The wall that is not quite a wall

A particle arrives at a step it has not the energy to climb. Classically it turns round, every time, with probability one. What the wave picture gives instead is a wave that does not stop at the wall.

The wave does not stop at the wall because it cannot. A wavefunction and its slope must both be continuous everywhere the potential is finite, and a function that is oscillating on one side and identically zero on the other cannot satisfy both conditions at the join. Something has to continue, and what continues is a decaying exponential.

Solving it rather than sketching it #

The calculation is short and worth setting out, because the shape of the answer explains everything that follows.

Outside the barrier the particle’s kinetic energy is positive and the wave oscillates with a wavelength set by its momentum, so with wavenumber k = √(2mE)/ħ. Inside, the kinetic energy would have to be negative, so the wavenumber is imaginary — and an imaginary wavenumber is a real exponential, with decay constant

$$\kappa = \frac{\sqrt{2m(V_0-E)}}{\hbar}.$$

Write the wave in the three regions with unknown coefficients, require the function and its slope to match at both boundaries, and four equations in four unknowns come out. Solving them gives the transmitted amplitude, and its square is the transmission probability:

$$T = \left[1 + \frac{V_0^2\sinh^2(\kappa a)}{4E(V_0-E)}\right]^{-1}.$$

The figure is drawn from the solved coefficients rather than from that formula, and the generator then computes the formula separately and refuses to draw anything if the two disagree by more than a part in a million. That is the site’s habit applied where it matters most: the matching problem is easy to get wrong in a way that still produces a plausible-looking wave, and comparing two independent routes is the only check that catches it.

The exponential, and why it dominates everything #

For a barrier more than a wavelength or so thick, sinh(κa) ≈ ½e^(κa), and the transmission becomes

$$T \approx \frac{16E(V_0-E)}{V_0^2}\,e^{-2\kappa a}.$$

The prefactor is a number between zero and four. The exponential is everything.

The straightness of that line is the practical content of the whole subject. A quantity that depends exponentially on a length is a quantity that can be used to measure the length with extraordinary precision, and it is also a quantity that is unmeasurably small the moment the length is macroscopic.

Put in ordinary numbers: a tennis ball approaching a wall has V₀ − E of order joules and a mass of 57 grams, giving κ around 10³⁴ per metre. For a wall one centimetre thick, 2κa is 10³², and e^(−10³²) is a number with no physical meaning. The effect is not merely unlikely for macroscopic objects; it does not happen.

What decides the decay constant #

κ = √(2m(V₀ − E))/ħ contains three levers and each has consequences.

The energy shortfall. Raising the particle’s energy toward the top of the barrier lowers κ and the transmission rises steeply. This is why the effect is a strong function of temperature in any setting where the particles have a thermal energy distribution — the ones in the tail do most of the crossing.

The mass. κ goes as √m — the same square root that puts a heavier particle’s wavelength below a lighter one’s — so a deuteron tunnels far less readily than a proton at the same energy. That is not an academic point: the difference in tunnelling rates between hydrogen and deuterium is measurable in enzyme kinetics, where a reaction slowed by a factor of seven or more on substituting deuterium is evidence that the hydrogen was tunnelling rather than climbing.

Planck’s constant. κ goes as 1/ħ, so the classical limit is ħ → 0 and the transmission goes to zero with it. Tunnelling has no classical counterpart at all; it is not a small correction to a classical process but a process with no classical version.

The same thing in optics, already #

The site has drawn this picture before, in a different subject, and the connection is the most useful one on the page.

Total internal reflection is a wave meeting a boundary beyond which it cannot propagate. It reflects completely — and the field beyond the boundary is not zero. It is an evanescent wave, decaying exponentially into the forbidden region with exactly the structure of the wave inside the barrier above.

Bring a second glass surface close, within a wavelength or so, and the evanescent field reaches it and re-establishes a propagating wave. Light crosses a gap it was totally reflected from. The phenomenon is called frustrated total internal reflection, it was described by Newton, and it is the identical mathematics: an oscillating solution, a decaying solution, and a second boundary close enough for the decay not to have finished.

Optics has had the same effect for two centuries. At a water–air boundary the critical angle is 48.8°, and past it no ray leaves — yet the field does not stop at the surface. It decays into the space beyond over a distance comparable to the wavelength, and a second medium brought inside that distance receives it, so light crosses a boundary it had been completely reflected from. Replace the light with an electron and the glass with a vacuum gap and this page’s figures result: the equation is the same equation, and only the interpretation of what is decaying has changed.

That correspondence is why tunnelling should not be filed under “quantum weirdness”. It is a wave phenomenon, it happens to water waves and radio waves and sound, and what is quantum about it is only that matter has a wave to do it with.

What the barrier does to the wave in front of it #

The left-hand side of every figure here is rippled, and the ripple is a measurement rather than an ornament.

An incident wave and a reflected wave of the same wavelength travelling in opposite directions add to a standing pattern whose contrast reports their relative amplitudes. If nothing were reflected the pattern would be flat; if everything were reflected the minima would go to zero. What the figures show is in between, and the depth of the ripple is a direct reading of R = 1 − T.

