Illustrated Physics
physics, drawn
Electromagnetism

The island that takes electrons one at a time

A capacitor holds charge in proportion to its voltage, and the proportion is a property of its shape — on any piece of metal large enough to see. Make the metal small enough and cold enough, and joined to the world only by tunnel junctions, and the proportion breaks into steps: one electron's charge costs more energy than the heat or the battery can supply, so no current flows at all until it can, and then the electrons cross one by one. A charge of a single electron becomes something a voltmeter can see.
16 min read 5 figures Only some values fitMade into an instrument

Assumes: How much charge a shape will hold, before anything is charged · Two walls that let more through than one

How much charge a shape will hold found that capacitance is settled by geometry before any charge arrives. Two pieces of metal have a number attached to them, and the charge they hold is that number times the voltage between them. The relation is linear and continuous: double the voltage, double the charge, and any charge in between is available for a voltage in between.

That cannot be quite right, because charge comes in whole electrons. On a capacitor of ordinary size the discreteness is invisible: a microfarad at a volt holds six million million electrons, and one more or less is a part in 101310^{13}. The continuous relation is an average over a number so large that its steps are smaller than any noise. The question this page follows is what happens when the number is small — when the capacitance is so tiny that adding one electron changes the voltage by more than anything else in the circuit can.

The energy of one electron

Putting a charge QQ on a capacitor CC takes energy Q2/2CQ^2/2C. For one electron, that is the charging energy

EC=e22C,E_C = \frac{e^2}{2C},

and it is the price of admission for each electron that steps onto an isolated piece of metal. Whether the price matters depends on what else is available to pay it. A battery supplies eVeV per electron; heat supplies about kTkT. If ECE_C is large compared with both, the electron cannot get on, and if it cannot get on, no current flows.

How small an island must be for one electron to matter. The energy needed to put one extra electron on an isolated metal sphere, e²/2C with C = 4πε₀r, expressed as a temperature, against the sphere's radius from a nanometre to a tenth of a millimetre, both on logarithmic axes; the horizontal lines are room temperature, liquid helium and a dilution refrigerator. One electron's charging energy beats ten times the thermal energy for spheres smaller than 2.8 nm at room temperature, 199 nm at 4.2 K and 41.8 µm at 20 mK. A real island, close to its leads and gate, has a larger capacitance than an isolated sphere and must be smaller still.
Fig. 1 The charging energy e2/2Ce^2/2C of an isolated metal sphere, C=4πε0rC = 4\pi\varepsilon_0 r, expressed as a temperature, against the sphere’s radius, both on logarithmic axes; horizontal, room temperature, liquid helium and a dilution refrigerator. The charging energy exceeds ten times kTkT for spheres below 2.8 nm at room temperature, 199 nm at 4.2 K and 41.8 µm at 20 mK.

For an isolated sphere the capacitance is 4πε0r4\pi\varepsilon_0 r — about a hundred attofarads for a sphere a micrometre across — and the charging energy is inversely proportional to the radius. At room temperature a sphere must be a few nanometres across, the size of a large molecule, for one electron’s charge to dominate. At the temperatures of a liquid-helium cryostat a fifth of a micrometre suffices, and in a dilution refrigerator a speck tens of micrometres across, large enough to make by ordinary lithography. Real islands sit close to their leads and to a gate electrode, which adds capacitance and lowers the charging energy, so in practice they must be several times smaller than these limits; but the scale is clear. Single-electron effects need no exotic physics, only a capacitance comparable to e2/kTe^2/kT, which is about 5×10−185\times10^{-18} farads at a kelvin.

