EXCESS SEMICONDUCTOR llOIJ'. TRANSPORT 415 



For the case of a surface recombination unintluenced by electron concen- 

 tration one obtains similarly, with R — v. 



— [same with v instead of V(\ (31b) 



5 = in -" (32b) 



V 



When ^Xe = 3^/, 2, as for germanium, (31a) and (31b) become respectively 



5/2 v^ 9 1 + 5V3 



+ 5j'o/3 ^ 1 + Sv,/Z "^ 4 " 1 + j/o 



] 



and 



— [same with v instead of j'ol (33a) 



r 1 VQ 



+ 51/0/3 ' 1 + 5t'o/3j 



— [same with v instead of vo] (33b) 



These can also be written, using (32a) and (32b), 



5/2 5/2 5 r (l + 5.o/3)(l + u) ! 



^ - ' + 1 + 5.0/3 ~ 1 + 5./3 + 4 ^ L(l + 5»'/3)(l + .o)J ^-^ ^ 



and 



/p -\- 3/5\ 1 1 ,^^^, 



^ - 5 + In , ' - ) - , , , ,^ + , , r n 34b) 



\vo + 3/3/ 1 + :)v/3 1 + 5^3 



Figures 5a and 5b show as a full curve the plot of eq. (33a) for the case 

 j/o = oc , and the full curve in Fig. 6 shows in the same way the plot of 

 (33b) for j/o = °o . Changing pq of course merely shifts either curve hori- 

 zontally. Note the very sharp increase of v for small |, which shows up in 

 pronounced manner on the expanded scale of Fig. 5b. The corresponding 

 values of s, computed from (32a) or (34a), are marked on the curve of Fig. 5; 

 the corresponding marks on the curve of Fig. 6 also represent values of s at 

 intervals of 0.2, but are not labeled with absolute values because (32b) is 

 infinite for I'd = 00 . 



For large ^, v becomes very small and it becomes legitimate to expand the 

 logarithms. The first few terms of the resulting asymptotic expression for ^ 

 are, for vo = °^ and the recombination function leading to (31a), 



^ ~ (--2 - 1) In (1 + //;, M.) - (1 + Me/MO - In P 



-i- (3 -f 2^Mn/^Jie)v (35a) 



