A defect is a level in the gap

A perfect crystal is a useful fiction and nothing is made of it, at any purity anyone has achieved. Every real solid has impurities, vacancies, atoms in the wrong place and surfaces, and an enormous amount of what materials are used for depends on those rather than on the perfection they interrupt.

The finite-matrix approach this field is built on handles a defect naturally, because a defect is one changed entry in a matrix. There is no need to restore periodicity artificially, no supercell, and no approximation: change the number and diagonalise.

That is a genuine advantage over the route through reciprocal space, and it is worth claiming rather than passing over. A calculation organised around a wavevector has to describe a defect by repeating it — building a large cell with one defect in it and imposing periodicity on that — which introduces an artificial interaction between a defect and its own images, and requires the cell to be large enough for the artefact to be small. Here there is nothing to repeat and nothing to check for, at the cost of the things a band with no structure in it sets out.

What one changed site does #

Take a uniform chain and raise the energy of one site by $h$, in units of $\beta$. Physically this is a substituted atom — a nitrogen among carbons, a phosphorus among silicons — with a different orbital energy from its neighbours.

Two things happen, and they are two aspects of one thing.

A level leaves the band. The uniform chain’s levels all lie between $-2\beta$ and $+2\beta$; after the change, one of them is outside that interval. It is at an energy no state of the pure material has, which is what “a level in the gap” means.

The state that level carries is localised. Its amplitude is largest at the modified site and falls off geometrically on either side. Where every state of the pure chain reaches across the whole system, this one occupies a region a few atoms wide.

The two go together necessarily. A state at an energy outside the band cannot propagate — there is nothing at that energy for it to propagate as — so it must decay away from wherever it is bound. That connection runs both ways and is worth keeping: a level outside a band is always a localised state, and a localised state always sits at an energy the extended states do not reach.

Measuring the localisation #

The measure used here is the participation ratio, which counts how many sites a state effectively lives on: one if it is confined to a single site, $n$ if it is spread evenly over all of them.

It is the same quantity used in the localisation transformation to show that methane’s localised bond orbitals live on essentially one hydrogen each while the canonical ones live on all four. The identical measurement applies to a chain, which is worth noticing: a surface state and a localised bond orbital are the same kind of object, measured the same way.

For a chain of sixty with one site raised by $1.2\beta$, the state that leaves the band lives on about five and a half sites. Its weight falls by a factor of $e$ every two and two-thirds sites, and that decay is geometric — the logarithm of the weight falls linearly with distance, which is what the fit in this site’s library returns.

The threshold, which turns out not to exist #

The obvious question is how strong the defect has to be. A potential well in three dimensions needs a minimum depth before it binds anything, so the expectation is a threshold below which nothing leaves the band.

Measured by bisection on chains of forty, eighty and a hundred and sixty sites, the apparent threshold comes out at $0.098$, $0.049$ and $0.025$. It halves with every doubling of the chain.

A quantity that halves when the system doubles is not a property of the physics; it is the finite chain’s own level spacing. The top level of a uniform chain sits slightly below $2\beta$, and a defect has to lift it past that before anything counts as “outside” — and the amount it sits below shrinks as the chain grows.

So the threshold is zero. A one-dimensional defect of any strength whatever binds a state.

That is a real and slightly startling result, and it is the one-dimensional counterpart of a standard fact about quantum mechanics: a one-dimensional attractive well always has a bound state, however shallow, while a three-dimensional one does not. The chain reproduces it out of a matrix with nothing in it about wells or dimensions.

The numbers, for the case drawn at the top #

The hero figure is a chain of sixty with its thirtieth site raised by $1.6\beta$, and its computed quantities are worth reading rather than glancing at.

The level that leaves the band sits at $2.561\beta$, comfortably outside the band’s edge at $2$. Its state lives on 2.30 sites — not five or ten, but slightly more than two — and its centre of mass is at site 30.0, which is exactly the site that was changed. Every one of those three numbers is a separate claim and each is checked.