There is a second condition, and it is about how the island is connected. If the leads touched the island with an ordinary metal contact, electrons would flow on and off freely, and the number on the island would not be defined at all. The island has to be joined by tunnel junctions — thin insulating barriers that electrons cross one at a time, by the quantum tunnelling that two walls that let more through than one followed through a double barrier — and the junctions must be opaque enough that an electron, once on the island, stays there long compared with the time its charge takes to be felt. The condition is that each junction’s resistance be large compared with h/e2h/e^2, 25.8 kilohms, the same combination of constants that the resistance counted in whole numbers found as the quantum of Hall resistance. With both conditions met, the number of electrons on the island is a whole number, and it changes one electron at a time.

The effect was seen before anyone set out to look for it. Thin metal films evaporated onto glass often form not a continuous layer but a mosaic of separate grains a few nanometres across, joined by gaps through which electrons tunnel, and in the 1950s and 1960s such films were found to become more resistive, not less, as they were cooled — the opposite of a metal. C. J. Gorter in 1951, and Charles Neugebauer and Michael Webb in 1962, explained it as the energy needed to charge a grain: at low temperature too few electrons have the thermal energy to hop, and conduction freezes out. In 1969 John Lambe and Robert Jaklevic saw the charging steps directly, in the capacitance of a layer of tiny metal particles embedded in an oxide. What was missing was a single island under control — one grain, made on purpose, with a gate — and that needed the lithography of the 1980s.

A current that waits for a whole electron

Put such an island between two leads, with a voltage VV across them, and ask when current can flow. An electron stepping from a lead onto the island changes the free energy of the whole circuit — counting the work the battery does — by the charging energy e2/2CΣe^2/2C_\Sigma, where CΣC_\Sigma is the island’s total capacitance to everything, minus the energy the electron gains by moving through the potential difference between the lead and the island. With no bias and no other charge nearby, every such step costs energy, and at a low enough temperature none happens. The island is blockaded.

A current that will not start until a whole electron can pay. Current against bias voltage through a metal island joined to two leads by identical tunnel junctions, at a temperature of a fiftieth of the charging energy, for three charges induced by a gate; the voltage is in units of e/CΣ and the current of e/CΣ(R₁ + R₂), with CΣ the island's total capacitance. The dashed line is the same two junctions with no charging energy. With no gate charge nothing flows until the bias reaches e/CΣ: no single electron can step onto the island without raising its energy by more than the bias supplies. A gate charge of e/4 halves the gap; at e/2 the states with zero and one extra electron cost the same and the gap closes. Far above the gap every curve runs parallel to the ohmic line, offset by 1.00 e/CΣ — the charging energy still being paid, one electron at a time.
Fig. 2 Current against bias through a metal island between two identical tunnel junctions, at a temperature of a fiftieth of the charging energy, for three charges induced by a gate; bias in units of e/CΣe/C_\Sigma, current in e/CΣ(R1+R2)e/C_\Sigma(R_1+R_2). Dashed: the same junctions with no charging energy. With no gate charge nothing flows until the bias reaches e/CΣe/C_\Sigma; a gate charge of e/4e/4 halves the gap, and at e/2e/2 it closes. Far above the gap every curve runs parallel to the dashed line, offset by e/CΣe/C_\Sigma.

As the bias rises, the energy an electron gains by stepping off the island onto the lower-potential lead grows, and when the bias reaches e/CΣe/C_\Sigma — split equally between the two junctions, half of it across each — that gain just pays the charging energy. One electron leaves, the island’s potential jumps, and the jump makes it favourable for an electron from the other lead to step on. The island cycles between two charge states and passes current, one electron per cycle.

The bookkeeping that gives the threshold needs care, because two kinds of energy change at once. The island’s electrostatic energy rises when an electron arrives; the battery, which holds the leads at fixed potentials, does work to replace the charge the lead has lost and to adjust the charges induced on every electrode by the new charge on the island. Counting both, an electron that steps from a lead at potential VkV_k onto an island at potential φ\varphi changes the free energy by e2/2CΣ+e(Vk−φ)e^2/2C_\Sigma + e(V_k - \varphi), where the second term is the electron’s own fall through the potential difference and the first is the price of the step. The figures compute the probability of every charge state from these energies, with a tunnelling rate for each step that grows linearly with the energy it releases and is exponentially small for a step that costs energy — the same Boltzmann suppression the exponential that decides everything found governing every thermally activated process. Below that bias the current is zero, not small: a flat stretch across the middle of the curve where an ordinary pair of junctions would conduct ohmically.

