FLOW OF HOLES AND ELECTRONS IN SEMICONDUCTORS 1203 



From these we verify the solution: 



(9 = h) V = h2 (133) 



for any harmonic hi and h2 satisfying (124). 

 The solutions given by (127) and (133) have the property 



grad (P-gradl) = 



and so may be considered, in a sense, complementary to the solutions in 

 Sections L and M for which 



grad (P X grad V = 0. - • 



P. Superposition of a Harmonic 3C Field, N. 5^ 



Inspection of the equation system [(33), (130)] reveals the following 

 superposition theorem for obtaining new solutions from some known solu- 

 tions for the case of no recombination or time variation: 



Theorem 12: If [ii, 5C] is a known solution and if h is any harmonic func- 

 tion such that grad il-grad h = 0, then [ix, 5q.-\- h]is also a solution. 



Or, in terms of (P and V : 



[Theorem 12': If [^, V] is a known solution and if h is any harmonic function 

 such that grad ^-grad h = 0, then [5, V -{- h]is also a solution. 



In the latter form it is evident from Section O that the theorem holds also 

 for iV = 0, but does not extend the results of Section O. 



Q. A Partial Differential Equation in Terms of 3C Alone, N 9^ 



For N = 0, (21) provides a differential equation involving only one de- 

 pendent variable — (P. We shall now derive an analogous — ^but vastly more 



complicated — differential equation for the case N 9^ 0, -— = 0. 



ot 



For this case (30) and (32) become 



div grad OC = - ^ (R(ai) 

 and 



div grad OC + - 01 grad (Ol -f 5C) = 0, 

 or in terms of a familiar vector symbolism 



