## Introduction to Energy Bands

In solids, electrons are not associated with individual atoms, but are spread throughout the material. Because of the close proximity of the atoms, the atomic energy levels overlap, forming continuous energy bands. The electron energy in solids is therefore not discrete, but rather exists within these energy bands. The conduction band is where electrons move freely.

## Definition of the Conduction Band

In terms of energy bands, the conduction band is defined as the range of energy levels, free or almost free of electrons, above the valence band (the band that includes the energy levels of the outer shell electrons involved in chemical bonding). The conduction band is separated from the valence band by an energy gap, referred to as the bandgap, denoted as .

## Energy and the Conduction Band

The energy of the conduction band can be described as . The bottom of the conduction band (minimum energy state in the conduction band where an electron can exist) can be mathematically represented as:

where is the energy at the top of the valence band, and is the bandgap energy.

## Conductivity and the Conduction Band

The electrical conductivity of a material depends on the population of electrons in the conduction band. If the conduction band has no electrons (as in an insulator), there is no current. If the conduction band has electrons (as in metals and semiconductors), a current can flow. The population of electrons in the conduction band can be given by the Fermi-Dirac distribution function, given as:

where is the probability that an energy state is occupied by an electron, is the Fermi energy, is the Boltzmann constant, and is the absolute temperature.

## Electronic Structure Calculations

Electronic structure calculations often include density of states (DOS) plots. The DOS at a specific energy level is proportional to the number of electron states per unit volume per unit energy at that level. For a one-dimensional free-electron system, it is given by:

where is the density of states, is the electron mass, and is the reduced Planck constant.

## Doping and the Conduction Band

Doping introduces impurity states in the bandgap that can alter the electron population in the conduction band. The number of impurity states introduced can be estimated by the effective mass approximation formula:

where is the donor density, is the effective density of states in the conduction band, is the donor ionization energy, is the Boltzmann constant, and is the absolute temperature.

## Conclusion

Understanding the concept of the conduction band and its mathematical descriptors is fundamental to solid-state physics and materials science. This concept is key to understanding and designing materials with desirable electronic properties, from semiconductors in electronics to optical materials in photonics.

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