Glossary

6.2 London equations

6 min read•august 21, 2024

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 and zero resistance.

The equations relate to and , explaining and . They're crucial for understanding superconductor behavior, from to high-temperature materials, and form the basis for applications like and .

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

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  • Characterized by zero electrical resistance and perfect diamagnetism below a (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 , 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 developed the equations in 1935 to explain superconductivity
  • Proposed a phenomenological approach based on 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: Js=−nse2mA\mathbf{J}_s = -\frac{n_s e^2}{m} \mathbf{A}
  • nsn_s represents superconducting electron density
  • ee denotes elementary charge
  • mm stands for electron mass
  • A\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: ∇×Js=−nse2mB\nabla \times \mathbf{J}_s = -\frac{n_s e^2}{m} \mathbf{B}
  • 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: ∇×Js=−nse2m(∇×A)=−nse2mB\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 λL=mμ0nse2\lambda_L = \sqrt{\frac{m}{μ_0 n_s e^2}}, where μ0μ_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 Φ0=h/(2e)\Phi_0 = h/(2e)
  • hh 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 and London equations

Ginzburg-Landau theory vs London 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 show perfect diamagnetism
  • Levitation experiments demonstrate field expulsion visually
  • Scanning SQUID microscopy maps field distribution at superconductor surface

Penetration depth experiments

  • techniques measure penetration depth accurately
  • (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 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
  • 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 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
  • Predicts novel phenomena like quantized magnetoelectric effect
  • Relevant for potential applications in topological quantum computing

Key Terms to Review (33)