The last of the three is worth its own sentence, because it is the one that could most easily have gone wrong. A state could perfectly well be outside the band and spread over the whole chain; being outside the band and being localised are different properties, and a computation that produced the first without the second would still look right in a level diagram. The centroid is what says the state belongs to the defect rather than to the chain, and it is the check that would fail if the modified entry had been put in the wrong place.

Compare with the weaker defect at $h = 1.2$, whose state lives on $5.55$ sites. Halving the strength roughly doubles the extent, which is the behaviour a bound state has: the more weakly bound, the further it reaches. In the limit of a vanishing defect the state reaches across the whole chain and stops being localised at all, which is the smooth version of the threshold result below.

Why a level in the gap matters so much #

A single defect among a hundred and sixty atoms sounds like something that should be undetectable. It is not, and the reason is that a level in the gap is at an energy where the pure material has nothing at all.

It absorbs light the pure material does not. This is the mechanism behind most of the colour in the mineral world. A transparent crystal has a gap larger than the visible range and passes light. A level inside that gap gives a transition at a smaller energy, which is inside the visible range, and the crystal is coloured. Almost every coloured gemstone works this way — corundum is colourless, and a trace of chromium in it makes a ruby.

It changes the conductivity by orders of magnitude. A level just below the empty band can donate an electron into it with a small thermal push; a level just above the filled band can accept one. That is doping, and it is why one phosphorus atom in $10^7$ silicons changes the conductivity of silicon by a factor of a thousand. The whole semiconductor industry is built on placing levels in gaps.

It provides somewhere for an excitation to go, on the way to somewhere else. An electron excited across a gap can drop into a defect level rather than back across, and whether it emits light on the way is a property of that level. This is what makes some defects useful — the colour centres in a laser crystal — and others ruinous, since a defect that lets an excitation decay without emitting is what kills the efficiency of a light-emitting device.

Three quite different technologies, and in every case the mechanism is the same: an energy at which the pure material has no states is an energy at which one defect has a monopoly, and a monopoly over a whole crystal’s worth of behaviour at that energy is worth a great deal more than one atom in a million suggests.

Above the band and below it #

There is a symmetry worth pointing out, and it is the reason a chemist’s intuition about donors and acceptors works.

Raising a site’s energy pushes a level out of the top of the band. Lowering it pushes one out of the bottom. The magnitude is the same and the direction follows the sign of the change, which is exactly what the arithmetic of the matrix requires.

In chemical terms: a substituent whose orbital lies above its neighbours’ contributes a level near the empty band, and one whose orbital lies below contributes a level near the filled one. The molecular version is the same calculation — Hückel with a heteroatom replaces a carbon with a nitrogen and watches the levels move, and the parameters used there are the same kind of quantity as the $h$ used here.

The molecular case also carries the caution that applies to this one. In a hydrocarbon Hückel calculation there is no fitted number at all — the matrix is ones and zeroes. The moment a different atom is introduced, a fitted parameter comes with it, and what the model can be trusted for changes accordingly.

What a defect does to the count of states #

There is a bookkeeping question that the level diagram answers neatly and that is worth following, because it says something about where the defect state came from.

A chain of $n$ sites has $n$ levels, and adding a defect does not change the number of sites. So the chain with a defect also has $n$ levels, and if one of them is now outside the band, the band has one fewer than it had.

The defect state was not added. It was taken from the band, pulled out of the top and away from the continuum, and what is left behind is a band with a level missing. That is why the defect state is spoken of as being “split off” rather than created, and it is a constraint worth remembering: a defect can move states around and cannot manufacture them.

It also bounds what one defect can do. One changed site removes at most one level from each band it touches, so a material with one defect per million atoms has at most one defect state per million band states. That the effect on the material’s properties is nonetheless enormous is entirely because of where the state sits rather than how many there are — an argument the essay has been making throughout and which the counting sharpens.

