p-n JUNCTIONS IN SEMICONDUCTORS 449 



corresponding to a condenser of thickness 2xm . It is evident thit formula 

 (2.44) will hold for a small change db<p superimposed on a Urge bi.is b^ pro- 

 vided that 2.v„, is the thickness of the space charge region under the condi- 

 tions when bip is applied. If i/'„o is the value of \p for 5.^ = 0, then ypm = 

 'An.o — V/2; and C will vary as 



C = /c[47r^a/3K'(^„.o - W2)]'/V87r (2.45) 



so that \IO should plot as a straight line versus b<p with slope 



- (87rA)'(3K/87r?a) = - i^ . (2.46) 



In addition to the holes which flow to account for the change in ^^ , the 

 concentration of holes in the 7^-region will be increased by a factor 

 expiqbip/kT). However, this increase does not lie in the transition region; 

 we shall consider it later, in Section 4, in connection with a-c. admittance. 



Comparison of lite Two Capacities 



It is instructive to compare the two capacities just derived. We suppose 

 that for one value of iii we have A^ » 1 so that the space charge solution is 

 good. For this case we choose Xa = —Xm and .r6 = +.r„» so as to bound the 

 space charge layer. We then imagine w, to be increased, either by raising 

 the temperature or by applying a potential difference 8if. The capacity then 

 changes from 



Csp.chg. = /c/STrXm to Cneut. = Sk/SttKLd (2.47) 



(i.e., from (2.44) to (2.42)) so that the ratio is 



Sj^ = ^ (2.48) 



For A' < 1, this ratio is large, both because of K in the denominator and 

 because .v„, > La so that Xm/Lo > Lj Ld = 1/2 K. 



In Section 4.4 w^e shall compare these capacities with that due to dilusion 

 of holes and electrons beyond the transition region. 



2.4 The Abrupt Transition 



For completeness we shall consider the case in which the impurity con- 

 centration changes abruptly from pp to Un at .t = 0. For this case the po- 

 tential in the space-charge layer will be of the parabolic type discussed by 

 Schottky, the potentials varying as 



^ = {2ir/K)q pp(x - Xpf + constant, x < (2.49) 



^ = -(27r/K)(/ Unix - Xnf + cottstant, .V > (2.50) 