BCS theory: BCS theory, named after its developers John Bardeen, Leon Cooper, and Robert Schrieffer, explains the phenomenon of superconductivity in certain materials at low temperatures. The theory describes how electrons form pairs, called Cooper pairs, that move through a lattice structure without scattering, which results in zero electrical resistance. This pairing and the resulting ground state are crucial for understanding various superconducting properties, including the Meissner effect and the formation of Type I and Type II superconductors.
Cooper pairs: Cooper pairs are pairs of electrons that are bound together at low temperatures in a superconductor, leading to the phenomenon of superconductivity. This pairing occurs due to attractive interactions mediated by lattice vibrations, allowing the electrons to move through the material without resistance. The concept of Cooper pairs is essential for understanding the underlying mechanisms of superconductivity and plays a crucial role in various effects observed in superconductors.
Critical Temperature: Critical temperature is the temperature above which a material cannot maintain a particular state, such as superconductivity or magnetism, regardless of pressure. It plays a crucial role in understanding phase transitions and the behaviors of materials as they approach these limits, influencing phenomena like superconductivity and structural transformations.
Cylindrical geometries: Cylindrical geometries refer to physical systems and structures that have a cylindrical shape, which significantly influences their physical properties and behaviors. In the context of superconductivity and the London equations, cylindrical geometries are important as they describe how magnetic fields interact with superconducting materials when shaped like cylinders, affecting the penetration of magnetic fields and the establishment of screening currents within these materials.
Finite element analysis: Finite element analysis (FEA) is a numerical method used to solve complex problems in engineering and physics by breaking down large systems into smaller, simpler parts called finite elements. This technique allows for detailed examination of the behavior of materials and structures under various conditions, making it particularly useful in the study of superconductors and the application of London equations, where electromagnetic properties need to be analyzed accurately.
Flux quantization: Flux quantization refers to the phenomenon where the magnetic flux passing through a superconducting loop can only take on discrete values, which are integer multiples of a fundamental unit of magnetic flux called the flux quantum. This behavior is intimately related to the quantum mechanical properties of superconductors and is essential in understanding how magnetic fields interact with superconducting materials, influencing concepts like the London equations and the Josephson effect.
Fritz London: Fritz London was a prominent physicist known for his foundational contributions to the understanding of superconductivity and quantum liquids. His work laid the groundwork for the London equations, which describe the electromagnetic properties of superconductors, and he was also instrumental in advancing the nearly free electron model, which provides insight into electron behavior in solids.
Ginzburg-Landau Theory: Ginzburg-Landau Theory is a theoretical framework used to describe superconductivity and phase transitions, focusing on the concept of an order parameter to characterize the state of a system. This theory connects critical phenomena, such as the Meissner effect and type I and type II superconductors, through a macroscopic wave function that represents the density of Cooper pairs in the material.
Heinz London: Heinz London refers to the physicists Fritz London and his brother Heinz London, who formulated the London equations in 1935. These equations describe how superconductors behave in the presence of magnetic fields, highlighting essential properties such as perfect diamagnetism and the Meissner effect. The significance of these equations lies in their ability to provide a theoretical framework for understanding the electromagnetic properties of superconductors.
High-temperature superconductors: High-temperature superconductors are materials that exhibit superconductivity at temperatures significantly higher than traditional superconductors, typically above 77 K. These materials often have complex crystal structures and are primarily composed of copper oxide compounds, which play a critical role in their unique electronic properties and the underlying mechanisms of superconductivity.
Josephson Junctions: Josephson junctions are quantum devices formed by two superconductors separated by a thin insulating barrier, allowing for the tunneling of Cooper pairs. These junctions exhibit unique properties like the Josephson effect, where a supercurrent can flow between the superconductors without any voltage applied, which is crucial for various applications in quantum computing and sensitive magnetometry. They are significant in understanding the underlying principles of superconductivity and its manifestations in both conventional and high-temperature superconductors.
London equations: The London equations are a set of two equations that describe the electromagnetic properties of superconductors, specifically how magnetic fields interact with them. These equations help to explain phenomena such as the Meissner effect, where a superconductor expels magnetic fields from its interior, and they are fundamental in understanding the macroscopic behavior of superconductors and their critical currents.
Magnetic field: A magnetic field is a vector field that represents the magnetic influence on moving electric charges, electric currents, and magnetic materials. It describes how a magnet interacts with its environment, influencing the force experienced by charged particles and the behavior of certain materials. Understanding magnetic fields is crucial in exploring phenomena such as diamagnetism and the behavior of superconductors as described by specific equations.
Magnetic field expulsion: Magnetic field expulsion refers to the phenomenon where a superconductor expels magnetic fields from its interior when it transitions into the superconducting state. This effect, described by the London equations, leads to the exclusion of magnetic flux lines, which is crucial in understanding how superconductors behave in the presence of magnetic fields.
Magnetic susceptibility: Magnetic susceptibility is a measure of how much a material will become magnetized in response to an applied magnetic field. It indicates the degree to which a material can be magnetized and is a key factor in understanding various magnetic behaviors, such as whether a material is attracted to or repelled from a magnetic field, which is important for distinguishing between different types of magnetism.
Maxwell's equations: Maxwell's equations are a set of four fundamental equations that describe how electric and magnetic fields interact and propagate. They unify electricity, magnetism, and optics, forming the foundation for classical electromagnetism. Understanding these equations is crucial for exploring phenomena like superconductivity, photonic crystal behavior, and the response of materials to electromagnetic fields.
Meissner effect: The Meissner effect is the phenomenon where a superconductor expels magnetic fields when it transitions into its superconducting state, resulting in zero magnetic flux density inside the material. This unique behavior connects to various concepts like the London equations, which describe electromagnetic properties of superconductors, and BCS theory, explaining how pairs of electrons form to enable superconductivity. The Meissner effect is fundamental in understanding the behavior of Cooper pairs and also plays a crucial role in the Josephson effect, influencing how superconductors interact with each other. Additionally, it differentiates between Type I and Type II superconductors and is an important consideration in the study of high-temperature superconductivity.
