Page 182 - Materials Science and Engineering An Introduction

154 • Chapter 5 / Diffusion

Using an interpolation technique as demonstrated in Example Problem 5.2 and the data presented
in Table 5.1, we find
x
= 0.4747 (5.10)
21Dt
4
The problem stipulates that x 0.75 mm 7.5 10 m. Therefore,
-4
7.5 * 10 m
= 0.4747
21Dt
This leads to
-7
Dt = 6.24 * 10 m 2
Furthermore, the diffusion coefficient depends on temperature according to Equation 5.8, and,
5
2
from Table 5.2 for the diffusion of carbon in g-iron, D 0 2.3 10 m /s and Q d 148,000 J/mol.
Hence,
Q d -7 2
Dt = D 0 expa - b(t) = 6.24 * 10 m
RT
148,000 J/mol -7 2
-5
2
(2.3 * 10 m /s) exp c - # d (t) = 6.24 * 10 m
(8.31 J/mol K)(T)
and, solving for the time t, we obtain
0.0271
t (in s) =
17,810
expa - b
T
Thus, the required diffusion time may be computed for some specified temperature (in K). The
following table gives t values for four different temperatures that lie within the range stipulated
in the problem.

Temperature Time
( C) s h
900 106,400 29.6
950 57,200 15.9
1000 32,300 9.0
1050 19,000 5.3

5.6 DIFFUSION IN SEMICONDUCTING MATERIALS
One technology that applies solid-state diffusion is the fabrication of semiconductor
integrated circuits (ICs) (Section 18.15). Each integrated circuit chip is a thin square
wafer having dimensions on the order of 6 mm 6 mm 0.4 mm; furthermore, millions
of interconnected electronic devices and circuits are embedded in one of the chip faces.
Single-crystal silicon is the base material for most ICs. In order for these IC devices to
function satisfactorily, very precise concentrations of an impurity (or impurities) must
be incorporated into minute spatial regions in a very intricate and detailed pattern on
the silicon chip; one way this is accomplished is by atomic diffusion.
Typically, two heat treatments are used in this process. In the first, or predeposition
step, impurity atoms are diffused into the silicon, often from a gas phase, the partial
pressure of which is maintained constant. Thus, the surface composition of the impurity