444 BELL SYSTEM TECHNICAL JOURNAL 



becomes a family of straight lines with "a" as a parameter. (Only a = 10 

 cm~ is shown for Si, all the other lines being for Go.) The half thickness 

 Xm {— iim/a) of the space-charge region is also shown. Solutions are obtained 

 when these lines cross the curves iim = fii exp (q\pm/kT), which are shown 

 for room temperature. The condition that the intersection lie well to the 

 right on the curve is equivalent to K » 1. For two Si samples cut from a 

 melt, a was determined from measurements of conductivity and was 

 about 10^^ to 10^^ cm"*. For these, the space change region has a half-width 

 Xm of more than 10~ cm. For other temperatures, the curves can be ap- 

 propriately translated.^ 



In Fig. 4(a) we show the limiting potential shapes: 



ax = Im sinh ^ for K « 1 (2.23) 



yp = (^Pm/2)(- {x/xmY + 3(x/xm)) for K » 1 (2.24) 



In Fig. 4(b) the charge densities are shown. For the space-charge case, 

 I Nd — Na I is greater than n or p. For a higher potential rise, i.e. larger 

 i/'„, , the discrepancy would be greater and Nd — Na would be unneutralized 

 except near Xm . 



2.3 The Transilion-Region Capacity 



When the voltage across the junction is changing, a flow of holes and 

 electrons is required to alter the space charge in the transition region. We 

 shall calculate the charge distribution in the transition region with the aid 

 of a pseudo-equilibrium model in which the following processes are imagined 

 to be prevented: (1) hole and electron recombination, (2) electron flow across 

 the /^-region contact at Xa (Fig. 1), (3) hole flow across the «-region boundary 

 at Xb . Under these conditions holes which flow in across Xa must remain in 

 the specimen. If a potential 5^ is applied at the p end, then holes will flow 

 into the specimen until ipp has increased by 5.^ so that the holes insiie are 

 in equilibrium with the contact which applies the potential. Since the speci- 

 men as a whole remains neutral, an equal electron flow will occur at Xb . 

 When the specimen arrives at its pseudo-equilibrium steady-state, the 

 potential distribution will be modified in the transition region and the num- 

 ber of holes in this region will be different from the number present under 

 conditions of true thermal equilibrium. The added number of holes is pro- 

 portional to bif for small values of bip and thus acts like the charge on a con- 

 denser. Our problem in this section is to calculate how this charge depends 



* Unpublished data of W. H. Brat tain and G. L. Pearson. 



' The effect of unionized donors and acceptors can also be included by letting Wi include 

 the properly weighted donor states and pi, the acceptor states. 



