Half filled is as bonded as it gets

Every band picture in this field so far has been drawn at one filling and asked what the filling decides about conductivity. There is a second question the same picture answers, and it is about cohesion: how much binding does a band supply, and how does that depend on how full it is?

The answer has a shape, the shape has a closed form, and one of its features is exact.

The sweep #

Take a ring of sixty sites with equal hopping, diagonalise it once, and then fill it with $2n f$ electrons for fractions $f$ from a thirtieth up to one. The occupied-level sum, divided by the number of sites, is the binding the band supplies per atom in this model.

Three features, and each is checked by computation rather than read off the curve.

The curve is symmetric about half filling. The value at $f$ and at $1 - f$ agree to the last bit — $0.1993$ at both $0.05$ and $0.95$, $0.9007$ at both $0.25$ and $0.75$. That follows from the band being symmetric about the atomic level, which follows from the hopping matrix having zero diagonal.

Nothing beats half filling. The maximum is at $f = 0.5$, and the check is written as no filling binds more rather than as an argmax, because a small ring can tie with it: a ring of twelve has a pair of levels at exactly zero, so ten electrons and twelve bind identically.

A full band binds nothing at all. At $f = 1$ the sum is zero to $10^{-15}$β per site.

Why a filled band is exactly zero #

The last of those is worth doing properly, because it is the one that surprises and because it is exact rather than approximate.

The sum of all the eigenvalues of a matrix is its trace. The tight-binding matrix here has zeros on its diagonal — every site has the same energy, taken as the origin — so the trace is zero, and therefore the eigenvalues sum to zero.

Filling every level means occupying every eigenvalue, so the total is twice the trace, which is nothing. Every bonding level below the atomic energy is matched by an antibonding level exactly as far above it, and filling both cancels the pair.

That is the extended-solid version of a statement about just two atoms. The antibonding level goes up more shows a filled bonding-and-antibonding pair coming out repulsive once the overlap is kept, which is why helium does not form a diatomic molecule. Here, with the overlap not kept, the filled band comes out at exactly zero — neither bonding nor antibonding — and the same conclusion follows in a weaker form: a filled band supplies no cohesion, and whatever holds such a solid together is not this.

The closed form #

The finite sum approaches an integral, and the integral has an elementary answer. For a one-dimensional band the occupied-level sum per site at filling $f$ is

$$E(f) = \frac{4}{\pi}\sin(\pi f)$$

in units of the hopping. At half filling that is $4/\pi = 1.2732$; the ring of sixty gives $1.2721$, short by $1.2\times10^{-3}$.

The departure is a finite-size effect and it falls as $1/n^2$: a ring of twelve departs by $2.9\times10^{-2}$, which is twenty-five times more for a ring five times smaller. The figure’s tolerance carries that scaling rather than a round number, so a small ring is held to a small ring’s standard instead of being excused or failed.

Nothing in the calculation was told the closed form. It is compared against the sum directly, which makes the agreement a test of the diagonalisation and of the filling routine at once.

The same curve read as an energy per electron #

Dividing by the number of sites gives the binding per atom, which is the quantity a cohesive energy is. Dividing by the number of electrons instead gives a different curve with a different message.

Per electron the curve has no maximum at all: it falls monotonically from just under $2$β at the bottom of the band to zero at the top. The first electrons into a band are the most useful ones, each collecting nearly the full $2$β the band edge offers, and every subsequent one collects less.

That is the same arithmetic as the ring calculation in what one pair can hold together, where a single pair in a ring of any size supplies exactly $4$β — which is $2$β per electron, the value the per-electron column approaches as the filling goes to zero.

So which quantity is being maximised decides where the answer is. A solid maximises binding per atom, because the atoms are what it is made of and the electron count follows from them; the maximum is at half filling. A hypothetical process free to choose how many electrons to spend would put them all at the bottom of the band, and there is no maximum to find.

What this says about real solids #

The model has no repulsion, no lattice, one orbital per site and a coordination number of two, so nothing here computes a cohesive energy for anything. What it supplies is a shape, and the shape is visible in measured data.

Cohesive energies across a transition series rise and then fall. The 3d, 4d and 5d series each show a maximum near the middle, with the highest melting points in the periodic table — tungsten, rhenium, tantalum — clustered in groups five to seven. The d band filling runs from empty at the left of each series to full at the right, so the parabola-like curve above is the qualitative account of that pattern.

