Until now the steepness of the exponential was a nuisance ── here it becomes a tool
An optical microscope cannot see anything finer than the wavelength of light (a few hundred nanometres). Electron microscopes broke through that by shortening the wavelength, but they still need a huge instrument and high voltages. Then in 1981, Binnig and Rohrer at IBM Zurich built a machine that sees atoms one at a time, on an entirely different principle. The mechanism is almost disappointingly simple ── bring a sharp needle right up to the surface and measure the current that slips through. That is all. Why does that let you see atoms? is this episode's subject. The answer is contained entirely in one line from Episode 1: tunneling rides on the exponent. So a tiny difference in distance becomes a huge difference in current. The steepness that was a nuisance turns into the finest tool there is.
The barrier height between tip and sample is set by the metal's work function (typically \(\phi\approx4\) eV). Insert into Episode 1's formula:
$$\kappa=\frac{\sqrt{2m\phi}}{\hbar}=5.12\sqrt{\phi[\mathrm{eV}]}\ \mathrm{nm^{-1}}\approx10.2\ \mathrm{nm^{-1}}$$ $$I\propto e^{-2\kappa d}\quad\Longrightarrow\quad \frac{I(d)}{I(d+0.1\,\mathrm{nm})}=e^{2\times10.2\times0.1}=e^{2.05}\approx \mathbf{7.8}$$Move 0.1 nm away and the current drops by about a factor of eight ── nearly an order of magnitude.
Atoms are 0.1–0.3 nm across. So the corrugation of the surface comes out directly as orders of magnitude in the current ── that is what atomic resolution really is.
This is the most interesting part. "Surely you need a tip one atom wide to see one atom?" ── You do not. Even a fairly blunt needle automatically gives one-atom resolution.
However crudely made, somewhere on the tip there is one atom that protrudes furthest. If the second atom sits 0.2 nm behind it, the fraction of current that atom carries is
$$e^{-2\kappa\times0.2\,\mathrm{nm}}=e^{-4.1}\approx 0.017$$So the frontmost atom carries 98% of the current. The second and beyond contribute almost nothing. The exponential, all by itself, narrows the effective tip down to a single atom.
This is what decisively distinguishes it from other microscopes. In the world of lenses, the precision of the instrument sets the limit on resolution. In the world of exponentials, the instrument can be sloppy and the physics does the narrowing.
The figure shows a tip scanning across a surface of atoms. The upper panel is a cross-section from the side (the tip's height and the surface's relief); the lower panel is the measurable current. Note that the vertical axis is logarithmic ── a relief of just 0.05 nm becomes peaks and valleys of many-fold in the current.
Watch the "share carried by the frontmost atom" as well. Widen the gap and it blurs; narrow it and it sharpens ── the resolution itself is decided on the exponent.
A real instrument does not measure the current; it adjusts the tip's height to hold the current constant (constant-current mode). Recording that up-and-down motion gives you the "map of atoms."
The reason is that the current is exponential and therefore spans too many orders to handle directly. Taking a logarithm and converting back to a height is what the feedback circuit does for you. The steepness of the exponential is both the source of the resolution and the source of the awkwardness.
| Quantity | typical value |
|---|---|
| tip-to-sample gap | 0.5–1 nm |
| bias voltage | 10 mV to a few V |
| tunneling current | 10 pA to 10 nA |
| in-plane resolution | about 0.1 nm |
| vertical resolution | about 0.01 nm (a thirtieth of an atom) |
Note how much better the vertical resolution is. That is because the exponential acts along the distance direction.
Then in 1993, Crommie, Lutz and Eigler arranged 48 iron atoms in a circle (a quantum corral) and photographed the standing-wave pattern of the surface electrons trapped inside. The textbook "particle in a box" wavefunction, taken as a photograph ── the picture from the sister series "Quantum That Clicks," made real.
An important caveat. An STM image is not a map of the surface's relief (topography). What is measured is the tunneling current, and that is proportional to the density of electronic states at that place and that energy (the Tersoff–Hamann approximation).
So this happens ── an atom that physically protrudes can appear dark. Conversely, a flat spot with concentrated electronic states appears bright. Change the bias voltage and even the image of the same place can change.
That is not a defect but also a feature: sweep the voltage while measuring (scanning tunneling spectroscopy) and you can read the energy levels of individual atoms. An STM is simultaneously a microscope and a spectrometer with single-atom resolution.
Established: the distance dependence \(I\propto e^{-2\kappa d}\) with \(\kappa=\sqrt{2m\phi}/\hbar\); that typical work functions (4–5 eV) give about an order of magnitude per 0.1 nm; that the frontmost atom carries most of the current; the invention of the STM by Binnig and Rohrer (1981, Nobel Prize in Physics 1986); constant-current operation; resolutions of about 0.1 nm in-plane and 0.01 nm vertically; atom manipulation by Eigler and Schweizer (1990) and the quantum corral of Crommie and colleagues (1993); and that what an STM measures is the local density of states (Tersoff–Hamann theory). All established physics and technology.
Points to note: (1) An STM image is not a topographic map. It is a contour of the local density of states and depends on the bias voltage; protrusions really can appear dark. (2) The simple \(I\propto e^{-2\kappa d}\) is an approximation treating the overlap of tip and sample wavefunctions to first order in perturbation theory (Bardeen / Tersoff–Hamann). Below about 0.4 nm the interaction can no longer be neglected and the picture breaks down. (3) The "frontmost atom carries 98%" estimate assumes a well-behaved tip shape. In practice, image doubling from a double tip is common. (4) The sample must be conducting (insulators are the territory of atomic force microscopy). (5) The figure is a one-dimensional cross-section, whereas real surfaces are two-dimensional; the surface shape is also idealised as a cosine. (6) Atom manipulation does not work for every atom on every surface; it is done under demanding conditions (low temperature, ultra-high vacuum) with particular combinations.
The principle of the scanning tunneling microscope is only Episode 1's \(I\propto e^{-2\kappa d}\). With a 4 eV work function \(\kappa\approx10.2\ \mathrm{nm^{-1}}\), so moving 0.1 nm away cuts the current by about a factor of eight. Atoms being 0.1–0.3 nm across, the surface's corrugation comes out directly as orders of magnitude in the current.
And the tip sharpens itself ── the single frontmost atom carries 98% of the current while an atom 0.2 nm behind contributes 1.7%. So a crudely made needle still gives atomic resolution. In the world of lenses the instrument's precision sets the limit; in the world of exponentials the physics narrows it for you.
And you can not only see but arrange (35 xenon atoms spelling "IBM" in 1990; the quantum corral photographing the standing waves of surface electrons in 1993). Though an STM image is not a topographic map but a map of the local density of states that changes with bias voltage ── which is not a defect but its function as a single-atom spectrometer.
Print / make a PDF: ⌘+P (Ctrl+P on Windows). On screen, move the gap and the work function to see how far the logarithmic current axis jumps. Even dropping the relief to 0.01 nm still shows clearly in the current. "See the answer" opens each solution.