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 t1 , 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 t1 and at a temperature of T1, then "beta" is calculated by

                                (3-14)

 

where D1 is diffusion coefficient at T1. The above relationship is used to determine time required to deposit a desired amount of dopant at a specified temperature.