User:John R. Brews/Draft

Fermi function

The Fermi function or, more completely, the Fermi-Dirac distribution function describes the occupancy of a electronic energy level in a system of electrons at equilibrium. The occupancy f(E) of an energy level of energy E at an absolute temperature T is given by:

$f(E)={\frac {1}{1+g(E)\exp \left({\frac {E-E_{F}}{k_{B}T}}\right)}}\ ,$

where g(E) is the degeneracy of the energy level, that is, the number of quantum states with that particular energy. Here E_F is called the Fermi energy and k_B is the Boltzmann constant.

Notice that for an energy level with E = E_F and a degeneracy factor g = 1, the occupancy is 1/2.

Dopant levels

Dopant impurities are used in semiconductors to adjust the conductivity of the material. They introduce energy levels for electrons, and if they are acceptors become negatively charged when occupied. Typically an acceptor provides an energy level related to the valence band structure of the host material. A common case is two levels, one from the "heavy" hole valence band and one from the "light" hole valence band. Thus, the number of negatively charge acceptors, compared to the total number of acceptors, is:

${\frac {N_{a}^{-}}{N_{a}}}={\frac {1}{1+4\exp \left({\frac {E-E_{F}}{k_{B}T}}\right)}}\ ,$

where the degeneracy factor of 4 stems from the possibility of either a spin-up or a spin-down electron occupying the level E, and the existence of two levels of energy E, one from the "heavy" hole band and one from the "light" hole band.

In contrast to acceptors, donors become positively charged and tend to give up an electron. The number of positive donors compared to the total number of donors is then:

${\frac {N_{d}^{+}}{N_{d}}}={\frac {1}{1+2\exp \left({\frac {E-E_{F}}{k_{B}T}}\right)}}\ ,$

where now the degeneracy factor is 2 (because of the spin-up or spin-down possibilities for occupancy) and there is typically only one energy level associated with the conduction band.