596 BELL SYSTEM TECHNICAL JOURNAL 



The solution of the problem is determined by/* and the three parameters 

 which specify the total currents: With these four quantities known, then, 

 from equations (56) or (57) in conjunction with (58) and the known 

 general solutions in the (AP, G)-plane which apply to the left and to the 

 right of the origin, all of the quantities P^, G~, G^,f~ and/"*" can be found 

 and the problem completely solved. 



The technique of obtaining the solution depends on a simple funda- 

 mental result which may be expressed as follows: 



For fixed /c and Ct , consider the sum of the magnitudes of the con- 

 centration gradients at a single common source from which holes flow 

 into a number of similar filaments in parallel, for any consistent distribu- 

 tion among the filaments of total currents, some of which may be pro- 

 duced by opposing fields. This sum is equal to the magnitude of the con- 

 centration gradient at the source if the entire flow, under the appropriate 

 aiding field, were confined to a single filament. 



The total magnitude of the concentration gradient, in this sense, is an 

 invariant for fixed fe and C^ . Specifically, for the ;/-type semiconductor, 

 it follows from equations (56) and (58) that 



(59) (-_cr=-* + (^+')^° 



1 -\- 2P° 

 Similarly, for the intrinsic semiconductor, 



(60) G--(r=- ^^ 



u- 



h+{h+ i)p 



]- 



/.- ' 



^+1. 



c.. 



The left-hand sides of these equations are the negative of the sum of the 

 magnitudes of the reduced concentration gradients, since G~ is always 

 positive and G+ always negative, and their right-hand sides are similar 

 in form to those of equations (56) and (57), with the quantities /« and Ci , 

 .characteristic of the source, replacing/" and C" , or/+ and C+ . 



The particular utility of these equations arises from their independence 

 of the unknowns /" and /+ . By means of equation (59) for the w-type 

 semiconductor the evaluation of the five unknown quantities can now be 

 effected as follows: With the current parameters known, the solutions in 

 the (P, G)-plane to the left and right of the X-origin are determined; 

 either both solutions are for field aiding, or else one is for field aiding and 

 the other for field opposing. From them, the sum of the magnitudes of 

 the reduced concentration gradients can be found as a function of P. 

 It is also given, for the origin, as a function of the unknown P" , by 

 equation (59). The values of the sum for the origin and of P" are ac- 

 cordingly found as those which satisfy both relationships. The value of 

 P" thus found determines both G~ and G+ from the respective solutions 



