---
title: Band Theory and Semiconductors
module: Semiconductors
moduleNumber: 7
lessonNumber: 1
order: 701
summary: >
  The periodic lattice splits atomic levels into allowed energy bands separated
  by forbidden gaps. Whether the highest occupied band is full or partly full,
  and how wide the gap above it is, sorts every solid into conductor, insulator,
  or semiconductor. Doping adds donor or acceptor levels inside the gap, and a
  p-n junction built from doped regions gives the diode, the solar cell, the LED,
  and the transistor.
topics: [Semiconductors]
draft: false
sources:
  - book: Tipler & Llewellyn
    ref: "Ch. 10 — Solid State Physics; §10-6 Band Theory of Solids, §10-7 Impurity Semiconductors, §10-8 Semiconductor Junctions and Devices, Hall Effect"
---

The [free-electron gas](/condensed-matter/free-electron-fermi-gas/free-electron-gas-and-conduction)
explains conduction but treats every metal alike; it cannot say why quartz
resists $10^{16}\ \Omega\cdot\text{m}$ while copper resists $10^{-8}\
\Omega\cdot\text{m}$, a span of twenty-four orders of magnitude. The missing
ingredient is the effect of the periodic lattice on the electron energies. Adding
it splits the continuous free-electron energies into **bands**, and the band
structure decides everything.

## Bands from a periodic potential

An electron in a crystal moves in the periodic potential of the ion lattice.
Bloch proved that the solutions of the Schrödinger equation in a potential of
period $L$ have the form

$$
\psi(x) = u_k(x)\,e^{ikx}, \qquad u_k(x) = u_k(x + L),
$$

a plane wave modulated by a function with the lattice periodicity. Solving the
Schrödinger equation for a periodic array of square wells (the **Kronig-Penney
model**) shows that traveling-wave solutions exist only for certain ranges of
energy — the **allowed bands** — separated by **forbidden gaps** where no
traveling wave can propagate.

> **Definition (Energy band and gap).** An **allowed band** is a continuous range
> of electron energies for which propagating states exist in the crystal; a
> **forbidden gap** $E_g$ is an energy interval between bands containing no
> propagating states. In a crystal of $N$ atoms each band holds exactly $N$
> levels (twice that counting spin).

The gaps appear at wave numbers satisfying $ka = n\pi$, which is the
[Bragg condition](/condensed-matter/crystal-structure/diffraction-and-structure-factors):
there the electron wave is Bragg-reflected by the lattice and forms a standing
wave. Two standing waves are possible, $\psi_1 \propto \sin(\pi x/a)$ concentrating
charge between the ions and $\psi_2 \propto \cos(\pi x/a)$ concentrating it on the
ions. They have different potential energies, and that difference is the gap.

$$
% caption: Energy versus wave number. A free electron follows the smooth
% parabola E = ℏ²k²/2m; in the periodic lattice, gaps open at the Brillouin-zone
% edges k_1 = π/a and k_2 = 2π/a, where Bragg reflection forms standing waves,
% breaking the parabola into allowed bands.
\begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (7.0,0) node[below] {$k$};
  \draw[->, black] (0,0) -- (0,4.2) node[left] {$E$};
  % free-electron parabola (dashed)
  \draw[black, dashed, domain=0:2.5, samples=60, variable=\x]
    plot ({\x*2.4},{0.55*\x*\x});
  \node[black, anchor=west, font=\scriptsize] at (5.4,3.6) {free electron};
  % band 1 (0 to pi/a=2.4)
  \draw[acc, very thick, domain=0:1.0, samples=40, variable=\x]
    plot ({\x*2.4},{0.55*\x*\x});
  % gap
  \draw[black, dashed] (2.4,1.32) -- (2.4,2.1);
  \node[anchor=west, font=\scriptsize] at (2.5,1.7) {$E_g$};
  % band 2 (pi/a to 2pi/a)
  \draw[acc, very thick, domain=1.0:2.0, samples=40, variable=\x]
    plot ({\x*2.4},{2.1 + 0.55*(\x-1.0)*(\x-1.0)});
  \draw[black] (2.4,0.08) -- (2.4,-0.08);
  \draw[black] (4.8,0.08) -- (4.8,-0.08);
  \node[anchor=north, font=\scriptsize] at (2.4,-0.1) {$k_1$};
  \node[anchor=north, font=\scriptsize] at (4.8,-0.1) {$k_2$};
\end{tikzpicture}
$$

