The resistance that is a length

Press two blocks of copper together and pass a current. Where does the resistance come from? The obvious answer is that it comes from the metal, and that it is the resistivity times the length of the path divided by the area of the path — the formula everything else in a circuit obeys. Both halves of the obvious answer are wrong, and the way in which they are wrong is more interesting than the answer that replaces them.

Two surfaces do not touch over their apparent area. They touch at the tops of the roughness, at a few spots, and the current has to converge into each of those spots and spread out again on the other side. There is no path length to speak of and no cross-section being crossed, because the current comes from all directions at once.

The problem is one of the classical exactly-solvable ones, and its answer is short. For a circular spot of radius $a$ between two conductors of resistivity $\rho$,

$$R = \frac{\rho}{2a}.$$

There is one length in it and it is the spot. Not the length of the conductors, not their cross-section, not the thickness of the surface layer, not the distance to where the current came from. A resistivity divided by a length, which is dimensionally correct and unlike every other resistance in a circuit.

Where the resistance is #

The reason nothing else appears in the answer is worth drawing, because it is the whole of the physics.

Resistance is dissipated where the current density is high, and the current density is high only where the current has been squeezed. A few spot radii away the current is spread over a hundred times the area, so the density is a hundredth and the dissipation ten thousandths. The integral converges fast, and it converges to something that can only depend on the geometry near the spot — which is a single length.

That is why the answer is what it is, and it also settles the practical question. A joint that is too resistive cannot be improved by using thicker busbar: ninety-four per cent of the trouble is within a few micrometres of the contact and the busbar is nowhere near it. It can be improved by making the spots bigger, which means pressing harder or making the surfaces smoother, and by making more of them.

There is a general habit here worth naming, because it recurs whenever a quantity is dominated by a small region. A capacitance is also a length times a constant, and for the same reason: the field of an isolated conductor is concentrated near it, so its capacitance depends on its size and hardly on its shape. Constriction resistance and self-capacitance are the same kind of quantity — a material constant divided by, or multiplied by, one length — and both are quantities that a formula built for a uniform path cannot produce.

The size at which it takes over #

The crossing point is the number that matters, and it is small. A contact spot a few micrometres across has the resistance of ten centimetres of ordinary hookup wire; a spot a tenth of a micrometre across has the resistance of ten metres of it. Real spots between pressed metal surfaces are of exactly this size, which is why contact resistance is a first-order quantity in a switch or a connector rather than a correction to one.

The scaling is the reason the effect refuses to go away as things get smaller. Bulk resistance falls as the cross-section grows, so a conductor scaled down by ten has a hundred times the resistance per unit length but is ten times shorter, giving ten times the resistance. A constriction scaled down by ten has ten times the resistance. Everything shrinks; the constriction shrinks least, in proportion, and so it takes over. In microelectronics the contacts stopped being a correction some decades ago and became the thing being designed.

The same argument extends past the point where classical conduction applies. Once the spot is smaller than the distance an electron travels between collisions, the picture of a current density diffusing through a resistive medium is wrong: electrons cross the constriction without scattering at all, and the resistance becomes a matter of counting how many quantum channels the aperture supports. The answer there is not $\rho/2a$ but a multiple of $h/2e^2$, a resistance made of nothing but fundamental constants — the same $h/e^2$ that turns up wherever conduction is counted rather than integrated. The classical result is the limit in which the spot holds a great many channels.

That the crossing hardly moves between copper and steel is worth pausing on, because it looks like a coincidence and is not. The contact’s resistance is $\rho/2a$ and the wire’s is $\rho \ell/\pi r^2$, so their ratio is $\pi r^2 / 2a\ell$ — a pure geometry, with the material gone. A poor conductor has a worse contact and a worse wire in exactly the same proportion. Choosing a better metal does not change which of the two is dominating; only changing a dimension does.

