Illustrated Physics
physics, drawn
Series

Bands — the series

8 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. One well, two wells, and the band they become. The energy levels of a chain of identical wells, for 1, 2, 3, 6, 12, 40 of them, with an on-site energy of -4 eV and a coupling of -0.9 eV between neighbours. One well has one level. Two split it into two, 1.80 eV apart. By 40 the levels have filled a band 3.59 eV wide, which is closing on the limit of four times the coupling, 3.60 eV — and no further widening happens however many more wells are added. The count of levels grows with the number of wells; the width of the band does not.

    What happens when the wells get close

    Two atoms brought together split one level into two. A thousand split it into a thousand, packed into a band whose width stops growing after the third. Whether that band is full or half full is the whole difference between a wire and a window.

    part 1 · quantum
  2. A band 3.60 eV wide, and the curvature is the whole story. Energy against wavenumber for one tight-binding band, α + 2β·cos(ka), with α = -4 eV, β = -0.9 eV and a repeat of 300 pm. The band is 3.60 eV from floor to ceiling and periodic in k, so the zone boundary at ka = π is not an edge of anything: it is where the curve turns over. Near the bottom it is a parabola, which is the free-electron dispersion with a different coefficient — a mass of 0.470 m_e rather than one — and near the top it is a parabola the other way up, a mass of -0.470. The inflection between them is at ka = π/2, and it is the point at which a constant force stops producing any acceleration.

    The mass a curve decides

    An electron in a solid answers a force with a mass that is nothing to do with the mass of an electron. It is set by how sharply the band bends, it is smaller than the free value in a wide band and larger in a narrow one, and near the top of any band it is negative — which is why aluminium's Hall voltage has the sign of a positive carrier and no adjustment to an electron count can repair it.

    part 2 · quantum
  3. Two levels that refuse to cross. On the left, the energies of a two-state system as one state is swept past the other, for couplings of 0, 0.05, 0.15, 0.35 electronvolts. With no coupling at all the two levels cross, which is the dotted pair. With any coupling whatever they do not: the eigenvalues are plus and minus half the square root of the squared detuning plus four times the squared coupling, so the closest approach is exactly twice the coupling — measured off the drawn curves as 0.000, 0.100, 0.300, 0.700 eV against 2V, agreeing to 0.0e+0. Far from the crossing the curves rejoin the uncoupled lines, so the repulsion is local: it is largest where the two states are degenerate and dies away as the square of the coupling divided by the detuning. On the right is what makes this more than a picture of two hyperbolas — the character of the upper state, meaning how much of the first basis state is in it. It swaps completely across a region whose width is set by the coupling, so the level that arrives as one thing leaves as the other. That exchange of identity is why the phrase avoided crossing is misleading: nothing is avoided, the labels are.

    The crossing that never happens

    Two energy levels swept past one another do not cross. Any coupling between them, however small, opens a gap of exactly twice the coupling — and the two levels exchange their identities across it, so the state that arrives as one thing leaves as the other while the labels are what avoided anything.

    part 3 · quantum
  4. The bands bend by the amount the Fermi levels differed. A junction between 1e+17 cm⁻³ p-type and 1e+16 cm⁻³ n-type silicon at equilibrium, with the Fermi level flat by construction — that is what equilibrium means — and the two band edges carrying the whole of the 0.774 V drop. The bending happens over 331.8 nm, and it is not symmetric: the depletion reaches 301.6 nm into the lightly doped side and only 30.2 nm into the heavily doped one, because the same exposed charge is reached sooner where there is more of it. An electron in the n-side conduction band therefore faces an uphill barrier of 0.774 V to reach the p side, while an electron already on the p side rolls downhill without any barrier at all — which is the asymmetry the whole device is, and it is drawn here before any current has been mentioned.

    One level, and the field that bends the bands

    Two pieces of the same crystal doped differently have their Fermi levels at different heights. Joining them cannot leave both, because a difference in electrochemical potential is precisely what makes charge move — and everything a diode does is the accounting of what had to happen for that one difference to reach zero.

    part 4 · quantum
  5. A barrier the metal was supposed to choose. The barrier height at a metal contact on n-type silicon against the metal's work function, from Bardeen's model with interface-state densities of none, 10¹² per cm² per eV, 10¹³ per cm² per eV, 10¹⁴ per cm² per eV, a 0.5 nm gap and a neutrality level a third of the gap above the valence band. With no interface states the barrier is Schottky and Mott's φₘ − χ, rising one for one with the work function: from aluminium to platinum, 0.23 to 1.60 eV. With 10¹² per cm² per eV the slope is 0.92 and the same metals give 0.27 to 1.53 eV; with 10¹³ per cm² per eV the slope is 0.53 and the same metals give 0.48 to 1.20 eV; with 10¹⁴ per cm² per eV the slope is 0.10 and the same metals give 0.71 to 0.84 eV. Dense enough interface states pin every barrier near 0.76 eV, and the metal hardly matters.

