FLOW OF HOLES AND ELECTRONS IN SEMICONDUCTORS 1183 



E. Addition of Arbitrary Time Functions to V and OC 



Since only the gradient of V appears in the basic equations (21) and (22), 

 it is evident that if 



V = V(x, y, z, /) 

 and 



(P = (P{x, y, z, 

 are a pair of functions satisfying (21) and (22), then so also are 



V = Vix, y, z, t) + mit) 

 and 



6> = (P(x, y, z, t) 



where m{t) is an arbitrary time function. 

 And since 1) = 01 + 5C, if 



5C = 3C(x, y, z, /) 

 and 



01 = Ol(a:, y, z, /) 



are a pair of functions satisfying (30)-(32), so also are 



dC = 3C(:r, y, z, /) + w(/) 



and 



01 = Ol(:r, y, z, t). 



These arbitrary additive functions with zero gradients are physically 

 trivial in that they merely reflect the arbitrariness of the reference voltage 

 level. They will, however, be retained for the sake of formal completeness 

 whenever they appear in the subsequent analyses. 



F. Summary of Solutions for No Recombination or Time Variation 



The next ten sections of this paj)er (Sections G-Q) contain a sequence 

 of detailed analyses in which is determined the existence or non-existence of 

 solution fields having certain prescribed formal properties. In most of these 

 studies time variability and recombination are admitted and the analysis 

 includes the establishment of the class of recombination rate functions 

 (R consistent with the property under consideration. In those cases where 

 solutions are found to exist, they are expressed in the simplest convenient 