Microwave cavity perturbation: Microwave cavity perturbation refers to the change in the electromagnetic field distribution and resonant frequency of a microwave cavity when a small object is introduced into it. This concept is crucial in understanding how materials interact with microwave radiation and is particularly relevant in the study of superconductivity as described by the London equations, where it helps to analyze changes in the electromagnetic properties of superconductors when they are exposed to microwaves.
Muon Spin Rotation: Muon spin rotation is a technique used in condensed matter physics to probe the magnetic properties of materials by measuring the precession of muons implanted in a sample. This method allows scientists to gain insights into the microscopic magnetic fields within materials, which is particularly important for understanding phenomena such as superconductivity and magnetism.
Penetration Depth: Penetration depth is a measure of how deeply a magnetic field can enter a superconductor before it is expelled, playing a crucial role in understanding the behavior of superconductors. This concept is essential in explaining how magnetic fields interact with superconducting materials and is quantitatively described by the London equations. It also plays a significant role in the study of high-temperature superconductivity, where penetration depth helps to reveal insights into the microscopic mechanisms that allow these materials to conduct electricity without resistance.
Pippard's Non-Local Generalization: Pippard's non-local generalization extends the classic London equations, which describe superconductivity, to include non-local effects that occur when the coherence length of a superconductor becomes comparable to its dimensions. This generalization helps explain the behavior of superconductors more accurately by accounting for the way magnetic fields penetrate and interact with the superconducting material over distances larger than the coherence length.
Quasiparticle excitations: Quasiparticle excitations are emergent phenomena that occur in many-body systems, acting like particles with well-defined properties that arise from the collective behavior of interacting particles. They help simplify complex interactions in condensed matter systems, allowing for a more intuitive understanding of phenomena like superconductivity, superfluidity, and the fractional quantum Hall effect.
Spherical superconductors: Spherical superconductors are materials that exhibit superconductivity and are shaped in a spherical form, allowing for unique electromagnetic properties. Their geometry leads to interesting implications when analyzing their magnetic response and behavior under external fields, especially in relation to the London equations, which describe how magnetic fields interact with superconductors.
Squids: Squids are a type of cephalopod mollusk known for their elongated bodies, large eyes, and tentacles. In the context of superconductivity, squids specifically refer to superconducting quantum interference devices (SQUIDs), which are highly sensitive magnetometers used to measure extremely small magnetic fields. These devices leverage the unique properties of superconductors, such as zero electrical resistance and the Meissner effect, making them critical in various applications, including medical imaging and quantum computing.
Superconducting magnets: Superconducting magnets are powerful magnets created using superconducting materials, which exhibit zero electrical resistance below a certain critical temperature. This property allows them to carry large electric currents without energy loss, leading to the generation of intense magnetic fields. These magnets are crucial in various applications, such as MRI machines and particle accelerators, where strong and stable magnetic fields are essential for operation.
Supercurrent density: Supercurrent density refers to the flow of electric current in a superconductor, characterized by its ability to carry current without any resistance. This phenomenon is essential for understanding the behavior of superconductors, especially in relation to the London equations, which describe how magnetic fields interact with superconductors and how supercurrents are maintained within them.
Thin films: Thin films are layers of material with a thickness in the nanometer to micrometer range, often used in various technological applications. These films can exhibit unique properties that differ from bulk materials due to their reduced dimensionality, such as altered electrical, optical, and mechanical characteristics. The behavior of thin films is particularly important when examining phenomena like superconductivity and magnetism.
Topological Superconductors: Topological superconductors are materials that exhibit superconductivity and possess topological order, meaning their quantum states are robust against local perturbations. This unique combination allows them to host exotic quasi-particles, such as Majorana fermions, which can have applications in fault-tolerant quantum computing. The interplay between the topological properties and the superconducting state creates intriguing phenomena that enhance our understanding of quantum mechanics and material science.
Two-fluid model: The two-fluid model describes the behavior of superfluid helium as comprising two interpenetrating fluids: a normal fluid and a superfluid. This model helps explain various phenomena observed in superconductors and superfluids, where each fluid can have distinct properties, such as differing responses to applied forces and temperatures.
Type I Superconductors: Type I superconductors are materials that exhibit superconductivity below a critical temperature and completely expel magnetic fields from their interior, a phenomenon known as the Meissner effect. These superconductors are characterized by their ability to transition to a superconducting state without any residual magnetism, leading to a perfect diamagnetism and the expulsion of all magnetic flux lines. This behavior is described by the London equations, which illustrate how these materials respond to electromagnetic fields.
Type II Superconductors: Type II superconductors are materials that can allow magnetic fields to partially penetrate them while remaining in the superconducting state. They exhibit a unique mixed state where magnetic flux lines can pass through the material in quantized units, allowing for higher critical magnetic fields compared to Type I superconductors. This behavior is critical for understanding the London equations and the Meissner effect.
Unconventional superconductivity: Unconventional superconductivity refers to a class of superconductors that do not conform to the traditional Bardeen-Cooper-Schrieffer (BCS) theory, which explains conventional superconductors. These materials often exhibit unique pairing mechanisms, such as spin-triplet pairing or d-wave symmetry, and their behaviors are influenced by strong electronic correlations and fluctuations. This leads to a range of fascinating phenomena that challenge our understanding of superconductivity.
Vector Potential: Vector potential is a vector field whose curl gives the magnetic field in electromagnetism. It serves as a useful mathematical tool in understanding magnetic phenomena, especially in situations where direct calculation of the magnetic field may be complex. In the context of superconductivity and the London equations, vector potential plays a crucial role in describing the behavior of superconducting materials under magnetic fields.
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