Two different lattices never coincide, and the question becomes how nearly

Everything in this field so far has been decidable. A form either closes or it does not; a twin law either is a symmetry of the lattice or it is not; a rotation either produces a coincidence lattice or it produces one shared point. Every question has been settled by integers with no tolerance anywhere, which has been this site’s whole method from the first essay onwards.

This rung is where that ends, and it ends for a reason that has nothing to do with the quality of the machinery.

Grow gallium arsenide on silicon. Their cubic cells are 5.653 Å and 5.431 Å, and the ratio is about 1.041. That ratio is not rational. No physical ratio is: a lattice parameter is a real number known to a few parts in ten thousand, and the rationals it is near are near it by accident and in unlimited supply.

Exact coincidence therefore does not exist between two different substances, and the question has to be replaced with a different one — exactly as the coincidence argument required rationality and got it for free when both lattices were the same.

The replacement question #

Given a ratio r, find integers p and q with p/q close to r and q not too large. The film then has p cells for every q of the substrate, the two are nearly in register at those points, and the leftover

$$f = \frac{p/q - r}{r}$$

is the misfit — the fractional strain the film has to carry to make the match exact.

That is a well-posed question with a computable answer, and the answer is a best rational approximation, which is the subject of continued fractions and has been since Euclid. What it is not is a decision. Two things have to be supplied from outside the arithmetic: how large a q is acceptable, and how large an f is acceptable. Neither is in the numbers. And they interact: a longer repeat always allows a smaller misfit, so the two are traded against each other, and where the trade lands depends on how far the film can maintain a correlation. A repeat of four cells is a structural feature; a repeat of forty is a statement about a coincidence that no atom in the film is in a position to notice.

What the film actually does #

The physics divides into two regimes and the boundary between them is the whole of thin-film growth.

Pseudomorphic growth. A thin film simply strains to match the substrate exactly, adopting the substrate’s spacing in the plane and adjusting perpendicular to it. That is a change of the film’s metric, and every interfacial angle in the strained film would report it. Every atom is in register, the interface is perfect, and the film carries an elastic energy proportional to f² and to its thickness.

Relaxed growth. Past a critical thickness the elastic energy exceeds what it costs to introduce misfit dislocations — line defects at the interface, one every 1/f cells, each absorbing one lattice vector of the accumulated mismatch. The film then relaxes to its own spacing and the mismatch lives in the dislocation array.

The critical thickness scales roughly as 1/f, so a misfit of a per cent gives a few tens of nanometres and a misfit of four per cent gives a few. Gallium arsenide on silicon is four per cent and relaxes almost immediately, which is why that particular combination — enormously desirable, for obvious reasons — is difficult.

None of that is symmetry. It is elasticity against dislocation energy, and the number that decides it is a modulus. What the arithmetic supplies is f.

Domain matching, where the arithmetic returns #

There is a middle regime, and it is where the coincidence idea comes back in a weakened form.

Instead of matching one cell to one cell, match m cells of the film to n cells of the substrate. The interface is then periodic with a repeat of n substrate cells, containing one extra or one missing film cell per period, which is a periodic array of dislocations built in by design rather than arriving by relaxation.

This is domain matching epitaxy, set out as a general scheme by Narayan and Larson in 2003, and it makes combinations work that one-to-one matching would rule out. Take a ratio of 1.28, a twenty-eight per cent mismatch and hopeless one-to-one. Five film cells on four substrate cells leaves 2.3 per cent; nine on seven leaves 0.4 per cent. A film that could not grow coherently at all in the naive sense grows with a periodic dislocation array and a residual strain small enough to live with.

The choice of m and n is exactly the best-rational-approximation problem, and here the answer is used rather than merely reported. The coincidence argument survives the loss of exactness by moving up a scale: what coincides is not the lattices but a superlattice of them, and the index of that superlattice plays the part Σ played on the previous rung.

What is different is that the superlattice is only approximate. A Σ3 grain boundary has one atom in three exactly shared; a four-on-five domain match has one atom in five nearly shared, with a residual that has to be strained away. The arithmetic is the same shape and the verdict has an error bar on it.

The worst possible match #

If the best rational approximations are what matters, it is fair to ask which ratio is hardest to approximate. The answer is famous and it lands this field back on the one next door.

A continued fraction approximates well when it has large partial quotients — a large denominator arriving early buys a great deal of accuracy. The number that approximates worst is the one whose partial quotients are all as small as possible, which means all equal to one, and that number is the golden ratio.

