568 BELL SYSTEM TECHNICAL JOURNAL 



well-known relationship of A. Einstein^^. In them, e denotes the magnitude 

 of the electronic charge; T is temperature in degrees absolute; and k is 

 Boltzmann's constant. With transport velocity defined as flow density- 

 divided by concentration, the product of the mobility and the quantity in 

 square brackets in the expression for Jp or J„ on the extreme right gives 

 the corresponding velocity potential, which is thus proportional to the 

 sum of an electrostatic potential and a diffusion potential. 



The next to last equation is Poisson's equation, which relates the di- 

 vergence of the field to the net electrostatic charge. Here e is the dielectric 

 constant; pa and «o are the concentrations of holes and electrons at 

 thermal equilibrium, in the normal semiconductor. The concentrations of 

 ionized donor and acceptor impurities at thermal equilibrium are repre- 

 sented by Dt and Aq , while Z)+ and A~ are dependent variables which 

 denote the respective concentrations in general of ionized donors and 

 acceptors in the semiconductor with added carriers. As shown in the 

 Appendix, variations in D^ and A~ may be neglected if the impurity 

 centers are substantially all ionized in the normal semiconductor, despite 

 the effect large concentrations of added carriers may have on the equilib- 

 ria'^. 



The expression of the electrostatic field as the gradient of a potential 

 according to the last equation is consistent with the circumstance that 

 the effects of magnetic fields, with none applied, are in general quite 

 negligible. 



Subtracting the first continuity equation from the second, it is found 

 that 



(2) diva, - ]n) = -lip - "), 



at 



since, with no trapping, p 't„ equals ;//r„ . Neglecting changes in the con- 

 centrations of ionized donors and acceptors, this equation and Poisson's 

 equation give 



W J„_J„ = J--^«-?; I„ + I„ = I-^^, 



Aire ai Air dt 



where J and I are solcnoidal vector point functions, in general time- 

 dependent. The latter is the total current density, and the term which 

 follows it in (3) gives the displacement current density. 



"-A. Kinstcin, Annulai dcr Phvsil; 17, 549-560 (1905); Muller-Pouillct, Lchrbach der , 

 Physilc, Braunschweig, 1933, IV (3), 316-319. 



'" II has l)t'i-n found I'roni measurements of the temperature clc])cn(lence of the con- ' 

 (luclivily and Hall coenUienl lliat the energN' of thermal ionization of the donors in H- 

 typc germanium of relalively iiigh jjurity is only about 10 -cI', whence most of tiie donors 

 are ionize<l at room temperature: G. L. Pearson and W. Shocklev, Pltvs. Rev. 71 (2), 142 

 (1947j. 



