FLOW OF HOLES AND ELECTRONS IN SEMICONDUCTORS 1177 



ried up to the point of actual substitution of numerical values in the 



formulae. 



Note: In the following, functions and constants described as "arbitrary" 

 are to be considered as being subject nevertheless to the restrictions 

 implied by (6). In any particular case it is an elementary matter to 

 determine these restrictions and we shall not usually carry out this 

 detail. Also, ''arbitrary" functions are subject to appropriate dif- 

 ferentiability conditions readily evident in any particular case. 



B. Some Properties of the Current Density Vector Fields 



o o 



Several interesting properties of the current density vector fields || p ,|| n , 



o 



and II are easily found from {S)-{5). 



It is evident that (3) and (4) can be rewritten as 



\\v = -envP grad (t) + ^ in p\ (8) 



and 



c / j^J- \ 



!|n = — ^Mn n grad [y — — In w j . (9) 



From (3), (4), and (7) we have 



II = -eiyLnfi + ixpp) grad V ^- kT grad (jinfi - Upp) (10) 



which because of (5) can be rewritten as 



fl = -e (Mn « + MP P) grad \v - ^"t^" In (m„ » + Mp p)\. (11) 



L ^ Mn I Mp J 



Now (8), (9) and (11) are all of the form 



U = <^ grad ^ 

 and hence obviously satisfy the condition 



U • curl u = 0. 

 Therefore we have 



CO o 



Theorem 1: || p , || n , and || are surface-normal vector fields. 

 From (8)-(10) we find, using (5) 



curl iIp = -e/zp grad p X grad V, (12) 



