HOT ELECTRONS IN GERMANIUM AND OHM'S LAW 1013 



The number of end states in dB d&2 space is thus^* 



(V/h^) lirmPi sin 6 dd dZi = p dd J82 (A2.2) 



(The density p introduced above is used below in calculating the transition 

 probability; since spin is conserved in the transitions of interest, the den- 

 sity of possible end states in phase space is 1/h^ instead of 2//^^) 



The transitions will occur between states of the entire system, electron 

 plus phonons, which conserve energy. The transition of the electron from 

 K to P2 requires a compensating change in the phonon field.^^ The con- 

 servation laws allow two possibilities: (I) phonon emission; the longitudinal 

 acoustical mode with 



P, ^ hk = -CP2 - K) (A2.3) 



undergoes a change 



«a -> ?Ja + 1 (A2.4) 



with a change in energy for the electron of 



82 - Si = -h coa = -h c/\ = -cPa (A2.5) 



where c is the velocity of the longitudinal phonons that are chiefly respon- 

 sible for the scattering. These relationships lead to conservation of the 

 sum of P for the electron plus Yl f^a Pa for the phonons. The other pos- 

 sibihty is (II) phonon absorption, for this case 



fp^kkp^ (K - K) (A2.6) 



np -^np- 1 (A2.7) 



£2 - Si = +cPp (A2.8) 



again with conservation of the sum of P vectors. 



If we denote by S the energy of the electron after collision plus the change 

 in phonon energy, then the requirement of equaUty for energy before and 

 after coUision gives 



S = S2 + dnycPy = Si (A2.9) 



where 8ny = +1 is the phonon emission or a surface and 8ny = — 1 is the 

 phonon absorption or jS surface of Fig. 4. 



^5 The notation in this appendix follows closely that of W. Shockley, "Electrons and 

 Holes in Semiconductors," D. van Nostrand (1950) to which we shall refer as E and H 

 in S. See page 253 for a similar treatment of p. 



1^ This condition is anaio'^ous to one for the conservation of momentum but has a 

 different interpretation. See for example E and H in S, p. 519 equation (15). 



