Superconductors: current with nothing in its way, and what warmth would change

Level 0 · The picture

Push electricity through an ordinary wire and some of it turns into heat. That is not a defect of the wire. The electrons knock into a lattice of atoms that is jiggling with warmth, and each knock takes a little energy out of the current and leaves it behind as heat. It is why a phone charger is warm, why power lines lose a share of what they carry, and why a strong electromagnet needs a cooling system more than it needs a power supply.

Now cool certain materials far enough and the knocking stops. Not "mostly stops" — stops. The current goes round and round with nothing taking anything out of it. Start a current in a closed loop of such a material and it keeps going.

That is the first thing. Here is the second, which is stranger and more useful. Bring a magnet near such a material and the magnetic field does not go through it. The material pushes the field out of its own interior and keeps it out. This is the Meissner effect, and it is what makes a magnet hang in the air over a cold ceramic disc.

Say this part carefully, because it is where people go wrong. A superconductor is not a source of energy. It is a road with no friction on it. Friction-free is not the same as free: you still have to put the car on the road, you still have to get it moving, and here you also have to pay — continuously — to keep the road cold. What a superconductor removes is a loss, and removing a loss is enormously valuable without being the same thing as making energy. A loop of superconducting wire with a current in it holds exactly the energy that was put into it, and not one joule more.

Ways to think about it

• Zero resistance is about what happens to a current. The Meissner effect is about what happens to a magnetic field. They are two sides of the same thing, and the second is the one that tells you a superconductor is genuinely a new state of matter and not just a very good conductor.
• A very good conductor would trap whatever field happened to be inside it when it got good. A superconductor throws the field out — even one that was already there. That is the difference, and it was the observation that settled the question.
• The cold is not part of the physics. It is the price of admission: warmth is disorder, and the paired, in-step state is an ordered one.

Level 1 · Foundations

A superconductor has a transition temperature: cool it past that and it superconducts, warm it past that and it does not. The number is a property of the material, and it is the number every popular account gives.

It is worth knowing early that it is not one number but a family of them, and that the family spreads enormously. Some materials need to be within a few degrees of absolute zero. The copper-oxide ceramics — the cuprates — need much less. Yttrium barium copper oxide, universally called YBCO, has a transition at about 92 K, which is above the boiling point of nitrogen, and that single fact is what turned superconductivity from a specialist's cryogenic art into something a university laboratory could do routinely: liquid nitrogen is cheap, and true Meissner repulsion became available to anyone who could pour it.

The two facts, said properly.

Zero resistance. A current started in a superconducting ring keeps circulating. There is no measured decay; experiments put upper bounds on it, and the bounds are absurd. This is not "resistance too small to see" in the way a very pure copper wire is — it is a different mechanism, and Level 3 says what it is.

The Meissner effect. Cool a piece of superconductor while a magnetic field is already threading it, and the field is expelled as the material crosses its transition. The expulsion is done by currents that spring up in a thin skin at the surface and circulate without loss, generating exactly the field needed to cancel the applied one inside. Those currents do not run down, so the cancellation does not either. When a superconducting shell was wrapped around a magnet so that the field could not escape, the proof that the shell had genuinely gone superconducting was that the field it contained came in whole packets and nothing leaked — the expulsion certified itself.

Why the cold. Warmth is the lattice jiggling and the electrons being knocked about by it. The paired state has a gap: it costs a definite minimum amount of energy to break a pair. While the typical thermal kick is smaller than that gap, the pairs survive and the state holds. Warm the material until the kicks are the size of the gap and the state comes apart. That is the whole of "why does it need to be cold", and it also tells you what a warmer superconductor would have to be: one with a bigger gap, or one whose pairing survives more disorder.

Level 2 · The three limits

This is the level most accounts skip, and skipping it is how magnets get destroyed.

A superconductor does not have one ceiling. It has three, and it stops superconducting when it meets any one of them:

• the critical temperature, above which the paired state does not form;
• the critical field, the magnetic field past which the material can no longer hold the field out;
• the critical current, the current past which the state cannot be sustained.

