The sheet resistance that forgets the shape

The inside of a conductor found that a conductor in equilibrium has no field inside it, and how much charge a shape will hold and the charge that has to be somewhere else turned that into capacitance and image charges. How far a field gets into metal let the field in when it changes fast, and the resistance that is a length followed a steady current as it was forced to converge into a small contact, where it found a resistance that depends on the size of the spot and nothing else. That argument ended by naming what comes next: the four-terminal measurement, which exists because a two-terminal one measures the contacts as well as the sample.

The four-terminal measurement has a remarkable special case. For a thin film — a layer of metal or semiconductor thin compared with its width — the quantity that matters is the sheet resistance, the resistance of any square piece of it between opposite edges, which is the resistivity divided by the thickness and is the same for a square of any size. It can be measured on a sample of any shape at all, with contacts anywhere on its edge, and the answer does not depend on either. The reason is one of the neatest applications of the mathematics of potentials in all of physics, and its one failure is as instructive as its success.

Why four contacts rather than two #

A resistance measured with two wires includes everything in the current’s path: the wires, the contacts, and the crowding of current into each contact, which the resistance that is a length found can easily exceed the resistance of the sample. Lord Kelvin’s remedy, from the 1860s, is to separate the jobs. Two contacts carry the current; two others, where no current flows, measure the voltage. A voltmeter that draws no current sees no drop in its own leads or contacts, so the reading is the potential difference inside the sample between the two voltage contacts, divided by the current.

For a long strip with the voltage contacts between the current contacts, the reading converts to a resistivity with the strip’s length, width and thickness. That needs a strip, cut accurately, with contacts at known places. Semiconductor wafers, thin films deposited on glass, flakes of new materials a few micrometres across: none of them comes as a neat strip, and cutting them into one changes them or destroys them.

The unit deserves a word, because it is what makes a film’s resistance a single number. A square of film carries current between two opposite edges with a resistance equal to its resistivity divided by its thickness, and that does not depend on how large the square is: doubling the side doubles the length the current travels and doubles the width it travels through, and the two cancel. A strip three squares long has three times the sheet resistance between its ends, whatever its width. Engineers therefore quote the resistance of films in ohms per square, and design a thin-film resistor by counting squares. The quantity every method is after is this one number, and in a strip it has to be disentangled from the geometry by measuring the length and width. Van der Pauw’s method never measures either.

Four contacts on any shape #

Van der Pauw’s method puts all four contacts on the edge of the film, anywhere, in order round the perimeter: A, B, C, D. Pass a current from A to B and measure the voltage from D to C; call the ratio $R_1$. Pass it from B to C and measure from A to D; call that $R_2$. Then, for a film of uniform thickness with no holes and contacts small enough to be points,

$$e^{-\pi R_1/R_s} + e^{-\pi R_2/R_s} = 1,$$

where $R_s$ is the sheet resistance. Given the two readings, there is exactly one $R_s$ that satisfies the equation, and it can be found in a moment.

The figure tests the claim without assuming it. Each film is laid out as a fine grid of identical resistors — a sheet of one ohm per square — and for each the current flow is solved directly, with a unit current fed in at one contact and taken out at another, and the voltage read between the other two. The square, with contacts at its corners, gives equal readings of 0.221 ohms. The disc, with its contacts bunched unevenly, gives 0.089 and 0.450. The irregular lamina gives 0.209 and 0.232. None of the readings means anything on its own, and all three pairs, put into van der Pauw’s relation, give the same sheet resistance to within the grid’s accuracy of a part in a thousand.

Where the current goes #

It helps to see the flow inside one of the films.

The figure draws lines of equal potential in the irregular film with current passing from A to B. Near each current contact the lines crowd into small semicircles, where the current converges, and the potential there changes steeply. Through the rest of the film they spread out, and the far part of the film, where C and D sit, lies in a gentle slope of potential whose difference between the two contacts is the reading. Every line meets the edge at right angles: no current crosses the edge, so the potential cannot change in the direction perpendicular to it.

