Glossary

2.1 London Theory

4 min read•august 14, 2024

London Theory lays the groundwork for understanding superconductivity. It introduces key concepts like and the London equations, which relate current density to electric and magnetic fields in superconductors.

The theory explains the and introduces the , a crucial parameter in superconductor behavior. While phenomenological, London Theory provides valuable insights into superconductivity's macroscopic properties, paving the way for more advanced microscopic theories.

Assumptions and Equations of London Theory

Key Assumptions

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  • Superconductivity arises from the presence of superconducting electrons not subject to resistance and can flow freely without energy loss
  • (J_s) is proportional to the vector potential (A) of the magnetic field
  • Superconducting electrons have a uniform density throughout the superconductor
  • Velocity of superconducting electrons is proportional to the vector potential (A)

London Equations

  • Relate the superconducting current density (J_s) to the electric field (E) and the magnetic field (B)
    • Js=−(1/μ0λL2)∗AJ_s = -(1/μ_0λ_L^2) * A
    • ∇×Js=−(1/μ0λL2)∗B∇ × J_s = -(1/μ_0λ_L^2) * B
  • Parameter λ_L is the London characterizing the distance an external magnetic field can penetrate into a superconductor

London Penetration Depth and Significance

Characteristics of London Penetration Depth

  • Characteristic length scale (λ_L) determining the extent an external magnetic field can penetrate into a superconductor
    • Given by the equation: λL=sqrt(ms/(μ0∗ns∗es2))λ_L = sqrt(m_s / (μ_0 * n_s * e_s^2)), where m_s is the mass of the superconducting electrons, n_s is their density, e_s is their charge, and μ_0 is the permeability of free space
    • For most superconductors, λ_L is on the order of tens to hundreds of nanometers, indicating strong screening of magnetic fields inside the superconductor
  • Temperature-dependent and diverges as temperature approaches the (T_c), signifying the breakdown of superconductivity

Significance in Superconductivity

  • Ratio of London penetration depth to coherence length determines whether a superconductor is classified as type-I or type-II
    • Type-I superconductors have a large λ_L compared to the coherence length, while type-II superconductors have a smaller λ_L
  • Plays a crucial role in understanding the Meissner effect, where magnetic fields are expelled from the interior of a superconductor

Magnetic Field Distributions in Superconductors

Calculating Magnetic Field Distributions

  • London equations used to calculate spatial distribution of magnetic field inside a superconductor
    • For a superconductor occupying the half-space x > 0 with an external magnetic field B_0 applied parallel to the surface, the magnetic field inside the superconductor decays exponentially as B(x)=B0∗exp(−x/λL)B(x) = B_0 * exp(-x/λ_L)
    • Screening current density (J_s) also decays exponentially with the same characteristic length scale λ_L
  • For a superconducting wire of radius R carrying a current I, the magnetic field inside the wire is given by B(r)=(μ0∗I/(2πR))∗(1−exp(−r/λL))B(r) = (μ_0 * I / (2πR)) * (1 - exp(-r/λ_L)), where r is the radial distance from the center of the wire

Meissner Effect

  • London theory predicts complete expulsion of magnetic field from the interior of a superconductor (Meissner effect) when the applied field is below a critical value
    • Demonstrates perfect diamagnetism, where the is -1
  • Meissner effect is a hallmark of superconductivity and is used to experimentally verify the superconducting state

Limitations of London Theory

Phenomenological Nature

  • London theory is a that does not provide a microscopic explanation for the origin of superconductivity
    • Assumes a uniform density of superconducting electrons and does not account for spatial variations or the presence of normal electrons
    • Does not explain the existence of a critical temperature (T_c) or the temperature dependence of superconducting properties

Lack of Microscopic Description

  • London theory does not address the formation of , the fundamental building blocks of the superconducting state according to the
    • Cooper pairs are bound states of two electrons with opposite spins and momenta, formed through an attractive interaction mediated by phonons
  • Fails to describe the behavior of type-II superconductors, which exhibit a mixed state with partial magnetic field penetration above a lower critical field

