THE FOKWAKD CHARACTERISTIC OF THE PIN DIODE 697 



fore we obtain the characteristic 



PV = ^n ( \_ + l) (56) 



which is vaHd until 7 approaches lo . Of course we would not have ob- 

 tained this ideal characteristic of a simple PN junction had w^e taken 

 recombination into account; our result depends upon the constancy of 

 n(x) and pix) in the middle region. For the case of no recombination in 

 the middle region (56) and (44) cover ranges (a), (b) and (c). Instead of 

 (44) the more exact expression (37) could be used requiring the evalu- 

 ation of y{I) from (31). It seems that the extra refinement is of no help 

 in understanding the device and unnecessary in treating experimental 

 data. Therefore, we shall adopt (44) and the approximations leading to 

 it as a model for treating the more complicated recombination case. 

 That is, we shall seek a generalization of (44) which takes recombination 

 into account in a sufficiently good approximation. 



Large Injection with Recombination 



We are interested in determining the effect of recombination in the 

 middle region upon the operating characteristics of the device. Therefore 

 we go immediately to the large injection case n = p. Equation (16) be- 

 come 



ny, = npe^^"" p^w = n„,(pivVpp)e^^"' 



(57) 

 no = ppe^^'^ noo = no(np'^/np)e^^° 



t which gives 

 /3(Fo + F„) = (nin^nolnl) (58) 



We shall assume that recombination is linear in the injected carrier 

 density to simplify the calculation. It will be possible, later to approxi- 

 mate bimolecular recombination by using an appropriate value for the 

 lifetime r corresponding to the injected carrier density. Therefore we 

 write 



O'i-n ui p n (rj(S\ 



dx dx T 



Eliminating In{x) by use of (13) gives the equation for n(x) 



(60) 



dn n 



dx^ L 



% 



