The resistance counted in whole numbers

On the night of 4 February 1980, at the high-magnetic-field laboratory in Grenoble, Klaus von Klitzing was measuring the Hall voltage of a silicon transistor cooled to a couple of kelvin in a field of fifteen tesla. The transistor’s channel was a sheet of electrons a few nanometres thick, and he was varying the voltage on its gate, which sets how many electrons are in the sheet. The Hall resistance — the voltage across the channel divided by the current along it — should have varied smoothly. It had flat steps, and the steps sat at values that did not depend on the sample’s size, shape or quality. One of them was 6,453.2 ohms. That is $h/4e^2$: Planck’s constant over four times the square of the electron’s charge.

The quantum Hall effect, as it became, is now the basis of the world’s standard of electrical resistance, and its plateaus have been compared between different materials and found to agree to parts in ten thousand million. That precision is the puzzle. A resistance in a real sample, with impurities, rough edges and contacts, measured to ten significant figures, comes out as a combination of fundamental constants with nothing else in it. Something in the measurement must be counting rather than measuring.

The classical Hall voltage, and what it should do #

A current flowing along a strip in a perpendicular magnetic field is pushed sideways by the magnetic force, charge piles up on one edge, and the pile-up makes a sideways electric field that balances the push. The sideways voltage divided by the current, the Hall resistance, comes out as $B/ne$, where $n$ is the number of carriers per unit area: it rises in proportion to the field, and its slope measures the density. The mass a curve decides used its sign to show that some metals conduct with holes rather than electrons. Nothing in the classical calculation suggests steps.

Levels that hold one electron per flux quantum #

A strong field makes the electrons’ motion in the plane quantised. The circling that comes in quanta found that an electron circling a magnetic field can only have energies in steps of $\hbar\omega_c$, the cyclotron energy, and in a two-dimensional layer that is all the motion it has: every electron in the layer must sit in one of the Landau levels at $\hbar\omega_c(n + \tfrac12)$. Each level holds a definite number of electrons per unit area, $eB/h$ — exactly one electron per flux quantum $h/e$ passing through the layer, the same unit of flux whose superconducting version the two in the flux quantum discussed.

The electrons that do this are not free electrons. They are the electrons at the bottom of a semiconductor’s conduction band — the band what happens when the wells get close built from the atoms’ own levels — which move through the crystal as if they were free particles with an effective mass set by the band’s curvature: 0.067 of an electron’s mass in gallium arsenide. That small mass is what makes the experiment possible. The cyclotron energy is inversely proportional to the mass, so the levels in gallium arsenide are fifteen times further apart than free electrons’ would be, and at ten tesla they are separated by far more than the thermal energy of liquid helium.

So the number of filled levels is the number of electrons per flux quantum, the filling factor

$$\nu = \frac{nh}{eB}.$$

As the field rises, each level holds more electrons, fewer levels are needed, and the Fermi energy — the energy of the highest occupied state — rides up the top occupied level until that level empties into the ones below, then drops.

At exactly integer filling, the Hall resistance $B/ne$ is $h/\nu e^2$: substitute $B = nh/\nu e$ and the density cancels. So the classical line passes through $h/e^2$, $h/2e^2$, $h/3e^2$ at integer filling. That explains where the quantised values occur, but only at isolated values of the field. It does not explain why the resistance stays at them over a wide range of fields — why there are plateaus rather than points.

Plateaus #

The measured curve is the figure’s solid line. Between plateaus the Hall resistance rises; on each plateau it is flat, exactly at $h/\nu e^2$, for a range of field a large fraction of the spacing between plateaus. And on every plateau the ordinary resistance — the voltage along the current — falls to zero. The layer conducts with no dissipation at all, though it is a dirty semiconductor at a finite temperature.

The answer to “why plateaus” is the dirt. In a perfectly clean layer every state in a Landau level has exactly the same energy, and the Fermi energy jumps from one level to the next as soon as one fills; there would be no plateaus, only the points. In a real layer the impurities and imperfections make the electrons’ potential energy vary from place to place, and each level is broadened into a band of energies. Most of the states in that band are confined: an electron in a field drifts along lines of constant potential, round the hills and dips of the disorder, as the drift that does not care what the charge is found for any charge in crossed fields, and a contour round a single hill or dip is a closed loop that goes nowhere. Only near the centre of each band do the contours wander from one side of the sample to the other. Those few states carry current; the rest are trapped.

