1278 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1956 



Let the density of fixed charge he N = Nd — Na = excess density 

 of donors over acceptors. N may be either positive or negative. In what 

 follows we shall assume that N is positive. So the structure is NvP 

 where v means weakly doped n-type. Equations (2.2) for the hole and 

 electron currents remain unchanged. Poisson's equation becomes 



^= aip-n-\- N) (7.1) 



ax 



We shall deal with the general case of arbitrary mobilities. As in Section 

 VI it is convenient to deal with a fictitious current q(Jp — Jn/b) and a 

 fictitious particle flow Jp + Jn/b. The extra term in (7.1) drops out by 

 differentiation when (7.1) is substituted into the equation for Jp —J„/'b 

 so (6.2) remains unchanged. However, instead of (6.1) we have 



So the fictitious particle flow is no longer the gradient of a potential 

 involving only E and s. 



No Recovibination 



As in Section VI the continuity equations can be immediately inte- 

 grated to give (6.6) and (6.7). Again / is given by (6.9) where the dif- 

 fusion term on the right can be neglected except at the junctions; so 

 again we have asE = 1(1 -\- ^x/L). Substituting this into (7.2) and com- 

 bining (7.2) and (6.6) gives a first order differential equation for E versus 

 X. It is convenient to again use dimensionless quantities with Ei , 2L, 

 and aEi as the units of field, length and current respectively. Then the 

 differential eriuation becomes 



!l 



dx 

 where 



= 2(.'c + aE) (7.3) 



I 



N 



and as before Ni = \/2ni£/Li , which is around 4 X 10'" in germanium 

 at room temperature. The solution of (7.3) contains one arbitrary con- 

 stant (which corresponds to A in the V = case). The lower limit on 

 the constant is determined by the necessity of joining onto a recombina- 

 tion solution \\hcre s approached unity. The positions of .r„ and Xp of 

 the Nv and vP junctions respectively are given by (6.5). 