And they are not independent. The current in a wire makes a magnetic field of its own, which eats into the field budget; a field pushed close to the ceiling leaves almost no current headroom; warming the material lowers both of the others. A conductor sitting comfortably below its transition temperature, in a field it is rated for, will still quench if the current is pushed past what the other two have left it.

The three limits, in one budget

A superconductor is usually given one number, its transition temperature, and that is the half of the story that loses magnets. There are three ceilings — temperature, magnetic field, and the current through the material — and the material stops superconducting when it meets any one of them. They also share a single budget: raising the field lowers the current the wire will carry, and warming it lowers both. Move the three controls and watch which one takes the largest share.

The three limits, in one budget

Temperature, as a fraction of the transition temperature

30%

Magnetic field, as a fraction of its own ceiling

20%

Current, as a fraction of its own ceiling

20%

Back to a cold, quiet, lightly loaded sample

• Envelope at the chosen current
• Envelope carrying no current
• Where this sample is being run

Budget used:

49%

Headroom left:

51%

temperature:

9%

magnetic field:

20%

current:

20%

State: Superconducting. Current flows with nothing in its way, and the magnetic field is held out of the interior.

Largest claim on the budget: magnetic field

Settled physics. That a superconductor has three ceilings rather than one, and that they trade against each other, is ordinary engineering practice and is why a magnet is specified at a temperature and a field together. The particular curve drawn here is the usual teaching form of that trade.

• Why a magnet is specified at a temperature and a field at once
• Where the field ceiling comes from: a superconductor holds magnetic field out until it cannot

The two kinds of superconductor. Some materials hold the field out completely and then, at one field, give up all at once. These are type I, and their critical fields are far too low to build a useful magnet from. Others do something much better. Past a first critical field they stop trying to hold the field out as one block and let it through as a countable set of thin tubes — each tube carrying exactly one quantum of magnetic flux, each with a small normal core — and the material goes on superconducting around them up to a second critical field many times higher. These are type II, and every strong superconducting magnet in the world is wound from one.

Flux pinning, and why it is the other half of the story. A flux tube in that mixed state is free to move, and a moving flux tube dissipates. Push a current through and the tubes are shoved sideways; the shoving costs energy; the "lossless" wire develops a voltage. The fix is not quantum mechanics but metallurgy — defects, grain boundaries, deliberate inclusions — anything that holds a tube where it is. Pinning is also what makes the levitation in the picture at Level 0 stable: pure expulsion repels a magnet but does nothing to stop it sliding away sideways, whereas pinned tubes have to be dragged before the magnet can move at all, so it hangs where it was left — above the sample, beside it, or underneath it.

Two kinds of superconductor, and why the magnets are all the second kind

Everyone is taught that a superconductor pushes magnetic field out. That is the whole story for the first kind, which gives up completely at a field far too low to build anything with. The second kind does something better: past a first critical field it stops holding the field out as one block and lets it through as a countable set of thin tubes, each carrying one quantum of flux, each with a small normal core, and it stays superconducting around them up to a much higher second field. Choose a kind, raise the field, and decide whether the flux is pinned in place.

Which kind of superconductor

Type I — holds the field out, then gives up all at onceType II — lets the field through as flux tubes and keeps working

Applied magnetic field, as a fraction of the highest field drawn

Applied magnetic field, as a fraction of the highest field drawn

45%

Current through the sample, as a fraction of what it could carry

40%

Pin the flux tubes in place (defects, grain boundaries, deliberate inclusions)Back to a lightly loaded sample in a small field

• Flux tubes
• Pinning sites

Field that has got inside:

35%

Flux tubes:

4 / 12

Which kind of superconductor:

15% — 100%

Inside the material: Mixed state. Flux tubes are through the material, each with a normal core, and the material is superconducting around them.

Carrying this current: With no voltage across it. This is the state a magnet is wound to work in.

