586 BELL SYSTEM TECHNICAL JOURNAL 



with the integral to be evaluated for the particular case it is desired to 

 consider. 



3.2 The extrinsic semiconductor: n-type germanium 



The evaluation of steady-state solutions for the extrinsic semiconductor 

 involves, as a first step, the determination of G as a function of P from 

 the differential equation (27), which is accomplished by numerical inte- 

 gration and by the use of series expansions. These variables are subse- 

 quently found in terms of X in the manner described in Section 2.4. The 

 series expansions, which are Maclaurin's series in P, and series in powers 

 of the current parameter, C, with coefficients functions of P, are given 

 explicitly for the «-type semiconductor in the Appendix; they readily 

 furnish the corresponding series for the /^-type semiconductor by means 

 of the transformation discussed at the end of Section 2.31. The Mac- 

 laurin's series in P are useful for starting the solutions at the {P, G)- 

 origin. As P increases, these series converge increasingly slowly, and it 

 becomes necessary to extend the solutions by other means. For the larger 

 values of C, however, the numerical integration for the important case of 

 field aiding becomes increasingly difficult, and it is advantageous to use 

 the appropriate series in the current parameter, which converges the more 

 rai)idly the larger is C. The first term alone in this series for field aiding 

 gives in closed form the solution for the case in which diffusion is neg- 

 lected; and the existence of the series itself was, in fact, originally sug- 

 gested by the form of the solution for this case^^. Series of this type are 

 given also for field opposing, and it seems probable that such series are 

 , obtainable for composite cases as well, though this has not been investi- 

 gated. 



Solutions were evaluated numerically for ;/-type germanium, by the 

 means described, using the value 1.5 for the mobility ratio'-^, b. For the 

 case of mass-action recombination, solutions for values of the current 

 parameter, C, up to 50, specifying | G | in terms of P, are given in Fig. 2, 

 both for field opposing and field aiding. These solutions in the (F, G)- 

 plane are given to permit the fitting of boundary conditions at a hole 

 source, according to a method described in Section 4. Solutions specifying 

 P in terms of A' for field aiding are given in Fig. 3, with the A'-origin 

 chosen more or less arbitrarily at P = 100. The corresponding solutions 

 for the reduced hole flow density, Cp , and the reduced field, F, are given 



2" The solution for this case was communicated by Conyers Herring and is given in his 

 paper of reference 12. 



" The hole mol)ility and the value 1.5 for the mobility ratio were determined by G. L. 

 Pearson from the temi)erature dei)endence of the conductivity and Hall coelTicicnt in 

 p-^yV^ germanium. J. R. Haynes has recently ()l)tained, from drift-velocity measurements, 

 the same hole mobility, but the larger value 2.1 for the ratio of electron mobility in «-type 

 germanium to hole mobility in /"-type: Pai)er L2 of the Chicago Meeting of the American 

 Physical Society, November 26, 1949. 



