Two lengths, and which one is longer

A superconductor has two lengths in it that are properties of the material rather than of the sample. The first is the penetration depth: a magnetic field is not excluded from a superconductor absolutely but decays into it over a distance of typically some tens to hundreds of nanometres. The second is the coherence length: the superconducting state is described by an order parameter, and that parameter cannot change abruptly — it has a stiffness, and the distance over which it can be built up or torn down is a few nanometres in some materials and micrometres in others.

Neither length by itself decides anything interesting. Their ratio decides everything.

The energy of a boundary #

Consider a flat boundary with normal metal on one side and superconductor on the other, at the field where the two are in equilibrium. Making that boundary costs and saves.

It costs, because the order parameter has to be brought from zero to full over a distance of about the coherence length, and over that distance the material is not enjoying the full condensation energy it would if it were properly superconducting. It saves, because the field leaks in over the penetration depth, and over that distance the material is not paying the cost of pushing the field out. Whether the boundary costs or saves is the question of which distance is longer.

The consequence of the sign is total. A material whose boundaries cost energy will have as few as possible: it keeps the field entirely out, up to a field at which the whole sample gives up at once. A material whose boundaries release energy will have as many as it can: it fills itself with them, which means letting the field in as finely divided as possible.

Nothing about the second behaviour is a defect or a compromise. It is a material doing what lowers its energy, and the fact that the answer is “make as much interface as possible” is the interesting part — because a system that gains energy by subdividing itself will subdivide until something else stops it, and what stops it here is quantisation.

Where each length comes from #

The two lengths are worth introducing properly, because they belong to different halves of the theory and are usually met a long way apart.

The penetration depth is the older and the more elementary. A superconductor is a medium in which charge accelerates freely rather than drifting against friction, and a medium like that expels a static magnetic field over a depth set by how many carriers there are and how heavy they are — the London depth, $\sqrt{m/\mu_0 n q^2}$. It is the same expression as the skin depth of a normal metal with the scattering taken out, which is why the two are the same length in a metal at very high frequency and quite different at low.

The coherence length belongs to the order parameter and has no counterpart in normal conduction at all. It is the shortest distance over which the density of the superconducting state can change without costing more in gradient energy than it saves in condensation energy, and it is what makes the transition a thermodynamic one with a stiffness rather than a switch. In a conventional superconductor it is essentially the size of a pair, which is why it is large — hundreds of nanometres in aluminium, because the binding is weak — and it is a few nanometres where the binding is strong.

The direction of that dependence is the useful thing to remember. Weak pairing gives a long coherence length and a small ratio; strong pairing and few carriers give a short one and a large ratio. So the materials that superconduct at high temperature, which is to say the ones with strong pairing, are automatically far above the changeover, and there is no such thing as a high-temperature superconductor of the first kind.

The changeover #

That the changeover sits at exactly $1/\sqrt2$ is a result rather than a definition, and the arithmetic above is one way of seeing it: the two profiles have a particular shape, and where their integrated contributions cancel is fixed. At that ratio something stronger is true — the two Ginzburg–Landau equations reduce to a pair of first-order ones and the surface energy vanishes identically at every field, not merely at one — but the numerical statement is enough to divide the materials.

The division is sharp and it is a division of the periodic table, not of the technology. Aluminium, tin, mercury and lead sit below the line; niobium, vanadium and every alloy and compound of interest sit above it. Abrikosov worked out what the second kind would do in 1953, was persuaded by Landau to hold the paper back, and published in 1957; the vortex lattice was seen directly in 1967.

There is one point of physics worth pulling out of the changeover, because it makes the ratio less arbitrary than it looks. The penetration depth is set by how many carriers there are — it is the same length that appears in the skin depth of a normal metal with the scattering removed — and the coherence length by how strongly they are bound. So the ratio is large in materials with few carriers and strong pairing, which is exactly the description of a high-temperature superconductor: its ratio is of order a hundred, and it is as far from the changeover as anything can be.

What the field does when it is let in #

Flux entering a type-II superconductor cannot enter smoothly. It enters as tubes, and each tube carries exactly one quantum — the same $h/2e$ that a superconducting ring traps and cannot let go of, for the same reason, which is that the phase of the order parameter must come back to itself round any closed path.

