The gap is not the band width

A band is usually described by two numbers, and the two are routinely confused because they are quoted in the same units and drawn on the same axis.

The width is how far the levels spread. The gap is how large a jump separates the last occupied level from the first empty one.

They are set by different things, changed by different changes, and neither constrains the other in either direction. A structure can have a very wide band and no gap at all, or a narrow band and a large gap.

The confusion has a visual source. Both are read off the same diagram — a shaded region with a break in it — and both are lengths measured along the same vertical axis in the same units. Nothing about the drawing suggests they are different kinds of quantity, and the drawing is where most people meet them.

Where each comes from #

The width is a statement about how much an orbital is spread out, and its exact form is a count. The width of a band is a count of neighbours shows that the mean of the squared level energies equals the mean coordination — exactly, for any structure — so the scale of a band is fixed by how many neighbours each atom has and how strongly it interacts with them.

The gap is a statement about how unequal the surroundings are. In the chains here it is opened by alternating the bonds, and its size is four times the alternation. In a real material it can be opened by alternating the atoms instead — a chain of two elements rather than one — or by any other change that makes one site or one bond differ from the next.

The distinction in one line: the width counts the neighbours and the gap measures the difference between them.

That is why a homonuclear chain with equal bonds has no gap however wide its band, and why a chain of alternating atoms has one however narrow.

The two are moved by different levers, computed #

The claim is easy to check because a tight-binding calculation separates the two quantities.

Take a chain and alternate its bonds by fifteen per cent. The gap opens to $0.60\beta$, from a value that was heading to zero. The second moment — the exact measure of the band’s scale — changes from $2$ to $2 + 2\delta^2 = 2.045$, a shift of about two per cent, and upward. A distortion that opened a large gap made the band very slightly broader rather than narrower.

Now take a chain and change the coordination instead, by attaching every site to two neighbours on each side rather than one. The second moment doubles and the band gets substantially wider. The gap does not open at all, because every site is still like every other.

Two changes, two effects, no overlap. The intuition that a gap is carved out of a band by narrowing it is a picture rather than an account.

What each one controls #

Having separated them, it is worth saying what each is good for, since both are useful and for different things.

The width controls how mobile an electron is. A wide band means the levels change rapidly with the state’s character, which in a periodic setting is exactly the statement that an electron moves easily. Narrow-band materials — the transition-metal oxides, the rare-earth compounds — have electrons that barely move even when there are plenty of empty states, and much of the interesting physics of correlated materials happens there. This is also, not coincidentally, where the one-electron picture fails worst.

The gap controls what light does. A material with a gap larger than the visible range is transparent and colourless; one with a gap inside the visible range is coloured, absorbing the high-energy end; one with no gap reflects everything and is a metal. Diamond’s gap is $5.5$ electronvolts and it is transparent; silicon’s is $1.1$ and it is opaque and grey; a metal’s is zero.

The gap also controls how conduction depends on temperature, since the number of carriers thermally excited across it goes as $\exp(-E_{\mathrm{g}}/2k_{\mathrm{B}}T)$. That exponential is why a small change in gap makes an enormous change in room-temperature conductivity, and why semiconductors span twenty orders of magnitude of resistivity across a modest range of gaps.

Nothing in that list is controlled by both quantities, which is the practical form of their independence.

Four ways to open a gap, none of which narrows the band #

Listing the mechanisms makes the independence concrete, because each of them changes the matrix somewhere the width does not live.

Alternate the bonds. Every other interaction stronger than the rest. This is the mechanism computed throughout this field, and the gap it opens is four times the fractional alternation. It is what happens spontaneously in a half-filled one-dimensional chain, for the reasons in a chain cannot stay even.

Alternate the atoms. Sites of two kinds with different energies, which is what a compound is. A chain of alternating $\alpha \pm h$ has a gap of $2h$ regardless of how strongly the sites interact, so a very ionic compound with weak overlap has a large gap and a narrow band — precisely the combination the wide-band rule says is impossible.

Close a shell in a ring. Benzene has a gap between its highest occupied and lowest empty π level because six electrons exactly fill three levels, and the gap is a consequence of the degeneracy pattern rather than of any inequality between the sites. Aromaticity as a computed shell closure computes it.