Far above the gap the island conducts like its two junctions in series, but the current stays offset from the ohmic line by e/CΣe/C_\Sigma of bias. Every electron that passes is still paying the charging energy, and the offset is the fee. The first clear measurement of this curve was made by Theodore Fulton and Gerald Dolan at Bell Labs in 1987, on an aluminium island a few hundred nanometres across at a kelvin, and the theory it confirmed had been worked out shortly before by Konstantin Likharev and Dmitri Averin in Moscow.

A gate charge that moves the price

The striking part of the curve is the effect of the gate. A third electrode near the island, connected to it by a small capacitance CgC_g and carrying no current at all, induces a charge CgVgC_gV_g on the island’s surface. That induced charge is not made of electrons; it is a continuous displacement of the island’s electron sea relative to its ions, and it can be any fraction of ee. It shifts the energy of every charge state.

That an electrode can put half an electron’s worth of charge on a piece of metal sounds like a contradiction of the very discreteness this page is about, and it is worth seeing why it is not. The charge that has to be somewhere else found that a charge brought near a conductor induces a surface charge on it, by moving the conductor’s electrons a little relative to its ions. The displacement is a shift of a whole sea of electrons by a tiny fraction of an atomic spacing, and the charge it produces at the surface is continuous, set by the field. The number of electrons on the island, counted as particles, is still an integer; what the gate controls is the polarisation of the island, and the energy of each integer depends on it.

With a gate charge of e/2e/2, the island with no extra electrons and the island with one extra electron have exactly the same energy, and an electron can step on and off at no cost. The blockade disappears, and the island conducts at any bias. Raise the gate charge to ee and the island is blockaded again, now holding one electron; the pattern repeats with period ee in gate charge.

The diamonds where the island's charge is fixed. The stability diagram of a single-electron island with identical junctions: bias voltage (units of e/CΣ) against the charge the gate induces (units of e). Inside each shaded diamond, labelled with the number of extra electrons the island holds, every possible tunnelling step costs energy and no current flows; outside, at least one step pays for itself and electrons pass one at a time. The diamonds repeat every time the gate charge rises by one electron, and touch at gate charges of e/2, 3e/2, … where two charge states cost the same and the island conducts at any bias. Their height is e/CΣ in each direction. Probing the model at three points outside the diamonds gives currents of 0.60, 0.25, 0.53; inside them, nothing.
Fig. 3 The stability diagram of an island with identical junctions: bias against the charge the gate induces. Inside each shaded diamond, labelled with the number of extra electrons the island holds, every tunnelling step costs energy and no current flows. The diamonds repeat every electron of gate charge, touch at e/2e/2, 3e/23e/2, …, and reach ±e/CΣ\pm e/C_\Sigma in bias.

Mapped against both bias and gate charge, the regions where no current flows form a row of diamonds, each holding a fixed number of electrons. Inside a diamond, the island’s charge is a definite integer; along its edges, one step becomes free; at the points where neighbouring diamonds touch, two charge states are degenerate and the island conducts at the smallest bias. The diamonds are the most direct picture of the quantisation of charge on a conductor: a capacitance that, instead of relating a continuous charge to a continuous voltage, sorts the plane of voltages into regions labelled by integers. In semiconductor quantum dots, where the island also has discrete energy levels of its own, the diamonds become irregular in size, and the pattern of their sizes is a spectroscopy of the levels; for a metal island, whose levels are too closely spaced to matter, they are all alike.

A staircase when the junctions are unequal

With identical junctions, the island spends its time in the two charge states nearest the blockade, and the current, once it starts, rises smoothly. If one junction lets electrons in much faster than the other lets them out, the island fills instead to as many electrons as the bias can hold, and current flows at the rate the slow junction allows — until the bias is high enough to admit one more electron, when the current jumps.

