Electrons and Holes as Species

Electrons have a spin of 1/2 and follow Fermi-Dirac statistics. The occupancy of a state of energy E is given by 

The change of units serves to emphasize the similarities between the analysis of semiconductors and electrolytic systems (see, e.g.. Chapter 5). The Fermi energy Ep is closely related to the electrochemical potential of electrons introduced in Section 5.2. Statistical mechanical arguments have been used to show that, under equilibrium conditions, the Fermi energy is equal to the electrochemical potential of electrons. At equilibrium, a single value of Fermi energy is sufficient to define the state of the system. Under nonequilibrium conditions, a separate Fermi energy can be defined for electrons and holes. 

The number of electrons within the conduction band can be expressed as an integral over the occupied states, i.e., 

The limit of oo in equation (12.4) could be replaced by the upper energy of the conduction band, and the limit of —oo in equation (12.6) could be replaced by the lower energy of the valence band. 

At moderate temperatures, electrons in the conduction band have energies close to Ec, and holes in the valence band have energies close to E . Under the assumption that the Fermi energy is not close to the band edges, the Fermi-Dirac distribution, equation (12.1), can be approximated by Boltzmann distribution functions. The concentration of conduction-band electrons can be expressed as 

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