FLOW OF ELECTRONS AND HOLES IN GERMANIUM 583 



In the case of the »-type semiconductor, / is not restricted in this way. 

 Consider, for this case, hole injection into the end of a semi-infinite 

 filament, to which the field-aiding solutions apply. As the total current is 

 increased indefinitely, the tangent to the solution in the (P, G)-plane at 

 the origin approaches the P-axis, as does the solution itself, and it is 

 evident from the boundary-condition curves of Fig. 1 that if f is less than 

 l/\b -\- I) the hole concentration P^ at the source approaches as a limit 

 the indicated abscissa of intersection of the appropriate boundary- 

 condition curve with the P-axis, or the value for which the quantity in 

 brackets vanishes. It is similarly evident that P" increases indefinitely 

 with total current in either semiconductor if / is greater than or equal 

 to l/{b + 1). This is a result otherwise to be expected from the qualitative 

 consideration that an extrinsic semiconductor becomes increasingly 

 intrinsic in its behavior as the concentration of injected carriers is in- 

 creased. 



Figure 1 serves also to facilitate a count of the number of degrees of 

 freedom which the steady-state solutions possess: Corresponding to values 

 of the concentration and concentration gradient at a point in a semi- 

 conductor filament in which added carriers flow, there is a point (P, G) 

 in the half-plane, P > 0, of the figure. If the total current density is speci- 

 fied in addition, the value of C and the solution through the point (P, G) 

 are determined. This solution applies in general to a composite case, 

 which therefore possesses three degrees of freedom. That is to say, at a 

 point in a filament, any given magnitudes of both concentration and con- 

 centration gradient can be realized for a preassigned total current density 

 by a suitable disposition of sources to the right and left. The cases of 

 field opposing or field aiding, however, possess only two degrees of freedom, 

 since the given concentration and gradient determine the total current 

 density and the solution, which must pass through the origin; and which 

 of the two cases applies depends on whether the point (P, G) lies to the 

 left or to the right of the curves, shown in the figure, for the zero-current 

 solution. Thus, in a filament with a single source of holes, for example, the 

 concentration, concentration gradient, total current density, and any 

 functions of these, such as hole flow density and electrostatic field, are 

 all quantities the specification of any two of which at a point completely 

 determines the solution for a source-free X-region which includes the 

 point. 



3. Solutions for the Steady State 



For a given value of the current parameter C, solutions for the steady 

 state of constant current in a single cartesian distance coordinate, specify- 

 ing G in terms of the relative hole concentration P, and P, the reduced hole 