A staircase of electrons. Current against bias for two islands of the same total capacitance at a temperature of a hundred-and-twenty-fifth of the charging energy: one with identical junctions (smooth) and one whose junctions differ fiftyfold in resistance and eightfold in capacitance (steps); units as before, the current in e/CΣ(R₁ + R₂). When one junction lets electrons in far faster than the other lets them out, the island fills to as many electrons as the bias can hold and waits at that number until the next one can be afforded. Each step in the current is one more electron on the island; the steps come every e/C₂ of bias — here 1.18 — the voltage that raises the slow junction's charge by one electron.
Fig. 4 Current against bias for two islands of the same total capacitance: one with identical junctions (grey) and one whose junctions differ fiftyfold in resistance and eightfold in capacitance (red), at a hundred-and-twenty-fifth of the charging energy. Each step in the red curve is one more electron on the island, every e/C2=1.18e/C_2 = 1.18 in units of e/CΣe/C_\Sigma.

The result is the Coulomb staircase: a current that rises in steps, each step marking one more electron held on the island. Seen with a scanning tunnelling microscope, whose tip forms one very thin junction above a metal nanoparticle lying on a thicker oxide that forms the other, the staircase can be measured at room temperature on particles a couple of nanometres across, where the charging energy is a tenth of an electronvolt. The steps of the staircase are the same quantisation as the diamonds, read along a single line of gate charge with unequal junctions to make them visible.

A thermometer and an electrometer

Warm the island, and the thermal energy begins to pay part of the charging energy. The blockade’s edges soften, then the gap fills in, and when kTkT is a sizeable fraction of ECE_C the island is an ordinary pair of resistors again.

Oscillations that a warm island washes out. The conductance of a single-electron island near zero bias, against the charge the gate induces, as a fraction of the conductance with no charging energy, at temperatures of 0.02, 0.08 and 0.3 times the charging energy e²/CΣ over k. Each peak is the gate charge at which two charge states cost the same and electrons can pass one at a time; between them the island is blocked. Warmed, the peaks broaden and the valleys fill: at kT = 0.02 the conductance swings from 0.50 to 0.000, at 0.3 only from 0.60 to 0.57. The island has turned back into an ordinary conductor whose capacitance is a smooth number.
Fig. 5 The conductance near zero bias against gate charge, as a fraction of its value with no charging energy, at temperatures of 0.02, 0.08 and 0.3 times e2/CΣe^2/C_\Sigma over kk. Cold, the island conducts only near gate charges of e/2e/2, swinging from 0.50 to nothing; at 0.3 the swing is from 0.60 to 0.57, and the capacitance is a smooth number again.

That washing-out is not only a limitation. The shape of the conductance near zero bias, in an array of junctions, depends on temperature in a way that can be calculated from first principles with no adjustable constant, and Coulomb-blockade thermometers built on it are used as primary thermometers in the millikelvin range — instruments whose reading needs no calibration against another thermometer, in the same spirit as half a kT in a piece of wire, where a resistor’s noise reads the temperature directly.

The cold oscillations, meanwhile, make the single-electron transistor the most sensitive electrometer known. Its conductance swings from maximum to zero as the gate charge changes by half an electron, so a charge of a thousandth of an electron near the island — on a defect, on a molecule, on a neighbouring quantum dot — moves the current measurably. The charge it senses is the induced, continuous kind, so the transistor reads fractions of an electron even though it passes electrons only whole; it is the instrument used to watch single electrons tunnelling on and off other devices, and to read out charge-based quantum bits. Operated at radio frequency, with the island’s conductance read through a resonant circuit rather than by a slow current, its sensitivity reaches about a hundred-thousandth of an electron’s charge in a second of averaging, and fast enough to follow individual tunnelling events as they happen. In that form it is the counter that sees the statistics of single-electron transport directly — the full distribution of how many electrons cross in a given time, rather than only its mean, which is the current.

