Callister - An introduction - 8th edition

is only an approximation), then the solution to Fick’s second law (Equation 5.4b)
takes the form
C1x, t2 Q 0 x2
exp a (5.11)
2pDt 4Dt b
Here Q 0 represents the total amount of impurities in the solid that were introduced
during the predeposition treatment (in number of impurity atoms per unit area);
all other parameters in this equation have the same meanings as previously. Furthermore,
it can be shown that
(5.12)
where C s is the surface concentration for the predeposition step (Figure 5.10), which
was held constant, D p is the diffusion coefficient, and t p is the predeposition treatment
time.
Another important diffusion parameter is junction depth, x j . It represents the
depth (i.e., value of x) at which the diffusing impurity concentration is just equal to
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:
(5.13)
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
5.6 Diffusion in Semiconducting Materials • 139
D p t p
Q 0 2C s
B p
x j c14D d t d 2ln a
bd
C B 2pD d t d
Boron atoms are to be diffused into a silicon wafer using both predeposition
and drive-in heat treatments; the background concentration of B in this silicon
material is known to be 1 10 20 atoms/m 3 . The predeposition treatment is to
be conducted at 900C for 30 minutes; the surface concentration of B is to be
maintained at a constant level of 3 10 26 atoms/m 3 . Drive-in diffusion will be
carried out at 1100C for a period of 2 h. For the diffusion coefficient of B in
Si, values of Q d and D 0 are 3.87 eV/atom and 2.4 10 3 m 2 /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 900C (1173 K)] using Equation 5.8.
(Note: For the gas constant R in Equation 5.8, we use Boltzmann’s constant
k, which has a value of 8.62 10 5 eV/atom#
K). Thus
Q 0
Q d
D p D 0 exp a b
kT p
1/2