So a film-to-substrate ratio of φ is the least accommodating value there is, in a sense that is worth stating carefully because the loose version of it is false. Every rational approximation to φ is as poor as an approximation of that denominator can be — the quantity q² times the error settles at one over the square root of five, the highest floor any irrational holds, where root two settles at one over two root two and is therefore approximated better for the same length of repeat. What is not true is that φ demands a longer repeat than any other ratio at a stated tolerance. A ratio that happens to sit near a low-denominator fraction beats it easily at small denominators: 1.042 is 521/500, and 25/24 brings it inside a third of a per mille at twenty-four cells, which φ cannot manage until thirty-four. The golden ratio is the worst value for length, not the longest wait.

That is the same φ that runs the Fibonacci chain and the inflation of a Penrose tiling, and it is the same property doing the work in both places. A quasicrystal is aperiodic because its two spacings are in the least-well-approximable ratio available, so no finite repeat ever brings them back into step — and an epitaxial system at that ratio is the least matchable there is, for exactly the same reason.

Two subjects that could hardly look less alike, and one continued fraction underneath both.

Continued fractions, which are the right tool #

The best rational approximations to a real number are not found by searching. They are the convergents of its continued fraction, and every one of them is better than any fraction with a smaller denominator — a theorem, and the reason continued fractions are the right machinery for this question rather than merely a way of doing it.

Write r as an integer plus a remainder, invert the remainder, repeat. For 1.041 that gives 1 + 1/(24 + 1/(2 + …)), and the convergents are 1/1, 25/24, 51/49 and so on. The large partial quotient of twenty-four arriving second is what makes 25/24 unusually good: a denominator of twenty-four brings the misfit to a few parts in ten thousand, where a general ratio would need a denominator of hundreds.

That is the arithmetic behind the jumps in the curve above. The misfit as a function of allowed repeat length is flat and then drops, flat and then drops, and the drops are at the records — the repeats that get nearer than every shorter one. Those are not quite the convergents. They are the best approximations of the first kind, which are the convergents together with certain semiconvergents lying between them, and at a ratio of 1.042 four of the eight thresholds below are crossed at a semiconvergent rather than at a convergent. The convergents are where the unusually good matches are; the semiconvergents fill in between. A search over every denominator up to some bound finds the same answers and does not explain them.

It also gives the rule for how large the jumps are. A partial quotient of a is worth roughly a factor of a² in accuracy at that step, so a ratio with a large partial quotient early has one excellent match and a ratio with only small ones has none — which is the golden ratio’s problem, stated in advance.

How far apart the dislocations sit #

The relaxed regime is described above as the one where misfit dislocations appear, and the arithmetic of how many is short enough to do and is the same best-approximation arithmetic in another guise.

A film relaxed onto a substrate has its own spacing back. The two lattices then drift out of step at a rate given by the fractional mismatch, and they come back into step whenever the accumulated drift is a whole lattice vector. So the register repeats every 1/f planes, with f the mismatch, and each repeat needs one extra plane on one side than on the other — which is a dislocation.

The numbers are unforgiving. A mismatch of one per cent puts a misfit dislocation every hundred planes; four per cent puts one every twenty-five; ten per cent puts one every ten, which is close enough together that their strain fields overlap and the interface is better described as a new structure than as a strained one.

That is the same jump-rather-than-drift behaviour the curve above shows. A ratio near a simple fraction has a long repeat and few dislocations; a ratio near nothing simple has a short repeat and many. And the spacing is not a fitted quantity: it is the denominator of the best approximation, measured in planes, and it is what a micrograph of an interface shows directly as a periodic array of dislocation cores.

Three ways a film can grow #

The essay’s three regimes are about mismatch. There is a second and independent classification of what a film does, it is about energies rather than lattices, and the two together decide what actually happens.

Layer by layer. If the film wets the substrate — if the substrate’s surface energy exceeds the film’s plus the interface’s — each layer completes before the next begins, and the film grows flat. That is the mode epitaxy wants.

Islands. If the film does not wet the substrate, material gathers into three-dimensional islands from the start, leaving bare substrate between them. No amount of lattice matching prevents it, because the decision is made by surface energies that need not have anything to do with the spacings.

