Semiconductor physics is of obvious importance, as it

Fermi-Dirac statistics. We assume only donor doping and neglect any acceptor impurities.
The concentration of neutral (not ionized) donor atoms is given by:
Nd 0 = Nd fFD (Ed) = Nd / (exp[(Ed-μ)/ kBT] +1),
In many situations the Fermi level is larger than Ed and the term unity in the denominator
can be neglected. However, we use below the full expression. From neutrality n = Nd + =
Nd - Nd 0 , we immediately obtain:
n = Nd / (exp[(μ-Ed)/ kBT] +1).
n/( Nd –n) = exp[(Ed-μ)/ kBT].
Or, by using the general expression for the electron concentration, H, eq.(7.11), we obtain,
(NOTE: This expression and its analogue for holes (eq. 7.12), are valid irrespective of
whether we have an intrinsic situation, or doping!)
n 2 /( Nd –n) = n0 exp[-(Eg-Ed)/ kBT].
3. If kBT>n and we arrive at the low temperature expression
n= ( Nd n0) 1/2 exp[-(Eg-Ed)/ 2kBT].
The electrical conductivity is schematically depicted in the figure below, where the different
regions are indicated.
One can also study the evolution of the Fermi level with temperature. We put the general
expression for n, H, eq. (7.11), equal to our expressions above. In the low temperature region
(4) we then obtain:
μ = Eg-(Eg-Ed)/2 + 1/2 kBT ln(Nd /n0).
It is seen that the Fermi level at absolute zero is situated half way between the donor levels
and the conduction band edge. As the temperature increases the Fermi level will sink towards