**4.2.2 Mathematical Model: Dopant Deposited**

In the pre-deposition step we found that the concentration profile of
dopant A was expressed as an error function. See Eqn (3-9). If the time
period of pre-deposition is t_{1 }, the amount of dopant that was
deposited will equal:

Flux of A at x = 0 is obtained from Fick's Equation:

We know the concentration profile of A at the end of pre-deposition step,
given by Eq (3-9). It can be rearranged to express dopant concentration
explicitly as

We can now take partial derivative of the above to obtain flux at x =
0.

(3-12)

Note that flux at the interface decreases with square root of time and
is directly proportional to concentration difference and square root of
diffusion coefficient. The amount of dopant deposited can now be calculated
by substituting the above in Eq (3-11). That is:

It is conventional to report amount of dopant deposited per unit area
rather than the total amount deposited. Therefore dividing the above by
the area term, we get the following design relationship:

(3-13)

If pre-deposition is carried out for a time period of t_{1} and
at a temperature of T_{1}, then "beta" is calculated by

(3-14)

where D_{1} is diffusion coefficient at T_{1}. The above
relationship is used to determine time required to deposit a desired amount
of dopant at a specified temperature.