Magnetism is a fascinating phenomenon that emerges from the quantum properties of electrons. Statistical mechanics provides a framework to understand how microscopic magnetic moments collectively give rise to macroscopic magnetic behavior in materials.

This topic explores the fundamentals of magnetism, from basic concepts like magnetic fields to more complex ideas like exchange interactions and magnetic ordering. We'll examine different types of magnetism and dive into the statistical mechanics of magnetic systems.

Fundamentals of magnetism

• Statistical mechanics provides a framework for understanding the collective behavior of magnetic systems at the microscopic level
• Magnetism emerges from the quantum mechanical properties of electrons and their interactions within materials
• Understanding the fundamentals of magnetism forms the basis for exploring more complex magnetic phenomena in statistical mechanics

Magnetic field and flux

• Magnetic field describes the region of influence around magnetic materials or moving electric charges
• Magnetic flux quantifies the amount of magnetic field passing through a given area
• Measured in units of tesla (T) for magnetic field strength and weber (Wb) for magnetic flux
• Magnetic field lines visualize the direction and strength of the magnetic field in space
• Relationship between magnetic field and flux given by the equation $\Phi = \mathbf{B} \cdot \mathbf{A}$

Magnetic dipole moment

• Fundamental quantity characterizing the magnetic strength and orientation of a magnetic dipole
• Represented by a vector pointing from the south pole to the north pole of a magnet
• Measured in units of ampere-square meter (A·m²) or joule per tesla (J/T)
• Torque experienced by a magnetic dipole in an external magnetic field given by $\boldsymbol{\tau} = \mathbf{m} \times \mathbf{B}$
• Potential energy of a magnetic dipole in an external magnetic field expressed as $U = -\mathbf{m} \cdot \mathbf{B}$

Diamagnetism vs paramagnetism

• Diamagnetism produces a weak magnetic field in opposition to an applied external magnetic field
  • Occurs in all materials but typically overshadowed by stronger magnetic effects
  • Results from the orbital motion of electrons in atoms
  • Examples include water, copper, and most organic compounds
• Paramagnetism generates a weak magnetic field aligned with an applied external magnetic field
  • Occurs in materials with unpaired electrons
  • Magnetic moments align partially with the external field
  • Examples include aluminum, platinum, and oxygen

Ferromagnetism and antiferromagnetism

• Ferromagnetism exhibits strong magnetic ordering with aligned magnetic moments
  • Spontaneous magnetization occurs even in the absence of an external field
  • Characterized by magnetic domains and hysteresis behavior
  • Examples include iron, cobalt, and nickel
• Antiferromagnetism displays magnetic ordering with alternating alignment of magnetic moments
  • Net magnetization is zero in the absence of an external field
  • Néel temperature marks the transition between antiferromagnetic and paramagnetic states
  • Examples include chromium, manganese oxide, and hematite

Microscopic origins of magnetism

• Statistical mechanics explores the collective behavior of magnetic moments at the atomic and molecular level
• Understanding microscopic origins helps explain macroscopic magnetic properties observed in materials
• Quantum mechanical effects play a crucial role in determining magnetic behavior at the microscopic scale

Electron spin and orbital motion

• Electron spin contributes to the intrinsic magnetic moment of an electron
  • Quantum mechanical property with no classical analogue
  • Spin angular momentum characterized by spin quantum number s = 1/2
• Orbital motion of electrons around the nucleus generates a magnetic field
  • Orbital angular momentum quantized by orbital quantum number l
  • Contributes to the total magnetic moment of an atom
• Total angular momentum J results from the coupling of spin and orbital angular momenta
  • Determines the magnetic properties of atoms and ions in materials

Exchange interaction

• Quantum mechanical effect responsible for magnetic ordering in materials
• Arises from the interplay between electrostatic repulsion and Pauli exclusion principle
• Direct exchange occurs between neighboring magnetic moments
• Indirect exchange mediated by conduction electrons in metals (RKKY interaction)
• Strength and sign of exchange interaction determine magnetic ordering type (ferromagnetic, antiferromagnetic)

