410 BELL SYSTEM T ECU NIC A L JOURNAL 



various points of the curve horizontally to the right with increasing time, 

 one must move them along a family of decreasing curves (cf. Figs. 4, 5, 

 and 6). The effect of diffusion can be described roughly as a migration of 

 each point from one of these curves to another. 



3. Complete Differential Equations of the Problem 



As was mentioned in Section 1 , the transport of electrons and holes along 

 a narrow filament can be described by one-dimensional equations even if 

 recombination at the surface of the filament causes the distribution of 

 electrons and holes to be non-uniform over its cross-section. In the equations 

 to follow, Uk and Ug will be understood to refer to averages, over the cross- 

 section, of the hole and electron densities, respectively; the electrostatic 

 field E can always be assumed uniform over the cross-section of the filament, 

 if the latter is thin. The as yet uncertain influence of the surface on the rate 

 of recombination of electrons and holes can be allowed for by writing the 

 recombination rate as «oi?(»;,/«o)/'r particles per unit volume per unit time, 

 where i? is a function which is asymptotically w/./»o as its argument — K), 

 and where r is the recombination time for small hole densities. For pure 

 volume recombination, R — nutie/iiQ = {nh/n^{\ -\- nh/no), while a con- 

 ceivable extreme of surface recombination would he R = Uh/jio . 



Using this function, the continuity equations for electrons and holes can 

 then be written, with inclusion of recombination and diffusion terms 



^ = -5- {EiJ-hm) - —R[-]-^ ^[Dh ^] (17) 



at ox T \iio/ ox \ dx / 



— - = - (£MeWj - —R\-] + ^[De-^] (18) 



ot ox T \iio/ ox \ ox J 



where the Z)'s are the diffusion constants, related to the mobilities /x by the 

 Einstein relation 



D/y. = kT/e (19) 



Using the neutrality condition Ue = «o + f^h , subtracting (17) from (18) 

 and integrating gives the equation of constancy of current, the generali- 

 zation of (3) : 



E[Qie + fih)nh + Me«o] + — (pe - mO t^ = JiO/e. (20) 



e ax 



Solving for E gives 



y — kT(jjie - ma) -^ 

 ^= r, , ^ , 1 (21) 