The same bands emerge from the other direction: as $N$ atoms are brought
together, each sharp atomic level splits into $N$ closely spaced levels, exactly
as the two atomic levels of
[H₂](/condensed-matter/molecules-and-bonding/bonding-mechanisms) split into bonding
and antibonding. For a macroscopic $N$ these merge into a quasi-continuous band.

$$
% caption: As the interatomic separation decreases, each discrete atomic level
% broadens into a band of N closely spaced levels; the equilibrium spacing R_0
% fixes which bands overlap and how wide the gap between them is.
\begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (7.0,0) node[below] {separation $R$};
  \draw[->, black] (0,0) -- (0,4.2) node[left] {$E$};
  % right: two discrete atomic levels
  \draw[black] (5.4,3.2) -- (6.4,3.2);
  \draw[black] (5.4,1.6) -- (6.4,1.6);
  \node[anchor=west, font=\scriptsize] at (6.45,3.2) {atomic level};
  \node[anchor=west, font=\scriptsize] at (6.45,1.6) {atomic level};
  % bands widening to the left
  \fill[acc!16, draw=acc] (1.0,2.6) -- (5.4,3.15) -- (5.4,3.25) -- (1.0,4.0) -- cycle;
  \fill[acc!16, draw=acc] (1.0,0.6) -- (5.4,1.55) -- (5.4,1.65) -- (1.0,1.9) -- cycle;
  \draw[black, dashed] (2.0,0) -- (2.0,4.0) node[above, font=\scriptsize] {$R_0$};
  \node[acc, anchor=west, font=\scriptsize] at (1.05,3.6) {band};
  \node[acc, anchor=west, font=\scriptsize] at (1.05,1.35) {band};
\end{tikzpicture}
$$

## Conductors, insulators, semiconductors

The band occupied by the outermost electrons is the **valence band**; the next
higher band is the **conduction band**. Whether a solid conducts depends on how
these are filled.

$$
% caption: Four band structures. A partly filled band (a) or overlapping bands
% (b) leave empty states just above the filled ones, giving a conductor; a wide
% gap (c) gives an insulator; a narrow gap (d) gives a semiconductor.
\begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  % (a) partly filled
  \begin{scope}
    \draw[acc, thick] (0,0) rectangle (1.2,1.0);
    \fill[acc!30] (0,0) rectangle (1.2,0.5);
    \node[anchor=north, font=\scriptsize] at (0.6,-0.15) {conductor};
  \end{scope}
  % (b) overlap
  \begin{scope}[xshift=2.6cm]
    \draw[acc, thick] (0,0) rectangle (1.2,0.8);
    \fill[acc!30] (0,0) rectangle (1.2,0.8);
    \draw[acc, thick] (0,0.6) rectangle (1.2,1.6);
    \node[anchor=north, font=\scriptsize] at (0.6,-0.15) {conductor};
  \end{scope}
  % (c) insulator
  \begin{scope}[xshift=5.2cm]
    \draw[acc, thick] (0,0) rectangle (1.2,0.8);
    \fill[acc!30] (0,0) rectangle (1.2,0.8);
    \draw[acc, thick] (0,2.2) rectangle (1.2,3.0);
    \node[anchor=north, font=\scriptsize] at (0.6,-0.15) {insulator};
    \node[font=\scriptsize] at (0.6,1.5) {gap $> 2$ eV};
  \end{scope}
  % (d) semiconductor
  \begin{scope}[xshift=7.8cm]
    \draw[acc, thick] (0,0) rectangle (1.2,0.8);
    \fill[acc!30] (0,0) rectangle (1.2,0.8);
    \draw[acc, thick] (0,1.3) rectangle (1.2,2.1);
    \node[anchor=north, font=\scriptsize] at (0.6,-0.15) {semiconductor};
    \node[font=\scriptsize] at (0.6,1.05) {small gap};
  \end{scope}
\end{tikzpicture}
$$

