In 1911, in Leiden, Heike Kamerlingh Onnes cooled a thread of solid mercury below 4.2\ \text{K} using the liquid helium he had only just learned to make, and measured its electrical resistance. It did not merely get small. It vanished — dropped to zero, abruptly, at a sharp temperature, as if a switch had been thrown. A current started in a superconducting ring will circulate for years without a battery and without measurably fading; experiments have bounded the decay time at longer than the age of the universe. And if you lay a small magnet above a piece of the cold metal, the magnet floats.

How can resistance be exactly zero — not tiny, but zero? How can a metal push a magnetic field out of itself entirely? These are not small refinements of ordinary conduction; they are a distinct state of matter, and explaining them took nearly half a century and a genuinely strange idea: that electrons, which repel one another, can nonetheless pair up. This page builds the picture from the free-electron model — the sea of electrons and its Fermi surface — up to the Cooper pair and the energy gap.

Zero resistance below a critical temperature

Cool a superconductor and nothing special happens to its resistance until, at a sharp critical temperature T_c, the resistance falls discontinuously to exactly zero. Above T_c it is an ordinary metal; below, it carries current with no dissipation at all. The values of T_c are low: 4.2\ \text{K} for mercury, 7.2\ \text{K} for lead, up to around 90\text{–}130\ \text{K} for the modern high-temperature cuprates.

"Exactly zero" is a strong claim, so it is worth saying what it means. In an ordinary wire, a current needs a voltage to sustain it (Ohm's law, V = IR); switch the driving voltage off and the current dies within nanoseconds as electrons scatter off the vibrating lattice. In a superconductor a current, once established, needs no voltage to keep it going and simply does not decay. That is the operational meaning of zero resistance: a truly persistent current.

The Meissner effect: expelling the field

The truly defining property is not zero resistance but what happens to a magnetic field. Cool a superconductor in a magnetic field and, as it crosses T_c, it actively pushes the field out of its interior: inside a superconductor, \mathbf{B} = 0. Screening currents spring up on the surface, exactly cancelling the field within. This is the Meissner effect (Meissner and Ochsenfeld, 1933), and it makes a superconductor a perfect diamagnet — \chi = -1, the strongest possible diamagnetic response. It is why a magnet levitates above a superconductor: the expelled field pushes back.

This field expulsion is what marks superconductivity as a genuine thermodynamic phase of matter, not just "a metal that happens to have no resistance". As the vignette below insists, a hypothetical perfect conductor would behave quite differently. The distinction is subtle and it is the heart of the physics.

Critical field and critical current

Superconductivity is fragile in a specific way: it can be destroyed by too strong a magnetic field or too large a current. Above a critical field H_c the field wins, forces its way in, and the material reverts to a normal metal. The critical field itself weakens as the temperature rises towards T_c, following a near-perfect parabola:

At T = 0 the material can withstand the full H_c(0); at T = T_c the critical field has fallen to zero, so an infinitesimal field destroys superconductivity. The curve traces the boundary between the superconducting and normal regions, plotted interactively below. Likewise a critical current exists: push too much current through and the magnetic field it generates at the surface exceeds H_c, quenching the state.

Worked example — critical field at a given temperature. A superconductor has H_c(0) = 0.08\ \text{T} and T_c = 7.2\ \text{K} (lead). At T = 3.6\ \text{K}, i.e. T/T_c = 0.5,

Three-quarters of the zero-temperature value survives at half the critical temperature — the parabola is flat near the top, so you keep most of the field tolerance until you get close to T_c.

Cooper pairs: electrons that attract

Here is the puzzle the theory had to solve. Electrons all carry negative charge and repel one another through the Coulomb force. So how can they team up? The answer, due to Leon Cooper in 1956, is that in a metal the electrons are not alone — they sit in a lattice of positive ions that can move.

Picture one electron speeding through the lattice. Its negative charge tugs the nearby positive ions very slightly inward, leaving behind a small, momentary region of excess positive charge — a ripple in the lattice, a phonon. Because the heavy ions respond sluggishly, that positive ripple lingers after the first electron has moved on, and a second electron is drawn toward it. One electron has attracted another — not directly, but through the deformation of the lattice it left in its wake. This phonon-mediated attraction can, for electrons right at the Fermi surface, just barely overcome the screened Coulomb repulsion, binding them into a Cooper pair.

The gap is the crux of zero resistance. In a normal metal an electron can scatter off an impurity or a phonon by any tiny amount, losing a little energy each time — that is resistance. In a superconductor the paired electrons move as one rigid condensate, and to disturb a single one you must supply at least 2\Delta to break its pair. At low temperatures the ambient thermal energy k_B T is simply too small to do that, so the condensate glides on undisturbed.

BCS theory and the gap relation

In 1957 John Bardeen, Leon Cooper and Robert Schrieffer turned Cooper's single-pair idea into a full microscopic theory of the superconducting ground state — BCS theory, one of the great achievements of twentieth-century physics. It predicts a whole condensate of overlapping pairs and an energy gap \Delta that opens at T_c and grows as the material is cooled. One of its cleanest predictions is a universal ratio tying the zero-temperature gap to the critical temperature:

This is remarkable: it says the gap and the transition temperature are two views of the same thing, related by a pure number that is (nearly) the same for every conventional superconductor. Measure T_c and you know \Delta, and vice versa.

Worked example — the gap of lead. Lead has T_c = 7.2\ \text{K}. Using 2\Delta = 3.53\,k_B T_c, the full gap in temperature units is 2\Delta/k_B = 3.53 \times 7.2 \approx 25.4\ \text{K}, so the half-gap is

A gap of about a millielectron-volt — tiny compared with the electron-volt scale of chemistry, which is exactly why superconductivity is a low-temperature affair: you must cool below the point where thermal energy can bridge the gap.

BCS also predicts that the gap \Delta(T) shrinks as the temperature rises and closes at T_c, where the pairs finally break apart and the material returns to normal — the superconducting analogue of a melting point.

Type-I, type-II, and the open puzzle

Superconductors come in two flavours according to how they meet a magnetic field. Type-I superconductors (most pure elemental metals) expel the field completely up to H_c, then abruptly go normal. Type-II superconductors (alloys, niobium compounds, the cuprates) do something cleverer: above a first critical field they let the field thread through in a lattice of quantised flux vortices, each a tiny normal-metal core carrying one quantum of flux, while the material around them stays superconducting. This lets type-II materials survive to enormous fields, which is why the powerful magnets in MRI scanners and particle accelerators are wound from type-II superconducting wire.

And the frontier: in 1986 a family of copper-oxide ceramics — the cuprates — was found to superconduct above the boiling point of liquid nitrogen (77\ \text{K}), far higher than BCS was thought to allow. Decades on, exactly why the high-temperature superconductors work — whether phonons, or magnetic interactions, or something else provides the pairing glue — remains one of the great unsolved problems of condensed-matter physics.