
We ask ourselves how the regular
atoms of a crystal diffuse. In the case of crystals with two or more different
atoms, we have to answer this question for each kind of atom separately.


The answer is easiest for a simple
(mono)vacancy mechanism in simple elemental cubic crystals. The
selfdiffusion coefficient
is given by g ·
a^{2} times the number of jumps per sec that the diffusing
particles make. 


Since only lattice atoms that have a vacancy as a
neighbor can jump, or, in other words, the number of lattice atoms jumping per
sec is identical to the number of vacancies jumping per sec, we obtain for the
diffusion coefficient of selfdiffusion by a simple vacancy mechanism the
following equations: 



D_{SD} 
= g · a^{2} · n
_{0} · exp – 
G_{m}
kT_{ } 
· exp – 
G_{F}
kT 





D_{SD} 
= g · a^{2} ·
n _{0} 
· exp 
S_{M}
k_{ } 
· exp – 
H_{m}
kT_{ } 
· exp 
S_{F}
k_{ } 
· exp – 
H_{F}
kT_{ } 

D_{SD} 
:= 
D* · exp – 
H_{m} + H_{F}
kT 




G_{m} is the free enthalpy
for a jump, i.e. the free enthalpy barrier that must be overcome between two
identical positions in the lattice. 


In words: All the material dependent constants
(including the migration and formation entropy) have been lumped together in
D*; and the exponential now contains the sum of the migration and formation energy of a
vacancy. 

Lets discuss this equation a
bit: 


As mentioned before, we need an entropy of
migration as a parameter of a point defect. In summary we need four parameters
correlated with an intrinsic point defect to describe its diffusion behavior
(if we discount the vibration frequency). 


But only two parameters, the formation energy and
the migration energy are of overwhelming importance. 


Everything else may be summarized in a (more or
less) constant preexponential factor D* which contains the
entropies. Since the entropies may be temperature dependent (for Si this
is probably the case), you must look at bit closer at your calculations if you
are interested in precise diffusion data. 


An
Arrheniusrepresentation (lg D
vs. 1/T) will give a straight line, the slope is given by
H_{M} + H_{F}. The preexponential factor
determines the intersection with the axis and is thus measurable. 


Since it is much easier to measure diffusion
coefficients compared to point defect densities, the sum H_{M} +
H_{F} for point defects is mostly much better known than the
individual energies. Some
values are given in the
link. 


Selfdiffusion vial selfinterstitials follows
essentially the same laws. 

For selfdiffusion in Si, we
find the following (rather small) values : D_{SD} =
(10^{–21} — 10^{–16}) m^{2}/s
in the relevant temperature regime.
Detailed data in an Arrhenius
plot for for selfdiffusion in Si can be found in the link; some
numbers for Si selfdiffusion as well as the migration parameters of
vacancies and interstitials and a few elements are
also illustrated. 