The last atom does all the seeing

The transmission through a barrier falls exponentially with its width. That is usually stated as a reason nothing happens: an electron faces a vacuum gap of a nanometre and the probability of getting across is a part in ten thousand million.

Turn the statement over and it is a specification. A quantity that changes by a decade when a distance changes by one atomic radius is the most sensitive length measurement available in any laboratory, and it needs no lenses, no wavelength and no vacuum better than a good one.

The number the exponential gives #

The transmission through a rectangular barrier of height $\phi$ above the electron’s energy goes as $e^{-2\kappa d}$ with

$$\kappa = \frac{\sqrt{2m\phi}}{\hbar}.$$

For a work function of $4.5$ electronvolts — copper, silver, most clean metals — that is $1.09\times10^{10}$ per metre, so $2\kappa$ is about $2.2$ per ångström and the current falls by $0.94$ decades for every ångström of extra gap.

Ten picometres, a tenth of an atomic radius, changes the current by $24$ per cent. Two picometres is measurable. Nothing else in a laboratory converts a distance into a signal that steeply.

The three curves in the figure differ only in work function, and the range they cover is the range real surfaces have: $3.5$ electronvolts for an alkali-covered surface, $5.5$ for a clean noble metal. The slope varies by twenty per cent over that range, which is the sensitivity of the instrument’s calibration to what it is looking at — and one of the reasons the height it reports is not quite a height.

Why the tip does not have to be sharp #

The resolution of an ordinary microscope is set by the size of its aperture compared with the wavelength. There is no wavelength here, and the aperture is a piece of tungsten wire that has been cut with scissors.

What decides the resolution is that the atoms of the tip are at different distances from the surface, and the exponential converts a small difference in distance into a large difference in current. An atom standing $2.5$ ångström behind the apex — which is a normal lattice spacing — carries $1/230$ of the apex’s current.

For the three tips computed, the apex carries $99.52$, $96.29$ and $81.92$ per cent of the total. The last of those is a very blunt tip indeed, with its neighbours less than an ångström behind, and it still puts four fifths of the current through one atom.

So the instrument’s resolution is supplied by physics rather than by manufacture. In practice a tip is prepared by dipping it into the surface, applying a voltage pulse, or simply crashing it gently, until some protrusion happens to stick out furthest — and then it works. The procedure would be absurd for any other kind of microscope.

The current cannot be the measurement #

An exponential that steep is unusable as a reading. A factor of two in current is nine picometres of height, and the current also depends on the work function, on the bias, and on what the tip picked up an hour ago.

The instrument’s measurement is the output of a feedback loop, so its speed, its noise and its stability are properties of the controller rather than of the tunnelling. That is worth being clear about: the current is not what is recorded. What is recorded is the correction the loop applied to keep the current constant, which is a height — and the exponential sensitivity that makes the technique work is also what makes the loop hard to tune.

The instrument is built the other way round. A feedback loop adjusts the tip’s height to keep the current at a set value, and what is recorded is the voltage the piezoelectric scanner needed. The tunnelling is used as a null detector, and everything about the precision then belongs to the feedback loop and to the mechanical stability.

That has three consequences worth naming.

The scan speed is limited by the loop, not by the physics. The tunnelling responds in femtoseconds; the loop responds in microseconds to milliseconds, and going faster means the tip does not have time to follow the surface.

Vibration is the enemy and it is a hard one. Picometre stability means isolating from building vibration by seventy decibels or more, which is why these instruments sit on stacks of plates and springs or on eddy-current-damped suspensions.

And the recorded surface is a contour of constant current, not of constant height. Where the surface’s electronic properties change — a different element, an adsorbed molecule — the loop moves the tip to keep the current constant and records an apparent height change that is not a height change at all.

The instrument’s other unusual feature follows from the same steepness: it works in air, in water, in oil and in ultra-high vacuum, because the barrier is whatever is in the gap and the argument only needs it to be an insulator. The work function is different in each case — a liquid lowers the effective barrier substantially — so the decay length changes by a factor of two or so, and the resolution with it. Nothing else about the method changes, which is why the technique spread into electrochemistry and biology within a few years of being invented for surface physics.

The picometre, and how anybody holds one #

A number that changes by a quarter for ten picometres is only useful if ten picometres can be held, and it is worth setting out what that requires, because it is the whole of the engineering.

The mechanical problem is the reverse of the usual one: nothing must be allowed to move by an amount that would be invisible in almost any other instrument. A picometre is a hundredth of an atomic diameter, and thermal drift, building vibration and acoustic noise all exceed it by orders of magnitude — so the microscope is mostly an exercise in isolation, and the physics of the junction is the easy part.

The scanner. Movement is by piezoelectric ceramic, which extends by a few nanometres per volt, so a millivolt of control is a few picometres of motion. No mechanism, no bearing, no lever: the material deforms, which is the only kind of motion available at this scale that has no stiction in it.

