The hydrogen atom inside a crystal

What happens when the wells get close found that bringing atoms together turns their levels into bands, and that a crystal with a full band separated by a gap from an empty one is an insulator or a semiconductor, transparent to light whose photons are too small to cross the gap. The mass a curve decides found that an electron in a band, and the hole left in a band when an electron is removed, each move as if they were free particles with a mass set by how sharply the band bends — a mass a tenth or a twentieth of an electron’s in common semiconductors.

Put those two facts together and something has been left out. When a photon is absorbed across the gap it creates an electron in the upper band and a hole in the lower one, at the same place, at the same moment. The electron has negative charge; the hole, being a missing electron, behaves as a positive one. They attract. Any honest account of the absorption has to include that attraction, and when it does, the crystal turns out to contain a scaled copy of the simplest atom there is.

Hydrogen, with two substitutions #

The hydrogen atom is an electron bound to a proton by the Coulomb attraction. Its levels are

$$E_n = -\frac{\text{Ry}}{n^2}, \qquad \text{Ry} = \frac{m_0 e^4}{2(4\pi\varepsilon_0)^2\hbar^2} = 13.6\ \text{eV},$$

and its size is the Bohr radius $a_0 = 0.053$ nm. Everything in those formulas comes from the electron’s mass and the strength of the attraction.

An electron and a hole in a crystal are the same problem with both changed. Their attraction is screened by the crystal: the atoms between them polarise and partly cancel the field, reducing it by the dielectric constant $\varepsilon$, which the field the matter takes away measured as the factor by which a material weakens a field inside it. And the two particles move with effective masses, so the mass in the formula is replaced by their reduced mass $\mu = m_e m_h/(m_e + m_h)$, which plays the role the electron’s mass plays in hydrogen because the proton there is almost infinitely heavy. With those substitutions

$$E_n = -\frac{\text{Ry}^*}{n^2}, \qquad \text{Ry}^* = \text{Ry}\,\frac{\mu/m_0}{\varepsilon^2}, \qquad a^* = a_0\,\frac{\varepsilon}{\mu/m_0}.$$

The two substitutions are not arbitrary corrections; each enters exactly where the hydrogen calculation used the corresponding quantity. Why an atom is the size it is found the Bohr radius as the outcome of a competition: confining an electron to a region of size $r$ costs kinetic energy $\hbar^2/2mr^2$, the attraction pays back $e^2/4\pi\varepsilon_0 r$, and the total is lowest at $a_0$. Lighten the mass and confinement costs more, so the minimum moves outward. Weaken the attraction by $\varepsilon$ and it pays back less, so it moves further out and gets shallower. The exciton is that competition re-run with the crystal’s numbers, and the binding energy falls twice — once through the mass, once through the square of the screening — because both halves of the competition have been weakened.

This is the Wannier–Mott exciton, after Gregory Wannier and Nevill Mott, who worked it out in the late 1930s. The two substitutions push in the same direction for energy and opposite directions for size, and both are large. In gallium arsenide the reduced mass is 0.059 electron masses and the dielectric constant is 12.9, so the exciton’s Rydberg is 13.6 eV × 0.059 / 166, or 4.8 millielectronvolts — nearly three thousand times shallower than hydrogen — and its Bohr radius is 0.053 nm × 12.9 / 0.059, or 11.6 nanometres, more than two hundred times larger.

The size that makes the model work #

That size is what justifies the model, and it is worth seeing why.

The dielectric constant and the effective masses are both properties of the crystal averaged over many unit cells. A dielectric constant describes the response of a material to a field that varies slowly on the scale of its atoms; an effective mass describes the motion of a wavepacket many cells wide. Neither means anything for a particle confined to one cell. The exciton in gallium arsenide spreads over about twenty lattice spacings in radius and contains some 140,000 atom pairs, so, from the pair’s point of view, the crystal really is a uniform medium with a dielectric constant, and the electron and hole really are free particles with light masses. The model is consistent with itself.

In crystals where it is not, it fails. In sodium chloride or solid argon the gap is large, the dielectric constant small and the masses heavy, and the predicted exciton radius comes out comparable to one unit cell. The exciton there is better described as an excited state of a single atom or ion, passed from site to site — a Frenkel exciton, after Yakov Frenkel’s 1931 picture — and the hydrogen series is not seen.

The lines below the gap #

The prediction that is easiest to test is in the absorption spectrum.