The maximum is not exactly at half filling in reality, and the departure is informative. Measured cohesion peaks a little past the middle of each series, and the reasons are all things this model omits: the s band overlapping the d band, magnetic effects in the 3d series, and the repulsion between filled cores that grows as the atoms are pulled together.

The two ends of each series are the weakly bound metals. The alkalis and the group-twelve metals — potassium, mercury, zinc — are soft and low-melting, and they sit where the relevant band is nearly empty or nearly full.

That much is a qualitative match between a curve computed from a chain of sixty and a table of measured melting points, and it should be read as no more than that. It is offered because the shape is a consequence rather than a fit: nothing in the model was adjusted to produce a maximum at the middle.

Two properties that coincide and need not #

That last coincidence deserves a paragraph, because it is easy to read as one fact when it is two.

Binding is maximal at half filling because that is where every bonding level is occupied and no antibonding level is.

Excitations are cheapest at half filling because the density of states of this band is largest at the middle, so the levels there are most closely spaced — which is what what a metal actually is measures as a level spacing falling toward zero with system size.

The two coincide here because the band is symmetric and the density of states peaks where the bonding-to-antibonding crossover is. Neither implies the other, and in a band with structure in it they can come apart: a system can be strongly bound and insulating, or weakly bound and metallic.

The simple solids treated here are built entirely on structureless bands, so it cannot show them coming apart. A band with no structure in it is the essay that says what that omission costs, and the coincidence above is a good example of the cost: two independent facts arriving at the same filling with no way to tell, from inside the model, whether that is a theorem or an accident of the shape.

A chain’s density of states is largest at its band edges, which is a peculiarity of one dimension — in three dimensions the shape is quite different. What survives the change of dimension is the position of the maximum in the curve above and not its shape.

Where the ends of the chain go #

The second trace in the first figure is the same sweep for a chain rather than a ring, and it sits below the ring’s everywhere.

The reason is the one what one pair can hold together computes at molecular sizes: a chain has ends, its states must vanish beyond them, and the resulting non-uniformity costs binding. The gap between the two curves falls as $1/n$, so the ends’ cost per site falls as the chain lengthens.

The end is the hardest place to bind measures the same quantity as a threshold, and the ring-against-chain difference computed elsewhere in this field settles on $0.712$β times $1/n$ — so a chain needs about seven hundred atoms before its ends cost under a thousandth of a β per site.

A ring’s band fills in from the edges as the ring grows and its lowest level never moves. That spectrum is the structure the filling curve is a sum over, and the fact that its edges are fixed is why the curve converges as fast as it does.

A band’s width goes as the square root of its coordination, because the second moment of the spectrum is the coordination exactly. So the maximum binding per atom does not go as the coordination, which is the arithmetic behind the whole of that essay and is assumed here.

What a gap does to the picture #

One case is worth adding because it is where the curve stops being smooth.

An alternating chain — one whose bonds are unequal, as a chain cannot stay even shows a half-filled chain making itself — has a gap in the middle of its band. Its filling curve is not the smooth arch above: the binding rises to the gap, and then filling past the gap means putting electrons above it, so the curve has a corner at half filling rather than a smooth maximum.

That corner is the whole of why the distortion happens. A system exactly at half filling gains from opening a gap, because every electron it has is below the gap and every level pushed up is empty; a system away from half filling gains nothing, because the levels the gap moves are ones it has already filled or will never fill.

The gap is not the band width computes the two quantities separately, and the sequences separate cleanly: a uniform chain’s gap falls from $0.299$ to $0.039$ across twenty to a hundred and sixty sites and is still halving, while an alternating one settles on four times the alternation and stays there.

The maximum sits at half filling because the chain has no triangles #

The symmetry of the curve about half filling has been checked, and it deserves a cause, because the cause turns out to name a fourth reason for the departure the measured metals show — one that survives even in a model with no s band, no magnetism and no core repulsion at all.

A chain and an even ring are bipartite: the sites can be coloured alternately with no bond joining two of the same colour. Every closed walk on such a structure has an even number of steps, so every odd moment of the spectrum vanishes, and a distribution all of whose odd moments are zero is symmetric about its mean. That is the pairing theorem, and it is why the filling curve comes back on itself and why the maximum is exactly in the middle.

