FLOW OF ELECTRONS AND HOLES IN GERMANIUM 565 



siderations and of certain interesting transient effects. Steady-state alter- 

 nating-current theory for relatively small total hole concentrations in the 

 w-type semiconductor has been used to describe the action of the filamen- 

 tary transistor''^ for which diffusion may in general be neglected.^ 



The steady-state solutions in one dimension apply to single-crystal semi- 

 conductor filaments, and for critical comparisons between theory and 

 experiment, the ideal one-dimensional geometry should be simulated as 

 closely as possible. Experimental estimates of hole concentrations and 

 flows are frequently obtained from measurements of potentials and con- 

 ductances of point contacts along a filament'. These estimates require a 

 knowledge of the dependence of the current-voltage characteristics of 

 point contacts on hole concentration. Theory for this dependence has been 

 presented by J. Bardeen", and the determination of hole concentrations by 

 means of the solutions here given should provide an essential adjunct to 

 this point contact theory for its comparison with experiment. 



2. General formulation 

 2 . 1 Outline 



The formulation of the general problem is initiated by writing the 

 fundamental equations for the time-dependent flow of holes and elec- 

 trons in a source-free region of a homogeneous semiconductor under the 

 assumption that there is no trapping. Conditions for their validity are 

 discussed. Neglecting changes in the concentrations of ionized donors and 

 acceptors, the fundamental equations are expressed in reduced or dimen- 

 sionless form by suitable transformations of the dependent and independ- 

 ent variables. They are simplified so that the general problem is formu- 

 lated by means of second-order partial differential equations in two de- 

 pendent variables, one for concentration and the other for electrostatic 

 potential; corresponding equations are derived for the intrinsic semi- 

 conductor. Various properties of the equations are adduced. For the flow 

 in one dimension, a differential equation in the hole concentration is 

 given for the n-type semiconductor, accompanied by expressions for the 

 electrostatic field and hole flow density, as well as by some boundary- 

 condition relationships involving specification of the latter. The equations 

 for this case are found to depend on three parameters: the ratio of elec- 

 tron to hole mobility; a reduced concentration of holes at thermal equilib- 

 rium ; and a parameter which fixes the total current density. 



The recombination of holes and electrons is specified by means of a 



1 loc. cit. 

 " loc. cit. 



"W. Shockley, G. L. Pearson, M. Sparks, and W. H. Brattain, Phvs. Rev. 76 (3), 

 459 (1949). 



