---
title: The Nearly-Free-Electron Model
module: Band Theory
moduleNumber: 6
lessonNumber: 2
order: 602
summary: >
  A weak periodic potential leaves the free-electron parabola almost intact
  except near Brillouin-zone boundaries, where two nearly degenerate plane
  waves mix. This lesson solves the resulting two-by-two secular problem, shows
  the gap of size twice the potential component opening at each boundary,
  identifies the two standing waves that pile charge on and between the ions,
  and works the exactly solvable Kronig–Penney model.
topics: [Band Theory]
draft: false
sources:
  - book: Ashcroft & Mermin
    ref: "Ch. 9 — Electrons in a Weak Periodic Potential"
  - book: Kittel
    ref: "Ch. 7 — Energy Bands"
  - book: Simon
    ref: "Ch. 15 — Electrons in a Periodic Potential"
---

[Bloch's theorem](/condensed-matter/band-theory/blochs-theorem-and-energy-bands)
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 $e^{i\vec q\cdot\vec r}$ with
energy $\hbar^2 q^2/2m$. Bloch's theorem is still satisfied trivially: any plane
wave is a Bloch wave with $u = \text{const}$, and its wavevector $\vec q$ can be
written $\vec k - \vec G$ for a unique $\vec k$ in the first zone and some
reciprocal-lattice vector $\vec G$. 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 $\vec k$
there is a ladder of energies

$$
E_{\vec G}^{0}(\vec k) = \frac{\hbar^2}{2m}\lvert \vec k - \vec G\rvert^{2},
$$

one for each $\vec G$. 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.

$$
% caption: The free-electron parabola (extended zone, faint) folds at the zone
% boundaries k = plus or minus pi/a into the reduced zone, producing the
% empty-lattice bands (solid). Distinct parabola segments become distinct band
% branches that touch at the boundary and at the zone centre.
\begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[black, ->] (-3.4,0) -- (3.6,0) node[below] {$k$};
  \draw[black, ->] (0,0) -- (0,3.4) node[left] {$E$};
  \draw[black, dashed] (1.6,0) -- (1.6,3.3) node[above, black, font=\scriptsize] {zone edge};
  \draw[black, dashed] (-1.6,0) -- (-1.6,3.3);
  % faint extended parabola
  \draw[black, thick, domain=-3.2:3.2, samples=120, variable=\x] plot ({\x},{0.32*\x*\x});
  % reduced: central band
  \draw[acc, very thick, domain=-1.6:1.6, samples=80, variable=\x] plot ({\x},{0.32*\x*\x});
  % reduced: second band folded (segments from |k| in [1.6,4.8] mapped back)
  \draw[acc, very thick, domain=0:1.6, samples=60, variable=\x] plot ({\x},{0.32*(\x-3.2)*(\x-3.2)});
  \draw[acc, very thick, domain=-1.6:0, samples=60, variable=\x] plot ({\x},{0.32*(\x+3.2)*(\x+3.2)});
  \node[acc, anchor=west, font=\scriptsize] at (0.15,2.7) {folded band};
\end{tikzpicture}
$$

## Degenerate perturbation theory at a zone boundary

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

$$
\left(\frac{\hbar^2}{2m}\lvert\vec k - \vec G\rvert^{2} - E\right) c_{\vec k - \vec G} + \sum_{\vec G'} U_{\vec G' - \vec G}\, c_{\vec k - \vec G'} = 0.
$$

Take the potential real with $U_0 = 0$ (a constant shifts every energy and can
be dropped). Away from any degeneracy, one plane wave $\vec k$ dominates and the
others enter only at second order, giving the small correction

$$
E(\vec k) = \frac{\hbar^2 k^2}{2m} + \sum_{\vec G \neq 0} \frac{\lvert U_{\vec G}\rvert^{2}}{\dfrac{\hbar^2 k^2}{2m} - \dfrac{\hbar^2}{2m}\lvert\vec k - \vec G\rvert^{2}} + \cdots,
$$

