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TimeDomain CVD, Inc. |
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In the Chamber: OverallIn a showerhead reactor, if we approximate the showerhead itself as an ideal porous material and treat the flow as incompressible, gases are dispensed uniformly from the showerhead but must move radially to exit at the perimeter.  Thus, at any radius, all the gas dispensed from inside must flow outward (ignoring changes due to reactions in the gas or at the surface). Since the area dispensing increases as the square of the radius, and the perimeter area is linear, the radial velocity of the gases must increase linearly with radial distance: the radial velocity should go as r/2Hc. With this in mind, let's see what can be learned from an overall examination of transport in the reactor, without taking into account the detailed nature of the gas flow. As usual we approximate the flows using the ideal gas law. In this case (as we'll see) the vertical temperature distribution is linear, so it is reasonable to treat thermal expansion by taking as the approximate chamber temperature the average of the wafer and ceiling temperatures. We assume transport properties for nitrogen as a representative gas. Here are the results: The diffusion length is much larger than the ceiling height, so we know that convection plays a weak role in transport in the axial (vertical) direction. The concentrations of precursors and products will be constant or linear in height. On the other hand, the diffusion length is comparable to the chamber radius: both diffusion and convection are important in radial transport. For the same reasons, the temperature in the gas phase will be linear in height from the wafer to the cooler ceiling. The residence time is very short: about 10 times less than in the comparable case we examined for a horizontal tube reactor. The combination of a short residence time and confinement of high temperatures to the region near the wafer makes showerhead reactors appropriate for use in systems where high rate gas phase reactions are important. The residence time is obviously linear in the ceiling height Hc; as we noted, variation of Hc is a simple way to influence operation of showerhead reactors.
A More Detailed Look I: Stagnation FlowHow do the gases actually make their way through a showerhead chamber? We'll first introduce and then to some extent debunk the relevance of a stagnation flow pattern (so named because the velocity of flow goes to zero in the middle of the flow); then we'll examine the boundary layer formation and see that our simplified treatment is actually very useful for most practical reactors. It turns out to be pretty simple to derive an analytic expression for the gas velocity in a showerhead-like configuration in the inviscid incompressible flow approximation: that is, we assume the gas is incompressible (also ignoring thermal expansion) so that volume in is the same as volume out, and we ignore the fluid viscosity. These are very similar assumptions to those underlying our plug flow treatment of cylindrical ducts. However, in this case we must account for the presence of an impermeable bottom boundary (the wafer and substrate holder): the vertical velocity must = 0 at z=0.
Let us examine the behavior of the solution. The vertical velocity is linear in height, the radial velocity similarly linear in radius. The total fluid velocity at any point is:
For r >> Hc, the velocity is dominated by the radial component. To make a picture of what this flow looks like, we can derive an equation for the streamlines (paths of test particles trapped in the flow):
The flow pattern looks like this, for a ceiling height of 1.5 cm and radius of 10 cm:
It is interesting to note that we can also calculate the time that a particle -- or a precursor gas molecule -- has spent in the stream since it entered the chamber, for any given position. The answer is
Since the time in the stream is only dependent on the height, the time for gas phase reactions is independent of the radial position. This may seem a bit puzzling: one is tempted to think that the gases have been around for longer as we move towards the outside of the showerhead, but the compression of the streamlines towards the bottom of the chamber compensates.
Details II: What About Viscosity?Can we really ignore viscosity? Since viscosity is the diffusion of momentum, and our overall analysis showed that diffusion dominates convection in the vertical direction, it seems likely that we can't. Let's take a look. The boundary layer thickness in stagnation flow is independent of the radial position: we have approximately [ Thin Film Deposition, D. Smith, p. 324] : Recalling that the kinematic viscosity increases as (1/P) and the inlet velocity has the same dependence, we can easily see that the boundary layer thickness is independent of the pressure, if the molar inlet flow is fixed. Thus we can re-express the boundary layer thickness in terms of the molar flow and molar volume at STP: For the example we examined at the beginning of this section the flow is 300 sccm (so VmFin = 300 cm3), the inlet area is about 700 cm2, and Hc = 4 cm: we find
The whole chamber is in the boundary layer. Just as we had derived in general from our overall analysis, convection plays little role in vertical transport of mass and momentum. The stagnation flow solution is not relevant to the chamber. On the other hand, if the flow is increased by e.g. a factor of 10 and the ceiling by a factor of 10, then the boundary layers would be thin compared to the chamber height and stagnation flow is relevant. Note that in this case, the ceiling height is comparable to the diameter! Diffusion-dominated:In this case, any given lateral region of the wafer essentially receives precursor from the showerhead above it. Uniform deposition will result if the gas is dispensed uniformly by the showerhead (ignoring all the other things that can go wrong, like edge effects, temperature variations, and plasma inhomgeneities). Stagnation flow: In reactors with large separations between the showerhead and the substrate-holder, operating at high flows, stagnation flow becomes relevant. Precursors are convected from the showerhead down and outward: when they reach the boundary layer height, they then proceed by diffusion to the wafer surface. The effective residence time for gases in the stream can be approximated by adding the time in the stream to reach the boundary layer to the time taken to diffuse across it: This time scales as (H/Vin), with the multiplier being a slowly-varying logarithmic term, and thus in practical cases doesn't differ very much from the simple residence time we calculated in our overall analysis based on chamber and gas flow volumes. (Exercise for the student: examine the second term in the equation for streamtime and verify that it also scales as H/Vin to within a factor of order 1.) To properly average over the whole chamber we would need to correct this expression for the near-edge gas flows, which exit the finite-sized chamber before they can reach the bottom boundary layer.
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with
a solution: 
which
is independent of the actual velocity Vin.
and
has the necessary and interesting feature than the time in the stream goes
to infinite as the height goes to zero: gas molecules can never actually
get to the wafer by pure convection! Diffusion is always necessary for
deposition.
