FLOW OF HOLES AND ELECTRONS IN SEMICONDUCTORS 1175 



lip = -Upe^p grad V + ~ grad p] (3) 



o r j^j- ~\ 



||n = — Mn e n grad V — — grad n (4) 



n — p = fiQ — po = N (a, constant) (5) 



n, p > (6) 



ll = ll+l|n (7) 



wherein 



n: concentration of negative carriers (electrons) 

 p: concentration of positive carriers (holes) 

 no: thermal equilibrium value of n 

 pQi thermal equilibrium value of p 



o 



lip! hole current density vector 



o 



||„: electron current density vector 



o 



1 1 : total current density vector 



/: time variable 



e: magnitude of electronic charge 



k: Boltzmann's constant 



Hpi hole mobility constant 

 * fin- electron mobility constant 



T: absolute temperature (assumed constant with time and uniform) 



V : potential of electrical intensity field 



(R: electron-hole recombination rate function (will usually be regarded 

 as depending on p — po and « — «o or equivalent variables). 



These relations have fundamental application to transistor electronics, 

 photoelectric effects, and related phenomena. Detailed discussions of their 

 physical bases will be found in References 1 and 3. In brief, (1) and (2) 

 are conservation conditions for the positive and negative carriers; (3) and 

 (4) express the dependence of the local current densities on the electrostatic 

 potential gradient and on the carrier concentration gradients (i.e., on con- 

 duction and diffusion); (5) expresses the condition of electrical neutraUty 

 under the assumption of a constant difference in concentrations of ionized 

 donor and acceptor centers; and (6) and (7) are self evident. 



The present study is directed toward the discovery of (1) general proper- 

 ties of the flow fields inside semiconductors and (2) families of exact solu- 

 tions to the flow equations. The approach to the latter objective is through 