When the Fermi energy lies in the trapped tails between two bands of extended states, changing the field or the density moves it through trapped states only. Electrons are added to or taken from loops that go nowhere, the current-carrying states below the Fermi energy stay exactly full, and the Hall resistance cannot change. The plateau lasts as long as the Fermi energy is among trapped states, and its width is set by how many of them there are — by how dirty the layer is. A cleaner layer has narrower plateaus. The disorder that would ruin any ordinary precise measurement is what makes this one possible.

Why the value is exact #

That explains the flatness but not the value. Why, on a plateau, is the Hall conductance exactly an integer times $e^2/h$, in a sample whose disorder has rearranged every state?

The cleanest answer is at the edges. In the interior of the sample an electron circles in place; at an edge, an electron whose circle meets the wall bounces off it and skips along, round after round, always in the same direction — one way along one edge, the other way along the opposite edge.

Quantum mechanically, each filled Landau level contributes exactly one such channel along each edge: a one-way lane at the sample’s boundary. A channel that carries electrons in one direction only has a fixed conductance, $e^2/h$, the conductance quantum, and $\nu$ filled levels give $\nu$ channels and a Hall conductance of $\nu e^2/h$. The disorder cannot change that number, because an electron in a one-way edge channel cannot be scattered backwards: the only states moving the other way are on the far edge of the sample, and on a plateau there are no extended states in the interior to carry it across. No backscattering means no dissipation, which is why the resistance along the current vanishes at the same time as the Hall resistance becomes exact.

There is a deeper version of the same answer. In 1981 Robert Laughlin showed that the Hall conductance follows from gauge invariance alone — the freedom the potentials that are not unique found in the vector potential, which a whole flux quantum leaves undetectable: threading one flux quantum through a loop of the material transfers exactly one electron per filled level from one edge to the other, whatever the disorder, and that transfer is the Hall conductance. A year later David Thouless and collaborators showed that the integer is a topological invariant of the filled states — a winding number, like the number of times a closed curve goes round a point, which cannot change under any smooth deformation of the material. Disorder is a smooth deformation. It can move the states about; it cannot change how many times they wind. The integer is a count, and counts are exact. The phase a magnet leaves on a path it never touched met the same kind of thing in the Aharonov–Bohm phase: a quantity fixed by the topology of the paths, not by the details along them.

How cold, and how clean #

The plateaus are exact only at low temperature, and the reason says how exact they are at any temperature. Dissipation needs an electron to be lifted from the filled extended states below the Fermi energy to the empty ones above, across the gap between Landau levels, or at least from trapped states into extended ones. The number of electrons thermally excited across a gap $\Delta$ falls as $e^{-\Delta/2kT}$, and the resistance along the current, and the departure of the Hall resistance from its quantised value, fall with it. In gallium arsenide at 10 tesla the cyclotron gap is 17 millielectronvolts, and at 1.5 kelvin the exponential is smaller than $10^{-25}$: the plateau’s error from thermal excitation is far below anything measurable, and the real limits come from the current used, which heats the electrons, and from imperfect contacts.

Cleanliness works the other way. A cleaner layer has electrons that move further between collisions, which is what experimentalists usually want — but a cleaner layer has fewer trapped states between the levels, so its plateaus are narrower, and in the limit of a perfect layer they shrink to points. The quantum Hall effect is one of the few phenomena in physics that needs a certain amount of disorder to exist, and the best metrology samples are not the cleanest ones made but the ones with plateaus wide and flat enough to measure on comfortably.

Graphene, and a plateau at room temperature #

In graphene, a single sheet of carbon atoms, the electrons near the Fermi energy behave as particles with no mass, their energy proportional to their momentum rather than to its square. Their Landau levels are not evenly spaced: the energies go as the square root of the level number and of the field, and there is a level at exactly zero energy shared equally between electrons and holes. The plateaus therefore come at a shifted sequence — $\nu = \pm2, \pm6, \pm10$, four times a half-integer, the four counting graphene’s two valleys and two spins — which was one of the first proofs, in 2005, that its electrons really are massless. And because the first gap, between the zero level and the next, is enormous at high field — over a hundred millielectronvolts at 45 tesla — a plateau in graphene survives to room temperature, as was shown in 2007, the only quantum effect of this precision seen at ordinary temperatures.

