One defect is a level, many are a band

Put one site of a chain at a different energy and a state leaves the band: an impurity level, localised on the impurity, at an energy outside the range the perfect chain allows.

The third put in two of them and watched the level split, by an amount that falls off exponentially with how far apart they are.

Adding many defects is the obvious next step, and the answer is not what the extrapolation from two suggests. Many defects do not give many split pairs. They give a band, its width is a measure of the concentration, and the count of levels stops being a count of defects at a concentration that turns out to be a distance.

One impurity, in closed form #

Before any concentration, the dilute case can be done exactly, and having a closed form is what makes everything after it checkable.

Write the bound state as an alternating decaying amplitude, $\psi_j = A(-1)^j y^{|j|}$ with $0 < y < 1$. Every site away from the impurity gives $x = -(y + 1/y)$; the impurity site itself gives $h = y - 1/y$. Square both, subtract, and

$$x = -\sqrt{h^2 + 4}, \qquad y = \tfrac{1}{2}\left(h + \sqrt{h^2+4}\right)$$

so the level and the decay length are both closed forms in the one parameter.

Two features of that figure are the whole of the dilute picture.

The level leaves the band immediately. There is no threshold depth below which nothing happens: any non-zero $h$ gives $\sqrt{h^2+4} > 2$, so a state is always pulled out. That is special to one dimension, and the end of a chain, where binding has a threshold, is about what changes when it is not.

The deeper the level, the more localised the state. A deep impurity binds tightly and holds its electron on itself; a shallow one holds a state that leaks over several sites. The two are the same statement, and the decay length falls from 2.08 sites to 0.69 across this figure.

Two, and then many #

A second impurity splits the level into two, by an amount that falls exponentially with the separation — because the splitting is the overlap of two states that decay exponentially.

The naive extrapolation from that picture is a set of pairs: put in twenty impurities and get twenty levels arranged in a pattern of splittings.

What actually happens is that every impurity has some amplitude at every other impurity’s site, the splittings are not pairwise, and the whole set of levels becomes a band.

Three things that figure says #

The width grows smoothly from zero. At one impurity in a hundred and sixty there is one level and no width. At two, the two impurities are far enough apart in every arrangement sampled that the levels are still degenerate to the resolution of the calculation. At four the width is 0.014; at eight, 0.29; at sixteen, 1.45.

There is no concentration at which the levels start interacting. They always interact, by an amount that is exponentially small in the typical separation, and the concentration at which anybody notices is set by their resolution rather than by the system.

The gap to the host band closes. The single-impurity level sits 1.61 below the band edge; at ten per cent the top of the impurity band is 0.69 from it, at twenty per cent 0.50, at thirty 0.25. Extrapolating, the two merge somewhere above forty per cent — at which point the phrase “impurity level” has stopped describing anything, because the impurities are a component of the structure rather than a perturbation on it.

And the states spread. The single impurity holds its state on 1.42 sites; at thirty per cent the average is 3.52. Still localised, and no longer on one atom.

The count that stops counting #

The second measurement in that study is the one that goes wrong in an interesting way.

Count how many levels lie below the host band’s edge and compare it with how many impurities were put in. The two agree exactly — sixteen levels for sixteen impurities, at one site in ten, in every arrangement sampled.

The shortfall has a cause that is visible in the two-impurity figure. Two impurities on neighbouring sites split their pair of levels strongly — one goes further down and one comes back up, and the one that comes up crosses the band edge and stops being an impurity level at all.

So the count is a count of isolated impurities, and it starts to fail when neighbouring pairs become common. At a concentration $c$ on a ring, the chance that a given impurity has a neighbour is about $2c$, and 2 × 0.15 is 0.30 — which is roughly the fraction of the shortfall at fifteen per cent, where three quarters of a level is missing out of twenty-four.

The crossover is a distance, not a number. What matters is whether two impurities are within the decay length of each other’s states, and that length is a property of the impurity depth. A shallower impurity has a longer decay length and its count fails at a lower concentration; a deeper one holds out further. Quoting a concentration without the depth would be quoting half a fact.

What sets the width #

The width of the impurity band is a measurable thing here, and where it comes from is worth deriving rather than describing, because the derivation says which parameter it depends on.

