Quantum Tunnelling

Press STM gap. The electron sits 4.5 eV below the top of the barrier, about the work function of a metal, and the gap is 0.5 nm. Widen it by 0.1 nm and T falls 8.789 times; the thick-barrier estimate, e^{2κ × 0.1 nm}, says 8.790. The ladder below the results shows the same step at every width from 0.4 nm out, to within 0.3%. Only the first steps are smaller, 7.293 from 0.1 to 0.2 nm, because a barrier that thin is not yet thick in units of 1/κ. Gerd Binnig, Heinrich Rohrer and their colleagues at IBM's Zürich research laboratory built the scanning tunnelling microscope on this steepness and reported it in 1982. Hold the current constant while the tip scans and the tip's height has to trace the surface; and because an atom 0.2 nm further back, two steps down the ladder, carries about 77 times less current, nearly all of it flows through the single atom at the apex. The 8.8 does not depend on E either. Change E and U together, keeping U − E at 4.5 eV, and it stays at 8.79, because κ = √(2m(U − E))/ħ contains only the difference.

Press Electron, 0.5 eV and then Proton, 0.5 eV: the same energy, the same 1 eV barrier, the same 0.5 nm. The electron gets through 10.14% of the time. The proton, 1,836 times heavier, gets through with a probability of 1.535×10⁻⁶⁷. Mass enters κ under a square root, so the proton's κa is only √1,836 ≈ 42.85 times the electron's, 77.62 against 1.811. But κa sits in an exponent, and e^−2κa turns a factor of 43 into 66 orders of magnitude. The muon, 206.8 electron masses, lands in between at 9.530×10⁻²³; the table under the ladder has all three. So tunnelling is not a property of small things in general. It belongs to light particles and thin walls: set the width to 0.05 nm and the proton's T climbs to 7.252×10⁻⁷.

Press 1.2 eV over 1 eV. The electron has 20% more energy than the barrier, so classically it would sail over every time. The page gives T = 53.64%. The reflection comes from the edges. At x = 0 the wavelength jumps, from 1.120 nm outside to a longer one inside, and any abrupt change in a wave's medium sends part of the wave back, as glass reflects a little of the light that enters it. The reflections from the front edge and the back edge interfere, and when the barrier holds a whole number of half-wavelengths, qa = nπ, they cancel exactly and T = 1. That happens at E − U = n²h²/(8ma²), which are the energy levels of a <a href="/phy/particle-in-a-box">particle in a box</a> as wide as the barrier. Press Box level: a 1 nm barrier, E − U = 0.376 eV, the ground level of an electron in a 1 nm box, and T = 100.00%. The energy plot marks these levels with dashed lines, and the curve touches 1 at each.