FLOW OF HOLES AND ELECTRONS IN SEMICONDUCTORS 1193 



or 



A = i ^ (grad hf = 1 = h\ (81c) 



The possibility h = x violates one defining condition for the present case 

 (i.e., grad (grad h)- ^ 0) and hence will be left for consideration in Case 3. 

 The remaining two possibilities lead respectively to the following forms of 

 ordinary differential equation for the determination of ai(A): 



or 



( S - 'g-u' V 



^ - "^"^^'V + «1^ «,(<,) =0. (82c) 



Given any 11 (A) satisfying one of these equations, the associated 3€(A) is 

 obtained by integration from (77): 



'^W = / (tT^) " + ' (/: arbitrary constant). ^^^^ 



It is evident from (72) that Case 1 does not exist if both recombination 

 and time variation are absent. 

 Case 2: 



In considering this case we shall exclude the condition -77- = because it 



on 



has been included in Section J. 



From the condition — — = we have 



3C = k{t)h + /(/) {k{t), l{t): arbitrary functions) (84) 



with kr^O. This shows that JC itself is a harmonic function and we can with- 

 out loss of generality use it in place of h. 



Equations (74) and (75) now yield the two conditions on ni(3C, /), (R(ni), 

 and3C(:*:, y^ z, t): 



^^'^ - "^ ax ^ . (85) 



11 + T 



