The Nearly-Free-Electron Model

Bloch's theorem guarantees that the eigenstates are plane waves modulated by a periodic envelope, but it says nothing about where the bands lie or how wide the gaps are. To get numbers one must solve the central equation, and the cleanest starting point is the limit in which the periodic potential is weak. In a metal the conduction electrons screen the ion cores so effectively that the residual potential seen by an electron near the Fermi energy is a small fraction of the Fermi energy itself. The nearly-free-electron model treats that residual potential as a perturbation on the free-electron gas and shows that even an infinitesimal periodic potential opens gaps at the zone boundaries, converting the single free-electron parabola into a set of bands.

The empty-lattice starting point

With $U = 0$ the eigenstates are plane waves $e^{i q \cdot r}$ with energy $ℏ^{2} q^{2} /2 m$ . Bloch's theorem is still satisfied trivially: any plane wave is a Bloch wave with $u = const$ , and its wavevector $q$ can be written $k - G$ for a unique $k$ in the first zone and some reciprocal-lattice vector $G$ . Folding every plane wave into the first zone this way gives the empty-lattice bands: the free-electron parabola cut into segments at the zone boundaries and translated back, so that at each $k$ there is a ladder of energies

E_{G}^{0} (k) = \frac{ℏ ^{2}}{2 m} ∣ k - G ∣^{2},

one for each $G$ . These are the free-electron energies relabeled; no gaps exist yet. The bands cross wherever two of them are degenerate, and those crossings are where a weak potential has its entire effect.

The free-electron parabola (extended zone, faint) folds at the zone boundaries k = plus or minus pi/a into the reduced zone, producing the empty-lattice bands (solid). Distinct parabola segments become distinct band branches that touch at the boundary and at the zone centre.

Degenerate perturbation theory at a zone boundary

Return to the central equation for the coefficients of a Bloch state of wavevector $k$ ,

(\frac{ℏ ^{2}}{2 m} ∣ k - G ∣^{2} - E) c_{k - G} + G^{'} \sum U_{G^{'} - G} c_{k - G^{'}} = 0.

Take the potential real with $U_{0} = 0$ (a constant shifts every energy and can be dropped). Away from any degeneracy, one plane wave $k$ dominates and the others enter only at second order, giving the small correction

E (k) = \frac{ℏ ^{2} k ^{2}}{2 m} + G \neq = 0 \sum \frac{∣ U _{G} ∣ ^{2}}{\frac{ℏ ^{2} k ^{2}}{2 m} - \frac{ℏ ^{2}}{2 m} ∣ k - G ∣ ^{2}} + \dots,

which merely bends the parabola slightly. The perturbation series diverges, however, exactly when the denominator vanishes — when two free-electron levels are degenerate,

∣ k ∣ = ∣ k - G ∣ .

Geometrically this is the condition that $k$ lie on the perpendicular bisector plane of $G$ : the Bragg plane, which is precisely a Brillouin -zone boundary. There the two coefficients $c_{k}$ and $c_{k - G}$ are equally important and must be treated together. Keeping only that pair, the central equation collapses to a two-by-two secular problem,

E_{k}^{0} - E U_{G}^{*} U_{G} E_{k - G}^{0} - E = 0, E_{k}^{0} = \frac{ℏ ^{2} k ^{2}}{2 m} .

Its roots are

E_{\pm} (k) = \frac{E _{k}^{0} + E _{k - G}^{0}}{2} \pm (\frac{E _{k}^{0} - E _{k - G}^{0}}{2})^{2} + ∣ U_{G} ∣^{2} .

The gap

Exactly on the boundary the two free-electron energies coincide, $E_{k}^{0} = E_{k - G}^{0} \equiv E^{0}$ , and the square root reduces to $∣ U_{G} ∣$ :

E_{\pm} = E^{0} \pm ∣ U_{G} ∣ .

The two levels, degenerate without the potential, split by

E_{gap} = E_{+} - E_{-} = 2 ∣ U_{G} ∣ .

A weak periodic potential opens a gap at every zone boundary equal to twice the magnitude of the corresponding Fourier component of the potential. No electronic state exists in the interval $(E^{0} - ∣ U_{G} ∣, E^{0} + ∣ U_{G} ∣)$ ; this is a band gap. Away from the boundary the square root is dominated by the kinetic difference, and $E_{\pm}$ rejoin the free-electron parabola, so the potential's effect is concentrated in a thin shell of $k$ around each Bragg plane.

