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THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1956 



a large amount, there can be no visible increment in solubility as a result 

 of pairing because P can never exceed N a which by definition is small. 

 The physical reason for these limitations is the following. Suppose Nd 

 is driven close to A''^ by the hole-electron equilibrium so that in terms of 

 carriers (holes and electrons) the specimen is very closely compensated. 

 Then if by the formation of pairs, additional donors are caused to enter 

 the crystal, the electrons they donate cannot be absorbed by holes be- 

 cause very few of the latter are present. Thus the following two sketched 

 equilibria will oppose each other 



Li (external) 



Li+ 



+ 

 Ga~ 



Ti 



[Li+Ga"] 



+ 



(A12) 



the one involving electrons attempting to drive lithium out of solution 

 because of the build-up of electron concentration, and the pairing equi- 

 librium attempting to bring lithium into solution in order to form pairs. 

 Thus the pairing process will not be as efficient a solubilizer as might 

 be thought at first. 



This point can be illustrated by considering a situation in which the 

 germanium crystal not only contains gallium to the level, A''^ but also 

 an immobile donor, to the level A'^ — 0.99 N a . Thus, the crystal is almost 

 compensated before any lithium has been added. Nevertheless, there are 

 still Na gallium ions so that even though the hole-electron equilibrium, 

 working on the differential, 0.01 Na , cannot increase the solubility of 

 lithium, the pairing process might. To investigate this situation equations 

 (A2) to (A7) can be adopted with the simple change that (A~ — N) re- 

 places A~ in (A7). 



Taking the situation covered by (AlO) at 300°K, Table All was com- 

 piled. Here again Nd* is the solubility for 12 = 0. 



If only the hole-electron effect were operative, then we could not ex 

 pect to drive Nd much beyond Na — N. In the 10^^ case Na — A" is 10 

 cm"^ and in the lO" case it is 10^^ cm"'. The values of Nd* in Table All 

 thus confirm this argument. Furthermore, Nd is in neither case much 

 greater than No* showing that despite the fact that there were, respec- 



14 



