What happens when the wells get close

One well, one level. Bring a second well close enough that the wave in each can leak into the other, and the single level becomes two — one where the two wells’ waves are in step and one where they are opposed.

Both halves of that are the result. The count grows without limit; the width does not. A solid therefore has an enormous number of states packed into a fixed energy range, and that combination — many states, finite range — is what makes a solid behave differently from a collection of atoms.

The chain, solved #

The model is the simplest one that shows the effect, and it has a closed form worth writing down because the closed form contains the saturation.

Take N identical wells in a row. Each has one level at energy α. Let a wave in one well leak into its neighbours with a coupling β, and ignore everything further away. The allowed energies are then

$$E_j = \alpha + 2\beta\cos\!\frac{j\pi}{N+1}, \qquad j = 1,\dots,N.$$

The cosine is where the saturation comes from. As j runs from 1 to N the cosine runs from just below 1 to just above −1, so the levels fill an interval of at most 4|β| whatever N is — and they fill it more and more densely.

That expression is not new to this site. It is the eigenvalue of a chain of N coupled oscillators, and it is exactly the condition that decides which shapes fit on a string, with j counting half-wavelengths along a chain of beads rather than along a continuous line. Nothing quantum is needed to produce the formula; what is quantum is that the levels are energies of one particle rather than frequencies of a vibration.

Why the width belongs to the coupling #

The saturation deserves a physical reading rather than a trigonometric one.

The lowest state of the chain has every well’s wave in step with its neighbours, which lowers the energy by the full coupling on each side — by 2β. The highest has every well opposed to its neighbours, raising it by 2β. Nothing can be more in step than fully in step, so no chain, however long, produces a state below the first or above the last.

That is why the band width is a local property. It is set by how strongly one well talks to the ones next to it, and it can be read off a two-well molecule: the splitting of the bonding and antibonding levels of H₂ is 2|β|, and it predicts the width of the band in a hydrogen chain of any length.

The consequence for materials is direct. Band widths report overlap. Tightly bound core electrons overlap almost not at all and their bands are millielectronvolts wide — effectively still atomic levels. Valence electrons overlap strongly and give bands several electronvolts wide. A material’s band diagram is a map of which of its electrons are sociable.

Two wells first, because the pattern is already there #

Everything on this page is visible in the simplest case, and it is worth doing before the chain because it has a chemical name.

Two identical wells, one level each. Coupling them produces two states: one in which the two waves add in the region between the wells, and one in which they cancel. The first has more amplitude between the wells, where the attraction from both is felt, and it lies lower by |β|; the second has a node there and lies higher by |β|.

Those are the bonding and antibonding states of a diatomic molecule. Putting two electrons into the lower one — the most the exclusion principle allows — lowers the total energy by 2|β| relative to two separate atoms, and that lowering is a covalent bond. Putting four in, two into each, gains nothing: helium does not form He₂ for exactly that reason, and the failure is a counting result rather than a chemical one.

The reason the chain matters rather than the pair is that a molecule’s levels stay far apart while a solid’s do not. Two levels 2.8 eV apart are two chemical states; 10²³ levels spread over 3.6 eV are a continuum, and a continuum can be occupied part way. Everything that distinguishes a solid from a large molecule is that difference.

What decides whether it conducts #

Levels alone do not decide anything. What decides it is filling, and filling is the exclusion principle doing arithmetic.

Each level in the band holds two electrons. N atoms each contributing one electron therefore fill exactly half the band, leaving the top half empty — and the highest occupied state has empty states immediately above it, a vanishing energy away. An electric field can then accelerate electrons into those states, and the material conducts. That is a metal.

N atoms each contributing two electrons fill the band exactly. The highest occupied state now has no empty state above it until the next band, which is a finite gap away. A small field cannot promote anything, no net current flows, and the material insulates.

So conduction is not about how many electrons a material has. It is about whether the count comes out at a band edge or in the middle of one, and that is a question about arithmetic and crystal structure rather than about the electrons themselves.

Four orders of magnitude in one number #

Once a gap exists, its size decides everything, and the dependence is exponential.

At temperature T — with the exponential weighting a thermal population carries — the fraction of electrons thermally promoted across a gap E_g goes roughly as e^(−E_g/2kT). At room temperature kT is 0.0259 eV, so the exponent is large for any gap of ordinary size and the answer moves violently.

That thirty-seven orders is the whole reason the electronics industry exists. A material with no gap cannot be switched off; a material with a large gap cannot be switched on. Silicon sits in the narrow band where thermal energy at room temperature is small compared with the gap and not negligibly small, so the population of carriers is tiny, controllable, and hugely sensitive to anything that shifts it — a voltage, a dopant atom, an arriving photon.

