8.3 Magnetic domains and hysteresis

Magnetic domains and hysteresis

Magnetic domains and hysteresis explain how magnetic materials respond to applied fields. Domains are regions where atomic magnetic moments align in the same direction, and hysteresis describes the nonlinear, irreversible magnetization curves these materials produce. Together, these concepts connect atomic-scale interactions to the macroscopic magnetic behavior you see in transformers, hard drives, and permanent magnets.

Magnetic domains

A magnetic domain is a region within a ferromagnetic or ferrimagnetic material where all the atomic magnetic moments point in the same direction. A bulk sample typically contains many domains pointing in different directions, so the material can appear unmagnetized even though each domain is internally magnetized. Domains exist because they lower the total magnetic energy of the system.

Origin of domains

Several competing energy contributions drive domain formation:

• Magnetostatic (demagnetizing) energy is the energy stored in the stray magnetic field outside the material. A single uniformly magnetized domain would create a large external field and high magnetostatic energy. Splitting into multiple domains with opposing orientations reduces this energy dramatically.
• Exchange energy favors parallel alignment of neighboring magnetic moments. It's the interaction responsible for ferromagnetism in the first place, and it resists the formation of domains because adjacent domains point in different directions.
• Magnetocrystalline anisotropy energy tends to lock moments along preferred crystallographic directions (the "easy axes").

The equilibrium domain structure is the configuration that minimizes the sum of all these energy terms. More domains reduce magnetostatic energy but increase the total domain wall area (and thus exchange and anisotropy energy), so the material settles on a compromise.

Domain walls

Domain walls are the thin transition regions between adjacent domains. Inside a wall, the magnetic moments gradually rotate from the direction of one domain to the next. The wall width is set by a balance: exchange energy wants a wide, gradual rotation (less misalignment between neighbors), while anisotropy energy wants a narrow wall (fewer moments pointing away from the easy axis).

A typical domain wall width in iron is on the order of tens of nanometers, and the wall energy per unit area is often expressed as:

$\sigma_w \approx \pi \sqrt{A K}$

where $A$ is the exchange stiffness and $K$ is the anisotropy constant.

Types of domain walls

• Bloch walls: The magnetization rotates out of the plane of the wall. These are the standard wall type in bulk materials, where there's no geometric constraint on out-of-plane rotation.
• Néel walls: The magnetization rotates within the plane of the wall. These are favored in thin films, where out-of-plane rotation would create large surface magnetic charges and raise the magnetostatic energy.
• Cross-tie walls: A hybrid structure combining segments of Bloch and Néel character, forming a periodic cross-tie pattern. These appear in thin films of intermediate thickness where neither pure Bloch nor pure Néel walls are energetically optimal.

Domain wall motion

When you apply an external magnetic field, domain walls move so that domains aligned with the field grow at the expense of those opposed to it. This is the primary mechanism of magnetization change in soft magnetic materials.

Wall motion is strongly influenced by pinning sites: crystal defects, impurities, grain boundaries, and internal stresses can trap domain walls. A wall remains pinned until the applied field is strong enough to overcome the pinning energy barrier, at which point it jumps to the next pinning site. This jerky motion is directly connected to hysteresis and the Barkhausen effect (discussed below).

Factors affecting domain structure

• Material composition and crystal structure (determines exchange and anisotropy energies)
• Sample geometry and size (controls magnetostatic energy; very small particles can be single-domain)
• Defects, impurities, and mechanical stress (create pinning sites and local anisotropy variations)
• Temperature (thermal energy competes with ordering; domains vanish above the Curie temperature)
• Applied magnetic field strength and direction

Observation of domains

• Magneto-optical Kerr effect (MOKE): Polarized light reflected from a magnetized surface undergoes a rotation in polarization proportional to the local magnetization. This gives surface-sensitive domain images.
• Magnetic force microscopy (MFM): A magnetized scanning probe tip detects stray field gradients above the sample, producing high-resolution maps of domain structure.
• Lorentz transmission electron microscopy (LTEM): Electrons passing through a thin sample are deflected by the Lorentz force from the sample's internal magnetic field, revealing domains and walls with nanometer-scale resolution.

Magnetic hysteresis

Hysteresis is the lag between the applied magnetic field and the resulting magnetization. If you cycle a ferromagnetic material through a complete field sweep, the magnetization traces out a loop rather than retracing the same curve. This irreversibility arises because domain wall motion and domain rotation involve overcoming energy barriers.

