HOT ELECTRONS IN GERMANIUM AND OHM'S LAW 1021 



A. 6 Approximate Treatments of Mobility in High Fields 



A correct treatment of mobility in high electric fields E is based upon 

 finding the steady state distribution function /(P, £, T) which satisfies the 

 Boltzmann equation, i.e. a function for which the rate of change due to 

 acceleration by E just balances that due to scattering. This method leads 

 at once to rather formidable mathematics which may tend to obscure 

 somewhat the physical forces at work. We shall in this section derive re- 

 lationships between drift velocity and E on the basis of simpler models 

 and shall compare the results with the exact treatment as given in the 

 literature for the case of a gas. 



For this purpose, we shall first suppose that the field in effect raises the 

 electrons to a temperature Te which is greater than the temperature T of the 

 phonon distribution. The electrons with temperature Te will have colli- 

 sions at a rate {Te/Ty^ greater than before and their mobility will be re- 

 duced from its equilibrium value /lo to a new value /x 



M= (r/ny/^Aio. (A6.1) 



The average rate at which the electric field does work on an electron is then 



(de,/dl){ioid = Force X Speed = qn,E\ (A6.2) 



For steady state conditions this must be equal to minus the average rate 

 at which an electron gains energy from the phonons. Denoting this by 

 id&fdl) phoaons we liave 



(de>/dl)no\d + (^S/^/)phonons = 0. (A6.3) 



In order to calculate the average rate of energy loss to the phonons, we 

 consider the average energy gain of an electron of momentum Pi. As given 

 by(A5.12): 



{8S)p, = 4wc2 [1 - (mv\/^kT)]. (A6.4) 



According to our assumption, the number of electrons in the velocity range 

 V to V -{- dv is N{v) dv = A exp {—mv /2kTe)v-dv. These electrons suffer 

 collision at a rate v/^. Hence the average rate of energy gain is 



WS/(//)phonon3 = / {5S)p{v/{)N{v) dv^ Jn{v) dv 



= (8/>/x)(mcV0[l - ive/vr?] 

 where we have introduced 



Vt = {2kT/myi\ Ve = {2kTe/myi\ (A6.6) 



(A6.5) 



