582 BELL SYSTEM TECHNICAL JOURNAL 



curves as well as boundary-condition curves for a source, for a given 

 positive value of C. Those for the intrinsic semiconductor differ only in 

 that the solution curves in the {P — \, G)-plane do not depend on C, all 

 being given by the ones for zero total current density, and the corre- 

 sponding boundary-condition curves are straight lines. 



Once a solution, G(P), for field opposing, field aiding, or a composite 

 case, specifying G as a function of P has been obtained, the dependence of 

 P on X is determined by evaluating 



in accordance with the definition of G, equation (26). For the general com- 

 posite case, G(P) is that one of the family of solutions for the given C such 

 that the integral between values of P determined by the intersections with 

 the boundary-condition relationships provides the correct interval in X. 

 If P° is determined by the condition that for X = 0, a fraction / of the 

 total current is carried by holes, then, from (19), P" is the point on the 

 solution curve which satisfies either 



.„^ ^0 b+{b+ 1)P° 



for the w-type semiconductor, or 



ib + lY 



f- 



b -\- (b + 1)P\ 



C 



m G" = - 



2b 



f- 



1 



6+ 1 



C 



for the intrinsic semiconductor, G being the corresponding value of G. 



From the manner of derivation of the boundary conditions (33) and 

 (34), it is evident that they are perfectly general, holding in particular 

 for the cases of field opposing and field aiding, and whatever be the sign 

 of C. The concentration gradient G may be seen to have the correct sign 

 for these cases if it is taken into account that /, defined as Cp/C or Ip/I, 

 may assume any positive or negative value, being positive for field aiding, 

 and negative for field opposing, for which the hole flow is opposite to the 

 applied field. For / negative, the quantities in brackets in equations (33) 

 and (34) are negative. The general principle that the sign of the con- 

 centration gradient G is such as to be consistent with the flow of holes 

 from a source requires also that the quantities in brackets be positive 

 for field aiding, or whenever/ is positive. For the intrinsic semiconductor, 

 this requires that/ for field aiding never be less than l/(b + 1). This is 

 clearly a consistent requirement which holds in all generality since, for 

 zero concentration of added holes, or for the normal semiconductor, G" 

 vanishes and the ratio of hole current to total current equals l/(b -f 1). 



