HOT ELECTRONS IN GERMANIUM AND OHM*S LAW 1029 



where Vi> is the velocity corresponding to hv: 



V, = {2hv/myf\ (A7.22) 



The rate of energy loss to the optical modes is simply 



hm/fov (A7.23) 



A. 7c. Approximate Steady State Treatment 



In order to test whether or not the role of optical modes can explain 

 Ryder's data, we shall use a very crude method. We shall assume that the 

 electrons all have the same energy and shall calculate their mobility on the 

 basis of the mean free time at that energy; from this we calculate the power 

 input. We shall also calculate the power loss in the same way. It is obvious 

 that this treatment is a very poor approximation to the actual situation. 

 An electron which loses energy to the optical modes will, under most cir- 

 cumstances, have only a small fraction of its energy left afterwards; thus to 

 assume a monoenergetic distribution is unrealistic. However, the treatment 

 does bring into the analytic expressions the principal mechanisms and, as 

 we shall show, appears to account for the main experimental features. 



The collision frequency or relaxation time for transitions involving the 

 optical modes is given in (A7.19) and the energy loss in (A7.23). We must 

 introduce corresponding expressions for the effect of the acoustical modes. 

 Since the single energy distribution is to be used over the entire range of 

 electric fields, we must introduce some approximations like those discussed 

 in connection with (A6.25) in order to make it converge on the correct be- 

 havior at £ = 0. The particular choice selected is a compromise between the 

 energy loss formulae for the Maxwellian and single energy distributions: 



idS>/dt) acous.phonons = (4 WcV/)[l " (^'lAr)"]. (A7.24) 



A simplified expression is also used for the mobihty: 



fx = qC/mvi = hoVt/v. (A7.25) 



The relationship between /zo and t given by this differs by 25 per cent from 

 the correct relationship (A6.7); since /zo is an adjustable parameter in the 

 comparison between theory and experiment, (A7.25) does not introduce any 

 error at low fields. Equations (A7.24) and (A7.25) cause fi to converge on /xo 

 and 8i on kT as E approaches zero. (It is probable that a slightly better fit 

 to the data would be obtained by using the procedure described with equa- 

 tion (A6.25) ; the calculations based on (A7.24) and (A7.25) were made be- 

 fore the (A6.25) procedure was worked out, however, and it was not consid- 

 ered worth while to rework them for this article.) 