Each tube has a normal core about a coherence length across, where the order parameter is driven to zero so that the phase can wind round it, surrounded by a circulating supercurrent extending out to about a penetration depth. It is a vortex, in the same sense and with the same structure as the quantised whirlpool in a superfluid — with the difference that the carriers here are charged, so the circulation is accompanied by a magnetic field and the tube carries flux rather than only angular momentum.

The tubes repel each other — parallel currents in the same sense attract, but the geometry here works the other way — and a set of mutually repelling parallel lines in a plane arranges itself into a triangular lattice, which is what is observed. The spacing is set entirely by the field and the flux quantum, with nothing about the material in it, which is why the lattice constant is a way of measuring the flux quantum and why the observed spacing was one of the confirmations that the quantum is $h/2e$ and not $h/e$.

The end of superconductivity now has a geometric meaning. Raise the field and the tubes crowd together; when the spacing reaches the size of a core, the normal centres touch and there is nothing superconducting left between them. That field, computed from the core size alone, comes out at thirteen tesla for these numbers — and it agrees with the thermodynamic route through the two lengths, which is a check on the picture rather than a restatement of it.

Three fields where there was one #

For a material of the first kind there is a single critical field, of order a few hundredths of a tesla, and it is where the condensation energy of the whole sample equals the energy of expelling the field. It is small because the condensation energy is small, and no such material can be used to make a magnet: the field the magnet is supposed to produce would destroy it.

For a type-II material that field still exists and marks nothing. What happens instead is that flux begins to enter at a much lower field and the material survives to a much higher one, and the ratio of the two to the thermodynamic value is set by the same $\kappa$. Niobium–titanium has $\kappa \approx 60$, so it holds out to about eighty-five times the thermodynamic field — fourteen tesla rather than a sixth of one — and that factor is the entire reason superconducting magnets exist.

The quantum, and why the tubes cannot be finer #

The subdivision has to stop somewhere, and what stops it is the same condition that traps flux in a ring.

The order parameter is a complex quantity with a phase, and the phase has to come back to itself round any closed path in the superconducting material. Draw a path round one flux tube and the phase winds by some whole number of turns; the flux enclosed is that number times $h/2e$. So a tube carrying half a quantum is not a tube with less flux in it — it is not a solution at all.

That is why the answer to “how finely can the material divide the field” is a number rather than “as finely as possible”. Every tube costs a core and saves an interface, and the material would keep splitting them if it could; the quantum is the floor. It is also why the vortices are all identical: there is no distribution of tube strengths to measure, because there is only one strength available.

The two-ness of the quantum is worth a line, since it is the one piece of evidence in this whole subject that comes from arithmetic rather than from measurement of a length. The flux quantum is $h/2e$ and not $h/e$, and the two is the charge of the thing whose phase is winding. What carries the current in a superconductor is a pair, and the vortex lattice’s spacing measures that pair’s charge every time it is imaged.

Why the good ones are dirty #

Here is the part that inverts the expectation, and it is the practical heart of the subject.

A vortex in a current-carrying superconductor feels a force. The current and the vortex’s flux give a force per unit length across the current’s direction, in exactly the way a current in a field feels a force, and there is nothing to resist it. So the lattice moves sideways. A moving flux tube is a changing flux through any circuit drawn round it, so it induces an emf; an emf along the current is a voltage; a voltage with a current is dissipation.

The conclusion is uncomfortable and correct: a perfect type-II superconductor has electrical resistance at any field above the lower critical one. Its resistivity is small but it is not zero, and it grows with the field. A wire made of an ideal, defect-free, chemically pure type-II material would be useless.

What makes such a wire useful is defects. A vortex core is a region of normal material, so it costs less energy to sit where the material was already not superconducting — at a precipitate, a dislocation tangle, a grain boundary. Those places pin the lattice, and while the pinning force exceeds the force the current exerts, the lattice does not move and there is no dissipation at all. The current at which the pinning gives way is the critical current, and it is a property of the microstructure rather than of the compound.