Incomplete Description of Superconducting Phenomena

  • Does not account for the quantization of magnetic flux in superconducting rings or the existence of vortices in type-II superconductors
    • Magnetic flux is quantized in units of the , Φ0=h/(2e)Φ_0 = h/(2e), where h is Planck's constant and e is the electron charge
    • Vortices are localized regions where the superconducting order parameter is suppressed, and magnetic field can penetrate the superconductor
  • While providing a useful description of some macroscopic properties of superconductors, a more comprehensive microscopic theory, such as the BCS theory, is needed to fully understand the phenomenon of superconductivity

Key Terms to Review (23)

BCS Theory: BCS Theory, named after its developers Bardeen, Cooper, and Schrieffer, is a theoretical framework that explains the phenomenon of superconductivity in conventional superconductors. It describes how electron pairs, known as Cooper pairs, form through attractive interactions mediated by lattice vibrations (phonons), leading to a state of zero electrical resistance below a critical temperature.
Cooper pairs: Cooper pairs are pairs of electrons that are bound together at low temperatures in a superconducting state, leading to zero electrical resistance. These pairs are crucial for understanding how superconductivity occurs, as they enable the flow of electric current without energy loss and form the basis of many theories about superconductivity.
Critical magnetic field: The critical magnetic field is the maximum magnetic field strength that a superconductor can withstand before it transitions into the normal (non-superconducting) state. This phenomenon is crucial for understanding how superconductors behave under external magnetic influences, and it is tightly linked to concepts such as critical temperature and current, coherence length, and penetration depth.
Critical Temperature: Critical temperature is the temperature below which a material exhibits superconductivity, meaning it can conduct electricity without resistance. This fundamental property defines the transition from a normal conductive state to a superconducting state and is crucial for understanding various aspects of superconductors, including their types and underlying theories.
Cryogenic cooling: Cryogenic cooling refers to the process of achieving extremely low temperatures, typically below -150°C (-238°F), to maintain superconductivity in materials. This cooling is crucial for various applications, including superconducting devices, as it allows for minimal electrical resistance and enhanced performance. Achieving these temperatures enables the functionality of devices that rely on the unique properties of superconductors, which are essential for advancements in technology and transportation systems.
Flux quantum: Flux quantum refers to the fundamental unit of magnetic flux that can pass through a superconducting loop. It is crucial in understanding the behavior of superconductors, particularly in the context of quantized magnetic flux, which plays a significant role in both theoretical models and practical applications involving superconductors. The presence of the flux quantum allows for phenomena like magnetic flux quantization and the operation of devices that leverage these properties.
Ginzburg-Landau Theory: The Ginzburg-Landau Theory is a theoretical framework used to describe superconductivity in terms of a complex order parameter, which captures the macroscopic quantum behavior of superconductors. This theory connects crucial concepts like the formation of Cooper pairs and the response of superconductors to magnetic fields, providing insights into phenomena such as flux quantization and the Meissner effect.
Heike Kamerlingh Onnes: Heike Kamerlingh Onnes was a Dutch physicist who is best known for his pioneering work in superconductivity. He discovered superconductivity in 1911 while experimenting with mercury at low temperatures, leading to the observation that mercury's electrical resistance dropped to zero below a certain critical temperature. This groundbreaking discovery laid the foundation for the study of superconductivity and influenced many subsequent theories and experiments, including the development of the London Theory.
John Bardeen: John Bardeen was a renowned American physicist who made significant contributions to the field of superconductivity and solid-state physics. He is best known for co-developing the BCS theory of superconductivity, which explains how certain materials exhibit zero electrical resistance at low temperatures, and for his role in the invention of the transistor, earning him two Nobel Prizes in Physics.
London Penetration Depth: London penetration depth is a characteristic length scale that describes how deeply magnetic fields can penetrate into a superconductor before they are expelled, which is key to understanding the behavior of superconductors in magnetic fields. This phenomenon is essential in describing how superconductors interact with external magnetic fields, especially in relation to perfect diamagnetism, and helps to explain the unique properties of type-II superconductors.
Magnetic susceptibility: Magnetic susceptibility is a measure of how much a material will become magnetized in an applied magnetic field. It indicates the degree to which a material can be magnetized, helping to classify materials into categories such as diamagnetic, paramagnetic, or ferromagnetic based on their response to magnetic fields. Understanding this property is crucial in the study of superconductivity and the behavior of materials under various magnetic conditions.