Fill the band completely. Two electrons per site leaves nothing empty at all, and the cheapest excitation is to whatever the next band is. This is the largest gap available and it comes from the electron count rather than from the matrix.

Only the first two are gaps in the sense a solid-state physicist means, and all four produce the same phenomenology: a finite cheapest excitation, a material that is transparent below it, no conduction without thermal help.

Why the confusion is so persistent #

The two get conflated for a reason that is worth naming rather than sneering at, because it is a genuine feature of the commonest examples.

In the series carbon, silicon, germanium, tin, the gap falls from $5.5$ electronvolts to $1.1$, $0.7$ and eventually to zero for the metallic form of tin. Over the same series the bands get wider, because the atoms get larger, their orbitals overlap more, and the interaction strength rises.

So in the single most-taught example, wider band really does go with smaller gap, and a rule extracted from it is right about that series for reasons that have nothing to do with a general relation. What is actually happening is that both quantities depend on the interaction strength — the width directly, and the gap through the fact that the two bands broaden towards each other until they meet.

The rule breaks the moment a case appears where the two levers move separately, and the alternating chain above is exactly such a case: the interaction strength is unchanged, the coordination is unchanged, and only the inequality between bonds has been altered.

Both quantities can be read off one small picture: benzene’s six π levels have a width from the top level to the bottom and a gap across the middle. On six atoms they are the same order of magnitude, which is exactly why the confusion survives — at sixty they are not.

The measured gap and the real one #

There is a further complication and it belongs here because it is where the number a reader is quoted comes from.

An optical measurement finds the lowest photon energy at which absorption begins. That equals the gap only if a transition between the states either side of the gap is allowed. If it is forbidden — because the two states have the same symmetry, or in a periodic material because they carry different wavevectors — the absorption begins higher up, and the measured onset overestimates the gap.

That is not a rare case. Silicon’s indirect gap is $1.12$ electronvolts and its direct gap is $3.4$, and an absorption measurement done carelessly returns the second. The difference is visible in the shape of the absorption edge rather than in its position, which is why the measurement is done properly by plotting the absorption in a way that separates the two.

Symmetry says why such a transition would be forbidden — selection rules are one theorem — and a structureless band model cannot say anything about the wavevector case, for the reason set out in a band with no structure in it. Both mechanisms produce the same discrepancy between the measured and the true gap.

An absorption measurement on a gapless chain begins at zero and rises; one on the same chain with alternating bonds begins at the gap. The density of states says where the states are and not whether anything can reach them, and the width is a property of the first.

The size of the gap, and where four times the alternation comes from #

The number the alternating chains settle on is worth deriving rather than reporting, because the derivation shows which feature of the structure it belongs to.

Alternating a chain’s bonds as $1 \pm \delta$ makes the repeat two atoms long, and the two levels that were adjacent at the middle of the band are pushed apart. The amount is the difference between the two interactions taken twice — the strong bond pulls one combination down and the weak bond fails to pull it as far, and the two effects add on both sides of the gap.

The result is $4|\delta\beta|$, and the computation confirms it: at $\delta = 0.15$ the gap settles at $0.6046\beta$ against $4\delta = 0.60$, converging from above as the chain lengthens through $0.759$, $0.655$, $0.617$, $0.605$.

Two things follow. The gap is linear in the alternation, so a small distortion opens a disproportionately large gap — a fifteen per cent change in bond strengths produces a gap that is fifteen per cent of the whole band width. And the gap depends on $\delta$ and $\beta$ together, so a material with weak interactions and strong alternation can have the same gap as one with strong interactions and weak alternation, while their bands differ by whatever factor separates their $\beta$.

That last observation is the independence stated arithmetically. The width goes as $\beta$ and the gap goes as $\delta\beta$, so their ratio is $\delta$ — a pure number describing how unequal the structure is, with the interaction strength divided out of it entirely.

The gap is not a bond energy either #

One more conflation is worth heading off, since it is the commonest way the number is misused outside physics.

