f>->i jiwcnoxs /.v sE.\fuo\i)rcT()Rs 451 



and excess electrons in a /'-lyi'c semicoiiducLor, respectively, have lifetimes 

 7-p aiul r„ given hy 



dp = —bpTp = —rn8p or T J, = \ rii ^ p g (3.2) 



and 



5/'/ = —b)r T„ = —rpbii or r„ = 1 r/? = ;/ ,(f. (3.3) 



We shall have occasion to use this interpretation later. (We later consider 

 the moditkations required when surface recombination occurs, Section 4.2, 

 Appendix V, and the effect of a localized region of high recombination rate, 

 Section 4.6, Ai){)endix III.) 



In principle, the steady-state solution can be obtained in terms of the 

 three potentials yp, ifp and ^„ . These must satisfy three simultaneous or- 

 dinary diflferential equations, which we shall derive. As discussed in Section 

 2, we consider all donors and acceptors to be ionized so that Poisson's equa- 

 tion becomes 



d^ = _1^ (^., + n^e"'"-''"''' - n,e"^-'^''''') (3.4) 



dx^ K 



an equation in which the unknowns are the three functions s?„ , s^- Ji'id \p. 

 The total current density, from left to right, is 



I ^ r,-^ h= -qn 



p 'Z*^''' + bn "^p 

 dx dx 



(^.5) 



The elimination of p and // by equation (2.4) results in an equation in- 

 volving the three unknown functions and /. The divergence of hole current, 

 equal to the net rate of generation of holes, is 



dip ^ _ r_^ (^_^\ _ ^ # (i<fi> I (^'fp l 



dx "^^^IkTydxJ kTdxdx^dx-'j (3.6) 



= gig - rnp) = qg{\ - e"'^'"-'"'"'), 



with p in the second term given by (2.4) so that (3.6) is also an equation 

 for the three unknown functions. The equation for dl,dx can be derived 

 from the last two and adds nothing new. These three equations can be used 

 to solve for d'xp r/.v", d'^pp dx' and dip,, dx in terms of lower derivatives and /. 

 They thus constitute a set of equations sufticient to solve the problem pro- 

 vided that physically meaningful boundary conditions are imposed. We 

 shall not, however, deal directly with these equations; the main reason for 

 deriving them was to show that the problem in question is, in principle, 

 completely formulated. Instead of attempting to solve the equations, we 

 shall discuss certain general features of the solutions for ^pp and (p„ , using 



