11.3 Magnetism

Magnetism arises from the quantum properties of electrons, and statistical mechanics gives us the tools to connect microscopic magnetic moments to the macroscopic behavior of materials. This topic covers the origins of magnetism, the key models (Ising, Heisenberg, mean field), phase transitions, and how all of this connects to real-world technology and thermodynamics.

Fundamentals of magnetism

Statistical mechanics treats magnetic systems as large collections of interacting magnetic moments. The collective behavior of these moments, governed by quantum mechanics and thermal fluctuations, determines whether a material is diamagnetic, paramagnetic, ferromagnetic, or something else entirely.

Magnetic field and flux

A magnetic field $\mathbf{B}$ describes the influence that magnetic materials or moving charges exert on their surroundings, measured in tesla (T). Magnetic flux $\Phi$ quantifies how much of that field passes through a given area, measured in weber (Wb). The two are related by:

$\Phi = \mathbf{B} \cdot \mathbf{A}$

Field lines point from north to south outside a magnet, and their density indicates field strength. This relationship becomes important when you analyze how magnetic moments interact with applied fields.

Magnetic dipole moment

The magnetic dipole moment $\mathbf{m}$ characterizes the strength and orientation of a magnetic source. It's a vector pointing from south to north, measured in A·m² (equivalently, J/T).

Two key results govern how a dipole responds to an external field $\mathbf{B}$:

• Torque: $\boldsymbol{\tau} = \mathbf{m} \times \mathbf{B}$ (tends to align the dipole with the field)
• Potential energy: $U = -\mathbf{m} \cdot \mathbf{B}$ (lowest when aligned, highest when anti-aligned)

These expressions show up repeatedly in partition function calculations for magnetic systems.

Diamagnetism vs paramagnetism

Diamagnetism produces a weak magnetization opposing an applied field. It's present in all materials but usually overshadowed by stronger effects. It arises from changes in electron orbital motion induced by the external field. Examples: water, copper, bismuth, most organic compounds.

Paramagnetism produces a weak magnetization aligned with an applied field. It requires unpaired electrons whose magnetic moments partially align with the field, competing against thermal disorder. Examples: aluminum, platinum, molecular oxygen.

The distinction matters for stat mech because paramagnetic systems are the starting point for partition function treatments of non-interacting spins, while diamagnetism is typically a small correction.

Ferromagnetism and antiferromagnetism

Ferromagnetism involves strong, cooperative alignment of magnetic moments due to exchange interactions. Key features:

• Spontaneous magnetization exists even without an external field (below the Curie temperature)
• Materials form magnetic domains and exhibit hysteresis
• Examples: iron, cobalt, nickel

Antiferromagnetism involves alternating alignment of neighboring moments, so the net magnetization cancels to zero. Key features:

• The Néel temperature $T_N$ marks the transition to a paramagnetic state
• The lattice can be thought of as two interpenetrating sublattices with opposite magnetization
• Examples: chromium, manganese oxide (MnO), hematite ($\alpha$-Fe$_2$O$_3$)

Microscopic origins of magnetism

The macroscopic magnetic behavior of a material traces back to what electrons are doing at the atomic scale. Quantum mechanics determines the magnetic moments; exchange interactions determine how those moments talk to each other.

Electron spin and orbital motion

Electrons contribute to magnetism in two ways:

• Spin: An intrinsic quantum property with spin quantum number $s = 1/2$, giving a magnetic moment of about one Bohr magneton ($\mu_B$). There's no classical analogue for spin.
• Orbital motion: Electrons orbiting the nucleus carry orbital angular momentum (quantized by quantum number $l$), which also generates a magnetic moment.

The total angular momentum $\mathbf{J} = \mathbf{L} + \mathbf{S}$ results from spin-orbit coupling and determines the net magnetic moment of an atom or ion. The way $\mathbf{L}$ and $\mathbf{S}$ combine (governed by Hund's rules and the relevant coupling scheme) dictates the magnetic properties of specific materials.