That is a general and useful consequence of the constriction resistance being proportional to the same resistivity as everything else. Contact resistance is not a separate material problem to be solved by plating. Plating helps for a different reason entirely — it stops the oxide — and the fact that gold-plated contacts are better than bare copper ones has almost nothing to do with gold’s conductivity.

Many spots, and a floor #

Real contacts are not one spot. They are a scatter of them over some patch, and the arithmetic of that is more interesting than a parallel sum.

Far away from the contact the current does not know that the patch is made of spots; it sees a region of radius $A$ that it must converge into, and pays $\rho/2A$ for the privilege. Only close in does it divide among the spots and pay $\rho/2na$ more. So the total has two terms, and subdividing a contact more finely buys less and less until it buys nothing.

The practical consequence is that a contact’s resistance is a measurement of two different things at once and can be made to give up both. A single measurement at one load gives a combination; measurements across a range of loads separate them, because pressing harder both enlarges the existing spots and creates new ones, and the two terms respond differently.

What the two terms say about a joint #

Reading the two terms apart is the whole of what a contact measurement can tell an engineer, and it is worth spelling out what each one means physically.

The spot term, $\rho/2na$, is about how much metal is actually touching. It falls when the load rises, because asperities flatten and new ones come into contact, and it falls when the surfaces are smoother, because more of them are near enough to touch at once. It is the term that responds to everything anybody does to a joint.

The cluster term, $\rho/2A$, is about how spread out the touching is. It does not respond to load at all once the patch has stopped growing, and it is the reason a heavily loaded joint stops improving. A contact designed to be many small spots over a wide area beats one designed to be a few large spots over a narrow one, at the same total real area — which is the opposite of what an intuition built on “area equals conductance” would suggest, and it is why contact surfaces are given a crown or a wipe rather than being made as flat as possible.

The ratio of the two terms is measurable and is $A/na$: the patch radius over the sum of the spot radii. For an ordinary pressed joint it is of order ten, which says that the spots occupy a per cent or so of the patch. That number is the real contact area fraction, obtained from nothing but two resistance measurements, and it is the same number the mechanics predicts.

The instrument this makes #

The reason to care about all of this is that it turns a resistance measurement into a measurement of something quite different: the real area over which two surfaces touch.

That quantity is the central one in the theory of friction that explains why the coefficient does not depend on the apparent area. Friction is proportional to the real contact area, which grows in proportion to the load because the asperities deform; the apparent area cancels out. The argument is compelling and its central quantity is not directly visible — the real area is buried between two opaque solids.

Electrical measurement reaches it. The constriction resistance depends on the same spots, through a different physics, and gives a number for $na$ and for the patch size without moving anything or taking the surfaces apart. Historically this was the check: Holm’s contact measurements and Bowden and Tabor’s friction experiments were done on the same materials in the same decades, and they agreed on how much of a surface actually touches — a percentage or less, at ordinary loads, for ordinary metals.

Two independent routes to a quantity that neither can see directly is the strongest kind of evidence a physical model gets. It is also worth noticing what makes it possible: the constriction resistance depends on the spots’ size rather than their area, and friction on their area. The two measurements are therefore differently weighted over the same population, and comparing them says something about the distribution of sizes as well as about the total.

Who worked it out, and from where #

The solution is older than the problem it is now used for, and its route into contact physics is a good example of a piece of mathematics arriving from somewhere else.

The potential of a charged conducting disc was solved in the nineteenth century as an electrostatics problem — the disc is one of the few shapes for which Laplace’s equation separates, in the oblate spheroidal coordinates the figures above are drawn in. Maxwell used it for capacitance. The observation that the same boundary-value problem describes a current crossing a circular aperture is a change of interpretation and not of mathematics: a conducting disc held at a potential, and a circular window between two conductors, have identical boundary conditions.