Doping, and the sensitivity that follows #

Silicon’s intrinsic carrier concentration at room temperature is about 10¹⁰ per cubic centimetre, against 5 × 10²² atoms. That is one carrier per five trillion atoms.

Adding one phosphorus atom per million silicon atoms — which is a purity level, not a concentration — introduces 5 × 10¹⁶ donors per cubic centimetre, raising the carrier count by a factor of five million. The conductivity moves with it.

Two facts follow, and both are consequences of the exponential above rather than of the doping.

The first is that semiconductor manufacture is an exercise in purity to a degree unusual in engineering: unintended impurities at the part-per-billion level change the material’s electrical behaviour measurably, which is why electronic-grade silicon is refined to nine nines and handled accordingly.

The second is that the same sensitivity makes the material a detector. A photon with energy above the gap creates a carrier pair, and against a background of 10¹⁰ intrinsic carriers a small number of created ones is a measurable signal. Every image sensor, solar cell and photodiode is that arithmetic.

The gap is not always where the arithmetic puts it #

The tidy story above assumes the bands do not overlap, and in real materials they often do.

Magnesium has two valence electrons per atom, so by the counting above its band should be exactly full and magnesium should be an insulator. It is a metal, because its 3s band and 3p band overlap in energy — the top of one lies above the bottom of the other, so the electrons spill into the second band before filling the first, and both end up partly occupied.

That is the general situation. Whether bands overlap depends on the crystal structure as much as on the atoms, which is why carbon is an insulator as diamond and a conductor as graphite with the same atoms in a different arrangement, and why tin is a metal in one crystalline form and a semiconductor in another below 13 °C.

The counting rule is therefore a necessary condition rather than a sufficient one: an odd number of electrons per unit cell guarantees a metal, and an even number permits either.

Why the same picture describes vibrations #

The chain solved above has an exact counterpart with no quantum mechanics in it, and noticing the correspondence is worth more than either case alone.

Replace the wells with masses and the coupling with springs. A chain of N masses has exactly N normal modes; their frequencies are ω_j = 2ω₀ sin(jπ/2(N+1)), which fills a band from zero up to 2ω₀ however long the chain is; and the band’s top is set by the spring constant and the mass, not by the length. Same count, same saturation, same reason.

A string is the continuous limit of the same construction — a chain of infinitely many coupled masses — and its modes are counted by how many half-waves fit between the ends. Its lowest modes look like the lowest states of the chain, node for node. The correspondence is not an analogy: the eigenvalue problem is the same one, with a second difference in place of a second derivative, and the difference is only what is being counted at the end.

The consequence is that a solid has two band structures, not one: an electronic band structure and a vibrational one, computed by the same method and measured by different instruments. The vibrational band’s quanta are phonons; its width sets the Debye temperature, which is where the silencing of stiff modes shows up in a solid’s heat capacity; and the same counting that decides whether a material conducts electricity decides how it conducts heat.

That doubling is why the tight-binding chain is worth learning as a shape rather than as a result about electrons. It appears wherever identical units are coupled in a line, and the answer is always the same: as many modes as units, filling a fixed interval, with the interval set by the coupling.

The chain built on purpose #

The model on this page is normally an idealisation of a solid. It can also be built directly, with every parameter under a dial, and doing so has turned several of its assumptions into experiments.

Shine two laser beams at one another and the standing wave they make is a periodic potential for atoms, with wells half a wavelength apart. Cold atoms loaded into it sit in exactly the arrangement of this page — identical wells, evenly spaced, coupled by tunnelling between neighbours — and the coupling is set by how deep the wells are, which is set by how bright the lasers are.

That makes β a knob. Turn the light up and the barrier between wells rises, the leakage falls exponentially, and the band narrows; turn it down and the band widens. The band structure itself is read out by switching the lattice off abruptly and photographing the momentum distribution as the cloud expands.

The interesting results are the ones about the assumptions the essay’s last section lists. Atoms in such a lattice do interact, and the strength of the interaction can be tuned independently of the coupling — so turning the interaction up while holding everything else fixed drives the system from a state where the atoms are spread over the whole lattice to one where each sits on its own site and cannot move. That is a Mott insulator, made deliberately in 2002, and it is the failure of “the electrons do not interact” performed on demand.

Disorder has been added the same way, by superimposing a random speckle pattern, and the localisation of the states that follows is what the chain does when its wells stop being identical.

So the four caveats at the end of this page are not merely acknowledged limits. Three of them are now laboratory parameters, and the model is studied by building it rather than by finding a material that approximates it.

Gaps for things that are not electrons #

The correspondence with vibrations has an engineering consequence that the essay’s own observation almost states: if a periodic array of coupled units produces a band with edges, then anything wave-like passing through a periodic structure has a range of frequencies it cannot propagate at.

