The field that is pushed out

Kamerlingh Onnes measured mercury’s resistance dropping to zero in 1911. For twenty-two years afterwards, superconductivity was understood as exactly that: a resistance that had become unmeasurably small.

In 1933 Meissner and Ochsenfeld measured what a superconductor does to a magnetic field and found something that zero resistance does not predict. The distinction is sharp, it is decided by a single experiment, and the experiment is a comparison of two histories.

What zero resistance would give #

Take a hypothetical perfect conductor — a normal metal whose resistivity has been set to zero — and ask what happens to a magnetic field in it.

Faraday’s law says a changing flux drives an electric field round a loop. Ohm’s law says a finite electric field in a zero-resistance material would drive an infinite current. So the electric field must be zero, so the flux through every loop inside the material is constant.

That is flux freezing, and it is the correct behaviour of a perfect conductor. The prediction it makes is about history: cool the sample in zero field and then apply one, and the field is kept out; apply the field first and then cool, and the field is kept in. Two different final states from two different orders of operations.

That prediction is testable and it is what Meissner and Ochsenfeld tested.

What a superconductor actually does #

A superconductor cooled in a field expels it. The final state does not depend on the order of operations: whatever the history, the interior ends up with no field in it.

That is a much stronger statement, and it means the state is a thermodynamic one rather than a consequence of a transport property. A state that depends only on temperature and field, and not on how they were reached, is a phase, and superconductivity is therefore a phase transition rather than a limiting case of conduction.

The practical consequence everybody has seen is that a magnet floats over a cooled superconductor. That would not happen for a perfect conductor cooled in the magnet’s field, which would keep the flux and feel no force; it happens because the superconductor expels the field it is cooled in, and the currents doing the expelling repel the magnet.

The distinction cost twenty-two years to notice, and it is worth asking why. Nobody had done the experiment in that order, because the prediction from zero resistance was so natural that the alternative was not obviously worth testing.

The depth that contains no frequency #

The field is not expelled from the very surface; it penetrates a short distance, and how short is the material’s own constant.

$$\lambda_{\text{L}} = \sqrt{\frac{m}{\mu_0 n e^2}}$$

for a carrier density $n$. That expression contains a mass, a charge, a density and two constants of nature — and no frequency. For a typical metallic density it gives about twenty-two nanometres.

Compare that with a normal metal’s skin depth, $\sqrt{2/\mu_0\sigma\omega}$, which has the frequency in the denominator. Copper’s skin depth is nine millimetres at fifty hertz, two micrometres at a gigahertz, and infinite at zero frequency: a steady field goes straight through a copper sheet, because a normal metal screens by dissipating and a steady field dissipates nothing.

The two lengths are equal at about nine terahertz, in the far infrared, and above that copper is the better screen. That is a useful corrective. A superconductor’s advantage is not that it screens harder; it is that its screening does not need the field to be changing.

The reason is that the screening currents cost no energy. In a normal metal the screening current dissipates, so it has to be continuously driven by a changing flux; in a superconductor it persists, so it can hold a static field out indefinitely.

The film that expels almost nothing #

The penetration depth is a length, and whether a sample is thick compared with it is a separate question from what the sample is made of.

The profiles in the second figure are the solution of London’s equation in a slab: a hyperbolic cosine, with the field at the centre reduced by the cosh of half the thickness in units of the depth. A slab forty depths thick has essentially nothing in the middle — four parts in a thousand million of the applied field — and expels 95 per cent of the flux that would otherwise have passed.

A slab one depth thick expels 7.6 per cent. Its centre still sits at 89 per cent of the applied field. It is superconducting in every sense — zero resistance, a critical temperature, a gap — and it does not expel the field, because there is not enough of it to screen with.

A perfect static screen has the field meeting its surface at right angles and the interior exactly empty. A bulk superconductor approaches that and a thin film does not — because expulsion costs energy, the screening currents flow in a layer of finite depth, and a film thinner than that depth has nowhere to put them. So the Meissner effect is not all-or-nothing: it is complete in bulk, partial in a film, and absent below a thickness the penetration depth sets.

That has a consequence which is used constantly. A film thinner than its own penetration depth carries much less screening current, so it costs much less energy to keep a field out — and it therefore survives to a much higher applied field before superconductivity is destroyed. The critical field of a thin film rises roughly as one over its thickness, which is why superconducting films are used in high-field applications where bulk material would quench.

The same length decides something else that is often stated as though it were separate. The field at a conductor’s surface meets it at a right angle because the conductor screens perfectly; a superconductor does the same, but only to the extent that it is thick compared with $\lambda$, so the boundary condition that seems geometrical turns out to have a length hidden in it. A microwave cavity plated with superconductor is a good cavity because the plating is many penetration depths thick, and a plating a few nanometres thick would be no better than the substrate.

It also means that “the Meissner effect” is not a property of a substance. It is a property of a sample, and the sample’s size relative to $\lambda$ is half of what decides it.

What the currents are doing #

The expulsion is not magic and it is not a new force. It is a current, flowing in the surface layer, whose own field cancels the applied one inside.