The two-by-two secular problem lifts the degeneracy at the zone boundary: the lower branch bends down and the upper branch bends up, opening a gap of 2 times the potential component. Far from the boundary both branches rejoin the free-electron parabola (faint).

The two standing waves

At the boundary the eigenstates are equal-weight superpositions of the two plane waves. For a real potential with $U_{G} > 0$ the symmetric and antisymmetric combinations, written for $G = (2 π / a) \overset{x}{^}$ at $k = π / a$ , are

ψ_{+} \propto cos \frac{π x}{a}, ψ_{-} \propto sin \frac{π x}{a},

with probability densities $∣ ψ_{+} ∣^{2} \propto cos^{2} (π x / a)$ and $∣ ψ_{-} ∣^{2} \propto sin^{2} (π x / a)$ . Both are standing waves — the Bragg condition reflects a right-moving wave into a left-moving one of equal amplitude, and their sum stands still, carrying no current. They differ in where they place the electron. The cosine state $ψ_{+}$ peaks at the ion sites ( $x = 0, a, 2 a, \dots$ ), where the attractive potential is deepest, and so has the lower energy $E_{-} = E^{0} - ∣ U_{G} ∣$ . The sine state $ψ_{-}$ has nodes at the ions and piles charge between them, sampling the potential where it is shallowest, giving the higher energy $E_{+} = E^{0} + ∣ U_{G} ∣$ . The gap is the electrostatic energy difference between concentrating the electron on the ions versus between them.

The two zone-boundary standing waves. The cosine density peaks on the ion cores (deepest potential, lower energy); the sine density peaks in the interstitial region (shallowest potential, higher energy). Their energy difference is the band gap.

The Kronig–Penney model

The nearly-free-electron gap can be seen in an exactly solvable one-dimensional model. Replace the potential by a periodic array of attractive delta functions of strength $P$ at the lattice sites,

U (x) = \frac{ℏ ^{2}}{ma} P n \sum δ (x - na) .

Between the spikes the electron is free with wavevector $q = 2 m E /ℏ$ . Matching the Bloch condition $ψ (x + a) = e^{ik a} ψ (x)$ across one delta function — continuity of $ψ$ and the delta-induced jump in $ψ^{'}$ — yields the dispersion relation

cos (k a) = cos (q a) + P \frac{sin ( q a )}{q a} .

The left side is bounded, $∣ cos (k a)∣ \leq 1$ , but the right side is an oscillating function of $q a$ (that is, of energy) whose envelope grows past unity. Wherever the right side exceeds $1$ in magnitude no real $k$ solves the equation: those energies are forbidden, the band gaps. Wherever it lies in $[- 1, 1]$ a Bloch state exists: those are the allowed bands. The allowed intervals appear where the free-particle energy $ℏ^{2} q^{2} /2 m$ crosses the values $q a = nπ$ , exactly the folded parabola crossings, and the gaps sit at the zone boundaries. As $P \to 0$ the gaps shrink to zero and the free -electron spectrum is recovered; as $P \to \infty$ the bands narrow to the discrete levels of an isolated well, the limit the tight-binding model starts from.

The Kronig–Penney dispersion function (right side of the equation) versus energy; the horizontal band at plus or minus one is the range accessible to cos(ka). Energies where the curve stays inside the band are allowed (bands); energies where it leaves are forbidden (gaps).

Sorting metals from insulators

The gaps do the sorting. Fill the bands with the crystal's electrons: each band holds $2 N$ states, so a monovalent crystal (one electron per cell) half-fills the lowest band and the Fermi level sits inside it, giving empty states just above and hence a metal. A divalent crystal supplies two electrons per cell, enough to fill the lowest band exactly; whether it is a metal or an insulator then depends on whether the gap to the next band is large compared to the band overlap, a question the empty-lattice bands cannot settle but the real band structure can. When a filled band is separated by a gap from the empty band above it, no state near the Fermi level can respond to a field and the crystal insulates. The Fermi-surface lesson makes this criterion precise; the semiconductor module takes up the case of a small gap, where thermal excitation across it produces the intermediate conductivity of a semiconductor.

The nearly-free-electron picture is quantitatively good for the simple metals — sodium, aluminium — whose valence electrons genuinely see a weak pseudopotential. Where the potential is strong, as in the transition metals with their tight $d$ -orbitals, the opposite starting point serves better, and that is the subject of the next lesson.