Magnetic domains

• Regions within a ferromagnetic material where magnetic moments align in the same direction
• Formation of domains minimizes the overall magnetic energy of the system
• Domain walls separate regions of different magnetization directions
  • Bloch walls involve rotation of magnetization perpendicular to the wall
  • Néel walls involve rotation of magnetization parallel to the wall
• Domain structure influences macroscopic magnetic properties and hysteresis behavior

Curie temperature

• Critical temperature above which ferromagnetic materials transition to a paramagnetic state
• Thermal energy overcomes exchange interaction, leading to disordered magnetic moments
• Characterized by a second-order phase transition in the magnetization
• Curie-Weiss law describes magnetic susceptibility above the Curie temperature
  • $\chi = \frac{C}{T - T_C}$, where C is the Curie constant and T_C is the Curie temperature

Statistical mechanics of magnetic systems

• Applies principles of statistical physics to understand the collective behavior of magnetic moments
• Enables the prediction of macroscopic properties from microscopic interactions
• Provides a framework for studying phase transitions and critical phenomena in magnetic systems

Ising model

• Simplified model of ferromagnetism in statistical mechanics
• Consists of discrete magnetic moments (spins) arranged on a lattice
• Each spin can have two states: up (+1) or down (-1)
• Hamiltonian of the Ising model given by $H = -J\sum_{<i,j>} s_i s_j - h\sum_i s_i$
  • J represents the exchange interaction strength
  • h denotes the external magnetic field
• Exhibits a phase transition between ordered and disordered states in dimensions d ≥ 2

Mean field theory

• Approximation method for studying many-body systems in statistical mechanics
• Replaces fluctuating interactions with an average effective field
• Simplifies the treatment of complex magnetic systems
• Mean field approximation for the Ising model:
  • Effective field $h_{eff} = h + zJm$, where z is the coordination number and m is the average magnetization
  • Self-consistent equation for magnetization: $m = \tanh(\beta(h + zJm))$
• Provides qualitative insights but often overestimates critical temperatures

Spin waves

• Collective excitations of magnetic moments in ordered magnetic systems
• Analogous to phonons in lattice vibrations
• Dispersion relation for ferromagnetic spin waves: $\omega(k) = Dk^2$, where D is the spin wave stiffness
• Contribute to the low-temperature specific heat and magnetization of ferromagnets
• Magnons represent quantized spin waves in the quantum mechanical description

Critical phenomena in magnets

• Study of behavior near phase transitions in magnetic systems
• Characterized by power-law divergences of physical quantities
• Critical exponents describe the scaling behavior near the critical point
  • Magnetization: $m \sim |t|^\beta$
  • Susceptibility: $\chi \sim |t|^{-\gamma}$
  • Correlation length: $\xi \sim |t|^{-\nu}$
  • t = (T - T_c) / T_c is the reduced temperature
• Universality classes group systems with similar critical behavior
• Renormalization group techniques provide a powerful framework for studying critical phenomena

Quantum theory of magnetism

• Incorporates quantum mechanical principles to describe magnetic behavior at the atomic and molecular level
• Explains phenomena not accounted for by classical theories of magnetism
• Provides a foundation for understanding complex magnetic materials and quantum magnets

Heisenberg model

• Quantum mechanical model of magnetism that considers vector spin operators
• Hamiltonian given by $H = -J\sum_{<i,j>} \mathbf{S}_i \cdot \mathbf{S}_j - h\sum_i S_i^z$
  • $\mathbf{S}_i$ represents the vector spin operator at site i
  • J is the exchange interaction strength
  • h denotes the external magnetic field
• Allows for continuous rotations of spins in three-dimensional space
• Reduces to the Ising model in the limit of strong anisotropy

Spin-1/2 systems

• Simplest quantum mechanical spin systems with two possible states
• Described by Pauli matrices: $\sigma_x = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \sigma_y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \sigma_z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$
• Spin operators given by $\mathbf{S} = \frac{\hbar}{2}\boldsymbol{\sigma}$
• Exhibit quantum phenomena such as superposition and entanglement
• Serve as building blocks for quantum computing and spintronics applications

