field as a function of angle, directly probing magnetic anisotropy.

Neutron Scattering

Neutrons carry a magnetic moment and no electric charge, making them ideal probes of magnetic structure and dynamics.

• Elastic neutron diffraction: reveals the magnetic unit cell, ordered moment directions, and long-range magnetic structure.
• Inelastic neutron scattering: measures the energy and momentum of spin wave excitations, mapping out the magnon dispersion relation.
• Polarized neutron reflectometry: determines the magnetization profile as a function of depth in thin films and multilayers.
• Small-angle neutron scattering (SANS): probes magnetic structures on the 1-100 nm scale, useful for studying domain walls, skyrmions, and magnetic nanoparticles.

Magnetic Force Microscopy

Magnetic force microscopy (MFM) images magnetic domain structures with spatial resolution down to tens of nanometers.

The technique works by scanning a magnetized tip over the sample surface:

1. First pass: the tip maps the surface topography in contact or tapping mode (like standard AFM).
2. Second pass (lift mode): the tip retraces the topography at a fixed height above the surface, sensing the magnetic force gradient from the sample's stray field.
3. The phase shift or frequency shift of the cantilever oscillation is recorded to produce a magnetic contrast image.

MFM is widely used to study domain patterns, domain wall structures, and recorded bit patterns in magnetic media. It can be combined with other scanning probe methods (AFM, STM) for correlative measurements.

Ferromagnetic Thin Films

When ferromagnetic materials are deposited as thin films (thicknesses from a few monolayers to hundreds of nanometers), their magnetic properties can differ significantly from the bulk. Reduced dimensionality, surface and interface effects, and strain all play a role.

Surface Effects

At a surface or interface, atoms have fewer magnetic neighbors than in the bulk. This reduced coordination changes the local exchange interactions and can modify the magnetic moment per atom.

• Surface atoms in transition metals often have enhanced magnetic moments compared to bulk atoms.
• Surface anisotropy (Néel surface anisotropy) can favor out-of-plane magnetization, even when the bulk prefers in-plane alignment.
• Oxidation, interdiffusion, and roughness at interfaces degrade magnetic properties and must be controlled during fabrication.
• Surface-sensitive techniques like magneto-optical Kerr effect (MOKE) and XMCD are used to probe these effects.

Magnetic Anisotropy in Films

Thin film geometry introduces several anisotropy contributions that compete with each other:

• Shape anisotropy: for a continuous thin film, the demagnetizing field strongly favors in-plane magnetization.
• Magnetocrystalline anisotropy: can be controlled through epitaxial growth on single-crystal substrates.
• Strain-induced anisotropy: lattice mismatch between the film and substrate creates strain, which couples to the magnetization through magnetostriction.
• Interface anisotropy: becomes dominant in ultrathin films (a few monolayers thick) and can overcome shape anisotropy to produce perpendicular magnetic anisotropy (PMA). PMA is essential for high-density magnetic recording and STT-MRAM.

The effective anisotropy of a thin film is often written as $K_{eff} = K_V + K_S/t$, where $K_V$ is the volume contribution, $K_S$ is the surface/interface contribution, and $t$ is the film thickness.

Exchange Bias

Exchange bias is a phenomenon that occurs when a ferromagnetic (FM) layer is in direct contact with an antiferromagnetic (AFM) layer. After cooling through the AFM's Néel temperature in an applied field, the hysteresis loop of the FM layer shifts along the field axis.

• The shift field is called the exchange bias field $H_{EB}$.
• Physically, the interfacial coupling between the FM and AFM layers creates a preferred magnetization direction in the FM, making it harder to reverse in one direction than the other.
• Exchange bias is used in spin valve structures, where it pins the magnetization of a reference layer while a free layer responds to external fields. This is the basis of GMR/TMR read heads in hard drives.
• The magnitude of $H_{EB}$ depends on interface quality, AFM grain size, cooling field, and temperature.
• It is studied using polarized neutron reflectometry, MOKE, and XMCD.