Why Nickel Oxide Refuses to Conduct: The Mott Insulator and the Cost of Double Occupancy — Epoche C1
Band theory — the standard quantum description of electrons in a crystal — comes with a famously simple rule. Because electrons in a periodic potential occupy states labelled by a wavevector confined to one Brillouin zone, and because each such state takes two electrons of opposite spin, every band holds exactly two electrons per unit cell, the smallest repeating block of the crystal. Count the electrons per cell and fill the bands from the bottom: a partly filled band means a metal, since electrons at the top of the filled region have empty states immediately above them and an applied field can accelerate them; only completely full or completely empty bands make an insulator. The rule works for copper, for silicon, for diamond. This essay takes a case where it fails spectacularly, identifies the assumption that breaks, and follows the repair — due chiefly to Nevill Mott and John Hubbard — through to its experimental tests. The problem: a predicted metal that insulates Nickel oxide, NiO, is the classic offender. The Ni 2+ ion holds eight electrons in its 3d shell, which offers ten states — five orbitals, two spins each. In the rocksalt crystal these atomic states broaden into a d band that is therefore filled to eight-tenths of its capacity, and the counting rule says "metal". A partly filled d band should conduct roughly like other transition-metal compounds, with a room-temperature resistivity of order $10^{-3}\ \Omega\,\mathrm{cm}$. Real NiO crystals, even imperfect ones, show resistivities of order $10^{7}\ \Omega\,\mathrm{cm}$ and higher. Subtracting the exponents of these two figures, $7-(-3)=10$: the prediction fails by at least ten orders of magnitude. The words "at least" are doing real work. NiO grown in air is systematically nickel-deficient, and each missing Ni 2+ must be charge-compensated by two holes elsewhere; those holes carry current, so measured resistivities vary by several orders of magnitude between samples and the quoted figure is a floor rather than the intrinsic value. This is not a defect in the evidence but part of it, since it is the first hint that removing electrons restores conduction — a point the last section turns into a test. De Boer and Verwey reported the anomaly in 1937, so band theory has carried this open wound almost from birth. Nor is the failure an artefact of the crude band calculations of the 1930s: density-functional calculations in the local density approximation, the workhorse of modern electronic structure theory, still place the Fermi level inside the partly filled d manifold and predict a metal. Ruling out the alternative explanation Before accepting that band theory needs repair, one must dispose of a rival that would save it, because band theory has a legitimate mechanism for opening a gap in a partly filled band: magnetic order. Slater pointed this out in 1951. NiO orders antiferromagnetically, with the magnetic moments on neighbouring nickel sublattices pointing in opposite directions, and an antiferromagnetic arrangement has a repeating unit twice the size of the chemical one. Doubling the unit cell halves the Brillouin zone, folds the band back on itself, and opens a gap at the new zone boundary. On this account NiO is a perfectly ordinary band insulator whose gap happens to be produced by spin order, and no new physics is required. The prediction that distinguishes the two accounts is unambiguous. If the magnetic order opens the gap, the gap must close when the order disappears. NiO's Néel temperature — the temperature above which the antiferromagnetic arrangement is destroyed by thermal disorder — is $523\ \mathrm{K}$. NiO remains an insulator far above it, up towards its melting point, with no transition of any kind at $523\ \mathrm{K}$ in its transport behaviour. The energies make the same point more sharply. The measured gap is about $4\ \mathrm{eV}$. The magnetic ordering temperature corresponds to an energy of $k_B \times 523\ \mathrm{K} = 0.045\ \mathrm{eV}$. The gap is some ninety times the energy scale of the magnetism that was supposed to have caused it, and an effect cannot exceed its cause by two orders of magnitude. Consistently, when antiferromagnetic order is imposed on a local-density calculation the gap it produces is a few tenths of an electronvolt — the right scale for the magnetism, and an order of magnitude short of the measurement. Slater's mechanism is real and operates in some materials; it is not what is happening in NiO. Diagnosis: which assumption breaks Band theory treats each electron as moving independently through a fixed, averaged potential; the repulsion between individual electrons is smeared into that average. The approximation is good for broad, overlapping orbitals, where two electrons rarely sit close together and the fluctuation about the average is small compared with the kinetic energy. It is poor in the compact 3d orbitals of nickel, and the size of the neglected quantity can be estimated directly. The quantity is the on-site Coulomb repulsion, written $U$: the energy cost of putting a second electron into an orbital that already holds one. Formally it is the Coulomb integral of the orbital density against itself, $$ U \;=\; \iint |\phi(\mathbf{r})|^{2}\,\frac{e^{2}}{4\pi\varepsilon_0\,|\mathbf{r}-\mathbf{r}'|}\,|\phi(\mathbf{r}')|^{2}\;\mathrm{d}^{3}r\,\mathrm{d}^{3}r' , $$ which is of order $e^{2}/4\pi\varepsilon_0 \langle r\rangle$ for an orbital of mean radius $\langle r\rangle$. In convenient units $e^{2}/4\pi\varepsilon_0 = 14.4\ \mathrm{eV}\,\text{Å}$, and a 3d orbital on a transition-metal ion has $\langle r\rangle$ of about $0.6\ \text{Å}$, giving roughly $24\ \mathrm{eV}$. That is the bare value for a free ion. In the solid the surrounding oxygen ions are polarisable and the remaining d electrons rearrange, both of which screen the added charge; the effective $U$ that governs the solid comes out around $8\ \mathrm{eV}$. Even after a threefold reduction it is an enormous energy on