CHEMICAL INTERACTIONS AMONG DEFECTS IN Ge AND Si 545 



j conductor phase, and to differentiate this with respect to No , keeping 

 I temperature and pressure fixed.* The result is 



' ^^ = juz,+° + kT in Nd 



(2 23) 

 j + F - kT In [1 + 2 Qx\^[- {Ed- F)/kT]] 



I in which it has been assumed that the concentration of impurity is 

 j sufficiently low so that the solution would be ideal if the impurity could 



not ionize. In the classical case the exponential in the logarithm is small 

 t compared to unity and (2.23) becomes identical with (2.19), as it should. 

 1 In the totally degenerate case the exponential dominates the unity and 



we have 



^^ = {^^+0 -{- Ed - kTin2] + kT (uNd 

 I (2.24) 



= fil -{- kTin Nd 



' which is the chemical potential of an un-ionized component of a dilute 



j * An interesting by-product of this derivation (discussed in Reference 5) is the 

 I fact that the Fermi level, F, is hardly ever the Gibbs free energy per electron for 

 the electron assembly, although it is always the electronic chemical potential, in 

 1 the sense that it measures the direction of flow of electrons. This arises because 

 I the Gibbs free energy is not alwa3-s a homogeneous function^^ of the first degree in 

 ; the mole numbers (electron numbers). Thus if the number of electrons in the as- 

 sembly is N, the Gibbs free energy, G, is given by 



G =^ NF + kT Z 



1 



T.N 



In ■- 

 hi 



where the sum is over all energy levels, j, referred to an invariant standard level. 

 ' V is the volume of the system, w/ is the total number of states at thejth level, and 

 , hj is the number of unoccupied states (holes) at the yth level. For F to be the free 

 ! energy per electron the term involving the sum must vanish so that 



But this can only happen when 



N 



CO; = KjV 



where K^ is independent of V. This requirement is formally met in the case of the 

 free electron gas where the electrons have been treated as independent particles 

 in a box so that 



CO,- = [8mo"' TT E dE/2h^V 



where mo is the electron mass, and h, Plank's constant. Since this is the case most 

 frequently dealt with in thermodynamic problems it has been customary to think 

 of F as the free energy per electron, although even here the truth of the contention 

 depends on the assumption of particle in the box behavior. 



At the other extreme, it is obvious that co, for a level corresponding to the deep 

 closed shell states of the atoms forming a solid cannot depend at all on the ex- 

 ternal volume since they are essentially localized. In computing the free energy 

 of the semiconductor phase it is necessary to understand carefully subtleties of 

 this nature. 