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

$$
\lvert\vec k\rvert = \lvert\vec k - \vec G\rvert.
$$

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

$$
\begin{vmatrix} E_{\vec k}^{0} - E & U_{\vec G} \\[2pt] U_{\vec G}^{\ast} & E_{\vec k - \vec G}^{0} - E \end{vmatrix} = 0,
\qquad E_{\vec k}^{0} = \frac{\hbar^2 k^2}{2m}.
$$

Its roots are

$$
E_{\pm}(\vec k) = \frac{E_{\vec k}^{0} + E_{\vec k - \vec G}^{0}}{2} \pm \sqrt{\left(\frac{E_{\vec k}^{0} - E_{\vec k - \vec G}^{0}}{2}\right)^{2} + \lvert U_{\vec G}\rvert^{2}}.
$$

## The gap

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

$$
E_{\pm} = E^{0} \pm \lvert U_{\vec G}\rvert.
$$

The two levels, degenerate without the potential, split by

$$
E_{\text{gap}} = E_{+} - E_{-} = 2\lvert U_{\vec G}\rvert.
$$

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 $(E^{0} - \lvert U_{\vec G}\rvert,\, E^{0} + \lvert U_{\vec G}\rvert)$;
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 $\vec k$ around each Bragg
plane.

$$
% caption: The two-by-two secular problem lifts the degeneracy at the zone
% boundary: the lower branch bends down and the upper branch bends up, opening
% a gap of 2 times the potential component. Far from the boundary both branches
% rejoin the free-electron parabola (faint).
\begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[black, ->] (0,0) -- (5.4,0) node[below] {$k$};
  \draw[black, ->] (0,0) -- (0,4.0) node[left] {$E$};
  \draw[black, dashed] (3.0,0) -- (3.0,3.9) node[above, black, font=\scriptsize] {zone edge};
  % free-electron parabola pieces (faint) around boundary
  \draw[black, thick, domain=0.2:4.9, samples=90, variable=\x] plot ({\x},{0.28*\x*\x});
  \draw[black, thick, domain=1.1:3.0, samples=60, variable=\x] plot ({\x},{0.28*(6-\x)*(6-\x)});
  % lower branch: bends below at k=3
  \draw[acc, very thick, domain=0.2:3.0, samples=80, variable=\x]
    plot ({\x},{0.28*\x*\x - 0.55*exp(-(\x-3)*(\x-3)/0.9)});
  % upper branch: bends above starting at k=3
  \draw[acc, very thick, domain=3.0:4.9, samples=80, variable=\x]
    plot ({\x},{0.28*\x*\x + 0.55*exp(-(\x-3)*(\x-3)/0.9)});
  % gap markers at boundary
  \fill[acc] (3.0,{0.28*9-0.55}) circle (1.8pt);
  \fill[acc] (3.0,{0.28*9+0.55}) circle (1.8pt);
  \draw[black, <->] (3.28,{0.28*9-0.55}) -- (3.28,{0.28*9+0.55}) node[midway, right, font=\scriptsize] {gap $2U_G$};
\end{tikzpicture}
$$

## The two standing waves

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

$$
\psi_{+} \propto \cos\!\frac{\pi x}{a}, \qquad \psi_{-} \propto \sin\!\frac{\pi x}{a},
$$

with probability densities $\lvert\psi_{+}\rvert^{2}\propto\cos^{2}(\pi x/a)$ and
$\lvert\psi_{-}\rvert^{2}\propto\sin^{2}(\pi x/a)$. 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 $\psi_{+}$ peaks at the ion sites
($x = 0, a, 2a, \dots$), where the attractive potential is deepest, and so has
the lower energy $E_{-} = E^{0} - \lvert U_{\vec G}\rvert$. The sine state
$\psi_{-}$ has nodes at the ions and piles charge between them, sampling the
potential where it is shallowest, giving the higher energy
$E_{+} = E^{0} + \lvert U_{\vec G}\rvert$. The gap is the electrostatic energy
difference between concentrating the electron on the ions versus between them.

