Dependence on Concentration of Diffusing Species

SOLUTION

We assume that we can use one of the solutions to Fick’s second law (i.e., Equation 5-7):

0 21Dt b

We will use concentrations in atoms cm 2 3 > , time in seconds, and D in cm s . Notice that the left-hand side is dimensionless. Therefore, as long as we use concentrations in the same units for c s ,c x , and c 0 , it does not matter what those units are.

10 atoms - 10 18 atoms

cm 3 cm 3 =

20 atoms

atoms

c s -c 0 10 cm 3 - 0 cm 3

182 CHAPTER 5

Atom and Ion Movements in Materials

= erf B x

cm 22 2 A 6.5 * 10 13

s B (3600 s) R

= erf x a

From the error function values in Table 5-3 (or from your calculator computer), >

If erf(z) ! 0.99, z ! 1.82, therefore,

x ! 1.76 * 10 - 4 cm

or

- 4 10 m m

x = (1.76 * 10 cm) a cm b

x = 1.76 mm

Note that we have expressed the final answer in micrometers since this is the length scale that is appropriate for this application. The main assumptions we made are (1) the D value does not change while phosphorus (P) gets incorporated in the silicon wafer and (2) the diffusion of P is only in one dimension (i.e., we ignore any lateral diffusion).

Limitations to Applying the Error-Function Solution