FLOW OF ELECTRONS AND HOLES IN GERMANIUM 599 



same in magnitude, while for C large compared with unity, 

 (70) 



Cp^ ~ ^J^c, 



Cp^^fiCf. 



Thus, according to this approximation, C should exceed about 10 if no 

 more than one per cent of the holes are to flow against the field. From 

 (75) in the Appendix, a value of 10 for C corresponds to a current density 

 of about 1.2 amp cm~^ in germanium of 10 ohm cm resistivity, with r 

 equal to 10 Aisec. This current density is moderately large among those 

 which have been employed in experiments with germanium filaments. 

 Experimentally, the ideal one-dimensional geometry postulated in the 

 present treatment of the problem of the single source in an infinite fila- 

 ment cannot easily be reahzed, hole injection generally being accomplished 

 through a point contact or a side arm on one side of the actual filament. 

 If suitable averages are employed, non-uniformity in P at the injection 

 cross-section does not, however, vitiate the approximate results for AP 

 large and AP small, since their applicability depends largely on the validity 

 over the injection cross-section of the approximation assumed. 



Acknowledgment 



The author is indebted to a number of his colleagues for their stimu- 

 lating interest and encouragement; to J. Bardeen and W. Shockley for a 

 number of valuable and helpful comments, as well as to W. H. Brattain, 

 J. R. Haynes, C. Herring, L. A. MacColl, G. L. Pearson, and R. C. Prim. 

 J. Bardeen also suggested the numerical analysis for «-type germanium 

 which constituted one of the initial points of attack, and aided materially 

 in its inception. The rather difficult numerical integrations and associated 

 problems were ably handled by R. W. Hamming, Mrs. G. V. Smith and 

 J. W. Tukey. 



5. APPENDIX 



5 . 1 The concentrations of ionized donors and acceptors 



While the donor and acceptor concentrations need not, of course, be 

 considered for the intrinsic semiconductor, for the extrinsic semicon- 

 ductor the fundamental equations, as they have been written, are in 

 principle incomplete: Two additional equations in the variables Z)+ and 

 A~ are required. One of the required equations is trivial, since changes in 

 the concentration of ionized centers which are compensated by those 

 which determine the conductivity type of the extrinsic semiconductor 



