FLOW OF HOLES AND ELECTRONS IN SEMICONDUCTORS 1201 



with F = 0. If time variation is absent it follows from (114) that recom- 

 bination is absent, too, and the results specialize to 



•0((P) = G + #ln(P (119) 



with (9{x, y, z) any harmonic function and G and R arbitrary constants. 

 These solutions play the same role for the intrinsic semiconductor {N = 0) 

 that (100) does for the extrinsic {N 9^ 0). 

 Case 3: 



^ F^ 0, grad (grad hf = 0. 



In this case it can be shown, just as in Case 3 of Section L, that no gen- 

 erality is lost by considering (P = (9{x, t) and V = V{x, I) in place of (P(A, /) 

 and V{h, t). Equations (107) and (109) then become 





and 



dv _ la^~^^^^J (121) 



dx (P 



Any solution of (120) when substituted into (121) gives an associated V 

 from 



(X) 



dx ^'^' ^ (122) 



-iw 



V(x, t) = q{t) +y f ^-^ dx 



If recombination is absent, (R((P) merely vanishes from (120). If time varia- 

 tion is absent, the functions g{t) and q(/) are replaced by arbitrary constants 

 and the standard change of variables 



U((P) for ^ 

 ax 



(123) 



u((P) -7:z for — 

 d(P dx 



leads to a solution of (120) in two quadratures. An equivalent solution is 

 given by W. van Roosbroeck in Reference 1. From (120) it follows that 

 recombination and time variation cannot simultaneously be absent for 

 Case 3. 



