1266 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1956 



for current becomes 





sE 



£2 



E^ + E 

 £2 



(5.2) 



This can be integrated at once. The boundary conditions are dE/dx = 01 

 when E = and E = Ej at x = L; the field Ej at the IP junction will 

 be determined by joining the solutions for the / and P regions. The 

 solution can best be expressed by parametric equations giving x and the 

 potential V as functions of E. 



L — X 



£ 



J E 



dE 



= £ 



Vj - F = £ 



E 



E Vl + E'-/2 



dE 2kT 



csch' 



E 



V2 



— csch 



-i^j 



Vl + E-'/2 



Q L 



sinh" 



1l 



V2 



sinh' 



— 1 



V2j 



E - 



V2_ 



(5.3) , 

 (5.4) 



where we have used the relation between dimensionless quantities 

 £ = ■\/2kT/q, which follows from (2.8) with Ei = 1. It will be more 

 convenient to express voltages in terms of kT/q rather than in terms of 

 the unit voltage 2EiLi ; then the ratio qV/kT is independent of the 

 units. For convenience we take the voltage as increasing in going toward 

 the IP junction with V = in the normal P material. The voltage Vj 

 at the junction is found by joining solutions. 



On the P side, let the excess acceptor density be P. Adding the term 

 — aP to the right hand side of Poisson's (2.1), and proceeding as before 

 we have, instead of (2.5) 



— s 



qT 

 '" kT, 



= J = 



where Sp = P/2ni . We shall assume that the P region is strongly ex- 

 trinsic so that n <K p. Then s = Sp in the normal p material, where 

 j^ = V = 0. Hence 



E' 



'qV 



'"'"Vkr \ 



(5.5) 



In the intrinsic material the corresponding solution is -E"" — s = — 1 . 

 Since both E and s are continuous at the junction, qVj/kT = 1 — 1/Sp 

 where l/sp can be neglected. Thus i?/ = Sj = Sp exp [ — (qVj/kT)] = 

 Sp/e where e = 2.72 is the base of the natural logarithms. 



Knowing Ej we can find the field and potential distributions in the 

 intrinsic material from (5.3) and (5.4). 



