Illustrated Physics
physics, drawn
Series

Tunnelling — the series

8 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. A wavefunction crossing a barrier it has not the energy for. An electron of 2 electronvolts meeting a 3 electronvolt barrier 0.3 nanometres wide, with the four matching conditions solved rather than sketched. Left of the barrier the incident and reflected waves add to a standing pattern; inside it the amplitude decays exponentially; to the right a travelling wave continues with amplitude 0.3912 of the incident one, so 15.3 per cent of the electrons get through. Classically none of them do.

    The wall that is not quite a wall

    A particle without enough energy to climb a barrier sometimes appears on the other side of it. The probability falls exponentially with the barrier's width, which is why the effect is invisible at ordinary scales and why it can be turned into a microscope.

    part 1 · quantum
  2. Twenty-four decades of lifetime from a factor of two in energy. The half-lives of 7 alpha emitters against the reciprocal square root of the alpha's energy — the Geiger–Nuttall coordinates — with the measured values as points and a one-line tunnelling model as the open ones. The energies span a factor of 2.2 and the half-lives span 24 decades, which is what an exponent does. The model has no fitted parameter in it and reproduces every lifetime to within 0.5 decades — bad arithmetic by any ordinary standard, and a hundred-thousandth of the range it is predicting.

    A wall that a factor of two makes impassable

    Polonium-212 lives three tenths of a microsecond. Thorium-232 lives fourteen billion years. The alpha particles they emit differ in energy by a factor of two, and the lifetimes differ by twenty-four decades — because the quantity that decides is not the energy but an exponent built from it, and an exponent is where small differences go to become enormous.

    part 2 · quantum
  3. A current that falls by a decade for every ångström. Tunnelling current against the width of a vacuum gap, on a logarithmic scale, for three work functions covering the range of clean metal surfaces. The curves are straight because the transmission is an exponential in the gap, and their slopes are 0.833, 0.944, 1.044 decades per ångström — fitted to the drawn curves and agreeing with 2κ/ln 10 to a part in a million. At 4.5 electronvolts a change of ten picometres, a tenth of an atomic radius, changes the current by 24 per cent. That is the sensitivity a scanning tunnelling microscope lives on, and it is why the instrument measures height by holding the current fixed and recording what the piezo had to do: the current is far too steep a function of height to be read as one.

    The last atom does all the seeing

    A tunnelling rate falls by a factor of eight for every tenth of a nanometre of extra barrier, which is normally quoted as the reason nothing ever tunnels anywhere. Read the other way it is a microscope: the second-nearest atom of a blunt metal tip carries a two-hundredth of the current the nearest one does, so a tip nobody sharpened resolves a single atom, and the resolution comes from an exponential rather than from any piece of engineering.

    part 3 · quantum
  4. 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.

    part 4 · quantum
  5. The delay that stops caring how thick the wall is. The Wigner phase time — how late the transmitted packet's peak arrives, compared with a free particle covering the same distance — for an electron under a 3 eV barrier, at 0.2 nm, 0.5 nm, 1 nm thick. 0.2 nm: 0.3728 fs at half the barrier height; 0.5 nm: 0.4372 fs at half the barrier height; 1 nm: 0.4388 fs at half the barrier height. The widths span a factor of 5.0 and the times span 1.18. A thicker barrier is exponentially harder to cross and the delay in crossing it barely moves — which is either a remarkable fact about tunnelling or a sign that the phase time is not a traversal time, and the rest of the essay is about which.

    How long the crossing takes

    Tunnelling has a probability, and asking how long it takes turns out to be a different kind of question. The delay a transmitted packet shows stops growing once the barrier is opaque, so a thicker wall is crossed in the same time — and several defensible clocks give several different answers.

    part 5 · quantum
  6. A temperature the circuit refuses to go below. The escape temperature — the temperature that would give the measured escape rate if escape were purely thermal — against the actual temperature of the junction, for a 10 µA, 5 pF junction at biases of 0.985 and 0.99 of its critical current. Hot, the two are equal: the phase is kicked over the barrier by thermal noise. Cooled below the crossover, 39 mK at 0.985 and 36 mK at 0.99, the escape temperature stops falling and levels off at 42 mK and 44 mK: the rate no longer depends on the temperature at all. This levelling-off, measured by Michel Devoret, John Martinis and John Clarke in 1985 with the junction's electrical environment carefully characterised, was the evidence that a macroscopic electrical variable tunnels.

    The circuit that goes through a wall

    Tunnelling is usually told about electrons and alpha particles — single, small, quantum things passing through barriers they cannot climb. In 1985 it was seen in an electrical circuit: the phase difference across a Josephson junction, a coordinate shared by all the electrons in two pieces of superconducting metal, trapped in a well and escaping from it through the barrier rather than over it. The evidence was a temperature below which the circuit's escape rate stopped caring how cold it was. The same circuit showed discrete energy levels, and its descendants are the qubits of superconducting quantum computers.

    part 6 · quantum
  7. A current that grows thirty decades with the field. The current density of electrons tunnelling out of a cold metal against the field at its surface, on a logarithmic scale, from the Fowler–Nordheim law, for work functions of 2.7 eV (lanthanum hexaboride), 4.5 eV (tungsten) and 5.6 eV (a surface barely willing to let electrons go). For tungsten the current is 0.009 A/m² at 2 GV/m, 5·10⁵ at 4 GV/m and 6·10⁹ at 8 GV/m — rising by more than twenty orders of magnitude between 1 and 4 GV/m. The work function sits in the exponent as a three-halves power, so lowering it from 4.5 to 2.7 eV raises the current at 3 GV/m by a factor of 2·10⁵. A current density of a million amperes per square metre, an ordinary working value for a field emitter, needs about 4.2 GV/m on tungsten.

    The electrons a field pulls from cold metal

    A metal holds its electrons behind a step a few electronvolts high, and to get them out the usual way is to heat the metal white-hot until some climb over it. Put a strong enough field outside instead and the step tilts into a triangle a couple of nanometres wide, and electrons leave a cold metal by tunnelling through it. The current grows thirty decades as the field rises tenfold, a sharp needle at a thousand volts reaches the field that a flat plate would need a million volts for, and the electrons that come out all have nearly the same energy — which is why the brightest, sharpest electron microscopes are fed from a single etched tungsten point.

    part 7 · quantum
  8. A hydrogen atom tilted by a laser field. The potential energy of hydrogen's electron along the direction of a laser's electric field at the field's peak, in atomic units (one unit of energy is 27.2 eV, one of length 0.053 nm), for peak intensities of 3 × 10¹³ and 10¹⁴ W/cm² and at the barrier-suppression intensity, 1.37 × 10¹⁴; the dashed line is the bound level, 13.6 eV below the ionisation limit. The field tips the Coulomb well into a barrier on one side. At 3 × 10¹³ W/cm² the barrier is 15 atomic units thick at the level — 0.8 nm — at 10¹⁴, 6; at the suppression intensity its top has fallen to the level and the electron can leave without tunnelling at all. The barrier is real only for the fraction of the laser's period that the field stays near its peak.

    The atom a light wave opens like a wall

    Red light cannot free an electron from hydrogen: each photon carries 1.55 electronvolts and the electron is bound by 13.6. Make the light intense enough, though, and the atom ionises anyway — by absorbing nine photons at once, or, at higher intensity, by having its potential tipped over by the light's electric field until the electron tunnels out through the side. Which of the two happens is decided by a single ratio, found by Leonid Keldysh in 1964: how long the electron needs to get through the barrier, against how long the field holds still. The tunnelling side of that ratio is where attosecond physics begins.

    part 8 · quantum

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