The current a superconductor stops with its own field

Heike Kamerlingh Onnes found superconductivity in mercury in 1911, and within two years he had a use for it. An electromagnet’s field is limited by the heat its current makes in the copper; a coil with no resistance would make no heat, and he calculated that a superconducting coil could reach a hundred thousand gauss — ten tesla — with no power at all. In 1913 and 1914 he wound coils of lead wire and tin wire, cooled them in liquid helium and passed current. Each stopped superconducting at a current far below what he needed, and at a field of a few hundredths of a tesla. The magnet he had planned was not built with superconductors for another half-century.

Onnes had found two limits — a critical field above which superconductivity was destroyed, and a critical current above which it was destroyed — and at first they looked like two separate properties of the metal. In 1916 Francis Silsbee, at the United States Bureau of Standards, suggested they were one. A current in a wire makes a magnetic field around it, and at the surface of the wire that field is, by Ampère’s law, the current divided by the circumference. The critical current is the current whose own field at the surface reaches the critical field:

$$I_c = 2\pi a H_c.$$

The rule is short, and it has consequences that the idea of a resistanceless conductor does not prepare one for.

The field a current makes at its own surface #

The field that wraps a current found that a straight current $I$ is circled by a magnetic field of strength $I/2\pi r$ at distance $r$ — Ampère’s law applied to a circle round the wire. At the wire’s own surface, $r = a$, the field is $I/2\pi a$. For a superconductor this field is not merely a by-product. A superconductor can remain superconducting only while the field at every point of its surface stays below the critical field $H_c$, the field at which the field that is pushed out found the expulsion of flux costs more energy than superconductivity saves. When the current’s own field reaches $H_c$ at the surface, the surface goes normal, and the current can no longer flow without resistance.

For lead the critical field is 80.3 millitesla at absolute zero, falling as $1 - (T/T_c)^2$ to zero at 7.19 kelvin. A lead wire a millimetre across in liquid helium at 4.2 K can therefore carry 132 amperes, and at absolute zero 201. Tin, with a critical field of 30.5 millitesla and a transition at 3.72 K, carries nothing at 4.2 K, because it is not superconducting there at all, and 64 amperes at 1.5 K. Onnes’s coils, wound from fine wire and immersed in their own field, failed at currents far below what a single straight wire could carry: the field inside a coil is the sum of every turn’s, far larger than one wire’s own field, and the coil’s field reached the critical value long before any wire’s did.

The rule ties the current to the temperature through the critical field and nothing else. Each curve is the critical field’s parabola scaled by $2\pi a$, falling to zero at the transition. No property of the metal’s conduction enters, which is the first sign that the current is not limited by the metal’s ability to carry it.

The current lives in the skin #

The proportionality to radius rather than area says where the current is. In a normal wire the current spreads uniformly through the cross-section, and the largest current a wire can carry without overheating grows roughly with its cross-section. In a superconductor the current is confined to a layer at the surface a penetration depth thick — about fifty nanometres in lead — for the same reason the field is: two lengths, and which one is longer found that a superconductor screens a magnetic field from its interior with surface currents decaying over the penetration depth, and a transport current, which comes with its own field, is screened in exactly the same way.

The London equation, which describes that screening, gives the profile across a cylinder exactly: the current density is proportional to a modified Bessel function of the distance from the axis measured in penetration depths. For a wire many penetration depths thick it is concentrated at the surface and falls exponentially inwards.

A wire a millimetre across has its current in a skin one twenty-thousandth of its radius. The interior carries nothing; it could be hollow. That is why the critical current depends on the circumference: the current is a surface current, and the field it makes at the surface is set by how much current there is per unit length of surface. Doubling the radius doubles the length of surface and doubles the current the surface can hold at the critical field — and multiplies the cross-section by four, three-quarters of which carries nothing.

Thick wires are worse #

So the critical current grows only as the diameter. A lead wire 1 mm across carries 192 amperes at 1.5 K, 244 amperes per square millimetre of its section. A wire 1 cm across carries ten times as much current, at a tenth of the density. A wire 10 micrometres across carries 1.9 amperes at 24,400 amperes per square millimetre. For type I superconductors the rule runs the opposite way to ordinary engineering, in which a thick conductor is the way to carry a large current.

