London Theory and the Meissner Effect

The phenomenology established two independent facts: below $T_{c}$ the resistance is exactly zero, and the magnetic flux is expelled. Zero resistance alone does not produce flux expulsion. A theory of the superconducting state must add a magnetic law to the electrical one, and the London brothers supplied the simplest such law in 1935. Its single new length scale, the penetration depth, is measured directly and fixes the density of the current-carrying electrons.

Perfect conductivity versus perfect diamagnetism

Take the electrical property at face value. Inside a perfect conductor a steady electric field would drive an ever-increasing current, so in the steady state $E = 0$ . Faraday's law then freezes the magnetic field,

\nabla \times E = - \frac{\partial B}{\partial t} = 0 ⟹ \frac{\partial B}{\partial t} = 0.

The interior field cannot change, but it can hold whatever value it had when the sample became perfectly conducting. Cool a perfect conductor in an applied field and the field stays trapped; cool it in zero field and then apply a field and the interior stays field-free. History decides the outcome.

A superconductor behaves differently. Cooled through $T_{c}$ in a field, it expels that field: the interior reaches $B = 0$ regardless of the path taken to get there. The final state is unique. This is a stronger statement than $\partial B / \partial t = 0$ , and no purely electrical property implies it.

Field-cooling contrast. A perfect conductor cooled in a field traps the flux it was cooled in; a superconductor cooled in the same field expels it, reaching B = 0 inside no matter what field it was cooled in.

The London equations

Model the superconducting electrons as a fraction with number density $n_{s}$ that moves without dissipation. Each carries charge $- e$ and mass $m$ and accelerates freely under a field,

m \frac{d v _{s}}{d t} = - e E .

With supercurrent density $J_{s} = - n_{s} e v_{s}$ , this is the first London equation,

\frac{\partial J _{s}}{\partial t} = \frac{n _{s} e ^{2}}{m} E .

It describes perfect conductivity: a static field is not needed to sustain a current, and any residual field accelerates the carriers. Taking the curl and inserting Faraday's law $\nabla \times E = - \partial B / \partial t$ ,

\frac{\partial}{\partial t} (\nabla \times J_{s}) = - \frac{n _{s} e ^{2}}{m} \frac{\partial B}{\partial t} ⟹ \frac{\partial}{\partial t} (\nabla \times J_{s} + \frac{n _{s} e ^{2}}{m} B) = 0.

A perfect conductor fixes only the time derivative of the bracketed quantity, so it permits any static field. The Londons imposed the stronger condition that the bracket vanish outright, not merely its rate of change. That choice is the second London equation,

\nabla \times J_{s} = - \frac{n _{s} e ^{2}}{m} B .

The penetration depth

Combine the second London equation with the static Ampère law $\nabla \times B = μ_{0} J_{s}$ and $\nabla \cdot B = 0$ . Taking the curl of Ampère's law,

\nabla \times (\nabla \times B) = μ_{0} \nabla \times J_{s} = - \frac{μ _{0} n _{s} e ^{2}}{m} B,

and using $\nabla \times (\nabla \times B) = \nabla (\nabla \cdot B) - \nabla^{2} B = - \nabla^{2} B$ , the field obeys a screened equation,

\nabla^{2} B = \frac{1}{λ _{L}^{2}} B, λ_{L} = \frac{m}{μ _{0} n _{s} e ^{2}} .

For a superconductor filling the half-space $x > 0$ with a field $B (0)$ applied parallel to the surface, the bounded solution decays exponentially,

B (x) = B (0) e^{- x / λ_{L}} .

The London penetration depth $λ_{L}$ is the distance over which an external field falls to $1/ e$ of its surface value. For typical metals $n_{s} \sim 1 0^{28} - 1 0^{29} m^{- 3}$ gives $λ_{L} \approx 20 - 50 nm$ . The field is not truly zero at the surface; it is excluded from the bulk by screening currents flowing in the same surface layer, since $J_{s}$ satisfies the identical equation $\nabla^{2} J_{s} = λ_{L}^{- 2} J_{s}$ .¹

An applied field parallel to the surface penetrates only a skin of thickness lambda_L, decaying as B(0) e^{-x/lambda}; the screening supercurrents occupy the same layer.