Two impurity states a distance $d$ apart split by roughly $2t\,y^{\,d}$, where $y$ is the decay factor from the closed form. Impurities at a concentration $c$ have a typical separation $1/c$. So the width goes as $y^{1/c}$ — exponential in the reciprocal of the concentration, which is why it is unmeasurably small at one per cent and a substantial fraction of the host band’s width at thirty.

That exponential is what makes the crossover look like a threshold while being nothing of the kind. A quantity that goes as $e^{-1/c}$ changes by orders of magnitude over a factor of two in $c$, so any fixed resolution finds a concentration at which it appears, and different resolutions find different ones.

The same shape of argument runs through this whole field. The width of the host band is a count of neighbours because the second moment of the levels is exactly the coordination; the width of the impurity band is a count of neighbours too, at a coordination that is fractional and set by the doping.

The levels move before the states do #

A second measurement separates the two things that are usually described with one word.

The levels spread as soon as there is more than one impurity, by an amount that is exponentially small and never zero. The states stay put much longer: the participation ratio goes from 1.42 sites for one impurity to 2.93 at ten per cent and 3.52 at thirty, which is to say that at thirty per cent impurity concentration a state still lives on three or four sites out of a hundred and sixty.

So a sample can have a fully formed impurity band — a continuous distribution of levels a spectroscopy would resolve as a band — and every state in it localised on a handful of atoms. Reading the band as a set of states that go somewhere is the error, and it is the same error a density of states invites in the host band, where a smooth distribution of levels says nothing about what any one of them looks like.

That distinction is the whole of the question of disorder in metals, where a chain with every site disordered has levels that crowd together exactly as a metal’s do and states that occupy ten sites out of two hundred.

What this is a model of #

The arrangement here — a fraction of the sites at a different energy, placed at random — is the standard model of a doped semiconductor, and the correspondence is worth stating carefully because the words in use are all borrowed.

The impurity band above is the impurity band of semiconductor physics, and its merging with the host band at high doping is the Mott transition’s structural half: at a high enough dopant concentration the impurity states overlap enough to conduct, and a material that was an insulator with isolated donor levels becomes a metal — which is the same crossing, from the other side, as a molecule ceasing to be one. The other half of that story is the one where repulsion rather than concentration is what decides.

Where the host band came from is a chain’s levels crowding together as it grows: two sites, four, eight, sixteen and forty, with the interval they occupy fixed and the spacing inside it closing. Everything in this essay is a perturbation of that spectrum, and the impurity states are the ones that leave it.

What the model does not have is the long-range Coulomb tail of a real dopant, which turns a single deepened site into a hydrogen-like series of levels, or the compensation and screening that decide how many of those levels are occupied.

Where the model stops #

A deepened site is not a missing one. Everything here is an impurity — a site whose energy has been changed — and the levels depend continuously on by how much. A vacancy is a different object: its levels sit at exactly zero for a reason that cannot be tuned, and a concentration of vacancies does something else entirely.

The impurities are placed at random and averaged over a handful of arrangements. Each point in the concentration figure is the mean of four placements on stated seeds, and two neighbouring concentrations can come out in the wrong order by a few hundredths because of it. The checks are therefore made across the range rather than between adjacent points — a fact about the sampling, and one a check has no business claiming past.

It is one dimension, where a state is always bound and where any disorder at all localises everything. The three-dimensional problem has a threshold in both respects.

And it is one electron. Nothing here counts the electrons the impurities brought or the repulsion between them, so this is a picture of levels and not of occupancy — and counting electrons in an extended structure is a separate exercise that decides which of these levels are filled.

How strong a perturbation has to be before a state leaves the band at all is a threshold that depends on where in the chain it sits. In one dimension a well in the middle binds at any depth whatever once the chain is long enough, so every impurity in this essay is above threshold and the question is only how far out its level goes.

And the limit this is heading towards is the case where every site is different rather than a few: no host band, no impurity band, and states that live on a handful of sites each. The impurity band is the intermediate object, and it stops existing when the impurities stop being a minority.

