Carsten Timm: Theory of superconductivity

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Electrodynamics of superconductors
Superconductors are defined by electrodynamic properties—ideal conduction and magnetic-field repulsion. It is
thus appropriate to ask how these materials can be described within the formal framework of electrodynamics.
5.1 London theory
In 1935, F. and H. London proposed a phenomenological theory for the electrodynamic properties of superconductors.
It is based on a two-fluid picture: For unspecified reasons, the electrons from a normal fluid of concentration
n n and a superfluid of concentration n s , where n n + n s = n = N/V . Such a picture seemed quite plausible based
on Einstein’s theory of Bose-Einstein condensation, although nobody understood how the fermionic electrons
could form a superfluid. The normal fluid is postulated to behave normally, i.e., to carry an ohmic current
governed by the Drude law
j n = σ n E (5.1)
σ n = e2 n n τ
m . (5.2)
The superfluid is assumed to be insensitive to scattering. As noted in section 4.2, this leads to a free acceleration
of the charges. With the supercurrent
j s = −e n s v s (5.3)
and Newton’s equation of motion
d
dt v s = F m = −eE m , (5.4)
we obtain
∂j s
∂t = e2 n s
E. (5.5)
m
This is the First London Equation. We are only interested in the stationary state, i.e., we assume n n and n s to be
uniform in space, this is the most serious restriction of London theory, which will be overcome in Ginzburg-Landau
theory.
Note that the curl of the First London Equation is
This can be integrated in time to give
∂
∂t ∇ × j s = e2 n s
m ∇ × E = n s
−e2 mc
∂B
∂t . (5.6)
∇ × j s = − e2 n s
B + C(r), (5.7)
mc
where the last term represents a constant of integration at each point r inside the superconductor. C(r) should
be determined from the initial conditions. If we start from a superconducting body in zero applied magnetic
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