The stripes a superconductor makes of a field

The field that is pushed out found the defining property of a superconductor: a magnetic field does not get in, beyond a penetration depth of tens of nanometres, because currents flow at the surface that cancel it inside. For a type I superconductor such as lead or tin there is a limit. Above the critical field $H_c$ — 0.080 tesla for lead near absolute zero — superconductivity is destroyed and the field enters. Below it, according to the simplest account, the field is expelled entirely.

The current a superconductor stops with its own field found the first crack in that account. A wire carrying a current larger than $2\pi a H_c$ cannot be wholly superconducting, because its own field at its surface exceeds the critical field, and it cannot be wholly normal either, because a normal wire carrying that current would have a field inside below the critical value. It settles into an intermediate state, part normal and part superconducting, and London’s model of it predicted the resistance correctly while leaving open what the pattern of normal and superconducting regions actually looks like. That question has an answer for a simpler geometry, a plate in a magnetic field, and the answer is a pattern with a definite size.

Why the field cannot simply be pushed out #

A superconductor that expels a field has to put the expelled flux somewhere, and it goes round the outside. For a long rod held along the field there is almost nothing to go round — the rod is thin across the flux — and the field at its surface is just the applied field. For any other shape the field lines have to crowd past the edges, and the field there is larger than the field applied. A superconducting sphere in a uniform field has, at its equator, a field of exactly 3/2 of the applied value. When the applied field reaches two-thirds of $H_c$, the equator reaches $H_c$ and the sphere can no longer remain entirely superconducting.

Nor can it go entirely normal. If it did, the field would enter, the crowding would vanish, and the field everywhere inside and on the surface would be only two-thirds of the critical field — low enough for superconductivity to return. The only consistent state has normal and superconducting regions side by side, with the normal regions holding exactly the critical field. The fraction that is normal is fixed by requiring that the total flux through the body be what the shape and the applied field demand, and it depends on the shape only through one number, the demagnetising factor $N$ that the magnet that has to fight its own field found deciding how strongly a magnetised body opposes itself. The first figure draws the result for four shapes: zero for a long rod along the field, a third for a sphere, a half for a long rod across the field, and one for an infinite plate across the field.

The arithmetic leaves a mark that can be measured without looking inside. A superconducting sphere’s magnetic moment grows in proportion to the applied field while it expels it completely, with the sphere behaving as a perfect diamagnet whose internal field, as the field that points against the magnet it is in found for any uniformly magnetised sphere, is reduced by its own magnetisation. At two-thirds of the critical field the growth stops and the moment falls, along a straight line, to zero at the critical field itself, as the normal fraction rises from nothing to everything. That ramp, rather than a sudden drop, is what measurements of superconducting spheres in the 1930s and 1940s found, and its slope fixed the demagnetising factor that had produced it. Shape alone, not any property of the metal, had spread a sharp transition over a third of the field range.

The plate is the extreme case. Laid flat across a field, it cannot expel any flux at all, because there is nowhere for the flux to go round: the field lines would have to travel sideways all the way to the plate’s edge. So from the smallest applied field, a fraction $H/H_c$ of the plate is normal, threaded by the critical field, and the rest is superconducting and threaded by nothing. The average over the plate is the applied field, as it must be. The question is how the normal metal is arranged.

Stripes, and the field that fans out of them #

Lev Landau proposed in 1937 that the normal metal forms flat sheets, laminae, running straight through the plate from one face to the other and alternating with superconducting sheets. Inside the plate each normal lamina carries the critical field, the superconducting laminae carry none, and the boundaries between them are walls running perpendicular to the faces. The second figure draws a cross-section near the top surface, with the field lines outside the plate computed from the pattern.

Outside, the field must become uniform, at the applied value, far from the surface. Just above the surface it is anything but uniform: it is concentrated where the normal laminae emerge and absent over the superconducting ones. The field lines fan out from each normal lamina and close up over the gaps, and the non-uniformity dies away with height. A pattern of period $a$ is a sum of Fourier components with wavelengths $a$, $a/2$, $a/3$ and so on, each of which decays above the surface as $e^{-2\pi n z/a}$, so the field is uniform to within a per cent at about a third of a period up. A distant observer measures only the applied field; the stripes are invisible beyond a fraction of their own width.

