Page 184 - Materials Science and Engineering An Introduction

156 • Chapter 5 / Diffusion

the background concentration of that impurity in the silicon (C B ) (Figure 5.10). For
drive-in diffusion x j may be computed using the following expression:
1/2
Q 0
x j = c (4D d t d )ln a b d (5.13)
C B 1pD d t d
Here, D d and t d represent, respectively, the diffusion coefficient and time for the drive-in
treatment.

EXAMPLE PROBLEM 5.6
Diffusion of Boron into Silicon

Boron atoms are to be diffused into a silicon wafer using both predeposition and drive-in heat
20
treatments; the background concentration of B in this silicon material is known to be 1 10
3
atoms/m . The predeposition treatment is to be conducted at 900 C for 30 min; the surface con-
3
centration of B is to be maintained at a constant level of 3 10 atoms/m . Drive-in diffusion
26
will be carried out at 1100 C for a period of 2 h. For the diffusion coefficient of B in Si, values
3
2
of Q d and D 0 are 3.87 eV/atom and 2.4 10 m /s, respectively.
(a) Calculate the value of Q 0 .
(b) Determine the value of x j for the drive-in diffusion treatment.
(c) Also for the drive-in treatment, compute the concentration of B atoms at a position 1
m
below the surface of the silicon wafer.
Solution
(a) The value of Q 0 is calculated using Equation 5.12. However, before this is possible, it is
first necessary to determine the value of D for the predeposition treatment [D p at T T p
900 C (1173 K)] using Equation 5.8. (Note: For the gas constant R in Equation 5.8, we use
#
5
Boltzmann’s constant k, which has a value of 8.62 10 eV/atom K). Thus,
Q d
D p = D 0 expa - b
kT p
3.87 eV/atom
-3 2
= (2.4 * 10 m /s) exp c - # d
-5
(8.62 * 10 eV/atom K)(1173 K)
2
= 5.73 * 10 -20 m /s
The value of Q 0 may be determined as follows:
D p t p
Q 0 = 2C s
B p
2
(5.73 * 10 -20 m /s)(30 min)(60 s/min)
26
3
= (2)(3 * 10 atoms/m )
B p
18
= 3.44 * 10 atoms/m 2
(b) Computation of the junction depth requires that we use Equation 5.13. However, before
this is possible, it is necessary to calculate D at the temperature of the drive-in treatment
[D d at 1100 C (1373 K)]. Thus,
3.87 eV/atom
-3 2
D d = (2.4 * 10 m /s)expc - # d
-5
(8.62 * 10 eV/atom K)(1373 K)
2
= 1.51 * 10 -17 m /s