CHEMICAL INTERACTIONS AMONG DEFECTS IN Ge AND Si 617 



lively, 10^^ and 10^^ cm~^ gallium ions available for pairing, the pairing 

 process did \'ery little to increase the solubility. 



If the constant 9, is exceedingly large as is probably the case for a 

 multiply charged acceptor, it is possible that ion paring will have a meas- 

 urable effect on solubility. 



Appendix B 

 concentration dependence of diffusivity in the presence of ion 



PAIRING 



In Section VIII it was mentioned that the diffusivity of a mobile donor 

 like lithium is concentration dependent when the donor participates in a 

 pairing equilibrium with an immobile acceptor. In this appendix we 

 propose to investigate the nature of the dependence. 



Consider a semiconductor, uniformly doped to the level, Na , with 

 acceptor. Let the local density of mobile donor be Noix), x being the 

 position coordinate. If P(x) is the local pair concentration, then the local 

 density of free diffusible ions is {No — P). The flux of these diffusing 

 ions then depends upon the gradient (assuming Fick's law^^) of {N d — 

 P). Thus, if Do is the diffusivity of free donor, i.e. the diffusivity in the 

 absence of pairing, then the flux density is 



/=-D.£(^[^) (Bl) 



dX 



If we apply (9.4) to the present case we can write 



O = ^ ^ - {No - P) + .V. 



{Na - P){Nu - P) [{Na - No) + (A^;, - P)]{No -P) ^ ^^ 



ifrom which it is possible to solve for {No — P). Thus 



Substitution of (B3) into (Bl) yields 

 Do 



^=-2 



\[Nn-NA + 



1 + 



0/ 



/I 





dNi 

 dx 



(B4) 



llf ion pairing was not thought of, the flux density would have been writ- 

 ten in terms of the gradient of the total concentration. No ■ 



f= -D^-p^ (B5) 



dx 



