FLOW OF HOLES AND ELECTRONS IN SEMICONDUCTORS 1179 



[Note: As is suggested by (18) and (19), the total carrier concentration 



(P = n-\- p = N + 2p = 2n- N {(P>\N\) 



will frequently appear as the "natural" concentration variable in 

 the relations with which we shall be working. Hence, expressions 

 involving p, or p and n will often be replaced in the sequel by their 

 equivalents in terms of the variable (P. It will be noted that 



grad (P = 2 grad p = 2 grad n.] 



Equations (17) and (19) yield at once the following theorems: 



Theorem 4: The vector field 



ll X ||„ = II X ||„ = ll X II 



is solenoidal. 



Theorem 5: The vector field ' 



o o 



/ IIp _ lU j jg irrotational with a potential (-eNV + kT(P). 

 Theorem 6: The vector field 



o o 



/ IIp _l_ Lb J is surface-normal (to the surfaces of constant V). 

 [Theorem 7: ||p, ||n, || , grad "0, and grad p are coplanar vectors. 



o o o 



Theorem 8: The flow lines of any two of the fields l|p ,||n , and || coincide 

 if and only if 



grad /> = {p = pit)) 



or grad '0 = fU = •U(/)) 



or V = V{p, /). 



Also, from (17) and (19) we obtain the curious relations: 



^grad(PX (^^+'^) (20a) 



2 \Mp M«/ 



If V 11? = 



Mp M» 



o o 



= -*?(Pcurl(i' + ^") (20b) 



= -f-[44:)]- 



