4()2 



BELL SYSTEM TECHNICAL JOURNAL 



through the mathematical details of Sections 3, 4, and 5. Section 3 contains 

 the complete differential equations of the problem, including diffusion and 

 recombination, and Section 4 gives the solution when only the diffusion 

 terms are neglected. Section 5 contains some order-of-magnitude estimates 

 regarding diffusion effects. Section 6 summarizes the capabilities of the 

 theory so far developed, presents some obvious generalizations, and dis- 

 cusses an interesting shock wave phenomenon which occurs whenever the 

 injected hole current is quickly decreased. 



1. Basic Assumptions and Boundary Conditions 



Consider the «-type semiconducting specimen shown in Fig. 1, having 

 electrodes at its two ends, x = —a and x = b, respectively, and an injection 

 electrode system at x = somewhere in between. Let a current of density ja 

 per unit area enter at the left-hand end, and let a current of density je be 

 injected at x = 0. To make the problem strictly one-dimensional, it will be 



Fig. 1 — Idealized experiment on hole transport in one dimension. 



supposed that this injection takes place uniformly over the plane cross- 

 section of the specimen at x = 0, instead of taking place at isolated points 

 of the surface, as is usually the case in experiments. This idealization will 

 presumably be justified if the thickness of the specimen is small compared 

 with lengths in the .^-direction which are significant in the experiment and 

 if the injected positive holes are able to spread themselves uniformly over 

 the cross-section before appreciable recombination has taken place. 



Unless otherwise stated, it will be supposed that je consists entirely of 

 positive holes, i.e., that the number of electrons withdrawn from the speci- 

 men by the electrode at re = is negligible compared with the number of 

 holes injected. The currents ja a,ndje need not be constant in time, although 

 most of the analysis to be given below will assume them constant after the 

 time of initiation of ;"« . 



One can set up differential equations for the variation with x and time 

 of the electron density, Ue , and the hole density, iii, . These equations will in 

 the general case involve migration due to electrostatic fields, diffusion, re- 

 combination, trapping, and thermal release of electrons and holes from 

 traps. It will be assumed, however, that trapping and thermal release from 

 traps can be neglected, or, more precisely stated, that creation of mobile 



