546 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1956 



solution, as it should be for the degenerate case in which ionization is 

 suppressed. Equating iid in (2.23) to n in (2.16) yields 



_ j H exp [(>■„/ - M° + Bo)/kT] \ . 



" - \ 1 + J/, exp [(£ - FVATJ / ^" ('-^S' 



which is identical Avith (2.22) if A'o is taken to be 



}i exp[(Mz>+° - M° + Eo)/kT] (2.26) 



Thus one arrives at the conclusion that the law of mass action remains 

 valid for the heterogeneous equilibrium even when it fails for the homo- 

 geneous internal equilibria. 



This is a fairly important result since it implies that solubilities can 

 give information on the behavior of the Fermi level and hence on the 

 distribution of electronic energy levels, even under conditions of de- 

 generacy. 



The chemical potential specified by (2.23) is of course important in 

 itself, for treating any equilibrium (external or internal) in which the 

 donor may participate. 



One last remark is in order. This concerns the treatment of heterogene- 

 ous equilibria involving some external phase, and the surface^^ rather than 

 the body of a semiconductor. In such treatments it has been customary 

 to compute the chemical potential of an ionizable adsorbed atom by 

 summing the ion chemical potential and the Fermi level, as in (2.19). 

 This is no more possible if the statistics of the surface states are non- 

 classical, then it is possible when considering non-classical situations 

 involving the body of the crystal. Care must therefore be exercised also 

 in the treatment of surface equilibria. 



The above discussion has shown that there are extensive ranges of 

 conditions under which holes and electrons obey the law of mass action, 

 and behave like chemical entities. In the next section some of the con- 

 sequences of this fact will be developed. 



III. APPLICATION OF THE MASS ACTION PRINCIPLE 



Equations (2.4) through (2.8) will now be used to determine how, in 

 the classical case, the solubility. No , of lithium in (2.1) depends upon 

 Na the concentration of boron in silicon. In the experiments to be de- 

 scribed, the systems are classical, and the donors and acceptors there- 

 fore so thoroughly ionized that No can be replaced by D and Na by 

 A~. Insertion of (2.4) into (2.5) yields 



D+n = aKoKo = K* (3.1) 



