A vacancy is not an impurity

The essays on defects so far are about impurities. A defect is a level in the gap changes one site’s energy in a chain of a hundred and sixty and watches a level leave the band; the end is the hardest place to bind does the same at a chain’s end and finds a threshold; two defects, and the level between them puts two of them at various separations and measures how their levels split.

Every one of those is about a site whose energy has been changed, by a stated amount h, and every answer depends on h. This essay is about the other kind of defect — a site that is simply absent — and the reason it deserves its own essay is that almost nothing carries over.

The word defect covers both, and covering both is the problem: the two objects behave so differently that a single word for them hides the more interesting one.

The obvious guess, and why it is not right #

The natural way to think of a missing atom is as an impurity taken to its limit. Raise a site’s energy far enough and no electron will go near it; the site is effectively removed; so a vacancy should be the h → ∞ end of the curve the earlier essays traced.

That reasoning is not wrong about what happens to the impurity’s own level — it does run away to infinity, exactly as the picture suggests. It is wrong about what is left behind, and what is left behind is the whole of the interesting part.

The structure the argument needs #

Before the theorem, the substrate, because a vacancy in the wrong structure has nothing to say and two of the obvious choices are the wrong structure.

Two choices in that figure are not decoration.

Not a chain. Removing a site from a chain cuts it, leaving two shorter chains that no longer interact at all. Whatever happens is a fact about two separate systems, and the interesting question — what a hole does to the material around it — does not arise.

Not a wrapped square structure of even side. Those have a large accidental degeneracy at the middle of the band: a 12 × 12 wrapped structure has twenty-two levels at zero before anything is removed, and the single level a vacancy contributes is invisible in the crowd.

A finite patch of six rows by five columns has neither problem. Its levels are 2cos(πp/7) + 2cos(πq/6), which vanishes only if p/7 + q/6 is exactly one, and no pair of integers in range does that. So the perfect patch has no level at zero at all, and any that appears afterwards is something the vacancy did.

The choice of a finite patch rather than a wrapped one is also the choice this field has made throughout — there is no lattice anywhere in the argument, only a matrix that is written down and diagonalised — and here it is not merely a preference. The accidental degeneracy that spoils the wrapped case is a genuine feature of a wrapped square structure, and the finite patch does not have it.

One vacancy, one level, and it is exactly at zero #

The distinction between a small number and zero has been made before and it is exactly the distinction here. The level is not near the middle of the band because the vacancy happens to be a weak perturbation. It is at zero because of a theorem, and the theorem is short enough to give in full.

The structure is bipartite: every bond joins a site of one colour to a site of the other. Number the sites so that all of one colour come first, and the matrix is entirely off-diagonal blocks — nothing joins a colour to itself. A matrix of that shape has a spectrum symmetric about zero, so its non-zero levels come in pairs. If there are more sites of one colour than of the other, some levels have nothing to pair with, and the only place an unpaired level can sit is at zero.

The count is therefore $|n_A - n_B|$, the difference between the numbers of sites of the two colours, and it is arithmetic rather than physics.

The eigenvector says the same thing #

The count is one reading of the theorem and the eigenvector is another, and the second is the one a picture can show.

A state at zero satisfies the equation with an eigenvalue of nothing, which means the sum of its amplitudes over the neighbours of every site vanishes. On the majority colour that is satisfiable with the minority colour’s amplitudes all zero; on the minority colour it is a constraint the majority’s amplitudes have to satisfy. So a zero state lives entirely on the majority colour — the other colour’s amplitudes are not small, they are zero — and that is checked here to 10⁻¹⁶ rather than eyeballed.

The disc areas in the first figure are that statement drawn. Every disc is on one colour and there are none at all on the other.

Two vacancies, and the answer is a colouring #

Here the contrast with an impurity becomes total.

Move one of the two vacancies to the neighbouring site and the sublattice bookkeeping changes rather than the distance: two removals from the same sublattice add, two from opposite ones cancel, and the count does not care how far apart they are.

The census below is the general statement, taken over every arrangement rather than over the three drawn here.

The row that matters most is the pair at different separations. Two defects, and the level between them measured two impurities splitting by a factor of 0.41421 for every site of separation — a number predicted from the isolated level’s energy and measured to six figures — and that essay’s whole subject is how the interaction dies away with distance.