The isolation. Building vibration is a few micrometres at a few hertz, which is a million times the tolerance. Isolation is achieved by stacking mass–spring stages, each attenuating above its own resonance, and the resonances have to be low — a fraction of a hertz — which means soft springs and a great deal of mass. Eddy-current damping is used because it adds damping without adding stiffness.

The thermal drift. A centimetre of steel expands by about a hundred nanometres per kelvin, so a millikelvin of drift moves the tip ten thousand times the tolerance. The answer is symmetry rather than temperature control: the tip and sample mounts are designed so that a uniform temperature change moves both by the same amount, and what remains is drift of a few picometres a minute rather than of nanometres.

None of this is exotic in isolation. What is unusual is that all three have to hold at once, and that the payoff for holding them is a signal so steep that nothing else needs to be precise at all.

What the image is actually of #

The last point is the one that turns the instrument from a camera into a spectrometer, and it is where the naive reading fails hardest.

The current tunnels between filled states on one side and empty ones on the other, so what is sampled is the states within a few tens of millivolts of the Fermi level. That is what the image is actually of: not the atoms, but the local density of electronic states at the energy the bias selects — which is why a molecule can image as a depression, and why changing the bias changes the picture without anything having moved.

The current comes from electrons tunnelling between filled states on one side of the gap and empty states on the other. The bias voltage decides which energies are involved: a positive sample bias lets electrons out of the tip into the sample’s empty states, a negative one lets them out of the sample’s filled states.

So the image is a map of the local density of electronic states near the Fermi level, at the energies the bias selects, convolved with whatever states the tip’s apex atom has — so it is a picture of where the electron probably is rather than of where the nucleus is.

Two consequences make this concrete rather than a caveat.

On graphite, half the atoms are missing. The two carbon sites in the surface unit cell are inequivalent — one has a neighbour directly below it in the next layer and the other does not — and their densities of states at the Fermi level differ enough that only one of them appears. Images of graphite show a triangular lattice where the atoms form a honeycomb.

On some semiconductor surfaces the image inverts with the bias. Filled and empty states sit in different places on a reconstructed silicon surface, so reversing the bias reverses which features are bright. Neither image is the atoms; both are true statements about the states.

That is not a limitation to be corrected. Taking the current-voltage curve at every point turns the instrument into a spectrometer that measures the local density of states as a function of energy, with atomic spatial resolution — which is what it is mostly used for now.

Moving atoms, which the same exponential makes possible #

An instrument that is exquisitely sensitive to the last atom is also exerting a force on it, and the two cannot be separated.

An atom on a surface sits in a landscape with two minima and a barrier between them, and bringing a tip close tilts it. That is how an atom is picked up, carried and put down — the same exponential that makes the microscope sensitive makes the tip’s influence local enough to move one atom without disturbing its neighbours, which is what made the IBM logo possible in 1989.

At the gaps used for imaging the interaction is weak. Bring the tip half an ångström closer and it is not: the tip–atom attraction becomes comparable with the atom’s binding to the surface, and the surface’s potential landscape is tilted enough that the atom will follow the tip.

That is how single atoms are positioned. Lower the tip until the adatom binds preferentially to it, move laterally, raise the tip and the atom is left behind. The whole procedure is a matter of choosing gaps a fraction of an ångström apart, and it is available because the same exponential that makes the current sensitive makes the force sensitive.

The demonstration everybody has seen — thirty-five xenon atoms arranged to spell a company’s name, in 1989 — is that operation repeated thirty-five times. What made it more than a stunt is what came next: arranging atoms into a closed ring on a copper surface produces a quantum corral, in which the surface electrons are confined and their standing-wave pattern is directly visible in the image. The interference pattern of the electrons inside is measured by the same instrument that built the enclosure.

That is a peculiar kind of experiment. The apparatus, the sample and the measurement are the same tip, and what is seen is a wavefunction’s amplitude on a surface, mapped point by point.

The barrier that is not a wall #

There is one more reading of the same exponential, and it connects this to a piece of classical optics.

The wavefunction inside the gap is not oscillating. It is a real exponential, decaying with length $1/\kappa$, which for a metal is about $0.5$ ångström. That is an evanescent wave and it is the same object that exists beyond a totally internally reflecting surface in optics.

The optical version has its own instrument — the near-field scanning optical microscope, where a probe is brought within a fraction of a wavelength of a surface to sample the evanescent field, and the resolution is set by the probe’s distance rather than by the wavelength. Both microscopes exploit the same fact: a field that decays exponentially carries information about a distance in its amplitude, and reading an amplitude is easier than resolving a wavelength.

The difference is one of scale. The optical decay length is a fraction of a wavelength — hundreds of nanometres — so the near-field probe must be positioned to nanometres. The electron decay length is half an ångström, so the tunnelling probe must be positioned to picometres, and gets a thousandfold better resolution for it.

