HOT ELECTRONS IN GERMANIUM AND OHM's LAW 1011 



It is possible, however, to explain the discrepancy by assuming that the 

 effective mass is not single valued. This assumption corresponds to the case 

 in which the surface in the Brillouin zone belonging to a single energy is not 

 a sphere but instead a complex surface of two or three sheets. Such surfaces 

 have been found as a result of numerical calculations for certain crystals^^ 

 and it has also been shown that such surfaces are to be expected in generaP^ 

 if the energy at the bottom of the conduction band is degenerate. It appears 

 necessary to assume that such complex surfaces occur in order to explain 

 magnetoresistance effects.^^ 



In terms of Fig. 4, this theory replaces the circular energy contours by 

 deeply re-entrant curves. Transitions from peak to peak of the curves result 

 in large energy transfers to the phonons and hence more effective energy 

 losses. This effect can occur without a compensating change in the effective 

 mass involved in the mobihty and, as a result, the critical field may be 

 increased by a large factor. A preHminary analysis indicates that in order 

 to increase the critical field by a factor of 3 a value of about 3 is also 

 required for the ratio of maximum to minimum momentum for the energy 

 surface. A similar analysis of magnetoresistance leads to a factor about 50% 

 larger in order to account for the increases in transverse resistance of about 

 7-fold observed by Suhl.^* At the time of writing, therefore, the author feels 

 that both the critical field data at low fields and the magnetoresistance data 

 require a modification of the effective mass picture and that the same modi- 

 fication may well explain both sets of data. 



I am indebted to E. J. Ryder, whose experimental results provoked the 

 analysis presented in this paper, to F. Seitz and J. Bardeen for several helpful 

 discussions, to Gregory Wannier for an introduction to the analogous case 

 in gas discharge theory and to Esther Conwell for help with the manuscript. 



I shall also take this opportunity to express my appreciation to C. J. 

 Davisson. The opportunity to work in his group was a large factor in my 

 decision to come to Bell Telephone Laboratories, where I enjoyed his stimu- 

 lating companionship while assigned to his group, and later as well. 



APPENDICES 



A.l Introduction and Notation 



The problem of energy exchange between the electrons and the phonons 

 requires a somewhat more sophisticated treatment than does the problem 

 of mobihty at low fields. In order to present the theory of energy exchange, 



II W. Shocklev, Phys. Rev. 50, 754 (1936). 

 12 W. Shockley, Phys. Rev. 78, 173 (1950). 

 WW. Shocklev, Phys. Rev. 79, 191 (1950). 



i*H. Suhl, Phys. Rev. 78, 646 (1950). Suhl finds increases in resistance in transverse 

 fields as high as 7-fold. 



