Two walls that let more through than one

A rectangular barrier taller than the energy of the particle meeting it transmits an exponentially small amount, and the exponential is the whole of what makes tunnelling remarkable: double the thickness and the transmission is squared.

The obvious extension is to ask what two such barriers do. The obvious answer is that the second barrier attenuates whatever got past the first, so the transmissions multiply and the result is very small indeed.

That answer is wrong by six hundred and forty times at the energies where it matters most.

What actually happens #

At particular energies the pair transmits everything. Not a large fraction, not more than expected — exactly one, to every digit the arithmetic carries, while each barrier alone passes about four per cent of what arrives.

The energies at which this happens are sharp, and between them the pair behaves as intuition says it should, transmitting less than either barrier alone. So the effect is not that two barriers are somehow better than one; it is that a pair of barriers has structure in energy that a single barrier does not.

The calculation is a transfer matrix and contains nothing exotic. Match the wavefunction and its derivative at each of the four interfaces, multiply the resulting $2\times2$ matrices in order, and read the transmission off. The complex arithmetic is unavoidable: below the barrier top the wavenumber inside is imaginary, and it is the interference between the two decaying-and-growing solutions in each barrier that produces everything interesting.

Amplitudes, not probabilities #

The error in the obvious answer is the most common error in quantum mechanics and is worth isolating.

A probability of getting through the first barrier and a probability of getting through the second would multiply, if the two events were independent and if what arrived at the second barrier were a definite number of particles per second. Neither is the case. What arrives at the second barrier is an amplitude, and what leaves is the sum of the amplitudes for every path: straight through; through, back, forth, through; through, back, forth, back, forth, through; and so on for ever.

Those amplitudes have phases. The phase acquired on a round trip between the barriers is $2kL$ plus whatever the reflections contribute, and when it is a multiple of $2\pi$ all the terms add constructively. Summing the geometric series then gives a transmission that can be one — and must be one for identical barriers, because unitarity forbids more and the constructive sum reaches the maximum.

Adding amplitudes rather than probabilities is what makes an interferometer’s output port dark, and it is exactly the same operation that makes two barriers transparent. In both cases there are several routes to the same place, each carrying a complex number, and the total is the sum of those numbers rather than the sum of their squares. A dark port and a transparent pair of barriers are the same arithmetic pointed at two different questions.

The structure is identical to an optical etalon: two partially reflecting mirrors, transmitting almost nothing except at the wavelengths where the round trip is a whole number of waves, where they transmit everything. Nobody finds the Fabry–Pérot surprising, and the quantum version is the same calculation with an imaginary wavenumber inside the mirrors.

It also explains a hundred-year-old anomaly in electron scattering. Slow electrons pass through argon almost without deflection at around one electronvolt — the Ramsauer–Townsend effect — because the atom’s potential well is a region in which the wave acquires just the right extra phase for the reflections from its two sides to cancel. That is the same resonance condition, arrived at by a well rather than by two barriers, and it was inexplicable in 1921 — before there was a wave to attribute to the electron — for exactly the reason the double-barrier result is surprising now: a scatterer that scatters nothing looks like an absence of a scatterer.

A quarter-wave coating makes a reflection vanish by arranging for two reflected amplitudes to cancel, and nobody in optics thinks it strange. The layer is a quarter of a wavelength thick so the round trip through it is half a wavelength, the two reflections come back exactly out of step, and the surface goes dark. That is the same arithmetic again, in a subject where it has been routine engineering for eighty years.

The antireflection coating is the third member of the family and the one everybody has met. A layer of the right thickness makes a surface transmit everything, by cancelling two reflected amplitudes against each other rather than by removing the surface.

The series, summed #

The geometric series is worth writing out, because it shows where the unity comes from and shows that it is not an accident of any particular barrier.

Let each barrier have transmission amplitude $t$ and reflection amplitude $r$ from the inside, with $|t|^2 + |r|^2 = 1$. The total transmitted amplitude is the sum over the number of round trips:

$$t_{\text{tot}} = t^2 e^{i\phi} \left(1 + r^2 e^{2i\phi} + r^4 e^{4i\phi} + \cdots\right) = \frac{t^2 e^{i\phi}}{1 - r^2 e^{2i\phi}},$$

with $\phi = kL$ the phase across the gap. Take the modulus squared and the transmission is

$$T_{\text{tot}} = \frac{|t|^4}{\left|1 - r^2 e^{2i\phi}\right|^2}.$$

When the round-trip phase makes $r^2e^{2i\phi}$ real and positive, the denominator is $(1-|r|^2)^2 = |t|^4$ and the whole expression is exactly one. The numerator is small — it is the product of the two transmissions, the naive answer — and the denominator is equally small at resonance, and the two smallnesses cancel completely.

That cancellation is the mechanism. Making the barriers worse makes the numerator smaller and the denominator smaller in the same proportion, which is why the peak height does not change and only the range of $\phi$ over which the cancellation is good gets narrower. The width of the resonance in phase is of order $|t|^2$, and in energy it is that times the level spacing.