Exchange interaction

The exchange interaction is the quantum mechanical effect responsible for magnetic ordering. It arises from the combination of electrostatic (Coulomb) repulsion and the Pauli exclusion principle, which together make the energy depend on the relative orientation of neighboring spins.

• Direct exchange: Overlap of electron wavefunctions between neighboring atoms
• Indirect exchange (RKKY): Mediated by conduction electrons in metals; oscillates in sign with distance, which is why some metallic systems show complex magnetic structures

The sign and magnitude of the exchange constant $J$ determine the type of ordering: $J > 0$ favors ferromagnetism (parallel alignment), while $J < 0$ favors antiferromagnetism (antiparallel alignment).

Magnetic domains

In a ferromagnet, the material breaks into domains, each a region where moments point in the same direction. Domains form because they minimize the total magnetic energy (reducing the stray field energy outside the material).

Domain walls separate regions of different magnetization:

• Bloch walls: Magnetization rotates perpendicular to the wall plane (common in bulk materials)
• Néel walls: Magnetization rotates within the wall plane (common in thin films)

Domain structure directly affects hysteresis, coercivity, and remanence, which are the properties that matter for applications like permanent magnets and recording media.

Curie temperature

The Curie temperature $T_C$ is the critical temperature above which a ferromagnet becomes paramagnetic. Below $T_C$, exchange interactions win over thermal disorder and the material has spontaneous magnetization. Above $T_C$, thermal fluctuations destroy long-range order.

This is a second-order (continuous) phase transition: the magnetization drops continuously to zero as $T \to T_C$ from below.

Above $T_C$, the magnetic susceptibility follows the Curie-Weiss law:

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

where $C$ is the Curie constant. Notice the susceptibility diverges as $T \to T_C^+$, signaling the phase transition.

Statistical mechanics of magnetic systems

This is where the core of the topic lives. Statistical mechanics lets you start from a microscopic Hamiltonian, compute the partition function, and derive all the thermodynamic properties of a magnetic system.

Ising model

The Ising model is the simplest non-trivial model of ferromagnetism. Discrete spins $s_i = \pm 1$ sit on a lattice and interact with their nearest neighbors.

The Hamiltonian is:

$H = -J\sum_{\langle i,j \rangle} s_i s_j - h\sum_i s_i$

• $J$ is the exchange coupling ($J > 0$ for ferromagnetism)
• $h$ is the external magnetic field
• The first sum runs over nearest-neighbor pairs only

The model exhibits a phase transition between ordered (ferromagnetic) and disordered (paramagnetic) states for dimensions $d \geq 2$. In 1D, there's no phase transition at finite temperature. The 2D Ising model was solved exactly by Onsager (1944), making it one of the few exactly solvable models with a phase transition.

Mean field theory

Mean field theory replaces the fluctuating field from all neighbors with a single average (mean) effective field. This turns a many-body problem into a single-spin problem.

For the Ising model, the approach works as follows:

1. Replace the interaction of spin $i$ with its neighbors by an effective field: $h_{\text{eff}} = h + zJm$, where $z$ is the coordination number (number of nearest neighbors) and $m = \langle s_i \rangle$ is the average magnetization per spin.
2. Solve the single-spin problem in this effective field, giving the self-consistency equation: $m = \tanh(\beta(h + zJm))$, where $\beta = 1/(k_BT)$.
3. Solve this equation (graphically or numerically) to find $m$ as a function of $T$ and $h$.

Mean field theory predicts a Curie temperature of $k_B T_C = zJ$. It captures the qualitative physics (spontaneous magnetization, phase transition, Curie-Weiss behavior) but overestimates $T_C$ because it ignores fluctuations. The approximation becomes more accurate in higher dimensions.

Spin waves

Spin waves are low-energy collective excitations in magnetically ordered systems. Think of them as small, wave-like deviations of spins from perfect alignment, analogous to how phonons are collective vibrations of atoms.

