Illustrated Physics
physics, drawn
Electromagnetism

The field a superconductor keeps after all

A superconductor is famous for throwing magnetic field out, and the experiment that showed it is the one that made superconductivity a state of matter. Most superconductors that are actually used do something closer to the opposite: they let the field in as threads, grip the threads where they are, and keep them when the field is taken away. The result is a slope of field inside the material that is as steep as the grip allows and no steeper, which is all that is needed to explain why such a superconductor has a hysteresis loop, heats up in an alternating field, can hang a magnet underneath itself, and can be made into a magnet stronger than any of iron.
16 min read 5 figures Made into an instrumentApproximations that lie

Assumes: The field that is pushed out · Two lengths, and which one is longer

The field that is pushed out told the story of the experiment that made superconductivity a state of matter rather than a very good conductor. A perfect conductor cooled in a magnetic field would keep the field inside it, because changing the field would need an electric field and a perfect conductor allows none. A superconductor cooled in a field throws it out. Walther Meissner and Robert Ochsenfeld saw that in 1933, and the difference between remembering and expelling has defined superconductivity ever since.

The materials that make the strong magnets of MRI scanners and particle accelerators do not behave like that. Niobium–titanium and niobium–tin wire, and the copper-oxide ceramics that superconduct in liquid nitrogen, are superconductors of the second kind described in two lengths, and which one is longer, which above a modest field let the field in as threads. And when the threads are held in place by defects in the material, the superconductor does something that looks far more like the perfect conductor of 1933 than the superconductor that replaced it. It keeps the field. Charles Bean worked out in 1962 how much it keeps, with an argument that needs one number and some straight lines.

Threads, and what holds them

A type-II superconductor in a field above its lower critical value lets flux in as vortices: thin threads each carrying exactly one quantum of flux, h/2eh/2e, circled by a whirl of supercurrent. Between the threads the material is superconducting; in each thread’s core, a few nanometres across, it is not.

A current flowing through the material pushes on the threads sideways, with a force per unit length equal to the current density times the flux quantum. In a perfect crystal nothing would resist that push, the threads would drift, and a drifting thread induces a voltage — so the material would have resistance after all, and carrying a current would not be possible. Real materials are not perfect. Grain boundaries, precipitates, missing atoms and deliberately added particles are places where a thread’s non-superconducting core costs less energy, and a thread sitting on one is held there. It moves only when the push exceeds the hold.

The largest current density the pinning can resist is the critical current density, JcJ_c. Below it the threads stay put and the current flows without loss. At it, threads start to move. Engineered pinning in modern conductors holds current densities of 10910^9 to 101010^{10} amperes per square metre, a thousand times what a copper wire carries before it overheats.

A slope as steep as the grip

Bean’s insight was that a pinned superconductor exposed to a changing field will always be at the edge of what its pinning holds. Raise the applied field a little and the threads at the surface are pushed inward by the field gradient — which by Ampère’s law is a current — until the gradient falls back to what pinning can resist. Deeper in, where the change has not arrived, nothing happens. So the field inside is the applied value at the surface and falls inward with a slope fixed at exactly the critical current: dB/dx=μ0JcdB/dx = \mu_0 J_c.

How a field enters a superconductor that pins it. The magnetic field inside a slab of type-II superconductor, against position across its thickness, as an applied field is raised from zero, in units of the full-penetration field H = Jc a. Each line is the profile at an applied field of 0.5 H, 1.0 H, 1.5 H, 2.0 H. The field enters from both faces as a wedge whose slope is the most current the pinned vortices can carry, Jc; at 0.5 H it reaches halfway to the middle, at H* it just reaches the middle, and beyond that the whole slab carries Jc and the profile rides up with the applied field. A type-I superconductor would show no field inside at all, and a normal metal the applied field everywhere.
Fig. 1 The field inside a slab of pinned superconductor across its thickness as the applied field is raised from zero, in units of the full-penetration field H∗=JcaH^* = J_c a. At 0.5 H* the field has entered halfway to the middle from each face; at H* it just reaches the middle; beyond that the whole slab carries the critical current and the profile rides up with the field.

