CHEMICAL INTERACTIONS AMONG DEFECTS IN Ge AND Si 547 



since a is maintained constant. Furthermore (2.7) can be written as 



np = /vi = ni (3.2) 



where Wj is obviously the concentration of holes or electrons under the 

 condition that the two are equal. It is called the intrinsic concentration 

 of holes or electrons. The values of rii in germanium and silicon have been 

 determined by Morin.^^' ^® Fig. 2 gives plots of the logarithms of n,- in 

 germanium and silicon versus the reciprocals of temperature. These re- 

 sults are necessary for subsequent calculations. 



Since A''^ and A" are assumed equal, we may dispense with (2.6). 

 The one remaining equation is then (2.8) which we adopt unchanged. 

 These three relations, (3.1), (3.2), and (2.8) are sufficient to determine 

 D^ or Nd as a function of A" or Na • The only undetermined parameter 

 in the set is K* and this can be evaluated by measuring the solubility, 

 D"^, in the absence of acceptor, i.e., under the condition that A~ is zero. 

 The symbol Do^ is used to designate this value of D'^. In Reference 6 it 

 is shown that 



Z)/ = K*/(K* + n^y 

 or 



K* = (Doy/2 + {{Doy/4: + ni'iDoy}'" (3.3) 



Eliminating K* by the use of this relation it is further shown in Ref- 

 erence 6 that 



A- ' 



1 + VI + (2n,/i)o+)^ ,^^, 



V/2 (^-4) 



D+ = 



+ 



_1 + Vl + {2ni/Do+y_ 



+\2\ 



+ (Do") 



which is the required relation between donor solubility and acceptor 

 concentration. 



Examination of (3.4) reveals several simple features, the more import- 

 ant of which we list below: 



(1) When A~ (the acceptor doping) is sufficiently large so that 

 {Do^Y in the second term can be ignored relative to the term in A~, 

 (3.4) reduces to that of a straight line with slope 



Knowledge of this slope is equivalent to knowledge of Do . 



(2) Wlicre the straight lino portion of the D^ \'ersus A~ curve is in- 



