580 



THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1956 



Table IV 



Experiments which will be described later indicate that in germanium, 

 gallium and lithium can approach as close as 1.7 X 10 cm, Usmg this 

 value of a, and k = 16, g = 4.77 X 10~^ statcoulombs, the values of Q 

 appearing in Table IV were computed from (9.2) 



With these values, P/Nd , the fraction of donors paired can be com- 

 puted from (9.5) as a function of temperature and N a for the simplest 

 case, i.e., the one for which A''^ = No . Fig. 15 contains plots showing 

 these dependences. It must be remembered that all other things remain- 

 ing the same P/Nd will be greater than the values shown in Fig. 15 

 when Nd < Na ' 



A rather important integral to which reference shall be made later is 



x^ exp {q/KkTx) 

 • 1 



dx 



(9.7) 



The integral appearing in (7.14) is a special case of (9.7) with ri = a, and 

 Ti = h. I(r2 , n) has been evaluated over a considerable range. To facili- 

 tate matters the transformation 



X = (q^'/KkT) X 



(9.8) 



has been employed. In this notation n and ro transform to pi and p2 , and 



I{r, , n) = {g'/KkTY r X' exp (1/X) dX = {q/KkTYHp^ , pi) (9.9) 



•'pi 



Figs. 16 and 17 contain plots of i{p2 , 0.05) out to p2 = 5. The choice 

 of pi equal to 0.05 was rather unfortunate since for k = 16, and T = 

 300°K it corresponds to pi = 2.5 X 10~^ cm. Since acceptors like gallium 

 possess values in respect to lithium as low as 1.7 X 10~ cm i(p2 , 0.05) 

 is not much use in these cases. The choice 0.05 was made before the ex- 

 perimental data on gallium was available. Below we shall describe a 

 method for extending t(p2 , pi) to cases where n is less than 2.5 X 10 

 cm. 



