A horizontal chemical vapor deposition (CVD) reactor for growth of gallium arsenide (GaAs) thin films is shown in Figure 30.2. In this process, arsine (AsH3), trimethyl gallium (Ga(CH3)3), and H2 gases are fed into the reactor. Inside the reactor, the silicon wafer rests on a heated plate called a

Question

A horizontal chemical vapor deposition (CVD) reactor for growth of gallium arsenide (GaAs) thin films is shown in Figure 30.2. In this process, arsine [latex]\left(\mathrm{AsH}_{3} ight)[/latex], trimethyl gallium [latex]\left(\mathrm{Ga}\left(\mathrm{CH}_{3} ight)_{3} ight)[/latex], and [latex]\mathrm{H}_{2}[/latex] gases are fed into the reactor. Inside the reactor, the silicon wafer rests on a heated plate called a susceptor. The reactant gases flow parallel to the surface of the wafer and deposit a GaAs thin film according to the simplified CVD reactions

[latex]2 \mathrm{AsH}_{3}(\mathrm{~g}) ightarrow 2 \mathrm{As}(\mathrm{s})+3 \mathrm{H}_{2}(\mathrm{~g}) ext { and } 2 \mathrm{Ga}\left(\mathrm{CH}_{3} ight)_{3}(\mathrm{~g})+3 \mathrm{H}_{2}(\mathrm{~g}) ightarrow 2 \mathrm{Ga}(\mathrm{s})+6 \mathrm{CH}_{4}(\mathrm{~g})[/latex]

If the process is considerably diluted in [latex]\mathrm{H}_{2}[/latex] gas, then the mass transfer of each species in the [latex]\mathrm{H}_{2}[/latex] carrier gas can be treated separately. The surface reaction is very rapid, and so the mass transfer of the gaseous reactants to the surface of the wafer limits the rate of GaAs thin film formation.

In the present process, the edge of a [latex]10-\mathrm{cm}[/latex] silicon wafer is positioned [latex]4 \mathrm{~cm}[/latex] downstream of the leading edge of the susceptor plate. The wafer is inset within this plate so that a contiguous flat surface is maintained. The process temperature is [latex]800 \mathrm{~K}[/latex], and the total system pressure [latex]101.3 \mathrm{kPa}[/latex] ( [latex]1 \mathrm{~atm}[/latex] ). Consider a limiting case where the flow rate of the [latex]\mathrm{H}_{2}[/latex]-rich feed gas to the reactor results in a bulk linear velocity of [latex]100 \mathrm{~cm} / \mathrm{s}[/latex], where trimethylgallium is present in dilute concentration. Determine the local mass-transfer coefficient [latex]\left(k_{c} ight)[/latex] for trimethylgallium in [latex]\mathrm{H}_{2}[/latex] gas at the center of the wafer using (a) boundary-layer theory and (b) film theory. The binary gas phase diffusion coefficient of trimethylgallium in [latex]\mathrm{H}_{2}[/latex] is [latex]1.55 \mathrm{~cm}^{2} / \mathrm{s}[/latex] at [latex]800 \mathrm{~K}[/latex] and [latex]1 \mathrm{~atm}[/latex].

Accepted Answer

At sufficiently high flowrate within the reactor, the physical system represents convective mass transfer over a flat plate, where the hydrodynamic boundary layer develops before the concentration boundary layer. Trimethylgallium diluted in [latex]\mathrm{H}_{2}[/latex] gas serves as the source for mass transfer, and the reaction at the boundary surface is the sink for trimethylgallium gas.

(a) First, the local Reynolds number is evaluated at [latex]x=9 \mathrm{~cm}[/latex], the distance from the edge of the susceptor [latex](4 \mathrm{~cm})[/latex] to the middle of the wafer [latex](5 \mathrm{~cm})[/latex]. The kinematic viscosity of the feed gas is approximated by the properties of the [latex]\mathrm{H}_{2}[/latex] gas, as trimethylgallium is present in only dilute concentration; from Appendix I, [latex]ν=5.686 \mathrm{~cm}^{2} / \mathrm{s}[/latex] at [latex]800 \mathrm{~K}[/latex] and [latex]1 \mathrm{~atm}[/latex] for [latex]\mathrm{H}_{2}[/latex] gas. Therefore

[latex]\operatorname{Re}_{x}=\frac{v_{\infty} x}{ν}=\frac{(100 \mathrm{~cm} / \mathrm{s})(9 \mathrm{~cm})}{5.686 \mathrm{~cm}^{2} / \mathrm{s}}=158.3[/latex]

and so the flow is laminar. The Schmidt number is

[latex]\mathrm{Sc}=\frac{ν}{D_{A B}}=\frac{5.686 \mathrm{~cm}^{2} / \mathrm{s}}{1.55 \mathrm{~cm}^{2} / \mathrm{s}}=3.67[/latex]

For the present system where the hydrodynamic boundary layer starts to develop before the concentration boundary layer, the local Sherwood number for laminar flow is

[latex]\operatorname{Sh}_{x}=0.332 \operatorname{Re}_{x}^{1 / 2}\left(\frac{\mathrm{Sc}}{1-\left(\frac{X}{x} ight)^{3 / 4}} ight)^{1 / 3}=0.332(158.3)^{1 / 2}\left(\frac{3.67}{1-\left(\frac{4 \mathrm{~cm}}{9 \mathrm{~cm}} ight)^{3 / 4}} ight)^{1 / 3}=8.375[/latex]