That is worth noticing because it means the transmission can be measured without going to the far side. An experiment that only has access to the incident region can still determine what fraction crossed, by measuring the standing-wave ratio — which is precisely how a transmission line’s load is characterised, in a subject with no quantum mechanics in it whatever.

There is a second feature of the incident region worth naming. The reflected wave is phase shifted, and the shift is not zero or π but something in between that depends on the barrier. The consequence is that the standing pattern’s nodes are not at the barrier face; they sit a short distance away, as though the wave were reflecting from a plane slightly inside the wall. That apparent displacement is the Goos–Hänchen shift in optics and the scattering length in nuclear physics, and it is one number that summarises everything about the barrier as seen from outside.

Where the energy is while it is inside #

A question the drawing invites and does not answer: what is the particle’s kinetic energy while it is inside the barrier, where its total energy is less than the potential?

The classical reading gives a negative kinetic energy, which is impossible, and the temptation is to conclude that the particle “borrows” energy briefly. That reading is common and wrong, and the reason it is wrong is the same one that applies to the energy–time relation: energy is conserved exactly, before, during and after.

The resolution is that a particle localised inside the barrier does not have the energy the incident beam had. Confining it to the barrier’s width forces a momentum spread, and that spread is large enough that a measurement finding the particle inside would also find it with enough energy to be there legitimately. The confinement required to ask the question supplies the energy the question presupposes was missing.

That is not a trick; it is the position–momentum trade doing its ordinary work, and it is a statement that “where is it and what is its energy” is not a question the state answers. What the state answers is “what fraction of a beam crosses”, and that number is definite, calculable and measured.

Made into a microscope #

The exponential sensitivity is a defect in almost every context and a gift in one.

A scanning tunnelling microscope holds a sharp metal tip a nanometre or so above a conducting surface and applies a small voltage. Current flows by tunnelling across the vacuum gap, whose barrier height is the material’s work function — a few electronvolts, giving κ ≈ 10¹⁰ per metre and about an order of magnitude of current change per ångström of gap.

Two consequences follow, and both are why the instrument works at all.

The current is dominated by the single atom at the very end of the tip, because any atom one ångström further back contributes ten times less. So the resolution is set by the exponential rather than by the sharpness of the tip, and a tip made by cutting a wire with scissors resolves individual atoms.

And a feedback loop that holds the current constant holds the gap constant to a few picometres, because a picometre of drift changes the current by a few per cent. Vertical resolution finer than a hundredth of an atomic diameter comes free with the exponential.

The same sensitivity in reverse is the tunnel junction’s use as a switch, a sensor and — in a superconducting version — the basis of the Josephson junction, whose voltage-to-frequency relation defines the volt.

Where it is doing the work in nature #

Three settings, in increasing order of how much depends on it.

Alpha decay. An alpha particle inside a nucleus is bound by the strong force and repelled by the Coulomb barrier outside it. Its energy is far below the barrier’s top — for uranium-238, 4.3 MeV against a barrier of about 30 MeV — so classically it can never leave. It leaves by tunnelling, and because the rate depends exponentially on the barrier, a factor of two in decay energy spans twenty-four orders of magnitude in half-life. That relation, the Geiger–Nuttall law, was an empirical curiosity for fifteen years and became Gamow’s 1928 explanation, the first successful application of quantum mechanics to the nucleus.

Fusion in stars. The Sun’s core is at 15 million kelvin, which gives protons a typical energy of about 1 keV against a Coulomb barrier of several hundred. Classically no fusion occurs; the Sun does not shine. What happens instead is that the small fraction of protons in the high-energy tail tunnel through, at a rate set by the same exponential, and the reaction rate is the product of a rising tunnelling probability and a falling thermal population — a narrow peak that decides the whole energy output.

Chemistry and biology. Proton transfer in enzymes — where the barrier is the top of an energy landscape rather than a square step, ammonia’s inversion, and the mobility of hydrogen in metals all proceed at rates far above what climbing the barrier would give. Tunnelling is not an exotic correction in these systems; below a certain temperature it is the only mechanism operating, which shows up as a reaction rate that stops depending on temperature at all.

When the barrier is a nucleus, the exponential in width becomes an exponential in time. Each alpha particle rattles against the Coulomb barrier some $10^{21}$ times a second with a fixed small probability of getting out, so the number remaining falls exponentially and the half-life is the tunnelling probability read as a rate. A factor of two in decay energy moves that probability by twenty-four orders of magnitude — which is why nuclear half-lives span from nanoseconds to longer than the age of the universe, and why the Geiger–Nuttall relation was an empirical curiosity for twenty years before it was an argument for quantum mechanics.

The memory that leaks on a schedule #

The scanning microscope uses the exponential to measure a distance. There is a device that uses it to store information, and its design is one number balanced against the same number evaluated at a different field.

A flash memory cell is a transistor with an extra gate that is completely surrounded by insulator — a conductor with no connection to anything, floating in a few nanometres of silicon dioxide. Putting electrons on it shifts the transistor’s threshold voltage, and reading the bit means asking which side of a threshold the transistor turns on.

