480 BELL SYSTEM TECHNICAL JOURNAL 



can be obtained by adding (22) and (23) and equating the result to zero. 

 This gives: 



The field vanishes for b = \, corresponding to equal mobilities for holes 

 and electrons. For b greater than unity and for equal concentration 

 gradients of holes and electrons, the diffusion current of electrons is 

 larger than that of holes. The field is such as to equate these currents by 

 increasing the flow of holes and decreasing the flow of electrons. 



If (25) is substituted into (22), the following equation is obtained for 



ip = —kTup 



^^-'^^ +llgrad^ (26) 



lNfb + p{b+ 1) 



If recombination is neglected, the hole current is conserved and 



div ip = 0. (27) 



Using this relation, an equation of the Laplace type can be obtained for 

 p which may be integrated subject to the appropriate boundary condi- 

 tions. This derivation is given in Appendix B. The results do not differ 

 significantly from those obtained below for p assumed small. 



Rather than continue with the general case, we shall at this point 

 assume that p <K Nf so that the first term in the parenthesis of Eq. 

 (26) is negligible in comparison with unity. This amounts to setting F = 

 in Eq. (3) and assuming that the holes move entirely by diffusion. 

 This is a very good approximation in most cases of practical interest and 

 is valid for small i as well as for i = 0. We then have 



ip = —kTupgrsid p. (28) 



The condition div ip = gives Laplace's equation for p: 



W = 0. (29) 



Equation (29) is to be solved subject to the appropriate boundary 

 conditions. For the model illustrated in Fig. 1 we can assume that p 

 depends only on the radial distance r and that 



p = p^titr = n, (30) 



p = piUtr = 00. (31) 



The solution of (29) which satisfies (31) is: 



P-- pi+ (Ip/2irkTfipr), (32) 



