818 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1954 



integral under the curve ti . This curve is zero for x < x(ti) and for 

 X > x(ti) it is 



8E(x, h) = (SQi/K) exp [- g(s, + ^0 + g(so)], (4.20) 



where 



X = x(so + t). (4.21) 



If the total transit time across A^ is *S so that 



x(S) = L, (4.22) 



then 



v(h) = f 8E(x, h) dx. (4.23) 



From this expression we can derive the desired formula for D(t). For 

 this purpose the integral over dx is replaced by an integral over s. At 

 time t the range of s is evidently from t to S and dx = u{s) ds. From 

 this we obtain: 



D(t) = vit)/5Qi , 



(4.24) 



= (1//0 [ exp [- g{s) + g{s - tMs) 



Js 



ds. 



From Fig. 4.3 we can see that there are competing tendencies in the 

 decay of D(t) some of which tend to produce the desired convex shape 

 discussed in Section 2 and others the concave shape. The effect of the 

 dielectric relaxation constant is adverse and tends to produce an ex- 

 ponential decay. On the other hand the advance of the pulse of holes 

 from left to right in Fig. 4.2 proceeds in an accelerated fashion with the 

 result that the range of x over which 8E is not zero decreases at an ac- 

 celerated rate. If the dielectric relaxation Avere zero, this would result 

 in the desired convex upwards shape. 



The resultant shape of the D{t) curve is thus sensitive to the exact 

 relationship of the transit time and dielectric relaxation. This can be 

 illustrated by giving the results of analysis for a p-n-p structure, neglect- 

 ing diffusion and considering n to be constant. The solutions of the dc 

 equations are readily obtained for this case and have been published. 

 For convenience we repeat them here: 



E = (J/m)(e"' - 1), (4.25) 



x(s) = iiJKinpf)-' (c"' - at - 1), (4.26) 