Finally, the local mass-transfer coefficient predicted by boundary-layer theory is

[latex]k_{c}=\frac{\mathrm{Sh}_{x}}{x} D_{A B}=\left(\frac{8.375}{9 \mathrm{~cm}} ight)\left(1.55 \frac{\mathrm{cm}^{2}}{\mathrm{~s}} ight)=1.44 \frac{\mathrm{cm}}{\mathrm{s}}[/latex]

(b) For film theory, recall that the local mass-transfer coefficient is defined as

[latex]k_{c}=\frac{D_{A B}}{\delta_{c}}[/latex]

For laminar flow, the hydrodynamic boundary-layer thickness is

[latex]\delta=\frac{5 x}{\sqrt{\operatorname{Re}_{x}}}=\frac{5(9 \mathrm{~cm})}{\sqrt{158.3}}=3.58 \mathrm{~cm}[/latex]

Now recall that the relationship between the hydrodynamic boundary-layer thickness and the concentration boundary-layer thickness is

[latex]\frac{\delta}{\delta_{c}}=\mathrm{Sc}^{1 / 3}[/latex] (28-18)

for laminar flow over a flat plate where the hydrodynamic and concentration boundary layers have the same starting point. For laminar flow over a flat plate where the hydrodynamic boundary layer is initiated before the concentration boundary layer, the combination of equations (28-21), (28-26), and (30-5)

[latex]\frac{\overline{{{k}}}_{c}x}{D_{A B}}=\mathrm{Sh}_{L}=0.664\,\mathrm{Re}_{L}^{1/2}\mathrm{Sc}^{1/3}[/latex] (28-21)

[latex]\begin{aligned} & \bar{k}_c=\frac{0.332 D_{A B}\left(\frac{v}{ν} ight)^{1 / 2}(\mathrm{Sc})^{1 / 3} \int_0^{L_t} x^{-1 / 2} d x+0.0292 D_{A B}\left(\frac{v}{ν} ight)^{4 / 5}(\mathrm{Sc})^{1 / 3} \int_{L_t}^L x^{-1 / 5} d x}{L} \ & \bar{k}_c=\frac{0.664 D_{A B}\left(\frac{v}{ν} ight)^{1 / 2}(\mathrm{Sc})^{1 / 3} L_t^{1 / 2}+0.0365 D_{A B}\left(\frac{v}{ν} ight)^{4 / 5}(\mathrm{Sc})^{1 / 3}\left[(L)^{4 / 5}-\left(L_t ight)^{4 / 5} ight]}{L} \ & \bar{k}_c=\frac{0.664 D_{A B}\left(\operatorname{Re}_t ight)^{1 / 2}(\mathrm{Sc})^{1 / 3}+0.0365 D_{A B}(\mathrm{Sc})^{1 / 3}\left[\left(\operatorname{Re}_L ight)^{4 / 5}-\left(\mathrm{Re}_t ight)^{4 / 5} ight]}{L} \end{aligned}[/latex] (28-26)

[latex]\mathrm{Sh}_{x}=0.332\,\mathrm{Re}_{x}^{1/2}\Bigg(\frac{\mathrm{Sc}}{1-\left(\frac{X}{x} ight)^{3/4}}\Bigg)^{1/3}[/latex] (30-5)

yields

[latex]\frac{\delta}{\delta_{c}}=\left(\frac{\mathrm{Sc}}{1-\left(\frac{X}{x} ight)^{3 / 4}} ight)^{1 / 3}[/latex] (30-6)

Substituting in the appropriate values, we obtain

[latex]\frac{\delta}{\delta_{c}}=\left(\frac{\mathrm{Sc}}{1-\left(\frac{X}{x} ight)^{3 / 4}} ight)^{1 / 3}=\left(\frac{3.67}{1-\left(\frac{4 \mathrm{~cm}}{9 \mathrm{~cm}} ight)^{3 / 4}} ight)^{1 / 3}=2.004[/latex]

and so

[latex]\delta_{c}=\frac{3.58 \mathrm{~cm}}{2.004}=1.79 \mathrm{~cm}[/latex]

Finally

[latex]k_{c}=\frac{D_{A B}}{\delta_{c}}=\frac{1.55 \mathrm{~cm}^{2} / \mathrm{s}}{1.79 \mathrm{~cm}}=0.868 \frac{\mathrm{cm}}{\mathrm{s}}[/latex]

We note that [latex]k_{c}[/latex] predicted by film theory is [latex]37 \%[/latex] lower than [latex]k_{c}[/latex] predicted by boundary-layer theory.

The preceding analysis is valid only in the limiting case where the thickness of the hydrodynamic boundary layer is significantly less than the height of the enclosure housing the susceptor plate. However, if the Reynolds number is sufficiently low, then the hydrodynamic boundary-layer thickness approaches the height of the enclosure. In this situation, the CVD reactor housing itself becomes the conduit for flow, and the analyses provided by Middleman and Hochberg [latex]{ }^{2}[/latex] and Middleman [latex]^{3}[/latex] are recommended.

Question 30.1: A horizontal chemical vapor deposition (CVD) reactor for gro...

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