- **Conductor.** The valence band is only partly filled (sodium: one $3s$
  electron per atom, so the $3s$ band is half full), or a filled band overlaps an
  empty one (magnesium: filled $3s$ overlapping empty $3p$). Empty states sit
  just above the filled ones, so a field accelerates the electrons freely.
- **Insulator.** The valence band is completely full and the gap to the empty
  conduction band exceeds about $2\ \text{eV}$. Diamond's gap is $7\ \text{eV}$;
  ordinary fields cannot lift an electron across it.
- **Semiconductor.** The valence band is full but the gap is small — silicon
  $1.1\ \text{eV}$, germanium $0.7\ \text{eV}$ — so thermal excitation puts a
  modest number of electrons into the conduction band.

Every electron promoted to the conduction band leaves a vacancy, or **hole**, in
the valence band. Neighboring valence electrons hop into the hole, so the hole
migrates in the direction of the field and acts like a positive charge carrier.

The number of carriers is set by the
[Fermi-Dirac distribution](/condensed-matter/free-electron-fermi-gas/sommerfeld-model-and-heat-capacity)
evaluated a half-gap above the Fermi level, which sits near the middle of the gap:

$$
f_{FD}(E) = \frac{1}{e^{(E-E_F)/kT} + 1}.
$$

| $E - E_F$ | multiple of $kT$ | $f_{FD}$ at 293 K |
| --- | --- | --- |
| $0.05\ \text{eV}$ | 2 | $1.2\times10^{-1}$ |
| $0.10\ \text{eV}$ | 4 | $1.8\times10^{-2}$ |
| $0.25\ \text{eV}$ | 10 | $5.1\times10^{-5}$ |
| $1.0\ \text{eV}$ | 40 | $6.5\times10^{-18}$ |

The occupation collapses as the gap widens: raising the gap from $0.25$ to $1.0\
\text{eV}$ drops the carrier fraction by twelve orders of magnitude. This also
explains the **negative temperature coefficient** of a semiconductor's
resistivity — heating creates far more carriers than it adds scattering, so
resistance falls as temperature rises, opposite to a metal.

Near a band edge the $E$-versus-$k$ curvature differs from a free electron, and
it is captured by an **effective mass** $m^*$ defined through $1/m^* = (1/\hbar^2)
\,\d^2 E/\d k^2$. For silicon $m^* \approx 0.2\,m_e$; effective masses can be a small
fraction of $m_e$ and can even be negative near the top of a band.

## Doping

Adding a controlled trace of impurity — **doping** — creates carriers without
thermal excitation across the full gap. Replacing a silicon atom (four valence
electrons) with **arsenic** (five) leaves one weakly bound extra electron. The
electron plus the arsenic ion core form a hydrogen-like system, but embedded in a
medium of dielectric constant $\kappa$ and with the electron's effective mass, so
its binding energy is scaled far down:

$$
E_1 = -\frac{1}{2}\left(\frac{ke^2}{\hbar}\right)^2\frac{m^*}{\kappa^2}\frac{1}{n^2}, \qquad \langle r_1 \rangle = a_0\,\frac{\kappa}{m^*/m_e}.
$$

For silicon ($m^* = 0.2\,m_e$, $\kappa = 11.8$) the ground state lies only
$0.020\ \text{eV}$ below the conduction band, with orbit radius $\sim 3\
\text{nm}$ — sixty Bohr radii. These **donor levels** sit just under the
conduction band and give up their electrons easily, making an **n-type**
semiconductor (negative carriers).