The dashed line in the figure is what a modern superconducting magnet wire does. Niobium–titanium, the alloy in the coils of nearly every hospital MRI scanner, carries about 3,000 amperes per square millimetre at 4.2 K in a field of 5 tesla, and the current it carries grows with its cross-section like an ordinary conductor’s. The two lines cross at 81 micrometres: in a thinner wire, lead’s surface current would beat the alloy’s bulk current. Above it the alloy wins, and wins by more the thicker the wire — and it does so in a field of 5 tesla, sixty times lead’s critical field, in which lead is an ordinary metal.

The difference is the subject of the field a superconductor keeps after all. Niobium–titanium is a type II superconductor: above a lower critical field it lets magnetic flux into its interior in quantised vortices, and the vortices are pinned by defects deliberately put into the alloy. The bulk can then carry current, each region holding the flux gradient its pinning allows, and the critical current is a current density times an area, as Onnes had hoped. Silsbee’s rule is the rule for the superconductors that cannot do this — the pure elements Onnes had, which expel flux entirely or not at all — and it is the reason type I superconductors have never been used to carry large currents.

Three limits, three exponents #

It is worth setting the rule beside the limits on other conductors, because each scales with the wire’s diameter by a different power, and the power says what the limit is made of.

A copper wire’s current is limited by heat. The heat it makes per metre is $I^2$ times its resistance per metre, which goes as $1/d^2$; the heat it can lose per metre goes as its surface, $d$, times a temperature rise it is allowed. Setting the two equal gives a current proportional to $d^{3/2}$ — a law electricians’ tables follow closely for bare wire, and one that the insulation that makes a wire lose more heat found bent by the insulation round a thin wire. A type II superconductor’s current is limited by how hard its pinning holds its vortices, a property of each cubic millimetre, and its current goes as $d^2$. A type I superconductor’s current is limited by the field at its surface, and goes as $d^1$.

The exponent is a fingerprint. Doubling the wire multiplies a copper wire’s capacity by 2.8, a niobium–titanium wire’s by 4 and a lead wire’s by 2. In each case the limit lives where the exponent says: in the copper’s balance between volume heating and surface cooling, in the alloy’s bulk, in the lead’s skin.

The skin is not peculiar to superconductors. How far a field gets into metal found that an alternating current in an ordinary metal also crowds into a surface layer, whose depth falls as the frequency rises; at a gigahertz the current in copper flows within two micrometres of the surface, and a thick conductor carries no more than a hollow tube of the same diameter. A superconductor does the same thing with a direct current. The penetration depth plays the part of the skin depth, but it does not depend on frequency: a superconductor’s skin is there at zero frequency, because the screening that produces it is not induced by changing flux but is a property of the state.

There is also a family resemblance to the current that squeezes what carries it, in which a current in a plasma is compressed by its own magnetic field — the pinch. In both cases a current’s field acts back on the current’s carrier, and in both the decisive quantity is the field at the carrier’s surface, set by Ampère’s law as $I/2\pi a$. In the plasma the field’s pressure squeezes the column; in the superconductor the field’s strength destroys the state that lets the current flow. Neither cares how the current is distributed inside, only how much of it is enclosed, which is the content of the law that is always true and rarely useful applied in one of the cases where it is useful.

What an applied field leaves #

A wire in a magnet is also in the magnet’s field, and the rule extends directly: the condition is that the total field at the surface stay below $H_c$, and the total depends on how the applied field is arranged relative to the wire.

A field along the wire is at right angles, everywhere on the surface, to the circling field of the current. The two add in quadrature, and the largest current is $2\pi a\sqrt{H_c^2 - H_0^2}$: a field half the critical value leaves 87 per cent of the current. A field across the wire is worse. A superconducting cylinder excludes it, and a field forced round a cylinder is doubled at the cylinder’s flanks — the same doubling that makes the flow of a fluid round a cylinder fastest at its sides. At the flanks the excluded field and the current’s field add directly, and the largest current is $2\pi a(H_c - 2H_0)$, which falls to nothing at half the critical field.

The transverse case is the one that defeated Onnes. The turns of a coil sit in each other’s fields, mostly across the wire, and the field at the inner turns of even a modest coil is several times one wire’s own surface field. The coil’s critical current is set by its worst turn, and in a coil of lead wire the worst turn reaches half the critical field — where the transverse curve hits zero — at a current far below what the wire alone could carry.