The screening currents form a sheet on the surface, and a solenoid’s field is produced by exactly such a sheet. A superconductor in an applied field carries one of precisely the size needed to cancel the field inside — so the expulsion is not a property of the material refusing the field but a current the material is running, and the energy to maintain it comes from nowhere because the resistance is zero.

A superconducting cylinder in an axial field carries a surface current that would, on its own, produce a field equal and opposite to the applied one inside. The arithmetic is identical to a solenoid’s, and the current density needed is set by the applied field and the penetration depth.

Two things follow immediately. There is a maximum field, because the current cannot exceed what the superconducting state can carry; beyond it the state is destroyed, and that is the critical field. And there is a force on the sample, because a current in a field feels one — which is what levitates the magnet and what has to be resisted mechanically in any large superconducting magnet.

The size of the force is not small. The magnetic pressure at one tesla is about 400 kilopascals, and at ten teslas it is 40 megapascals, so a large magnet’s windings have to be restrained against stresses comparable with a pressure vessel’s. That is one of the two engineering problems in a superconducting magnet; the other is what happens when part of it goes normal and has to absorb the stored energy.

The experiment as it was actually done #

The measurement deserves its own description, because the way it was made explains why it took so long to be made at all.

What Meissner and Ochsenfeld measured was the field outside their samples, which is the same information from the other side: a body that expels flux crowds it into the space around itself, and that crowding is what a magnetometer sees. The experiment’s importance is in the order of operations — they cooled the sample in a field, and the field came out, which no amount of perfect conductivity would produce.

Meissner and Ochsenfeld did not have a probe that could be put inside a piece of tin. What they measured was the field outside two long cylinders placed side by side, using a search coil in the gap between them, as the pair was cooled through the transition in an applied field.

If the flux had stayed inside, the external field would not have changed. If it was expelled, the flux had to go somewhere, and where it goes is into the space between the samples — so the field in the gap should rise as the samples become superconducting. It rose.

That indirect route is why the experiment is harder than it sounds and why the result was not obvious in advance. It also introduced a complication that had to be argued away: the samples were tin and lead, which are soft, and a change in the field outside could in principle have come from a change in their shape or position. The published paper spends more space on that than on the physics.

The modern version of the same measurement — cool a sample in a field and watch the magnetisation, then warm it and repeat — is called field-cooled and zero-field-cooled magnetometry, and the difference between the two curves is the standard diagnostic for whether a new material is a superconductor or merely a very good conductor. It is exactly the comparison of histories described above, made routine.

Where the model runs out #

London’s equation is phenomenological. It was written down to reproduce the Meissner effect and it does, but it says nothing about why the electrons behave that way, and it fails in ways that were informative.

Where the model runs out is at quantisation. Flux through a superconducting ring is conserved for the reason above, and it is additionally required to be a whole multiple of $h/2e$ — a constraint with no classical counterpart, arising because the condensate has a single phase that must come back to itself around the loop. The ring therefore holds a discrete ladder of allowed states rather than a continuum, and a flux forced between two rungs is carried by a vortex instead — which is where the whole of type-II behaviour comes from.

The penetration depth it predicts is too small. Measured depths in simple superconductors are a few times the London value, because the current at a point depends on the field over a region — the coherence length — rather than at the point. Pippard’s non-local generalisation fixes it and introduces the second length that the whole classification of superconductors rests on.

It does not predict flux quantisation. A superconducting ring encloses flux in multiples of $h/2e$, which is a statement about a wavefunction’s phase and has no place in a classical equation. The factor of two in that quantum is the direct evidence that the carriers are pairs.

It says nothing about the transition itself. The equation describes the superconducting state and has no temperature in it, so it cannot say at what temperature that state appears, how the specific heat behaves there, or what kind of ordering has occurred. Those are questions for a thermodynamic theory, and Ginzburg and Landau supplied one before the microscopic account arrived.

And it does not distinguish the two types. Type II superconductors do not expel flux completely; above a lower critical field they admit it as an array of quantised vortices, each carrying one flux quantum, and remain superconducting up to a much higher field. Every practical superconducting magnet is made of type II material, so the effect this essay is about is precisely what the useful superconductors do not do.

That last point is worth sitting with. The Meissner effect is the defining experiment and the reason superconductivity is understood as a phase, and it is also the behaviour that had to be given up to make the technology work.

What the state actually is #

The account that explains rather than describes arrived in 1957, and its shape can be indicated even if its details cannot.

Superconductivity opens a gap at the Fermi level, and the existence of that gap is what makes the condensate rigid against small perturbations. A gap means the lowest excitation costs a finite energy, so a weak disturbance cannot excite anything at all — and a state that cannot be disturbed is a state whose current does not decay. The rigidity and the zero resistance are the same fact.

Electrons near the Fermi surface bind into pairs through an attraction mediated by the lattice, and the pairs — being composite bosons — occupy a single quantum state described by one wavefunction with one phase. That condensate has a gap: exciting it costs a finite energy, so small perturbations cannot scatter it, which is where the zero resistance comes from.

