
Lets quickly recount the
essentials: 


The concentrations of electrons or holes in the
conduction or valence bands are 


n_{e}^{–} 
= 
N_{eff} ·
f(E_{C},E_{F},T) 




n_{h}^{+} 
= 
N_{eff} · [1
– f(E_{V},E_{F},T)] 




The density of states for the donors and
acceptors is simply their concentration N_{D} and
N_{A}, we thus have : 

Concentration of nonionized
(neutral) donors N_{D}^{o} (i.e. the electron is
still sitting on the donor level) 


N_{D}^{o} 
= 
N_{D} ·
f(E_{D},E_{F},T) 




Concentration of ionized donors
N_{D}^{+} (i.e. the electron is in the conduction
band). 


N_{D}^{+} 
= 
N_{D} · [1 –
f(E,E_{F},T)] 




Concentration of ionized acceptors
N_{A}^{–} (i.e. an electron from the valence
band is sitting on the acceptor level):. 


N_{A}^{–} 
= 
N_{A} ·
f(E_{A},E_{F},T) 




Finally, the concentration of
neutral acceptors N_{A}^{0}; it is 


N_{A}^{o} 
= 
N_{A} · [1 –
f(E_{A},E_{F},T)] 



This can be easily
envisioned in a simple drawing 



The Fermi energy results from
equating the sum of all negative charges with the sum of all positive
charges. 


The resulting equation is easily written down,
but cannot be solved analytically. 


That is why we do it numerically. 