1260 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1956 



made in the no-recombination case. It will be an even better approxima- 

 tion in the recombination region, where the second derivative of E is less. 



(2) The particle flow is by diffusion. That is, E'^/E-C can be neglected 

 in comparison with s. 



(3) The ratio of recombination rate r to generation rate g is propor- 

 tional to ^ — r; that is ^ — r = ^(1 — s~). 



All of these simplifying assumptions can be justified by substituting 

 the resulting solution into the original expressions and showing that the 

 neglected terms are small when recombination is important. If the 

 solution is substituted into (4.3) and (2.6) the neglected terms will 

 turn out to be negligible — and therefore assumptions (1) and (3), 

 justified — when s^ is large compared to £/L, . Assumption (2) follows 

 from (1) and the fact that IjaEx is small compared to unity. 



Assumptions (2) and (3) may also be justified by the discussion fol- 

 lowing (3.7) in the following way: Where recombination is important s 

 must be near unity. So the cubic will begin to break down when s = 

 II<jE becomes near to unity, or when E approaches I la. However, if 

 E is approximately I /a then cE^/IEi is approximately {I/aEif, which, 

 as we saw in the Section III, is small compared to unity in practical 

 cases. Thus recombination becomes important and the solution joins 

 onto the cubic in the range where E'^/Ei is small compared to I/aEi . 

 In this range the particle flow is by diffusion and p — n is small compared 

 to p -f n. As we move toward the center of the intrinsic region s increases 

 and E and dE/dx decrease. Therefore, since assumptions (2) and (3) 

 are good where the solution joins onto the cubic, they are good through- 

 out the region where recombination is important. 



The Recombinafion Solution 

 The differential ecjuation (4.2) now takes the form 



d's (1 - s') 



dx" 2L,2 



(4.4) 



The solution for s in the recombination range is seen to be the same for 

 all values of the current. When s has been found E is found from E = 

 I/as. 



For small disturbances in normal carrier concentration, s is only 

 slightly different from unit}' and (4.4) takes the familiar form 



d' , . I - s 



-r^ (1 - s) = 



dx' L 



2 



