The Fermi level determines the probability of electron occupancy at different energy levels. The closer the Fermi level is to the conduction band energy, the easier it will be for electrons in the valence band to transition into the conduction band.

Electrons settle into the lowest available energy states at absolute zero temperature and build a "Fermi sea" of electron energy states. The Fermi Level (with Fermi energy *E _{f}*) is the “surface” of this sea where electrons will not have enough energy to rise above the surface. It is the energy level which is occupied by the highest electron orbital at 0 Kelvin (absolute zero temperature) and a parameter of the Fermi-Dirac distribution:

where *T* is the absolute temperature and *k* is Boltzmann’s constant. This function gives the probability *f(E)* of an electron to occupy a state with energy *E*. The distribution function value will have to be nonzero in the conduction band for electrical conductivity to be possible. For solids, the probability of occupancy with the energy level determines the ease of conductivity; whether a material is an insulator, semiconductor, or conductor.

Different orbitals have different corresponding energies. The lower energy orbitals form a band called the valence electron band, and the higher energy orbitals form a band called the conduction band. There is a gap between the valence and conduction band called the energy gap; the larger the energy gap, the more energy is required to transfer the electron from the valence band to the conduction band. At the Fermi level (when *E*=*E _{f}*), the probability simplifies to ½ and thus

*E*lies halfway between the valence and conduction band, or in the middle of the energy gap (

_{f}*E*/2).

_{gap}For insulators, the energy gap is large enough so that the Fermi level lies far from the conduction band or states that can carry current. The opposite is true for conductors where the Fermi level is within a band or nearby states that readily carry current.

Impurities and temperature can affect the Fermi level. Semiconductor atoms are closely grouped together in a crystal lattice and so they have very few free electrons to be good conductors. The ability of semiconductors to conduct electricity can be greatly improved by introducing donor or acceptor atoms to the crystalline structure, either producing more free electrons or more holes. This process is called “doping” and as the semiconductor material is no longer pure, these donor and acceptor atoms are collectively referred to as “impurities”. By carefully designing the doping process, semiconductor crystals can be modified into one of two distinct types of semiconductors: N-type or P-type. In both semiconductor types, the position of the Fermi level relative to the band structure can be controlled to a significant degree by doping. Introducing impurities to atoms will bring the Fermi level up and when it is brought high enough, part of the tail will go over to the conduction band. This makes it easier for electrons to travel to the conduction band and thus conductivity will improve. From the distribution function, temperature directly affects how the energy states are occupied. The tail of the function also gets longer and wider at higher temperatures, stretching out to the conduction band.

It is to be noted that although the probability function has a value in the energy gap, there are no available energy states within the interval (hence energy gap, with zero density of states) and therefore no electrons populate the gap. When both the probability distribution function and density of states have nonzero values in the conduction band, there is a finite number of electrons that participate in conduction. For semiconductors, electrons populating the conduction band level means capability of conducting electricity with energy. At higher temperatures, more electrons can bridge the energy gap and contribute to electrical conduction.

## Summary

The Fermi level represents the electron population at energy levels and consequently the conductivity of materials. Semiconductor doping and higher temperatures can greatly improve the conductivity of the pure semiconductor material. The effect of impurities and higher temperatures on conductivity can be visualized by the band gap and a relative shift in the Fermi level in the band diagram. This shift allows electrons to reach the conduction band and determines how easily materials conduct electricity.