1026 THE BELL SYSTEM TECHNICAL JOURNAL, OCTOBER 1951 



On the other hand, if the band is degenerate and has energy surfaces 

 consisting, for example, of three sheets, then an optical displacement may 

 split the degeneracy and the shift in energy for Pi = can be linear in the 

 displacement. (For example consider one wave function with angular de- 

 pendence of the form (cos 6 -\- sin cos (p + sin 6 sin (p) i.e. a p-type wave 

 function with its axis along [111]. Its energy should certainly change for 

 relative displacements of the two sublattices along the [111] direction since 

 positive and negative displacements are unsymmetrical in their distortion 

 in respect to this Hne.) 



Chiefly from evidence on magneto-resistance, the writer is convinced 

 that the electron energy band is complex in form. Thus it would be expected 

 that the optical modes would have matrix elements for low values of Pi. 

 We shall assume that this is the case but shall not endeavor to deal with a 

 non-spherical band. This an inherently inconsistent approach, but should 

 give at least a semiquantitative agreement with an exact theory. 



In order to illustrate the effect of the optical modes, we shall assume for 

 the moment that for high fields they are the dominant mechanism of scat- 

 tering and that the scattering is isotropic. If the mean free time between 

 colhsions is t, then the power input is 



(J8/J0fieid = q'rE^/m. (A7.4) 



If the temperature is so low that the optical modes are only slightly ex- 

 cited, the transitions will in general absorb an optical phonon so that 



{de,/dt)op = hv/r. (A7.5) 



The steady state condition leads to the surprising but simple result that 



va = qrE/m = {hp/mY'' (A7.6) 



so that the drift velocity is independent of E. 



If we insert k 520° K for hv and the free electron mass for m, Vd becomes the 

 velocity of an electron with k 260°K of energy giving 



Vd = 0.88 X 10^ cm/sec. (A7.7) 



The limiting value on Fig. 2 for 298°i^ corresponds to extrapolating Ohm's 

 law to about 1500 volts/cm or a drift velocity of 1500 X 3600 = 0.54 X 10^. 

 The limiting value for 77°K is about twice as high. These values are seen 

 to be in reasonable agreement with the predicted value. 



It should be pointed out, however, that the answer obtained for Vd de- 

 pends implicitly on the assumption that a relaxation time may be used in 

 the simple way employed above. To illustrate that the result is not com- 

 pletely general we refer the reader to equation (4.4) which was obtained 



