CHEMICAL INTERACTIONS AMONG DEFECTS IN Ge AND Si 615 



The value of Ui is taken from Figure 2, of Do'^, from Figure 5, and of fi, 

 from Table IV. Using (AlO) together with (A9) and (A8) leads to the 

 results tabulated in Table AI. In this table, Nd* represents the solu- 

 bility for the case = 0, i.e., the solubility if there were no ion pairing. 

 The main feature to be obtained from the Table is that Nd is not very 

 much larger than Nd*, no matter how large the value of A^^ . This is 

 true in spite of the fact that the last column which lists P shows that at 

 Na = 10^^ cm~^ P is about 98 % of A''^ so that pairing of the donor is 

 virtually complete. 



The result is not limited to the special conditions of doping and tem- 

 perature chosen in compiling Table AI, but must be quite general. One 

 can arrive at this conclusion in the following way. 



By subtracting (A3) from (A2) we obtain 



Na- Nd = A~ - Z)+. 



(All) 



^+ 



The quantities A~ and D appear in equations (A4) and (A7), while n 

 and p, appearing in (A4) and (A7) are related by (A5) . These three equa- 

 tions are sufficient for the determination oi D^ in its dependence on A". 



' That this is the case is immediately obvious when (A4), (A5), and (A7) 

 are recognized as reproductions of (3.1), (3.2) and (2.8). In fact this 

 means that the desired relationship between D^ and A~ is nothing more 

 than equation (3.4) which itself is predicated on (3.1), (3.2), and (2.8). 

 Hut then it is known according to (3.6), that D'^ can at the most be 

 slightly greater than A~, although most likely less. This assumes of course 

 that we deal with dopings sufficiently high so that (3.5) applies. On 

 the other hand at low dopings (3.4) tells us that Z)"^ will be Do^. There- 

 fore if we work with a system in which in the absence of pairing the elec- 

 1 ion-hole equilibrium has driven the value of Nd close to Na (as it has 

 ill this system — see Nd*) the introduction of pairing cannot drive it 

 much higher, since according to (All) if D^ cannot get higher than A~, 

 \'d cannot exceed Na • This is evident in Table AI where Nd comes very 



I close to Na but never exceeds it. 



When A''^ is very small so that D"*" equals Do'^ and does exceed A~ by 



