4.2.1 Mathematical Model: Pre-Deposition

Consider the pre-deposition step. The wafer is placed in an atmosphere in which dopant atoms are deposited on the surface by a gas that bears the dopant atom. Let the concentration of dopant A at the surface be CAs, given in atoms per cm3. The surface concentration (Cas) will remain constant during the pre-deposition step. The value of CAs depends strongly on temperature and on the dopant. For example, Boron's value at 1223 K is 4.5 x 1020 atoms/cm3 and increases by about 35% as temperature is increased to 1423 K.

Referring to Figure 3-5, consider a control volume of thickness dx and length and depth of unit distance.



Fig 3-5. Diffusion Geometry.


The amount of A diffusing in at x is given by Fick's Law,

and the amount of A diffusing out at x+Dx is

 

The difference is accumulated within the control volume and is given by

 

Dividing the above by dx and then taking the limit as dx goes to zero, one gets the classical one dimensional transient diffusion equation shown below.

                                              (3-7)

 

The above is a partial differential equation whose solution depends on the boundary conditions we impose. When we maintain a constant surface concentration of CAs during pre-deposition, the following boundary values are applicable.

                                (3-8)

 

The first two equations above state that the surface concentration and concentration far-far away from the surface are constant values. The distance of infinity is not to be interpreted literally; instead one visualizes any distance farther than the region wherein concentration changes as being infinite distance. For example, in Figure 3-3 and 3-4, a distance of 2 µm can be treated as infinite distance within the time frame considered. The third condition imposed above is the initial condition, namely the concentration profile within the wafer before the pre-deposition was initiated. Here the statement is to be taken as the wafer having a constant dopant concentration of CA0. In many practical applications, CA0 will be zero. Here, we will use a variable in our formulation to maintain a greater level of generality.

The solution to Equation (3-7) subjected to boundary conditions in (3-8) is given below.

                                              (3-9)

 

The abbreviation "erf" refers to error function. Error function of a variable y is defined as:

                                              (3-10)

 

As shown in Figure 3-6, error function increases monotonically from 0 to 1 as its argument increases; it is essentially 1 when argument is 2. Error function is an odd function, meaning that its value when its argument is negative is negative. that is: erf(-y) = - erf(y).



Figure 3-6 Error Function Values