Quantum phase transitions

• Phase transitions driven by quantum fluctuations at zero temperature
• Occur when varying a non-thermal parameter (magnetic field, pressure) in the ground state
• Characterized by a quantum critical point separating different quantum phases
• Quantum Ising model in a transverse field exhibits a quantum phase transition
  • Hamiltonian: $H = -J\sum_{<i,j>} \sigma_i^z \sigma_j^z - h\sum_i \sigma_i^x$
  • Critical point occurs at $h_c = J$ in one dimension
• Quantum critical behavior influences finite-temperature properties in the quantum critical region

Magnetic ordering

• Describes the arrangement of magnetic moments in materials
• Results from the interplay between various interactions and thermal fluctuations
• Different types of magnetic ordering lead to distinct macroscopic magnetic properties

Spontaneous magnetization

• Emergence of a net magnetic moment in ferromagnetic materials below the Curie temperature
• Occurs without the application of an external magnetic field
• Results from the alignment of magnetic moments due to exchange interactions
• Magnetization curve M(H) exhibits hysteresis in ferromagnetic materials
  • Remanent magnetization: residual magnetization when external field is removed
  • Coercive field: field required to reduce magnetization to zero

Magnetic anisotropy

• Preference for magnetic moments to align along specific crystallographic directions
• Types of magnetic anisotropy:
  • Magnetocrystalline anisotropy: due to spin-orbit coupling and crystal field effects
  • Shape anisotropy: arises from the demagnetizing field in non-spherical samples
  • Stress anisotropy: induced by applied or residual mechanical stress
• Energy associated with magnetic anisotropy often expressed as $E = K\sin^2\theta$
  • K is the anisotropy constant
  • θ is the angle between the magnetization and the easy axis

Magnetostriction

• Change in physical dimensions of a material in response to an applied magnetic field
• Caused by the reorientation of magnetic domains
• Joule magnetostriction: change in length parallel to the applied field
• Volume magnetostriction: change in volume of the material
• Magnetostrictive strain typically on the order of 10^-5 to 10^-3
• Applications in sensors, actuators, and energy harvesting devices

Experimental techniques

• Methods used to investigate magnetic properties and structures of materials
• Provide insights into microscopic and macroscopic magnetic behavior
• Essential for validating theoretical models and discovering new magnetic phenomena

Magnetometry methods

• Techniques for measuring magnetic properties of materials
• Vibrating Sample Magnetometer (VSM):
  • Measures magnetic moment by vibrating a sample in a uniform magnetic field
  • Based on Faraday's law of electromagnetic induction
  • Sensitivity on the order of 10^-6 emu
• Superconducting Quantum Interference Device (SQUID) magnetometer:
  • Utilizes superconducting loops and Josephson junctions
  • Extremely sensitive, capable of detecting magnetic fields as small as 10^-15 T
  • Measures magnetic flux through a superconducting loop
• Alternating Gradient Magnetometer (AGM):
  • Measures force on a sample in an alternating magnetic field gradient
  • High sensitivity and fast measurement times
  • Suitable for thin films and small samples

Neutron scattering

• Powerful technique for probing magnetic structures and excitations
• Neutrons interact with unpaired electron spins in materials
• Elastic neutron scattering:
  • Reveals magnetic ordering and spin structures
  • Bragg peaks provide information on magnetic unit cell and moment directions
• Inelastic neutron scattering:
  • Measures magnetic excitations such as spin waves and magnons
  • Provides information on exchange interactions and magnetic anisotropy
• Polarized neutron scattering allows separation of magnetic and nuclear scattering

Magnetic resonance techniques

• Methods based on the interaction of magnetic moments with electromagnetic radiation
• Nuclear Magnetic Resonance (NMR):
  • Probes local magnetic fields experienced by atomic nuclei
  • Provides information on electronic and magnetic structure of materials
  • Resonance condition: $\omega = \gamma B$, where γ is the gyromagnetic ratio
• Electron Spin Resonance (ESR) or Electron Paramagnetic Resonance (EPR):
  • Studies unpaired electrons in paramagnetic materials
  • Reveals information about electronic structure and magnetic interactions
  • Typically operates at microwave frequencies (9-10 GHz for X-band ESR)
• Ferromagnetic Resonance (FMR):
  • Investigates collective magnetic excitations in ferromagnetic materials
  • Provides information on magnetic anisotropy and damping mechanisms
  • Resonance condition influenced by shape anisotropy and demagnetizing fields

