564 BELL SYSTEM TECHNICAL JOURNAL 



divided by the local increased conductivity. With electrons more mobile 

 than holes, this ohmic field is modified by a contribution which is directed 

 away from a hole source and proportional to the magnitude of the con- 

 centration gradient divided by the local conductivity. This contribution 

 gives a non-vanishing electrostatic field for zero total current. 



The intrinsic semiconductor has, as the result of a conductivity which 

 is everywhere proportional to the concentration of carriers of either type, 

 the property that the flow in it is as if the added carriers were actuated 

 entirely by diffusion, with only the carriers normally present drifting 

 under a field equal to the unmodulated applied field. The extrinsic semi- 

 conductor becomes in effect intrinsic if the concentrations of carriers are 

 sufficiently increased, by whatever means, the ohmic contribution to the 

 current density of either electrons or holes then becoming proportional to 

 the total current density and, in this case, negligible compared with the 

 contribution due to diffusion. It may, for example, be expected that the 

 transport velocity of added carriers in an extrinsic semiconductor can be 

 increased by an increase in the applied field only if the consequent joule 

 heating does not unduly modify the semiconductor in the intrinsic 

 direction. 



General solutions for the steady state in one dimension are obtainable 

 analytically in closed form for a number of important special cases. Aside 

 from that for which diffusion is neglected, they include the general cases 

 for no recombination, for the intrinsic semiconductor, and for zero total 

 current, and the limiting cases of small and of large concentrations of 

 added carriers. W. Shockley has made use of small-concentration theory 

 in an analysis oi p — n junctions^. J. Bardeen and W. H. Brattain have 

 given solutions for the steady-state hole flow in three dimensions, neg- 

 lecting recombination, in the neighborhood of a point-contact emitter.^' " 

 Transient solutions are obtainable analytically for the intrinsic semi- 

 conductor for constant mean lifetime, and for the extrinsic semiconductor 

 if the concentrations of added carriers are sufficiently small that the 

 change in conductivity is negligible. For concentrations unrestricted in 

 magnitude, Conyers Herring has described a general method for graphical 

 or numerical construction of transient solutions in one dimension from a 

 first-order partial differential equation appropriate to the case for which 

 diffusion is neglected in the extrinsic semi-conductor, and has given some 

 solutions so obtained, with estimates of the effect of diffusion. Reference 

 might be made to his paper^^ also for discussion of various physical con- 



' loc. cit. 



"• loc. cit. 



" See the paper of J. Bardeen in this issue. 



"Conyers Herring, B. S. T. J. 28 (3), 401-427 (1949). 



