HOT ELECTRONS IN GERMANIUM AND OHM S LAW 



1001 



necessarily be correct. In a cubic semiconductor, the electron waves can 

 be "refracted" as are the longitudinal and transverse acoustical waves. 

 The deviations from Ohm's law of Fig. 2 furnish evidence that the simpli- 

 fied assumption of equation (3.6) must in fact be replaced by the more 

 general possibility. We shall return briefly to this point in Section 5. 



In addition to the conservation of energy and momentum, there are two 

 other approximate selection rules which, while not exact, are so nearly 

 fulfilled that no appreciable error is introduced by using them: 



PHONON 



absgrptionN 



PHONON N 



EMISSION' 



CONSTANT 

 /•'ENERGY 



AVERAGE ENERGY 

 'AFTER SCATTERING 



ENERGY AFTER 

 ^SCATTERING 



-6 -4 -2 2 4* 6 SmC 



Px 

 (a) SCATTERING BY PHONONS 



6 smc 



(b) SCATTERING BY ELASTIC 

 COLLISION OF MASSES 



Fig. 4— Comparison of the scattering by acoustical phonons with the scattering of a 

 small mass in elastic collision with a larger mass. 



Only bUy = zbl is allowed. 



For the acoustical modes, only the longitudinal modes interact with 

 electrons. (This restriction does not apply to optical phonons.) 



Figure 4 shows the allowed transitions for an electron with initial mo- 

 mentum Pi in the x-direction. If the energy of the phonons were zero, the 

 allowed transitions would be to points on the sphere (or circle in Fig. 4) 

 with Pi = P\ . Since the energy of a phonon is 



hvy = hc/\ = cPy = c\P2- Pi 



(3.7) 



however, the end points lie on the surfaces shown. 



These surfaces do not differ much from the sphere, as may be seen by 

 considering the final energy for an electron that reverses its motion by 



