HOT ELECTRONS IN GERMANIUM AND OHM's LAW 1025 



A .7 The Effect of the Optical Modes 

 A .7 a. Introduction 



The treatment presented above is based entirely upon interaction with 

 the longitudinal acoustical modes of the crystal. Since the diamond struc- 

 ture has two atoms per unit cell, it is also possible to have "optical modes" 

 in which the two atoms vibrate in opposite directions. Such modes may 

 have long wave lengths; for example, do the same thing in each unit cell, 

 and thus correspond to small values of P^. Hence they may interact with 

 the electron waves with P-y — P\. Their frequencies correspond roughly to 

 a wave length of about (|) the lattice constant in the [100] direction and 

 hence to a frequency of about 



(4.92 X 105)((i)5.6 X 10-V (2/7r) = 1.12 X 10^^ ^^^~i (a7.i) 



The last factor of (2/7r) is a crude allowance for dispersion, which leads to 

 a decreasing phase velocity at short wave lengths. This corresponds to an 

 energy 



hv = k 520°K. (A7.2) 



These optical phonons thus contain much more energy than the acousti- 

 cal phonons; the latter having at room temperature an energy of about 



cPi = 2{c/vi) {viPi/2) = (l/10)yfe 300° = k 30°K (A7.3) 



[or about k 100°i^ if we use the higher effective value of c discussed in 

 Section 4]. Furthermore, the optical phonons will be only shghtly excited; 

 thus collisions with them will in general involve phonon emission so that a 

 coUision will on the average result in an energy loss of nearly 500^. This is 

 very large compared, for example, to the average loss of about 8 mc^ — k 

 12° per collision for electrons with energies of 4kT. 



A difficulty with the optical modes is that for the approximation of 

 spherical energy bands, which we have used in earlier parts of this paper, 

 they should have matrix elements which vanish when Pi = 0. This con- 

 clusion is reached by considering the deformation potentials corresponding 

 to an optical displacement: this may be thought of as moving one of the 

 face-centered cubic sublattices of the diamond structure in respect to the 

 other. Since the initial position is one of tetrahedral symmetry, there can 

 be no first order change in the energy &c at the bottom of the conduction 

 band. There may be a distortion of the energy spheres for higher energies, 

 however, and this can lead to matrix elements proportional to Pi and transi- 



4 25 



tion probabilities proportional to Pi . 



25 This view is in disagreement with the position stated by F. Seitz in his two papers 

 on mobility. Pliys. Rev. 73, 550 (1948) and 76, 1376 (1949). See for example the text 

 between equations (14) and (15) of the latter paper. 