Graphene’s plateaus give the same $h/e^2$ as gallium arsenide’s, compared directly, to better than a part in ten thousand million. Two materials with different electrons, different band structures and different disorder agree, because neither is measuring anything about itself: each is counting edge channels, and a channel in graphene conducts exactly as a channel in gallium arsenide does.

A standard made of whole numbers #

The precision follows. The Hall resistance on the $\nu = 1$ plateau is $h/e^2 = 25{,}812.807$ ohms, the von Klitzing constant, and it has been measured equal in silicon transistors, gallium-arsenide heterostructures and graphene to better than a part in a thousand million, in samples of different sizes, shapes and qualities. In 1990 it became the practical standard of resistance; since 2019, when the International System of Units fixed the values of $h$ and $e$ exactly, it is not merely a standard but the definition. A resistance is calibrated by comparing it with a quantum Hall sample, as a voltage is calibrated against the Josephson effect that the voltage that is a frequency described, which turns a frequency into a voltage through $2e/h$. Between them the two effects tie the volt, the ohm and therefore the ampere to the fixed values of two constants.

The count also turns the effect into a measuring instrument for the layer itself. The plateaus come at integer filling, $\nu = nh/eB$, so their index increases by one each time the inverse field increases by $e/nh$: they are evenly spaced in $1/B$, and the spacing measures the density without any knowledge of the sample’s dimensions.

In a national metrology laboratory the quantum Hall sample sits in a cryostat at about 1.5 kelvin and 10 tesla, on its $\nu = 2$ plateau, and the resistor to be calibrated is compared with it by a cryogenic current comparator — a superconducting transformer that balances the currents through the two so that the ratio of their voltages is set by a ratio of turns. The comparison reaches a few parts in a thousand million in an afternoon. Before 1990 resistance standards were wire-wound resistors kept in oil baths, drifting by parts in ten million a year and compared between countries by carrying them across the world; a quantum Hall sample in any laboratory now gives the same answer as one in any other, with nothing carried.

That is the same periodicity in $1/B$ that appears in the oscillating magnetoresistance of three-dimensional metals, used for decades to map their Fermi surfaces. In two dimensions the oscillations become plateaus because the levels are fully separated and the states between them trapped.

What is left out #

The figures use a simple model. Each Landau level is counted once, with equal spacing; in real gallium arsenide each orbital level is split in two by the electrons’ spin, by an energy much smaller than the cyclotron spacing, so the odd-numbered plateaus are narrower and appear only at lower temperatures or higher fields than the model suggests. The levels are given Gaussian widths, and the extended states a fixed fraction of each, chosen to look like measured curves rather than derived from a model of the disorder; how the extended states really narrow towards the band centre, and the critical behaviour of the transitions between plateaus, are subjects of their own. The longitudinal resistance is drawn on an arbitrary scale. And the edge-state picture is classical, for illustration; the quantum edge channels and the localised interior states are what the argument rests on. The domain is the integer effect at temperatures low enough that thermal excitation across the level spacing is negligible.

Still open: the fractions #

In 1982 Daniel Tsui, Horst Störmer and Arthur Gossard, measuring cleaner gallium-arsenide layers at lower temperatures and higher fields, found a plateau at $\nu = 1/3$: three times $h/e^2$. Integer filling could not explain it. Laughlin showed that it comes from the electrons’ repulsion: at a third filling the electrons form a correlated liquid whose excitations carry a third of an electron’s charge, and many more fractional plateaus — 2/5, 3/7, 5/2 and others — followed. Some of the fractional states are predicted to have excitations whose exchange is neither bosonic nor fermionic, and whose braiding could store quantum information in a form immune to local disturbance; experiments since 2020 have found evidence of such anyonic exchange, but whether the most exotic of them, at $\nu = 5/2$, behaves as predicted, and whether it can be used, is unsettled.

The integer effect is settled. Electrons in a plane in a field fill Landau levels holding $eB/h$ each; disorder traps most of each level’s states and leaves only its centre extended, so while the Fermi energy sits among trapped states the Hall resistance stays flat at $h/\nu e^2$ — 25,812.807 ohms over $\nu$ — with no resistance along the current, because each filled level is one one-way channel along each edge that nothing can reverse. The resistance is exact because it is not a property of the material but a count of its edges, and a count is the one measurement no impurity can spoil.