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.
At its defaults
Drawn even though every essay passes options, because a default nothing exercises is a trap for the next essay to call this with none — which has happened here twice.
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.
One well, two wells, and the band they become
The options are the ones No two in the same state, and why matter has volume passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.
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 bands bend by the amount the Fermi levels differed
The options are the ones One level, and the field that bends the bands passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.
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.
Two Fermi levels, and the difference that has to go
The options are the ones One level, and the field that bends the bands passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.
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.
The same picture three times, with the gap changed
The options are the ones One level, and the field that bends the bands passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.
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.
One solution, drawn three times
The options are the ones One level, and the field that bends the bands passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.
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.
What checks it
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.
Where it is called
Changing this generator changes every figure on this list. That is what makes the list worth publishing rather than keeping in a check script.
No two in the same state, and why matter has volume
Nothing in the energy levels of an atom says how many electrons may occupy each one. The answer is one per state, it is not derived from any force, and it is the reason a table holds a cup up.
QuantumOne 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.
QuantumThe 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.
WavesThe chirp a staircase sings back
A hand clap at the foot of the great stair of El Castillo at Chichén Itzá is answered by a chirp — a falling whistle that some have compared to the call of the quetzal bird. Nothing in the stone is tuned. Each of the ninety-one steps returns its own copy of the clap, the copies arrive as a train of clicks, and the ear hears a regular train of clicks as a pitch. The pitch falls because the steps are not evenly spaced as the listener sees them: low steps are seen almost level, a tread apart, and high ones almost along the slope, a whole step's diagonal apart. It is Bragg reflection from a grating whose period, seen from one spot, changes along its length.
QuantumThe 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.
WavesThe end that knows how the middle was cut
A chain whose links alternate, strong and weak, has the same bands whichever kind of link is counted as inside a cell. The infinite chain cannot tell the two choices apart. A finite chain can: cut it so that a weak link is outermost and each end holds a state at exactly zero energy, in the middle of the gap; cut it the other way and it holds none. What decides is not anything at the ends but a whole number counted from the bulk — how many times a loop winds round a point.
WavesThe gap a repeat opens
Stack two transparent materials in alternating layers and there is a band of frequencies the stack will not carry — not weakly, not with loss, but not at all. Nothing has been absorbed and neither material has a resonance there. What forbids those frequencies is the repeat itself, and the width of the band has a closed form containing only the ratio of the two indices.
QuantumThe 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.
WavesThe light that spreads in two beams
Shine light into one of a row of identical optical fibres laid side by side and close enough to leak into each other, and it does not spread out into a widening glow as it would in a slab of glass. It splits into two beams that run outward at a fixed speed, bright at their leading edges, leaving the middle almost dark. The reason is that light hopping between neighbours has a fastest possible sideways speed, and piles up at it. The same periodicity gives the row two tricks no uniform medium has: a direction in which a beam travels without spreading at all, and a direction in which diffraction runs backwards, so that a spread beam can be gathered up again.
QuantumThe 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.
WavesThe mirror that works from every direction
A stack of alternating transparent layers reflects nearly all the light of one colour arriving straight on, and less as the light tilts, because tilting moves the forbidden band. A structure that repeats in only one direction ought therefore to be a mirror for only a range of directions. It is not, if the layers differ enough: light arriving from air cannot bring enough sideways momentum to escape the forbidden band at any angle, for either polarisation, and a flat stack of plastic and tellurium reflects every angle over a band of frequencies almost half as wide as its centre.
WavesThe mode that lives in the mistake
A perfect stack of alternating layers refuses a whole band of frequencies — not weakly, not with loss, but not at all. Break the repeat once, by inserting a single layer of the wrong thickness, and exactly one frequency inside that band passes through the whole stack with a transmittance of one. The useful thing about a forbidden band turns out to be not what it excludes but what a single flaw is thereby allowed to hold.
WavesThe pile of glass that stops every colour
A sheet of window glass reflects about eight per cent of the light that falls on it. A pile of a hundred sheets, George Stokes calculated in 1862, should still let through a tenth, because each extra sheet only adds a little more reflection to the total. That is right for sunlight, whose colours are too mixed to interfere. For a single colour, with every sheet and every gap a slightly different thickness, the same pile lets through a thousandth, and a pile of four hundred a hundred-millionth of a millionth. Interference with random phases does not average away into the intensity sum. In one dimension it traps every wave, at every wavelength, however weak each reflection is.
QuantumThe 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.
QuantumThe 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.
WavesThe sandbars that reflect the sea
A stack of alternating layers reflects the one colour whose half-wavelength matches the repeat. A row of sandbars on a gently sloping seabed is the same arrangement for ocean waves, and it does the same thing. Waves twice as long as the bar spacing are sent back out to sea, one small reflection per bar, all adding in step. A patch of eight bars can return most of the swell that meets it. The reflected wave, added to the incoming one, makes a standing pattern whose nodes are spaced exactly like the bars. Sand gathers in that pattern, so a field of bars can build its own next bar.
QuantumWhat 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.