448 BELL SYSTEM TECHNICAL JOURNAL 



p3nsated by holes. The total number of holes can be expressed in terms of 

 8<p through the dependence of iii on 8^. The second term is thus 



(»i/a)[ln(«6/"i) + In {pa/ui)] 



= (nVa)e'''"''' ■ [\n{,u,pjn]) - qb^/kT] (2.3S) 



Hence for a small change db(p in b(p, the change in charge dQ ^ qdP and the 

 capacity C are given by 



C = ^ = i ^' [\n{n,pj^d - iqbp/kT) - 1]. (2.39) 



This capacity can be reexpressed in terms of the difference in \p between 

 Xa and Xb : When 5^ = 0, corresponding to the thermal equilibrium case, 

 we have 



Panb = nW^'"^"'"" (2.40) 



Using this together with the definitions of Ld and La we obtain 



, k[9(^6 - ^g - K^ ) /kT - 1] e"''"' 



^ - M2LI/U ^-''^ 



In this expression ^o and ^6 are the potentials when 8<p = 0; so that 



^6 — (lAo + V) 



is thus the increase in potential in going from Xa to Xb when 8ip is applied. 



For thermal equilibrium, 8ip = and, as discussed above, the term in 



^6 ~ ^o will be about 10. Hence, using the definition K = LojlLa , we have 



C S K/M^KLo/m) (2.42) 



For K <K I, the case for which this formula is valid, C will be the capacity 

 of a condenser whose dielectric layer is much less than Ld thick. 



Capacily for Space Charge Case, K ^ 1 



As discussed in connection with (2.30) and (2.31), the applied potenlid 

 8ip reduces the increase (= 2i/'„,) in \p between the /^-region and the ;i-region 

 by 8ip/2 on each side of x — 0. This is accomplished by a narrowing of the 

 space charge layer by 5.v,„ on each side where (according to (2.20)) 



5i/'m = —8ip/2 = 4TqaXm8xm/K (2.43) 



The decrease in width 8x,n brings with it an increase in number of holes 

 — ax 8xm per unit area of the junction on the p-s'ile and an equal number of 

 electrons on the w-side. Thus a charge of holes per unit area of 53 = — qaXm8xm 

 must flow in from the left. The capacity per unit area is, therefore, 



C = 8Q/8ip = qaxm8xm/8ip = K/Air2xm (2.44) 