Half a resistance, all at once #

What happens above the critical current is not what the word “critical” suggests. If the wire simply went normal it would switch from zero resistance to its full normal resistance. It cannot do that, for a reason that comes from the same field. Suppose the outer layer goes normal. The current it was carrying must now flow further in, through the still-superconducting core — but a current confined to a smaller radius makes a larger field at its own surface, larger than $H_c$, so the core’s surface goes normal too. Suppose instead the whole wire goes normal. Then the current spreads uniformly, and the field inside a uniformly carrying wire falls linearly to zero at the axis — below $H_c$ in an inner region, which should then be superconducting.

Neither state is consistent. Fritz London showed in 1937 that the wire settles into an intermediate state: an arrangement of normal and superconducting regions in which the field everywhere in the core is held at exactly $H_c$. His model gives the resistance as

$$\frac{R}{R_n} = \frac12\left[1 + \sqrt{1 - \left(\frac{I_c}{I}\right)^2}\right],$$

which jumps from zero to one half the instant the critical current is passed.

The jump to a half is a definite prediction about a resistance, made from nothing but the field geometry and the critical field, and measurements on wires of tin and indium found the resistance appearing abruptly at the critical current at a sizeable fraction of the normal value — usually somewhat more than London’s half, because the real intermediate structure is not his idealised one and its normal regions take up more of the wire than the minimum. The jump itself, discontinuous and partial, is what distinguishes a type I wire from a type II, whose resistance rises smoothly from zero as vortices begin to move.

The rule run in reverse #

Silsbee’s rule is used now mainly in reverse, as a diagnosis. When a thin film or a fine wire of a superconductor carries less current than it should, the first question is whether the limit is the field at the surface — set by geometry, by Silsbee — or the material’s own ability to carry current, set by pinning or by the depairing of its electrons. In films a few micrometres wide the edges carry most of the current, for the same reason the surface of a wire does, and the field at the edges is enhanced by the film’s flatness; a film’s critical current is then limited by its edges, and the remedy is geometrical rather than metallurgical.

The rule also explains why type I materials survive in a few places where the current is small and the geometry favourable: the aluminium in superconducting qubits, the circuit that forgets its charge, carries currents of nanoamperes in films a fraction of a micrometre thick, far below anything the surface field could limit. At those currents the limit is not the field but the junctions.

What the figures leave out #

The figures use the bulk critical field of each metal and treat the wire as a long, straight, uniform cylinder. For wires thinner than a few penetration depths the current spreads through the section, as the thinnest profile shows, and the critical field itself rises above its bulk value, so the rule underestimates a very fine wire’s current. The parabolic law for the critical field’s temperature dependence is an approximation that is good to a few per cent for these metals. Real wires have grain boundaries, surface roughness that concentrates field, and contacts at which the current enters through a normal metal; and the intermediate state’s resistance depends on how the wire is cooled and on its history. The domain of the drawings is a clean type I wire, much thicker than its penetration depth, carrying a steady current in a uniform or zero applied field.

Still open: how the intermediate state arranges itself #

What London’s model leaves unspecified is the actual arrangement of normal and superconducting regions inside a wire above its critical current. He proposed concentric shells; later experiments and calculations suggested regions shaped like lenses or discs stacked along the axis, moving along the wire as the current flows, and magneto-optical imaging of current-carrying strips has shown patterns that change with current and with the sample’s history. How the pattern is selected — by the balance of field energy and the energy of the walls between normal and superconducting regions, by the dynamics of how it forms, or by pinning on defects — is not fully settled, and the same questions arise in the intermediate state of a type I plate in a perpendicular field, where the patterns of normal and superconducting regions form labyrinths and bubbles of a kind seen in other systems with competing long- and short-range forces.

Silsbee’s own rule is exact. A type I superconducting wire stops superconducting when its current’s own field at the surface, $I/2\pi a$, reaches the critical field, so $I_c = 2\pi a H_c$ — 132 A for a 1 mm lead wire in liquid helium — proportional to the radius because the current flows in a surface skin fifty nanometres thick; above it the wire goes into an intermediate state at half its normal resistance. A conductor with no resistance is still limited, and the limit is written in the field of the very current it carries.