The result is a wedge. At a small applied field the field enters a little way from each face and the middle is untouched — screened, as in a Meissner state, but by currents spread over a depth set by the pinning rather than squeezed into the penetration depth. The depth is H/JcH/J_c, and for a field of a tesla and a critical current of 10910^9 amperes per square metre it is close to a millimetre, against a penetration depth of a tenth of a micrometre. As the field rises the wedge deepens until it meets itself in the middle, at the full-penetration field H∗=JcaH^* = J_c a, where aa is half the slab’s thickness. Above that the whole slab is in the critical state and the profile simply rises with the applied field, keeping its V shape.

There is a close mechanical analogy, and it is worth having because it makes every later result predictable. Sand poured onto a table builds a heap whose slope is the angle of repose: steeper than that and grains slide, shallower and nothing moves. Add sand and it avalanches down the surface until the slope is back at the angle. The field inside a pinned superconductor is a heap of flux threads, the critical current is its angle of repose, and changing the applied field adds or removes threads at the edge. Pierre-Gilles de Gennes drew the comparison in the 1960s, and it holds right down to the avalanches: flux enters real superconductors in sudden jumps, as sand does.

The loop

What a magnetometer measures is the slab’s magnetisation — the average field inside minus the applied field. A superconductor that expels the field entirely has a magnetisation of minus the applied field, a straight line through the origin, with no memory. The critical state is different, because the profile depends on the history, not just on the present field.

The hysteresis loop of a pinned superconductor. The magnetisation of the slab — its average field minus the applied field — as the applied field is raised from zero to 3 H, lowered to −3 H and raised again, in units of H. On the first rise the magnetisation falls as −H + H²/2H and then sits at −H/2 once the slab is fully penetrated. Reversing the field reverses every current in the slab, which takes a swing of 2H, after which the magnetisation sits at +H/2. The loop's width is set by Jc alone. At zero applied field on the way down the slab still holds an average field of 0.50 H: a trapped field, held by nothing but pinning.
Fig. 2 The slab’s magnetisation as the applied field rises to 3 H*, falls to −3 H* and rises again, in units of H*. The first rise (dashed) follows −H+H2/2H∗-H + H^2/2H^* and then sits at −H*/2; reversing the field takes a swing of 2H* to reverse every current, after which the magnetisation sits at +H*/2. At zero applied field on the way down, the slab still holds an average of H*/2.

On the first rise the magnetisation starts along the perfect-screening line and curves away from it as the wedges deepen, following −H+H2/2H∗-H + H^2/2H^* until full penetration, after which it stays at −H∗/2-H^*/2. Now lower the field. At the surface the field falls, but the threads inside are held, so the profile does not simply drop: a new front of opposite slope enters from each face, the critical state rebuilt in reverse. It takes a fall of 2H∗2H^* for the new front to sweep the whole slab, after which the magnetisation is +H∗/2+H^*/2 and stays there. The loop has a width set by JcJ_c and nothing else.

Hysteresis of this shape is what a ferromagnet’s loop looks like too, and for a related reason: in both cases something is stuck, and the system follows the applied field only when pushed past what holds it. In iron it is domain walls held on defects; here it is flux threads held on defects. The superconducting version is the cleaner of the two, because the critical current density is close to a single material constant and the field in each thread is fixed exactly by quantum mechanics.

What it remembers

At zero applied field on the way down, the slab is still magnetised. The profile is a roof, highest in the middle, with the critical slope on each side, and the field inside it has nowhere to go because the threads carrying it are pinned.

