London equations

As a material drops below its superconducting critical temperature, magnetic fields within the material are expelled via theMeissner effect.The London equations give a quantitative explanation of this effect.

TheLondon equations,developed by brothersFritzandHeinz Londonin 1935,areconstitutive relationsfor asuperconductorrelating its superconducting current toelectromagnetic fieldsin and around it. WhereasOhm's lawis the simplest constitutive relation for an ordinaryconductor,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 theMeissner 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:

${\frac {\partial \mathbf {j} _{\rm {s}}}{\partial t}}={\frac {n_{\rm {s}}e^{2}}{m}}\mathbf {E},\qquad \mathbf {\nabla } \times \mathbf {j} _{\rm {s}}=-{\frac {n_{\rm {s}}e^{2}}{m}}\mathbf {B}.$

Here${\mathbf {j} }_{\rm {s}}$is the (superconducting)current density,EandBare respectively the electric and magnetic fields within the superconductor, $e\,$ is the charge of an electron or proton, $m\,$ is electron mass, and $n_{\rm {s}}\,$ is a phenomenological constant loosely associated with anumber densityof superconducting carriers.

The two equations can be combined into a single "London Equation" in terms of a specificvector potential$\mathbf {A} _{\rm {s}}$which has beengauge fixedto the "London gauge", giving:

$\mathbf {j} _{s}=-{\frac {n_{\rm {s}}e^{2}}{m}}\mathbf {A} _{\rm {s}}.$

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

• $\nabla \cdot \mathbf {A} _{\rm {s}}=0,$
• $\mathbf {A} _{\rm {s}}=0$in the superconductor bulk,
• $\mathbf {A} _{\rm {s}}\cdot {\hat {\mathbf {n} }}=0,$where${\hat {\mathbf {n} }}$is thenormal vectorat the surface of the superconductor.

The first requirement, also known asCoulomb gaugecondition, leads to the constant superconducting electron density${\dot {\rho }}_{\rm {s}}=0$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$\mathbf {A}$by simply defining$\mathbf {A} _{\rm {s}}=(\mathbf {A} +\nabla \phi )$,where$\phi$is a scalar function and$\nabla \phi$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

Main article:London penetration depth

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

$\nabla \times \mathbf {B} =\mu _{0}\mathbf {j}$,

then it can be turned into theHelmholtz equationfor magnetic field:

$\nabla ^{2}\mathbf {B} ={\frac {1}{\lambda _{\rm {s}}^{2}}}\mathbf {B}$

where the inverse of thelaplacianeigenvalue:

$\lambda _{\rm {s}}\equiv {\sqrt {\frac {m}{\mu _{0}n_{\rm {s}}e^{2}}}}$

is the characteristic length scale,$\lambda _{\rm {s}}$,over which external magnetic fields are exponentially suppressed: it is called theLondon penetration depth:typical values are from 50 to 500nm.

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 thezdirection. Ifxleads perpendicular to the boundary then the solution inside the superconductor may be shown to be

$B_{z}(x)=B_{0}e^{-x/\lambda _{\rm {s}}}.\,$

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

Rationale

Original arguments

While it is important to note that 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 toOhm'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 theLorentz force law

$\mathbf {F} =m{\dot {\mathbf {v} }}=-e\mathbf {E} -e\mathbf {v} \times \mathbf {B}$

these electrons should encounter a uniform force, and thus they should in fact accelerate uniformly. Assume that the electrons in the superconductor are now driven by an electric field, then according to the definition of current density$\mathbf {j} _{\rm {s}}=-n_{\rm {s}}e\mathbf {v} _{\rm {s}}$we should have

${\frac {\partial \mathbf {j} _{s}}{\partial t}}=-n_{\rm {s}}e{\frac {\partial \mathbf {v} }{\partial t}}={\frac {n_{\rm {s}}e^{2}}{m}}\mathbf {E}$

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

$\nabla \times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}$,

to obtain

${\frac {\partial }{\partial t}}\left(\nabla \times \mathbf {j} _{\rm {s}}+{\frac {n_{\rm {s}}e^{2}}{m}}\mathbf {B} \right)=0.$

