562 BELL SYSTEM TECHNICAL JOURNAL 



which retains, wherever convenient, such generality as is instructive per se 

 or of manifest utiHty, is speciaHzed so as to apply to the steady state of 

 constant current in a single cartesian distance coordinate. For the in- 

 trinsic semiconductor, general analytical solutions are obtainable in 

 closed form, and such solutions are given, as well as general solutions 

 obtained numerically for /z-type germanium in which the hole concentra- 

 tion at thermal equilibrium may be neglected compared to the electron 

 concentration. Solutions for these cases are given explicitly for each of 

 two recombination laws: recombination according to a mass-action law, 

 and recombination such that the mean lifetime of the added carriers is 

 constant. Methods are described for the fitting of boundary conditions, 

 and the following relatively simple boundary-value problems are con- 

 sidered: a source at the end of a semi-infinite semi-conductor filament; 

 and a single source in a doubly-infinite filament. 



To indicate the presumed scope and application of the results obtained, 

 it may suffice to outline briefly the principal assumptions on which they are 

 based and the approximations employed: The assumption is made at the 

 outset that trapping effects may be neglected, which provides the im- 

 portant simplification that the recombination rates of holes and electrons 

 are equal at all times. One justification for this is the circumstance that 

 the fairly high hole mobilities found by G. L. Pearson from Hall-effect 

 and conductivity measurements' are no larger than those found by J. R. 

 Haynes from transit times under pulse conditions^ With hole trapping, 

 holes injected in a pulse would initially fill traps; and if there were subse- 

 quent relatively slow release of the holes from the traps, an apparent 

 reduction of mobility would be manifest. It is further assumed that sub- 

 stantially all donor and acceptor impurities are ionized. With the assump- 

 tion that the semi-conductor is homogeneous in its bulk, and free from 

 grain boundaries* or rectifying barriers, the assumption of the electrical i 

 neutrality of the semiconductor, or of the neglect of space charge, is in 

 general an excellent approximation: Small departures from electrical 

 neutrality in the volume would vanish rapidly, with time constant equal 

 to that for the dielectric relaxation of charge, which for germanium 

 equals 1.5 -10"^- sec per ohm cm of resistivity^ and is in general small 

 compared with the mean lifetime of added carriers. A uniform local de- 

 parture from electrical neutrality in germanium of only one per cent in 

 relative concentration would produce api)reciable changes in field in a 



' G. L. Pearson. PItvs. Rro. 76 (1), 179-18T (19 tJ). 



"G. L. Pearsun, P/ivs. Rro. 76 (3), 459 (1949); W. E. T.ivl )r a-i 1 H. Y. Fan, piper 

 0A5, and \. H. Odell and H. Y. Fan, paper 0A2 of the 1950 Annual Mssting of the 

 American Physical Society, February 3, 1950. 



'■".A value of 16.6 for the dielectric constant of germanium is o!)tained from optical 

 data of H. B. Briggs: Phys. Rev. 77 (2), 287 (1950). 



