One well, two wells, and the band they become

At its defaults it draws 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.

band-formation is one function in lib/figures/atomic.js — what happens once one is bound by the other. Everything below came out of it during this build, at parameters taken from the essays rather than invented for this page. A figure here is the figure a reader meets in an essay, and if the generator changes, this page changes with it.

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.

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.

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.

The same crystal doped two ways, before the two are joined. Doping does not move the band edges — the gap is 1.12 eV in both — but it moves the Fermi level, which is where the electrochemical potential of an electron sits: up towards the conduction band where donors have been added (0.357 V above the intrinsic level at 1e+16 cm⁻³) and down towards the valence band where acceptors have (-0.417 V). The difference is 0.774 V. Joining the two cannot leave both, because a difference in electrochemical potential is exactly what makes charge move, and it moves until the difference is gone. Everything a junction does is the accounting of what had to happen for that one number to reach zero: the bands bend by exactly this much, and the barrier a carrier meets is exactly this high.

Valence and conduction bands for copper, silicon, diamond, with the gap between them drawn to scale in electronvolts and the fraction of electrons thermally promoted across it at room temperature printed underneath: copper, gap 0 eV, no barrier at all; silicon, gap 1.12 eV, 3.9e-10; diamond, gap 5.47 eV, 1.1e-46. Nothing about the three drawings differs except the height of one white band, and that one number is the difference between a wire, a transistor and a window.

Charge density, electric field and electrostatic potential across the same junction. They are not three facts: the second is the integral of the first and the third is the integral of the second, which is Poisson's equation written as a picture. The exposed dopants make two rectangles of opposite sign whose areas are equal — 1e+17 cm⁻³ over 30.2 nm against 1e+16 cm⁻³ over 301.6 nm — because the junction as a whole is neutral. Integrating them gives a triangular field peaking at 46.65 × 10⁵ V/m at the metallurgical junction, and the area of that triangle is 0.7738 V, which is the barrier. That last equality is the check: a field profile drawn to look right would not integrate to the potential the Fermi levels demand, and the width is whatever makes it do so.

physicscheck asserts something about band-formation that could fail — it draws it and measures the result against a value reached some other way.

Across the library: 100 interrogated, 2 exercised only, 1 untouched, of 103. Read out of the gate's source by the gate's own two patterns — and the gate's last claim fails the build if that read disagrees with what it was handed while running.