Superconductivity, the Meissner Effect and BCS Theory

By the end you should be able to: Define transition temperature and critical field, use the parabolic critical-field law, explain why the Meissner effect proves a superconductor is more than a perfect conductor, distinguish Type I from Type II, and outline BCS theory qualitatively.

In 1911 Kamerlingh Onnes cooled mercury below 4.2 K and its resistance did not merely become small — it became unmeasurable. Currents started in superconducting rings have been watched for years without detectable decay.

Transition temperature

Below a sharp transition temperature (or critical temperature) $T_c$ , characteristic of the material, the resistance falls to zero. The transition is abrupt, not gradual: over a few thousandths of a kelvin the resistance drops by many orders of magnitude.

Some values: mercury 4.2 K, lead 7.2 K, niobium 9.2 K, and among the high-temperature ceramics YBCO at about 92 K — above the boiling point of liquid nitrogen (77 K), which is why YBCO matters commercially. Note that the best ordinary conductors, copper and silver, never become superconducting at all.

Critical field

A magnetic field destroys superconductivity. Above a critical field $H_c$ the material reverts to normal, however cold it is. The critical field itself depends on temperature:

\boxed{H_c(T) = H_c(0)\left[1 - \left(\frac{T}{T_c}\right)^{2}\right]}

A parabola: maximum at absolute zero, falling to zero at $T_c$ . The two ways to break superconductivity — warming it and applying a field — are therefore not independent. As you approach $T_c$ , an ever weaker field suffices.

Critical current follows from this. A current in a wire makes its own magnetic field, and when that field reaches $H_c$ at the surface, superconductivity is destroyed. For a wire of radius $r$ , Silsbee's rule gives

I_c = 2\pi r H_c

So a superconductor is not an unlimited conductor. Exceed the critical current and it stops being a superconductor at all.

The Meissner effect — and why it is the important one

Cool a superconductor below $T_c$ in a magnetic field and it expels the field from its interior. Inside the material $B = 0$ .

It is tempting to think this follows from zero resistance. It does not, and the distinction is the physics of this topic.

Consider a hypothetical perfect conductor — zero resistance, nothing more. Faraday's law says a changing flux induces an emf; with zero resistance, any change in flux would drive an infinite current. So a perfect conductor keeps the flux through it constant. If you cooled it in a field, the field would be trapped inside, not expelled.

A superconductor expels the field regardless of history. Cool it in a field, then measure: the field is gone. Apply the field after cooling: still gone. The final state does not depend on the order of operations.

That is a genuinely different property, and it means superconductivity is a distinct thermodynamic phase, not merely a limit of very good conduction. A superconductor is a perfect diamagnet, with susceptibility

\chi = -1, \qquad \mu_r = 0

The expulsion is achieved by persistent screening currents in a thin surface layer — the London penetration depth, typically tens of nanometres — that generate a field exactly cancelling the applied one inside.

Levitating a magnet above a cooled superconductor is the Meissner effect made visible: the expelled field pushes back.

Type I and Type II

Type I. One critical field. Below $H_c$ , complete expulsion; above it, an abrupt return to normal. These are mostly pure elemental metals — lead, mercury, tin — and their critical fields are low, of order $10^4$ to $10^5$ A m⁻¹. Too low for magnets, which is why Type I superconductors are of little practical use for high fields.

Type II. Two critical fields, $H_{c1}$ and $H_{c2}$ .

Below $H_{c1}$ : complete expulsion, exactly as Type I.
Between $H_{c1}$ and $H_{c2}$ : the mixed or vortex state. The field penetrates in quantised flux tubes, each with a normal core, while the material between them stays superconducting and still carries current without resistance.
Above $H_{c2}$ : normal.

$H_{c2}$ can be enormous — tens of teslas. Every superconducting magnet ever built, including every MRI scanner, uses a Type II material (typically niobium-titanium or niobium-tin), because only Type II survives the fields such magnets themselves produce. These are alloys and compounds rather than pure elements, and all the high-temperature ceramics are Type II.

BCS theory — qualitatively

Bardeen, Cooper and Schrieffer explained the mechanism in 1957.

The puzzle is that electrons repel. BCS shows that in a lattice they can nonetheless attract each other indirectly. An electron moving through the lattice pulls the positive ions slightly towards it, leaving a momentary region of excess positive charge in its wake. A second electron is attracted to that region. The lattice distortion — a phonon — mediates an attraction between the two electrons.

The attraction is feeble, which is why it survives only at low temperature where thermal agitation cannot break it.

Two electrons bound this way form a Cooper pair, with equal and opposite momenta and opposite spins. The pair has integer spin, so it behaves as a boson, and bosons are not subject to the exclusion principle: all the pairs can and do condense into a single quantum state described by one wavefunction across the whole sample.

Resistance requires scattering an individual electron. To scatter one member of a pair you must first break the pair, which costs the energy gap — and at low temperature no scattering event has that much energy available. The pairs therefore move without dissipation. Zero resistance.

Supporting evidence: the isotope effect, where $T_c \propto M^{-1/2}$ for different isotopes of the same element. Since the ion mass sets the lattice vibration frequency, the fact that $T_c$ depends on it at all is direct evidence that phonons mediate the pairing.

BCS explains the conventional low-temperature superconductors well. It does not satisfactorily explain the high-temperature cuprates, whose mechanism remains an open problem.

Applications

MRI scanners — the largest commercial use by far. Superconducting magnets sustain a strong, extremely stable field with no ongoing power cost once energised.
Maglev trains — levitation and guidance by superconducting magnets.
SQUIDs — superconducting quantum interference devices, the most sensitive magnetometers known, sensitive enough to detect the magnetic fields of brain activity.
Particle accelerator magnets — the LHC's beam-steering dipoles.
Lossless power transmission and fault current limiters, in limited trials; cryogenic cost is the obstacle.