A band gap is not a bond energy

Four quantities describing the same material are routinely quoted in electronvolts, drawn on the same kind of diagram, and treated as interchangeable. They are not, and confusing them produces conclusions that are wrong by factors of two or more — which is worse than it sounds, because a factor of two in an exponent is a factor of thousands in a rate.

The band gap is the energy needed to move one electron from the highest filled level to the lowest empty one, with the atoms staying exactly where they are.

The bond energy is the energy needed to pull two atoms apart, taking their shared electrons with them and letting everything relax.

The ionisation energy is the energy needed to remove one electron from the material entirely.

The HOMO–LUMO gap is the first of these applied to a molecule, and its relation to the first is subtler than the shared definition suggests.

All four are differences between energies of the same system in different states, and that is the whole of what they have in common. Which states, and what has been done to get from one to the other, differs in every case.

The gap and the bond energy #

The commonest of the four confusions is the first pair, and the numbers make it plain.

Silicon’s band gap is $1.12$ electronvolts. Its Si–Si bond enthalpy is about $2.3$ electronvolts. Diamond’s gap is $5.5$ and its C–C bond is about $3.6$. The ratio is two in one direction for silicon and about two-thirds in the other for carbon, so there is not even a consistent factor to convert between them.

The reason is that they are answers to different questions about different processes.

Exciting an electron across the gap keeps the structure intact. One electron leaves a bonding level and arrives in an antibonding one; the net bonding is reduced by one electron’s worth, the atoms stay where they are, and the cost is the distance between two levels.

Breaking a bond removes a pair and lets the atoms leave. Both electrons come out of the bonding level, the atoms separate to infinity, and everything left behind relaxes into a new arrangement. The cost includes the level difference, the relaxation, and the change in every repulsion term in the material.

In this field’s own arithmetic, the level difference at the top of a band is a fraction of a $\beta$ and the cost of removing a pair from a bonding level is around $2\beta$ — a difference of an order of magnitude, arising from the fact that one process moves an electron and the other removes two and rearranges the neighbourhood.

Where the confusion comes from #

The two get conflated because both are described as “the energy it takes to break something”, and because a diagram of levels makes them look like the same kind of vertical distance.

There is also a genuine correlation that encourages it. Materials with strong bonds do generally have large gaps, for a shared reason: both depend on the interaction strength between neighbouring orbitals. A strong interaction pushes the bonding level far down and the antibonding level far up, which makes both the bond energy and the level separation large.

That correlation is real and it is not a conversion. Comparing carbon and silicon shows why: silicon’s weaker interaction gives it both a smaller gap and a weaker bond, but the ratios are different because the two quantities also depend on other things — the antibonding level’s position for the gap, the relaxation energy for the bond.

The gap and the ionisation energy #

The second confusion is between the gap and the energy to remove an electron entirely, and it is worth separating because photoelectron spectroscopy measures the second and is often used to discuss the first.

Removing an electron entirely takes it to a state at rest infinitely far away — the vacuum level. The energy required is the ionisation energy for a molecule, or the work function for a metal.

The gap is between two states inside the material, and both are far below the vacuum level.

So the gap is a difference of two energies both measured from the same place, and the ionisation energy is one energy measured from somewhere else. A material can perfectly well have a large gap and a small work function, or the reverse; the two are set by different features of the electronic structure.

What a photoelectron spectrum measures works through what the experiment actually returns for a molecule, and the caution there applies to solids with more force: the measured quantity is the difference between the total energy of the system before and after removing the electron, and reading it as an orbital energy assumes the rest of the electrons did not notice.

The HOMO–LUMO gap and the band gap #

The fourth quantity is the subtlest, because the definitions really are the same and the behaviour is not.

A molecule’s HOMO–LUMO gap and a solid’s band gap are both the distance from the highest occupied level to the lowest empty one. The difference is what happens as the system grows.

For a chain with a closed shell — alternating bonds, an even number of electrons per repeat — the sequence converges. Measured at $\delta = 0.15$ on chains of twenty, forty, eighty and a hundred and sixty, the gap goes $0.759$, $0.655$, $0.617$, $0.605$, converging on $4\delta = 0.60$. That limit is the band gap, and the molecular quantity approaches it.

