Electrons in Solids: Band Theory
Why one solid conducts electricity and the next does not is the oldest quantitative question in the physics of matter, and the answer — energy bands — is the most consequential single application of quantum mechanics to anything larger than a molecule. This chapter builds it in the order in which the evidence forced it. Drude modelled a metal as a classical gas of electrons scattering off the ions [Drude:1900a] [Drude:1900b], recovered Ohm's law and the Wiedemann–Franz ratio measured half a century earlier [Wiedemann:1853], and failed by two orders of magnitude on the electronic specific heat. Sommerfeld replaced Maxwell–Boltzmann by the Fermi–Dirac statistics of Quantum Statistics and repaired the thermal properties at a stroke [Sommerfeld:1928]. Bloch then put the electrons into the periodic potential of the lattice of Phonons and Lattice Dynamics and proved the theorem [Bloch:1929] from which bands, gaps and Wilson's criterion separating metals from insulators [Wilson:1931a] [Wilson:1931b] all follow.
The chapter's second half is about evidence, because a band structure is a falsifiable curve \(E_{n}(\vect{k})\) and not a picture. The Fermi surface is measured by the de Haas–van Alphen oscillations [deHaas:1930] read through Onsager's relation [Onsager:1952], by cyclotron resonance [Dresselhaus:1955], and today directly by angle-resolved photoemission [Damascelli:2003]. The chapter closes honestly on where the one-electron picture breaks: on the Mott insulators that band theory declares to be metals [Mott:1949], and on disorder [Anderson:1958]. Its neighbours are the lattice dynamics of Phonons and Lattice Dynamics before it and, after it, the doped band insulators of Semiconductors, the quantized magnetotransport of Experiment: The Quantum Hall Effect, the ordered moments of Magnetism in Matter and the paired condensate of Superconductivity and Superfluidity. The standard modern treatment is [Ashcroft:1976].
Electrons in Solids: Band Theory: all derivations of this chapter are pending.
The free-electron metal
Drude's kinetic model
[Reserved: the metal as a classical gas of conduction electrons scattering off fixed ions, introduced by Drude [Drude:1900a]; the relaxation-time equation of motion, the direct-current conductivity \(\sigma=ne^{2}\tau/m\) recovering Ohm's law, the frequency-dependent Drude response and the plasma frequency; the galvanomagnetic and thermomagnetic sequel [Drude:1900b], in which the same model delivers the Hall coefficient \(R_{H}=-1/ne\) for the effect Hall had discovered [Hall:1879]; relaxation times of order \(10^{-14}\,\mathrm{s}\) at room temperature, inferred rather than assumed; the kinetic-theory machinery imported from Kinetic Theory of Gases.]
Two facts about metallic conduction are simple enough to look uninformative and are not. First, the current density in a metal is strictly proportional to the applied electric field over many decades of field strength, with a conductivity that depends on temperature and purity but not on the field: a metal has no memory of how hard it is being driven. Second, a current \(j_{x}\) crossed with a magnetic field \(B\) along \(z\) develops a steady transverse electric field \(E_{y}\), in the ratio
which Hall found to be independent of the field, of the current and of the sample dimensions [Hall:1879]. For the alkali metals \(R_{H}\) is negative and its magnitude corresponds to about one mobile carrier of charge \(-e\) per atom [Ashcroft:1976]. That a solid should hand back its carrier density to a voltmeter is the first quantitative measurement of the electron gas.
