chemical interactions among defects in ge and si 623 



Appendix D 



minimization of the diffusion potential 



In Section V the statement was made that equation (11.2) was a valid 

 approximation everywhere within a p type region, provided that No 

 did not fluctuate through ranges of order A^^ in shorter distances than 



= 4/^ (Dl) 



This statement will now be proved. 

 The electrostatic potential is determined by the space charge equation 



31 



dx^ 



where we assume that the material is everywhere p-type so that the elec- 

 tron density, n, does not enter the right side of (D2). Furthermore, the 

 mobility of holes is so much greater than that of donor ions that the for- 

 mer may be considered to always be at equilibrium with respect to the 

 distribution of the latter. Boltzmann's law^^ may then be applied to p. 

 The result is 



p = Na exp [-qV/kT] (D3) 



where the potential is taken to be zero when p = A^^ . 



Choose an arbitrary point, Xo , where the potential is Vo and investi- 

 gate (D2) in its neighborhood. We wish to determine the conditions under 

 which the right side of (D2) may be approximated by zero, i.e., the "no- 

 space-charge condition," in this neighborhood. The limits of the neigh- 

 borhood will be defined such that 



\V - Vol = \n\ ^ kT/2q (D4) 



so that, in it, the exponential in (D3) can be linearized 



p = Na exp [- gVo/kT] (l - ||) (D5) 



jThen (D2) becomes 



i~ = ^ Ina [1 - exp (- qVo/kT)] - Noix) 





+ 



w ^^p ^~ ^^°/^^^ 



u\ 



(D6) 



