Below the gap, where there is nothing to absorb

Where absorption acquires thresholds an absorption edge is a discontinuity, and the figures there draw it as a vertical line. That is a fair drawing of a core-level threshold: the K electron of iodine has a binding energy, the photon either carries it or does not, and the width of the transition is set by the femtosecond lifetime of the hole left behind — an electronvolt, invisible on an axis three decades wide.

A semiconductor’s own edge is a different object, and looking at it on an axis a few tens of millielectronvolts wide shows that it has no step in it anywhere.

A straight line on a logarithmic axis is an exponential, and an exponential in energy has an energy in its exponent. That energy is the whole subject here. It is a few millielectronvolts, it is reproducible to a fraction of itself, it depends on temperature in a way that can be predicted, and it was noticed in 1953 as an empirical regularity holding across materials that have very little else in common — as flat a piece of phenomenology as the fixed loss per cycle that opens this subject, and as durable — alkali halides, semiconductors, glasses, organic crystals.

The obvious explanation is wrong, and saying why is the fastest route to the right one.

It is not absorption by states in the gap #

The obvious explanation is that the material has defects, the defects put electronic states inside the gap, and a sub-gap photon is absorbed by exciting an electron out of one of them. Every part of that happens. It is not what the tail is.

Three things rule it out. The tail’s slope depends on temperature, and a fixed population of defects does not change as a sample is cooled. The tail is present in materials made as perfectly as anybody knows how, and it does not go away as the purity improves — it approaches a floor. And the slope is nearly the same for absorption measured at a given temperature whether the sample is a crystal, a polycrystal or a melt, which is not how a defect concentration behaves.

The description that does produce it contains no states in the gap at all. It works like this.

An electron and a hole are created at the band edges, and where those edges sit depends on the positions of the atoms — a gap is what a repeat opens, so moving the atoms moves it. A lattice at a finite temperature is vibrating — and it is the anharmonic part of that vibration that also makes the mean gap move as the crystal is heated — so at any instant the gap is slightly different in one region from another — narrower where the atoms happen to be squeezed together, wider where they are apart. A photon with less energy than the mean gap can still be absorbed, provided it arrives somewhere the gap is momentarily narrower than its own energy.

How often is a region momentarily narrow by a given amount? For a fluctuation built from a large number of independent contributions, rarely, and with a probability falling roughly exponentially in the size of the fluctuation. So the absorption falls exponentially below the mean gap, and the scale of the exponential is the size of a typical fluctuation.

That is the Urbach energy: not a density of states, not a defect concentration, but a measure of how much a material’s band gap is moving about.

Why an exponential rather than a Gaussian #

The argument above has a gap in it that is worth opening, because closing it is where the physics is and because the obvious closing gives the wrong answer.

If the gap fluctuates because a large number of independent lattice modes are displaced, then by the usual argument about sums of independent things the fluctuation should be Gaussian — and a Gaussian tail is $\exp(-\Delta^2/2\sigma^2)$, which on a logarithmic axis is a downward parabola, not a straight line. The measured tails are straight over decades. Something is wrong with the obvious argument.

What is wrong is that the absorption does not simply count how often the gap is narrow. It counts how often the gap is narrow over a region large enough to hold an electron.

A deep local narrowing of the gap is a potential well, and an electron put into it is confined. Confinement costs kinetic energy, by the same trade that fixes the size of an atom: a well of width $L$ imposes a kinetic energy of order $\hbar^2/2mL^2$ on whatever sits in it. So a very deep and very small fluctuation is useless — it is rare, and what it offers is cancelled by the confinement it demands.

The absorption at a given energy below the gap is therefore dominated by the fluctuation that optimises that trade: deep enough to bring the gap down to the photon’s energy, wide enough that the electron does not pay too much to be localised in it, and no rarer than it has to be. Working the optimisation through turns the Gaussian cost of a fluctuation into something much closer to linear in the depth over the range that matters, and an exponential tail comes out.

This is the optimal-fluctuation account, and its standing should be stated plainly: it explains why the tail is close to exponential rather than Gaussian, it gets the order of magnitude of the slope right, and it does not derive the empirical rule exactly for any real material. The rule is older than the explanation by twenty years and is still better established than it.

The habit underneath it is worth more than the result. When a rate is a product of how rare a fluctuation is and how useful it would be, the answer is set by neither factor alone but by where their trade is best — which is the same structure as a nucleation barrier, where a droplet too small to survive and a droplet too large to form leave one size that decides everything.

The slope is a frozen part plus a shaking part #

If the fluctuations are thermal then the Urbach energy should follow the temperature, and if some of the disorder is built into the material then part of it should survive to absolute zero.

The hyperbolic cotangent is the same function that gives the mean energy of a quantum oscillator, and it is here for the same reason: the mean square displacement of a lattice mode at temperature $T$ goes as $\coth(\hbar\omega/2k_BT)$, tending to $2k_BT/\hbar\omega$ when the mode is hot and to one when it is cold. So the lattice contribution to the Urbach energy is proportional to temperature at high temperature and saturates at low.