Derivation. Drude's model treats the conduction electrons as free particles of charge \(-e\) and mass \(m\), accelerated by the applied fields and randomized by collisions at a mean rate \(1/\tau\), so that on average the collisions remove momentum at the rate \(m\vect{v}/\tau\) [Drude:1900a]. The steady-state equation of motion is
With \(\vect{B}=0\) this gives \(\vect{v}=-e\tau\vect{E}/m\) and hence
which is Ohm's law: the response is linear because the drag term in Equation (126.2) is linear, and the conductivity contains no reference to \(\vect{E}\). Now take \(\vect{B}=B\hat{\vect{z}}\) and a sample whose sides are insulating, so that in the steady state \(v_{y}=0\). The \(y\) component of Equation (126.2) then reads \(-e(E_{y}-v_{x}B)=0\), so \(E_{y}=v_{x}B\); and \(j_{x}=-nev_{x}\) gives \(v_{x}=-j_{x}/ne\). Substituting into Equation (126.1),
independent of \(\tau\), of \(m\) and of the field — which is why the measurement is a carrier count and not a fit. Two things should be noted about what has not been derived. The sign in Equation (126.4) is fixed by the assumption that the carriers are electrons, and it is contradicted in several metals (Section 126.1.3); and \(\tau\) has been postulated, not computed, so Equation (126.3) explains the linearity of Ohm's law but not the magnitude or the temperature dependence of \(\sigma\) (Section 126.6).
∎The Wiedemann–Franz law
[Reserved: the empirical proportionality of thermal to electrical conductivity at fixed temperature, measured across the metals by Wiedemann and Franz [Wiedemann:1853]; the Lorenz ratio \(L=\kappa/\sigma T\) and its near-universal value close to \(2.44\times 10^{-8}\,\mathrm{W}\,\mathrm{\Omega}/\mathrm{K}^{2}\); why Drude obtained it almost exactly right through two compensating errors of order one hundred, in the electronic heat capacity and in the mean square speed [Drude:1900a] [Ashcroft:1976]; the regimes in which the law is observed to fail, and what inelastic scattering has to do with it.]
At ordinary temperatures the thermal conductivity \(\kappa\) and the electrical conductivity \(\sigma\) of a metal are proportional, and the constant of proportionality is very nearly the same for every metal: the Lorenz ratio
clusters about \(2.44\times 10^{-8}\,\mathrm{W}\,\mathrm{\Omega}/\mathrm{K}^{2}\) across metals differing by orders of magnitude in \(\sigma\) itself [Wiedemann:1853] [Ashcroft:1976]. A ratio that survives when its two members do not is a strong statement: the same carriers, scattered by the same mechanism, must be responsible for both currents. The law fails, in a way that is itself diagnostic, at intermediate temperatures, where the scattering that degrades a heat current and the scattering that degrades a charge current cease to be the same.
Derivation. Kinetic theory gives the thermal conductivity of a gas of carriers as \(\kappa=\tfrac{1}{3}c_{V}v^{2}\tau\), with \(c_{V}\) the heat capacity per unit volume of the carriers and \(v\) their typical speed. For a degenerate electron gas both quantities are those of the Fermi surface, not of a classical gas: from Quantum Statistics, \(c_{V}=\tfrac{1}{2}\pi^{2}nk_{B}(T/T_{F})\), and the carriers move at the Fermi speed, \(\tfrac{1}{2}mv_{F}^{2}=k_{B}T_{F}\). Hence
Dividing by \(\sigma T\) with \(\sigma=ne^{2}\tau/m\) from Equation (126.3), the carrier density, the mass and — the essential point — the relaxation time all cancel:
a combination of fundamental constants alone, which is why it is the same for every metal. The cancellation of \(\tau\) is what makes Equation (126.7) robust, and its failure is what makes the observed deviations informative: the derivation assumed one relaxation time for both currents, whereas a small-angle inelastic collision with a phonon can relax an energy current while barely relaxing a charge current. Drude obtained almost this number classically because his two errors were reciprocal: the classical heat capacity is too large by about \(T_{F}/T\sim10^{2}\) and the classical mean square speed too small by about the same factor, so their product survived while neither factor did [Drude:1900a] [Ashcroft:1976].
