The end is the hardest place to bind

Surfaces are where almost everything interesting about a solid happens — catalysis, corrosion, adhesion, every measurement made with a beam that does not penetrate. So it matters whether the electronic structure of a surface is special, and in what way.

An atom at the end of a chain has one neighbour where every other atom has two. It is undercoordinated, its bonding is incomplete, and every intuition says it should be the most special place in the structure — the easiest place to trap something, the most reactive site, the obvious home for a state outside the band.

Half of that is right. The end site is special, and it is the most reactive place. But it is the hardest place to bind a state, not the easiest, and the difference is not marginal: a site in the middle binds for free and a site at the end demands a full $\beta$.

The measurement #

The procedure is a bisection. Take a chain, raise one site’s energy by $h$, diagonalise, and ask whether any level has left the band. Then search for the smallest $h$ for which the answer is yes.

Run at the middle of the chain, the answer is $0.098$ for forty sites, $0.049$ for eighty and $0.025$ for a hundred and sixty. Each doubling halves it, so this is the chain’s own level spacing rather than a threshold, and the true value is zero.

Run at the end, the answer is $1.025$, $1.013$ and $1.006$. It falls, but it is converging on one rather than on zero, and it approaches from above.

Two sites in the same chain, differing only in position, and one of them requires forty times what the other does at the largest size tested — with the ratio growing without limit as the chain lengthens.

The bisection itself is worth a sentence, since it is the whole measurement. The test at each candidate strength is simply whether any eigenvalue lies outside the interval the unmodified chain occupies, which is a question with a yes or a no and no fitting in it. Forty halvings of the interval from zero to three give the answer to better than a part in ten thousand, and the cost is forty diagonalisations.

Why the intuition fails #

The intuition comes from thinking of the end site as a place with something missing, and therefore as a place where an electron has less to hold it. That reasoning would be right if the question were about binding energy. It is the wrong question.

The question is whether a state can be pushed out of the band, and what a band’s edges are is decided by the whole system. Here is the difference.

A state bound at a site in the middle has to compete with a band whose top level is at $2\beta$ — the state that has the same amplitude and the same sign on every site, which is the most bonding arrangement available. Beating it requires only that the modified site be modified at all, because the modified site’s own energy adds directly to whatever the band offers.

A state bound at the end has to compete with the same band, but the end site itself contributes much less to the topmost band state. In a chain the top state’s amplitude tapers to nothing at the ends — that is what having ends does — so the end site is already, in a sense, weakly represented at the band edge. The state trying to localise there starts from a weaker position, and it takes a full $\beta$ of extra site energy to make up the difference.

Put in one sentence: the end is where a state is least able to borrow from the band, which makes it hard to bind rather than easy.

The same statement from the other end #

There is a second way of seeing it that some readers will find more convincing, and it turns on what the unmodified chain already does at its ends.

A uniform chain’s top level is at $2\cos(\pi/(n+1))\beta$, slightly below the band edge, and the state carrying it is a half-wave: zero at both ends, largest in the middle. The next state down is a full wave, and so on. Every one of them vanishes at the ends, because the ends are where the chain stops and there is nowhere for amplitude to be supported from.

So the ends of an unmodified chain are already places where every state is small. A defect works by taking a state and concentrating it somewhere; at the end, the states available to concentrate are the ones that were smallest there to begin with.

The middle is the opposite. The topmost band state has its largest amplitude in the middle, so a defect there is intervening at the point where the band is strongest and gets the most return for a given change.

That reading also explains the direction of the convergence. As the chain lengthens, the ends become a smaller fraction of it and the taper becomes sharper relative to the chain’s length, so the disadvantage of sitting at the end becomes cleaner and the threshold settles on its limiting value from above rather than below.

The value is exactly one, and that is a known result #

The threshold converging on exactly $1\beta$ is not a coincidence of these particular chains. It is the condition Tamm derived in 1932 for a surface state in the simplest model of a semi-infinite chain, and the fact that it comes out here to three decimal places from a $160 \times 160$ matrix with nothing about surfaces in it is the sort of thing worth showing.

The convergence is worth noticing too: $1.025$, $1.013$, $1.006$, halving its excess with every doubling. So the finite chain’s threshold approaches the semi-infinite one as $1/n$, from above, which is the same $1/n$ end effect that governs the energy per site — where a molecule stops being one works through why per-site quantities converge at that rate.

The check requires both behaviours and is written so that swapping them fails. An end threshold falling towards zero is refused by one test, a middle threshold settling on any finite value by the other. Neither test could pass by accident.

What it means for real surfaces #

The threshold is a real condition, and real surfaces mostly satisfy it — which is why surface states are common rather than rare. But the fact that there is a condition at all changes how the subject should be thought about.

A missing neighbour does not by itself produce a state in the gap. It produces one if the surface atom’s energy differs from a bulk atom’s by more than about a $\beta$, and whether it does is a chemical question about that particular surface. A cleaved silicon surface has dangling bonds whose energy differs enormously and duly has states deep in the gap; a surface that has reacted with hydrogen has those bonds satisfied and very few.

This is why surface passivation works. Attaching something to a dangling bond brings the surface atom’s energy back towards the bulk value, and once it is within a $\beta$, the state it carried retreats into the band and stops being available. Hydrogen on silicon, an oxide layer, a ligand shell on a nanocrystal — all of them are ways of moving a number back below a threshold.

And it is why surface reconstruction happens. A surface with states in the gap is a surface with electrons in high-energy levels, and rearranging the atoms to pair those states up lowers the energy. Silicon’s surfaces famously do this, and the driving force is exactly the one this argument identifies.