The saturation value is not zero, and that is the part worth stopping on. A crystal at absolute zero is still vibrating, because a quantum oscillator cannot be brought to rest, and its zero-point motion smears the gap just as thermal motion does. The cleanest crystal at the lowest temperature still has an exponential tail, and its slope is a statement about the material’s zero-point energy.

The frozen part is different in kind. It is static disorder — alloy composition varying from place to place, strain, the absence of any lattice at all — and no amount of cooling removes it. Hydrogenated amorphous silicon has fifty millielectronvolts of it and crystalline gallium arsenide has four and a half, and the difference between those two numbers is most of the difference between the two materials as devices.

The two pictures put the difference plainly. In the crystal, cooling from three hundred kelvin to ten steepens the tail by a third and pulls the absorption at a fixed energy down by orders of magnitude. In the glass, cooling does almost nothing, because there was almost nothing thermal in the tail to begin with.

The surprising part of the comparison is the dashed line in the previous figure. Room temperature is 25.9 millielectronvolts, and the two best materials there have Urbach energies well below it while the amorphous one is at twice it. A tail shallower than $k_BT$ means the material’s gap is better defined than the temperature of the sample it is in — which is not a contradiction, because the gap fluctuation and the thermal energy are different quantities, but it is a genuinely demanding condition and very few materials meet it. The ones that do are the ones photovoltaics is built out of.

What the tail does to the number in the table #

A band gap is quoted in every table of semiconductor properties to three decimal places. The tail makes that precision partly a matter of convention, and the convention is usually not stated.

The construction being used here is the standard one. For an allowed direct transition the band-to-band absorption satisfies $\alpha E \propto (E - E_g)^{1/2}$, so plotting $(\alpha E)^2$ against $E$ gives a straight line whose intercept on the energy axis is the gap. For an indirect transition the exponent is two rather than a half, and $(\alpha E)^{1/2}$ is the quantity to plot.

The trouble is that the straight line is straight only above the point where the band-to-band absorption takes over from the tail. That point is not marked on a measurement; it has to be inferred, and the honest way to infer it is that the logarithmic slopes of the two pieces agree there, which puts it at $E_g + E_U/2$ for a direct edge and $E_g + 2E_U$ for an indirect one. Below it the plotted curve bends downward, and a least-squares line fitted through any of the bend comes out shallower and intercepts lower.

So a measurement that could not reach high enough in absorption — because the sample was thick, or the detector saturated, or the film was too thin to absorb enough — returns a gap that is too small, and nothing in the data says so.

There is a fifth reading, printed in the legend of each of those two figures and not drawn: what happens if the wrong exponent is used. Applying the indirect construction to a direct material, or the reverse, returns a gap that is wrong by hundreds of millielectronvolts rather than tens. That is not a subtlety and it is a common error, because the exponent is chosen by assuming an answer to the question the measurement was supposed to settle.

The tail is why nothing is quite transparent #

The absorption coefficient turns into a length by inversion, and a length can be compared with a slab.

Two consequences follow from that spread, and they point in opposite directions.

For a solar cell, a steep tail is worth having. Every photon absorbed below the gap produces a carrier pair with less energy than the gap, and the cell’s voltage is set by the population of carriers rather than by the photon that made them, so sub-gap absorption is a loss of voltage without a corresponding gain in current. A material with a fifty-millielectronvolt tail carries that loss and one with a seven-millielectronvolt tail nearly does not, and the difference shows up in the open-circuit voltage of otherwise identical devices.

For a window it is the reverse: the tail is what a substrate manufacturer is selling. Crystalline silicon’s indirect edge is so weak that a wafer of it is nearly transparent at its own band gap — a hundred and eighty micrometres of silicon absorbs less than one per cent of light at 1.12 electronvolts — which is exactly why silicon is a usable infrared window and why a silicon solar cell has to be textured, mirrored on the back and made thick in order to catch light that a direct-gap material would absorb in a micrometre.

And the general form of that observation is worth separating from semiconductors entirely. A transparent material is one whose absorption is exponentially small rather than zero, the exponent is a ratio of an energy to a fluctuation scale, and the fluctuation scale is usually a temperature. The same arithmetic decides how much of a chemical reaction goes over a barrier it cannot afford, how much of a gas escapes a planet whose escape velocity it does not have, and how often a magnetised grain forgets which way it was pointing. In every case the answer is not “none” but “a number with a large negative exponent”, and whether that matters depends entirely on how many chances the process gets.

The same curve, read twice #

These two essays have now read two different kinds of structure off the same measurement, and the pair is worth putting side by side because the readings are opposites.

The ripple above a core-level edge takes a few hundred electronvolts above a core-level edge and finds a ripple of a few per cent whose period is a distance. That works because a core level is sharp: the threshold is a property of one atom, the photoelectron leaves with a definite energy, and every feature above the edge belongs to what the electron met on its way out.