The Meissner effect comes from the same rigidity. A magnetic field would change the phase gradient of the condensate, and the condensate resists, in a way that has no analogue in a normal metal because a normal metal has no single phase to be rigid about. That is why zero resistance and flux expulsion are two consequences of one thing rather than one being a consequence of the other.

And the pairing explains the factor of two in the flux quantum, the isotope effect on the critical temperature, and the exponential temperature dependence of the heat capacity — three quite separate measurements that no phenomenological equation had connected.

The equation, read as a mass #

London’s result can be written as a statement about the field alone:

$$\nabla^2\mathbf{B} = \frac{\mathbf{B}}{\lambda_{\text{L}}^2},$$

whose solutions decay exponentially over the length $\lambda_{\text{L}}$. That is not a form peculiar to superconductors. It is exactly the static equation obeyed by a field whose quantum has a mass, with the range of the field being the inverse of that mass — the same relation that makes a short-ranged nuclear force correspond to a heavy exchanged particle and an infinite-ranged electromagnetic one to a massless photon.

So inside a superconductor the photon behaves as though it were massive. Putting the numbers in, a penetration depth of twenty-two nanometres corresponds to a mass of a few electronvolts — light by particle standards and enormous compared with the photon’s usual value of zero. The magnetic field is excluded for precisely the reason the nuclear force does not reach across a room: a massive field has a finite range.

That reading is not a curiosity. Anderson pointed out in 1963 that a superconductor is the worked example of how a gauge field can acquire a mass without the underlying theory losing its gauge symmetry — the symmetry is not broken by the equations but by the state, exactly as this essay’s condensate picks a phase. Higgs and others carried the argument into particle physics the following year, and it is how the carriers of the weak interaction are understood to be heavy while the photon is not.

The condensed-matter version came first and is the one that can be held in a hand. A superconductor is a piece of metal in which the electromagnetic field has a mass, and the Meissner effect is what a massive field’s finite range looks like from outside.

The diagnostic, and what it has recently caught #

The comparison of two cooling histories is not only how superconductivity was first distinguished from perfect conduction. It is how every claim of a new superconductor is checked, and it has been doing strenuous work.

The critical temperature climbed slowly for seventy-five years — mercury at 4 kelvin in 1911, niobium alloys at 23 by the 1970s — and then jumped. A copper oxide at 35 kelvin in 1986, another above the boiling point of nitrogen the following year, and a subject that had been quiet became the most crowded in physics overnight. More recently, hydrogen-rich compounds squeezed to a substantial fraction of a million atmospheres have been reported superconducting at temperatures approaching room temperature.

A claim like that is extraordinary and the checking is correspondingly severe, because zero measured resistance on its own is weak evidence: a short circuit, a contact artefact or a filament of some other phase can all produce it. What is asked for is the magnetic signature — cool in a field and cool in zero field, and show that the two give different magnetisation curves in the way this page describes, with the field-cooled state expelling flux it started with.

Several recent claims have failed exactly there. A material reported in 2023 as a room-temperature superconductor was found to contain a ferromagnetic impurity that made a fragment tip up over a magnet in a way that looked like levitation and was not, and its magnetic response never showed the expulsion. Two higher-profile papers on hydrides were retracted after the magnetisation data behind them could not be reproduced or accounted for.

Ninety years on, the experiment that separated a phase from a transport property is still the one that decides.

What the pictures cannot show #

The London depth drawn is the zero-temperature one. It grows as the temperature approaches the critical one, diverging there, so a sample thick compared with $\lambda$ at low temperature becomes thin compared with it near $T_c$ and stops expelling.

Everything is one-dimensional. A real sample has edges, where the field concentrates and the local value can far exceed the applied one — which is why a thin plate perpendicular to a field goes normal at its rim long before its middle.

No temperature dependence appears at all. Both the depth and the critical field are strong functions of temperature, and every practical question about a superconductor is a question about where it sits in the temperature–field plane.

The carrier density is a chosen number. Twenty-two nanometres follows from assuming one carrier per atom at metallic density; real superconductors give measured depths from about 40 nanometres for lead to several hundred for the high-temperature materials, and the discrepancy is informative — it says the density of paired carriers is not the density of electrons.

And the high-frequency comparison ignores the gap. Above a photon energy equal to the gap the pairs break, the superconductor becomes an ordinary metal, and the flat line in the first figure turns into a sloping one. For a conventional superconductor that happens in the far infrared — which is, coincidentally, near where the two curves cross anyway.

The ladder from here #

Later rungs on this anchor: flux quantisation and the ring experiment that measured the factor of two; type II superconductors and the vortex lattice that lets them survive to high fields; the Josephson effect, where the condensate’s phase becomes directly measurable; and the critical current, which is the practical limit on every superconducting device and is set by vortex motion rather than by pair breaking.

The neighbouring ladders are how far a field gets into metal, which is the normal-metal screening this is compared against, the inside of a conductor, which is the static limit a superconductor reaches and a normal metal does not, and the field that cannot get out, where flux freezing is the correct answer rather than the wrong one.