For a chain with a half-filled band the sequence does not converge to anything. The gap goes $0.299$, $0.153$, $0.078$, $0.039$, halving with every doubling, and its limit is zero. There is no band gap for it to converge to, because there is no band gap.

So a HOMO–LUMO gap is a band gap in the limit only when there is one, and computing the gap of one cluster tells a reader nothing about which case they are in. Where a molecule stops being one is about exactly this sort of question and its answer is always a sequence rather than a number.

The optical gap, which is a fifth quantity #

There is a fifth number that belongs on the list and is usually spoken of as though it were the first.

An optical gap is the lowest photon energy at which a material begins to absorb. It equals the band gap only when a transition between the states either side of the gap is allowed, and there are two ways it can fail to be.

Symmetry can forbid it. If the two states have the same behaviour under an operation the structure possesses, the transition moment between them is exactly zero — not small, zero, as selection rules are one theorem establishes and as is computed elsewhere at $10^{-17}$. Absorption then begins at the next allowed transition, higher up.

Momentum can forbid it. In a periodic material the two states may carry different wavevectors, and a photon carries essentially none, so the transition needs a lattice vibration as well. Silicon is the standard case: its lowest gap is $1.12$ electronvolts and its lowest direct gap is $3.4$, and a careless absorption measurement returns something between the two.

There is a third failure in the opposite direction. An electron and the hole it leaves behind attract each other, and the bound pair — an exciton — sits slightly below the gap. So the optical onset can be lower than the band gap as well as higher, by the exciton binding energy, which in a molecular crystal can be several tenths of an electronvolt.

So the optical gap can be above the band gap, below it, or equal to it, depending on effects the band structure does not contain. That it is nonetheless the most-quoted of all these numbers, because it is the easiest to measure, is worth knowing when reading a table.

The four in one material #

Putting the numbers for one material side by side makes the independence concrete. For silicon:

• Band gap $1.12$ electronvolts.
• Direct gap $3.4$ electronvolts, three times the first.
• Si–Si bond enthalpy about $2.3$ electronvolts.
• Work function about $4.8$ electronvolts.
• First ionisation energy of the free atom $8.15$ electronvolts.

Five numbers, all in electronvolts, all describing silicon, spanning a factor of seven. No two of them are convertible into each other, and each answers a different question about a different process. A sentence containing any two of them and the word “so” is almost certainly wrong.

Four binding energies on a logarithmic scale are all pair quantities, in which atoms are separated rather than electrons promoted. None of the four is a gap and none is comparable with one — which is the whole of the objection, and the reason a table of bond energies and a table of band gaps cannot be read across.

The practical damage #

It is fair to ask whether any of this matters outside a pedantic argument, and it does, in a way that shows up in published work.

The commonest concrete error is estimating a material’s thermal stability from its band gap. The number of carriers excited across a gap goes as $\exp(-E_{\mathrm{g}}/2k_{\mathrm{B}}T)$, and the rate of a bond-breaking process goes as $\exp(-E_{\mathrm{bond}}/k_{\mathrm{B}}T)$. Using the wrong energy in the wrong exponent, with a factor of two also at stake, produces predictions wrong by many orders of magnitude — and both exponentials look identical on a page.

The second is inferring photostability from a gap. A material that absorbs a photon of a given energy has an excited state at that energy; whether that state decays harmlessly or breaks a bond depends on where the excited state’s energy surface leads, which is a question about geometry and not about the gap at all.

The third is comparing a computed HOMO–LUMO gap with a measured optical onset and concluding a calculation is accurate or inaccurate. Those are two different quantities before any question of accuracy arises, and the discrepancy between them is physics rather than error.

What a computed number is a number for #

The general discipline this essay is an instance of is worth stating separately, since it applies well past band gaps.

Every quantity in a computation is the answer to a specific question about a specific process, and the unit does not identify the question. Electronvolts are used for level separations, total energies, differences of total energies between charge states, dissociation energies, thermal energies and photon energies, and mixing them is the commonest error in the subject.

Three questions distinguish them reliably.

Does the process change the number of electrons in the system? Ionisation and electron affinity do, excitation across a gap does not. Quantities that do are differences between states of different charge, and the relaxation of the remaining electrons is part of them.

