London equations are the backbone of superconductivity theory, describing how these materials expel magnetic fields and conduct electricity without resistance. They bridge classical electromagnetism and quantum mechanics, providing a mathematical framework for the Meissner effect and zero resistance.

The equations relate supercurrent density to vector potential and magnetic field, explaining magnetic field expulsion and penetration depth. They're crucial for understanding superconductor behavior, from thin films to high-temperature materials, and form the basis for applications like SQUIDs and superconducting magnets.

Foundations of London equations

• London equations form the cornerstone of phenomenological theories in superconductivity, describing macroscopic electromagnetic properties of superconductors
• These equations provide crucial insights into the behavior of superconducting materials, bridging classical electromagnetism and quantum mechanics in condensed matter systems

Superconductivity basics

• Characterized by zero electrical resistance and perfect diamagnetism below a critical temperature (Tc)
• Exhibits Meissner effect, expelling magnetic fields from the superconductor's interior
• Occurs in various materials (metals, ceramics, organic compounds) under specific conditions
• Governed by quantum mechanical effects at the microscopic level

Two-fluid model

• Describes superconductors as a mixture of normal electrons and superconducting electrons
• Normal electrons behave like those in ordinary metals, contributing to resistivity
• Superconducting electrons form Cooper pairs, carrying current without resistance
• Ratio of normal to superconducting electrons depends on temperature
• Explains thermal and electromagnetic properties of superconductors

London brothers' contribution

• Fritz and Heinz London developed the equations in 1935 to explain superconductivity
• Proposed a phenomenological approach based on Maxwell's equations and quantum mechanics
• Introduced the concept of a superconducting electron fluid with quantum mechanical properties
• Provided a mathematical framework to describe Meissner effect and zero resistance

Mathematical formulation

First London equation

• Relates supercurrent density to vector potential: $\mathbf{J}_s = -\frac{n_s e^2}{m} \mathbf{A}$
• $n_s$ represents superconducting electron density
• $e$ denotes elementary charge
• $m$ stands for electron mass
• $\mathbf{A}$ is the magnetic vector potential
• Describes acceleration of superconducting electrons in an electric field

Second London equation

• Connects curl of supercurrent density to magnetic field: $\nabla \times \mathbf{J}_s = -\frac{n_s e^2}{m} \mathbf{B}$
• $\mathbf{B}$ represents the magnetic field
• Explains Meissner effect and magnetic field expulsion
• Leads to exponential decay of magnetic fields inside superconductors

Curl of current density

• Derived from the first London equation: $\nabla \times \mathbf{J}_s = -\frac{n_s e^2}{m} (\nabla \times \mathbf{A}) = -\frac{n_s e^2}{m} \mathbf{B}$
• Shows relationship between supercurrent and magnetic field
• Demonstrates how supercurrents generate screening currents to expel magnetic fields
• Crucial for understanding magnetic field behavior in superconductors

Physical implications

Meissner effect explanation

• London equations predict complete expulsion of magnetic fields from superconductor interior
• Supercurrents generate opposing magnetic fields, canceling external fields
• Occurs regardless of whether field is applied before or after cooling below Tc
• Distinguishes superconductors from perfect conductors

Magnetic field penetration depth

• Characteristic length scale for magnetic field decay inside superconductor
• Defined as $\lambda_L = \sqrt{\frac{m}{μ_0 n_s e^2}}$, where $μ_0$ is vacuum permeability
• Typically ranges from 10 to 100 nanometers for conventional superconductors
• Depends on material properties and temperature
• Crucial for understanding thin film superconductors and multilayer structures

Flux quantization

• Magnetic flux through a superconducting loop is quantized in units of $\Phi_0 = h/(2e)$
• $h$ represents Planck's constant
• Results from quantum mechanical nature of Cooper pairs
• Explains behavior of superconducting rings and SQUIDs (Superconducting Quantum Interference Devices)
• Provides evidence for charge carriers being paired electrons

Limitations and extensions

Validity in local limit

• London equations assume local relationship between current and field
• Accurate for superconductors with coherence length much smaller than penetration depth
• Breaks down in clean, low-temperature superconductors (non-local effects become important)
• Works well for type-II superconductors and dirty type-I superconductors

Pippard's non-local generalization

• Addresses limitations of London theory in clean superconductors
• Introduces non-local relationship between current and vector potential
• Incorporates coherence length as a fundamental parameter
• Explains anomalous skin effect in superconductors
• Bridges gap between microscopic BCS theory and London equations

Ginzburg-Landau theory vs London theory

• Ginzburg-Landau theory provides more comprehensive description of superconductors
• Introduces order parameter to describe superconducting state
• Applicable near critical temperature and in presence of strong fields
• Reduces to London equations in appropriate limits
• Explains type-I and type-II superconductors, including mixed state