Two vacancies on the same colour do not split at all. At two sites apart and at four sites apart the two levels are at zero to within 3 × 10⁻¹⁷, which is the same number twice. There is nothing to plot against separation, because separation is not one of the quantities the answer depends on.

What the impurity limit actually gives #

The guess this essay opened with can now be repaired rather than discarded, because there is a sense in which it is right.

An impurity pair behaves in exactly the way a vacancy pair does not, and the contrast is worth drawing rather than stating.

The other quantity an impurity has and a vacancy does not is a threshold: a site’s energy difference has to reach a certain size before it traps a state at all, and at the end of a chain that threshold is about one β while in the middle it goes to zero as the chain grows. Every number in that statement is a function of a parameter. The vacancy count is a function of none.

Take an impurity’s strength to infinity and two things happen. Its own level runs off to infinity with it, which is what the picture predicts. And the remaining spectrum converges on the spectrum of the structure with that site deleted — which is the vacancy’s spectrum, zero level and all.

So a vacancy is the limit of a strong impurity in the sense that the leftovers agree. What it is not is a strong version of the same phenomenon: the impurity’s level is a continuous function of a parameter and the vacancy’s is fixed by a count. The limit is a limit of the background, not of the level everyone is looking at.

The count survives the structure being wrong #

One further property of the theorem is worth drawing out, because it is what makes a count useful in a case where a calculation would not be.

Nothing in the argument used the shape of the patch, the equality of the bonds, or even the dimension. What it used was that every bond joins one colour to the other. So the same count applies to a patch with a ragged edge, to one whose bonds are all different strengths, to a chain, to a honeycomb sheet, and to a three-dimensional structure — and it gives the same answer in each without a single eigenvalue being computed.

That is a strong claim and it is the kind that ought to be tested against a case where it could fail. The obvious candidate is the end of a chain, which is the geometry that was hardest for the impurity argument: a site at the end demands a whole β of energy difference before it traps anything, where a site in the middle traps a state for any difference however small.

A vacancy at a corner is no different from one in the middle. The census above includes it — the corner site of the patch, removed — and it gives one level at zero, exactly as an interior vacancy does, because the two sites are the same colour and the count does not know where they are. The geometry that made the impurity argument delicate makes no difference to this one at all.

Why this is a different kind of statement #

The standing distinction is between quantities that are computed and quantities that are decided by symmetry, and defects now supply one of each.

An impurity level is computed. It sits at −√(h² + 4), which is an arithmetic function of a parameter, and every essay about it is a measurement. Change the model — add a second-neighbour interaction, make the impurity’s neighbours slightly different, put the structure under strain — and the number moves.

A vacancy level is decided. Its position is fixed by the colouring alone, so every change that leaves the structure bipartite leaves the level exactly where it was. Unequal bond strengths, a distortion, a second vacancy elsewhere, the shape of the patch: none of them moves it by a thousandth, because none of them puts a matrix element between two sites of the same colour.

That is the same kind of robustness symmetry gives elsewhere — a statement that survives every change except the one that breaks the symmetry it rests on. And the change that breaks this one is easy to name: any interaction between two sites of the same colour. A second-neighbour hopping, however weak, joins colour to colour, and the levels move off zero immediately.

What it means for a real material #

Two consequences follow and both are worth having, with the caution that this is a one-electron model of a graph rather than a calculation of any substance.

A level at the middle of the band is the worst possible place for it, if the material is wanted as an insulator. A level near a band edge holds an electron loosely and gives it up at a modest temperature; a level in the middle of the gap is a trap, and traps are what stop a semiconductor working. The theorem says that vacancies in a bipartite structure produce exactly that, and produce it whatever else is done to the material.

The second consequence is a lever. If a structure has as many vacancies of one colour as of the other, there are no such levels at all — the pairs annihilate, in the sense that a matched pair of missing atoms leaves the spectrum with nothing unpaired. A material with a balanced set of vacancies is, on this argument, better behaved than one with a single vacancy in it — and the balance is a statement about which sublattice the missing atoms came from, which in a compound of two elements is a statement about which element is missing.

