CHEMICAL INTERACTIONS AMONG DEFECTS IN Ge AND Si 593 



gions develop positive potentials and a field exists tending to expel 

 lithium. This causes the lithium to drift as well as diffuse so that Fick's 

 laW^ is no longer valid. 



The most that can be done toward the elimination of diffusion poten- 

 tials is to minimize them so that no local space charge exists. At equilib- 

 rium, this corresponds to the condition^^ 



Nd - Na = 2ni smh(qV/kT) (11.1) 



where V is the local electrostatic potential. It is always permissible to 

 assume that fast moving electrons and holes are in equilibrium relative 

 to diffusing ions. If a material which is p-type everywhere is being con- 

 sidered, (11.1) can be simplified to 



Na - Nj, = Ui exp [-qV/kT] (11.2) 



In Appendix D it is proved that (11.2) will be valid everywhere within 

 a region where N a is constant and greater than Nd , provided that No 

 does not fluctuate through ranges of the order Na in a distance less than 



(11.3) 



Under most conditions of experiment I will be of the order of 10~ cm. 

 Unfortunately many of the experiments described in this section (par- 

 ticularly those performed at 25°C.) involve diffusion layers as thin as 

 10"^ cm. As a result space charge will exist and the diffusion potential 

 will not always be minimized. Even if it is minimized so that (11.2) is 

 satisfied the residual field will still aid diffusion and lead to higher ap- 

 parent diffusivities. Therefore the effect cannot be ignored even when 

 minimization has been achieved. 



In the absence of space charge the drift component of flux density 

 due to the field is easily computed. It will be given by 



-M ^ No (11.4) 



dx 



According to (11.2) 



_dV _ kT dNp 



dx ~ q{NA - Nd) dx 

 so that (11.4) becomes 



nkT / Nd \ ONd ^ _fiokT / _ _P \ / Nd \ dNp 

 ^'a - No) dx q \ Nd) \Na - Nd) dx 



Nd \ dNp 

 - Nd) dx 



(11.5) 



(ii.rO 