A current standard made of counting

The inverse use is to make a current out of a count. If a gate voltage cycles the island through its charge states at a frequency ff, carefully enough that exactly one electron crosses per cycle, the current is exactly efef. Devices of this kind — turnstiles and pumps built from chains of junctions, and later from semiconductor islands whose entrance and exit barriers are opened and closed in turn — have been made to transfer electrons with errors of parts in ten million at gigahertz rates, about a hundred picoamperes. Since 2019 the ampere has been defined by fixing the value of ee, and a single-electron pump is a direct realisation of that definition: a current that is a frequency times a constant of nature, with no resistor or voltage in the chain. Light arrives in lumps found the discreteness of light in the statistics of detector clicks; here the discreteness of charge has become a unit of measurement.

Where the orthodox picture stops

The figures use the simplest consistent theory of the island, called the orthodox theory: tunnelling events are independent and instantaneous, each with a rate set by the energy it releases and the junction’s resistance, and the island’s charge is the only variable. It assumes junction resistances much larger than h/e2h/e^2; as they approach it, electrons can cross by two-step processes through virtual states of the island — cotunnelling — which leak a small current through the blockade, with a conductance that rises as the square of the bias, and which set the accuracy of the simplest pumps. It assumes the metal’s own energy levels are a continuum; in very small islands, and in semiconductor dots, discrete levels add structure inside each diamond. It assumes that the island’s surroundings do not fluctuate; in practice, charges hopping in defects of the substrate shift the gate charge at random by small fractions of ee, and that background charge noise, which the electrometer is so good at detecting, is the main limitation of devices built from many islands.

What the pictures cannot show is the moment of tunnelling itself. The orthodox theory treats an electron as being on one side of a barrier and then on the other, with a rate and nothing in between, and the question of how long the crossing takes — how long the crossing takes followed it for a single barrier — does not arise in it. Nor do the diamonds show that superconducting islands behave differently: with a gap in their spectrum, they prefer even numbers of electrons, carry Cooper pairs one pair at a time, and lead to the charge qubit. The domain of the drawings is a normal-metal island in the orthodox regime, with charging energies from a few to a hundred times the thermal energy.

Still open: how well an electron can be counted

The single-electron pump reached the accuracy needed for metrology only recently, and the outstanding question is whether it can go further. Closing the metrological triangle — checking that a current made by counting electrons, passed through a resistance set by the quantum Hall effect, gives a voltage equal to one set by the Josephson effect — tests the claim that ee and hh enter all three exactly, and every factor of ten in the pump’s accuracy tightens that test. How far errors from cotunnelling, from heating and from charge noise can be pushed down, and whether a pump can be both accurate to a part in 10910^9 and deliver a current large enough to compare, are active questions in national laboratories.

The underlying physics is the one this page began with. Adding one electron to an island costs e2/2Ce^2/2C, and when that beats kT and the island is joined only through junctions opaque compared with 25.8 kΩ, its charge is a whole number: no current flows until the bias reaches e/CΣ, the blocked regions form diamonds that repeat every electron of gate charge, unequal junctions turn the current into a staircase, and warming to kT≈ECkT \approx E_C turns the island back into a capacitor with a smooth number. The capacitance of a shape is an average, and on a small enough island the average breaks into the electrons it was made of.