The rigid condensate wave function

The second London equation reads as an ad-hoc postulate, but it follows from a single macroscopic wave function for the superconducting condensate, $ψ = n_{s} e^{i θ (r)}$ , whose carriers have charge $q$ and mass $m^{*}$ . The gauge-invariant current of such a wave function is

J_{s} = \frac{n _{s} q}{m ^{*}} (ℏ\nabla θ - q A),

where $A$ is the vector potential, $B = \nabla \times A$ . Taking the curl removes the phase gradient, since $\nabla \times \nabla θ = 0$ , and leaves

\nabla \times J_{s} = - \frac{n _{s} q ^{2}}{m ^{*}} B,

the second London equation with $q \to - e$ , $m^{*} \to m$ . The physical content is rigidity: the condensate phase resists deformation, so an applied field generates exactly the screening current that cancels it in the interior. When the pairing charge $q = - 2 e$ and mass $m^{*} = 2 m$ are used, as Ginzburg–Landau theory requires, the penetration depth becomes $λ_{L} = m^{*} / (μ_{0} n_{s} q^{2}) = m / (μ_{0} n_{p} (2 e)^{2} /2)$ with $n_{p} = n_{s} /2$ the pair density, numerically unchanged.²

Thermodynamics of the transition

The superconducting state is the thermodynamically stable phase below $T_{c}$ because its Gibbs free energy is lower. Expelling a field costs energy, so the free energy of the superconductor rises with applied field $H_{a}$ . Per unit volume, holding out a uniform field costs the magnetic work density $\frac{1}{2} μ_{0} H_{a}^{2}$ ,

g_{s} (T, H_{a}) = g_{s} (T, 0) + \frac{1}{2} μ_{0} H_{a}^{2} .

The normal-state free energy is nearly field-independent (its susceptibility is tiny). At the critical field the two phases are in equilibrium, $g_{s} (T, H_{c}) = g_{n} (T)$ , which fixes the condensation energy — the free-energy gain of the superconducting state at zero field,

g_{n} (T) - g_{s} (T, 0) = \frac{1}{2} μ_{0} H_{c}^{2} (T) .

The critical field is measured to follow a near-parabolic law,

H_{c} (T) = H_{c} (0) [1 - (\frac{T}{T _{c}})^{2}],

which separates the superconducting and normal regions of the $(T, H)$ plane.

The critical-field phase boundary. Below the parabola H_c(T) the material is superconducting; above it, normal. H_c falls from H_c(0) at T = 0 to zero at T_c.

The entropy of each phase is $S = - \partial g / \partial T$ . Differentiating the condensation-energy relation,

S_{n} - S_{s} = - μ_{0} H_{c} \frac{d H _{c}}{d T} .

Because $d H_{c} / d T < 0$ , the superconducting state has the lower entropy: it is the more ordered phase, as an ordered condensate should be. The latent heat of the field-driven transition is

L = T (S_{n} - S_{s}) = - μ_{0} T H_{c} \frac{d H _{c}}{d T} .

At $T_{c}$ in zero field $H_{c} = 0$ , so $L = 0$ : the zero-field transition releases no latent heat and is second order. In a finite field, where the transition occurs at $T < T_{c}$ with $H_{c} \neq = 0$ , latent heat appears and the transition is first order.

Differentiating once more gives the heat capacities, $C = T \partial S / \partial T$ ,

C_{s} - C_{n} = μ_{0} T [(\frac{d H _{c}}{d T})^{2} + H_{c} \frac{d ^{2} H _{c}}{d T ^{2}}] .

At $T_{c}$ the second term drops ( $H_{c} = 0$ ) and the heat capacity jumps by a finite amount even though the latent heat vanishes — the signature of a second-order transition. This is Rutgers's formula,

Δ C ∣_{T_{c}} = C_{s} - C_{n} = μ_{0} T_{c} (\frac{d H _{c}}{d T})_{T_{c}}^{2} .

Heat capacity through the transition. The superconducting branch jumps above the normal gamma-T line at T_c, then falls exponentially below it as the gap freezes out excitations; the jump with no latent heat marks a second-order transition.

The intermediate state

The parabola describes a long, thin sample aligned with the field, where the field at the surface equals the applied field. For any other shape the field crowds around the sample and the local surface field exceeds $H_{a}$ . A sphere has demagnetizing factor $N = \frac{1}{3}$ , so the field at its equator is $H_{a} / (1 - N) = \frac{3}{2} H_{a}$ . The equator reaches $H_{c}$ while the applied field is still only $\frac{2}{3} H_{c}$ , and the sample cannot remain wholly superconducting, yet the interior field is too low to go entirely normal.

The resolution is the intermediate state: the sample breaks into alternating laminae of normal and superconducting material, each thin enough that the field in the normal sheets is exactly $H_{c}$ . This geometric subdivision of a type I superconductor is distinct from the vortex mixed state of a type II material, which arises from the sign of the surface energy between the phases and is treated in the next lesson.

The intermediate state of a type I superconductor: alternating normal and superconducting laminae, with the field passing through the normal sheets at exactly H_c and expelled from the superconducting ones.

London theory captures the electrodynamics of the superconducting state with one empirical constant, $λ_{L}$ , and the thermodynamics with one measured function, $H_{c} (T)$ . It says nothing about why the state forms or how deep the condensation energy is. The order parameter $ψ$ introduced here as a device becomes the central object of the phenomenological theory taken up next, and the microscopic origin of the rigidity waits for BCS theory.

Kittel, Ch. 10, London equation and penetration depth. ↩
Ashcroft & Mermin, Ch. 34, on the London rigidity and the macroscopic wave function. ↩