Meissner Effect: The Meissner Effect is the phenomenon where a superconducting material expels magnetic fields as it transitions into the superconducting state, allowing it to exhibit perfect diamagnetism. This effect is fundamental to understanding how superconductors interact with magnetic fields and is crucial for applications like magnetic levitation.
Niobium-titanium: Niobium-titanium is an alloy that exhibits superconducting properties at low temperatures, making it a key material for various applications in superconducting devices. Its unique characteristics allow it to function effectively in both Type II superconductors and in high magnetic field environments, which is crucial for many advanced technologies, including medical imaging and scientific research.
Penetration Depth: Penetration depth is a fundamental characteristic of superconductors that describes how deep a magnetic field can penetrate into a superconductor before being expelled. This depth varies depending on the type of superconductor, with Type I superconductors completely expelling magnetic fields, while Type II superconductors allow some field lines to penetrate in quantized vortices. Understanding penetration depth is crucial for grasping the behavior of superconductors under magnetic fields and their practical applications.
Phenomenological Theory: Phenomenological theory refers to an approach in physics that seeks to describe the macroscopic behavior of systems without delving into the underlying microscopic details. This theory focuses on observable phenomena and provides a framework for understanding complex behaviors through empirical observations and mathematical descriptions, particularly useful in fields like superconductivity.
Quantum Levitation: Quantum levitation refers to the phenomenon where a superconductor can effectively float above a magnetic field due to the principles of quantum mechanics. This occurs when a superconductor enters a state known as the Meissner effect, expelling magnetic fields from its interior and allowing it to hover or 'levitate' while maintaining a fixed position relative to the magnetic source. This captivating behavior highlights the interplay between superconductivity and magnetic fields, illustrating key concepts in understanding superconducting materials.
Superconducting Current Density: Superconducting current density refers to the amount of electric current flowing per unit area in a superconducting material. This concept is crucial in understanding how superconductors can carry electricity without resistance, leading to significant energy efficiency and unique electromagnetic properties. The current density plays a pivotal role in determining the behavior of superconductors, including their critical magnetic fields and how they respond to external magnetic influences.
Superconducting electrons: Superconducting electrons are pairs of electrons that move through a superconducting material without resistance, forming what is known as Cooper pairs. These pairs are a fundamental aspect of superconductivity, where below a certain critical temperature, the material exhibits zero electrical resistance and expels magnetic fields, leading to unique electromagnetic properties.
Type I Superconductor: Type I superconductors are materials that exhibit superconductivity below a specific critical temperature and completely expel magnetic fields due to the Meissner effect. They can be classified based on their response to magnetic fields, showing a complete transition to the superconducting state without any mixed phase or magnetic flux penetration.
Type II Superconductor: Type II superconductors are materials that exhibit superconductivity at higher magnetic fields and temperatures than Type I superconductors, allowing them to penetrate magnetic fields partially. They display a mixed state where magnetic flux lines can partially enter the material, leading to unique properties like magnetic levitation and persistent currents.
Vortex state: The vortex state refers to a unique magnetic state that occurs in type-II superconductors when subjected to an external magnetic field. In this state, magnetic flux lines penetrate the superconductor in the form of quantized vortices, each carrying a quantum of magnetic flux. This phenomenon plays a critical role in understanding the behavior of superconductors and is deeply connected to concepts such as magnetic field penetration and the Ginzburg-Landau and London theories.
X-ray Diffraction: X-ray diffraction is a technique used to study the structure of materials by directing X-rays at a sample and measuring the angles and intensities of the scattered X-rays. This process reveals the atomic arrangement within a material, making it a powerful tool in material science, including the study of superconducting devices. By analyzing diffraction patterns, researchers can gain insights into crystallographic structures and the properties of materials at the atomic level.
Yttrium Barium Copper Oxide: Yttrium barium copper oxide (YBCO) is a high-temperature superconductor characterized by its ability to conduct electricity without resistance at temperatures significantly higher than conventional superconductors. It is a type II superconductor and plays a crucial role in understanding magnetic properties and applications in superconducting circuits, making it a key material in the field of superconductivity.
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