Will it hold a magnet at a fixed distance? Yes. Pinned tubes have to be dragged before the magnet can move, so it hangs where it was left — above the sample, beside it, or underneath it.

Settled physics. The mixed state, the quantisation of the flux each tube carries, and the loss that follows when flux moves are standard superconductivity and are the reason large magnets are wound from type II conductors.

• Flux through a superconductor comes in whole quanta, and the expulsion is what proves it
• Large bulk superconductor discs, made to specification and run under levitation fields

Ways to think about it

• "Superconductors expel magnetic field" is the type I sentence. The type II sentence is: they expel it until it is cheaper to let it in as countable tubes, and then they keep working around the tubes.
• A superconducting magnet is specified at a temperature and a field and a current, always together, because each one is spending the same budget.
• A quench — the sudden loss of superconductivity in a working magnet — is what happens when a small warm spot eats the last of that budget locally, and the stored energy of the magnet then has to go somewhere.

Level 3 · Undergraduate

One wave, not many particles. Below the transition the charge carriers are not individual electrons but Cooper pairs, and every pair occupies the same quantum state. The whole condensate is then described by a single complex function — an amplitude and a phase — that extends across the entire piece of metal. Phase, which for one electron is a private bookkeeping quantity, has become a property of an object you can pick up.

Flux quantisation falls straight out of it. Go once around a superconducting ring and the wave must come back to itself: the phase can only change by a whole number of turns. Because the phase responds to the vector potential, that requirement forces the magnetic flux threaded through the ring to come in whole multiples of a single quantum, h/2e. The two in the denominator is the charge of a pair, and its presence is itself the evidence that the carriers are paired. When a toroidal magnet was sealed inside a superconducting layer and read out by electron holography, the relative phase between the two electron beams came out as either nothing or exactly half a turn and never anything between — an odd or an even number of trapped quanta, and no other option on the menu.

The interferometer with the field switched off

Split an electron beam, send the two halves either side of a magnetic flux that is completely confined and completely shielded, and bring them back together. On the paths themselves the magnetic field is exactly zero — the shaded corridor below is field-free the whole way. The vector potential there is not zero, and the fringes move. Dial the enclosed flux and watch the pattern slide sideways by exactly one fringe for every flux quantum. Then try the second control. It changes the gauge — the potential itself, at every point on both paths, as the arrows show — and the fringes do not move at all. Both controls are live; only one of them changes anything a detector can see.

Two controls: one moves the fringes, one cannot

Enclosed flux, in flux quanta h/e

0.00

Advance by one whole flux quantumReturn to zero enclosed flux

Gauge choice λ — the freedom in the potential itself

0.00

Return to the textbook gauge

• Upper path
• Lower path
• Confined flux
• Vector potential on the paths

What is happening: The enclosed flux is a whole number of quanta, so the phase difference is a whole number of turns and the pattern sits exactly where it started. This is the periodicity that makes the effect a measurement rather than a drift.

Magnetic field on the electron paths

zero, at every setting

Vector potential on the upper path, in this gauge

0.00 nWb/m

Vector potential on the lower path, in this gauge

0.00 nWb/m

Phase the gauge adds to each half

0.00 radians — the same for both halves, so it cancels

Phase difference between the two halves

0.00 radians

Fringe shift

0.00 fringe widths

What no choice of gauge can change

the field on the paths, the phase difference, the fringe shift

The precision rule this instrument exists to protect: the vector potential controls phase. No energy is stored in it, and none needs to be — the phase difference is set by the flux the two paths enclose between them, which is exactly the quantity you are dialling. And the folklore reading — that the effect proves the potential is physically real at a point — is exactly what the second control is here to foreclose: the honest observable is the loop integral, never the potential at a point.

Nothing happening is the result here, not a broken control. The potential is not unique: for any smooth function χ you may replace A by A + ∇χ, which changes the potential at every point and changes the magnetic field nowhere, because the curl of a gradient is zero. This instrument performs that substitution for real and then integrates the full potential, gauge term included, around the closed loop the two halves of the beam make between them. Each half does pick up a phase from the gauge term — exactly 2λ radians in this family — and both halves pick up the same one, so it cancels out of their difference. What survives is the loop integral, which is the enclosed flux, and that is the observable.