∎What the classical electron gas cannot do
[Reserved: the missing electronic specific heat — classically \(\tfrac{3}{2}k_{B}\) per electron, observed near room temperature to be smaller by about two orders of magnitude; the sign of the Hall coefficient, positive in aluminium, zinc and beryllium where [Drude:1900b] demands it be negative; the complete absence of any account of why a solid should be an insulator at all; each of these a symptom of applying Maxwell–Boltzmann statistics to a degenerate fermion gas, and each resolved in Quantum Statistics.]
In aluminium, zinc, beryllium and cadmium the measured Hall coefficient Equation (126.1) is positive, and in several metals its magnitude does not correspond to any plausible carrier density [Hall:1879] [Ashcroft:1976]. Equation (126.4) admits no such value: within the free-electron picture the sign of \(R_{H}\) is the sign of the carrier charge, and the carriers are electrons. The observation is therefore not a quantitative discrepancy but a qualitative refutation, and the fact that it occurs in ordinary, well-characterized, chemically simple metals rules out an explanation by impurities or by a second species of ion.
The dynamics of a nearly filled band described through its vacancies: that the total current of a filled band vanishes, so that the current of an almost-filled band equals minus the current of the missing states; that a wave packet near a band maximum has negative band curvature and therefore responds as a particle of positive charge and positive mass; and the resulting positive Hall coefficient, with the multi-band and open-orbit corrections that govern its magnitude
The Sommerfeld model
[Reserved: Sommerfeld's substitution of the Fermi–Dirac distribution into Drude's kinetic picture [Sommerfeld:1928]; the Fermi sphere, the Fermi energy \(E_{F}=\hbar^{2}k_{F}^{2}/2m\) of a few \(\mathrm{eV}\) and the Fermi temperature of order \(10^{4}\,\mathrm{K}\) that makes every metal a degenerate gas at room temperature; the Sommerfeld expansion, the linear electronic heat capacity \(c=\gamma T\) and the measured coefficients \(\gamma\) of the alkali metals; Pauli paramagnetism [Pauli:1927a] and Landau diamagnetism [Landau:1930] as the magnetic counterparts, taken up in Magnetism in Matter; exactly what of Drude survives.]
Electrons in a periodic potential
Bloch's theorem
[Reserved: the theorem that eigenfunctions of a Hamiltonian invariant under the lattice translations of Phonons and Lattice Dynamics take the form \(\psi_{n\vect{k}}(\vect{r})=\ee^{\ii\vect{k}\cdot\vect{r}} u_{n\vect{k}}(\vect{r})\) with \(u_{n\vect{k}}\) lattice periodic, proved by Bloch [Bloch:1929]; crystal momentum \(\hbar\vect{k}\) as a label of an irreducible representation of the discrete translation group and not a true momentum; the band index \(n\); Born–von Karman boundary conditions and the counting of allowed \(\vect{k}\) that gives \(N\) states per band per spin; the group-theoretic reading via Linear Algebra and Representation Theory.]
Reciprocal lattice and Brillouin zones
[Reserved: the reciprocal lattice and the equivalence \(\vect{k}\sim\vect{k}+\vect{G}\) that makes every band a periodic function on a torus; the first Brillouin zone as the Wigner–Seitz cell of the reciprocal lattice, introduced together with the reading of zone boundaries as Bragg planes by Brillouin [Brillouin:1930]; the consequent link between a band gap and the diffraction condition measured in Phonons and Lattice Dynamics; reduced, extended and repeated zone schemes; high-symmetry points and the conventional paths along which band structures are plotted.]
Nearly free electrons
[Reserved: degenerate perturbation theory (Approximation Methods) applied to a weak periodic potential; the gap \(2\abs{V_{\vect{G}}}\) that opens at each Bragg plane, the distortion of the free-electron parabola and the origin of the very idea of a forbidden band [Bloch:1929] [Ashcroft:1976]; why the weak-potential limit describes real sp-bonded metals far better than the strength of the ionic potential suggests, which is the pseudopotential argument of Section 126.3.3 [Phillips:1959].]