What a pinned superconductor remembers of a field. The field left inside the slab after it was cooled into the superconducting state in an applied field and the field was then removed, against position, for cooling fields of 0.5 H (average left 0.37 H), 1.0 H (average left 0.50 H), 2.0 H (average left 0.50 H). A perfect conductor would keep the whole field it was cooled in, flat across the slab; a type-I superconductor would keep none, having expelled it on cooling. The pinned superconductor keeps what its critical current can hold: a roof whose slopes are set by Jc, and never more than H* at the middle however strong the field it was cooled in.
Fig. 3 The field left inside the slab after cooling into the superconducting state in a field and then removing the field, for cooling fields of 0.5, 1 and 2 H*. Dotted: the field it was cooled in, which a perfect conductor would keep entire. The pinned slab keeps a roof of critical slope, never more than H* at the middle; the two larger cooling fields leave the same roof.

This is the experiment that separated superconductors from perfect conductors, repeated on a pinned type-II material. Cool the slab in a field and switch the field off. A perfect conductor would keep all of it, flat across the slab. A Meissner superconductor would have expelled it on cooling and keep none. The pinned superconductor keeps a roof: everything the critical current can hold and nothing more. Cooled in half of H∗H^*, it keeps a flat top at that value with critical slopes at the faces; cooled in H∗H^* or anything stronger, it keeps the same roof, peaking at H∗H^* in the middle, because the critical current cannot support a steeper one.

So on the question that defined superconductivity, a hard superconductor sits between the two answers, and the position between them is set by JcJ_c. Strong pinning makes it nearly a perfect conductor — the flux frozen in, as it is in a collapsing star’s plasma — and weak pinning lets it return towards the Meissner state. The Meissner effect remains the defining property, because the expulsion is what the material would do if the threads were free. It is just that in most materials anyone uses, they are not.

The roof does not last for ever. Pinning is not absolute at any temperature above zero: thermal agitation lets threads hop from one pinning site to the next, preferentially down the slope, and the trapped field decays logarithmically in time, a few per cent per decade of time in high-temperature superconductors near 77 kelvin. That is the superconducting version of the slow decay every magnetisation suffers, and it is why persistent-current magnets that must hold their field for years are built from materials and run at temperatures where the hopping is negligible.

The price of an alternating field

A superconductor carrying a steady current, or sitting in a steady field, dissipates nothing at all. Change the field and the threads must move, and threads moving against pinning dissipate energy in the same way that dragging a block across a table does: the grip that holds them is released in jerks, and the work done against it becomes heat. In an alternating field the threads are dragged in and out every cycle, and the heat per cycle is exactly the area of the hysteresis loop.

The heat a pinned superconductor makes in an alternating field. The energy turned into heat per cycle of an alternating applied field, per unit volume of slab, in units of μ₀H², against the field's amplitude in units of H, both on logarithmic axes — the area of the hysteresis loop, integrated from the computed profiles. 0.1 H: 6.8·10⁻⁴; 0.3 H: 0.0181; 1.0 H: 0.665; 3.0 H: 4.65. Below full penetration the loss rises as the cube of the amplitude, 2H³/3H, because a wider front sweeps across more flux; above it, linearly, as 2H − 4H/3. A superconductor carries direct current with no loss at all and alternating current with this one, which is why the wire in an alternating-current cable is split into fine filaments, each small enough that its own H* is low.
Fig. 4 Heat per cycle of an alternating field, per unit volume, in units of μ₀H², against the field’s amplitude in units of H, on logarithmic axes. The line is Bean’s closed form; the dots are the loop areas integrated from the computed profiles. Below H* the loss rises as the cube of the amplitude, above it linearly.

Below full penetration the front sweeps in and out a distance proportional to the amplitude, through a field also proportional to it, and the loss rises as the cube: 2μ0H3/3H∗2\mu_0 H^3/3H^* per unit volume per cycle. Above full penetration the whole slab is swept and the loss rises linearly, as 2μ0H∗H−4μ0H∗2/32\mu_0 H^* H - 4\mu_0 H^{*2}/3. The dots in the figure are the loop areas computed by integrating the magnetisation round the cycle, and they land on the formulas.

