484 BELL SYSTEM TELllMCALJOlRSAL 



jmred to v,-, Vm iuul w , a situation wliich insures local equilibrium between 

 p and pa . Under these conditions we obtain 



Dy' = ico[\ -\- a] -j- V r -\- OiVra 



Dividing by (1 + ex) gives 



pVr + PaVra 



[D/(l + a) It' = [Dp/ip + pJh" - 'CO + 



P + pa 



The interpretation is that the holes diffuse as if their diffusion constant were 

 reduced by the fraction of the time p/(p + pa) they are free to move and 

 recombine with a properly weighted average of u,- and Vra • 



APPENDIX VII 



Solutions of the Space Charge Equation 



\\"c shall I'lrst show that the space charge equation (2.11) has a unique 

 solution for the one dimensional case. For simplicity we write (2.11) in 

 the form 



^J^ = sinh u - fix) (A7.1) 



ax- 



to which it can be readily reduced. We shall deal with the case for which 



f = fa for .V < .v„ (A7.2) 



/ = fh for .V > Xh > Xa (A7.3) 



so that the interval (.r„ , Xb) is bounded by semi-infinite blocks of uniform 

 semiconductor. We shall require that u be finite at .v = ± cc . This boundary 

 condition requires that for large values of | .v | 



w = Ua + AaC^'"'' x-^ -^ (A7.4) 



u = Ub + Abe~'"''' a; ^ + «) (A7.5) 



where 



sinh Ua = fa , sinh Ub = fh 



7„ = I (cosh Ma)"" I , 7'- = 1 (cosh UhY'' I 



(If the opposite signs of the7's were present, the boundary conditions would 

 not be satisfied.) The exponential solutions are valid for | u — Ua \ or 

 I « — 2^6 I « 1. For larger values, however, solutions exist which are ob- 

 tained by integrating (A7.1) to larger or smaller values of x. 



For these extended solutions the values of nix, Aa) and u'{x, .1„) (= du V/.v) 