Without the attraction, a direct-gap crystal would absorb nothing below its gap and would start absorbing at the gap with an absorption rising as the square root of the excess energy, because that is how fast the number of available pairs of states grows. With the attraction, there are bound states below the gap, and light can create the pair directly in one of them. The result, worked out by Roger Elliott in 1957, is a series of sharp lines at $E_g - \text{Ry}^*/n^2$, crowding together towards the gap exactly like the Lyman series of hydrogen, with strengths falling as $1/n^3$ because the pair must be created at the same point and the higher states have less probability of being found there.

The series is not a curiosity of theory. Evgenii Gross and N. A. Karryev photographed the first members of it in cuprous oxide in 1951, cooled in liquid nitrogen, as a row of faint dark lines on the blue edge of the crystal’s absorption, and fitted their positions to a Rydberg formula with a constant of about a tenth of an electronvolt. It was the first direct evidence that the pair created by light is a bound object rather than two independent particles, and the fit was good enough that nobody doubted the hydrogen analogy afterwards — only how far it could be pushed. In cuprous oxide, a red crystal whose exciton Rydberg is about 92 millielectronvolts, lines have been resolved up to the twenty-fifth, in a 2014 experiment by Tomasz Kazimierczuk and colleagues on a natural crystal from a Namibian mine. An exciton in its twenty-fifth state is over a micrometre across. It is the solid-state counterpart of the atom the size of a bacterium, with the same exaggerated sensitivity to fields and to neighbours that makes Rydberg atoms useful, in a crystal sitting on a laboratory bench.

A step where there should be a slope #

The attraction changes the spectrum above the gap as well, and that change is less often noticed.

An electron and hole created with more than enough energy to escape each other are still pulled together while they are close, and the probability of finding them at the same point — which is what an optical transition needs — is raised. The factor by which it is raised was first computed by Arnold Sommerfeld in 1931 for a different problem, the capture of electrons by nuclei, and it depends only on how the energy compares with the Rydberg. Far above the gap it tends to one: a fast pair barely notices the attraction. Near the gap it grows as the inverse square root of the excess energy, exactly cancelling the square-root fall of the number of states, so their product tends to a constant. The absorption reaches the gap not as a slope from zero but as a step, and below it the lines carry on at the same average strength, crowded so closely that a broadened spectrum shows a plateau running straight through the gap.

That is why the absorption edge of a direct-gap semiconductor at low temperature is so sharp, and why measuring a band gap from the point where absorption “starts” gives the wrong answer by about one exciton Rydberg. Below the gap where there is nothing to absorb found the edge broadened at higher temperature into an exponential tail; the excitons are what that tail is broadened from.

Which crystals keep their excitons #

The binding energy is small, and the crystal is not cold. At room temperature the lattice vibrates with typical energies of $kT = 25.9$ millielectronvolts, and an exciton bound much more weakly than that is broken apart almost as soon as it forms.

The points fall on a diagonal because for fixed dielectric constant the binding energy and the radius are inversely proportional — their product is $\text{Ry}\,a_0/\varepsilon$, independent of the mass — and the dielectric constants of these crystals differ by less than a factor of two. The crystals with light carriers and strong screening, gallium arsenide and indium phosphide, bind their excitons by about five millielectronvolts, far below $kT$; their excitons are seen clearly only at liquid-helium temperatures. Silicon’s sit at fifteen. Gallium nitride and cadmium sulphide are near $kT$. Zinc oxide and cuprous oxide are well above it, and their excitons survive at room temperature, which is why zinc oxide has been studied for decades as a material for ultraviolet lasers that would work by exciton recombination.

The agreement between the filled and open points is good for the weakly bound excitons and poorer for zinc oxide, whose measured binding of 60 millielectronvolts is nearly twice the prediction. The failure is instructive. The dielectric constant used is the static one, which includes the slow response of the crystal’s ions as well as the fast response of its electrons. The constant that depends on how fast it is asked found that a material’s permittivity falls as the field oscillates faster than its constituents can follow. An exciton bound by sixty millielectronvolts orbits faster than the ions of zinc oxide can respond, so it feels a screening partway between the static value and the smaller electronic one, and is bound more strongly than the static constant predicts. The model is not wrong; it has been fed the wrong number for the regime it is in.

Squeezed into a plane #

Confining an exciton to a plane — in a semiconductor layer a few nanometres thick, sandwiched between materials with a larger gap — changes the hydrogen problem from three dimensions to two. The two-dimensional hydrogen atom can be solved exactly, and its levels are $-\text{Ry}/(n - \tfrac{1}{2})^2$. The ground state is bound four times as strongly as in three dimensions, because the pair, unable to spread out of the plane, is held at a shorter average distance where the attraction is larger. Quantum-well excitons in gallium arsenide are bound by about ten millielectronvolts instead of four, and the strengthened, sharpened exciton lines are what quantum-well modulators use to switch light.

