Two defects, and the level between them

A single site with an energy different from its neighbours pulls a level out of the band and holds a state that decays away from it. That is a defect is a level in the gap, and it is the whole of what one impurity does.

Two impurities do something a single one cannot, and the something is what makes doping work — the process by which a solid is a molecule that did not stop acquires the carriers that make it useful.

The isolated level, exactly #

Take a chain with equal hopping between neighbours, and give one site an energy $h$ below the rest. The bound state’s energy has a closed form:

$$E = -\sqrt{h^2 + 4t^2}$$

For $h = -2$ and $t = 1$ that is $-2\sqrt{2} = -2.828427$, and diagonalising a chain of sixty-one with one such site returns exactly that.

The state is localised: its amplitude falls away from the impurity by a constant factor per site. That factor, $\kappa$, follows from the energy alone. A state at energy $E$ outside a band of half-width $2t$ decays with $\kappa + 1/\kappa = |E|/t$, so

$$\kappa = \frac{|E| - \sqrt{E^2 - 4t^2}}{2}$$

which for $E = -2.828427$ gives $\kappa = 0.414214$. That is $\sqrt2 - 1$, and it will reappear in a moment from a completely different calculation.

Two impurities share a pair of levels #

Now put two impurities of the same strength in the same chain, a stated number of sites apart, and diagonalise again.

Exactly two levels come out below the band, and they are not degenerate. They are the symmetric and antisymmetric combinations of the two bound states, split by an amount that measures how much the two overlap — precisely the two-level problem overlap decides sets up for two atoms, arriving here for two defects in a solid.

Two things happen at once as the pair separates.

The splitting collapses geometrically. Each extra site of separation multiplies it by about $0.414$. From two sites to fourteen the splitting falls by a factor of forty-three thousand, and the fall is a straight line on the logarithmic axis of the figure — which is the signature of a geometric decay rather than a power law, and is the distinction that decides whether two defects a long way apart interact at all.

The two levels close on the isolated value. Their mean approaches $-2.828427$, which is $-2\sqrt2$ — so far apart, each impurity behaves as if the other were not there, which is the definition of isolated made quantitative.

The count of levels outside the band #

Before the splitting is worth measuring, the number of levels to measure has to be right, and that number is a check rather than an observation.

Two impurities pull exactly two levels out of the band, at every separation from two sites to fourteen. Not one, not three. It is worth checking at every separation, because a count of the levels pulled from the band catches a badly conditioned diagonalisation long before any physics is at stake.

The count follows from the structure of the problem: each impurity supplies one bound state, the two mix, and mixing two states gives two. It survives the strong-coupling case where the two are on top of each other and the weak-coupling case where they cannot feel each other, which is why it is a good invariant to check against.

It also fails informatively. Weaken the impurities enough and one or both levels are drawn back inside the band, at which point the count drops — and the strength at which that happens is exactly what the end is the hardest place to bind measures, with an end site needing a full β and a middle site needing nothing at all in the limit of a long chain.

How localised is localised #

The participation ratio answers the question the word localised leaves open: on how many sites does the state actually live?

For the single impurity at $h = -2$ in a chain of sixty-one, it is $1.89$. The state occupies about two atoms — the impurity and a little of each neighbour — and there is essentially nothing of it four sites away.

That figure and the decay constant are the same statement twice. A state falling as $\kappa = 0.414$ per site has squared amplitudes falling as $0.171$, so a geometric sum gives an effective extent of about two sites. And it is why the splittings above collapse so fast: two states each living on two atoms, four atoms apart, have almost nothing to overlap with.

A shallower impurity spreads further, and that is the case worth drawing, because it is the one a real dopant is.

The decay factor, predicted rather than fitted #

The ratio of successive splittings, measured from the last two points, is $0.414213$. The decay constant computed from the isolated bound state’s energy is $0.414214$.

Those are two routes with nothing in common. One is the ratio of two eigenvalues of a $61 \times 61$ matrix at two different separations; the other is a root of a quadratic in the energy. They agree to six figures, and the agreement is checked rather than merely displayed.

The reason they must agree is worth stating, because it explains what the splitting is. Two localised states overlap by an amount proportional to the tail of one evaluated where the other is centred; the tail falls as $\kappa$ per site; so the overlap, and with it the splitting, falls as $\kappa^d$ for separation $d$. The energy of one state fixes how fast its tail decays, so the energy of one impurity predicts the interaction between two.

A deeper impurity interacts over a shorter range. Making $h$ more negative deepens the level, which makes $|E|$ larger, which makes $\kappa$ smaller — so a strongly bound state is a tightly held one and reaches less far.