Part 8 of 8

This essay is one argument about Conductors. The others:

Three shapes of film, one sheet resistance. Three thin films of the same material, of sheet resistance one ohm per square, with four small contacts on each edge labelled A to D in order. For each, the two readings van der Pauw's method takes — the voltage between D and C per unit current from A to B, and between A and D per unit current from B to C — computed by solving for the current flow on a fine grid. For a square, contacts at the corners: 0.221 and 0.221 Ω, giving 1.000 Ω per square; a disc, contacts unevenly spaced: 0.089 and 0.450 Ω, giving 0.999 Ω per square; an irregular lamina: 0.209 and 0.232 Ω, giving 0.999 Ω per square. The readings change with the shape and with where the contacts are; the sheet resistance recovered from them by exp(−πR₁/Rₛ) + exp(−πR₂/Rₛ) = 1 does not. The sheet resistance that forgets the shape Part 7 — To measure how well a thin film conducts, the obvious method is to cut it into a neat strip and pass a current along it. In 1958 Leo van der Pauw, an engineer at Philips, showed that the strip is unnecessary. Put four small contacts anywhere on the edge of a film of any shape, take two readings, and one equation returns the film's resistance per square — the shape and the spacing of the contacts drop out entirely. The proof is a single fact about logarithms and the maps that preserve angles, and it fails for exactly one kind of film: one with a hole in it.

Where this goes next

Two plates 0.20 plate-widths apart, with the field traced. The electric field between two oppositely charged plates separated by 0.20 of their own width, traced by following the field of 26 discrete charges on each plate rather than drawn as parallel lines. In the middle the lines are straight and evenly spaced; near the ends they bow outward. The field nine-tenths of the way to the edge is 80 per cent of the field at the centre. How much charge a shape will hold, before anything is charged Capacitance is decided by geometry alone. Two pieces of metal have a number attached to them, fixed by their shape and their separation, and it is settled before any charge arrives. Two barriers, and the energies at which they stop being barriers. Transmission against energy on a logarithmic scale, for one barrier of width 0.9 and height 20, and for two of them separated by a well of width 2. Below the top of the barrier the single one transmits an exponentially small amount and does so smoothly. The pair does not: at particular energies the transmission climbs by orders of magnitude and reaches 1.000000 — one, to every digit the arithmetic has — even though each barrier on its own passes 3.9e-2 of what arrives. Multiplying the two single-barrier transmissions would give 1.5e-3, which is wrong by 6e+2: amplitudes are what compose, not probabilities, and the region between the barriers is a box whose quasi-bound levels are where the amplitudes reinforce. The resonances sit at 6.438, 13.915 and the levels of an infinite well of the same width are at 2.467, 9.870 — near, and not equal, because a well with leaky walls has states that are not quite bound. Two walls that let more through than one Put a second barrier behind the first and the transmission does not fall — at particular energies it rises to exactly one, through a pair of walls each of which stops all but a few per cent. Probabilities cannot do that. Amplitudes can, and the region between the barriers is a box whose levels say where. A resistance counted in whole numbers. The Hall resistance of a gallium-arsenide layer with 3 × 10¹⁵ electrons per square metre, against the perpendicular field (solid), beside the classical B/ne (dotted); and the resistance along the current, on an arbitrary scale (red). Classically the Hall resistance rises in proportion to the field. In the layer it climbs in steps and stays on flat plateaus at h/νe² — 25813, 12906, 8604, 6453 ohms — each centred where the classical line crosses it, at integer filling. On each plateau the resistance along the current falls to zero: the current flows with no dissipation at all. Between plateaus both change. The model drawn here gives each Landau level a width 0.18 of the level spacing and lets only the middle 0.16 of it carry current across the sample. The resistance counted in whole numbers Pass a current along a thin layer of electrons in a strong magnetic field and measure the voltage across it. Classically the ratio rises steadily with the field. In fact it climbs in steps, and on each step it sits at the fundamental constant h/e² divided by a whole number — 25,812.807 ohms, then half that, then a third — the same to better than a part in a thousand million in every sample, every material and every laboratory, however dirty the layer and however irregular its shape. The steps exist because of the dirt, and they are exact because of a count that disorder cannot change.

The objects named here

The third axis, after the field and the reading path: the things themselves, and every essay that touches each one.

CapacitanceCharging energyCoulomb blockadeElectrometerMetrologyQuantisationThermal energyTunnelling