One of the fundamental achievements in the understanding of point-defect
diffusion mechanisms in silicon was the realization that both types of
point-defects, interstitials and vacancies, are involved in the
self-diffusion process [See68] [Fra84]. The formation of intrinsic
point defects is governed by the principle of minimizing the total Gibbs
free energy of the silicon crystal. Any destruction of the perfect silicon
lattice increases the Gibbs free energy. The existence of point-defects on
the other hand increases the randomness of the crystal and hence can
decrease the Gibbs free energy via entropy. The thermal equilibrium
concentrations of the point-defects 
and 
are given by a
balance of entropy and enthalpy (3.1-1) - (3.1-3), where H and
S are the formation enthalpy and entropy, respectively, T is the lattice
temperature and k the Boltzmann constant. Quantity 
gives the number
the equivalent defect locations.
Up to now there is still a lack of determing these equilibrium
concentrations by experiments. The most common technique for measuring the
equilibrium concentrations is to use the silicon self-diffusion phenomenon,
where the diffusion of radioactive silicon isotopes is traced until
equilibrium is reached. By assuming that both interstitials and vacancies
diffuse, the diffusivity of silicon is given by (3.1-4), where 
and 
are the diffusivities of interstitials and vacancies,
respectively, 
is the number of lattice sites, and 
and
are correlation factors of diffusion for each mechanism involved
[Hu85].
The temperature dependence of the self-diffusion coefficient 
shows good agreement which an Arrhenius expression (3.1-5). However,
the tracer experiments cannot distinguish between the contribution of
interstitials and vacancies to the self-diffusion. It was found that dopants
which diffuse mainly by an interstitial interchange process can be used to
monitor the self-diffusion of interstitials. Some researches performed
experiments using gold [Sto84] [Zim89] and others used platinum
in-diffusion for silicon self-diffusion studies [Zim92]
[Zim93]. These experiments lead to a fairly rough estimation for 
(3.1-6). Data for the vacancy concentrations
(3.1-7) came from the positron annihilation study of Dannefaer
[Dan86] or tracer experiments [Mor83].
It should be pointed out that no measurement technique has directly
extracted the equilibrium concentrations of interstitials and vacancies in
silicon. It is still facing research today to determine and
over the range of temperature involved in nowadays thermal
processing.
Point-defects exist in silicon either charged or uncharged. Even multiple
charge states are possible, e.g. by a doubly charged vacancy 
or
during precipitation and clustering processes (see Section
3.1.5). The energy associated with the presence or absence
of an electron is directly related to the background energy of the
crystal. Therefore, each charged species has different equilibrium
values. The equilibrium concentration of the charged species depends upon the
Fermi level and can be expressed, e.g. for doubly positively charged
vacancies, by (3.1-8), where 
is the Fermi energy and
the respective energy level in the band gap for the charged
species.
It is possible for a non-degenerate semiconductor, to sum up the r charged intrinsic point defect states to one equilibrium level (3.1-9),
where 
gives the intrinsic carrier density, n and p the electron
and hole concentration, respectively.
Each charge state may also be related to a different diffusivity
. Like the equilibrium concentrations, we can also lump the
diffusivities of the different charged point defects together to one
effective diffusivity 
(3.1-10).
(3.1-10) implies that electronic interactions are occurring on a much faster time scale than chemical diffusion, hence, charge neutrality is accomplished.