Tonomura's 1986 experiment answered the obvious objection by construction: the flux sat inside a toroidal magnet wrapped in a superconducting shield, so stray field was not merely small on the electron wave but excluded from it. The shift was still there, and it was still one fringe per flux quantum.

Settled physics. Scope: a drawn two-path interferometer with a perfectly confined flux, which is the geometry of the published experiment. Beam energy, path length and magnet size are not modelled, and no absolute intensity is asserted. Φ₀ = h/e ≈ 4.136e-15 Wb.

The source sheet: Tonomura et al. 1986, Evidence for Aharonov–Bohm effect with magnetic field completely shielded from electron wave

What this mount teaches that its original does not. On the vector-potential course this interferometer makes the point that the potential, not the field, is what an electron wave responds to. Here the same instrument is read the other way round: impose the single-valuedness of a macroscopic wave on the same loop and the phase is no longer free to take any value — it is forced onto a ladder. The Aharonov–Bohm phase is the mechanism; flux quantisation is what that mechanism does to an object big enough to hold.

The energy gap. Breaking a Cooper pair costs a minimum energy. Nothing can be exchanged with the condensate below that amount, which is why small thermal knocks, small impurities and small mechanical disturbances do nothing at all. The gap is the reason the resistance is exactly zero rather than merely small, and it is also a hard ceiling in frequency: drive a superconductor above the frequency whose quantum matches the gap and the pairs break and the superconductivity is destroyed. It also has a subtler consequence worth carrying forward — the gap protects the condensate from decoherence, so the pairs stay delocalised across the whole sample in a way the ordinary lattice ions do not, and the two therefore do not have to move together.

Two lengths. A superconductor has two characteristic distances, and almost everything about a given material follows from which is larger.

• The penetration depth is how far a magnetic field reaches into the surface before the screening currents have cancelled it. The Meissner effect is not a discontinuity at the face of the metal; it is an exponential decay over this length.
• The coherence length is the distance over which the paired state can change — how sharply a boundary between superconducting and normal regions can be drawn.

A material whose coherence length is the larger of the two is type I. One whose penetration depth is the larger is type II, because a boundary between normal core and superconducting surroundings then costs the material nothing and it is happy to make many of them. That is the whole reason the two kinds exist: one ratio of two lengths.

The London equations. The oldest description, and still the most useful one to hold in your head, is a pair of relations that replace Ohm's law inside a superconductor. Instead of "electric field drives a steady current", one says "electric field accelerates the current" — which is what a carrier with no friction does. Combine the second with Maxwell's equations and the expulsion of magnetic field comes out as an exponential decay over the penetration depth. This is a phenomenological description: it says what the condensate does, not why it exists. Its shape is durable enough to be rewritten for other fields entirely — the same relation, written for gravity instead of electromagnetism, produces a gravitational penetration depth in exactly the same form.

The London result: the Meissner effect as an equation(1)

$$\nabla^2 \mathbf{B} = \frac{1}{\lambda_L^2}\,\mathbf{B}, \qquad \lambda_L = \sqrt{\frac{m}{\mu_0 n_s q^2}}$$

Definitive Zero resistance, the Meissner effect, the two kinds, flux quantisation in units of h/2e and the energy gap are measured, reproduced and used in industrial hardware.

Level 4 · Graduate

Ginzburg–Landau: the condensate as an order parameter. Rather than starting from the electrons, write the free energy of the material as a series in a complex field — the order parameter — that is zero above the transition and grows below it. Its magnitude squared is the density of the condensate; its phase is the phase of Level 3. Minimising that free energy gives two coupled equations whose solutions contain the penetration depth, the coherence length, their ratio, and therefore the type I / type II distinction as a single dimensionless number. It is the description you reach for whenever the condensate is not uniform: near a surface, near a normal region, inside a flux tube, or close to the transition itself, where the order parameter is small and its response to an external field is at its most sensitive.