The chemistry of a surface, in the model’s terms #

It is worth translating the whole result into the vocabulary a chemist uses for surfaces, because the two accounts are usually kept apart and are the same account.

A dangling bond is an orbital on a surface atom that has no partner. In this model it is the end site with an energy that differs from the bulk, and the energy differs because the orbital that would have overlapped a neighbour is instead pointing at nothing. Whether it produces a state in the gap is the threshold question.

Passivation is giving that orbital a partner — a hydrogen, an oxygen, a ligand. The site’s energy returns towards the bulk value, and once the difference falls below a $\beta$ the state retreats into the band.

Reconstruction is the surface atoms rearranging so that their unpaired orbitals pair with each other rather than with an adsorbate. Silicon’s (100) surface does it by forming dimers, so two dangling bonds become one bond and one antibonding level, and the pair of gap states becomes one filled and one empty level further apart. That is a Jahn–Teller argument on a surface, and it is the same argument a chain cannot stay even makes about a bulk chain.

Adsorption is the same operation performed by something arriving from outside, and the energy released is what makes the surface catalytically useful. That energy is the $0.71\beta$ of missing binding rather than anything to do with gap states, which is the separation the next section is about.

The reactivity, which is a different question #

None of the above contradicts the observation that surface atoms are reactive, and it is worth separating the two claims carefully because they concern different quantities.

Reactivity is about how much energy is available from making a new bond, and an undercoordinated atom has a great deal — the two ends of a chain are worth about $0.71\beta$ between them in missing binding, as a solid is a molecule that did not stop measures. That number is a property of the ground state and it does not shrink as the chain grows.

Binding a state in the gap is about whether a level can be pushed outside a continuum, and it is a property of the spectrum rather than of the total energy.

Those are different questions with different answers, and conflating them is the source of the intuition this essay is about. An end atom is energetically hungry and spectroscopically hard to distinguish, at least until something is done to it.

The surface state this chain cannot have #

The threshold measured here says a surface site must differ from the bulk by a whole β before it holds a state. That is a real result about a real mechanism, and it is not the only mechanism, which matters because clean metal surfaces carry states without any impurity on them at all.

The state computed here is of the kind that requires the surface atom to be different — a changed site energy, from a missing neighbour or an adsorbed atom — and it is that requirement which the threshold prices. There is a second kind that requires nothing to be different. It arises from the truncation alone: a chain cut at a particular place, with every site energy identical to the bulk’s, can still support a state in a gap, because the boundary condition at the cut cannot be satisfied by any propagating solution and a decaying one appears instead.

That mechanism needs a gap to put the state in, and a gap needs more than one orbital per site. A chain with one orbital per site has a single band and nowhere for such a state to live, so this model cannot host one by construction — which is why the threshold measured above is a complete answer within the model and a partial one about surfaces.

The distinction is not academic. The states seen on clean, unreconstructed noble-metal surfaces — the ones measured routinely by angle-resolved photoemission on copper, silver and gold — are of the second kind. Nothing about those atoms differs chemically from the bulk; the surface exists, the band has a gap in the right place, and a state appears in it for free.

So the honest summary has two clauses. Where a surface state must be paid for by a changed site energy, the end of a chain is the most expensive place in the structure, which is the measurement and it is exactly one β. And where a surface state comes from the termination rather than from a difference, no payment is required at all, and whether that is available is decided by the band structure rather than by the chemistry — which is the one property not computed here.

The intuition that gets it backwards #

The result is easy to write down backwards, and the backwards version is worth stating because it is the argument for computing things.

The natural expectation is that a middle site would need about one $\beta$ because it has two neighbours holding it in place, and an end site would need nothing because it is already special. That is a reasonable-sounding argument, it is widely held, and it is exactly wrong in both halves.

The bisection returns $0.098$, $0.049$, $0.025$ for the middle and $1.025$, $1.013$, $1.006$ for the end. A check that requires both behaviours, in the directions measured, is what keeps the result from quietly reverting to the intuition.

Nothing about the backwards version would show up anywhere else. A figure drawn from the arithmetic would be correct, its labels would fit, the numbers on it would be right — because the numbers are always whatever the arithmetic produces — and only the sentence describing them would be wrong, which is the failure mode that survives everything except reading the words against the calculation.

That is the general shape of the habit worth keeping. A claim is stated, the claim is computed, and a claim that disagrees with the arithmetic is refused. What that cannot protect against is a claim nobody thought to test, which is why the interesting work is deciding what to check.

What the two thresholds have in common #

The pair of results — zero in the middle, one at the end — is worth holding as a single statement rather than as two facts.

Both are answers to the question how much must this site differ before a state belongs to it rather than to the whole system, and both are decided by how much the site was contributing to the band edge in the first place. A middle site contributes fully, so any change makes it stand out. An end site contributes little, so it has to overcome its own weak position first.

The general form applies past this model: the threshold for localisation is set by what the site was already doing, not by what it is missing. That is why the interior of a crystal is easy to dope and a surface is not, why an impurity next to a vacancy behaves differently from one in clean material, and why any statement of the form “undercoordinated sites bind things” needs to say which quantity is meant.

The result also transfers past this model, which is worth saying because a one-dimensional chain is a thin thing to draw a conclusion from. The mechanism — a site’s ability to hold a state being set by how much it contributed to the band edge — has nothing one-dimensional in it, and the corresponding statement for a real surface is that a surface state requires the surface atom’s energy to differ from the bulk by more than the amount the surface atom was contributing. The threshold in three dimensions is a different number and the structure of the argument is the same.

It is also a compact demonstration of the value of computing rather than reasoning. The intuition here is strong, widely held and backwards, and nothing but the arithmetic would catch it — the first version anyone writes down claims the opposite, and the bisection refuses it.