The state between the barriers #

The energies at which the resonances occur are not arbitrary, and the reason is worth drawing.

The state doing the work is recognisably a level of the box between the barriers. With impenetrable walls those levels are sharp and the particle never leaves; make the walls penetrable and each level acquires a width and shifts a little, becoming a resonance rather than a bound state. Nothing else changes — the wavefunctions are the same shapes, the spacing is nearly the same spacing — and the width is the only new quantity.

The region between the barriers is a well. If the barriers were infinitely high it would have bound states at definite energies, and a particle placed in one would stay for ever. The barriers here are finite, so those states are not bound: a particle placed in one leaks out on both sides after a while.

Such a state is a quasi-bound state, and it is precisely what a resonance is. Sending a particle in at the energy of a quasi-bound state is sending it in at the energy the well is prepared to hold, and the amplitude builds up inside — many times larger than the incoming amplitude — until the leakage out of the far side matches the arrival at the near one. In steady state that means everything that arrives leaves at the far side, which is transmission of one.

The resonances here sit at $6.44$ and $13.92$ where the infinite-well levels of the same gap are at $2.47$ and $9.87$. They are shifted upward because a state that penetrates its walls is effectively in a slightly wider box and — more importantly here — because the well is not infinitely deep, so the low-lying infinite-well levels have no counterpart at all.

The width is a rate #

Every peak reaches one, whatever the barriers are like. What changes when the barriers are made better is the width of the peak, and it changes fast: going from a barrier width of $0.6$ to $1.2$ narrows the resonance by a factor of fourteen.

That is not two facts but one. The state trapped between the barriers leaks out at a rate proportional to how well each barrier transmits, so a better barrier means a longer-lived state. And a longer-lived state has a narrower line, because a linewidth is an inverse lifetime — the relation is the same one that governs an unstable nucleus, a laser mode and a struck bell.

That width is the same Lorentzian a driven resonance has, and the correspondence is exact rather than suggestive: the half-width of the transmission peak is the reciprocal of the time the trapped state survives. A narrow peak means a long-lived state and a slow device; a wide one means a short-lived state and a fast device. There is no setting at which both are had.

So the double barrier is a filter with a trade built into it that cannot be engineered away. A sharper filter is a slower one: the transmission peak takes correspondingly longer to build up, because building it up means accumulating amplitude inside the well through a barrier that has been made harder to cross. Anybody who has met the same trade in an electrical filter or an optical cavity will recognise it, and the reason is the same in all three — the quality factor of a resonance is a lifetime measured in cycles, whatever the resonance is made of.

Populate the quasi-bound state between two barriers and leave it alone, and it decays exponentially, with a decay constant that is the linewidth of the transmission peak. Two measurements — a frequency scan and a stopwatch — return the same number, which is what makes the identification a statement about the system rather than a way of speaking about a graph.

The check that keeps the arithmetic honest #

A transmission calculation that comes out greater than one is a calculation with a bug in it, and a resonance peak that comes out at $0.9996$ may be a physical result or may be a sampling artefact. Both traps were live here and both are worth naming.

The first is guarded by testing the whole scanned curve against unity: no energy, for either the single or the double barrier, may transmit more than one. A lossless barrier cannot amplify, so an excess is a sign that a sign or a matching condition is wrong, and the check catches it wherever in the energy range it happens rather than only at the peak.

The second is subtler. A resonance a hundredth of the plotted range wide, scanned on a grid of two thousand points, is located to a few parts in a thousand — so the highest sampled point sits slightly off the true peak and reports a transmission below one. The first version of this figure said $0.9996$ and claimed it was one, which is the kind of small dishonesty that survives review because it looks like rounding. The peak is now refined by golden-section search on the function itself before being reported, and it comes out as $1.000000$.

The two failures are opposite in character and both would have produced a plausible picture. One is a physical impossibility that a reader might not notice; the other is a numerical shortfall that a reader would take as a physical statement about imperfect resonance. Neither is visible in the drawing.

The device this became #

The arrangement is not a thought experiment. Tsu and Esaki proposed it in 1973 as a semiconductor structure, and the first working resonant tunnelling diode was built the following year.

The barriers are thin layers of a wide-gap semiconductor — aluminium gallium arsenide — grown either side of a narrow-gap well of gallium arsenide a few nanometres across, by molecular beam epitaxy, which is the technique that made growing layers of a definite number of atoms possible in the first place. Electrons approach with an energy set by the applied bias.

Bring wells together and their levels split, spreading into a band whose width is set by the coupling. A double barrier is the two-well case of that construction, with the well between the barriers holding the state that does the work — and a superlattice, which is what these devices are grown as, is the many-well case, with a genuine miniband where the resonance was.