In a crystal only one atomic layer thick, such as molybdenum disulphide, a second effect adds to the first. The field lines between electron and hole run mostly outside the layer, through vacuum or a substrate with little polarisability, so the screening that weakened the attraction in the bulk is largely missing. Measured binding energies in such monolayers reach several hundred millielectronvolts, an order of magnitude above $kT$, and the screening no longer follows a single dielectric constant at all: the attraction is weakened at short distances and nearly bare at long ones, and the series departs from any form of $1/n^2$.

Why excitons decide how a solar cell is built #

The binding energy matters for more than spectroscopy, because it decides whether light absorbed in a material becomes current by itself. A photon absorbed in silicon creates an exciton bound by fifteen millielectronvolts, and at room temperature the lattice’s vibrations break it apart within picoseconds into a free electron and a free hole, which the field that bends the bands at a junction then sweeps in opposite directions. The exciton is a brief detour that costs almost nothing.

In an organic semiconductor — a film of conjugated molecules or polymers, the material of flexible solar cells and displays — the dielectric constant is about three and the carriers are heavy, so the exciton is bound by several tenths of an electronvolt, ten times $kT$, and is small enough to sit on one or two molecules. Nothing thermal will split it. The pair lives for about a nanosecond, diffuses ten nanometres or so, and then recombines, wasting the photon, unless it meets an interface with a second material whose energy levels are offset enough to pull the electron across while the hole stays behind. That is why organic solar cells are built as intimate blends of two materials, intermixed on a scale of tens of nanometres: the architecture is dictated by the exciton’s binding energy and the distance it can travel before it dies.

The new perovskite semiconductors, which have reached silicon’s efficiency in a decade, owe part of their success to sitting at the lucky end of the same arithmetic: light carriers and a large, slowly responding dielectric constant give excitons of a few tens of millielectronvolts at most, which separate thermally as silicon’s do.

Where the picture stops #

The masses are not simple. The reduced mass in the formulas assumes both bands are parabolic and the same in every direction. Silicon’s conduction band is not: its electrons are light in one direction and heavy in two others, and the hole bands of most semiconductors come in light and heavy versions that mix. Each introduces corrections of tens of per cent and splits the hydrogen levels that the simple model makes degenerate.

Silicon’s gap is indirect. In silicon the lowest conduction-band states and the highest valence-band states are at different momenta, so a photon alone cannot create a pair at rest; a lattice vibration must take up the difference. The exciton still exists and is still bound by about fifteen millielectronvolts, but it shows in absorption as a step rather than a line, and the spectrum drawn here, which is for a direct gap, does not apply.

Excitons interact. At high density, excitons bind into molecules — biexcitons, the counterpart of molecular hydrogen — and at higher density still the screening of each pair by the others dissolves the binding altogether, the Mott transition into an electron–hole plasma. Excitons are bosons, and below a critical temperature a dense gas of them should condense into a single state, as the atoms of the liquid that will not slow down do; whether excitons in bulk crystals have done so convincingly has been argued for decades.

The lifetime is finite. An exciton ends when the electron falls back into the hole, emitting the photon that created it. In a direct-gap crystal that takes about a nanosecond; in cuprous oxide, whose lowest exciton is optically forbidden to first order, microseconds. The width of each line in the spectrum is set by that lifetime and by collisions with lattice vibrations, and the high-$n$ lines vanish into the continuum as soon as their width exceeds their spacing.

Still open: how far the hydrogen copy can be pushed #

Cuprous oxide’s series reached $n = 25$ in 2014 and has since been pushed further in better samples, with excitons several micrometres across. At that size they interact with each other over distances of micrometres, and each prevents others from forming within a sphere around it, exactly as Rydberg atoms do in the cold-atom experiments that use that blockade to build quantum logic. Whether excitons in a solid can be controlled well enough for the same purpose — whether their lines are sharp enough, their lifetimes long enough and their interactions clean enough — is being worked out. The crystal is both an advantage and an obstacle: it holds the excitons in place without a trap, and it supplies the vibrations, impurities and strains that broaden their lines.

What has been settled is that the copy is a copy. The hydrogen atom’s spectrum, scaled by the crystal’s reduced mass and dielectric constant, predicts where the lines of a semiconductor’s absorption lie to within a few per cent for crystals where the model’s own assumption — an exciton much larger than a unit cell — holds, and departs in a direction that can be understood where it does not. The simplest atom in physics turns out to live inside most of the semiconductors in use, a few thousand times shallower than itself and a few hundred times larger, every time a photon is absorbed near the gap.