That fanning field is where the pattern’s scale comes from. A magnetic field stores energy, $B^2/2\mu_0$ per unit volume, and the non-uniform field above the surface stores more than a uniform field of the same flux would. The extra energy is proportional to the depth to which the non-uniformity extends, which is proportional to the period. Coarse stripes leave a deep layer of non-uniform field above each face; fine stripes leave a shallow one. On this account alone the plate would divide its normal metal ever more finely.

The price of a wall #

The opposing cost is the walls. A boundary between normal and superconducting metal in a type I superconductor costs energy, because the superconducting order cannot switch on abruptly. It rises over a coherence length, tens to hundreds of nanometres, while the magnetic field is excluded over the shorter penetration depth, and in the layer between the two the metal pays for exclusion without having fully gained the energy of condensation. Two lengths, and which one is longer found that the sign of this wall energy is what distinguishes type I from type II: when the coherence length is the longer, walls cost energy and the metal keeps them few; when the penetration depth is longer, walls are profitable and the field enters as a lattice of separate quantised vortices instead. The wall energy is a surface tension in the strict sense that the skin that is not a skin defined for a liquid — an energy per unit area of boundary, which the system lowers by having less boundary — and the intermediate state is what a surface tension does when something else is paying for more boundary. In a type I metal the wall energy per unit area can be written as $\tfrac12\mu_0 H_c^2\,\delta$, the condensation energy of a slab of thickness $\delta$, where $\delta$ is a length of the order of the coherence length.

A plate of thickness $d$ divided into laminae of period $a$ has two walls per period, each of area $d$ per unit length, so its wall energy per unit area of plate is $\mu_0 H_c^2\,\delta d/a$: falling as the stripes widen. The stray field’s energy rises as $a$. In the simple model drawn here — stripes running straight to the surface, the field outside built from their Fourier components — the total energy per unit area is

$$e(a) = \mu_0 H_c^2\left[\frac{\delta d}{a} + \frac{a\,S(h)}{\pi^3}\right], \qquad S(h) = \sum_n \frac{\sin^2(\pi n h)}{n^3},$$

where $h$ is the normal fraction. The third figure plots it for a lead plate a millimetre thick, half normal, with $\delta$ a tenth of a micrometre. The minimum is at $a = \sqrt{\pi^3\delta d/S(h)}$, 54 micrometres, where the two contributions are equal. It is a shallow minimum. Periods from 38 to 76 micrometres cost at most six per cent more energy, which is part of why real intermediate-state patterns are wavy, branched and irregular rather than ruled, and why they depend on how the field was applied.

Widest where either phase is scarce #

The fourth figure varies the applied field. When the plate is half normal the stray field is as strong as it can be, the stripes’ contrast greatest, and the period smallest. When the normal laminae are thin, at weak applied field, or the superconducting ones are thin, near the critical field, the field emerging from the surface is nearly uniform already, the stray energy is small for a given period, and the walls win: the period grows. In this model it diverges at both ends, symmetrically.

Real plates do not follow the model to the ends. When one phase is scarce it is cheaper, for the same amount of wall, to arrange it as separate tubes or islands than as long sheets, and that is what experiments show: near zero field the flux threads the plate in isolated normal tubes, and near the critical field the superconducting regions shrink to isolated islands. Which arrangement appears depends on the sample’s history as well as the field — whether the field was raised from zero or lowered from above $H_c$ — because once a pattern has formed, rearranging it means moving walls through a metal with defects that pin them.

The square root of two lengths #

The last figure shows the property of the pattern that does not depend on the details. The period goes as the square root of the plate’s thickness times the wall length. Neither length alone sets the scale: $\delta$ is a fraction of a micrometre and $d$ is a millimetre, and the stripes are tens of micrometres wide, at the geometric mean. Quadruple the thickness and the stripes double in width. The critical field, which sets how much of the plate is normal, drops out of the period altogether: both the wall energy and the stray-field energy are proportional to $\mu_0 H_c^2$, so a metal with a weak critical field, such as tin at 0.03 tesla, and one with a stronger field, such as lead at 0.08, make stripes of the same width if their wall lengths and thicknesses are equal. What distinguishes metals is $\delta$, and through it the coherence length, which is why measuring the period of the laminae in plates of known thickness became one of the ways of measuring the wall energy, and so the structure of the boundary, of a superconductor. The result follows from the form of the competition alone — a cost proportional to thickness over period, against a cost proportional to period — and the details of the stray field enter only through the constant in front.