The Josephson relations. Put two superconductors within a few atoms of each other and their two waves lock together through the barrier. The current across the gap is set by the difference of their phases alone, with no voltage anywhere; and a voltage across the gap makes that phase difference wind at a rate fixed by fundamental constants only. This is the point at which the macroscopic phase becomes a thing you can wire to a battery, and it has a whole course of its own on this site — the junction, the volt, the SQUID and the qubit — so here it is only the bridge.

Turn the voltage, watch the frequency follow

A bias voltage across the gap makes the phase difference wind round, and the current oscillates at a rate the voltage alone sets — about 484 megahertz for every microvolt, with nothing about the metal, the barrier or the temperature in it. The marks along the axis are the eleven operating points Stephenson tabulates for his junction array, and the line through them is built from one of his own rows. Move the dial and the frequency, the wavelength and the emitter that would radiate it all follow.

Bias voltage across the junction

Bias voltage across the junction

49.73

Return to the band the paper selects

Jump to a published operating point

1.296 GHz (2.68 µV)2.32 GHz (4.80 µV)3.4 GHz (7.03 µV)5.76 GHz (11.91 µV)10.368 GHz (21.44 µV)24.048 GHz — the band the paper selects (49.73 µV)47.088 GHz (97.37 µV)76.233 GHz (157.64 µV)120 GHz (248.14 µV)

• The rate the relation sets
• the band the paper selects

Josephson frequency:

24.048 GHz

The rate the relation sets:

483.571 MHz/µV

Free-space wavelength:

1.247 cm

Quarter-wave emitter pad:

0.312 cm

Full-wavelength spacings across a 150 mm wafer:

12.0

What the junction is doing: The dial is standing on a published row of the table.

Where the line comes from. It is not a constant copied in. The slope is one published row divided by itself — 24.048 gigahertz over 49.73 microvolts — and the other ten rows of the same table land on that line to better than a tenth of a percent. That agreement is the whole point: a relation built only from the charge of a pair and Planck's constant has no room in it for the metal, the barrier or the temperature.

Why this band. The paper spaces its emitter pads by full wavelength increments to keep them in phase, so the wavelength decides how many can share a wafer. Below about four gigahertz fewer than two spacings fit across a 150 millimetre wafer; at the selected band twelve do. Run the dial down and watch the count fall — the author's own reason for the choice, reached from his own figures.

Scope: the second Josephson relation drawn as a line, with a published table of operating points laid on it. Critical current, junction materials, linewidth and emitted power are not modelled, and nothing about gravitational radiation is asserted or measured here.

Settled physics · Designed, not yet built. Two different standings, and they are not the same. The voltage-to-frequency conversion is the settled part: it has defined the volt since 1990. The junction array that would put it to work as a gravitational-wave emitter is a paper design with an open problem its own author names, and the course takes that up at Level 5.

The source sheet: Stephenson 2026, Stimulated Emission of Gravitational Waves via Dissimilar Superconducting Josephson Junctions — Section 5.1, Table 1

What this mount teaches that its original does not. On the junction course this dial is the junction's own defining rule. Here it is the answer to the question Level 3 leaves hanging: what is a macroscopic phase actually good for? It is good for this. Two of the things a phase can be — locked, or winding — are a current source and a frequency source, and the frequency has no property of the metal in it.

The pairing theory, and its boundaries. The microscopic account explains how an attraction between electrons can exist at all despite their mutual repulsion, why the resulting state has a gap, how the transition temperature depends on the strength of that coupling, and why the carriers have charge 2e. It is one of the most quantitatively successful theories in condensed matter, and it explains the low-temperature superconductors extremely well.