What is quoted, and what is computed #

Nothing is quoted. The chain, the ring, the impurity depths and the concentrations are all parameters of the model; every level, width, gap, count and participation ratio is computed by diagonalisation, and the single-impurity level is computed twice — once by diagonalisation and once from the closed form derived above — with the two required to agree to a millionth.

What was checked #

The single-impurity level matches −√(h² + 4) at six depths, to a millionth, and the deeper impurity holds its state on fewer sites at every step.

Exactly one level leaves the band for one impurity. A check on the position alone would pass on a calculation that had lost a state.

The impurity band widens with concentration and closes on the host band, across the range and at its midpoint, rather than between every adjacent pair — because the sampling cannot support the stronger claim.

At the lowest concentration every impurity level is at the same energy, to a part in a billion. That is what makes the widening a measurement rather than an artefact of the placement.

And the count of levels stops matching the count of impurities above some concentration, while matching exactly below it. Both halves are checked; the first alone would pass on a calculation that never counted right.

What a measurement would see #

Three consequences follow for anybody reading a spectrum of a doped material, and all three are visible in the figures above.

A single quoted impurity level is a dilute-limit number. Two samples of the same material at different doping have impurity levels at different energies, and the difference is not a measurement error.

The band’s width is a measurement of the doping, and a better one than a level position: the width goes exponentially with the reciprocal concentration, so it changes by a lot where the position changes by a little.

And a count of levels undercounts the defects above a concentration set by how localised the states are. A technique that counts absorbing centres is counting isolated ones.

None of that is news to a semiconductor physicist, and all of it is usually presented as phenomenology. What the arithmetic here adds is that every one of the three follows from a chain of a hundred and sixty sites with one number changed on some of its diagonal entries, and that the crossover between them is not a threshold at all.

When the two bands meet, the material stops being a semiconductor #

The figures above track a gap closing: the impurity levels spread into a band, and that band’s edge moves towards the host’s until only 0.69 separates them. It is worth saying what happens when the remaining gap goes to zero, because the answer is a material with a different name.

A lightly doped semiconductor has its impurity levels in the gap, isolated from the host band. An electron in one of them has to be promoted across the remaining gap to conduct, so conduction is thermally activated: cold enough, and the carriers freeze out onto their donors and the material stops conducting. That freeze-out is the standard behaviour of a doped semiconductor and it is what the dilute limit of this model describes.

Dope it more heavily and the impurity band broadens until it touches the host band and merges with it. There is then no gap to be promoted across. The carriers are in a continuum at all temperatures, conduction stops being activated, and the resistance behaves like a metal’s — falling as the sample is cooled rather than rising. Such a material is called degenerate, and the crossover in silicon happens above roughly $10^{19}$ impurities per cubic centimetre, which is about one atom in five thousand.

That is a great many for a defect and a very small number for a material. One atom in five thousand, of a different element, is enough to remove a gap that the other four thousand nine hundred and ninety-nine had between them — which is a fair summary of what defects in solids are about.

The merged state has an optical signature that is worth knowing because it runs the opposite way from the intuition. Heavy doping fills the bottom of the host band, and a filled state cannot be excited into, so the lowest-energy optical transition available is no longer across the gap but from the valence band up to the first empty state above the filled ones. The absorption edge therefore moves to higher energy as more carriers are added, and by an amount that measures how many there are.

So the same process that closes the gap in the electronic structure opens it in the optical measurement, and the two are not in conflict: one is the energy needed to move a carrier, and the other is the energy needed to create a pair in a band whose bottom is already occupied. The distinction is one worth keeping — a gap is a difference between two states, and which two depends on what is being done to the material.

Still open: when there is no host #

One impurity in a chain, two, and a concentration of them have now been treated. The direction not yet taken is the one where the defects stop being a perturbation on a host: at fifty per cent there is no host, and what the model describes is an alloy of two components in which neither is the impurity.

The interesting question there is not the level structure but whether the alloy’s band is the average of the two components’ or something else. The average is what a simple picture predicts and the answer is known to be more interesting than that — for a large enough energy difference the two components’ levels stay separate and the band splits in two, and for a small enough one they merge. Where the crossing is, and whether it is sharp, is a computation the same model can already do, and it is the same question as whether two bands overlap asked about disorder rather than about symmetry.