Replacing silicon with **gallium** (three valence electrons) does the opposite:
the gallium accepts an electron from the valence band, creating a hole. Its
**acceptor levels** sit just above the valence band, making a **p-type**
semiconductor (positive carriers).

$$
% caption: Doping introduces levels inside the gap. Donor levels (n-type) lie
% just below the conduction band and release electrons upward; acceptor levels
% (p-type) lie just above the valence band and capture electrons, leaving holes.
\begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  % --- n-type ---
  \begin{scope}
    \draw[acc, thick, fill=acc!8] (0,2.4) rectangle (2.4,3.2);
    \node[font=\scriptsize] at (1.2,2.8) {conduction};
    \draw[acc, thick, fill=acc!30] (0,0) rectangle (2.4,0.8);
    \node[font=\scriptsize] at (1.2,0.4) {valence};
    % donor levels
    \foreach \x in {0.4,1.0,1.6,2.0}
      \draw[black] (\x,2.1) -- (\x+0.25,2.1);
    \node[anchor=north, font=\scriptsize] at (1.2,1.95) {donor levels};
    \node[anchor=north] at (1.2,-0.1) {n-type};
  \end{scope}
  % --- p-type ---
  \begin{scope}[xshift=4.6cm]
    \draw[acc, thick, fill=acc!8] (0,2.4) rectangle (2.4,3.2);
    \node[font=\scriptsize] at (1.2,2.8) {conduction};
    \draw[acc, thick, fill=acc!30] (0,0) rectangle (2.4,0.8);
    \node[font=\scriptsize] at (1.2,0.4) {valence};
    % acceptor levels
    \foreach \x in {0.4,1.0,1.6,2.0}
      \draw[black] (\x,1.1) -- (\x+0.25,1.1);
    \node[anchor=south, font=\scriptsize] at (1.2,1.2) {acceptor levels};
    \node[anchor=north] at (1.2,-0.1) {p-type};
  \end{scope}
\end{tikzpicture}
$$

Doping is potent: one impurity atom per million can raise the conductivity by
several orders of magnitude.

## The p-n junction

Join an n-type and a p-type region — in practice a single silicon crystal doped
differently on its two sides. Electrons diffuse from the n side to the p side and
holes the other way, until the resulting charge double layer builds a **contact
potential** that stops further diffusion. The junction region, swept clear of
carriers, is the high-resistance **depletion region**.

$$
% caption: The p-n junction in equilibrium: diffusion of electrons and holes
% leaves a depletion region of fixed charge and a contact potential that halts
% further diffusion; the n side sits at higher potential than the p side.
\begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  % p region
  \draw[acc, thick, fill=acc!8] (0,0) rectangle (2.4,1.4);
  \node[acc] at (1.2,0.7) {p side};
  % n region
  \draw[acc, thick, fill=acc!8] (4.2,0) rectangle (6.6,1.4);
  \node[acc] at (5.4,0.7) {n side};
  % depletion region
  \draw[black, thick, fill=black!6] (2.4,0) rectangle (4.2,1.4);
  % fixed charges
  \foreach \y in {0.35,0.75,1.15}{
    \node[font=\scriptsize] at (2.9,\y) {neg};
    \node[font=\scriptsize] at (3.7,\y) {pos};
  }
  \draw[black, ->] (3.3,2.05) -- (3.3,1.45);
  \node[black, font=\scriptsize, anchor=south] at (3.3,2.05) {depletion region};
\end{tikzpicture}
$$

Apply an external voltage. **Forward bias** (battery's positive terminal on the p
side) lowers the barrier and floods the junction with diffusing carriers, giving
a large current. **Reverse bias** raises the barrier and blocks diffusion; only a
tiny minority-carrier current flows. The junction conducts essentially one way —
it is a **rectifier**. Treating the carriers with a Boltzmann tail gives the
**diode equation**,

$$
I = I_0\left(e^{eV_b/kT} - 1\right),
$$

where $I_0$ is the small reverse saturation current.

