CHEMICAL INTERACTIONS AMONG DEFECTS IN Ge AND Si 543 



respectively. Moreover, from (2.11) 



i)+ ^ Nn[l - 2e"'^e-^'"'^] = Nu (2.13) 



where the second term in brackets is ignored for the same reason as unity 

 in the denominators of (2.9) and (2.10). Substituting (2.11) through 

 (2.13) into (2.5) yields 



D^n _ ?^-- (2.14) 



in which the right side is truly independent of composition, since F has 

 cancelled out of the expression. Similar arguments hold for (2.6) and 

 (2.7). Therefore in the classical limit the law of mass action is valid, at 

 least insofar as internal equilibria are concerned. 



We have next to examine the validity of (2.4) which is really the law 

 of mass action applied to the heterogeneous equilibrium between phases. 

 Substitution of (2.11) into (2.4) leads to the prediction 



a = ^"^^ {e"''}No = K{e"''}Nu (2.15) 



in the classical case, if (2.4) is valid. In order to confirm (2.15) it is neces- 

 sary to evaluate the chemical potentials of the donor in the external 

 phase and in the semiconductor, and equate the two. The resulting ex- 

 pression should be equivalent to (2.15). 



Since a is the activity of the donor in the external phase its chemical 

 potential in that phase is, by definition, 



M = fl'iT, p) + kT in a (2.16) 



where /i°, the chemical potential in the standard state, may depend on 

 temperature and pressure, but not on composition. To compute the chem- 

 ical potential in the semiconductor statistical methods must once more 

 be invoked. Thus, according to (2.13), donor atoms are nearly totally 

 ionized in the classical case, so that the addition of a donor atom to the 

 semiconductor amounts to addition of two separate particles, the donor 

 ion and the electron. The chemical potential of the added atom is there- 

 fore the sum of the potentials of the ion and the electron separately. 

 Since the ions are supposedly present in low concentration the latter 

 can serve as an activity, ^^ and in analogy to (2.16) we obtain for the 

 ionic chemical potential 



MD+ = hd+\T, p) -f kT (n Z)+ (2.17) 