That condition at the edge, together with the fact that the potential inside a uniform sheet with no current sources obeys Laplace’s equation — the equation one number for every point found governing any potential in empty space — is all the physics there is. The rest is geometry.

The proof in a half-plane #

The simplest shape to compute is the least practical one: a sheet filling half of an infinite plane, with its four contacts on the straight edge.

A current $I$ fed into a half-plane at a point on its edge spreads out in semicircles, and the potential falls as the logarithm of distance, because the current crosses semicircles whose length grows in proportion to their radius. With a source at A and a sink at B, the potential along the edge is a difference of two logarithms, which the figure draws: it runs to plus infinity at the source and minus infinity at the sink and settles smoothly elsewhere. The reading $R_1$ is the potential difference between D and C, and it comes out as $1/\pi$ times the logarithm of a combination of the four positions called their cross-ratio. The second reading is $1/\pi$ times the logarithm of a second cross-ratio. For any four points in order on a line, the two cross-ratios in question add to exactly one, and that identity, with the logarithms undone, is van der Pauw’s relation.

The step to any other shape uses a theorem of Riemann’s: any simply connected region of the plane can be mapped onto a half-plane by a transformation that preserves angles, a conformal map. Such a map carries solutions of Laplace’s equation into solutions, point sources into point sources of the same strength, and edges where no current crosses into edges where no current crosses. It moves the four contacts to four new points on the straight edge, and the readings of the original film are the readings of the half-plane with the contacts at those points. Whatever the four new points are, their cross-ratios add to one. The shape has disappeared into the positions of the contacts, and the relation does not care about the positions.

The correction factor #

In practice the relation is written in a form that shows how far from ideal a particular placement of contacts is.

When the two readings are equal, as they are for a square with contacts at its corners or a disc with contacts evenly spaced, the relation solves at once: the sheet resistance is $\pi/\ln 2$, 4.532, times the reading. When they differ, the sheet resistance is 4.532 times their mean, multiplied by a factor $f$ that depends only on their ratio and falls from one as the ratio grows. A ratio of ten needs $f = 0.70$. A large ratio means the contacts are unevenly placed, so that one pair sits close together and its reading is small; the correction then carries much of the answer, and a small error in the smaller reading becomes a large error in the result. Laboratories therefore place contacts as symmetrically as the sample allows, check that the ratio is near one, and then repeat each reading with the current reversed and the roles of the contacts exchanged, averaging out thermoelectric voltages at the contacts and any small asymmetry.

The same four contacts, with a magnetic field applied perpendicular to the film and the current passed between opposite contacts rather than adjacent ones, measure the Hall voltage, which gives the density of charge carriers and their sign. Van der Pauw’s 1958 paper treated both. The pair of measurements — sheet resistance and Hall coefficient on an arbitrary flake — is the standard characterisation of a new semiconductor, and it is what the barrier the metal cannot choose would have needed to know about the doping of its silicon.

What a hole does #

The argument used one fact about the film beyond its uniformity: that it could be mapped onto a half-plane. That needs a film with no holes in it.

The figure cuts a circular hole out of the middle of the square and repeats the calculation. The current, which had spread through the whole film, now has to go round the hole, and both readings rise. The relation, which cannot know about the hole, interprets the higher readings as a more resistive material: a hole a tenth of the side across gives a sheet resistance six per cent too high, and one four-tenths across gives eighty per cent too high. The error has no warning sign. The readings still satisfy the relation for some value of $R_s$, and nothing in them says the value is wrong.

A region with a hole is not simply connected: a loop round the hole cannot be shrunk to a point without leaving the film, and no conformal map can carry it onto a half-plane, which has no holes. A potential that does not come back to itself met the same topological fact in magnetostatics, where a hole in the region is what lets a scalar potential be multivalued. Here it is what breaks a measurement. In practice holes are cracks, voids, or regions where a film failed to deposit, and a sample that gives inconsistent results when its contacts are rotated round the edge is the usual first sign of one.

