Hot Electrons in Germanium and Ohm's Law 



By W. SHOCKLEY 



The data of E. J. Ryder on the mobility of electrons in electric fields up to 

 40,000 volts per cm are analyzed. The mobility decreases many fold due to the 

 influence of scattering by optical modes and due to increases of electron energy. 

 It is estimated that electron "temperatures" as high as 4000°K have been pro- 

 duced in specimens having temperatures of atomic vibration of 300° K. The 

 critical drift velocity above which there are deviations from Ohm's law is about 

 2.6 X 10^ cm/sec. This is three times higher than the elementary theory 

 and an explanation in terms of complex energy surfaces is proposed. 



Table of Contents 



1. Introduction: Fundamental Deviations from Ohm's Law 



2. E. J. Ryder's Experimental Results 



3. Theory of Deviations From Ohm's Law 



a. Electrons in »-Type Germanium 



b. The Phonons 



c. The Selection Rules 



d. Energy Exchange and the Equivalent Sphere Problem 



e. Acoustical Phonons and Electric Fields 



4. Comparison Between Theory and Experiment 



a. Discrepancy in Critical Field 



b. The Effect of the Optical Modes 



c. Electron "Temperatures" 



5. An Explanation of the Low Field Discrepancy 



Appendices 



AL Introduction and Notation 



A2. The Probability of Transition into Energy Range Sea 



A3. The Allowed Ranges for Py 



A4. The Matrix Element and the Mean Free Path 



A5, Approximate Equivalence to Elastic Sphere Model 



A6. Approximate Treatments of MobiUty in High Fields 



A7. The Effect of the Optical Modes 



1. Introduction: Fundamental Deviations from Ohm's Law 



^TT^HE starting point of many branches of physics is a linear relation. 

 -■- Among the most prominent of these are Hooke's law, which relates 

 stress and strain for solid bodies, Newton's second law of motion F = 

 ma and Ohm's law. In all of these cases, the linear relation is only an ap- 

 proximation that may be regarded as the first term in a Taylor's expansion 

 of the functional relationship between the two variables. Important physi- 

 cal principles are brought to attention when the nonlinear range is reached. 



Of the three laws mentioned, Newton's is, of course, the one in which the 

 failure of linearity is the most significant representing as it does the en- 

 trance of relativistic effects into the laws of motion. 



The failure of Hooke's law may be of either a primary or secondary form. 



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