566 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1956 



concentration of pairs since tlie latter possess effectively no fields. In any 

 event when pairing occurs the Debye-Hiickel effects are relatively second 

 order, since, even normally, they represent quite small deviations from 

 ideal solution behavior. Under pairing conditions it is desirable, in the 

 first approximation, to focus one's attention on the pairing interaction. 



While developing the aqueous solution analogy inherent in our semi- 

 conductor model it is natural to inquire whether or not a system like 

 (2.1), in which at least one of the ions can move, will show effects due to 

 coulomb interaction. A preliminary calculation using (6.1) indicates 

 that if coulomb effects are to be observed they are likely to be of the ion 

 pairing variety rather than of the Debye-Huckel type because the dielec- 

 tric constants of semiconductors are low relative to that of water, e.g., 

 12 for silicon^^ and 16 for germanium^'' as against 80 for water .^^ The 

 dominance of ion pairing stems, as it will become clear later, from still 

 another feature peculiar to semiconductors. This is the closeness with 

 which two ions of opposite sign can approach one another in semicon- 

 ductors. In any event experiments are not yet at the stage of sensitivity 

 necessary for the accurate measurement of the small Debye-Hiickel 

 effects so that we are virtually compelled to ignore such phenomena. 



Fig. 10 is a picture of an ion pair in boron-doped silicon. Corresponding 

 to this process one may sketch in another vertical equilibrium in (2.1) 

 to yield (ignoring un-ionized Li) 



Li (external) ^ Li'^ + e~ 



+ + 



B- + e"^ (6.2) 



u u 



[Li-^B-] eV 



where [Li'^B~] stands for the ion pair in which the individual ions main- 

 tain their polar identities and the binding energy is coulombic. The ion 

 pair is a compound in a statistical sense since as will be seen later the dis 

 tance between the ions of a pair is distributed over a range of values. The ■ 

 interaction between Li'^ and B~ is to be distinguished from that sho\\ii 

 in (5.2). The latter occurs at high temperatures whereas the former is 

 presumably limited to low temperatures, below 300°C. 



The quantitative aspects of ion pairing were first considered by Bjer- _ 

 rum and later by Fuoss^^ who placed Bjerrum's theory on a somewhat 

 more acceptable basis. Fuoss's theory, however, suffers from some of the 

 same limitations as Bjerrum's. Nevertheless the Bjerrum-Fuoss theory is 

 capable of satisfying experimental data over broad ranges of conditions. 

 In the next section w^e present a brief resume of this theory together with 

 relevant criticism and its relation to a more refined theory due to Reiss. 



