🔬Condensed Matter Physics Unit 5 – Magnetism and Magnetic Materials

Fundamentals of Magnetism

• Magnetism arises from the motion and spin of electrons in atoms and their interactions with each other
• Magnetic dipole moments are the fundamental building blocks of magnetism and represent the strength and orientation of a magnetic field source
• Magnetic fields are vector fields that describe the force experienced by a moving charge or magnetic dipole at any point in space
• Magnetic fields can be generated by electric currents (Ampère's law) and changing electric fields (Faraday's law)
• Magnetic fields exert forces on moving charges (Lorentz force) and other magnetic dipoles, leading to attraction or repulsion
  • The Lorentz force is given by $\vec{F} = q\vec{v} \times \vec{B}$, where $q$ is the charge, $\vec{v}$ is the velocity, and $\vec{B}$ is the magnetic field
• Magnetic fields are characterized by field lines, which represent the direction and strength of the field at each point
• Magnetic monopoles, isolated north or south poles, have not been observed in nature, unlike electric monopoles (individual positive or negative charges)

Magnetic Field and Flux

• Magnetic flux ($\Phi_B$) is a scalar quantity that measures the total magnetic field passing through a surface
  • Mathematically, magnetic flux is defined as the surface integral of the normal component of the magnetic field over a given area: $\Phi_B = \int \vec{B} \cdot d\vec{A}$
• Gauss's law for magnetism states that the net magnetic flux through any closed surface is always zero, implying that magnetic field lines always form closed loops
• Faraday's law of induction describes how a changing magnetic flux induces an electromotive force (EMF) in a conductor
  • The induced EMF is given by $\mathcal{E} = -\frac{d\Phi_B}{dt}$, where $\Phi_B$ is the magnetic flux and $t$ is time
• Lenz's law states that the direction of the induced EMF is such that it opposes the change in magnetic flux that caused it
• The Biot-Savart law relates the magnetic field generated by a current-carrying conductor to the current and the geometry of the conductor
  • The magnetic field at a point due to a current element is given by $d\vec{B} = \frac{\mu_0}{4\pi} \frac{I d\vec{l} \times \hat{r}}{r^2}$, where $\mu_0$ is the permeability of free space, $I$ is the current, $d\vec{l}$ is the current element, and $\hat{r}$ is the unit vector pointing from the current element to the point of interest
• Ampère's circuital law relates the magnetic field around a closed loop to the electric current passing through the loop
  • In its integral form, Ampère's law is given by $\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{enc}$, where $I_{enc}$ is the total current enclosed by the loop

Types of Magnetic Materials

• Diamagnetic materials have a weak, negative magnetic susceptibility and are slightly repelled by magnetic fields
  • Diamagnetism arises from the induced magnetic moments that oppose the applied field, following Lenz's law
  • Examples of diamagnetic materials include water, copper, and bismuth
• Paramagnetic materials have a small, positive magnetic susceptibility and are weakly attracted to magnetic fields
  • Paramagnetism occurs due to the alignment of magnetic moments with the applied field, but thermal agitation disrupts perfect alignment
  • Examples of paramagnetic materials include aluminum, platinum, and oxygen gas
• Ferromagnetic materials have a large, positive magnetic susceptibility and exhibit strong attraction to magnetic fields
  • Ferromagnetism arises from the alignment of magnetic moments in domains, leading to a net magnetization even in the absence of an applied field
  • Examples of ferromagnetic materials include iron, nickel, and cobalt
• Antiferromagnetic materials have magnetic moments that align antiparallel to each other, resulting in a net zero magnetization
  • Antiferromagnetism occurs due to the exchange interaction between neighboring magnetic moments, favoring antiparallel alignment
  • Examples of antiferromagnetic materials include chromium, manganese oxide, and nickel oxide
• Ferrimagnetic materials have magnetic moments that align antiparallel but with unequal magnitudes, resulting in a net magnetization
  • Ferrimagnetism is observed in compounds with different types of magnetic ions occupying distinct sublattices
  • Examples of ferrimagnetic materials include magnetite (Fe3O4) and yttrium iron garnet (YIG)