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:

$\nabla \times \mathbf {j} _{\rm {s}}+{\frac {n_{\rm {s}}e^{2}}{m}}\mathbf {B} =0$

This results in the second London equation and$\mathbf {j} _{s}=-{\frac {n_{\rm {s}}e^{2}}{m}}\mathbf {A} _{\rm {s}}$(up to a gauge transformation which is fixed by choosing "London gauge" ) since the magnetic field is defined through$B=\nabla \times A_{\rm {s}}.$

Additionally, according to Ampere's law$\nabla \times \mathbf {B} =\mu _{0}\mathbf {j} _{\rm {s}}$,one may derive that:$\nabla \times (\nabla \times \mathbf {B} )=\nabla \times \mu _{0}\mathbf {j} _{\rm {s}}=-{\frac {\mu _{0}n_{\rm {s}}e^{2}}{m}}\mathbf {B}.$

On the other hand, since$\nabla \cdot \mathbf {B} =0$,we have$\nabla \times (\nabla \times \mathbf {B} )=-\nabla ^{2}\mathbf {B}$,which leads to the spatial distribution of magnetic field obeys:

$\nabla ^{2}\mathbf {B} ={\frac {1}{\lambda _{\rm {s}}^{2}}}\mathbf {B}$

with penetration depth$\lambda _{\rm {s}}={\sqrt {\frac {m}{\mu _{0}n_{\rm {s}}e^{2}}}}$.In one dimension, suchHelmholtz equationhas the solution form$B_{z}(x)=B_{0}e^{-x/\lambda _{\rm {s}}}.\,$

Inside the superconductor$(x>0)$,the magnetic field exponetially decay, which well explains the Meissner effect. With the magnetic field distribution, we can use Ampere's law$\nabla \times \mathbf {B} =\mu _{0}\mathbf {j} _{\rm {s}}$again to see that the supercurrent$\mathbf {j} _{\rm {s}}$also flows near the surface of superconductor, as expected from the requirement for interpreting$\mathbf {j} _{\rm {s}}$as physical current.

While the above rationale holds for superconductor, one may also argue in the same way for a perfect conductor. However, one important fact that distinguishes the superconductor from perfect conductor is that perfect conductor does not exhibit Meissner effect for$T<T_{c}$.In fact, the postulation$\nabla \times \mathbf {j} _{\rm {s}}+{\frac {n_{\rm {s}}e^{2}}{m}}\mathbf {B} =0$does not hold for a perfect conductor. Instead, the time derivative must be kept and cannot be simply removed. This results in the fact that the time derivative of$\mathbf {B}$field (instead of$\mathbf {B}$field) obeys:

$\nabla ^{2}{\frac {\partial \mathbf {B} }{\partial t}}={\frac {1}{\lambda _{\rm {s}}^{2}}}{\frac {\partial \mathbf {B} }{\partial t}}.$

For$T<T_{c}$,deep inside a perfect conductor we have${\dot {\mathbf {B} }}=0$rather than$\mathbf {B} =0$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

$\mathbf {j} _{\rm {s}}=-n_{\rm {s}}e\mathbf {v} _{\rm {s}}.$

Taking this expression from a classical description to a quantum mechanical one, we must replace values$\mathbf {j} _{\rm {s}}$and$\mathbf {v} _{\rm {s}}$by the expectation values of their operators. The velocity operator

$\mathbf {v} _{\rm {s}}={\frac {1}{m}}\left(\mathbf {p} +e\mathbf {A} _{\rm {s}}\right)$

is defined by dividing the gauge-invariant, kinematic momentum operator by the particle massm.Note we are using$-e$as the electron charge. We may then make this replacement in the equation above. However, an important assumption from themicroscopic theory of superconductivityis 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 momentumpis zero. This leaves

$\mathbf {j} =-{\frac {n_{\rm {s}}e^{2}}{m}}\mathbf {A} _{\rm {s}},$

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