    The barrier the metal cannot choose

    Put a metal on a semiconductor and a barrier forms at the interface. The obvious theory says its height is the difference between the metal's work function and the semiconductor's electron affinity. Platinum and aluminium differ by nearly an electronvolt and a half in work function, so they should make wildly different contacts. On silicon and gallium arsenide they do not: nearly every metal gives nearly the same barrier. Bardeen explained why in 1947. A thin layer of electronic states at the interface screens the metal, and it takes only a few such states per hundred surface atoms to pin the barrier wherever they put it.

    part 5 · quantum
  6. Absorption below and above the gap. The light absorbed by a direct-gap crystal against photon energy measured from the gap, in units of the exciton's Rydberg, from Elliott's formula with each line broadened to a half-width of 0.06 Rydberg (solid), beside the absorption the same crystal would have if electron and hole did not attract (dashed), which rises from zero as the square root of the energy above the gap. Below the gap the attraction adds a series of lines at −1, −1/4, −1/9 … Rydberg, the hydrogen series scaled down, with strengths falling as 1/n³, crowding towards the gap. Above it the continuum is lifted: the attraction pulls each free pair together, and at the gap itself the absorption is a finite step rather than zero.

    The hydrogen atom inside a crystal

    Light absorbed by a semiconductor lifts an electron across the gap and leaves a hole behind, and the two attract. In an ordinary crystal they bind into a copy of the hydrogen atom, with the same series of levels and the same 1/n² — but the crystal screens their attraction and lends them light masses, so the copy is a few thousand times shallower and several hundred times larger. That rescaling is the whole theory, it predicts lines below the gap and a finite step at it where no attraction would give a smooth rise from zero, and it says which crystals keep their excitons at room temperature.

    part 6 · quantum
  7. A resistance counted in whole numbers. The Hall resistance of a gallium-arsenide layer with 3 × 10¹⁵ electrons per square metre, against the perpendicular field (solid), beside the classical B/ne (dotted); and the resistance along the current, on an arbitrary scale (red). Classically the Hall resistance rises in proportion to the field. In the layer it climbs in steps and stays on flat plateaus at h/νe² — 25813, 12906, 8604, 6453 ohms — each centred where the classical line crosses it, at integer filling. On each plateau the resistance along the current falls to zero: the current flows with no dissipation at all. Between plateaus both change. The model drawn here gives each Landau level a width 0.18 of the level spacing and lets only the middle 0.16 of it carry current across the sample.

    The resistance counted in whole numbers

    Pass a current along a thin layer of electrons in a strong magnetic field and measure the voltage across it. Classically the ratio rises steadily with the field. In fact it climbs in steps, and on each step it sits at the fundamental constant h/e² divided by a whole number — 25,812.807 ohms, then half that, then a third — the same to better than a part in a thousand million in every sample, every material and every laboratory, however dirty the layer and however irregular its shape. The steps exist because of the dirt, and they are exact because of a count that disorder cannot change.

    part 7 · quantum
  8. The push that makes an electron swing. An electron in a band of width 4J, pushed by a constant force F = 0.25 J per lattice spacing, over two Bloch periods h/Fa. Above: its crystal momentum, which grows steadily as Newton's law says it should, and is folded back each time it reaches the edge of the Brillouin zone, ±π/a. Below: its position, in lattice spacings, which does not grow steadily at all. The electron's velocity is the slope of the band, and the slope reverses at the zone edge, so it moves forward, slows, stops 16 lattice spacings along — 4J/F — and comes back, oscillating instead of accelerating. A free particle with the band-bottom effective mass, pushed by the same force, is dashed: the two agree only for the first fraction of a period. The electron is not stopped by anything; it is turned round by the lattice, which takes the momentum the force gives it.

    The push that makes an electron swing

    Push a free electron with a steady force and it accelerates for as long as the force lasts. Push an electron in a crystal with the same force and, according to the band theory every transistor depends on, it should do something else entirely: move forward, slow, stop, come back and do it again, oscillating on the spot at a frequency set by the force and the lattice spacing. Felix Bloch's oscillation was predicted in 1928 and not seen for sixty-four years, because in a real metal the electron is knocked off course ten thousand times before it completes one swing. It was found in crystals built with a thirty-times-larger spacing, and in lattices made of light where nothing scatters at all.

    part 8 · quantum

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