Hysteresis loop

The hysteresis loop (also called the B-H or M-H loop) plots magnetization $M$ versus applied field $H$. Key features to identify on the loop:

1. Start from a demagnetized state ($M = 0$, $H = 0$).
2. Increase $H$: domains aligned with the field grow via wall motion, then moments rotate into the field direction. $M$ rises and eventually saturates.
3. Decrease $H$ back to zero: $M$ doesn't return to zero but instead retains a value called the remanence.
4. Reverse $H$: you must apply a reverse field equal to the coercivity before $M$ reaches zero.
5. Continue increasing the reverse field until saturation in the opposite direction, then sweep back to complete the loop.

The area enclosed by the loop equals the energy dissipated per unit volume per cycle. This is why transformer designers want narrow loops (low loss) and permanent magnet designers want wide loops (high energy storage).

Saturation magnetization

Saturation magnetization ($M_s$) is the maximum magnetization a material can achieve, reached when every magnetic moment in the sample is aligned with the field. It's an intrinsic property determined by the number and magnitude of atomic moments per unit volume. For example, iron has $M_s \approx 1.7 \times 10^6$ A/m at room temperature. $M_s$ decreases with increasing temperature and drops to zero at the Curie temperature.

Remanent magnetization

Remanent magnetization ($M_r$) is the magnetization that remains after the external field is removed from a saturated sample. It exists because domain walls stay pinned in their displaced positions rather than returning to the demagnetized configuration.

The squareness ratio $M_r / M_s$ indicates how "square" the loop is. A ratio near 1 means the material retains almost all its saturation magnetization, which is desirable for permanent magnets and magnetic recording media.

Coercivity

Coercivity ($H_c$) is the reverse field needed to bring the magnetization back to zero after saturation. It quantifies how resistant a material is to demagnetization.

• Soft magnetic materials: $H_c$ typically below ~1000 A/m (easy to demagnetize)
• Hard magnetic materials: $H_c$ can exceed $10^6$ A/m (very resistant to demagnetization)

Coercivity depends on microstructure (grain size, defect density), anisotropy, and domain wall pinning strength.

Hysteresis in soft vs. hard magnetic materials

Magnetic anisotropy

Magnetic anisotropy means the magnetic properties of a material depend on direction. It determines which directions are "easy" (low energy, preferred magnetization direction) and which are "hard" (high energy, magnetization resists pointing this way). Anisotropy directly shapes the hysteresis loop because it controls how much field is needed to rotate moments away from easy axes.

Shape anisotropy

Shape anisotropy comes from the demagnetizing field, which depends on sample geometry. The demagnetizing field is weaker along long dimensions and stronger along short ones, so:

• An elongated rod has its easy axis along its length.
• A thin film has its easy axis in the plane of the film.

Shape anisotropy can be engineered by controlling the geometry of magnetic elements, which is particularly important in patterned nanostructures for data storage.

Magnetocrystalline anisotropy

This is an intrinsic property arising from spin-orbit coupling, which links the orientation of magnetic moments to the crystal lattice. In BCC iron, the easy axes are along $\langle 100 \rangle$ directions; in HCP cobalt, the easy axis is along the $c$-axis. The anisotropy energy for a uniaxial crystal can be written as:

$E_a = K_1 \sin^2\theta + K_2 \sin^4\theta + \ldots$

where $\theta$ is the angle between the magnetization and the easy axis.

Anisotropy constants

The constants $K_1$, $K_2$, etc., quantify the strength of magnetocrystalline anisotropy in units of J/m$^3$. They are material-specific and temperature-dependent (generally decreasing as temperature rises toward the Curie point). For most practical purposes, $K_1$ dominates, but higher-order terms matter in materials with complex crystal symmetry or under strong fields.

Magnetostriction

Magnetostriction is the small change in a material's physical dimensions when it's magnetized. It originates from the same spin-orbit coupling responsible for magnetocrystalline anisotropy. The fractional change in length is characterized by the magnetostriction coefficient $\lambda$:

• $\lambda > 0$: the material expands along the magnetization direction (e.g., iron along $\langle 100 \rangle$)
• $\lambda < 0$: the material contracts along the magnetization direction (e.g., nickel)

Magnetostriction matters in transformer cores (it causes audible hum) and is exploited in magnetostrictive sensors and actuators.

Factors affecting hysteresis

• Material composition and crystal structure
• Defect density, impurities, and grain boundaries (control pinning strength)
• Temperature (thermal energy helps moments overcome barriers, reducing coercivity)
• External stress (modifies anisotropy through magnetoelastic coupling)
• Domain structure and magnetic anisotropy

Applications of hysteresis

• Permanent magnets: motors, generators, actuators (need wide loops, high $H_c$ and $M_r$)
• Magnetic recording media: hard disk drives, magnetic tapes (need moderate $H_c$, high squareness)
• Transformer cores and inductors: power electronics (need narrow loops, low energy loss)
• Magnetic sensors and switches: reed switches, Hall effect sensors

Magnetization processes

When a magnetic field is applied to a demagnetized ferromagnet, the magnetization increases through two main mechanisms: domain wall motion and domain rotation. Which process dominates depends on the field strength and the material's anisotropy.

