HOT ELECTRONS IN GERMANIUM AND OHM'S LAW 1009 



is used. This value is 3.2 times larger than the value given by equation 

 (3.36) using c = 5.4 X 10^ cm/sec, the value appropriate for longitudinal 

 phonons.^'^ The interpretation of this discrepancy, which we refer to as the 

 'low field" discrepancy, is discussed in Section 5. It does not, of course, 

 imply an error in the value of the sound velocity, but instead an error in the 

 theory leading to the formula for critical velocity in terms of sound velocity. 



Although an exact theory along the lines discussed in Section 5 has not 

 been developed, it appears evident that its chief effect will be to increase 

 energy interchange with the phonons by a factor of 3.2 squared or approxi- 

 mately 10. This increase can be effected in a mathematically equivalent way 

 by introducing an ejfeclive velocity for dealing with phonon energies which is 

 3.2 times larger than the true velocity of longitudinal waves. The approxi- 

 mate theory in the Appendices uses this procedure. 



We may remark in passing that only two constants were arbitrarily chosen 

 to fit the curves to the data. One of these was the effective velocity c = 1.73 

 X 10^ cm/sec. which is 3.2 times larger than the speed of longitudinal waves. 

 The other was the mobihty of electrons at room temperature. Three other 

 constants were chosen from independent estimates of the properties of the 

 crystal. One of these is Jiv for the optical modes for which a value of ^520°K 

 was used; another is the effective electron mass, for which the free electron 

 mass was used; and a third was the interaction constant for optical modes, 

 which was set equal to that for the acoustical modes. The meaning of these 

 terms is discussed in the Appendices. 



4b. The Effect of the Optical Modes 



We shall next discuss briefly the role of the optical modes before remark- 

 ing on a theory of the low field discrepancy. 



As discussed above the optical modes can act only if 8i > hvop . Theory 

 indicates, however, that when they do come into action they are much more 

 effective than the acoustical modes. On the basis of these ideas, we can see 

 how they can act to give a limiting drift velocity that does not increase 

 with increasing electric field. For purposes of this illustration we shall imagine 

 that so high an electric field is applied that an electron may be accelerated 

 from Pi = to P2 = {Irnhvox^y^i at which its energy equals //I'op, in so short 

 a time that it is not scattered by acoustical modes. As soon as it reaches 

 P2 , we assume that it is scattered by the optical modes, loses all its energy 

 and returns to zero energy. This process then repeats, the period being 



1" This is the velocity of longitudinal vaves in the fl 10] direction as reported by W. L. 

 Bond, VV. P. Mason, H. J. McSkimin, K. M. Olsen and G. K. Teal, Pliys. Rev. 73, 549 (1948>. 

 See "Electrons and Holes in Semiconductors," page 528, for the reason for using this wave. 