$$
% caption: The two zone-boundary standing waves. The cosine density peaks on the
% ion cores (deepest potential, lower energy); the sine density peaks in the
% interstitial region (shallowest potential, higher energy). Their energy
% difference is the band gap.
\begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[black] (-0.2,0) -- (6.4,0);
  \foreach \x in {0,2,4,6}{ \fill[black] (\x,0) circle (2.4pt);
    \node[black, anchor=north, font=\scriptsize] at (\x,-0.1) {ion}; }
  % cos^2 density (peaks on ions)
  \draw[acc, very thick, domain=0:6.2, samples=200, variable=\x]
    plot ({\x},{0.9*cos(180*\x/2)*cos(180*\x/2)});
  \node[acc, anchor=south west, font=\scriptsize] at (0.1,0.95) {cosine: charge on ions};
  % sin^2 density (peaks between), shifted up
  \draw[acc, thick, dashed, domain=0:6.2, samples=200, variable=\x]
    plot ({\x},{1.7 + 0.9*sin(180*\x/2)*sin(180*\x/2)});
  \node[acc, anchor=south west, font=\scriptsize] at (0.1,2.65) {sine: charge between};
\end{tikzpicture}
$$

## 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 $\mathcal{P}$ at the lattice sites,

$$
U(x) = \frac{\hbar^{2}}{ma}\,\mathcal{P}\sum_{n} \delta(x - na).
$$

Between the spikes the electron is free with wavevector $q = \sqrt{2mE}/\hbar$.
Matching the Bloch condition $\psi(x + a) = e^{ika}\psi(x)$ across one delta
function — continuity of $\psi$ and the delta-induced jump in $\psi'$ — yields
the dispersion relation

$$
\cos(ka) = \cos(qa) + \mathcal{P}\,\frac{\sin(qa)}{qa}.
$$

The left side is bounded, $\lvert\cos(ka)\rvert \le 1$, but the right side is an
oscillating function of $qa$ (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 $\hbar^2 q^2/2m$ crosses the
values $qa = n\pi$, exactly the folded parabola crossings, and the gaps sit at
the zone boundaries. As $\mathcal{P} \to 0$ the gaps shrink to zero and the free
-electron spectrum is recovered; as $\mathcal{P} \to \infty$ the bands narrow to
the discrete levels of an isolated well, the limit the
[tight-binding model](/condensed-matter/band-theory/tight-binding-method)
starts from.

$$
% caption: The Kronig–Penney dispersion function (right side of the equation)
% versus energy; the horizontal band at plus or minus one is the range
% accessible to cos(ka). Energies where the curve stays inside the band are
% allowed (bands); energies where it leaves are forbidden (gaps).
\begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[black, ->] (-0.2,0) -- (9.2,0) node[below right] {energy};
  \draw[black, ->] (0,-2.3) -- (0,2.5) node[left] {$f$};
  % allowed strip |f|<=1
  \fill[acc!10] (0,-1) rectangle (9.0,1);
  \draw[black] (0,1) -- (8.4,1) node[above, font=\scriptsize, black] {$+1$};
  \draw[black] (0,-1) -- (8.4,-1) node[below, font=\scriptsize, black] {lower bound};
  % oscillating growing-envelope curve: cos(qa)+P sin(qa)/(qa) proxy
  \draw[acc, very thick, domain=0.35:8.8, samples=300, variable=\x]
    plot ({\x},{cos(200*\x) + 1.4*sin(200*\x)/(0.6*\x)});
  % mark a forbidden region
  \node[black, anchor=south, font=\scriptsize] at (1.6,1.35) {gap};
  \node[black, anchor=north, font=\scriptsize] at (3.0,-1.35) {band};
\end{tikzpicture}
$$

## Sorting metals from insulators

The gaps do the sorting. Fill the bands with the crystal's electrons: each band
holds [$2N$ states](/condensed-matter/band-theory/blochs-theorem-and-energy-bands),
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](/condensed-matter/band-theory/fermi-surfaces-and-semiclassical-dynamics)
makes this criterion precise; the
[semiconductor module](/condensed-matter/semiconductors/semiconductor-bands-and-junctions)
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.

[^am]: Ashcroft & Mermin, Ch. 9.
[^kittel]: Kittel, Ch. 7.
[^simon]: Simon, Ch. 15.