Does the process change where the atoms are? Dissociation does, vertical excitation does not. Quantities that do include an elastic and a structural term that a level diagram cannot show.

Is the process one electron or a pair? A bond holds two, an excitation moves one, and the factor of two is the least of the differences between them.

Answering those three about any quoted energy is enough to say which of the four it is, and to know which comparisons are legitimate.

The molecular version, which is older and equally muddled #

None of this is new with solids. The identical confusion exists for molecules and has been around longer, in the form of the relation between a HOMO–LUMO gap and a bond dissociation energy.

Ethene’s π to π* transition is at about $7.6$ electronvolts. Its π bond is worth about $2.7$. The ratio is nearly three, and for the same reasons: the excitation moves one electron from bonding to antibonding at fixed geometry, and the dissociation removes a pair and lets the fragments relax.

There is a further wrinkle in the molecular case that carries over to solids and is usually skipped. The antibonding level goes up by more than the bonding level goes down — the asymmetry is computed rather than assumed in the antibonding level goes up more — so the excitation energy is not twice the bonding stabilisation but somewhat more than twice. A reader who assumed symmetry would be wrong in a predictable direction, which is the most useful kind of wrong to know about.

That same asymmetry is why filling an antibonding level costs more than emptying a bonding one gains, why a molecule with one electron promoted is often dissociative, and why the excited state of a solid relaxes into a different geometry from its ground state. All three are the same fact about where two levels go when two orbitals interact.

And the gap is not a constant of the material either #

The four quantities have been separated by asking what process each one describes. There is a further separation to make within one of them, because the band gap of a material is not a single number even after the process has been specified.

Silicon’s gap is 1.17 electronvolts at absolute zero and 1.12 at room temperature. Germanium’s runs from 0.74 to 0.66, gallium arsenide’s from 1.52 to 1.42. The shifts are four to ten per cent, which is far larger than the precision any of them is quoted to, and they are not a broadening — the gap genuinely moves.

Two mechanisms do it and neither is thermal excitation. The lattice expands, so the atoms are further apart, the interactions between them weaken and the levels close up. And the atoms vibrate, which shifts every level by an amount that depends on how the level responds to a displacement — an average of a quantity over a distribution rather than its value at the mean, which is the correction the atoms are not at the points computes for a bond angle and which applies to an energy level in exactly the same way.

The second mechanism does not switch off at absolute zero, and that is the part worth carrying. A lattice at zero temperature is still moving, so even the 1.17 electronvolts is a renormalised number: the gap a perfectly static silicon lattice would have is larger by something of order sixty millielectronvolts, and for diamond, whose atoms are lighter and whose vibrations are faster, by several hundred. The gap of a motionless crystal is not a measurable quantity of any crystal, because no crystal is motionless.

So a band gap carries a temperature the way a delocalisation energy carries a reference. Quoting one without it is the same species of half-number, and the difference between a calculated gap and a measured one is partly this rather than entirely the calculation’s error — a computed value from a static lattice is being compared against a material that never held still.

What this field’s numbers are #

Applying the test to this field’s own output, for the sake of the practice.

The gaps computed here — the $0.60\beta$ of an alternating chain, the shrinking sequence of a uniform one — are one-electron level separations at fixed geometry. No electron count changes, no atom moves, and one electron is involved.

The energy per site used in the Peierls calculation is a total energy summed over occupied levels. Atoms move, no electrons are added or removed, and it is a sum over pairs rather than a single-electron quantity. It cannot be compared with a gap directly and the essay that uses it does not.

The band width is a range of level energies and not an energy at all in the process sense — nothing is being done to the system, and it is a property of a spectrum.

The cohesion figures are pair binding energies, so atoms move and the electron count does not change. They are comparable with bond energies and not with gaps, which is why what holds a solid together puts $\beta$ on its scale and no gap.

Four quantities from one field, all in the same units, three of which cannot legitimately be compared with each other. That is the ordinary situation rather than a peculiarity of this subject, and the habit of asking which process a number describes is the only defence against it.

It is also why a figure should name the quantity it draws rather than labelling an axis “energy”. An axis labelled in electronvolts with no statement of what is being done to the system is an invitation to the comparison this essay exists to prevent, and the invitation is usually accepted.