Experimental verification

Magnetic field expulsion measurements

• Direct observation of Meissner effect using sensitive magnetometers
• Measurements of magnetic susceptibility show perfect diamagnetism
• Levitation experiments demonstrate field expulsion visually
• Scanning SQUID microscopy maps field distribution at superconductor surface

Penetration depth experiments

• Microwave cavity perturbation techniques measure penetration depth accurately
• Muon spin rotation (μSR) probes local magnetic field distribution
• Scanning tunneling spectroscopy maps spatial variation of superconducting gap
• Temperature dependence of penetration depth provides information on gap symmetry

Flux quantization observations

• Electron interference experiments in superconducting rings
• Direct imaging of fluxoids in type-II superconductors using magnetic decoration
• SQUID measurements of quantized flux in superconducting loops
• Josephson junction arrays demonstrate macroscopic quantum coherence

Applications in superconductivity

Josephson junctions

• Consist of two superconductors separated by thin insulating barrier
• Exhibit quantum tunneling of Cooper pairs
• Used in ultra-sensitive magnetometers and high-speed digital circuits
• Form basis for superconducting qubits in quantum computing

SQUID devices

• Superconducting Quantum Interference Devices measure extremely weak magnetic fields
• Combine Josephson junctions with superconducting loops
• Applications include biomagnetism (magnetoencephalography, magnetocardiography)
• Used in geophysical surveys and materials characterization

Superconducting magnets

• Generate strong, stable magnetic fields with minimal power consumption
• Used in MRI machines, particle accelerators, and fusion reactors
• Employ type-II superconductors (NbTi, Nb3Sn) to achieve high critical fields
• Require cryogenic cooling systems to maintain superconducting state

London equations in different geometries

Thin films

• London equations modified to account for reduced dimensionality
• Penetration depth can exceed film thickness, leading to different field distributions
• Important for superconducting electronics and quantum computing applications
• Proximity effects between superconducting and normal layers become significant

Cylindrical geometries

• Relevant for superconducting wires and cables
• London equations solved in cylindrical coordinates
• Field and current distributions depend on radius and applied field orientation
• Critical for understanding ac losses in superconducting power transmission lines

Spherical superconductors

• London equations applied to spherical symmetry
• Useful for studying superconducting nanoparticles and grains
• Meissner state and vortex configurations differ from bulk superconductors
• Provides insights into size effects on superconductivity

Connection to microscopic theory

BCS theory relationship

• London equations emerge as long-wavelength limit of BCS theory
• BCS provides microscopic justification for London equations' phenomenological approach
• Relates London penetration depth to microscopic parameters (energy gap, Fermi velocity)
• Explains temperature dependence of penetration depth and critical field

Quasiparticle excitations

• London equations describe superfluid component, neglecting quasiparticle contributions
• Quasiparticles affect electromagnetic response at finite temperatures
• Two-fluid model incorporates both superfluid and normal fluid components
• Important for understanding thermodynamic and transport properties of superconductors

Cooper pairs and London equations

• London equations describe collective behavior of Cooper pairs
• Cooper pair wavefunction relates to superconducting order parameter
• London equations capture macroscopic quantum coherence of Cooper pairs
• Explain flux quantization and Josephson effects in terms of Cooper pair dynamics

Numerical methods

Finite element analysis

• Solves London equations in complex geometries
• Used to model field and current distributions in superconducting devices
• Incorporates material properties and boundary conditions
• Enables optimization of superconductor shapes for specific applications

Computational challenges

• Multiscale nature of superconductivity requires careful meshing strategies
• Nonlinear behavior near critical current complicates simulations
• Time-dependent problems (ac losses, flux dynamics) demand efficient algorithms
• Coupling between electromagnetic and thermal problems in practical devices

Simulation of superconducting systems

• Combines London equations with Maxwell's equations for complete electromagnetic analysis
• Incorporates material models for critical current density and flux pinning
• Used to design superconducting magnets, RF cavities, and quantum circuits
• Enables virtual prototyping of superconducting devices, reducing development costs

London equations in modern research

High-temperature superconductors

• London theory extended to describe anisotropic and layered superconductors
• Modified penetration depth tensor accounts for crystalline anisotropy
• Explains unusual electromagnetic properties of cuprate and iron-based superconductors
• Challenges arise from strong thermal fluctuations and short coherence lengths

Unconventional superconductivity

• London equations adapted for non-s-wave pairing symmetries (d-wave, p-wave)
• Incorporates nodal structure of superconducting gap
• Describes exotic vortex states in unconventional superconductors
• Provides framework for studying topological aspects of superconductivity

Topological superconductors

• London equations modified to include Berry curvature effects
• Describes electromagnetic response of topological superconductors
• Predicts novel phenomena like quantized magnetoelectric effect
• Relevant for potential applications in topological quantum computing