This essay takes a few tens of millielectronvolts below a band edge and finds an exponential whose slope is a fluctuation. That works for the opposite reason: a band gap is not a property of one atom, it is a property of an arrangement of very many, and it is therefore the sort of quantity that has an instantaneous value different from its mean.

So a sharp threshold hands over information about geometry, and a soft one hands over information about motion. Both are carried by absorption, both are invisible at the resolution of the figures two essays back, and neither is a defect.

There is a practical consequence for anybody comparing materials. The Urbach energy has become the standard single number for how good a semiconductor is, quoted for every new photovoltaic absorber within months of its first synthesis, precisely because it is one number, it is measured on a film rather than a device, and it correlates with almost everything anybody cares about afterwards. It is doing the job that a linewidth does elsewhere: standing in for a distribution nobody can measure directly, and being useful in proportion to how little else is needed to obtain it.

No exciton, a fixed pivot, and a floor the tail runs into #

The two-piece model has no exciton in it. An electron and a hole attract one another, so there are bound pairs just below the gap — in gallium arsenide, by four millielectronvolts — and at low temperature they appear as a sharp absorption peak sitting exactly where this model draws an exponential. Below about fifty kelvin the peak dominates the edge, and the ten-kelvin curve drawn here is a picture of the wrong thing. The model becomes fair once the exciton is thermally broken up, which for gallium arsenide is above roughly a hundred kelvin.

The lines are drawn pivoting about a fixed point. Real spectra also shift bodily, because a band gap narrows as a crystal is heated by an amount much larger than the change in the tail slope — some seventy millielectronvolts for gallium arsenide between ten kelvin and room temperature. The figures here remove that shift so that the fan is the tail’s doing, and a measurement has to remove it too before an Urbach energy can be extracted. The convergence of real tails onto a single focus is approximate and is itself an empirical observation rather than a theorem.

The exponential is not exact over unlimited range. Several decades down, the tail of a real material runs into whatever defect absorption it has, which is flatter, and the curve levels off onto a shoulder. The figures continue the exponential past where any measurement supports it, and a tail drawn for seven decades is drawn over a range only the best samples and the most patient techniques can reach.

And the amorphous case is being treated as though disorder had a single scale. It does not. A glass has a distribution of local environments with its own structure, and the resulting tail is exponential over a useful range for reasons that are better established empirically than derived. The frozen contribution here is one number standing in for a whole distribution.

Seven decades of absorption, and what measuring them costs #

They cannot show what a measurement of seven decades costs. The absorption at the bottom of the gallium arsenide curve corresponds to a metre of material, so a millimetre sample transmits all but a part in a thousand of the beam, and measuring that requires either a photothermal technique that detects the heat deposited rather than the light lost, or a photocurrent measurement in a device. Both are indirect, both need to be put on an absolute scale by matching to a transmission measurement higher up, and the join between them is where a published tail usually has its kink.

Nor can they show that the tail is a local statement. The exponential describes absorption averaged over a sample, and what is happening underneath is that different regions have different gaps — so a sub-gap photon is not absorbed weakly everywhere but strongly in a few rare places. Anything that depends on where the carriers were made, which includes most of what a device does with them, needs that distinction, and an absorption coefficient is exactly the quantity that has averaged it away.

And they cannot show emission. Absorption and emission at a band edge are tied together by a relation as strict as the one tying absorption to refraction, so a material with an exponential absorption tail has a corresponding tail on the low-energy side of its luminescence, and measuring that is often easier than measuring the absorption. The two measurements are different views of one function, and only one of them is drawn here.

Still open: whether the floor is a property or a limit #

The frozen part of the Urbach energy has fallen, material by material, as growth has improved, and in the best crystals it is now comparable to the zero-point contribution — which cannot fall. Whether a given material has a genuine floor set by its own lattice dynamics, and whether the record values reported for the best perovskites and the best gallium arsenide are at that floor or merely at the current limit of what growth achieves, is not settled, and it matters because the floor sets how close a solar cell of that material can come to the thermodynamic bound on its voltage.

The question is hard to answer because the two contributions are separated by extrapolating the temperature dependence to zero, and at low temperature the exciton takes over the edge and the extrapolation is made through the region where the model is least trustworthy. A measurement that could separate frozen disorder from zero-point motion without extrapolating through the exciton would settle several arguments at once.

The habit worth carrying away is about thresholds that are not sharp. A threshold in a collective property is smeared by whatever makes that property fluctuate, and the shape of the smearing reports the fluctuation rather than the threshold. A core level belongs to one atom and its edge is sharp; a band gap belongs to the whole crystal and its edge is a slope. Reading the slope as a defect, when it is the gap’s own restlessness, is the mistake this essay exists to prevent — and the general form of it is to attribute a soft edge to the population of states beyond it, when it is the edge that is moving.