For a ferromagnet, the dispersion relation at long wavelengths is:

$\omega(k) = Dk^2$

where $D$ is the spin wave stiffness. This quadratic dispersion (contrast with the linear dispersion of acoustic phonons) has important consequences:

• The density of states differs from phonons, leading to a $T^{3/2}$ correction to the magnetization at low temperatures (Bloch's $T^{3/2}$ law)
• Spin waves also contribute to the low-temperature specific heat

Magnons are the quantized excitations of spin waves, just as phonons are quantized lattice vibrations.

Critical phenomena in magnets

Near the Curie temperature, magnetic systems exhibit critical phenomena: physical quantities diverge or vanish as power laws in the reduced temperature $t = (T - T_C)/T_C$.

The key critical exponents are:

• Magnetization: $m \sim |t|^{\beta}$ (for $T < T_C$, $h = 0$)
• Susceptibility: $\chi \sim |t|^{-\gamma}$
• Correlation length: $\xi \sim |t|^{-\nu}$

A remarkable result is universality: systems with the same dimensionality and symmetry of the order parameter share the same critical exponents, regardless of microscopic details. The 3D Ising model, for example, has $\beta \approx 0.326$, $\gamma \approx 1.237$, $\nu \approx 0.630$, which differ from the mean field values ($\beta = 1/2$, $\gamma = 1$, $\nu = 1/2$).

Renormalization group (RG) techniques provide the theoretical framework for understanding universality and computing critical exponents systematically.

Quantum theory of magnetism

Classical and semiclassical pictures only go so far. A fully quantum treatment is needed to capture phenomena like entanglement between spins, quantum phase transitions, and the behavior of low-spin systems.

Heisenberg model

The Heisenberg model generalizes the Ising model by treating spins as quantum mechanical vector operators rather than scalars:

$H = -J\sum_{\langle i,j \rangle} \mathbf{S}_i \cdot \mathbf{S}_j - h\sum_i S_i^z$

Here $\mathbf{S}_i$ is the spin operator at site $i$, with components $S_i^x, S_i^y, S_i^z$. The dot product $\mathbf{S}_i \cdot \mathbf{S}_j = S_i^x S_j^x + S_i^y S_j^y + S_i^z S_j^z$ means spins can point in any direction in 3D space, not just up or down.

The Ising model is recovered in the limit of strong anisotropy, where only the $S^z S^z$ term survives. The isotropic Heisenberg model has full rotational symmetry, which has deep consequences for the nature of its phase transitions (Mermin-Wagner theorem forbids spontaneous magnetization in 1D and 2D at finite temperature for the isotropic case).

Spin-1/2 systems

The simplest quantum spins have $s = 1/2$, with just two states: $|\uparrow\rangle$ and $|\downarrow\rangle$. The spin operators are expressed using Pauli matrices:

$\sigma_x = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \quad \sigma_y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \quad \sigma_z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$

with $\mathbf{S} = \frac{\hbar}{2}\boldsymbol{\sigma}$.

These two-level systems exhibit genuinely quantum behavior: superposition of up and down states, entanglement between pairs of spins, and non-commuting observables ($[S_x, S_y] = i\hbar S_z$). Spin-1/2 chains and lattices are also the foundation for quantum computing (qubits) and spintronics.

Quantum phase transitions

Unlike thermal phase transitions (driven by temperature), quantum phase transitions occur at $T = 0$ and are driven by varying a non-thermal parameter like magnetic field or pressure. Quantum fluctuations, rather than thermal fluctuations, destroy the ordered phase.

The textbook example is the transverse-field Ising model:

$H = -J\sum_{\langle i,j \rangle} \sigma_i^z \sigma_j^z - h\sum_i \sigma_i^x$

The transverse field $h$ competes with the Ising coupling $J$. In 1D, the quantum critical point occurs at $h_c = J$:

• For $h < J$: the ground state is ferromagnetically ordered
• For $h > J$: the transverse field dominates and spins align along $x$, destroying ferromagnetic order

Even though the transition happens at $T = 0$, the quantum critical point influences thermodynamic behavior at finite temperatures in a fan-shaped quantum critical region above the critical point.

Magnetic ordering

Different types of magnetic order arise from the competition between exchange interactions, anisotropy, thermal fluctuations, and applied fields.

