422 BELL SYSTEM TECHNICAL JOURNAL 



it is desired that the ratio (43) be less than about 0.1 in the steady state 

 for values of v as high as 0.3, the upper curve of Fig. 9 shows that the current 



density used must be large enough to make ( - j < ryy. i-C-, j ^ 11-7 /, 



where / is given by (25) and (26). 



An approximate evaluation of (40) and (41) in the transient region can 

 be performed by graphical or numerical differentiation of a curve such as 

 that of Fig. 7. For example, a rough calculation based on Fig. 7 gives, in 

 the middle of the transient portion (^ = 0.75), 



div. diffusion curren 



div. drift current 



■'l-tO' 



More important and also more difficult to estimate is the effect of difi"usion 

 in rounding off the front and rear edges of the transient. Various ways can 

 be devised to estimate a rough upper limit to the amount of rounding off 

 to be expected. One such is to compute what the diffusive flux just behind 

 the front of the advancing disturbance would be if the distribution of holes 

 were the same as in the absence of diffusion. Under conditions where diffusion 

 is not too serious the time integral of this diffusive flux between any two 

 times can be equated to the increase in rounding of the front, as measured 

 by the area between an ideal curve such as that of Fig. 7 and the actual 

 curve of v against ^ for the same time s. The integration cannot be extended 

 back to time zero, however, since the integral of the flux diverges logarith- 

 mically. The fact that the diffusive flux is actually finite instead of mfinite 

 of course arises from the fact that at small times the concentration gradient 

 a short distance behind the front can no longer be approximated by the 

 gradient which would obtain in the absence of diffusion, but instead is very 

 much less. This suggests that an upper limit to the total diffusive flux passing 

 into the region of the front from time to time 5 can be obtained by taking 

 the flux computed as described above between the times Sa and s, and adding 

 to it the total number of holes which have left the injection electrode be- 

 tween time and time 5o . Since this gives an upper limit for any Sq , one 

 may use the minimum of the resulting sum as So is varied. 



The results of some sample calculations of this sort are shown in Fig. 10, 

 which refers to the same time, currents, and recombination function as Fig. 7, 

 viz., s = 1/t = l.O, je/ja = 2/13, ideal volume recombination. The full 

 curve is the transient portion of Fig. 7 replotted on a larger scale. The lower 

 dotted curve is a curve drawn in by hand in such a way as to make the 

 area between it and the full curve equal the upper limit computed in the 

 manner just described, for the case; = 1007. The upper curve was drawn 