Magnetic Domains and Hysteresis

• Magnetic domains are regions within a ferromagnetic or ferrimagnetic material where the magnetic moments are aligned in the same direction
• Domain walls are the boundaries between adjacent magnetic domains, across which the orientation of the magnetic moments gradually changes
• The formation of magnetic domains minimizes the overall magnetic energy of the system, including exchange energy, magnetocrystalline anisotropy energy, and magnetostatic energy
• Magnetization processes involve the motion of domain walls and the rotation of magnetic moments within domains in response to an applied magnetic field
• Magnetic hysteresis is the dependence of a material's magnetization on its magnetic history, characterized by a hysteresis loop
  • The hysteresis loop plots the magnetization (M) versus the applied magnetic field (H) and exhibits key features such as saturation magnetization, remanent magnetization, and coercivity
• Saturation magnetization (Ms) is the maximum magnetization achieved when all magnetic moments are aligned with the applied field
• Remanent magnetization (Mr) is the remaining magnetization when the applied field is removed after saturation
• Coercivity (Hc) is the magnitude of the reverse magnetic field required to reduce the magnetization to zero after saturation
• The area enclosed by the hysteresis loop represents the energy dissipated during a complete magnetization cycle, which is related to magnetic losses and determines the suitability of a material for specific applications

Quantum Origins of Magnetism

• Magnetism has its origins in the quantum mechanical properties of electrons, specifically their intrinsic spin and orbital angular momentum
• The Bohr magneton ($\mu_B$) is the fundamental unit of magnetic moment for an electron, given by $\mu_B = \frac{e\hbar}{2m_e}$, where $e$ is the electron charge, $\hbar$ is the reduced Planck's constant, and $m_e$ is the electron mass
• The magnetic moment of an atom is determined by the vector sum of the spin and orbital magnetic moments of its electrons, following Hund's rules
  • Hund's first rule: Electrons maximize their total spin angular momentum (S) while obeying the Pauli exclusion principle
  • Hund's second rule: Electrons maximize their total orbital angular momentum (L) while maintaining the total spin angular momentum
  • Hund's third rule: The total angular momentum (J) is given by $J = |L - S|$ for less than half-filled shells and $J = L + S$ for more than half-filled shells
• The Zeeman effect describes the splitting of atomic energy levels in the presence of an external magnetic field due to the interaction between the magnetic field and the magnetic moments of the electrons
• The Stern-Gerlach experiment demonstrated the quantization of spin angular momentum and the existence of intrinsic electron spin
• The exchange interaction is a quantum mechanical effect that arises from the Coulomb interaction and the Pauli exclusion principle, leading to the alignment of magnetic moments in ferromagnetic and antiferromagnetic materials
  • The Heisenberg Hamiltonian describes the exchange interaction between localized spins: $H = -\sum_{i,j} J_{ij} \vec{S}_i \cdot \vec{S}_j$, where $J_{ij}$ is the exchange constant and $\vec{S}_i$ and $\vec{S}_j$ are the spin operators for atoms $i$ and $j$

Magnetic Ordering and Phase Transitions

• Magnetic ordering refers to the long-range, cooperative alignment of magnetic moments in a material below a critical temperature
• The exchange interaction between magnetic moments leads to different types of magnetic ordering, such as ferromagnetism, antiferromagnetism, and ferrimagnetism
• The Curie temperature (Tc) is the critical temperature above which a ferromagnetic material becomes paramagnetic
  • At the Curie temperature, thermal fluctuations overcome the exchange interaction, disrupting the long-range magnetic order
• The Néel temperature (TN) is the critical temperature above which an antiferromagnetic material becomes paramagnetic
• Magnetic phase transitions occur when a material undergoes a change in its magnetic ordering as a function of temperature, magnetic field, or other external parameters
• The order parameter for a magnetic phase transition is the magnetization (M), which is zero in the high-temperature paramagnetic phase and non-zero in the low-temperature ordered phase
• Landau's theory of phase transitions describes the behavior of the order parameter near a continuous phase transition using a phenomenological free energy expansion
• Critical exponents characterize the power-law behavior of various thermodynamic quantities near a continuous phase transition, such as the magnetization, susceptibility, and specific heat
• The Ising model is a simple statistical mechanical model that captures the essential features of magnetic phase transitions and provides insights into critical behavior
  • In the Ising model, spins are represented as discrete variables ($\sigma_i = \pm 1$) on a lattice, and the energy of the system is determined by the interactions between neighboring spins and an external magnetic field

