9.2 London equations

The London equations provide a crucial framework for understanding superconductivity. They explain perfect conductivity and diamagnetism in superconductors, linking supercurrent density to electromagnetic fields. These equations form the basis for describing the Meissner effect and penetration depth.

The London theory, while phenomenological, accurately predicts key superconductor behaviors. It introduces important concepts like the London penetration depth and coherence length, which help classify superconductors into types I and II. Despite limitations, the theory remains fundamental to our understanding of superconductivity.

London equations for superconductors

• The London equations provide a phenomenological description of superconductivity by relating the supercurrent density to the electromagnetic fields
• These equations capture the essential macroscopic properties of superconductors, namely perfect conductivity and perfect diamagnetism
• The London equations are derived from the basic assumptions of the two-fluid model and the concept of macroscopic quantum coherence

First London equation for perfect conductivity

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• States that the electric field $\vec{E}$ inside a superconductor causes the supercurrent density $\vec{J}_s$ to accelerate without resistance: $\frac{\partial \vec{J}_s}{\partial t} = \frac{n_s e^2}{m_e} \vec{E}$
• Here, $n_s$ is the density of superconducting electrons, $e$ is the electron charge, and $m_e$ is the electron mass
• Implies that a constant electric field would lead to an infinite current, which is unphysical, so the electric field must be zero in the steady state (perfect conductivity)
• Explains the absence of electrical resistance in superconductors and the persistence of supercurrents without any applied voltage

Second London equation for perfect diamagnetism

• Relates the curl of the supercurrent density $\vec{J}_s$ to the magnetic field $\vec{B}$ inside a superconductor: $\nabla \times \vec{J}_s = -\frac{n_s e^2}{m_e} \vec{B}$
• Implies that magnetic fields are expelled from the interior of a superconductor (Meissner effect) and that superconductors exhibit perfect diamagnetism
• Leads to the concept of London penetration depth, which characterizes the distance over which magnetic fields can penetrate into a superconductor
• Together with the first London equation, provides a complete description of the electrodynamics of superconductors in the London limit

Relation between current density and vector potential

• The supercurrent density $\vec{J}_s$ can be expressed in terms of the magnetic vector potential $\vec{A}$ as: $\vec{J}_s = -\frac{n_s e^2}{m_e} \vec{A}$
• This relation is obtained by combining the two London equations and using the definition of the magnetic field in terms of the vector potential: $\vec{B} = \nabla \times \vec{A}$
• Provides a convenient way to calculate the distribution of supercurrents and magnetic fields in superconductors
• Allows for the derivation of the London equation in the Coulomb gauge, which is often used in theoretical calculations

Consequences of London equations

• Explain the zero electrical resistance and perfect diamagnetism of superconductors
• Predict the existence of the Meissner effect and the expulsion of magnetic fields from superconductors
• Lead to the concept of London penetration depth and its relation to the superconducting electron density
• Provide a basis for understanding the behavior of type I and type II superconductors in magnetic fields
• Have been experimentally verified through measurements of magnetic flux expulsion, penetration depth, and the observation of the Meissner effect

Meissner effect in superconductors

• The Meissner effect is the complete expulsion of magnetic fields from the interior of a superconductor, regardless of the field's presence during the superconducting transition
• It is a fundamental property of superconductors and a direct consequence of the London equations
• The Meissner effect demonstrates that superconductors are not just perfect conductors but also perfect diamagnets

Expulsion of magnetic fields

• When a superconductor is cooled below its critical temperature in the presence of a weak magnetic field, the field is expelled from the interior of the superconductor
• This expulsion occurs due to the generation of surface supercurrents that create a magnetization opposing the applied field
• The expulsion of magnetic fields is a thermodynamic effect and minimizes the free energy of the superconducting state
• The Meissner effect is a hallmark of superconductivity and distinguishes it from perfect conductivity