Ragnar Holm spent the 1920s to the 1960s turning that into the physics of electrical contacts, and the reason the subject needed a book rather than a paper is everything the clean solution leaves out — the films, the heating, the mechanical settling, the way a spot’s size changes while current passes through it. What survives untouched from the clean solution is the scaling, and it is what the whole subject is organised around: a resistance proportional to one over a length means that everything depends on how big the touching regions are and on nothing else, so the entire engineering problem becomes the mechanics of how surfaces meet.

That is a recurring pattern worth naming. A solvable idealisation is often useful not because it is accurate but because it identifies which quantity the messy reality has to be described in terms of. Nobody computes a real contact from ρ/2a. Everybody describes a real contact by the effective spot size that formula implies, and that number is measurable, comparable between laboratories, and connected to the mechanics.

Where the model stops #

The two conductors are assumed clean. They are not. Copper grows an oxide within seconds of exposure, and the oxide is a far worse conductor than the metal, so a real contact’s resistance is usually dominated by a surface film rather than by the constriction. Pressing hard enough fractures the film at the asperities and metal touches metal — which is why contact resistance falls sharply above a threshold load and why a connector specifies a minimum contact force rather than a maximum.

The current is assumed to spread diffusively. That fails when the spot is smaller than the electron mean free path, which in copper at room temperature is about forty nanometres and at low temperature can be micrometres. Below that the resistance is smaller than the classical formula gives, because electrons cross ballistically, and the crossover between the two regimes has its own well-studied form.

Heating is ignored, and it is the thing that fails. All the dissipation happens in a volume of order a spot radius cubed, so the power density at a contact is enormous, and the temperature rise turns out to depend only on the voltage across the constriction rather than on the current or the material — a result with the same flavour as the resistance’s, and for the same reason, since heat spreads out of the spot by the same equation the current is obeying with a conductivity in the place of a conductivity. A few tenths of a volt across a contact softens most metals whatever they are made of.

A softened contact changes while it is measured. A spot at the softening temperature flattens under the load, which enlarges it, which lowers the resistance and the temperature with it — so a contact carrying a large current settles into a state where the constriction voltage sits near the softening value and stays there. That is not always a fault: it is how a contact beds in. It is also how one welds shut, and the boundary between the two is a matter of how far past the softening point the melting point is.

And the spots are treated as circular, coplanar and independent. None is exactly true. What survives is the scaling — a resistance proportional to a resistivity over a length — because that follows from the shape of the problem rather than from the shape of the spot, and a non-circular spot of the same size differs by a factor near one.

What the pictures cannot show #

The section through the contact draws equipotentials and current lines in one plane and cannot show what the axial symmetry is doing. The equipotential surfaces are oblate spheroids, flattened almost into discs near the spot, and the current lines are hyperboloids of one sheet; the drawing is a slice, and the crowding it shows is milder than the real crowding, which happens in three dimensions.

None of the figures shows the current density, which is the quantity an engineer would want and which is what actually decides whether the contact survives. It diverges at the rim of the spot — the exact solution has an inverse square root singularity there, in the same way that the charge on a conducting disc piles up at its edge — so the model predicts an infinite density at a sharp rim. What happens in a real contact is that the rim is not sharp, and the divergence is cut off by whatever radius it actually has.

Where the ladder goes next #

The conductors ladder began with the field inside a conductor being exactly nothing, went through how much charge a shape will hold, the charge that has to be somewhere else, how far a field gets into metal and the pressure a charge puts on its own metal. This rung is about where a current is forced to converge. The rungs after it: the four-terminal measurement, which exists precisely because a two-terminal one measures the contacts as well as the sample; thermal constriction, which is the identical mathematics with a temperature in the place of a potential and is why the same joint is a thermal bottleneck; and conduction through a single atom, where the count of channels replaces the geometry.

The habit worth carrying away is dimensional. When a resistance comes out as a resistivity divided by a length rather than by a length over an area, the current is converging rather than flowing along. That form is a signature, it identifies the geometry from the units alone, and it says immediately that the answer cannot depend on how big the conductors are.