So the manufacture of a superconducting wire is largely metallurgy: niobium–titanium is drawn down and heat-treated repeatedly to precipitate a fine dispersion of a normal phase, at a spacing chosen to match the vortex lattice at the intended operating field. The best conductor is the one with the most carefully arranged mess in it. That is an unusual relationship between purity and performance, and it comes directly from the sign of a surface energy.

The number a magnet is actually limited by #

Putting the pieces together gives the limit on a superconducting magnet, and it is not the one the phase diagram suggests.

The upper critical field says where superconductivity ends. The critical current says how much current the wire will carry before the vortex lattice tears loose, and it falls to zero at the upper critical field but is already small well below it. And the magnet’s own field acts on its own wire, so a coil is always operating in the field it is making — which means the design point is where the critical-current curve crosses the field the coil produces, and neither number alone gives it.

There is a third limit that is mechanical rather than electromagnetic. A coil carrying current in its own field experiences an outward force, and at high field that force is enormous: the magnetic pressure at fourteen tesla is about eighty megapascals, which is a substantial fraction of the yield strength of the metals available. Large magnets are therefore structures as much as circuits, and the reinforcement is a large part of what is being bought.

And a fourth, which is the one that actually destroys them. All the energy stored in the field is stored in the bore, and if any part of the winding becomes normal — from a wire moving a few micrometres and rubbing, which is enough — that part becomes resistive, heats, and drives its neighbours normal in turn. The stored energy then arrives in whatever length of wire has gone normal. Managing that is the central engineering problem, and it is a consequence of the field being where the energy is rather than of anything about superconductivity.

Where the model stops #

Ginzburg–Landau theory is an expansion near the transition temperature. It assumes the order parameter is small and varies slowly, which is excellent just below the critical temperature and progressively worse at low temperature. Both lengths diverge at the transition and the ratio does not, which is why $\kappa$ is a useful label; but the profiles drawn here are quantitatively right only near the top of the phase diagram.

The order parameter is treated as a single complex number, which is a choice about the material. In a conventional superconductor it is right. In one whose pairing has a lower symmetry there are several components, the vortex core has internal structure, and the surface energy is not a single number — so the sharp division at $1/\sqrt2$ is a statement about the simplest case.

The lattice is assumed rigid, and it is soft. Its shear modulus vanishes as the upper critical field is approached, so near that field the lattice melts into a vortex liquid, which flows past any pinning and dissipates. In high-temperature superconductors that region is large, and the practical limit on a magnet is the melting line rather than the upper critical field — a phase boundary inside the superconducting state that has no counterpart in the low-temperature materials.

And thermal activation is ignored entirely. A pinned vortex can be shaken loose by thermal energy, so the critical current is not sharp and the voltage does not appear abruptly; it creeps. At four kelvin the creep is negligible and at seventy-seven it is not, which is one of the practical differences between the two families of superconductor.

What the pictures cannot show #

The boundary figure draws two profiles along a line and cannot show that they belong to a two-dimensional structure. A vortex is not a boundary between half-spaces; it is a tube, and the order parameter has to vanish on its axis for a topological reason rather than an energetic one — the phase winds by a whole turn round the core, and a non-zero order parameter with a winding phase would be many-valued at the centre. The flat-boundary calculation gets the energetics right and says nothing about why the core exists.

The lattice figure draws a spacing and not a lattice. What it cannot show is that the tubes interact, that the arrangement is triangular rather than square by a margin of about two per cent in energy, and that a real lattice is full of dislocations and grain boundaries of its own — a crystal inside a crystal, with its own defects, its own melting and its own elasticity.

Where the ladder goes next #

The superconductivity ladder began with the field that is pushed out and the two in the flux quantum. This rung asks what decides how a superconductor meets a field. The rungs after it: flux creep and the critical state, where the current distribution in a wire is set by how much flux has been pushed in and never comes to equilibrium; the Josephson junction, where two superconductors separated by a barrier carry a current set by the difference of their phases; and the quenching of a magnet, where a millimetre of wire going normal releases the stored energy of the whole coil into itself.

The habit worth carrying away is to look for the ratio. When a system has two lengths in it, the physics is usually in which is larger rather than in either. Here the ratio splits a family of materials in two, at a value that is a pure number, and every practical consequence — how much field can be held, how much current can be carried, and by what kind of metal — follows from which side of it a material sits on.