A band gap of $1.1$ electronvolts does not mean that breaking a bond in silicon costs $1.1$ electronvolts. Silicon’s Si–Si bond enthalpy is about $2.3$ electronvolts, roughly twice the gap, and the two quantities are answering different questions: the gap is the cost of moving one electron from a bonding level to an antibonding one, and the bond energy is the cost of removing a pair from a bonding level entirely and letting the atoms separate.

Those differ by the antibonding level’s position, by the change in geometry that accompanies separation, and by every electron-repulsion term the one-electron picture omits. A band gap is not a bond energy sets the whole comparison out, because it is a large enough mistake to deserve its own treatment.

The pair of numbers as a description of a material #

Taken together the two numbers are a reasonable first description of an electronic structure, and it is worth saying what a reader can and cannot infer from a pair.

A wide band and no gap is a good metal: mobile electrons and states available at every energy. The alkali metals, copper, aluminium.

A wide band and a moderate gap is a covalent semiconductor: silicon, germanium, gallium arsenide. The width means that once a carrier is present it moves easily, which is what makes these materials useful in devices.

A narrow band and a large gap is an ionic insulator: the alkali halides, most oxides. Little overlap between neighbouring orbitals, and a large difference between the sites.

A narrow band and no gap is the interesting and difficult case, and it is where the one-electron picture stops being trustworthy. The electrons are barely mobile and there is nothing to stop them moving, so whether the material conducts is decided by how strongly they repel one another rather than by anything in the band. A half-filled band is not always a metal is about exactly this corner.

Three of those four are describable by band theory and the fourth is not, which is a fair summary of the theory’s reach. Notice also that the four cells are populated — every combination of wide and narrow with gapped and gapless exists in real materials, which is the strongest possible statement of the independence this essay has been arguing for.

The two numbers control two different transport properties #

The width and the gap being independent is an arithmetic statement, and it has a practical form that says why both are worth measuring: they decide two different things about how a material conducts, and neither decides the other’s.

The gap sets how many carriers there are. Making a carrier means promoting an electron across it, so the carrier count falls off exponentially with the gap divided by the temperature. A tenth of an electronvolt either way changes it by orders of magnitude.

The width sets how fast each carrier moves. A carrier’s effective mass goes inversely with the band width — a wide band has a strongly curved dispersion and light, mobile carriers; a narrow one has heavy, sluggish ones. The gap does not enter.

So a material’s conductivity is a product of two factors controlled by two independent numbers, and knowing one of them predicts nothing about the other.

The comparison that makes it concrete is between two families of semiconductors with similar gaps. Silicon has a gap near 1.1 electronvolts and a valence band around a dozen electronvolts wide: modest carrier count, very light carriers, high mobility. A molecular organic semiconductor can have a gap in the same range and a band width of a few tenths of an electronvolt, because its molecules interact weakly with each other — the same carrier count, carriers heavier by more than an order of magnitude, and a mobility lower by as much.

That is the essay’s independence turned into a design statement. Widening a band improves mobility without touching the carrier count; narrowing a gap improves the carrier count without touching the mobility, and a material engineer working on one of the two is working on a quantity the other measurement cannot see.

One more reading is worth having, because it says what a shorter sequence would have concluded.

What holds across the whole field #

Two summary statements survive everything above.

The width is a count and the gap is a difference. The width is exactly determined by the coordination through an identity that requires no limit; the gap requires that something distinguishes one site or bond from the next, and its size measures how much.

Neither is a spectrum. A width and a gap are properties of a set of energies, and what an experiment returns is a property of transitions between them. The two coincide often enough for the habit of quoting a gap as though it were a measured onset to survive, and diverge often enough for that habit to be worth breaking.

Both statements are versions of the same discipline this site applies everywhere: name which quantity is being computed, say what would change it, and do not let two numbers that share an axis share a meaning.

There is a practical test that separates them on any computed spectrum, and it costs nothing. Compute the second moment and compare it with the counted coordination — that is the width, exactly, and it is available without looking at the filling. Then find where the filling stops and measure the distance to the next level — that is the gap, and it required the electron count that the second moment never saw. Two measurements, two different inputs, and a computation that reports only one of them has answered only one of the questions.