$$
% caption: The diode current-voltage curve. Forward bias gives an exponentially
% rising current; reverse bias gives only the tiny saturation current I_0 until
% breakdown, so the junction passes current in one direction.
\begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (-3.4,0) -- (3.0,0) node[below] {$V$};
  \draw[->, black] (0,-1.2) -- (0,3.2) node[left] {$I$};
  % forward: exponential
  \draw[acc, very thick, domain=0:1.6, samples=60, variable=\x]
    plot ({\x*1.4},{0.28*(exp(\x*1.9)-1)});
  % reverse: flat small negative
  \draw[acc, very thick] (0,0) -- (-2.6,-0.28);
  % breakdown drop
  \draw[acc, very thick] (-2.6,-0.28) -- (-3.0,-1.1);
  \node[acc, anchor=south, font=\scriptsize] at (1.7,2.4) {forward};
  \node[acc, anchor=north, font=\scriptsize] at (-1.6,-0.32) {reverse};
  \node[black, anchor=east, font=\scriptsize] at (-2.9,-0.8) {breakdown};
\end{tikzpicture}
$$

Variations on the junction give a family of devices:

- **Tunnel diode.** Doping both sides so heavily that the bands overlap lets
  electrons tunnel across the thin depletion region; over part of the curve
  raising the voltage _decreases_ the current (negative resistance), useful for
  fast switching.
- **Solar cell.** Photons with energy above the gap create electron-hole pairs;
  the junction field sweeps them apart, producing a photovoltage of about $0.6\
  \text{V}$.
- **Light-emitting diode.** Under large forward bias, electrons and holes
  recombine across the gap and emit photons; a resonant cavity makes it a diode
  laser, with light-out to power-in efficiency above 50%.
- **Transistor.** Two junctions in series (npn or pnp) with a thin base between
  emitter and collector; a small base signal controls a large collector current,
  the basis of amplification and logic.

## The Hall effect

Which sign of carrier actually moves is settled by the **Hall effect**. Pass a
current through a thin doped strip in a perpendicular magnetic field $B$. The
magnetic force $q\vec{v}_d \times \vec{B}$ pushes carriers to one edge
regardless of their sign, building a transverse **Hall voltage** until the
electric force balances the magnetic one:

$$
V_H = v_d B w,
$$

for strip width $w$. The **sign** of $V_H$ reveals whether the carriers are
negative (n-type) or positive (p-type), and its magnitude gives $v_d$, hence the
carrier density $n$ from $j = nqv_d$.

$$
% caption: Hall geometry: a current along the strip in a perpendicular field B
% deflects the carriers to one edge, and the resulting transverse Hall voltage
% V_H reveals the sign and density of the charge carriers.
\begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[acc, thick, fill=acc!8] (0,0) rectangle (5.0,1.6);
  % B field into page, marked inside the strip as small crosses
  \foreach \x in {1.4,2.6,3.8}{
    \draw[black] (\x-0.09,0.71) -- (\x+0.09,0.89);
    \draw[black] (\x-0.09,0.89) -- (\x+0.09,0.71);
  }
  \node[black, font=\scriptsize, anchor=south] at (2.6,1.75) {$B$ into page};
  % current
  \draw[black, ->, very thick] (-1.1,0.8) -- (-0.1,0.8);
  \node[anchor=east, font=\scriptsize] at (-1.15,0.8) {current $I$};
  \draw[black, ->, very thick] (5.1,0.8) -- (6.1,0.8);
  % Hall voltage across width
  \node[anchor=west, font=\scriptsize] at (5.2,1.5) {pos edge};
  \node[anchor=west, font=\scriptsize] at (5.2,0.1) {neg edge};
  \draw[black, <->] (0.6,0.15) -- (0.6,1.45) node[midway, right, font=\scriptsize] {$V_H$};
\end{tikzpicture}
$$

A closing note on **magnetism in solids**: it too comes from electron spins.
Atoms with unpaired spins carry magnetic moments; when neighboring moments align
spontaneously the solid is **ferromagnetic**, when they merely respond to an
applied field it is **paramagnetic**, and when the response opposes the field —
as in a [superconductor](/condensed-matter/superconductivity/superconductivity-phenomenology)
— it is **diamagnetic**. The next lessons of this module first make the carrier
count quantitative, then follow the p-n junction into the devices it builds.