Applications in Condensed Matter Systems

• Magnetic materials play a crucial role in various technological applications, such as data storage, sensors, and power generation
• Hard disk drives (HDDs) utilize the magnetization of thin ferromagnetic films to store and retrieve digital information
  • The read/write head of an HDD uses the giant magnetoresistance (GMR) effect or the tunneling magnetoresistance (TMR) effect to detect changes in the magnetic field associated with the stored data
• Magnetic random access memory (MRAM) is a non-volatile memory technology that uses the magnetization of ferromagnetic elements to store information
  • MRAM offers fast read/write speeds, high endurance, and low power consumption compared to conventional memory technologies
• Magnetoresistive sensors, such as GMR and TMR sensors, exploit the change in electrical resistance of magnetic multilayers in response to external magnetic fields
  • These sensors find applications in position and motion detection, current sensing, and bio-magnetic field measurements
• Magnetoelectric coupling in multiferroic materials allows for the control of magnetic properties using electric fields and vice versa
  • Multiferroic materials exhibit coexisting ferroelectric and magnetic order, enabling the development of novel devices for information processing and sensing
• Magnetic nanoparticles and ferrofluids have diverse applications in biomedicine, including targeted drug delivery, hyperthermia treatment, and contrast enhancement in magnetic resonance imaging (MRI)
• Magnetocaloric materials exhibit a significant temperature change when exposed to a varying magnetic field, making them promising for solid-state refrigeration and heat pumps
  • The magnetocaloric effect is based on the entropy change associated with the alignment and misalignment of magnetic moments in response to an applied field

Advanced Topics and Current Research

• Spintronics is an emerging field that exploits the spin degree of freedom of electrons for information processing and storage
  • Spin-polarized currents and spin-dependent transport phenomena, such as spin injection and detection, form the basis of spintronic devices
  • Examples of spintronic devices include spin valves, magnetic tunnel junctions, and spin-transfer torque magnetic random access memory (STT-MRAM)
• Topological magnetic materials, such as magnetic skyrmions and chiral magnets, exhibit novel spin textures and transport properties arising from their non-trivial topology
  • Magnetic skyrmions are nanoscale, topologically protected spin configurations that can be manipulated using low-density currents, making them promising for energy-efficient data storage and processing
• Frustrated magnets are systems in which the competing interactions between magnetic moments cannot be simultaneously satisfied, leading to highly degenerate ground states and exotic magnetic phases
  • Examples of frustrated magnetic systems include spin ices, spin liquids, and kagome lattices
  • Frustrated magnets offer a platform for studying unconventional magnetic phenomena and the emergence of novel quasi-particles, such as magnetic monopoles and spinons
• Quantum magnetism explores the behavior of magnetic systems in the quantum regime, where quantum fluctuations and entanglement play a significant role
  • Quantum spin liquids are exotic states of matter that exhibit long-range entanglement and fractionalized excitations, such as spinons and visons
  • Quantum magnetism is closely connected to the study of high-temperature superconductivity, as many unconventional superconductors arise from doping antiferromagnetic Mott insulators
• Ultrafast magnetism investigates the dynamics of magnetic systems on femtosecond to picosecond timescales, using advanced experimental techniques such as time-resolved magneto-optical Kerr effect (TR-MOKE) and ultrafast X-ray spectroscopy
  • Ultrafast magnetization processes, such as demagnetization, spin-flip scattering, and spin transport, are crucial for the development of high-speed magnetic devices and the understanding of non-equilibrium magnetic phenomena
• Magnetic 2D materials, such as magnetic graphene and transition metal dichalcogenides (TMDs), exhibit unique magnetic and electronic properties that arise from their reduced dimensionality and symmetry
  • 2D magnets offer opportunities for studying fundamental magnetic interactions and developing novel spintronic devices, such as spin valves and spin-logic circuits, that can be integrated with existing 2D material platforms