566 BELL SYSTEM TECHNICAL JOURNAL 



suitable function of the concentration of the added carrier, whose form is 

 specified for two recombination laws: recombination according to a mass- 

 action law, and recombination characterized by constant mean lifetime. 

 It is shown that essentially the same reduced equations apply to the case 

 for which recombination is neglected. 



Second-order differential equations in the hole concentration for the 

 «-type semiconductor with the thermal-equilibrium value of the hole 

 concentration assumed negligible compared to the electron concentration, 

 and for the intrinsic semiconductor, are then written for the steady state 

 of constant current in one dimension. These are converted into first- 

 order equations which have, as dependent variable a reduced concentra- 

 tion gradient G, and as independent variable a reduced concentration of 

 added holes, AP. Boundary conditions are expressed as relationships 

 between these variables. Properties of the general solutions and of the 

 boundary conditions are accordingly examined in the (AP, G)-plane. It is 

 found that there are two intersecting solutions through the (AP, G)- 

 origin, which is a saddle-point of the differential equation, and that these 

 are the solutions for field directed respectively towards and away from 

 sources in semi-infinite regions which have sources only to one side. They 

 are called field-opposing and field-aiding solutions, and possess two degrees 

 of freedom. Solutions which do not intersect at the origin are asymptotic 

 to these, possess three degrees of freedom, and are called solutions of the 

 composite type. This is the general type, and applies to a finite region in 

 distance at both ends of which boundary conditions are specified. The 

 region may, for example, be one between a source and either another 

 source, a sink, a non-rectifying electrode, or a surface upon which re- 

 combination takes place. While the analysis of composite cases is straight- 

 forward, the present treatment is confined to the simpler cases of field 

 opposing and field aiding, the latter being the one most generally appli- 

 cable to experiments in hole injection. Also, where the differential equations 

 involved are linear, solutions for composite cases can be written as linear 

 combinations of field-aiding and field-opposing solutions. 



From the properties of the curves in the (AP, G)-plane is determined the 

 qualitative behavior of the hole concentration at a hole source at the 

 end of a semi-infinite filament as the total current is indefinitely in- 

 creased. 



2 . 2 Fundamental equations for the flow of electrons and holes 



The equations for the flow in three dimensions of electrons and holes 

 in a homogeneous semiconductor contain, as principal dependent vari- 

 ables, the hole and electron concentrations, p and //, the flow densities 

 J,, and J„ , and ihc- electrostatic field, E, or potential, V. With no tra])i)ing, 