What it does not do is settle the cuprates. The theory says: give me the coupling and I will give you the transition temperature. It does not say which couplings are available in a given material, nor does it promise that the electron–lattice route is the only one. In the copper oxides the transition temperatures are far above what that route comfortably accounts for, the state is anisotropic in a way the simple version is not, and the normal state above the transition is itself unlike an ordinary metal. This is the honest boundary between "solved" and "open", and stating it as a boundary rather than a failure is the accurate description. The tools for probing it now include reshaping the electromagnetic environment a sample sits in — a way to modify a material and, at the same time, a way to ask which lattice vibrations actually matter in a given compound, including the ones whose microscopic mechanism is still open.

Strong Ginzburg–Landau and the Josephson relations are standard and are used to design hardware. The pairing mechanism of the high-temperature families is not settled, and the site's word for that is below.

Level 5 · Research frontier

Read this level as a landscape, not a league table. The useful question is not which material holds a record. It is: what does a transition temperature actually depend on, and which of those dependencies can be reached from outside? Four of them, each with published work behind it.

The coupling. The transition temperature rises with the strength of whatever binds the pairs. Anything that stiffens or softens the relevant lattice vibrations moves it. In one experiment the coupling was altered without touching the sample at all, by putting it in a plasmonic environment whose vacuum electromagnetic field was squeezed into a very small volume and letting a polymer with matching vibrations relay the coupling: a rubidium-doped fullerene moved from 30 K to 45 K, and YBCO moved the other way, from 92 K to 86 K. The direction of the shift is the interesting part. An effect with a sign is a mechanism; an effect that only ever improves things is a measurement artefact.

Dimensionality and the layers. The cuprates are stacks of copper-oxide planes, and how well the planes talk to each other is a separate question from what happens inside one. That is why interlayer coherence is a thing you can watch appear on its own: a stripe-ordered cuprate that does not superconduct at all was hit with a mid-infrared pulse and became a transient three-dimensional superconductor, the coherent tunnelling between planes announcing itself as a Josephson plasma resonance and assembling within one to two picoseconds. Nothing about that is a room-temperature device. What it is, is a clean demonstration that the superconducting state can be assembled by driving a material rather than only by cooling it, and that the assembly is fast.

Pressure. Squeezing a material shortens its bonds, stiffens its lattice and rearranges its electronic structure, and all three feed the transition temperature. The physics of why pressure helps is the same physics as why cold helps, read from the other end: cold protects a delicate ordered state from being shaken apart, and pressure makes the state less delicate in the first place by strengthening the interaction that builds it. The rungs that matter for this course are the mechanism and the trade — a state that needs enormous pressure to exist has moved its difficulty rather than removed it, in the same way that a state needing enormous cold has. This site's library carries no sheet on pressure-induced superconductivity, so no pressure, no temperature and no material is named here; the mechanism above is stated and the numbers are not, which is the house rule when the shelf cannot support them.

The vacuum the material sits in. The thread this site follows arrives here. The electromagnetic environment around a sample is not passive, and the plasmonic result above is one demonstration that it can be used as a control knob without any illumination at all. Chapter 2's subject — that the vacuum has structure and that structure is adjustable — meets chapter 11's subject here, in a measurement that already exists.

Suggestive Modifying a transition temperature by reshaping a sample's electromagnetic environment is published, with controls, and has a sign. What it is not yet is a route to a material anyone can wind into a magnet.

What to watch

1. A modified transition temperature reproduced by a second group, in a second material family, with the control samples reported alongside — the same shape of evidence the plasmonic result already reports for itself.
2. Transient superconductivity driven into a material lasting long enough to carry a measurable current, rather than long enough to shift an optical response.
3. A statement of the pairing mechanism in the copper oxides that predicts a transition temperature for a material nobody has made yet, and then that material being made.

Applications · What it is all for

Everything above is paid for here. Each of these is running hardware, and each has a cooling plant attached to it.

Magnets, and therefore scanners. The first and still the largest use. A superconducting winding carries a current that does not decay into a magnetic field that does not need topping up, and the field it reaches is limited by the three ceilings of Level 2 rather than by how much heat you can remove from a resistive coil. Medical imaging, research magnets, and the confinement magnets of a fusion machine are all the same engineering problem. The list of what pushes the materials forward is short and revealing: the sensitivity of magnetometer circuits, the wish for better scanners, energy storage, transformers and grid delivery, and motors and generators — and the obstacle is just as concrete, because the ceramics with the best temperatures are brittle and hard to form into a coil, while the metals that form beautifully into coils need far more cooling.

