FLOW OF ELECTRONS AND HOLES IN GERMANIUM 



567 



the equations may be written in a symmetrical form, so that they are 

 apphcable to either an w-type, a ^-type, or an intrinsic semiconductor, 

 as follows: 



dp 



! (1) 



— = - [p/Tp - go] - div ]p 



— = - [n/jn - go] - div J„ 

 at 



J«-ji> 



Mp 



pt, — — grad p 



= -f^pp grad 



F + — log /> 

 e 



In = Mn 



e 



■iwe 



kT 



iiE grad n 



e 



— M^wgrad 



— V-\- — log n 

 e J 



div E = — [(/> - po) - (n - m) + (/)+ - Dt) - U~ - Ao)] 



e 



E = - grad V. 



In the first two equations, which are the continuity equations for holes 

 and electrons written for a region free from external sources, go is a con- 

 stant which represents the thermal rate of generation of hole-electron 

 pairs per unit volume; for cases in which hole-electron pairs are produced 

 also by penetrating radiation, appropriate source terms in the form of 

 identical functions of the space and time coordinates can be included on 

 the right in the respective equations. The mean lifetimes of holes and 

 electrons, Tp and r„ , are in general considered to be concentration-de- 

 pendent and, since trapping is neglected, the quantities p/rp and w/r„ 

 are equal, being the rate at which holes and electrons recombine. Evalu- 

 ated for the normal semiconductor, or the semiconductor at thermal 

 equilibrium with no injected carriers, they equal go . 



The equations for Jp and J„ , which are vectors whose magnitudes equal, 

 respectively, the numbers of holes and of electrons which traverse unit 

 area in unit time, are diffusion equations of M. von Smoluchowski, 

 written for hole flow and for electron flow'^ Of the type frequently em- 

 ployed, after C. Wagner, in theories of rectification, each expresses the 

 dependence of the flow density on the electrostatic field and on the con- 

 centration gradient, the diffusion constant for holes or electrons having 

 been expressed in terms of the mobility, fjtp or Hn , in accordance with the 



» S. Chandrasekhar, Rev. Mod. Phys. 15, 1-89 (1943). 