Applications of magnetism

• Utilizes magnetic properties and phenomena for practical purposes
• Spans a wide range of technological and scientific fields
• Continues to drive innovation in various industries and research areas

Magnetic materials in technology

• Permanent magnets:
  • Used in electric motors, generators, and speakers
  • Materials include rare-earth magnets (NdFeB, SmCo) and ferrites
• Soft magnetic materials:
  • Used in transformers, inductors, and magnetic shielding
  • Examples include silicon steel and permalloy
• Magnetic recording media:
  • Hard disk drives utilize magnetic thin films for data storage
  • Magnetic tape still used for archival storage
• Magnetic sensors:
  • Hall effect sensors for position and current sensing
  • Magnetoresistive sensors for navigation and automotive applications
• Magnetic levitation:
  • Used in high-speed trains (maglev) and frictionless bearings
  • Employs superconducting magnets or permanent magnets

Spintronics

• Utilizes electron spin for information processing and storage
• Giant Magnetoresistance (GMR):
  • Significant change in electrical resistance with applied magnetic field
  • Used in read heads of hard disk drives
  • Discovered by Albert Fert and Peter Grünberg (Nobel Prize 2007)
• Tunnel Magnetoresistance (TMR):
  • Based on spin-dependent tunneling through a thin insulating barrier
  • Higher magnetoresistance ratio compared to GMR
  • Used in Magnetic Random Access Memory (MRAM)
• Spin-transfer torque:
  • Manipulation of magnetization using spin-polarized currents
  • Enables efficient writing in MRAM devices
• Spin-orbit torque:
  • Utilizes spin-orbit coupling for magnetization switching
  • Promises lower power consumption and faster operation

Magnetic data storage

• Hard Disk Drives (HDD):
  • Store data on rotating magnetic disks
  • Use GMR or TMR read heads and inductive write heads
  • Areal density exceeding 1 Tb/in^2 achieved through perpendicular magnetic recording
• Magnetic tape:
  • Used for archival storage and backup systems
  • High capacity and low cost per bit
  • Employs particulate or thin film magnetic coatings
• Magnetic Random Access Memory (MRAM):
  • Non-volatile memory technology based on magnetic tunnel junctions
  • Combines speed of SRAM with non-volatility of flash memory
  • Spin-transfer torque MRAM (STT-MRAM) offers improved scalability and energy efficiency

Thermodynamics of magnetic systems

• Applies principles of thermodynamics to magnetic materials and phenomena
• Explores the relationship between magnetic properties and thermodynamic variables
• Provides insights into energy conversion and cooling processes involving magnetic materials

Magnetic entropy

• Measure of disorder in the magnetic system
• Contributes to the total entropy of magnetic materials
• Magnetic entropy change associated with ordering/disordering of magnetic moments
• Entropy change during magnetization process given by Maxwell relation:
  • $\left(\frac{\partial S}{\partial H}\right)_T = \left(\frac{\partial M}{\partial T}\right)_H$
• Plays a crucial role in magnetocaloric effect and magnetic refrigeration

Magnetocaloric effect

• Temperature change of a magnetic material when subjected to a changing magnetic field
• Based on the coupling between magnetic and lattice degrees of freedom
• Adiabatic magnetization leads to heating of the material
• Adiabatic demagnetization results in cooling of the material
• Magnetic entropy change and adiabatic temperature change characterize the magnetocaloric effect
• Giant magnetocaloric effect observed in materials near first-order magnetic phase transitions
  • Examples include Gd5(Si2Ge2) and La(Fe,Si)13 compounds

Adiabatic demagnetization

• Cooling technique based on the magnetocaloric effect
• Process:
  1. Magnetize the material isothermally, removing heat to maintain constant temperature
  2. Thermally isolate the material (adiabatic conditions)
  3. Slowly reduce the magnetic field, causing the material to cool
• Effective for reaching very low temperatures (below 1 K)
• Used in low-temperature physics research and specialized cooling applications
• Paramagnetic salts (cerium magnesium nitrate) used for cooling to millikelvin temperatures
• Nuclear demagnetization can achieve temperatures in the microkelvin range