Conduction Band

Table of Contents

Introduction to Energy Bands

In a solid, electrons are not bound to individual atoms but are shared across the whole material. Because the atoms sit close together, their discrete atomic energy levels split and merge into continuous energy bands: the Pauli exclusion principle forbids identical electron states, so each atomic level fans out into a dense band of allowed energies. Electron energies in a solid therefore lie within these bands, separated by forbidden gaps. The conduction band is the band in which electrons can move freely and carry current.

Definition of the Conduction Band

The conduction band is the range of energy levels immediately above the valence band — the band that holds the outer-shell electrons involved in chemical bonding. In a semiconductor or insulator at absolute zero the conduction band is empty; electrons reach it only when they gain enough energy to cross the forbidden region. That region, separating the top of the valence band from the bottom of the conduction band, is the bandgap, denoted $E_g$ .

Energy and the Conduction Band

Writing $E_c$ for the energy at the bottom of the conduction band (the lowest state an electron can occupy there) and $E_v$ for the energy at the top of the valence band, the two are related by the bandgap:

$E_c = E_v + E_g$

An electron promoted into the conduction band leaves a vacancy — a positively charged hole — in the valence band, so charge carriers are always created in electron-hole pairs by thermal or optical excitation across $E_g$ .

Occupation: the Fermi-Dirac Distribution

The electrical conductivity of a material depends on how many electrons occupy the conduction band. An insulator has essentially none, so no current flows; metals and semiconductors have enough for current to flow. The probability that a state of energy $E$ is occupied is set by the Fermi-Dirac distribution,

$f(E) = \dfrac{1}{1 + e^{(E - E_F)/k_B T}}$

where $E_F$ is the Fermi energy, $k_B$ is the Boltzmann constant, and $T$ is the absolute temperature. Because the conduction-band states lie well above $E_F$ in a semiconductor, the exponential dominates and the distribution reduces to the Boltzmann factor $e^{-(E - E_F)/k_B T}$ .

Density of States

The density of states (DOS) measures how many electron states are available per unit volume per unit energy. For a three-dimensional, free-electron-like band — a good model for the bottom of the conduction band — it grows as the square root of the energy measured from the band edge:

$D(E) = \dfrac{1}{2\pi^2} \left(\dfrac{2m_e^*}{\hbar^2}\right)^{3/2} \sqrt{E - E_c}, \qquad E \ge E_c$

where $m_e^*$ is the electron effective mass and $\hbar$ is the reduced Planck constant. (The characteristic $\sqrt{E}$ behaviour and the exponent $3/2$ are specific to three dimensions; a one-dimensional band instead diverges as $E^{-1/2}$ , and a two-dimensional band gives a constant DOS.)

Carrier Concentration in the Conduction Band

Integrating the density of states weighted by the occupation probability gives the number of electrons per unit volume in the conduction band. For a non-degenerate semiconductor this evaluates to

$n = N_c\, e^{-(E_c - E_F)/k_B T}, \qquad N_c = 2\left(\dfrac{2\pi m_e^* k_B T}{h^2}\right)^{3/2}$

where $N_c$ is the effective density of states at the conduction-band edge and $h$ is the Planck constant. The factor $e^{-(E_c - E_F)/k_B T}$ shows directly how raising the temperature or lowering the energy distance to $E_F$ rapidly increases the conduction-electron population.

Doping and the Conduction Band

Doping deliberately adds impurity atoms that introduce energy levels inside the bandgap. In n-type doping, donor atoms (such as phosphorus or arsenic in silicon) create a filled level a small distance $E_c - E_d$ below the conduction band — typically only a few hundredths of an electronvolt. Because this ionization energy is comparable to $k_B T$ at room temperature, almost every donor gives up its electron to the conduction band. The conduction-electron concentration is then set by the donor concentration $N_D$ itself,

$n \approx N_D \quad \text{(donors fully ionized)}$

and the Fermi level shifts upward, toward $E_c$ . Doping thus controls conductivity over many orders of magnitude without changing the host material’s bands — the mechanism at the core of every transistor and diode.

A Concrete Example

For silicon at room temperature, $E_g \approx 1.12\ \text{eV}$ and $k_B T \approx 0.0259\ \text{eV}$ . Thermal excitation across the gap is governed by $e^{-E_g/2k_B T}$ , an exponent of about $21.6$ , so the intrinsic carrier concentration is only $n_i \sim 10^{10}\ \text{cm}^{-3}$ — tiny compared with the roughly $10^{22}\ \text{cm}^{-3}$ atoms present. Doping silicon with a donor concentration of, say, $N_D = 10^{16}\ \text{cm}^{-3}$ raises the conduction-electron population by six orders of magnitude, transforming a nearly insulating crystal into a useful conductor. This exponential leverage is what makes the conduction band so central to device engineering.

Conclusion

The conduction band, and the population of electrons within it, controls whether and how a material conducts. Its position relative to the valence band sets the bandgap; the Fermi-Dirac distribution and the density of states fix how many electrons reach it; and doping lets us tune that population at will. Together these ideas form the backbone of solid-state physics and underpin the design of semiconductors, photonic materials, and the electronic devices built from them.

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