544 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1956 



Furthermore, it is well established'" that the Fermi level plays the role 

 of chemical potential, ju* , for the electron 



ile = F (2.18) 



Thus the chemical potential for the donor atom is 



y^D^ + M. = MB+' + kTfnD^ + F 



(2.19) 



= Mz)+° + kT inNn + F = /x/>+" + kT In [e''"'^]Nn 



where (2.13) has been used to replace D'^ by Nd . We note that the ac-| 

 tivity of the donor atom must be 



{e^^'^\Nn (2.20)1 



with e^""^ playing the role of an activity coefficient." 



Equating ixd given by (2.19) to n in (2.16) results in the equation 



a = exp[(M.>+° - ix')/kT]{e'^"]Nu (2.21)| 



which can be made identical to (2.15) by identifying 



exp[(Mz.+° - n')/kT] 



with K of that expression. Thus in the classical case the law of mass] 

 action is applicable to the heterogeneous equilibrium. 



When classical statistics no longer apply it is still possible to evaluatei 

 Nd — D'^, using the full expression (2.9). Therefore the solubility Nd J 

 of the donor can still be determined if (2.4) remains valid. To decidef 

 this question it is necessary to evaluate hd , the chemical potential of j 

 the donor in the semiconductor under non-classical conditions. Thisl 

 problem is not as simple as those treated above, but it can be solved,™ 

 and the detailed arguments can be found in Reference 5. Here we shall 

 be content with quoting the results. However, before doing this the non- 

 classical counterpart of (2.15) will be written by combining (2.9) with! 

 (2.4). The result is 



a = [K,/{1 + yi exp[(^„ - F)/kT\\]ND (2.22), 



and if (2.4) is valid (2.22) should be derivable by equating n to the| 

 proper value of (Xd . 



Since in the non-classical case a finite portion of the donor states are'' 

 occupied by electrons, the introduction of an additional average donoi 

 atom is no longer equivalent to adding two independent particles whose 

 chemical potentials can be summed. In the statistical derivation of ni 

 it is therefore necessary to evaluate the total free energy of the semi- 