The formulas carry the engineering. At a fixed amplitude, the loss below full penetration is inversely proportional to H∗H^*, and at large amplitudes it is proportional to H∗=JcaH^* = J_c a — so for a conductor in an alternating field, the loss falls if the superconductor is divided into fine pieces, each with a small aa. Every superconducting wire meant for alternating current is made that way, with thousands of filaments a few micrometres across embedded in copper, and twisted so that the filaments do not couple through the copper into one large conductor again. A superconducting power cable is lossless for direct current and loses a small, calculable amount of power for alternating current, and that amount is the area of a loop like this one.

Magnets that are not magnets

If a pinned superconductor can keep a field, it can be a magnet. Cool a disc in a strong field, remove the field, and the roof of trapped flux remains. For a long cylinder magnetised to the critical state the field at the centre is μ0Jca\mu_0 J_c a, proportional to the radius: unlike an iron magnet, whose field is limited by the saturation of iron at about two teslas whatever its size, a trapped-field magnet gets stronger as it gets bigger.

The field a pinned superconductor can trap, by size. The field held at the centre of a long superconducting cylinder magnetised to its critical state, μ₀J_c a, in tesla against the cylinder's radius in millimetres, for three critical current densities. YBCO at 77 K, Jc 10⁸ A/m²: 1.6 T at 12.5 mm; YBCO at 77 K, Jc 5 × 10⁸ A/m²: 7.9 T at 12.5 mm; GdBCO at 26 K, Jc 1.2 × 10⁹ A/m²: 18.8 T at 12.5 mm. The record, 17.6 T at 26 K between two 24 mm discs of gadolinium barium copper oxide (dot), lies close to the long-cylinder line for 1.2 × 10⁹ A/m², which gives 18.1 T. Because the trapped field grows with size at fixed Jc, a larger disc is a stronger magnet — until the magnetic stress, B²/2μ₀, 120 MPa at 17.6 T, cracks it; the record discs were wrapped in steel.
Fig. 5 The field at the centre of a long cylinder magnetised to its critical state, μ0Jca\mu_0 J_c a, against the radius, for three critical current densities: yttrium barium copper oxide at 77 K at 10⁸ and 5 × 10⁸ A/m², and gadolinium barium copper oxide at 26 K at 1.2 × 10⁹ A/m². The dot is the record: 17.6 T at 26 K, between two 24 mm discs, close to the 18.8 T the long-cylinder line gives.

At 77 kelvin, in liquid nitrogen, discs of yttrium barium copper oxide a few centimetres across trap one to two teslas, limited by their critical current at that temperature and by their thinness, which lets the field return round the edge. Colder, the critical current rises steeply. In 2014 a group in Cambridge trapped 17.6 teslas at 26 kelvin between two discs of gadolinium barium copper oxide 24 millimetres across, close to what the long-cylinder formula gives for a critical current of 1.2×1091.2 \times 10^9 amperes per square metre. The record was limited not by the superconductor but by its strength. The magnetic field pushes outward on the currents that make it with a stress of B2/2μ0B^2/2\mu_0, 123 megapascals at 17.6 teslas, more than the ceramic can bear in tension, and the discs survived only because they were shrink-fitted inside steel rings.

The same trapped flux makes the most striking demonstration in the subject. A magnet held above a pinned superconductor that has been cooled while the magnet was nearby stays where it was put — above the superconductor, beside it or underneath it. No arrangement of static fields can hold a magnet stably, but this is not an arrangement of static fields. It is a frozen configuration: any motion of the magnet would change the flux through the superconductor, the pinned threads resist the change, and the magnet is held where it was as surely as if it were embedded in the flux. That is the perfect conductor’s behaviour once more, the one the Meissner experiment was supposed to have ruled out.

Where the model stops

Bean’s model assumes a single critical current density, independent of the field. Real critical currents fall as the field rises, often steeply, and the profiles curve: shallow where the field is strong, steep where it is weak. Kim’s model and its successors replace the constant by a function and keep everything else, and the loop changes shape but not character.