Tight binding and Wannier functions
[Reserved: the opposite limit of weakly overlapping atomic orbitals; the band \(E(\vect{k})=\epsilon-2t\sum_{i}\cos(k_{i}a)\) of a simple cubic lattice, with bandwidth set by the hopping integral \(t\) and centre by the on-site energy \(\epsilon\); the Slater–Koster parametrization of the two-centre integrals that made the method practical [Slater:1954]; Wannier functions as the localized unitary dual of the Bloch states [Wannier:1937]; why two limits that could hardly differ more agree on the qualitative band picture.]
The Kronig–Penney model
[Reserved: the exactly solvable periodic square-well chain of Kronig and Penney [Kronig:1931]; the transfer-matrix condition \(\cos(ka)=\cos(\alpha a)+P\sin(\alpha a)/(\alpha a)\), whose failure to admit a real \(k\) over finite energy intervals is a band gap in closed form; the delta-comb limit and the width of the bands as a function of barrier strength; what the model proves — that periodicity alone, and no special property of atoms, produces gaps.]
Band structure and the classification of solids
Filling and Wilson's criterion
[Reserved: the count of \(2N\) states per band including spin, and the resulting rule that a crystal with an odd number of electrons per primitive cell must be a metal; Wilson's identification of an insulator as a solid whose bands are exactly filled, and of a semiconductor as an insulator whose gap is small enough to be bridged thermally [Wilson:1931a] [Wilson:1931b]; the divalent metals, saved by band overlap; the criterion's known counterexamples, deferred honestly to Section 126.7.2.]
The room-temperature electrical resistivity of solids spans more than twenty orders of magnitude, from about \(10^{-8}\,\mathrm{\Omega}\,\mathrm{m}\) in a good metal to above \(10^{16}\,\mathrm{\Omega}\,\mathrm{m}\) in a good insulator, with no bunching in between that would mark a natural boundary. What does mark one is the sign of the temperature dependence: on cooling, the resistivity of a metal falls and tends to a finite residual value, while that of an insulator or semiconductor rises without limit. The same distinction appears in the low-temperature heat capacity, which carries the electronic term \(\gamma T\) of Quantum Statistics in a metal and does not in an insulator. Empirically the class a crystal belongs to correlates with the number of electrons in its primitive cell rather than with any property of the individual atoms: an odd number of electrons per cell has never yielded an insulator [Wilson:1931a] [Wilson:1931b].
Bloch's theorem for a periodic potential and the counting that follows from it: that the allowed wavevectors in one Brillouin zone number the primitive cells of the crystal, so that each band holds exactly two electrons per cell; that a crystal whose bands are either completely full or completely empty has no states available at infinitesimal excitation energy and therefore carries no current at zero temperature; and the consequent identification of a semiconductor as an insulator whose gap is comparable with the thermal energy
Semimetals, semiconductors and the gap scale
[Reserved: the measured gaps that set the scale of all electronics — about \(1.12\,\mathrm{eV}\) in silicon, \(0.67\,\mathrm{eV}\) in germanium, \(1.42\,\mathrm{eV}\) in gallium arsenide, against about \(5.5\,\mathrm{eV}\) in diamond [Madelung:2004]; direct against indirect gaps and the phonon that must be supplied in the second case; the semimetals bismuth, antimony and graphite, with band overlaps of \(\mathrm{meV}\) and carrier densities four orders below a normal metal; handover to Semiconductors.]
Computing band structures
[Reserved: the cellular method and the first quantitative band calculation for a real solid, metallic sodium, by Wigner and Seitz [Wigner:1933]; orthogonalized plane waves [Herring:1940]; the pseudopotential construction that explains why nearly-free-electron bands work [Phillips:1959]; the augmented-plane-wave and \(k\cdot p\) families [Ashcroft:1976]; what “ab initio” does and does not mean here, and the discipline that every computed band is judged against the measurements of Section 126.5.]
