FLOW OF HOLES AND ELECTRONS IN SEMICONDUCTORS 1199 



Since we do not here allow grad h = 0, (108) implies 



.'[S-*'] 



dV ' Idh ^^^J ^kT .... ,. . . . (109) 



rr = :;^ y = KgW ' arbitrary function) 



on (y ae 



Case 1: 



9^ 0, grad (grad hY 7^ 0. 



In this case, as in the associated case in Section L, the implications of 

 (107) together with 



div grad h{x^ y, z, /) = 



are not known when time variation is present. 



When time variation is absent, we work with the conditions 



(P = (9{h) and V = V{h) 



with 



div grad h{x, y, z) = 



and arrive at counterparts of (107) and (108): 



(P"-(grad/>)' = ^(R((P) (110) 



((P' - i (P^'V = 0. (Ill) 



Proceeding as in the analysis of Case 1 of Section L, we infer that h must 

 be of the kind given by (81b) or (81c). The associated second-order differ- 

 ential equations restricting (9{h) are then, respectively: 



and 



and 



(P"-^r^(R((P) =0 (112a) 



Kl ft 



The V{h) associated with any solution of (112) can be obtained by integra- 

 tion from 



V{h) = C + f y^ ~ ^ dh (C, 5: arbitrary constants). (113) 

 J iy 