Critical magnetic field strength

• The Meissner effect persists only up to a certain critical magnetic field strength, denoted as $H_c$
• Above $H_c$, the superconducting state becomes energetically unfavorable, and the material reverts to the normal state
• The critical field strength depends on the superconducting material and decreases with increasing temperature
• The temperature dependence of $H_c$ is given by the empirical relation: $H_c(T) = H_c(0) [1 - (T/T_c)^2]$, where $T_c$ is the critical temperature

Type I vs type II superconductors

• Superconductors can be classified into two types based on their behavior in magnetic fields
• Type I superconductors (pure metals like Pb, Hg) exhibit a complete Meissner effect up to a single critical field $H_c$, above which they abruptly transition to the normal state
• Type II superconductors (alloys, compounds like Nb-Ti, YBCO) have two critical fields: a lower critical field $H_{c1}$ and an upper critical field $H_{c2}$
• Between $H_{c1}$ and $H_{c2}$, type II superconductors exist in a mixed state (vortex state) where magnetic flux partially penetrates the material in the form of quantized vortices
• Type II superconductors are technologically important due to their higher critical fields and ability to carry high currents in the presence of strong magnetic fields

Penetration depth in superconductors

• The London penetration depth, denoted as $\lambda_L$, characterizes the distance over which magnetic fields can penetrate into a superconductor
• It is a fundamental length scale in superconductivity and arises from the second London equation
• The penetration depth is related to the screening of magnetic fields by the superconducting electrons

Definition of London penetration depth

• The London penetration depth is defined as: $\lambda_L = \sqrt{\frac{m_e}{\mu_0 n_s e^2}}$, where $m_e$ is the electron mass, $\mu_0$ is the vacuum permeability, $n_s$ is the superconducting electron density, and $e$ is the electron charge
• It represents the characteristic length scale over which the magnetic field and supercurrent density decay exponentially from the surface of a superconductor
• Typical values of $\lambda_L$ range from a few tens to a few hundred nanometers, depending on the superconducting material

Temperature dependence of penetration depth

• The London penetration depth is temperature-dependent and diverges as the temperature approaches the critical temperature $T_c$
• The temperature dependence of $\lambda_L$ is given by: $\lambda_L(T) = \lambda_L(0) [1 - (T/T_c)^4]^{-1/2}$, where $\lambda_L(0)$ is the penetration depth at absolute zero
• The divergence of $\lambda_L$ near $T_c$ is related to the decrease in the superconducting electron density as the thermal energy disrupts the Cooper pairs
• Measurements of the temperature dependence of $\lambda_L$ provide valuable information about the superconducting state and the pairing mechanism

Relation to superconducting electron density

• The London penetration depth is inversely proportional to the square root of the superconducting electron density $n_s$
• A higher $n_s$ leads to a shorter $\lambda_L$, indicating a stronger screening of magnetic fields and a more robust superconducting state
• Changes in $n_s$ due to temperature, impurities, or other factors directly affect the penetration depth
• The relation between $\lambda_L$ and $n_s$ allows for the estimation of the superconducting electron density from measurements of the penetration depth

Coherence length in superconductors

• The coherence length, denoted as $\xi$, is another fundamental length scale in superconductivity
• It characterizes the spatial extent of the Cooper pair wavefunction and the distance over which the superconducting order parameter can vary
• The coherence length is related to the superconducting energy gap and the Fermi velocity of the electrons

Definition of coherence length

• The coherence length is defined as: $\xi = \frac{\hbar v_F}{\pi \Delta}$, where $\hbar$ is the reduced Planck constant, $v_F$ is the Fermi velocity, and $\Delta$ is the superconducting energy gap
• It represents the size of a Cooper pair and the minimum length scale over which the superconducting order parameter can change significantly
• Typical values of $\xi$ range from a few nanometers to a few micrometers, depending on the superconducting material

Relation to superconducting energy gap

• The coherence length is inversely proportional to the superconducting energy gap $\Delta$
• A larger energy gap leads to a shorter coherence length, indicating a more tightly bound Cooper pair and a more localized superconducting state
• The temperature dependence of $\xi$ is related to the temperature dependence of $\Delta$, which vanishes at the critical temperature $T_c$
• Measurements of the coherence length provide information about the strength of the pairing interaction and the nature of the superconducting state

