580 BELL SYSTEM TECHNICAL JOURNAL 



since R is close to unity for P small, whence 



Similarly, for the intrinsic semiconductor, for P—\ small, 



^^^ d{p -\) p - \ "^ y 2b ' 



There are thus, in each case, two solutions through the (AP, G)-origin, 

 one with a positive derivative and the other with a negative derivative. 

 Consider now the doubly-infinite region with a source at X = 0. Then, 

 for X > 0, the negative derivatives apply, since the concentration gra- 

 dient G is negative. Similarly, for X < 0, the positive derivatives apply. 

 Now, the value of the current parameter C will be substantially the same 

 in both regions, since it has been assumed that AP is small. For C posi- 

 tive, equation (30) for the 7^-type semiconductor indicates that the 

 magnitude of dG/dP for X^ < exceeds that for X" > 0, and the situation 

 is reversed if the sign of C is changed. That is, the magnitude of the 

 concentration gradient increases more slowly with concentration for 

 field directed away from a source than for field directed towards a source, 

 which is otherwise plausible. For the intrinsic semiconductor, on the 

 other hand, equation (31) shows that corresponding magnitudes of the 

 concentration gradient are equal and entirely independent of C, a result 

 which the differential equation (28) establishes in general. 



It thus appears that a differential equation for the steady state possesses 

 two. solutions through the (AP, G)-origin, and that one of the solutions 

 corresponds to the case of field directed towards a source, the other to the 

 case of field directed away from a source. Field directed towards a source 

 is called field opposing, while field directed away from a source is called 

 field aiding, the latter being the one commonly dealt with in hole-injec- 

 tion experiments. It should be noted that the cases of field opposing or 

 field aiding can be realized in a given X-region only if it adjoins a semi- 

 infinite region free from sources and sinks. In the region between two 

 sources, neither of these cases applies. L. A. MacColl has shown, through a 

 more detailed consideration of the singularity at the (AP, G)-origin, that 

 the two solutions through this point are the only ones through it. The 

 origin is thus a saddle-point of the differential equation, and there exist 

 families of nonintersecting solutions in the (AP, G)-plane for which the 

 solutions which intersect at the origin are asymptotes. A solution for an 

 A'-region between two sources, for e.xample, is a member of such a family, 

 as is in general any solution determined by boundary conditions at the 

 ends of a finite region in X. Such a solution will be called a solution for a 

 comj)osite case; it approaches asymptotically both a field-opposing and a 