Layers and then islands. The mixed case is the common one, and it is where the two classifications meet: the first few layers wet the substrate and grow flat while strained, the accumulated strain energy grows with thickness, and past some point islanding becomes cheaper than continuing flat. The film relieves its mismatch by roughening rather than by dislocating.

That third mode is why the arithmetic on this page is necessary and not sufficient. A perfectly matched pair can still island, if the wetting is wrong; a badly matched pair can grow flat for a few layers, if the wetting is right and the thickness is small. The lattice decides what a coherent interface would cost, and something else entirely decides whether the film builds one.

Where this leaves the field #

The applied field opened with a question about crystal faces that was settled by integers, and it closes with a question about interfaces that is not settled at all. That progression is worth stating plainly, because it is the honest shape of what a symmetry argument buys.

Decidable, exactly. Which faces a class requires together. Whether a form closes. How many twin laws a class has on a given lattice. How many domain states a transition produces. Which rotations of a lattice give a coincidence, and at what index. Every one of these has an answer in integers, with no threshold and no measurement.

Not decidable, and honestly so. Which faces a crystal will grow. Whether it will twin. How large the domains are. Whether a boundary is cheap enough to be common. Whether a film will grow coherently. Every one of these needs an energy, and no group has one.

And in between, the tolerance cases, which are the interesting ones. Pseudo-merohedral twinning, near-symmetry, domain matching: places where an exact criterion is nearly satisfied and whether that counts is decided by an instrument or by an elastic modulus rather than by the arithmetic. The near-symmetry essay named the pattern; this field has now met it four times.

Three regimes, and which one a pair falls into #

The framework has a shape worth stating compactly, because the three cases are governed by three different quantities.

Small mismatch — under about one per cent. Pseudomorphic growth to substantial thickness. Aluminium arsenide on gallium arsenide is 0.14 per cent, and layers of it grow to any thickness anybody needs. The relevant quantity is the misfit itself.

Moderate mismatch — a few per cent. Pseudomorphic below a critical thickness, relaxed above it, and the interesting things happen at the boundary. Indium arsenide on gallium arsenide is about seven per cent and relieves its strain by breaking into islands rather than by dislocating, which is how self-assembled quantum dots are made. The relevant quantity is the critical thickness, and the mechanism of relief is a competition between surface energy and strain energy.

Large mismatch — tens of per cent. No coherent one-to-one growth at all, and domain matching or nothing. The relevant quantity is the best rational approximation at a repeat length the interface can sustain.

Each regime uses a different piece of the arithmetic, and in none of them does the arithmetic decide the outcome by itself. What it does in all three is say what the options are, which is the same service the coset counting performed for twins and domains.

What a boundary between two substances actually looks like #

It is worth ending on a picture rather than a principle.

An interface between two different crystals has, in general, no shared points at all. It is not a defect and it is not a mistake: it is two lattices that have no reason to agree, brought together by a growth process that cares about chemistry rather than about arithmetic. What holds it together is bonding across the interface, and the mismatch is accommodated by strain, by dislocations, by a rearranged layer or two, or by all three.

The strain that accommodates a mismatch is not free of consequences elsewhere, either. A film strained in the plane is stretched or compressed perpendicular to it by an amount its elastic constants decide, and those constants are themselves a property symmetry constrains — twenty-one independent components in a triclinic crystal and three in a cubic one. So the response to a lattice mismatch is filtered through the same classification that decided which properties the material could have at all.

The remarkable thing is not that such interfaces are difficult. It is that so many of them are good — that a semiconductor laser has a dozen interfaces between different alloys, every one of them coherent, and works for years. That is an achievement of composition control rather than of symmetry: the alloys are tuned so that their lattice parameters match, which is a chemical solution to an arithmetic problem.

Where the field goes next #

Four anchors have been built on one observation: the questions mineralogy, structural crystallography, ferroelectrics and metallurgy each developed separately are all questions about orbits and cosets of groups this site already had.

What has not been touched is the interior of a crystal — the defects that are not interfaces, the dislocations that carry plastic flow, the point defects that carry diffusion, and the modulated structures whose periodicity is real and irrational. Each of those has a symmetry description, and two of them need a group of a kind this site has not built — the same honest boundary the cut-and-project construction drew when it needed six dimensions to index a five-fold pattern: dislocations are classified by the fundamental group of the space around them, and modulated structures need superspace, which is a periodic group in more dimensions than the crystal has.

Both are proper extensions rather than applications, and both would need a fourth round trip before an essay here could claim anything about them.