It assumes a slab, infinitely wide, in a field parallel to its faces. A thin disc in a perpendicular field — the shape of most bulk magnets and of every superconducting tape — behaves differently, because the field bends round the edges and the screening currents produce fields of their own far larger than the applied one near the edges. The geometry changes the full-penetration field and the trapped field by factors that depend on the aspect ratio, which is why a thin disc traps much less than the long-cylinder formula in the last figure promises.

And it ignores the reversible part of the magnetisation, the lower critical field below which no threads enter at all, the surface barrier that can keep them out even above it, and the creep that slowly relaxes every profile. Each is a correction, important in some materials and conditions, and none of them alters the picture that the field inside a pinned superconductor stands at the steepest slope its pinning allows.

What the pictures cannot show

The profiles are straight lines because the model says so. A measured profile — obtained by scanning a tiny field sensor across a cut surface, or by magneto-optical imaging, in which a film laid on the superconductor rotates polarised light in proportion to the field — shows the same wedges and roofs, with rounded corners, a bulge where the critical current falls in the strongest field, and fine fingers where flux has avalanched in. The figures draw what the critical state is, not what a sample looks like.

They also draw the field as continuous. It is not: it is made of individual flux quanta, about 5 × 10¹⁴ of them per square metre for every tesla, and the smooth slope is an average over millions of threads, each pinned on its own defect. The model works because the threads are so many and so small.

The domain of the argument is a type-II superconductor with pinning, at fields between the lower and upper critical fields, changed slowly enough that the threads keep pace with the critical state and not so slowly that creep matters. Inside that domain the field stands at the critical slope. Outside it — a clean sample of the first kind, or one without pinning — the Meissner picture returns.

Still open: what sets the critical current

The critical current density is the one number this whole essay runs on, and it cannot be calculated from first principles for any real material. It depends on how many pinning sites there are, how strongly each holds a thread, and how the threads, which repel one another and form a lattice, share the load — a collective problem in which a lattice of elastic threads is held by a random landscape of defects, much as a heap of grains is held by a random network of contacts. Theories of collective pinning give the right scaling in some regimes and fail in others, and the best conductors are still improved by trial: adding nanorods of one oxide to films of another, irradiating with heavy ions to make columnar tracks, and measuring what results. How high JcJ_c can go before the pinning sites themselves begin to spoil the superconductor, and whether the theoretical ceiling set by the pair-breaking current can be approached, is an open question with a direct line to every superconducting magnet that will be built.

The superconductor that throws out magnetic field is real and is the one that defines the state; the one that keeps it is the one that does the work. In a pinned type-II superconductor the field stands inside at the steepest slope its pinning holds, μ0Jc\mu_0 J_c, so it enters as wedges, fills the slab at H∗=JcaH^* = J_c a, leaves a roof of up to H behind when the field is removed, dissipates the loop’s area in every alternating cycle, and in a disc two and a half centimetres across can hold 17.6 teslas.* It remembers the field as a perfect conductor would, to exactly the extent its threads are held.

Part 7 of 7

This essay is one argument about Superconductivity. The others:

Rotation is a magnetic field, and a superconductor cancels it. Magnetic fields along the spin axis inside a spinning metal, in units of 2mω/e, as seen by its electrons in the frame turning with it. In that frame the Coriolis force on an electron is exactly the force a magnetic field of 2mω/e would exert, pointing against the spin for a negative charge (Larmor's theorem). A normal metal spun in no field has no real field inside, and its electrons feel the rotation's equivalent field in full. A superconductor will not let its carriers feel any field in their interior — that is the Meissner effect — so it generates a real field of exactly the same size pointing the other way, and the sum vanishes. The real field is the London moment. The field a spinning superconductor makes of itself Part 6 — Spin a lump of superconductor with no magnet anywhere near it, and a magnetic field appears inside it, pointing along the axis of spin: about eleven trillionths of a tesla for every radian per second, the same in lead, niobium or a copper-oxide ceramic. The field is not left over from anything. In a turning frame the Coriolis force on a moving charge is indistinguishable from a magnetic force, a superconductor will not let its carriers feel any field inside it, and so it makes a real field to cancel the one rotation imitates. The size depends on nothing but the carriers' mass and charge, which makes the field a balance for weighing them — and the one time it was read to a part in a hundred thousand, it gave an answer theory has not matched.

