HOT ELECTRONS IN GERMANIUM AND OHM's LAW 097 



of donors: chemical impurities such as arsenic or antimony. These donors 

 substitute themselves for germanium atoms in the crystal structure, form 

 electron-pair bonds with their four neighbors and release their fifth valence 

 electron to the conduction band. The density of donors is about lO'Vcm 

 or one per cube 10~^ cm = 1000 A on an edge. The donors are fixed positive 

 charges and do not move in electric fields. Their charges neutralize those 

 of the electrons. The electrostatic energy of interaction between electrons 

 and donors leads to a deflection of the electron's motion. For the tempera- 

 tures of Fig. 2, however, this effect is unimportant compared to the effect 

 of thermal vibrations of the atoms. 



The electrons in the conduction band move in accordance with a wave 

 equation. They may, however, be thought of as particles. The justification 

 is that, under many experimental conditions, the wave functions will actu- 

 ally be wave packets. These wave packets, it can be shown, behave much 

 as particles and can be dealt with as particles, at least provided the ]5he- 

 nomcna considered do not involve distances smaller than the size of the 

 wave packet. 



Under conditions of thermal equilibrium we may think of the 10'^ elec- 

 trons in each cubic centimeter as an electron gas with the electrons (as 

 wave packets) movhig at random with an average kinetic energy of motion 

 of (3/2)^r. 



If the atoms of the crystal were held rigidly at rest in a perfectly regular 

 crystal structure, an electron wave, and a wave packet too, would be trans- 

 mitted through it with no scattering. At 300°K the vibrations are such that 

 the wave packet moves for only 2500 A before being scattered. At low tem- 

 peratures, the mean free path is longer and at liquid hydrogen tempera- 

 tures it is so long that thermal vibrations are less important than the 

 fields of the ionized donors. As stated above, however, we may neglect this 

 scattering by ions over the temperature range of Fig. 2. 



Before proceeding with the discussion of the interactions of electrons 

 and thermal vibration we shall point out that two problems must be solved 

 before the dependence of mobility upon electric field can be explained: 



First, the mechanisms of the individual processes must be analyzed. 

 This is the basic physical problem. In order to solve it we must apply quan- 

 tum mechanics to the model representing the electron moving in the crystal 

 and determine the probabiHties of various types of transitions and some 

 appropriate averages. 



Second, the statistical consequences must be worked out. On the basis 

 of the individual processes, the statistics of the assemblage of electrons 

 must be analyzed and a steady state solution found. 



The first problem poses the more physical problems and is given the most 