Domain wall motion

At low to moderate fields, magnetization changes primarily through domain wall motion. Walls shift so that favorably oriented domains expand. This is a relatively low-energy process and produces large changes in magnetization for small changes in field. It dominates in soft magnetic materials with weak pinning.

Domain rotation

At higher fields, or in materials with strong anisotropy and few mobile walls, the moments within domains rotate coherently toward the applied field direction. This process requires more energy than wall motion because it works against the anisotropy energy. Domain rotation is the dominant mechanism in hard magnetic materials and in single-domain particles.

Reversible vs. irreversible magnetization

• Reversible processes: Small, elastic displacements of domain walls or slight rotations of moments that return to their original state when the field is removed. These occur at very low fields.
• Irreversible processes: Domain walls jump past pinning sites and don't return when the field is removed. These jumps are what create hysteresis. The stronger the pinning and anisotropy, the more irreversible the magnetization process.

The initial slope of the M-H curve (at low fields) is mostly reversible, while the steep middle portion involves large irreversible wall jumps.

Barkhausen effect

The Barkhausen effect is the observation that magnetization doesn't change smoothly but in discrete, jerky steps. Each step corresponds to a domain wall suddenly breaking free from a pinning site and jumping to the next one. If you wrap a coil around the sample, these jumps induce measurable voltage pulses. The Barkhausen effect provides direct experimental evidence that magnetization proceeds through discontinuous domain wall motion.

Magnetic after-effect

After a sudden change in applied field, the magnetization continues to drift slowly toward equilibrium over time. This magnetic after-effect (also called magnetic relaxation) occurs because some domain walls are trapped in metastable states and need thermal energy to overcome their pinning barriers. It's more pronounced in materials with strong pinning and at low temperatures where thermal activation is limited.

Magnetic viscosity

Magnetic viscosity describes the time-dependent response of magnetization to a changing field. It's closely related to the magnetic after-effect and is characterized by a viscosity coefficient $S$:

$M(t) = M_0 + S \ln(t/t_0)$

where $M_0$ is the initial magnetization and $t_0$ is a reference time. Magnetic viscosity is relevant in applications with rapidly varying fields (transformers, inductors) and affects the long-term stability of permanent magnets and recorded data.

Magnetic materials

Different classes of magnetic materials are distinguished by how their atomic moments interact and order. The choice of material for a given application depends on whether you need easy magnetization reversal (soft) or permanent magnetization (hard).

Soft magnetic materials

Soft magnetic materials have low coercivity (typically $H_c < 1000$ A/m) and narrow hysteresis loops, meaning they magnetize and demagnetize with minimal energy loss.

• Examples: pure iron, silicon steel (Fe-Si), permalloy (Ni$_{80}$Fe$_{20}$), supermalloy
• Key properties: high permeability, low coercivity, low hysteresis loss
• Applications: transformer cores, inductors, electromagnetic shielding, magnetic read heads

The low coercivity comes from minimal pinning (few defects, low anisotropy) and easy domain wall motion.

Hard magnetic materials

Hard magnetic materials have high coercivity and wide hysteresis loops. They retain strong magnetization after the external field is removed.

• Examples: alnico alloys, barium/strontium ferrites, SmCo$_5$, Nd$_2$Fe$_{14}$B
• Key properties: high coercivity ($H_c > 10^5$ A/m for rare-earth magnets), high remanence, high energy product $(BH)_{max}$
• Applications: permanent magnets in motors and generators, magnetic recording media, magnetic latches

Their high coercivity arises from strong magnetocrystalline anisotropy and microstructures that impede domain wall motion (single-domain grains, grain boundary phases).

Ferromagnetic materials

Ferromagnetic materials exhibit spontaneous magnetization below their Curie temperature ($T_C$). All atomic moments within a domain align parallel due to the exchange interaction.

• Examples: Fe ($T_C = 1043$ K), Co ($T_C = 1394$ K), Ni ($T_C = 631$ K), and their alloys
• Above $T_C$, thermal energy overcomes exchange coupling and the material becomes paramagnetic.

Ferrimagnetic materials

Ferrimagnetic materials have two magnetic sublattices with antiparallel moments of unequal magnitude, producing a net spontaneous magnetization. They show hysteresis just like ferromagnets.

