542 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1956 



In (2.5), (2.6), and (2.7) all the i^'s are independent of composition. To 

 these equations is added the charge neutrality condition, 



D+ + p = A~ + 7i (2.8) 



Equations (2.4) through (2.8) are enough to determine No in its de- 

 pendence on Na , «, and the various K's. Together they represent the 

 mass action approach. To demonstrate their validity it is necessary to 

 appeal to statistical considerations. 



Thus Nd — D^, the concentration of un-ionized donor is really the 

 density of electrons in the donor level of the energy diagram for the semi- 

 conductor. According to Fermi statistics this density is given by 



No- D+ = No/{l + M exp \{Eu - F)/kT]} . (2.9) 



in which Ed is the energy of the donor level, F is the Fermi level, k, 

 the Boltzmann constant, and T, the temperature. Furthermore, accord- 

 ing to Fermi statistics, n, the total density of electrons in the conduction 

 band is 



n 



= E ^y {1 + exp [{Ei - F)/kT]} (2.10) 



where Qi is the density of levels of energy, Ei , in the conduction band, 

 and the sum extends over all states in that band. Similar expressions are 

 available for the occupation of the acceptor level and the valence band. 

 F is usually determined by summing over all expressions like (2.9) and 

 (2.10) and equating the result to the total number of electrons in the 

 system. This operation corresponds exactly to applying the conserva- 

 tion condition, (2.8). It is obvious from the manner of its determina- 

 tion that F depends upon No — D^y n, etc. 



If we now form the expression on the left of (2.5) by substituting for 

 each factor in it from (2.9) and (2.10), it is obvious that the result de- 

 pends in a very complicated fashion upon F, and so cannot be the con- 

 stant, Kd , independent of composition, since in the last paragraph F 

 was shown to depend on composition. On the other hand if attention is 

 confined to the limit in which classical statistics apply^ the unities in 

 the denominators of (2.9) and (2.10) can be disregarded in comparison 

 to the exponentials, and those equations become 



1 



No - /)+ = 2Noe''"\-'''"'' (2.11); 



and 



n = e 



I 



^"'' Z 9ie~"'"" (2.12) 



I 



