If C2 = 0 (Figure 11.5), then all three solutions, that is, Equation 11.13, become, for positive values of x,
C x t C erf x Dt
C erfc x ( , )= − Dt
=
1 1
2 1
4 2 4
which is rather interesting because this is the same result that was obtained for the semi-infinite boundary conditions, but in this case C1/2 takes the place of CS. In other words, the concentration at C(0, t) = C1/2 for all values of time and C(x, t) values for positive values of x are the same as for the fixed-surface concentration, which is now C1/2. This result suggests that infinite boundary condi- tion results can be used to solve other semi-infinite boundary condition problems.
−100 −80 −60 −40 −20 0 20 40 60 80 100 0.0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Gold concentration (mole/cm3)
Distance (μm)
t = 0, original interface t = 100 s t = 500 s
t = 2000 s t = 1000 s
Gold (Au) Silver (Ag)
C=1 erfc C1 2
x
√4Dt
FIGURE 11.6 Interdiffusion of pure silver and gold at 950°C for various times illustrating the infinite boundary condition solution and that the concentration at x = 0, C(0, t) = 0.5 m/o gold. In this case, the self-diffusion coefficients are virtually identical so the infinite boundary condition solution is reasonably accurate. Also note that the diffusion profile for silver diffusing into gold is occurring at the same time, and is the exact mirror opposite of that for gold.
11.5.2 d
IffusIono
utofas
emI-I
nfInItes
labThe use of the infinite boundary condition solution can be applied to diffusion out of a semi-infinite slab at x = 0. For Equation 11.13 in the form
C x t C C
erf x Dt
C C
( , )= −
+ +
2 1 2 1
2 4 2
if C1 = −C2 as shown if Figure 11.7, which fixes the concentration at the interface to zero, C(0, t) = 0.
The solution for positive x becomes simply
C x t C erf x ( , )= Dt .
2 4 (11.15)
11.5.3 I
ntrInsIcG
etterInGIns
IlIconA very practical application of out-diffusion via Equation 11.15 is intrinsic gettering in silicon that is used for integrated circuits. The term gettering had been used for a long time in vacuum technology for processes that removed small amounts of residual gas molecules from the vacuum. This termi- nology was applied to materials processing to describe processes that remove unwanted impurities—
particularly those that could interfere with integrated circuit device functions—from solid solution.
The undesirable impurities in silicon used for integrated circuits are particularly rapidly diffusing interstitial transition elements such as copper and nickel. These impurities tend to segregate to the stress fields of dislocations* and create highly conducting paths along the dislocation line. If such a dislocation intersects a diode or transistor on the silicon chip, it can short-circuit the device and make it non functional. There are several processes that are used for gettering impurities in silicon, and most of them generate dislocations far from the surface of the wafer where the integrated cir- cuits are being processed: only the top few microns of the wafer surface are used for devices and the more than 600 μm of the rest of the wafer are just there for mechanical support. The basic concept of gettering is to generate dislocations far from the surface that will attract and essentially immobilize the unwanted impurities and keep them away from the devices on the surface.
Intrinsic gettering is a unique process that uses the properties and composition of a grown sili- con single crystal wafer to get rid of the impurities (Wolf and Tauber 2000; Campbell 2001). Normal silicon wafers, although very pure, have about 1018 oxygen atoms per cm3 in solid solution—about 20 ppm—because the silicon crystals are grown from molten silicon in silicon dioxide, SiO2, crucibles.
At the elevated temperatures where many of the integrated circuit processing steps are performed, such as the diffusion of dopants to form p-n junctions (Section 11.2.2), residual oxygen precipitates as small SiO2 particles. These particles, because of a molar volume difference, generate stresses and dislocations in the silicon that can trap fast-diffusing impurities. However, it is undesirable to have the impurities near the surface where the integrated circuits are made. Therefore, intrinsic gettering
* This process is called decorating the dislocation and allows it to be observed more easily in a transmission electron microscope (TEM).
C1=−C2
C2
−∞ 0 x ∞
Concentration
C = C2erf x 4Dt
FIGURE 11.7 Infinite boundary condition solution in which C1(x, 0) =−C2(x, 0) and the concentration at the original interface remains C(0, t) = 0. This solution can be used for the semi-infinite boundary condition problem where C(0, t) = 0: out-diffusion from a semi-infinite bar.
11.5 Application to Semi-Infinite Boundary Conditions 393
involves an initial high temperature heat treatment, about 1200°C, in an inert gas to allow dissolved oxygen near the surface to diffuse out so its concentration will drop below the concentration at which precipitates form, 12 ppm or less (about 6 × 1017 cm−3). Typically, the diffusion time is such that at 1200°C, a zone free of precipitates, the so-called denuded zone, is about 20 μm, shown schemati- cally in Figure 11.8. After the surface oxygen removal heat treatment, the wafer is then given a low temperature anneal, about 800°C, to generate a high nucleation rate and a large number of oxide pre- cipitates below the surface. Finally, a third heat treatment near 1000°C is used to grow the nucleated SiO2 precipitates large enough to generate dislocations needed to trap the transition metal impurities.
For oxygen diffusion in silicon, D= ×7 10−2exp( ((− 235 460, J mole RT/ ) / )) (Mayer and Lau 1990) so D
(
1200οC)
=3 12 10. × −10cm s2 −1. Figure 11.9 is a plot of Equation 11.15 at 1200°C for0 5 10 15 20 25 30 35 40 45 50
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Concentration (C/C2)
Distance from surface (μm) 100 s
1000 s 3000 s
10,000 s
T = 1200°C C = C2erf 4Dtx
FIGURE 11.9 Diffusion of oxygen from the silicon surface at 1200°C showing that after about 3 hours (104 seconds) the oxygen concentration has dropped from an initial concentration of 1018 cm−3 below about 6 × 1017 cm−3 over the first 20 μm or so from the surface. This is sufficiently low so that SiO2 will not precipi- tate in this denuded region during following heat treatments.
20 μm
Denuded zone (precipitate-free) Surface
Silicon water
SiO2 precipitates
FIGURE 11.8 Schematic microstructure near the surface of a silicon wafer that has undergone the process of intrinsic gettering. The wafer is given three heat treatments: one at high temperatures to diffuse oxygen out near the surface; a second at low temperatures to nucleate SiO2 precipitates in the region still rich in oxygen;
and a third high temperature treatment to grow the precipitates, generate dislocations, and trap rapidly diffu- sion impurities. Precipitation does not occur in the denuded zone near the surface where the oxygen content has been reduced below the SiO2 solubility required for precipitation during the first heat treatment.
various times showing that the oxygen concentration will drop to about 6 × 1017 cm3 at 20 μm depth, as required, in about 104 seconds, or about 3 hours.