Density functional theory
[Reserved: the Hohenberg–Kohn theorems — the ground-state density fixes the external potential up to a constant, and the energy is a variational functional of that density [Hohenberg:1964]; the Kohn–Sham construction trading the interacting problem for non-interacting particles in a self-consistent effective potential [Kohn:1965]; the local-density and generalized-gradient approximations [Perdew:1996]; and the honest caveats — that Kohn–Sham eigenvalues are not excitation energies, that computed gaps come out low by tens of percent, and that the many-body \(GW\) self-energy [Hedin:1965] is what brings them into agreement with the photoemission of Section 126.5.4.]
Semiclassical dynamics of Bloch electrons
Group velocity and effective mass
[Reserved: the wave-packet velocity \(\vect{v}_{n}=\hbar^{-1}\nabla_{\vect{k}}E_{n}(\vect{k})\) and the semiclassical force law \(\hbar\dot{\vect{k}}=-e(\vect{E}+\vect{v}\times \vect{B})\), in which the lattice force is absorbed entirely into the band; the inverse effective-mass tensor \((m^{*})^{-1}_{ij}=\hbar^{-2}\pp^{2}E/\pp k_{i}\pp k_{j}\), introduced with the theory of galvanomagnetic effects by Peierls [Peierls:1929a]; measured values spanning \(0.067\,m_{e}\) in gallium arsenide to tens of \(m_{e}\) in heavy-fermion metals; the interband conditions under which the semiclassical description is legitimate.]
Holes
[Reserved: an almost-full band described by its vacancies; the hole's positive charge, positive effective mass near a band maximum, and hence the positive Hall coefficient observed in aluminium, zinc and beryllium — precisely the anomaly of Section 126.1.3; Peierls' introduction of the concept [Peierls:1929a]; holes as the majority carriers of p-type material in Semiconductors, and the care needed not to confuse this collective vacancy with the antiparticle of The Dirac Equation.]
Bloch oscillations and Zener breakdown
[Reserved: a static field sweeps \(\vect{k}\) uniformly across the zone, so a perfect crystal should oscillate at \(eEa/\hbar\) rather than conduct [Bloch:1929]; why scattering destroys the effect in bulk metals and why it is nevertheless observed in semiconductor superlattices, whose small zone raises the frequency above the scattering rate [Feldmann:1992]; interband tunnelling at large field, computed by Zener [Zener:1934], which returns as the tunnel diode of Semiconductors.]
Magnetotransport and the Hall coefficient
[Reserved: Hall's discovery that a field transverse to a current produces a transverse voltage [Hall:1879]; the free-electron result \(R_{H}=-1/ne\) and its everyday use as a carrier-density measurement; magnetoresistance, closed and open orbits, and the saturation criterion that reads Fermi-surface topology directly off a magnetoresistance curve [Ashcroft:1976]; the same geometry taken to two dimensions, high field and low temperature becomes the quantized effect of Experiment: The Quantum Hall Effect.]
The Fermi surface and how it is measured
Landau levels and quantum oscillations
[Reserved: the quantization of cyclotron orbits into Landau levels [Landau:1930], with degeneracy \(eB/h\) per unit area, and the passage of those levels through the Fermi energy periodically in \(1/B\); the Lifshitz–Kosevich amplitude formula with its temperature and Dingle damping factors [Lifshitz:1956]; why oscillations demand \(\omega_{c}\tau\gg1\), hence high-purity crystals, fields of order \(10\,\mathrm{T}\) and temperatures of order \(1\,\mathrm{K}\).]
The de Haas–van Alphen effect
[Reserved: the oscillation of the magnetic susceptibility of bismuth periodic in \(1/B\), discovered by de Haas and van Alphen [deHaas:1930]; Onsager's relation \(\Delta(1/B)=2\pi e/\hbar A_{\mathrm{ext}}\), which ties the observed period to an extremal cross-sectional area of the Fermi surface and thereby converted a curiosity into the primary instrument of the field [Onsager:1952]; the reconstructed Fermi surfaces of copper, gold and aluminium, with their necks touching the zone boundary [Shoenberg:1984]; Pippard's independent determination for copper from the anomalous skin effect [Pippard:1957].]