It is worth noting how recent all of this is. The exponential has been understood since 1928; the instrument dates from 1981, and the reason for the gap is not the physics but the mechanics. Nothing about tunnelling had to be discovered to build a scanning tunnelling microscope. What had to be arranged was a way of holding two pieces of metal a nanometre apart, stably, while moving one of them in picometre steps — and that is a problem about vibration isolation and piezoelectric ceramics rather than about quantum mechanics.

The image that settled a twenty-four-year argument #

The instrument’s reception was cool, and the thing that changed it was one picture.

A clean silicon surface does not keep the arrangement a bulk crystal would give it. The atoms rearrange, and on the (111) face the rearrangement has a unit cell seven times the bulk spacing in each direction — a result obtained by electron diffraction in 1959 and then unexplained for a generation. Diffraction gives the periodicity directly and the positions only through a model: a structure is proposed, its diffraction pattern computed, and the fit assessed. Dozens of models for the seven-by-seven cell were proposed over two decades and none was decisive, because several fitted about as well.

In 1983 Binnig and Rohrer’s group put a tip over that surface and simply looked. The image showed twelve bright maxima arranged within a rhombic cell of the right size, with a deep hole at each corner — a real-space map, requiring no model to read, of where the states were. Every proposed structure that did not have twelve protrusions and a corner hole was eliminated at a stroke, and the model that was eventually built to fit it has stood since.

That is what the technique added, and it is worth stating as a general point rather than as a piece of history. Diffraction measures a transform and needs a model to invert it; a scanning probe measures the thing itself, one point at a time. The first is enormously more efficient where a model is available and the second is decisive where one is not. The Nobel Prize followed three years later.

A vibrational spectrum of one molecule #

The steepness that gives the instrument its resolution has one more use, and it turns the microscope into a chemical identifier.

Put a single molecule in the junction and raise the bias slowly. Below a certain voltage the electrons can only tunnel elastically, arriving with the energy they left with. Once $eV$ exceeds the energy of one of the molecule’s vibrational quanta, a new route opens: an electron can cross and leave a quantum of vibration behind. That extra channel adds a per cent or so to the conductance, and it switches on at one voltage — so the second derivative of current with respect to voltage has a peak at the vibrational energy.

Doing that at every point in a scan gives a vibrational spectrum of an individual molecule, positioned where the image says it is. It was first done in 1998 on a single acetylene molecule on copper, whose carbon–hydrogen stretch appeared at 358 millielectronvolts.

The confirmation is the part worth admiring. Replace the hydrogens with deuterium and the same molecule’s stretch should drop by roughly the square root of the mass ratio, since the frequency of a spring depends on what is hanging from it. The measurement returned 266 millielectronvolts, which is what that arithmetic predicts — from one molecule, identified by substituting its atoms.

The requirement is temperature. A vibrational step is smeared by the thermal width of the electron distributions on both sides, which is a few times $kT$: about a hundred millielectronvolts at room temperature and two at four kelvin. So the spectroscopy lives in a liquid-helium cryostat while the imaging does not, which is why the two capabilities arrived seventeen years apart.

Where the model runs out #

The barrier is not rectangular. The image charge of the electron lowers the barrier near both surfaces, rounding the top and reducing the effective work function by an amount that grows as the gap shrinks. At the gaps used the correction is tens of per cent, and it is why the measured apparent barrier height is usually below the true work function.

The tip’s own states are treated as featureless. The usual theory assumes the apex has a spherically symmetric state, which is convenient and untrue: a tungsten tip’s dangling orbital is directional, and a tip that has picked up a molecule images with that molecule’s orbital instead. That is now done deliberately — a carbon monoxide molecule on the apex sharpens the image dramatically — and it means the tip is part of the measurement rather than an instrument pointed at it.

Nothing here is at a temperature. The states either side are sharp only at low temperature; at room temperature the Fermi edge is smeared over about $100$ millielectronvolts, which limits spectroscopy far more than it limits imaging.

And the whole account is one-dimensional. The current is computed as though the electron faced a slab of vacuum, when it actually faces a three-dimensional geometry in which the lateral spreading of the wavefunction matters. Bardeen’s tunnelling formalism and Tersoff and Hamann’s treatment of it are the honest versions, and they change the prefactor and the interpretation rather than the exponential — which is why an essay about the exponential can get away with the slab.

The ladder from here #

Later rungs on this anchor: resonant tunnelling through a double barrier, where two barriers in series transmit better than one at particular energies; the Josephson junction, where the tunnelling particle is a pair and the current depends on a phase rather than on a bias; tunnelling spectroscopy as a measurement of the superconducting gap, which is where the technique began; and macroscopic quantum tunnelling, where the coordinate that gets through the barrier is a current in a circuit rather than a particle.

The neighbouring ladders are the wall that is not quite a wall, which is the barrier problem itself, a wall that a factor of two makes impassable, where the same exponential produces twenty-four decades of nuclear lifetime, and the reflection that happens where the glass is not, which is the optical evanescent field this one is the electronic version of.