992 THE BELL SYSTEM TECHNICAL JOURNAL, OCTOBER 1951 



perimental techniques and results briefly in the next section and shall then 

 present some aspects of the quantitative theory that explains them, leaving 

 the bulk of the mathematical manipulations for the appendices. 



Before discussing Ryder's results, we may indicate why his procedure 

 succeeded whereas previous attempts, of which there have apparently been 

 a number, largely unpubhshed, have failed. Ryder's work takes advantage 

 of three factors: (1) the availabihty of electrical pulses of microsecond dura- 

 tion, (2) the high resistivity of germanium, and (3) the high mobihty of 

 electrons in germanium. Because of (3), it is possible to dehver energy to 

 electrons at relatively high rates by electric fields. In effect this "heats" 

 the electrons above the temperature of the crystal and lowers their mobility. 

 The generalized equation is then 



V, = n{T, E)E (1.2) 



where the fact that /x depends on E represents the fundamental nonlinear- 

 ity. We shall show that Ryder's techniques raise the "temperature" of the 

 electrons by a factor of about thirteen fold to above 4000°K. Since the re- 

 sistivity is high, say 10 ohm cm, the power delivered to the specimen is 

 sufficiently low that the heating in one pulse is negligible. The pulse repeti- 

 tion rate is then kept so low that accumulated heat is negligible also. 



These conditions are enormously more favorable than those met with 

 in metals. In a metal the average electron energy is several electron volts; 

 in order to double this energy, each electron would acquire an added energy 

 roughly equal to the cohesive energy per atom of the crystal. Furthermore, 

 in a metal there is about one conduction electron per atom, compared with 

 10"^ per atom in Ryder's samples. Thus the stored energy due to "hot" 

 electrons in a metal would be enough to vaporize it, whereas in germanium, 

 or a similar semiconductor, a temperature of 10,000°K for the electrons 

 would be enough to raise the crystal less than 0.01°K. From this reasoning 

 it appears that it will be extremely difficult, if not impossible, to produce 

 significant fundamental deviations from Ohm's law in metals and certainly 

 impossible to produce effects of the magnitude described below. 



It should be pointed out that the behavior of electrons in crystals in 

 fields so high that equation (1) fails have been subject to both experimental 

 and theoretical investigation in connection with dielectric breakdown.^ 

 The work does not apply to cases in which the specimens obey Ohm's law 

 at low fields, however, and the experiments do not permit accurate deter- 



2 See, for example, H. Frfthlich and F. Seitz, Phys. Rev. 79, 526 (1950) and F. Seitz» 

 Phys. Rev. 76, 1376 (1950). Much of the treatment presented in the Appendices is essen- 

 tially equivalent to that given in Seitz. In our Appendices, however, we give much more 

 emphasis to the low field case. The Seitz paper also contains a review of the literature 

 to which the reader is referred. 