Patterns of that size are visible under a microscope, and they have been seen since the 1940s, first by Alexander Meshkovsky and Alexander Shalnikov scanning a fine bismuth wire across a gap in a tin sphere, then by sprinkling superconducting powder, which is repelled by the field and settles on the superconducting regions, and later through magneto-optical films that rotate the polarisation of light in proportion to the field beneath them. The photographs show laminae of roughly the predicted width in thick plates of tin and lead, often bent into labyrinths and fingerprints, and they confirm the square-root dependence on thickness that the model predicts. The wall energy deduced from them agrees with what other measurements give for $\delta$.

The same square root appears in ferromagnets, and for the same reason. A ferromagnetic plate magnetised perpendicular to its faces would produce a large stray field outside; it divides into domains of alternating magnetisation, each costing a wall, and Charles Kittel showed in 1946 that the domain width is the geometric mean of the plate’s thickness and a length set by the wall energy. The first length that belongs to the substance found that wall’s own width, $\sqrt{A/K}$, in exactly this competition one level down. The physics differs — a ferromagnet makes its own field and a superconductor refuses an applied one — but the accounting is identical: walls on one side, a leaking field on the other, and a pattern at the mean.

Where the straight stripes are an idealisation #

Landau himself recognised that straight laminae running unchanged to the surface are not the lowest-energy arrangement. The stray field can be reduced if the normal laminae widen as they approach the surface, letting their flux spread out inside the metal instead of above it, and if they branch into finer laminae near the faces. Landau’s later branching model gives periods that vary with depth, coarse in the middle of the plate and fine at the surfaces, and lowers the energy somewhat. The magnetostatic model drawn here keeps the laminae straight and computes only the field outside, which gives the right square-root scaling and a period within a factor of order one of the more refined treatments. The constant 5.43 is a property of the model, not of lead.

The model also treats the plate as infinite. A real plate has edges, and near them the flux can escape sideways and the demagnetising factor is less than one, so the normal fraction near an edge differs from that in the middle. It ignores the pinning of walls by defects, which makes the pattern depend on history, and it treats $\delta$ as a constant, although it depends on temperature, diverging as the critical temperature is approached because the coherence length does.

The pattern a drawing of stripes leaves out #

The figures show a static, periodic arrangement in a cross-section, and the real pattern is neither static nor periodic. Seen from above, the normal regions form labyrinths, spirals and arrays of tubes, depending on the field and how it was reached, and they move when a current is passed through the plate: the current exerts a force on the flux in the normal regions and drives the whole pattern across the sample, the effect behind the moving discs and lenses that the wire carrying a supercurrent was found to contain. None of that is in a model of straight, stationary laminae. The figures also say nothing about the microscopic structure of a wall, whose width and energy come from the theory of superconductivity itself; they take $\delta$ as given.

Still open: what selects the shape of the pattern #

The period is understood; the topology is not, or not fully. Whether a plate in a given field settles into laminae, a labyrinth, or a lattice of tubes depends on small differences in energy between arrangements that the simple models cannot resolve, on the dynamics by which flux enters, and on hysteresis. Experiments since 2000 with fast magneto-optical imaging have shown that flux entering a type I plate forms tubes when the field rises and laminae when it falls, a topological hysteresis in which the same field can hold either pattern depending on the direction it was approached from. The bubble that grows by counting its sides found a froth’s cells evolving by rules that depend only on topology; whether the intermediate state’s labyrinths obey comparable rules as they coarsen, and what determines which arrangement a given history produces, is still being worked out.

The habit worth carrying away is to ask what a refusal costs. A superconductor that cannot expel a field without exceeding $H_c$ at its surface — a sphere from 2/3 of $H_c$, a flat plate from the first trace of field — divides into normal laminae holding exactly $H_c$ and superconducting ones holding none, and the laminae’s width is fixed by balancing wall energy against the field fanning out of the surface: $\sqrt{\pi^3\delta d/S}$, 54 micrometres for a millimetre of lead half normal, the geometric mean of a fraction of a micrometre and a millimetre. Wherever a wall cost competes with a cost that grows with the size of the regions it separates, the pattern lands at the mean of the two lengths.