for almost all
dopants. The chemical process of diffusion can be easily described by
continuum theory using Fick's diffusion laws (3.1-13) and
(3.1-14), where 
denotes the diffusion flux of the particles,
C the concentration of the species and D the diffusion coefficient or
diffusivity.
It is possible by the continuum theory to describe the diffusion mechanism for almost all dopant at intrinsic doping conditions. The appropriate diffusion coefficients have been extracted from measurements [Sha75] [Jai75] [Fai81]. Because of using higher doping concentrations and lower thermal budgets for annealing, it was found quickly, that the Fickian laws are not accurate enough for the arising anomalous diffusion behavior. It was necessary to focus on the atomistic level of diffusion to get insight on the interaction of dopants with lattice atoms. There is no doubt, that diffusion of impurities can only occur within the crystal lattice due to the presence of neighboring point-defects. Three possible atomic diffusion mechanisms were established in the past in the literature [Sze88] [Fah89] [Tay93]. The first one is the direct mechanism, where impurities which have small ionic radii can travel directly from one interstitial site to another one (see Fig. 3.1-1). Especially Group-I and Group-VIII elements are diffusing mainly directly and are therefore fast diffusers.
Figure 3.1-1: Direct diffusion
mechanism within crystal targets. The diffuser can travel exclusively
on interstitial sites.
Substitutional dopants migrate through the host lattice by moving via adjacent vacant lattice sites. This mechanism is known as vacancy mechanism, where an substitutional dopant exchanges its position with the neighboring vacancy (see Fig. 3.1-2a). To avoid oscillations of this exchange procedure, the vacancy must move at least to a third-neighbor site away from the dopant. Thus, the vacancy or another vacancy within the diffusion regime can return on a different path within the silicon's diamond lattice. Then long-range migration of the dopant will take place (see Fig. 3.1-2b). Energetic details of this complicated exchange can be found in, e.g. [Fah89].
Figure 3.1-2: Dopant diffusion by vacancy
mechanism. Vacancy must move to the third-neighbor within the diamond
lattice for persist dopant migration.
The counterpart of the vacancy mechanism is the interstitialcy mechanism (see Fig. 3.1-3). Thereby dopant diffusion takes place when a dopant at a substitutional site is approached by a silicon interstitial (Fig. 3.1-3a). The dopant is kicked out by the interstitial to reside at interstitial position (Fig. 3.1-3b), while the original self-interstitial has disappeared by occupying the regular lattice site. Now this interstitial dopant is able to move towards an adjacent lattice site to re-form a silicon interstitial by the same ``kick-out'' process, like the interstitial dopant itself was generated (Fig. 3.1-3c).
Figure 3.1-3: Dopant diffusion via a substitutional-interstitialcy
interchange. Dopant at a substitutional site is pushed to an interstitial
site by the ``kick-out'' reaction.
It is common to denote the diffusing species involved by one of the above
mechanism as dopant-defect pairs. In the case of the vacancy mechanism, we
label a dopant A which is in the vicinity of a vacancy V as
AV-pair. If the dopant resides on a substitutional lattice position, it is
referred to as A or 
. When the dopant A is adjacent to an
interstitial I, it will be written as AI-pair and as 
when it rests
at an interstitial site (e.g after implantation). Note, that we make no
difference whether the dopant or the point-defect is at an interstice
position, both cases are treated as AI-pairs. The rate equations for the
basic atomistic diffusion mechanisms are:
Summarizing, dopants can diffuse within a crystal lattice using point-defects as diffusion vehicle either by the vacancy mechanism (3.1-15) or the interstitialcy mechanism (3.1-16,3.1-17) or a combination of both. The most common p-type dopant boron diffuses mainly via interstitials, where the n-type dopant phosphorus shows at intrinsic concentrations interstitial dominated diffusion and for high concentrations a dual mechanism with a strong vacancy component. Arsenic is also diffusing via interstitials and vacancies, but the interstitialcy diffusion is at least limited due to the relatively large ionic radii. Large dopants like Antimony will find no stable position between lattice sites, so they can only diffuse via vacancies.
Permanent transport of dopants within the lattice is not only controlled by the diffusion mechanism, there are also recombination processes between dopants and point-defects occurring, which can disturb or even prevent dopants from diffusion. These recombinations are always possible when dopant-defect pairs are approaching lattice defects of the opposite type. The basic rate equations for AI and AV pair recombinations are given by (3.1-18) and (3.1-19).
The first equation (3.1-18) gives the Frank-Turnbull recombination, which was found to be very efficient for phosphorus diffusion [Dun92] [Bac92a] [Van92], where the opposite recombination mechanism is known as dissolution process (3.1-19). In both cases the involved point-defect disappears at the recombination site, and a dopant at a substitutional position remains. The dopants are paired with point-defects during the diffusion process. It is possible now for the dopants to get rid of the defect-binding by means of these recombination processes. Furthermore, as the dopants are electrically active at substitutional sites, the recombination process can be used for activation of prior inactive dopants. If there are enough recombination centers for the dopant-defect pairs at a certain lattice position, the whole diffusion process is stopped, because there are no point-defects in the vicinity of the substitutional dopants available to act as diffusion vehicle.
The previously defined diffusion current can now be expressed with
respect to point-defect interactions by (3.1-20), where 
and
are the diffusivities of AI and AV pairs, respectively.
If the dopant concentration level 
is significantly below the intrinsic
carrier concentration 
, then we have intrinsic doping
conditions. The intrinsic carrier concentration 
for dopants within the
silicon lattice is calculated using approximations to Morin and
Maita's empirical models (3.1-21), where 
is the band gap energy
of silicon [Mor54].
Under intrinsic conditions the diffusivities of the dopants show no
dependence on the doping level. Furthermore, the point-defects and the
dopants are in local thermal equilibrium. By applying the mass action law
and equilibrium conditions for the point-defects, 
, we can accomplish to express the diffusion flux in terms of
the dopant concentration by (3.1-22), where 
is the intrinsic
diffusion coefficient of the dopant (3.1-23).
The temperature dependence of the diffusivities is commonly given by an Arrhenius expression. Figure 3.1-4 depicts the intrinsic diffusion coefficients for the most common dopants in silicon.
Figure 3.1-4: Intrinsic diffusion coefficients for the most common diffusers in
silicon. Data for B,P,As,Sb from [Sel84],
Al from [EK95], V from [Bro87], and
I,V from [Fah89].
