Question 24.4: In the chemical vapor deposition of silane (SiH4) on a silic...
In the chemical vapor deposition of silane \left(\mathrm{SiH}_{4}\right) on a silicon wafer, a process gas stream rich in an inert nitrogen \left(\mathrm{N}_{2}\right) carrier gas has the following composition:
y_{\mathrm{SIH}_{4}}=0.0075, \quad y_{\mathrm{H}_{2}}=0.015, \quad y_{\mathrm{N}_{2}}=0.9775The gas mixture is maintained at 900 K and 100 Pa total system pressure. Determine the diffusivity of silane through the gas mixture. The Lennard–Jones constants for silane are \varepsilon_{A} / \kappa=207.6 \mathrm{~K} and \sigma_{A}=4.08 Å
The binary diffusion coefficients at 900 K and 100 Pa total system pressure estimated by the Hirschfelder equation (24-33) are
D_{A B}=\frac{0.001858 T^{3 / 2}\left[\frac{1}{M_{A}}+\frac{1}{M_{B}}\right]^{1 / 2}}{P \sigma_{A B}^{2} \Omega_{D}} (24-33)
D_{\mathrm{SiH}_{4}-\mathrm{N}_{2}}=1.09 \times 10^{3} \mathrm{~cm}^{2} / \mathrm{s} \quad \text { and } \quad D_{\mathrm{SiH}_{4}-\mathrm{H}_{2}}=4.06 \times 10^{3} \mathrm{~cm}^{2} / \mathrm{s}
The binary diffusion coefficients are relatively high because the temperature is high and the total system pressure is low. The composition of nitrogen and hydrogen on a silane-free basis are
y_{N_{2}}^{\prime}=\frac{0.9775}{1-0.0075}=0.9849 \quad \text { and } \quad y_{H_{2}}^{\prime}=\frac{0.015}{1-0.0075}=0.0151Upon substituting these values into the Wilke equation (24-49), we obtain
D_{1-\text { mixture }}=\frac{1}{y_{2}^{\prime} / D_{1-2}+y_{3}^{\prime} / D_{1-3}+\cdots+y_{n}^{\prime} / D_{1-n}} (24-49)
D_{\mathrm{SiH}_{4} \text {-mixture }}=\frac{1}{\frac{y_{\mathrm{N}_{2}}^{\prime}}{D_{\mathrm{SiH}_{4}-\mathrm{N}_{2}}}+\frac{y_{\mathrm{H}_{2}}^{\prime}}{D_{\mathrm{SiH}_{4}-\mathrm{H}_{2}}}}=\frac{1}{\frac{0.9849}{1.09 \times 10^{3}}+\frac{0.0151}{4.06 \times 10^{3}}}=1.10 \times 10^{3} \frac{\mathrm{cm}^{2}}{\mathrm{~s}}
This example verifies that for a dilute multicomponent gas mixture, the diffusion coefficient of the diffusing species in the gas mixture is approximated by the binary diffusion coefficient of the diffusing species in the carrier gas.