• Examples: magnetite (Fe$_3$O$_4$), nickel ferrite (NiFe$_2$O$_4$), yttrium iron garnet (Y$_3$Fe$_5$O$_{12}$)
• Applications: microwave devices (ferrite isolators, circulators), magnetic recording media, high-frequency inductors
• Ferrites are electrically insulating, which suppresses eddy currents and makes them ideal for high-frequency use.

Antiferromagnetic materials

Antiferromagnetic materials have two sublattices with equal and opposite moments, so the net magnetization is zero. They order below the Néel temperature ($T_N$).

• Examples: Cr, MnO, NiO, FeO
• While they don't produce a net magnetic field on their own, they're technologically important as pinning layers in spin valves and magnetic tunnel junctions, where they pin the magnetization of an adjacent ferromagnetic layer through exchange bias.

Amorphous magnetic materials

Amorphous (non-crystalline) magnetic materials lack long-range atomic order. This eliminates magnetocrystalline anisotropy, resulting in very soft magnetic behavior.

• Examples: metallic glasses such as Fe$_{80}$Si$_{10}$B$_{10}$, Co-Fe-Si-B alloys, amorphous ribbons produced by rapid quenching
• Key properties: very low coercivity, high permeability, low core losses
• Applications: high-efficiency transformer cores, magnetic sensors, electromagnetic shielding

Nanostructured magnetic materials

When at least one dimension of a magnetic structure is reduced to the nanometer scale, new phenomena emerge:

• Single-domain behavior: Below a critical size (typically 10-100 nm depending on the material), particles can't support domain walls and become single-domain, maximizing coercivity.
• Superparamagnetism: If the particle is small enough that thermal energy can flip the entire moment, the particle behaves like a paramagnet with a giant moment. This sets a lower limit on bit size in magnetic recording.
• Examples: Fe$_3$O$_4$ and $\gamma$-Fe$_2$O$_3$ nanoparticles, Co/Pt multilayer thin films, Ni and Co nanowires
• Applications: high-density magnetic recording, magnetic biosensors, targeted drug delivery, MRI contrast agents

Measurement techniques

Characterizing magnetic materials requires instruments that can measure magnetization, map domain structure, and probe anisotropy. Here are the most common techniques you'll encounter.

Vibrating sample magnetometer (VSM)

A VSM works by vibrating the sample mechanically in a uniform magnetic field. The oscillating magnetic moment induces a voltage in nearby pickup coils, and this voltage is proportional to the sample's magnetic moment. By sweeping the applied field, you trace out the full hysteresis loop. VSM is a workhorse technique for bulk samples, offering good sensitivity and straightforward operation.

Superconducting quantum interference device (SQUID)

SQUID magnetometry is the most sensitive technique for measuring magnetic moments, capable of detecting changes as small as $\sim 10^{-17}$ T. The sensor is a superconducting loop interrupted by one or two Josephson junctions. Magnetic flux threading the loop modulates the supercurrent, and this modulation is measured with extreme precision. SQUID is essential for studying weakly magnetic samples, nanostructures, thin films, and biological specimens.

Magneto-optical Kerr effect (MOKE)

MOKE measures magnetization by detecting changes in the polarization of light reflected from a magnetic surface. The Kerr rotation angle is proportional to the surface magnetization. Three geometries are used depending on which magnetization component you want to probe:

• Polar MOKE: sensitive to out-of-plane magnetization
• Longitudinal MOKE: sensitive to in-plane magnetization parallel to the plane of incidence
• Transverse MOKE: sensitive to in-plane magnetization perpendicular to the plane of incidence

MOKE is surface-sensitive and widely used for thin film and multilayer studies, as well as domain imaging.

Magnetic force microscopy (MFM)

MFM is a variant of atomic force microscopy that uses a magnetized tip. The tip scans across the sample surface at a fixed height, and the magnetic force gradient between tip and sample shifts the cantilever's resonance frequency. This produces a spatial map of the stray field above the surface, revealing domain patterns, domain walls, and individual magnetic nanostructures with resolution down to ~25 nm. MFM works in ambient conditions and is non-destructive.

Lorentz transmission electron microscopy (LTEM)

In LTEM, an electron beam passes through a thin magnetic sample. The Lorentz force ($\vec{F} = -e\vec{v} \times \vec{B}$) deflects electrons differently depending on the local magnetization direction. Two main imaging modes exist:

• Fresnel mode: defocused imaging where domain walls appear as bright or dark lines
• Foucault mode: an aperture blocks electrons deflected in one direction, making domains with different orientations appear with different contrast

LTEM provides the highest spatial resolution of any domain imaging technique and reveals the internal magnetic structure of thin specimens.