The magnetic susceptibility of a pure crystal of bismuth at low temperature does not vary smoothly with the applied field but oscillates, and the oscillations are periodic not in \(B\) but in \(1/B\) [deHaas:1930]. The same periodicity appears in the resistivity [Shubnikov:1930] and, as instrumentation improved, in every clean metal examined. The observed frequencies are properties of the material alone; there may be several of them, and they change in a characteristic way as the crystal is rotated in the field. Read through Onsager's relation [Onsager:1952], each frequency measures an extremal cross-sectional area of the Fermi surface normal to the field, which converts the effect into a tomograph: the Fermi surfaces of copper, gold and aluminium, complete with the necks by which they touch the zone boundary, were reconstructed this way [Shoenberg:1984] and confirmed independently for copper from the anomalous skin effect [Pippard:1957]. The requirement is a cyclotron period short compared with the scattering time, hence high purity, fields of order \(10\,\mathrm{T}\) and temperatures of order \(1\,\mathrm{K}\).
Semiclassical quantization of a closed cyclotron orbit, giving orbits of quantized area in reciprocal space evenly spaced in the magnetic field; the resulting oscillation of the free energy as successive quantized tubes cross the Fermi surface, periodic in the inverse field with a frequency proportional to the extremal cross-sectional area; and the Lifshitz–Kosevich amplitude with its temperature and scattering damping factors
Shubnikov–de Haas and cyclotron resonance
[Reserved: the same quantization seen in the resistivity rather than the susceptibility [Shubnikov:1930], the transport analogue that returns in Experiment: The Quantum Hall Effect; cyclotron resonance, in which microwave power is absorbed at \(\omega=eB/m^{*}\), used by Dresselhaus, Kip and Kittel to measure the effective-mass tensors of silicon and germanium directly [Dresselhaus:1955] — the measurement that established the many-valley conduction bands underlying Semiconductors.]
Photoemission
[Reserved: angle-resolved photoemission as the only probe that maps \(E_{n}(\vect{k})\) directly, using the photoelectric effect of The Photon: Photoelectric and Compton Effects together with energy and in-plane momentum conservation at the surface; what is actually measured — the one-particle spectral function, hence self-energies, lifetimes and mass renormalizations — and the surface sensitivity and final-state uncertainty that limit it [Damascelli:2003]; present resolutions of a few \(\mathrm{meV}\) and the direct imaging of Fermi surfaces, kinks and gaps [Sobota:2021].]
Transport and scattering
Residual resistivity and Matthiessen's rule
[Reserved: the striking consequence of Bloch's theorem that a perfect periodic crystal has infinite conductivity, so that all resistance is a defect of periodicity; impurity and defect scattering, the temperature-independent residual resistivity, and the residual resistance ratio as the standard purity figure of merit; the additivity of scattering rates measured by Matthiessen and Vogt in alloys [Matthiessen:1864] and the conditions under which it fails; the Boltzmann transport equation that derives it, developed in Nonequilibrium Thermodynamics and Transport.]
The resistivity of a metal is, to a good approximation, a sum of two independent contributions,
in which \(\rho_{0}\) is temperature-independent and depends only on the concentration of impurities and defects, while \(\rho_{\mathrm{i}}(T)\) depends only on the temperature and is the same function for every sample of the metal, however pure — the rule Matthiessen and Vogt established on alloy series [Matthiessen:1864]. Two consequences are measured. As \(T\to0\) the resistivity saturates at \(\rho_{0}\) instead of vanishing, so the residual resistance ratio is the standard figure of merit for crystal purity; and \(\rho_{\mathrm{i}}\) is linear in \(T\) well above the Debye temperature and falls as a high power of \(T\) well below it, following the form Bloch derived and Grüneisen fitted across the metals [Bloch:1930a] [Gruneisen:1933]. The striking part is \(\rho_{0}\): a perfectly periodic crystal would have none, so every ohm a metal presents is a defect of its periodicity.
