Two major theories of diffusion have been proposed. These are continuum theory of Fick and the atomistic theory of defects and vacancies. Let us consider Fick's approach first. Fickian theory states that flux of a species across a plane is proportional to concentration gradient. That is, as illustrated in Fig 3-2, the number of atoms of A that
will cross the X-plane per unit area and per unit time in the direction of X is proportional to the first derivative of concentration. That is
We use partial differential to express concentration gradient because
concentration in general can vary with the other directions y and z. Flux
in X- direction, JAx has the units of number of atoms
of A diffusing per unit area (m2) per unit time (s).
The above can be written as an equation by introducing a proportionality
constant called diffusion coefficient, D. That is
If x is in measured in meters and concentration, CA
is in number of atoms per cubic meters, the diffusion coefficient will
have the units of square meters per second (m2 s-1)
If we limit the discussion to only one dimensional transport, which
means that concentration of A does not change in y and z directions in
the example we are considering, we can convert the partial differentials
to total differentials, and the above equation becomes
Let us consider the region in which A is diffusing as consisting of
pure B. (In our applications, B will be silicon.) We can mathematically
indicate this by labeling the proportionality constant as DA
diffusing in B, and for short we will use the symbol DAB.
(Some use the notation, DA-B. to indicate the same).
That is, the diffusion equation now takes the form
If concentration of A decreases along x from left to right as shown
in the figures, we would say that A is diffusing from left to right, because
diffusion occurs from a region of higher concentration to a region of lower
concentration. This is analogous to heat flowing from higher temperature
to lower temperature and fluid flowing from higher pressure to lower pressure.
In the present example, will
be negative because A decreases from left to right. In order to preserve
our common sense understanding of diffusion direction (similar to heat
flow direction), we would have to include a negative sign on the right
hand side. That is
In this form, we have defined Fickian theory of diffusion (stated earlier
descriptively) in a precise mathematical form. If we were to consider diffusion
in three dimensions, we would write the above equation as
the gradient operator and i, j and k are unit vectors in s, y and z. In
this form concentration is allowed to be a function of x, y and z.
Although the general 3-dimensional form is more complete, in many situations one dimensional form is adequate to describe process behavior. Let us now consider the one-dimensional model. The diffusion coefficient can be considered constant when A is diffusing through essentially pure B. That is, A is present in B in dilute concentrations. As we stated earlier, the diffusion coefficient does depend on the nature of medium in which the diffusion occurs. In fact this is the reason why we included the subscript AB on the diffusion coefficient. It is therefore not a surprise that DAB will vary with concentration of B. In gas phase, average distance between molecules are so far apart compared to liquid and solid, and compared to distances over which the inter-molecular forces of attraction and repulsion occur that we usually approve DAB as a constant in gas phase. In liquids and in solids, and the current application in semiconductor processing, we cannot make such an approximation without some loss in accuracy.