Why this is what doping is #

A semiconductor doped at one part in $10^6$ has impurities perhaps a hundred lattice spacings apart on average. A semiconductor doped at one part in $10^3$ has them ten apart.

At a hundred spacings the splitting between any two impurity levels is $\kappa^{100}$, which for the values here is about $10^{-38}$ — zero for every practical purpose. Each impurity has its own level at its own energy, and they do not talk to each other.

At ten spacings the splitting is $\kappa^{10} \approx 1.5\times10^{-4}$ of the hopping. Still small, but with a great many impurities each interacting with several neighbours, the isolated level broadens into a band of states.

So the doping concentration at which a set of levels becomes a band is set by an exponential, and an exponential moves fast. Changing the concentration by a factor of ten changes the separation by a factor of about two, and squaring an exponentially small number is a much smaller number. That is why the transition from isolated donors to an impurity band happens over a narrow range of concentration rather than gradually.

Nothing here computes where that transition falls for any real material, and it should not be read as trying to. What the model supplies is the mechanism and the functional form: geometric decay of the state, geometric decay of the interaction, and a threshold that is therefore sharp.

How strong an impurity has to be before it binds anything at all has an answer that depends on where it sits: a defect is a level in the gap computes the threshold at three chain lengths and finds an end site converging on exactly 1β and a middle site on zero. Everything in this essay assumes an impurity comfortably past whichever threshold applies to it, which is why the shallow case above is at 1.2 rather than at 1.0.

What happens when the pair is close #

The two-site case in the table is worth a second look, because it does not behave like a perturbed pair of isolated levels.

At two sites apart the levels are $-2.500$ and $-3.035$, and their mean is $-2.768$ — not the isolated $-2.828$. The two impurities at that distance are not two impurities; they are one defect complex two sites wide, with its own bound states.

The lower state is the symmetric combination, concentrated between the two impurities and below what either could produce alone; the upper is antisymmetric, with a node between them, and lies above. That is the same structure as a bonding and antibonding pair, which is what it is.

The transition between the two regimes is smooth and it has no particular place. At six sites apart the mean is $-2.828283$ and the splitting is $0.0143$; by ten the mean is right to six figures. Isolated is a limit approached geometrically rather than a state of affairs that begins somewhere.

The −2.500 in that pair is a number the chain produces in one other place, and the coincidence is not one. Put the same impurity at the end of the chain, where it has one neighbour instead of two, and its level comes out at −2.500 as well. A site with a neighbour missing and a site whose neighbour is occupied by another impurity are short of the same thing, and the arithmetic does not distinguish them.

The band the levels come out of #

Nothing above says anything about the band itself, and the band is what makes the levels visible as levels.

The band those levels come out of is the ordinary one: a uniform chain of two thousand has its density of states given by 1/(π√(4−x²)), drawn in where the states pile up, and every level of it lies between −2β and +2β. The bound states in this essay sit below −2β, outside that interval, which is the whole of what “outside the band” means and is why they are visible as separate levels rather than as features of a distribution.

A state at $-2.828$ is $0.828$ below the band edge at $-2$, and the edge itself is a computed quantity rather than a boundary drawn on a diagram — the width of a band is a count of neighbours derives where it falls. That gap is what makes the state localised: inside the band, a state at any energy has propagating solutions to mix with and cannot stay put, while outside it there is nothing to mix with and the amplitude must decay.

The decay constant $\kappa$ is a measure of how far outside the state is. A level just barely outside the band has $\kappa$ close to $1$ and decays slowly, spreading over many sites; a deep level has $\kappa$ small and sits on one or two. That is the same relation as the one between a bound state’s energy and its extent in any potential well, arriving here as arithmetic on a tridiagonal matrix.

The background they sit against tightens as the chain grows — a solid is a molecule that did not stop draws uniform chains at three sizes, with the spacing near the middle of the band falling as the chain lengthens — so a bound state below the band edge becomes easier to distinguish from the band, not harder, as the system gets larger.

What the model leaves out #

Three things, each of them large.

There is no repulsion. Two electrons on the same impurity cost nothing here, and in a real doped semiconductor that cost is the reason a donor level is singly occupied and the reason an impurity band can be a Mott insulator rather than a metal. The insulator band theory cannot see is the essay about what that omission costs.

There is no lattice and no dimension. This is a chain, and this collection’s solids field has no lattice anywhere in it by design — a band with no structure in it says what that omission costs. Real impurities sit in three dimensions where the number of neighbours at a given distance grows and the statistics of separations matter. The functional form survives; the counting does not.

The impurity is a change of one number. A real substitutional impurity changes the site energy, the hopping to its neighbours, and the positions of the atoms around it. Only the first is here.