That a Bloch state is a stationary state of the perfect periodic potential and therefore propagates without attenuation, so that resistance requires a departure from periodicity; the Boltzmann equation in the relaxation-time approximation and the additivity of scattering rates from independent channels that yields Matthiessen's rule, together with the conditions under which the additivity fails; and the electron–phonon calculation giving the linear high-temperature and the steep low-temperature resistivity
Phonon scattering and the Bloch–Grüneisen law
[Reserved: electron–phonon scattering off the lattice modes of Phonons and Lattice Dynamics; the linear high-temperature resistivity and the \(T^{5}\) low-temperature law derived by Bloch [Bloch:1930a] and fitted across the metals by Grüneisen [Gruneisen:1933]; normal against umklapp processes and why only the latter degrade a current [Ashcroft:1976]; the electron–phonon coupling constant that reappears as the pairing interaction of Superconductivity and Superfluidity; thermopower, the Peltier effect and thermal transport deferred to Nonequilibrium Thermodynamics and Transport.]
Where the one-electron picture fails
Fermi-liquid theory
[Reserved: the question the whole chapter has been begging — why a dense, strongly interacting electron liquid behaves like a gas of independent particles at all; Landau's answer in terms of quasiparticles adiabatically connected to the free states, carrying a renormalized mass and a lifetime diverging as \((E-E_{F})^{-2}\) at the Fermi surface [Landau:1957b]; the experimental signatures — enhanced but still linear specific heat, \(T^{2}\) resistivity, and the sharp quasiparticle peak seen in Section 126.5.4; the measured metals that violate all three.]
Mott insulators and the Hubbard model
[Reserved: the solids band theory declares to be metals and that are measured to be insulators — nickel oxide, cobalt oxide, the transition-metal monoxides — and Mott's argument that on-site Coulomb repulsion rather than band filling localizes their electrons [Mott:1949] [Mott:1968]; the Hubbard Hamiltonian, hopping \(t\) against on-site repulsion \(U\), and the transition near \(U\sim W\) [Hubbard:1963]; the antiferromagnetic order that usually accompanies it, in Magnetism in Matter; recorded as an open problem of What We Observe but Do Not Understand rather than as a solved one.]
Nickel oxide, cobalt oxide and the other transition-metal monoxides have an odd number of \(d\) electrons per primitive cell and a partially filled band, and by the counting of Phenomenon 126.4 must therefore be metals. They are measured to be insulators, and good ones, with activated conduction and resistivities many orders of magnitude above any metal [Mott:1949] [Mott:1968]. The failure is not marginal and cannot be repaired by a better band calculation, because it is not a failure of approximation but of the premise: the one-electron picture treats the Coulomb repulsion between electrons as an average potential, and in a narrow band the cost of putting two electrons on the same site exceeds the energy gained by letting them move.
The Hubbard Hamiltonian, in which a hopping amplitude competes with an on-site repulsion, and the localization of one electron per site when the repulsion exceeds the bandwidth — a transition to an insulating state that occurs at fixed, partial band filling and therefore has no description in band theory at all; noting that no controlled solution of this model in three dimensions exists, so the derivation available is a limiting-case argument and not a theory
Disorder and localization
[Reserved: Anderson's demonstration that sufficiently strong random disorder localizes the single-particle eigenstates outright, so that diffusion stops and conduction ceases with no gap anywhere in the spectrum [Anderson:1958]; the mobility edge separating localized from extended states; weak localization and its characteristic negative magnetoresistance; and the strongest quantitative evidence for the whole picture — localization is what gives the quantum Hall plateaux of Experiment: The Quantum Hall Effect a finite width.]