FLOW OF ELECTRONS AND HOLES IN GERMANIUM 563 



mean free path for the carriers, equal to 1.1 • 10~' cm at room temperature, 

 which would even preclude the applicability of the fundamental equations 

 employed. In qualitative terms, the conductivity of the semiconductor is 

 sufficiently large that the currents which commonly occur are produced 

 by moderate fields whose maximum gradients are relatively small. Space 

 charge may persist in the steady state, but then only in surface regions 

 whose thickness^" in germanium is generally less than about 10^* cm and 

 whose effects may be dealt with through suitable boundary conditions. 



The steady-state solutions, in their qualitative aspects, are illustrative 

 of the phenomena taken into consideration. In an extrinsic semiconductor, 

 if the concentrations of added carriers are not too large, the solutions 

 for moderate and large fields are in general approximately ohmic in their 

 local behavior. The effect of diffusion is then comparatively small, and 

 the added carriers largely drift under a field which varies with distance 

 through the increased conductivity which these recombining carriers 

 themselves produce. Diffusion effects are incident in addition to this 

 behavior, and become pronounced for large concentrations or small ap- 

 plied fields. For example, solutions which specify the concentrations of 

 added holes as functions of distance, for different total currents or applied 

 fields in a source-free region, all approach a common solution for large 

 hole concentrations, regardless of applied field; those for the hole cur- 

 rent and the electrostatic field behave similarly. This behavior results from 

 diffusion in conjunction with the increase in conductivity. Another example 

 is that of the solutions for zero total current: As the result of diffusion in 

 conjunction with recombination, a flow of added holes can occur along a 

 semi-conductor filament with no flow of current. It is, of course, accom- 

 panied by an equal electron flow, so that the hole and electron currents 

 cancel, and occurs in any open-circuited semi-conductor filament which 

 adjoins a region in which added holes flow. It can also be realized by suit- 

 able irradiation of an end of a filament, with no applied field. A closely 

 related effect is illustrated in the flow of holes injected through a point- 

 contact emitter into a semi-conductor filament along which a sweeping 

 field is applied: Some of the holes will flow against the field, an appre- 

 ciable proportion, unless the current in the filament is sufficiently large. 

 As a further example, if the mobilities of holes and electrons were equal, 

 the electrostatic field would be given by Ohm's law as the total current 



'" The (largest) distance over which the increment in electrostatic potential exceeds 

 kT/e may be expressed in units of the length Ld = {kTe/&Trn,e~)\ where lu is the thermal- 

 equilibrium concentration of electrons (or holes) in the intrinsic semiconductor; see the 

 paper of reference 3, also W. Schottky and E. Spenke, Wiss. Verijjf. Siemcns-Werken 18 

 (3), 1-67 (1939). This distance increases with resistivity, never exceeding the value 1.4 

 Ld for the intrinsic semiconductor. In high back voltage w-type germanium, it exceeds 

 about 0.5 Ld, and Ld for germanium is about 7.4-10^' cm at room temperature. 