Make a material whose refractive index varies periodically on the scale of a wavelength of light and it acquires a photonic band gap: a range of colours that cannot travel through it in any direction, for the same reason an electron cannot travel through a semiconductor at an energy inside its gap. Such a structure reflects those colours completely while being made entirely of transparent material.

Nature got there first. An opal is a close-packed array of silica spheres a few hundred nanometres across, and its colour is a band gap rather than a pigment; so are the blues of many butterflies and beetles. Structural colour of that kind does not fade, because there is no dye to bleach — the colour is in the arrangement.

The engineered version has a use the electronic case cannot match. A fibre with a hollow core surrounded by a periodic lattice of holes guides light in air, because the cladding has a gap at the operating wavelength and the light has nowhere else to go. That is guidance without total internal reflection, and it lets a fibre carry powers and wavelengths that would destroy or be absorbed by glass.

The same construction works for sound. A periodic array of scatterers in an elastic medium has a phononic gap, and structures built to have one are used to isolate vibration at a chosen frequency by geometry rather than by damping — which is the vibrational band structure of this page, arranged rather than inherited.

Where the model stops #

The chain used here is the simplest tight-binding model, and four of its assumptions are doing work.

Only nearest neighbours are coupled. Including further neighbours changes the band’s shape — the cosine acquires harmonics — without changing the count or the qualitative width.

One level per well. A real atom contributes many, each broadening into its own band, and the bands can overlap as described above. The single-level chain cannot show a gap at all; the gap in the figure above is drawn rather than derived.

The wells are identical and evenly spaced. Disorder breaks the picture severely: strongly disordered chains localise their states, and the band stops being a set of extended waves. That is Anderson localisation, and it is why a band diagram is a statement about a crystal and not about a material.

The electrons do not interact. Everything here treats each electron as moving in a fixed potential. In materials where the interaction dominates, band theory can predict a metal and the material is an insulator — the Mott insulators, of which several high-temperature superconductor parent compounds are examples.

The single well the whole construction starts from has exact levels and localised states, and the chain’s are neither. What the coupling does is take a state confined to one well and spread it over the whole chain — which is why a band’s states carry current and an atom’s do not, and why the transition between the two is worth a page of its own. A localised state has no direction to go in; a delocalised one has nothing else to do.

Where the numbers come from #

Two of the numbers used above deserve their provenance stated, because the figures are drawn at them and a reader ought to know which are measurements and which are choices.

The gaps are measurements: silicon 1.12 eV, diamond 5.47 eV, both at 300 K and both known to three figures. They shrink slightly as temperature rises, because the lattice expands and the coupling weakens — silicon’s is 1.17 eV extrapolated to absolute zero — and every device specification carries that dependence.

The coupling β in the chain figures is a choice. Real couplings are extracted by fitting a band structure computed from first principles, or measured by photoemission, which maps the occupied bands directly by measuring the energy and angle of ejected electrons. Values of a fraction of an electronvolt to a few electronvolts are typical, and the figures use 0.9 to 1.4 because that produces bands of a recognisable width.

The distinction matters for how the drawings should be read. The gap figure is a picture of three real materials and the numbers on it can be checked against a handbook. The chain figures are pictures of a model, exact for the model and illustrative of a real solid, and the quantity to take from them is the shape of the dependence — count linear in N, width independent of it — rather than any particular energy.

What the picture cannot show #

The band diagram plots energies against nothing. Real band structures are plotted against crystal momentum, and the dependence of energy on direction in the crystal is most of the useful information — it decides the effective mass of a carrier, whether the gap is direct or indirect, and therefore whether the material can emit light. Silicon’s indirect gap — a mismatch in crystal momentum that a photon cannot supply — is why it makes excellent transistors and poor lamps, and no drawing on this page could distinguish it from a direct-gap material.

The figure also draws discrete lines up to forty wells and then asks the reader to extrapolate to 10²³. At that count the lines are separated by 10⁻²³ eV and no drawing can show them; what is drawn instead is a continuum, and every practical calculation replaces the count with a density of states, which is the quantity that actually enters every prediction.

And nothing here shows temperature. The occupancy of states near a band edge is a Fermi–Dirac distribution whose width is a few kT, and every conductivity, every carrier concentration and every device characteristic depends on that smearing.

Where the ladder goes next #

The rungs from here: Bloch’s theorem, which replaces the finite chain with a periodic infinite one and gives the band structure as a function of crystal momentum; the density of states, and why it diverges at band edges in one dimension and does not in three; effective mass, and why an electron in a crystal responds to a field as though it had a different mass; the p–n junction, which is two doped regions and the whole of modern electronics; and the exclusion principle at 10²³, where the Fermi energy becomes the organising quantity.

The claim to carry forward is that nothing new was introduced. A band is what a set of levels becomes when the boxes holding them are brought close together, the count is the number of boxes, the width is the coupling, and the difference between a wire and a window is which of those levels are occupied.