12G8 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1956 



X curve can now bo found from (5.8) and 



X - Xo = I -r 



L - 



\dx/ 



=/:©"'- 



(5.9) 



In general we will be intrested in cases where the junction solution 

 holds over a length L — .To that is small compared to L, so we can take 

 .To = L in (5.7). It will also be valid to let Eo in (5.7) and (5.8) be the 

 value Ec on the cubic at x = L. Putting Ec = Eo in equation (5.6) then 

 gives Ej in terms of Eo and Sp = P/2ni , where P is the majority carrier 

 concentration in the extrinsic region. In what follows we shall use these 

 approximations. It will be convenient to express .To = L in (5.7) in 

 terms of / using / = \/2Z/<£. We continue to use dimensionless quan- 

 tities with El , 2Li and aEi as the units of field, length and current re- 

 spectively, and 2LiEi as the unit of voltage. In general however we can 

 express voltages in terms of kT/g. 



When Eo^ is either large or small compared to /, the junction solution 

 takes a simple form and the field and potential distributions can be 

 found analytically. We next consider two approximations that hold in 

 those two cases respectively. Relatively good agreement between the 

 two solutions at Eo = I indicates that each solution may be used up to 

 this point. 



Case of Eo Large Compared to I 

 From (5.7) to (5.9) 



X — Xo = -x/ScC / 



J E 



l_ 



3\2 



+ {E' - Eo') 



-1/2 



dE 



(5.10) 



This can be solved in the following two overlapping ranges where the 

 integrand has a simple form: 



Range 1. Here E — Eo'is small compared to 2Eo , so (5.10) becomes 



£ 



X — Xo = 



\/2Ec 



sinh 



-1 



\e - Eo) ?f-'j 



(5.11) 



Since E and Eo are almost equal, we have for the voltage drop in this 

 range 



V - Vo = Eo(x - Xo) (5.12) 



Range 2. Here E'^ — Eo' is large compared to 2(£L/Eof, so (5.10) 

 gives 