Comparison with London penetration depth

• The ratio of the London penetration depth $\lambda_L$ to the coherence length $\xi$ is known as the Ginzburg-Landau parameter, denoted as $\kappa = \lambda_L / \xi$
• The value of $\kappa$ determines whether a superconductor is type I ($\kappa < 1/\sqrt{2}$) or type II ($\kappa > 1/\sqrt{2}$)
• In type I superconductors, $\xi > \lambda_L$, meaning that the superconducting order parameter varies slowly compared to the screening of magnetic fields
• In type II superconductors, $\xi < \lambda_L$, indicating that the superconducting order parameter varies rapidly compared to the screening of magnetic fields, allowing for the formation of vortices in the mixed state

Limitations of London theory

• While the London equations provide a successful phenomenological description of superconductivity, they have several limitations
• These limitations arise from the simplifying assumptions made in the derivation of the equations and their inability to capture certain aspects of superconductivity
• Understanding these limitations is crucial for developing a more complete theory of superconductivity

Inability to explain microscopic origin

• The London theory does not provide a microscopic explanation for the origin of superconductivity
• It does not address the formation of Cooper pairs or the mechanism behind the attractive interaction between electrons
• The equations are based on macroscopic considerations and do not derive from a microscopic theory of superconductivity
• A complete understanding of superconductivity requires a microscopic theory, such as the BCS theory, which explains the pairing mechanism and the superconducting energy gap

Assumption of local electrodynamics

• The London equations assume that the relationship between the supercurrent density and the electromagnetic fields is local
• This assumption implies that the response of the superconductor at a given point depends only on the fields at that point
• However, in real superconductors, there can be non-local effects, especially when the coherence length is comparable to or larger than the penetration depth
• Non-local electrodynamics becomes important in describing the behavior of superconductors in the presence of strong fields or at interfaces

Failure at high frequencies and short wavelengths

• The London theory is valid only in the limit of low frequencies and long wavelengths
• At high frequencies or short wavelengths, the assumption of local electrodynamics breaks down, and the London equations no longer provide an accurate description
• The non-local effects and the finite response time of the superconducting electrons become significant in this regime
• A more general theory, such as the Mattis-Bardeen theory or the Eliashberg theory, is needed to describe the high-frequency and short-wavelength behavior of superconductors

Experimental verification of London equations

• The London equations have been extensively tested through various experimental techniques
• These experiments provide strong evidence for the validity of the London theory and its predictions
• The successful experimental verification of the London equations has established them as a cornerstone of the phenomenological description of superconductivity

Magnetic flux expulsion experiments

• Experiments have directly demonstrated the expulsion of magnetic fields from superconductors, as predicted by the Meissner effect
• Techniques such as magnetic susceptibility measurements and magnetic levitation have been used to observe the diamagnetic response of superconductors
• The complete expulsion of magnetic fields up to a critical field strength has been verified for type I superconductors
• The partial flux expulsion and the mixed state have been observed in type II superconductors, in agreement with the predictions of the London theory

Measurement of penetration depth

• The London penetration depth has been measured using various experimental methods, such as muon spin rotation ($\mu$SR), neutron scattering, and microwave cavity perturbation
• These techniques probe the spatial variation of the magnetic field inside the superconductor and allow for the determination of $\lambda_L$
• The temperature dependence of $\lambda_L$ has been studied, confirming the divergence near the critical temperature as predicted by the London theory
• Measurements of $\lambda_L$ have provided valuable information about the superconducting electron density and the strength of the screening currents

Observation of Meissner effect

• The Meissner effect has been directly observed through magnetic levitation experiments
• A superconductor can be made to levitate above a strong magnet due to the complete expulsion of the magnetic field
• The stability of the levitation depends on the type of superconductor and the strength of the applied field
• Meissner effect demonstrations have become a classic illustration of the unique properties of superconductors and the validity of the London equations
• The observation of the Meissner effect has also been crucial in the development of superconducting magnets and other technological applications