What the model gets right and what it leaves out #

The localisation, the level’s position outside the band, the geometric decay and the absence of a threshold are all consequences of the matrix and are as reliable as the matrix is.

What is left out is substantial and worth listing.

The lattice relaxes. A real defect pulls its neighbours towards it or pushes them away, which changes the interactions as well as the site energy. The effect can be large enough to move the level substantially, and this model has no positions to relax.

The defect has its own charge. A donor that has given up an electron is positively charged, and the resulting long-range Coulomb attraction binds the electron in a shallow orbit spread over many lattice spacings — an entirely different object from the tightly bound state computed here, and one that needs a distance-dependent interaction the model does not have.

Defects interact. At a concentration high enough for their states to overlap, the individual levels broaden into an impurity band, and above a critical concentration the material conducts through it. That is a real transition with a name, and it needs many defects rather than one.

Each of those is a limitation of the same kind: this model knows connectivity and not geometry. What it does know, it computes exactly, and the exact part is enough to establish that one atom in a hundred and sixty changes the states rather than merely the count.

The division is a useful one to hold when reading anything about defects. Whether a level appears in the gap and roughly where it sits is decided by the chemistry — which atom, with what orbital energy, replacing what. How far the state extends and how strongly it couples to everything else is decided by the geometry, and needs distances. The first question is often the one that matters and is the cheaper one to answer, which is why a table of substituents and the levels they contribute is a more useful object than it has any right to be.

The other kind of defect level, which is the useful one #

The second item on the list above deserves more than a line, because the defect that matters commercially is that one, and it is the opposite of everything computed here in every respect.

A donor in silicon has given up an electron and is left positively charged. The electron it gave up is not bound by a changed site energy; it is bound by the Coulomb attraction of that charge, screened by the material around it. So the problem is a hydrogen atom with two substitutions: the electron’s mass becomes the effective mass it has in the band, and the vacuum’s permittivity becomes the crystal’s.

Both substitutions push the same way and they are not small. Hydrogen’s binding energy of 13.6 electronvolts is scaled by the effective mass ratio and divided by the square of the dielectric constant, which for silicon — a mass ratio near 0.26 and a dielectric constant of 11.7 — gives about 26 millielectronvolts. Measured donor levels in silicon run from 43 to 54, so the estimate is the right size and low by rather less than a factor of two; the difference is the short-range part of the potential, which depends on which atom the donor actually is and is the only place the chemistry enters.

The radius scales the other way. Hydrogen’s half-ångström becomes the dielectric constant divided by the mass ratio times a half-ångström, which is 24 ångström — a sphere holding a few thousand silicon atoms.

Put the two objects side by side. The state computed in this essay lives on 1.42 sites and sits an appreciable fraction of a band width outside the band. A donor state in silicon lives on a few thousand atoms and sits twenty-six millielectronvolts below the band edge, which is close enough that room temperature ionises it — which is the entire point, since a dopant that held its electron would not dope anything.

Deep and shallow are not two ends of one scale. They are two mechanisms: a short-range perturbation that traps a state on a handful of atoms, and a long-range one that binds a hydrogenic orbit over thousands. This model has the first exactly and cannot represent the second at all, because the second needs a potential that reaches, and every interaction here stops at a neighbour.

Where this goes next #

Two directions lead out of this essay and the field takes both.

The first is what happens when the changed site is not in the middle. An end site is missing a neighbour rather than carrying a different atom, and that turns out to behave in the opposite way from everything above: instead of binding for free, it demands a full $\beta$ before it binds anything at all. The end is the hardest place to bind measures the threshold and works out why.

The second is what happens when there are enough defects for their states to overlap and interact. That is beyond a single-defect calculation, and the reason is worth stating: two defect states that overlap mix, two levels become two split levels, and many become a band of their own — which is the same argument this whole field is built on, applied to the defects instead of to the atoms.