The same relation for anything that obeys Laplace’s equation #

Nothing in the proof used the fact that the thing flowing was electric charge. It used a quantity that is conserved as it flows, a flux proportional to the gradient of a potential, and an edge across which nothing flows. Heat in a thin plate obeys exactly those rules, with temperature for potential and thermal conductivity for conductivity, and the relation holds with thermal resistances in place of electrical ones: heat a plate at one edge point, cool it at another, read the temperature difference between two further edge points, and the plate’s sheet thermal conductance follows from two such readings whatever its shape. Methods based on this are used for thin films where the thermal conductivity is the property of interest, with laser spots or micro-heaters as the contacts.

The same mathematics also has a probabilistic reading. The potential is where the wanderers stop found that the solution of Laplace’s equation at a point is an average over where random walkers starting there first reach the boundary. In a film with current contacts, the potential difference between two edge points measures how differently a walker released from each of them is drawn towards the source rather than the sink, and the edge where no current flows is a wall off which walkers bounce. The conformal invariance that proves van der Pauw’s theorem is, in that language, the statement that the paths of a two-dimensional random walk look the same, statistically, after any angle-preserving distortion of the plane — a property of planar Brownian motion established by Paul Lévy and now central to the modern theory of random curves.

And the potential inside the film obeys the maximum principle that nothing can be held still by a static field derived for electrostatics: with no sources inside, the potential has no peak or pit in the interior, and its extremes lie at the current contacts. That is why every line of equal potential in the figure of the irregular film runs from edge to edge, and why none closes on itself in the middle.

Contacts that are not points #

The other assumption is that the contacts are points on the edge. Real contacts have a size, and if they are placed a little inside the edge rather than on it, current can flow round them. Van der Pauw estimated the errors: for a disc with contacts of length $\ell$ on its circumference, the sheet resistance comes out low by about $\ell^2/(16 D^2 \ln 2)$ of itself, where $D$ is the diameter — a contact a tenth of the diameter long gives an error under a tenth of a per cent. Contacts set back from the edge are worse, because current passes behind them. For this reason test structures on chips are often made in the shape of a cloverleaf or a Greek cross: the contacts sit at the ends of four arms, far from the central region the current actually samples, and their size stops mattering.

The grid calculations in the figures have a related limitation of their own. A contact on a grid is a single cell, and the current converging into it sees the grid’s spacing rather than a true point, which is why the recovered sheet resistances are 0.999 rather than exactly one. The error shrinks as the grid is refined, and it is the numerical counterpart of a finite contact.

Where the model stops #

The method assumes a film of uniform thickness and uniform resistivity. A film whose thickness varies across it returns an average weighted by where the current flows, which depends on the placement of the contacts, and rotating the contacts then changes the answer — which is, again, a useful diagnostic. An anisotropic film, one that conducts better in one direction than another, can be handled by stretching the coordinates to make it isotropic, and a related method of Montgomery’s measures both principal resistivities. And the method measures a sheet resistance, not a resistivity: the thickness has to be known separately to convert one into the other, and for films a few atoms thick the idea of a resistivity may not apply at all.

Still open: what the flake is doing inside #

Van der Pauw’s method has become the standard way to characterise materials that exist only as small, irregular flakes: graphene, the transition-metal dichalcogenides, twisted bilayers whose electrical properties depend on the angle between their layers. Those samples are often not uniform, with folds, bubbles, and domains of different twist angle, and the readings from four edge contacts average over whatever is inside. How much of the variation reported between laboratories for the same nominal material reflects the materials and how much reflects the averaging is often unclear, and scanning methods that map the local conductivity directly, rather than inferring it from the edge, are increasingly used to check. The method is exact for the film it assumes; whether the film under test is that film is a question it cannot answer.

The habit worth carrying away is to ask what a measurement is invariant under. The potentials of a flat conductor are unchanged in character by any map that preserves angles, and that map can carry any film without holes onto a half-plane, where the two readings are logarithms of cross-ratios that always add to one — so the shape of the film and the placement of the contacts both drop out, and a single hole puts them back. Four contacts on the edge of a flake measure what the flake is made of, and nothing about how it was cut.