The London equations, developed by brothers Fritz and Heinz London in 1935, are constitutive relations for a superconductor relating its superconducting current to electromagnetic fields in and around it. Whereas Ohm's law is the simplest constitutive relation for an ordinary conductor, the London equations are the simplest meaningful description of superconducting phenomena, and form the genesis of almost any modern introductory text on the subject. A major triumph of the equations is their ability to explain the Meissner effect, wherein a material exponentially expels all internal magnetic fields as it crosses the superconducting threshold.

Description

There are two London equations when expressed in terms of measurable fields:

is the (superconducting) current density, E and B is a phenomenological constant loosely associated with a number density of superconducting carriers.

The two equations can be combined into a single "London Equation" in terms of a specific vector potential which has been gauge fixed to the "London gauge", giving:

In the London gauge, the vector potential obeys the following requirements, ensuring that it can be interpreted as a current density:

• in the superconductor bulk,

• is the normal vector at the surface of the superconductor.

The first requirement, also known as Coulomb gauge as expected from the continuity equation. The second requirement is consistent with the fact that supercurrent flows near the surface. The third requirement ensures no accumulation of superconducting electrons on the surface. These requirements do away with all gauge freedom and uniquely determine the vector potential. One can also write the London equation in terms of an arbitrary gauge is the change in gauge which shifts the arbitrary gauge to the London gauge. The vector potential expression holds for magnetic fields that vary slowly in space.

London penetration depth

If the second of London's equations is manipulated by applying Ampere's law,

1. ,

then it can be turned into the Helmholtz equation for magnetic field:

where the inverse of the Laplacian eigenvalue:

, over which external magnetic fields are exponentially suppressed: it is called the London penetration depth: typical values are from 50 to 500 nm.

For example, consider a superconductor within free space where the magnetic field outside the superconductor is a constant value pointed parallel to the superconducting boundary plane in the z direction. If x leads perpendicular to the boundary then the solution inside the superconductor may be shown to be

From here the physical meaning of the London penetration depth can perhaps most easily be discerned.

Rationale

Original arguments

While the above equations cannot be formally derived, the Londons did follow a certain intuitive logic in the formulation of their theory. Substances across a stunningly wide range of composition behave roughly according to Ohm's law, which states that current is proportional to electric field. However, such a linear relationship is impossible in a superconductor for, almost by definition, the electrons in a superconductor flow with no resistance whatsoever. To this end, the London brothers imagined electrons as if they were free electrons under the influence of a uniform external electric field. According to the Lorentz force law

we should have

This is the first London equation. To obtain the second equation, take the curl of the first London equation and apply Faraday's law,

1. ,

to obtain

As it currently stands, this equation permits both constant and exponentially decaying solutions. The Londons recognized from the Meissner effect that constant nonzero solutions were nonphysical, and thus postulated that not only was the time derivative of the above expression equal to zero, but also that the expression in the parentheses must be identically zero:

, which leads to the spatial distribution of magnetic field obeys :

. In one dimension, such Helmholtz equation

as physical current.

field) obeys:

as the superconductor. Consequently, whether the magnetic flux inside a perfect conductor will vanish depends on the initial condition (whether it's zero-field cooled or not).

Canonical momentum arguments

It is also possible to justify the London equations by other means. Current density is defined according to the equation

by the expectation values of their operators. The velocity operator

is defined by dividing the gauge-invariant, kinematic momentum operator by the particle mass m. as the electron charge. We may then make this replacement in the equation above. However, an important assumption from the microscopic theory of superconductivity is that the superconducting state of a system is the ground state, and according to a theorem of Bloch's, in such a state the canonical momentum p is zero. This leaves

which is the London equation according to the second formulation above.

Flux quantization

The quantization of the magnetic flux can be predicted using the London equations. For this, it is assumed that the density of superconducting electrons is governed by a bosonic wavefunction, which satisfies

can vary over space. Thus, it is possible to write

Using the canonical momentum, the velocity is written in terms of the wavefunction as

such that the current is

Flux quantization can be observed in a hollow superconducting cylinder, where an external magnetic field is applied inside the hole, and the cylinder itself is free of magnetic fields and currents due to the Meissner effect, so the phase is governed by the equation

around the hole, the first term becomes

. By Stokes' theorem, the second term is

is the magnetic flux inside the hole. Since these two terms must be equal, there is the requirement that

.

References