The first odd moment that could break it is the third, and it has a completely concrete meaning:

$$\operatorname{Tr}(A^3) = 6 \times (\text{number of triangles}),$$

because a closed walk of three steps is a triangle traversed one way or the other from any of its three vertices. The skew of a band is a count of its three-membered rings, in exactly the way the width is a count of its neighbours.

A chain has none, so its band is symmetric. Close-packed metals are made of almost nothing else: in a face-centred cubic or hexagonal lattice every atom sits in a mesh of triangles, and the third moment is large and positive.

A positive third moment puts a long tail at the top of the band and piles the weight toward the bottom, so more than half the levels lie below the mean. The binding is largest when the filling has just exhausted the levels below the mean and taken none above it, so the maximum moves to a filling greater than one half.

That is the direction the measurements go. Cohesion across each transition series peaks a little past the middle, and the essay above attributes the shift to s–d overlap, to magnetism and to core repulsion — all real, all omitted here. The triangle count is a fourth cause of the same shift, it is present in a model with one orbital per site and nothing else in it, and it is the only one of the four that can be evaluated by looking at the lattice rather than by computing anything.

Two things follow that are worth separating.

The exact zero at complete filling survives. It comes from the trace of the matrix rather than from any symmetry of the spectrum, so it holds for a triangular lattice, a face-centred cubic one and a random network alike. A filled band supplies no binding whatever the shape of the band is.

The maximum at exactly half filling does not. It is a theorem about bipartite structures quoted as though it were a theorem about bands, which is the same kind of over-extension the band-width formula suffers from — and the honest version has the scope in it: for a structure with no odd rings, the binding is largest at half filling.

Which is a satisfying place for the arithmetic of filling to arrive at, since it means the two most-quoted facts about a band’s filling are protected by two quite different things: one by a trace, and one by a colouring.

Where the model stops #

There is no repulsion between the atoms. A real solid is held apart by the overlap repulsion of filled cores, and the equilibrium spacing is where that balances the attraction computed here. Without it, this model predicts a solid of zero size.

There is no electron-electron repulsion either. A half-filled band is not always a metal is the essay about what that omission costs, and it costs most exactly at half filling, which is where this essay’s maximum sits.

There is one orbital per site. Real cohesion in a transition metal involves an s band and a d band overlapping, with the s band nearly free-electron-like and the d band narrow, and the total is not the sum of two independent contributions.

And the units are β. Every number here is in units of a hopping integral that is fitted rather than computed, so the curve has a shape and no scale.

What actually holds solids together spans four kinds of binding, and the band term computed here is the covalent and metallic column. The ionic column is a lattice sum and the dispersion column is a different calculation again — this curve is a statement about one of the four.

Who found it, and when #

The band-filling account of cohesion in transition metals is Jacques Friedel’s, from the 1960s, and it is usually presented with a rectangular density of states — which gives a parabola in the filling and the same maximum at the middle. The refinement into a quantitative theory is Pettifor’s and others’ through the 1970s and 1980s.

The exact statement about a filled band is much older and belongs to the molecular side of the subject. Hund and Mulliken’s molecular orbital treatment of the 1920s gives the same result for two atoms — a filled bonding-and-antibonding pair supplies no net bonding — and it was the explanation for why the noble gases are monatomic before there were bands to apply it to.

That the two are the same statement at different sizes is the thread this whole field is strung on. A band is what a set of molecular orbitals becomes when there are enough of them, and the sum rules that hold for two atoms hold for two thousand because they come from the trace of a matrix rather than from anything about size.

Still open: the repulsive half of cohesion #

The band picture starts from a solid as a molecule that did not stop, goes through the width of a band as a count of neighbours and the density of states as something other than a spectrum, and reaches the question of where a molecule stops being one. Filling the band up finds a maximum in the middle and an exact zero at the end.

What it points at, and cannot do, is the other half of cohesion: the repulsive term. What holds a solid together compares four kinds of binding by magnitude and finds an ionic pair at 5.14 eV against a covalent β of about 2.5 — and the comparison turned out to be the defect rather than the ordering, since an ionic solid’s binding is a lattice sum and a covalent solid’s is not. The same caution applies to everything in this essay: a curve with a shape and no scale is a statement about which filling, and never about how much.