688 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1956 



nation follows some power v of the injected density 



the forward characteristic of a simple PN junction is of the form 



exp W^^iv + 1)F] (6) 



Thus nonlinear recombination can account for the observation that in 

 silicon diodes the slope of V versus log / is usually much less than /3. 

 Our purpose here is not to study this interesting effect, but to study those 

 effects which are due to the presence of the middle region. Therefore, we 

 assume linear recombination for the sake of simplicity. In the last sec- 

 tion we give a brief consideration of what to expect in the case of non- 

 linear recombination in the contacts. Recombination in the middle 

 region will also be assumed to be linear in the injected carrier density, 

 but this assumption is not critical, since it turns out that a moderate 

 amount of recombination in the middle region does not change the quali- 

 tative behavior of the device. 



BASIC EQUATIONS 



Fig. 1(b) shows the electrostatic potential V{x) for the equilibrium 

 case 7 = 0. The potential is constant except in the space charge layers. 

 If w^e call the potential of the middle region zero, the P^ and N^ contacts 

 are at the potentials — Vi and Vi respectively, where 



/3Fi - In (P^/pp) 



(7) 

 ^F2 = {n (N^/np) 



Figure Ic shows the potential when a forward current I flows and a 

 forward bias F is produced across the device. We shall define the poten- 

 tial so that the A^"^ contact remains at V2 , which puts the P"^ contact at 

 potential F — Fi . The potential at a point x is then given by 



V{x) = V2- r E{x) dx (8) 



"WW 



where E{x) is the electric field assumed zero in the contact regions x > 

 WW and x < 00. The applied bias F consists of three terms 



F = Vo + Vp + F,„ (9) 



' This potential distribution has been discussed b^y A. Herlet and E. Sp(MiI<o, 

 Zeits. f. Ang. Phys., B7, H3, p. 149, 1955. 