Levitation and maglev. Flux pinning from Level 2, at scale. A pinned type II sample does not merely repel a magnet, it holds position relative to it, which is what turns levitation from a party trick into a bearing or a guideway. The same pinned bulk material is dual-use in a way worth noticing: the large ceramic discs made for gravity-modification experiments are the same components as a power-storage flywheel, because a rotor that floats on pinned flux has no bearing to wear out.

Magnetometry. Two junctions in a superconducting ring make an interferometer for the condensate's own phase, and because the phase is tied to the flux through the ring, the device counts flux quanta. That is the most sensitive magnetic measurement there is, and it is why these circuits appear at the head of the list of what drives the materials forward. It is also the workhorse instrument of the field itself — the plasmonic transition-temperature shifts of Level 5 were read out on one.

The volt. Drive a junction with microwaves and its voltage locks to exact multiples set by the drive frequency and fundamental constants, with no property of the metal anywhere in the relation. That is not a good way to measure a volt; it is what the volt is. It is the cleanest example on this site of a quantum effect promoted to the status of a definition, and the junction course carries it in full.

Quantum processors. A junction is a nonlinear circuit element, and a nonlinear circuit has unevenly spaced energy levels, so its lowest two can be addressed on their own. Shunt it with a large capacitor and the sensitivity to stray charge falls away exponentially while the unevenness you need falls only slowly — a lopsided trade in the designer's favour, and the design underneath essentially every superconducting quantum processor in service.

Resonant cavities and accelerators. A cavity's usefulness is how many times a wave can bounce inside it before the walls have absorbed it. Superconducting walls absorb so little that the stored field survives orders of magnitude longer, which is why particle accelerators use superconducting cavities and why the same cavities are the detectors of choice for very weak high-frequency signals.

Lossless transport and fault-current limiters. A cable that carries current without loss is the obvious application and the hardest one, because the cable has to be cold along its entire length. The less obvious application is better: a superconductor's transition is itself a switch. A device sitting comfortably below its critical current is invisible; a fault current takes it past that ceiling in a fraction of a cycle, it becomes resistive, and the fault is limited by the material's own physics rather than by something having to notice and act.

And this site's own thread. Both of the gravitational-wave source designs this site follows are superconducting hardware. One puts an array of Josephson junctions between dissimilar superconductors — a conventional one joined to a cuprate — on a wafer and drives it at 24 gigahertz; the other uses the gap between the two pairing symmetries of a cuprate as the two levels of the emitter, pumped electrically through a junction from a low-temperature superconductor. Neither is exotic in its superconductivity: both are built from Level 2's type II ceramics, Level 3's condensate and Level 4's junction. What is open in them is the coupling to gravity, not the materials.

Speculative Junction arrays as gravitational-wave sources: fully specified designs with named open questions and first experiments; no signal yet.

What warmth would change

Now the question everyone actually asks, asked the way it can be answered: hold the physics of the levels above fixed, remove the cooling requirement, and go down the list.

The cooling plant is rarely the biggest item; it is almost always the item that decides the shape of the machine. A scanner is built around its cryostat; the magnet is a smaller object than the vessel that keeps it cold. So:

• Magnets and scanners. The field would not get stronger — that is set by the critical field and the critical current, not by the cooling. What changes is size, weight, siting and service. A magnet that needs no cryogenic plant can go where the plant cannot, and a machine whose cold vessel is removed is a fraction of its former volume.
• Levitation and bearings. The clearest win, because the application is mechanical and the cryostat is directly in the way of the moving part. A pinned bearing needs a cold rotor near a warm shaft, which is an engineering compromise at every point; take the cold away and the compromise goes.
• Magnetometry and processors. Least changed, and this is the counter-intuitive one. A quantum processor is cold not only to be superconducting but to be quiet — thermal noise would swamp a microwave-frequency qubit long before it broke a pair. A warm superconductor removes one reason for the refrigerator and leaves the other standing.
• Transport and fault limiters. The largest change of all, because the cost here scales with length. A cable that needs cooling along every metre is an economic problem that does not get better with scale; one that does not is ordinary infrastructure.
• Cavities and accelerators. Partly changed. The cryogenics is a large share of an accelerator's operating cost. But cavity performance depends on surface quality at a level that cold currently helps deliver, so the gain is real and not total.

