FLOW OF ELECTRONS AND HOLES LY GERMANIUM 569 



It may be well to point out that the validity of the diffusion equations 

 depends on two assumptions, which, while hardly restrictive in general 

 for homogeneous semiconductors, indicate the nature of the generaliza- 

 tions which might otherwise be necessary". The first assumption is that 

 there are no appreciable time changes in the dependent variables in the 

 relaxation time for the conductivity, or the time of the elementary fluctua- 

 tions. This is tantamount to the requirement that the carriers undergo 

 many colhsions in the time intervals of interest. The second assumption is 

 that the changes in the carriers' electrostatic potential energy over 

 distances equal to the mean free path are small compared with the aver- 

 age thermal energy. In accordance with this assumption, very large fields 

 in the electrically neutral semiconductor for which the carriers are not 

 substantially in thermal equilibrium with the lattice are ruled out. The 

 neglect of space charge then in general validates the two assumptions, if 

 the resistivity is not too small, since the neglect of changes in the de- 

 pendent variables which occur in the dielectric relaxation time obviates 

 their change in the relaxation time for conductivity; and the neglect in 

 the steady state of appreciable variations in electrostatic potential, and 

 thus in the other dependent variables, in the distance^" Ld , obviates their 

 variation in a mean free path. The dielectric relaxation time for ger- 

 manium, 1.5 -10"^^- sec per ohm cm of resistivity, in high back voltage 

 material exceeds the relaxation time for conductivity, which is about 

 1.0- 10^^' sec; and in semi-conductors in which the mobilities and the 

 conductivity are smaller than the comparatively large values for ger- 

 manium, the dielectric relaxation time may be appreciably larger than the 

 relaxation time for conductivity. Similarly, Ld for germanium is about 7 

 times the mean free path, and this ratio, which is essentially inversely 

 proportional to the square root of the product of mobility and intrinsic 

 conductivity, may be appreciably larger for other semiconductors. 



If, on the other hand, it should be desired to consider space-charge 

 effects in germanium, the diffusion equations may be of rather marginal 

 applicability, and the use of their appropriate generalization indicated, 

 since with Ld equal to 7 mean free paths, appreciable space-charge varia- 

 tion of potential, corresponding to a field which is not small compared 

 with the free-path thermal-energy equivalent of about 3500 volt cm~\ 

 may occur in at least one of the free paths. For example, diode theory, 

 rather than diffusion theory, provides the better approximation for the 

 characteristics of germanium point-contact rectifiers, and is particularly 

 applicable to those from low resistivity material for which the potential 

 variation is largely confined to one mean free path or less". 



'" loc. cit. 

 " loc. cit. 

 " H. C. Torrey and C. A. Whitmer, "Crystal Rectifiers," New York, 1948, Sec. 4.3. 