There is a third contrast and it is the sharpest. Every other quantity in this collection that lives near a boundary has a reach — a distance over which it settles, measured as the size at which a property stops changing. A vacancy’s zero level is not on that scale at all. Its position is fixed by a count rather than by a length, so it does not converge to anything as the structure grows: it is already exact at every size, including the smallest.

That last reading is the one worth carrying away, because it turns an abstract colouring into something a chemist controls. In a structure of alternating A and B atoms the two colours are the two elements. Removing an A and a B leaves nothing at zero; removing two A atoms leaves two states pinned at the middle of the gap. So the question of whether a material’s vacancies are electrically active is, in this model, the question of whether they are balanced between the elements — and nothing about how far apart they are enters it.

Half a level, and what it is worth #

A count of states says nothing about whether they are occupied, and for a defect level that is most of the chemistry.

At half filling — one electron per site, which is the case this whole field works at — the levels below zero are full, the levels above are empty, and a level at zero sits exactly on the boundary. It holds one electron rather than two or none, which is what a level at the Fermi energy always does, and which is why a structure with an odd number of missing atoms of one colour is a very different object from one with none.

An electron in such a state is neither bound below the gap nor promoted above it: it is at the energy where adding or removing it costs nothing at all, which is the definition of a metal applied to a single state. So the zero level makes an otherwise gapped material locally metallic, in a region set by how far the state spreads, and that region is decided by the structure rather than by any parameter.

The obvious next question — what the repulsion between two such half-filled states does — is exactly the question the smallest many-electron calculation is built to ask, and no diagonalisation in this essay answers it. A theorem does, and the section below is what it says.

The count becomes a spin, by another theorem with the same input #

The question the section above leaves open — what electron repulsion does to a set of half-filled zero levels — is not answerable inside this model, and it is answerable. There is a theorem for exactly this case, it takes the same colouring as its input, and it turns a count of levels into a magnetic moment.

For a connected bipartite structure at half filling, with a repulsion paid whenever two electrons share a site, the ground state’s total spin is

$$S = \tfrac{1}{2}\,\bigl|N_A - N_B\bigr|$$

for any repulsion greater than zero, however small. The right-hand side is the sublattice imbalance this essay counts. The left-hand side is a magnetic moment.

So the colouring argument does more than say how many levels sit at zero. It says how many unpaired spins the material has, and it says so without a calculation, without a parameter and without any dependence on how strong the repulsion is — the same insensitivity that makes the zero levels themselves untunable.

The prediction is not abstract. Remove one atom from a honeycomb sheet and the two sublattices differ by one, so the imbalance is one, the zero-level count is one, and the spin is one half — a magnetic moment on a defect in a material made entirely of carbon, which has no partly filled d shell anywhere in it. That is observed: single vacancies in graphene carry a local moment, seen directly with a scanning probe and in transport measurements, and the moment is a property of the missing atom rather than of anything added.

It also explains a rule that looks arbitrary from the outside. Two vacancies on opposite sublattices leave no imbalance, so no zero levels and no moment; two on the same sublattice leave an imbalance of two, two zero levels and a spin of one. Two defects the same distance apart, differing only in which colour of site was removed, are magnetic or not — and no measurement of the separation distinguishes them.

Which is the sharpest available statement of this essay’s own theme. An impurity’s level depends on a strength and a vacancy’s depends on a colouring, and the colouring survives being asked a many-electron question that the strength could not have survived.

What is left #

Three limits, stated rather than glossed.

The model is a graph with equal bonds and no repulsion, and the insulator band theory cannot see is the standing account of what the second omission costs. A zero-energy level half-filled by one electron is exactly the situation where repulsion between electrons matters most, and nothing here can say what happens to it.

The structure is bipartite, which is a strong condition and a common one — a chain, a square net, a honeycomb, a cubic structure and every alternant hydrocarbon are bipartite; a triangular net and a five-membered ring are not. In a structure with odd cycles the theorem has nothing to say and the levels a vacancy produces are computed rather than counted.

And the amplitudes are unnormalised in one specific respect: a zero state is only determined up to mixing with the other zero states when there is more than one, so the picture of a two-vacancy state above is one member of a two-dimensional space rather than the state. What is basis-independent is the count, the position and the colour — which is what has been checked.