The honest summary is that the cooling requirement is a multiplier on deployment, not a multiplier on performance. The physics of what a superconductor does — the zero resistance, the three ceilings, the flux quantum, the gap, the junction relations — is the same physics at any temperature. Nothing in any of the levels above would need rewriting. What would change is where the machines can go, which is a different and in most cases larger prize.

And the sentence from Level 0 stands at every temperature: a superconductor removes a loss. It is not a source. A warm one would make the removal cheap to deploy, and it would still be a road rather than an engine.

Teaching aids

Three demonstrations you can run

1. The disc and the magnet. A cuprate pellet in a shallow dish of liquid nitrogen with a small rare-earth magnet above it. Cool it with the magnet held away and the magnet is repelled and slides off — that is expulsion. Cool it with the magnet held at a fixed height and it stays at that height, and can be turned upside down without falling — that is pinning. The difference between the two runs is the whole of Level 2.
2. The persistent current. A closed loop of superconducting wire, a current induced in it, and a compass or a magnetometer nearby. Nothing is connected to it. The needle does not move over a lesson, over a day, over a week.
3. The three limits, on paper. Take any conductor's published rating and try to hold two numbers fixed while raising the third. It cannot be done: every table is a surface, and reading it as three independent numbers is the mistake the second instrument on this page exists to prevent.

Self-check (answers below)

1. What distinguishes a superconductor from a hypothetical perfect conductor, in one observation?
2. Why does the charge 2e, rather than e, appear in the flux quantum?
3. A wire is well below its transition temperature and in a field well below its critical field, and it quenches. What happened?
4. Which of the two characteristic lengths has to be the larger for a material to be type II, and why does that make strong magnets possible?
5. A warm superconductor is discovered. Which of the applications on this page changes least, and why?

Answers. (1) Cool both with a magnetic field already through them: the perfect conductor traps the field, the superconductor expels it. (2) The carriers are pairs, and the quantisation condition is set by the carrier's charge. (3) The current exceeded what the temperature and field had left it — the third ceiling, which the other two had already eaten into. (4) The penetration depth. When it exceeds the coherence length, a boundary between a normal core and superconducting surroundings costs the material nothing, so it makes many of them and keeps working in fields far above where a type I sample gives up. (5) The quantum processor: it is cold to be quiet as much as to be superconducting, and only the second reason would go away.

One-page summary for the wall

• Two facts: current with no resistance, and magnetic field held out of the interior. The second is what makes it a state of matter rather than a very good metal.
• Three ceilings, one budget: temperature, field, current. Meet any one and it is over; raise any one and the other two shrink.
• Two kinds: type I gives up at one low field; type II admits countable flux tubes and keeps working to a much higher one. Pin the tubes or they move, and moving flux dissipates.
• One wave: paired carriers in a single macroscopic state. Single-valuedness round a loop gives the flux quantum h/2e; the pairing gives the gap; the gap gives the exact zero and a frequency ceiling.
• Two lengths: penetration depth and coherence length. Their ratio is the type.
• Two descriptions: London and Ginzburg–Landau for what the condensate does, the pairing theory for why it exists — settled for the low-temperature metals, open for the copper oxides.
• What it is for: magnets, scanners, bearings, magnetometers, the volt, qubits, cavities, cables, fault limiters — and the two gravitational-wave source designs this site follows.
• What warmth would change: where the machines can go, not what they can do. And a superconductor removes a loss; it is never a source.