P-ii jrxcTioxs rx sK.\f[coxDrcrnRs 447 



p- aiul //-regions. In the following calculations, we shall consitler a unit area 

 of the junction so that values of P and of capacity are on a unit area bases. 

 The value oi P is obtained by integrating p dx making use of the neu- 

 trality condition to establish the functional relationship between p and .v. 

 The neutrality condition can be written as 



ax = 2,u sinh ^^^--^'^ = 2//, sinh u (2.32) 



where ii = ij{\p — <^i ) kT and 



p = me'^"'"^^"'' = me-'' (2.33) 



;/ = »i e " (2.34) 



so that the value of /-• can be obtained by changing variables from .v to u: 



P = p (Ix = / p{2ni/a) cosh u dn 



J X,, '' »,, 



(2.35) 

 = («l/fl) ["' [1 + e"--"] du = (nl/a)[u, - «« + (r""" - r'"^)/2] 



J Ua 



For the cases of practical interest, the value of p at .v = Xa , denoted by 

 pa , and the value of 11 at x = Xb , denoted by iib , will both be large compared 

 to ;/i . Consequently, we conclude that 



Ha = —ln{pa/)ii) and Ub = In nb/ni 



are both larger than unity in absolute value but probably less than twenty 

 for a reasonable variation of impurity between Xa and Xb . (For example for 

 a change in potential of 0.2 volts such as would occur between p- and //-type 

 germanium, //„ and iib would each be about 4 in magnitude.) Hence we ob- 

 tain for P, 



P = (nl/2a)(2iub - Ua) + (A, ;';/i)' - (ni/ubf) 



^ pl/2a + Oh/a)(t(b — Ua) (2.36) 



where we have neglected (ni/nb)' which is «1 and the negligible compared 

 to Kb — Ua ■ The term pa/2a is simply the integrated acceptor-miiuis-donor 

 density in the /^-region, as may be seen as follows: 



[ (Xa ^ X,) dx = f (-a.v) dx = axl/2 = pl/2a. (2.37) 



The second term in (2.36) is essentially the sum of the holes of the right 

 of .V = plus the electrons to the left of .v = 0, whose charge is also com- 