Where this goes next

The screening that survives to zero frequency. How far a magnetic field gets into copper against how fast it is changing, drawn beside the London depth of a superconductor with a carrier density of 6.0e+28 per cubic metre. The metal's curve falls as one over the square root of the frequency — measured on the drawn curve as -0.5000 against −½ — and it is a straight line on these axes with no bottom: at a hundred hertz the field reaches nine millimetres in, and at zero frequency it reaches all the way through, because a normal metal screens by dissipating and a steady field dissipates nothing. The superconductor's line is flat at 21.7 nanometres. The frequency does not appear in the expression for it, so there is nothing for it to depend on, and the screening is as complete at zero frequency as at any other. The two lines cross at 9.0e+12 hertz, in the far infrared, and above that the ordinary metal is actually the better screen — which is a useful corrective, because a superconductor's advantage is not that it screens harder but that it does not need the field to be changing. What the picture cannot show is where the flat line stops: above the energy gap the pairs break, the superconductor becomes an ordinary metal, and the flat line turns into a sloping one. The field that is pushed out A perfect conductor keeps whatever field was inside it when its resistance vanished. A superconductor expels the field either way, and the difference between remembering and expelling is the experiment that showed superconductivity is a state of matter rather than a very good conductor. Circulation against how fast the bucket turns. The circulation round the rim of a bucket of radius 1 mm, against the angular velocity it is spun at. An ordinary liquid ends up rotating with the bucket, and its circulation is 2Ω times the area — the straight dashed line, continuous in Ω and with no special value anywhere on it. A superfluid's velocity is the gradient of a phase, so it can carry circulation only in whole units of h/m = 9.969e-8 m²/s. Below 0.256 radians per second it carries none at all: the bucket turns and the liquid does not, which is what Hess and Fairbank measured. Above it the circulation is a staircase of 13 steps, each exactly one quantum high and each 1.59e-2 radians per second wide. The staircase runs below the classical line by the ln(R/a) quanta the threshold costs, a fixed lag: at 102 radians per second the two agree to 0.24 per cent, which is why a rotating superfluid looks like a rotating liquid at any speed a bucket is normally spun at. The whirlpool that comes in one size Spin a bucket of ordinary liquid and it ends up turning with the bucket. Spin a bucket of superfluid helium slowly and it does not turn at all. Spin it faster and it does not turn either — until a threshold, at which a single line of circulation appears, carrying not some amount but exactly h/m. There is nothing in between, because the velocity is the gradient of a phase and a phase has to come back to itself. What a collapse does to a field. A body of radius 700,000 km carrying a field of 0.01 T, collapsing to 10 km. The flux through every comoving loop is fixed, so B goes as 1/R² and the field reaches 4.9·10⁷ T — a compression of 7·10⁴ in radius bought a factor of 4.9·10⁹ in field. The line has slope exactly −2 and that is the only claim being made: what a real object ends up with also depends on how well the flux was held, and on what generated it. The field that cannot get out Squeeze a lump of conducting fluid and its magnetic field comes with it, because the flux through any loop that moves with the material cannot change. Halve the radius and the field goes up fourfold; collapse by a factor of seventy thousand and it goes up by five thousand million.

The objects named here

The third axis, after the field and the reading path: the things themselves, and every essay that touches each one.

Critical stateEddy currentFlux freezingFlux quantumHysteresisMagnetisationMeissner effectSuperconductivityType ii superconductor