Electron and Hole Concentrations at Equilibrium
The Fermi distribution function can be used to calculate the concentrations of electrons and holes in a semiconductor, if the densities of available states
in the valence and conduction bands are known.
Electron and hole concentrations at equilibrium:
For example, the concentration of electrons in the conduction band is
where N(E)dE is the density of states (cm-3) in the energy range dE. The subscript 0 used with the electron and hole concentration symbols (n0,p0) indicates
equilibrium conditions. The number of electrons per unit volume in the energy range dE is the product of the density of states and the probability of occupancy f(E). Thus the total electron concentration is the integral over the entire conduction band, as in Eq (1)
The result of the integration of Eq. (1) is the same as that obtained if we represent all of the distributed electron states in the conduction band by an effective density of states Nc located at the conduction band edge Ec. Therefore, the conduction band electron concentration is simply the effective density of states at Ec times the probability of occupancy at Ec :
In this expression we assume the Fermi level Ef lies at least several kT below the conduction band. Then the exponential term is large compared with unity, and the Fermi function f(Ec) can be simplified as
Since kT at room temperature is only 0.026 eV, this is generally a good approximation. For this condition the concentration of electrons in the conduction band is
The effective density of states Nc is shown
In Eq. (5), m* is the density-of-states effective mass for electrons, let us consider the 6 equivalent conduction band minima along the Z-directions for Si.
There is a longitudinal effective mass ml along the major axis of the ellipsoid, and the transverse effective mass mt, along the two minor axes. Since we have (m*)3/2 appearing in the density-of-states expression Eq. (5), by using dimensional equivalence and adding contributions from all 6 valleys, we get
It can be seen that this is the geometric mean of the effective masses.
By similar arguments, the concentration of holes in the valence band is
where Nv is the effective density of states in the valence band.