Getting the electrons on and off is the interesting part. There is no wire; the only route is through the oxide, which presents a barrier over three electronvolts high and some seven to ten nanometres thick. At zero applied voltage that barrier is essentially impassable — which is the requirement, since the charge has to stay for ten years.

Apply ten or twenty volts across it and the barrier is no longer rectangular. The field tilts it into a triangle, and a triangular barrier is thinner at the particle’s energy than a square one of the same height. The effective width collapses, the exponential responds as exponentials do, and the same oxide that leaks nothing in a decade is crossed in microseconds.

The whole design is that one exponential evaluated twice. It must be astronomically small at zero field and usefully large at twenty volts, and the ratio between the two is what makes a non-volatile memory possible at all. Nothing else in the cell is remarkable; the physics is entirely in the sensitivity of a tunnelling rate to the shape of a barrier.

Two consequences follow and both are visible to anybody who owns a storage device.

It wears out. Every write drives electrons through the oxide, and a fraction of them are trapped in it. Trapped charge distorts the barrier and eventually makes it leaky, so a cell fails after somewhere between a thousand and a hundred thousand write cycles. That is why solid-state drives spread writes across the whole device rather than rewriting the same cells, and why their lifetime is specified in bytes written rather than in years.

And it stopped shrinking. Below about six or seven nanometres of oxide the zero-field tunnelling is no longer negligible: the stored charge leaks away in months rather than decades, and the memory stops being non-volatile. That is a hard floor set by an exponential, and it is why flash memory stopped getting smaller in the plane and started being stacked instead — modern devices are hundreds of layers deep, built upward because the barrier could not be made thinner.

The exponential that stopped a transistor shrinking #

The same barrier, in a neighbouring part of the same chip, produced the most consequential engineering limit of the last thirty years.

A transistor’s gate is separated from the channel it controls by a thin insulator, and the thinner that insulator the more strongly the gate controls the channel — more capacitance, more drive current, a faster switch. For four decades the whole programme of making computers faster consisted in large part of making that layer thinner, and it worked.

Below about two nanometres it stopped working, for the reason this page is about. Electrons tunnel straight through the gate oxide, and the leakage current rises by roughly a factor of ten for every two or three tenths of a nanometre removed. At about 1.2 nanometres — five atomic layers of silicon dioxide — the leakage through the gate became comparable with the current the transistor was supposed to be switching.

A chip in which every transistor leaks as much as it conducts does not merely waste power; it cannot be cooled. The exponential had turned a design variable into a wall, and the wall arrived within a couple of atomic layers of where the geometry was.

The escape was to stop treating the layer’s thickness as one quantity. What the transistor needs is capacitance, which depends on the thickness divided by the material’s permittivity; what the tunnelling cares about is the physical thickness alone. Replacing silicon dioxide with a hafnium-based oxide of five or six times the permittivity gives the same capacitance from a layer three times thicker — so the electrical thickness stays where it is wanted and the barrier the electrons face triples.

Three times the width, in an exponential that costs a factor of ten per two tenths of a nanometre, is a reduction in leakage of several orders of magnitude. The change was introduced in production in 2007, it required abandoning the silicon–silicon-dioxide interface that the entire industry had been built on, and it was done because there was no alternative that an exponential would permit.

What the picture cannot show #

The figure draws a standing wave on the left, a decay in the middle and a travelling wave on the right, and every part of that is a steady state. Nothing in it arrives or departs; it is the time-independent solution, which describes a continuous beam rather than one particle crossing.

That matters because it makes the obvious question unanswerable in this picture: how long does a particle take to get through? The steady-state solution has no answer, and the question turns out to be genuinely subtle — several different definitions of tunnelling time exist, they disagree, and experiments distinguishing them have only recently become possible. The figure quietly declines to say.

The drawing also shows Re ψ, and a wave to the right of the barrier is complex. The real part alone looks like a standing wave when it is a travelling one, which is why the magnitude is drawn as a dashed envelope alongside — flat on the right, because a travelling wave’s magnitude does not vary, and rippled on the left, because there the incident and reflected waves genuinely do interfere.

And nothing on the figure indicates that the barrier is an idealisation. A real barrier has a shape — the Coulomb funnel of a decaying nucleus rather than a rectangle — and the transmission through an arbitrary shape is not this formula but an integral of κ along the path — the WKB approximation, which reduces to this when the barrier is square and is what every application above actually uses.

Where the ladder goes next #

The rungs from here: the WKB approximation, and tunnelling through a barrier that is not rectangular; resonant tunnelling, where two barriers in series transmit perfectly at certain energies because the region between them is a box with levels in it; the tunnelling time problem and the attosecond experiments that address it; the Josephson effect, where what tunnels is a many-particle phase rather than a particle; and radioactive decay, which is the exponential on this page turned into an exponential in time.

The claim to carry forward is what the exponential does to the argument. Tunnelling is not rare because it is forbidden; it is common in every system whose barriers are thin on the scale of a wavelength, and unobservable in every system whose barriers are not. The transition between those two worlds is not gradual — it is exponential, and that is why one page can describe both a microscope that resolves single atoms and a wall no object will ever pass through.