For contrast there is one other way a level can appear where none should be, and it is a different mechanism entirely: a chain cannot stay even plots a uniform chain’s highest-occupied to lowest-empty spacing falling towards zero with size against an alternating chain’s settling on a finite gap. That is a gap made by the structure, shared by every site; the levels here are made by two sites and live on them.

What the geometric decay predicts about a real doped crystal #

The finding here — that the interaction between two localised states falls by a fixed factor per site, at a rate their own energy fixes — is the ingredient a much-quoted criterion is built from, and it is worth following through to a number, because the number is checkable against a measured transition.

The argument is a competition. Two impurity states that barely overlap give two levels split by an amount too small to matter, and an electron sitting on one of them stays there. Make the impurities closer and the splitting grows geometrically. When it grows past the energy it costs to put two electrons on one impurity, an electron can move, and the material stops being an insulator.

Turning that into a concentration takes one dimensionless number. Both the splitting and the repulsion scale with the same length — the radius of the bound state — so the criterion cannot depend on anything but the ratio of the impurity spacing to that radius, and the transition occurs at

$$n_c^{1/3}\, a^* \approx 0.26,$$

with $n_c$ the impurity concentration and $a^*$ the bound state’s radius. The constant is the only fitted thing in it, and it was fitted once, across materials spanning nine orders of magnitude in $n_c$.

Silicon doped with phosphorus is the standard test. The radius of the donor state depends on which estimate of the binding energy is used, and both are available: the effective-mass value of 26 millielectronvolts gives a radius of 23.6 ångström, and the measured 45 millielectronvolts gives 13.7. Putting each into the criterion gives

and the measured transition in phosphorus-doped silicon is at $3.7 \times 10^{18}$ — between the two, and within a factor of three of either.

That is a good deal better than the argument deserves. Everything about it is crude: a hydrogenic bound state, a single fitted constant, no disorder in the impurity positions, and a competition sketched rather than solved. What survives is the structure of it, and the structure is what is measured here — an overlap that falls geometrically with separation, against an energy that does not.

It also says which of the two quantities carries the uncertainty. The criterion’s constant is known to a few per cent across many materials; the radius is not, and the factor of five between the two predictions above is entirely a factor of 1.7 in the radius, cubed. A criterion that depends on the cube of a length is a criterion that needs the length to three times the accuracy of its answer, which is why the useful form of it is a bracket rather than a value, and why the measurement sitting inside the bracket is the honest way to report the agreement.

One further consequence is worth drawing out, because it is what makes the transition sharp rather than gradual. The splitting measured in this essay falls by a constant factor for every site of separation, so the coupling between two impurities is exponential in their distance while their number density is only a power of it. An exponential against a power law crosses once and crosses steeply: a small change in concentration changes the typical coupling by orders of magnitude, which is why doping a semiconductor a little more heavily does not make it a little more conducting near the threshold but takes it across a transition. The geometric decay measured here is the reason the criterion has a value at all rather than describing a slow crossover.

Who found it, and when #

The single-impurity problem in a tight-binding chain is old and has been rediscovered repeatedly; the closed form above is elementary enough to appear as an exercise. Its physical importance dates from the 1930s work on colour centres in alkali halides, where a missing ion traps an electron in a level within the gap and the crystal acquires a colour.

The pair problem is the beginning of a much larger story. Nevill Mott’s argument that a set of impurity levels becomes metallic at a critical concentration — the Mott transition, with its criterion relating the concentration to the effective Bohr radius — is from 1949 onward and rests on exactly the competition sketched here, between the overlap that broadens the levels into a band and the repulsion that keeps them apart. Philip Anderson’s 1958 work on localisation in a disordered chain is the other descendant: with impurities of random strengths rather than two identical ones, the states stay localised however much they overlap.

Neither of those is solved here, and both need what this model omits — repulsion for the first, disorder and much larger systems for the second; the section above takes Mott’s criterion as far as a scaling argument and a measured transition, which is not a solution of it. What is supplied here is the piece both rest on: the interaction between two localised states falls geometrically with their separation, at a rate their own energy fixes.

The same shape wherever localisation appears #

A single impurity gives one level, and the end of a chain turns out to be the hardest place to bind a state. Two impurities in one chain show that the isolated answer is a limit rather than a fact.

That pattern — a quantity that is exact for one object and approached geometrically when there are two — recurs wherever localisation does. It is the same shape as the ring-against-chain difference falling as $1/n$, and the same shape as the overlap between two atomic orbitals falling exponentially with distance. The subject keeps producing it because the underlying equations keep producing exponentials outside a band and power laws inside one.