Spontaneous magnetization

Below $T_C$, a ferromagnet develops a net magnetization $M$ without any applied field. This spontaneous magnetization results from exchange interactions aligning moments cooperatively.

The magnetization curve $M(H)$ for a ferromagnet shows hysteresis:

• Remanent magnetization $M_r$: the magnetization remaining after the external field is removed
• Coercive field $H_c$: the reverse field needed to bring the magnetization back to zero

The area enclosed by the hysteresis loop represents energy dissipated per cycle, which matters for applications. Hard magnets (large $H_c$) are good for permanent magnets; soft magnets (small $H_c$) are good for transformers.

Magnetic anisotropy

Magnetic anisotropy means the energy depends on the direction of magnetization relative to the crystal axes. The main types are:

• Magnetocrystalline anisotropy: From spin-orbit coupling and crystal field effects. The simplest (uniaxial) form is $E = K\sin^2\theta$, where $K$ is the anisotropy constant and $\theta$ is the angle from the easy axis.
• Shape anisotropy: From the demagnetizing field in non-spherical samples. A long needle-shaped sample prefers magnetization along its length.
• Stress anisotropy (magnetoelastic): Induced by mechanical stress, coupling the magnetic and elastic degrees of freedom.

Anisotropy determines the easy axis, the domain wall width, and the coercivity. It's a critical parameter in designing magnetic materials for specific applications.

Magnetostriction

Magnetostriction is the change in a material's physical dimensions when it's magnetized. As domains reorient in an applied field, the lattice distorts slightly.

• Joule magnetostriction: Change in length parallel to the applied field
• Volume magnetostriction: Change in overall volume

Typical magnetostrictive strains are on the order of $10^{-5}$ to $10^{-3}$. Terfenol-D (a rare-earth alloy) has particularly large magnetostriction and is used in sensors, actuators, and sonar transducers.

Experimental techniques

These are the main methods used to probe magnetic properties and validate the theoretical models discussed above.

Magnetometry methods

Vibrating Sample Magnetometer (VSM): A sample vibrates in a uniform field, and the changing flux induces a voltage in pickup coils (Faraday's law). Sensitivity is around $10^{-6}$ emu. Workhorse technique for measuring $M(H)$ and $M(T)$ curves.

SQUID Magnetometer: Uses superconducting loops with Josephson junctions to detect tiny changes in magnetic flux. Sensitivity reaches $\sim 10^{-15}$ T, making it the most sensitive magnetometry technique available. Essential for studying weakly magnetic samples.

Alternating Gradient Magnetometer (AGM): Measures the force on a sample in an oscillating field gradient. Fast and sensitive, particularly suited for thin films and small samples.

Neutron scattering

Neutrons carry a magnetic moment but no charge, making them ideal probes of magnetic structure.

• Elastic neutron scattering: Reveals magnetic ordering patterns. Magnetic Bragg peaks tell you the magnetic unit cell, moment directions, and sublattice structure.
• Inelastic neutron scattering: Measures spin wave dispersions and other magnetic excitations. This is how exchange constants and anisotropy parameters are experimentally determined.
• Polarized neutron scattering: Separates magnetic from nuclear contributions to the scattering cross section.

Magnetic resonance techniques

Nuclear Magnetic Resonance (NMR): Probes the local magnetic field at nuclear sites. The resonance condition is $\omega = \gamma B$, where $\gamma$ is the nuclear gyromagnetic ratio. Provides information on local electronic and magnetic environments.

Electron Spin Resonance (ESR/EPR): Studies unpaired electrons in paramagnetic materials. Typically operates at microwave frequencies (around 9-10 GHz for X-band). Reveals g-factors, hyperfine interactions, and local symmetry.

Ferromagnetic Resonance (FMR): Probes collective precession of magnetization in ferromagnets. Gives information on magnetic anisotropy, exchange stiffness, and damping (Gilbert damping parameter $\alpha$).