The current–voltage curve then has a feature nothing classical produces. As the bias rises, the incoming electron energy sweeps through the resonance: the current rises to a peak and then falls, because past the resonance the structure has gone back to being two barriers in series. A region of falling current with rising voltage is a negative differential resistance, which is exactly what an oscillator needs, and resonant tunnelling diodes are among the fastest electronic devices there are — they have oscillated above a terahertz, because the response time is the resonance’s lifetime and that is femtoseconds.

The amplitude inside, which is the part that can break things #

There is a quantity the transmission curve does not show and that anybody building such a device has to know: how large the wave gets in the well.

On resonance the amplitude between the barriers is larger than the incoming amplitude by roughly $1/|t|$, so the intensity there exceeds the incident intensity by $1/|t|^2$ — which for the barriers drawn here is a factor of about twenty-five, and for good barriers can be thousands. The energy is not appearing from anywhere: it is stored, and the steady state is reached when the leakage out matches the arrival in, exactly as in a resonant cavity being driven.

The optical version of this is routine and consequential. A Fabry–Pérot cavity with mirrors of $99.9$ per cent reflectivity has a circulating intensity a thousand times the incident one, which is how a modest laser damages a good mirror and why the highest-power optics in a laboratory are usually inside a cavity rather than outside it. The quantum version has the same structure with electron density in place of intensity, and it is why a resonant tunnelling structure driven hard develops space-charge effects that shift its own resonance — a nonlinearity with no nonlinear material anywhere in it.

Where it stops #

The phase has to survive the round trip. Everything here depends on the amplitudes for many traversals adding coherently, which requires the electron to keep a definite phase relationship between them. An inelastic collision inside the well — with a phonon, with another electron — destroys that, and the transmission collapses toward the incoherent answer, which is the product of probabilities that was wrong to begin with. That is why the device works at low temperature more cleanly than at room temperature, and why the well must be a few nanometres rather than a few hundred.

Nothing here is about a particle bouncing about between the barriers. The picture of a trapped object rattling to and fro and occasionally escaping gets the lifetime roughly right and the mechanism entirely wrong: it would predict the product of probabilities, since each attempt would be independent. What actually happens is that a single stationary state extends across the whole structure, with a large amplitude in the well and matching amplitudes outside, and the transmission is a property of that state rather than a history of attempts. The same difference distinguishes an interference pattern from a sequence of arrivals.

The barriers have to be identical for the peak to reach one. With unequal barriers the round-trip amplitudes still add in phase but the incoming and outgoing rates no longer match, and the peak transmission falls to $4T_1T_2/(T_1+T_2)^2$ — still enormously larger than $T_1T_2$, and no longer unity. The symmetric case is the special one, and it is another instance of the impedance-matching condition that appears wherever two couplings have to balance.

The incoming state has to be broad enough in time to build the resonance. A transmission of one is a steady-state statement: it describes what happens once the amplitude in the well has accumulated. A pulse shorter than the resonance’s lifetime never gets there, and its transmission is much closer to the naive product. So the same structure is transparent to a narrow-band beam and opaque to a short pulse of the same central energy, which is the time-domain face of the width–lifetime relation and is a genuine design constraint rather than a subtlety.

And a rectangular barrier is a caricature. Real barriers have shaped tops, image-force rounding and interface roughness, all of which shift and broaden the resonances. The qualitative structure survives every one of those; the quantitative positions do not, which is why a real device is characterised by measurement rather than by the formula.

What the resonance is worth as a measurement #

Because the peak position depends on the well’s width and depth, and the peak width on the barriers’, the transmission curve of such a structure is a complete report on its own geometry.

A shift of the resonance by a measurable amount corresponds to a change in the well’s width of a fraction of a monolayer, which makes the curve a way of characterising a growth process at atomic precision after the fact. A broadening beyond what the barrier thicknesses predict is a report of scattering inside the well, and its temperature dependence separates phonon scattering from interface roughness — the first grows with temperature and the second does not.

The same logic runs in nuclear and particle physics, where it was worked out first. A resonance in a scattering cross-section has a position that says where a quasi-bound state sits and a width that says how fast it decays, and reading both off a measured curve is how the properties of an unstable particle are determined without ever holding one. The double barrier is the simplest system in which that whole apparatus can be derived from scratch in an afternoon, which is a large part of why it is worth building.

The ladder from here #

Later rungs on this anchor: the Breit–Wigner form of a resonance, which is the general shape of the transmission near a quasi-bound state and appears identically in nuclear scattering; superlattices, where many barriers in series turn isolated resonances into bands and the structure becomes an artificial crystal; the tunnelling time problem, which asks how long the traversal takes and has resisted a clean answer for sixty years; and the Coulomb blockade, where the well between the barriers is small enough that putting a single electron in it costs a measurable energy and the resonance structure changes character.

The neighbouring ladders are the wall that is not quite a wall, which is the single barrier; the layer that makes a reflection vanish, which is the optical version of the same interference; and the width that is a lifetime, which is the relation the sharpening of these peaks obeys.