594 BELL SYSTEM TECHNICAL JOURNAL 



if there is mass-action recombination present, so that a 5^ 0. The depend- 

 ence of P on A' for these approximations is readily obtained by integrating 

 the differential equations which result from writing in place of G, its 

 definition, dP/dX\ constant mean lifetime gives an exponential depend- 

 ence. An examination of (54) and (55) in conjunction with the general 

 differential equation (27) shows that, for P large, the dominant term in the 

 differential equation is independent of C. It follows that solutions for all 

 values of C approach a common solution for P large, which is given by 

 (54) or (55). The solutions run together appreciably for P sufl5ciently 

 large that P and M are substantially proportional, that is, for P large 

 compared with h/{h + 1), which is of order unity. It is to be expected 

 that the approximations (54) and (55) should apply equally well to the 

 intrinsic semiconductor, and this expectation is easily verified by evalu- 

 ating the integral in equation (35) for the intrinsic semiconductor, for P 

 large, for the two recombination cases here considered. 



4. Solutions of Simple Boundary-Value Problems for a Single 



Source 



Among the boundary-value problems whose solutions are useful in 

 the interpretation of data from experiments in hole injection are the 

 following: the semi-infinite filament for field aiding, with holes injected at 

 the end, which constitutes a relatively simple case; and the doubly- 

 infinite filament with a single plane source, with which this section will be 

 primarily concerned. 



Consider first the semi-infinite filament, and suppose that it starts at the 

 X-origin and extends over positive A^, so that the current parameter is 

 positive for field aiding. If two quantities are specified, namely the current 

 parameter and the fraction /« of the current carried by holes at the origin 

 or injection point, then the solution of the boundary-value problem is 

 completely determined. It is merely necessary to select the general field- 

 aiding solution for P or Cp in terms of A", for the particular value of the 

 current parameter, and then to determine the A'-origin, corresponding to 

 the source, which is simply the X at which the ratio /of Cp to C equals /« . 



Use in the boundary-condition equations {i2>) and (34) of the approxi- 

 mate expressions given in (54) and (55) for G in terms of P, for large P, 

 permits the complete analytical determination of the dependence ofP" 

 on total current as this current is indefinitely increased. It was shown in 

 Section 2.4 that, if /^ is less than \/{b + 1) for the w-type semiconductor, 

 P" approaches as a limit the value for which G" vanishes according to the 

 boundary-condition equation (33); in all other cases for the ;/-type semi- 

 conductor, or if/, exceeds \/{b + 1) for the intrinsic semiconductor, P° 

 increases indefinitely with C. For/, > \/{b -f- 1), it is readily seen that 