Applications of magnetism

Magnetic materials in technology

• Permanent magnets: NdFeB and SmCo rare-earth magnets are used in motors, generators, and speakers. Ferrite magnets serve lower-cost applications.
• Soft magnetic materials: Silicon steel (transformer cores) and permalloy (magnetic shielding) have low coercivity and high permeability.
• Magnetic recording media: Hard disk drives use thin magnetic films; magnetic tape remains common for archival storage.
• Magnetic sensors: Hall effect sensors detect position and current; magnetoresistive sensors are used in navigation and automotive systems.
• Magnetic levitation: Maglev trains use superconducting or permanent magnets for frictionless transport.

Spintronics

Spintronics exploits the electron's spin degree of freedom for information processing.

Giant Magnetoresistance (GMR): The electrical resistance of a multilayer structure (alternating ferromagnetic and non-magnetic layers) changes significantly depending on whether adjacent ferromagnetic layers are aligned parallel or antiparallel. Discovered by Fert and Grünberg (2007 Nobel Prize), GMR enabled modern hard drive read heads.

Tunnel Magnetoresistance (TMR): Spin-dependent tunneling through a thin insulating barrier (e.g., MgO) produces even larger resistance changes than GMR. TMR is the basis for Magnetic Random Access Memory (MRAM).

Spin-transfer torque (STT): A spin-polarized current can flip the magnetization of a thin layer, enabling efficient writing in STT-MRAM devices.

Spin-orbit torque (SOT): Uses spin-orbit coupling to switch magnetization, promising faster operation and lower power consumption than STT.

Magnetic data storage

• Hard Disk Drives (HDD): Data stored on rotating magnetic disks using perpendicular magnetic recording. Areal densities exceed 1 Tb/in². Read heads use GMR or TMR sensors.
• Magnetic tape: High capacity and low cost per bit. Used for archival storage and backup.
• MRAM: Non-volatile memory combining the speed of SRAM with the persistence of flash. STT-MRAM offers improved scalability and energy efficiency compared to earlier toggle MRAM designs.

Thermodynamics of magnetic systems

Thermodynamics connects the magnetic degrees of freedom to measurable quantities like entropy, heat capacity, and temperature changes during magnetization processes.

Magnetic entropy

The magnetic entropy $S_{\text{mag}}$ measures the disorder in the spin system. For a system of $N$ non-interacting spin-1/2 moments, the maximum magnetic entropy is $Nk_B \ln 2$ (when spins are fully disordered).

The key thermodynamic relation connecting magnetization and entropy is the Maxwell relation:

$\left(\frac{\partial S}{\partial H}\right)_T = \left(\frac{\partial M}{\partial T}\right)_H$

This relation is central to understanding the magnetocaloric effect: it tells you that a material whose magnetization changes rapidly with temperature will also have a large entropy change when you apply or remove a field.

Magnetocaloric effect

The magnetocaloric effect (MCE) is the temperature change a magnetic material undergoes when the applied field changes. The physics is straightforward:

• Applying a field aligns spins, reducing magnetic entropy
• If the process is adiabatic (total entropy constant), the lattice entropy must increase to compensate, so the material heats up
• Removing the field reverses the process: spins disorder, magnetic entropy increases, and the material cools

Materials with a giant magnetocaloric effect (large entropy change near a first-order magnetic transition) are promising for magnetic refrigeration. Examples include Gd$_5$(Si$_2$Ge$_2$) and La(Fe,Si)$_{13}$ compounds.

Adiabatic demagnetization

Adiabatic demagnetization is a cooling technique that exploits the magnetocaloric effect to reach very low temperatures. The process has three steps:

1. Isothermal magnetization: Apply a strong magnetic field while the sample is in thermal contact with a heat bath. The spins align and the released heat flows into the bath.
2. Thermal isolation: Disconnect the sample from the heat bath (adiabatic conditions).
3. Slow demagnetization: Gradually reduce the field. The spins disorder, absorbing energy from the lattice, and the temperature drops.

Using paramagnetic salts like cerium magnesium nitrate (CMN), temperatures below 1 mK can be reached. Nuclear demagnetization pushes even further, achieving microkelvin temperatures by exploiting the much smaller nuclear magnetic moments.