Transport in Showerhead Reactors: Simplified Analysis

In 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 linear in the ceiling height Hc; as we noted, variation of Hc is a simple way to influence operation of showerhead reactors.

More Details Part 1: Stagnation Flow

How 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 equal 0 at z=0.

We treat the showerhead as a continuous source of gas, ignoring the individual dispensing holes: thus the vertical velocity is fixed at z=(ceiling height). Incompressibility means that the divergence of the velocity field must be zero everywhere: this is a potential flow.

The divergence equation can be written out in cylindrical coordinates (assuming no azimuthal flow):

We assert that the solutions for the vertical and radial velocity are:

These expressions satisfy the boundary conditions: the vertical velocity is equal to -(the inlet velocity) at z=Hc, and is 0 at z=0. The radial velocity is just what we derived at the beginning of this section, by conserving total gas volume.

To prove that this is the correct solution we verify that the divergence is zero, using the chain rule to differentiate the r*v term:

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, and is just linear in the radial distance, as one would expect.

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 path a particle follows is determined by the ratio of the vertical and radial velocities at each point:

We can integrate the equation, since we know the velocities, and find:

which is independent of the inlet velocity vin. 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

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.

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 for larger r compensates.

Details Part 2: Convection + Diffusion = Confusion?

Hopefully not. In the analysis so far we haven’t explicitly accounted for the balance between convection and diffusion inside the chamber. To do so in a (somewhat) tractable fashion, we make use of a very nice result from Howling et. al.’s 2012 paper [1]: because the concentrations of species are uniform radially in an ideal reactor, transport in the vertical direction is decoupled from transport in the radial direction. This is a somewhat counter-intuitive result: we can treat transport in the z-direction only, despite the fact that the streamlines become almost horizontal near the reactor exit. This is true because, as long as the concentration is uniform in r, the stuff carried in along a streamline from an inner point has the same concentration as the stuff that is leaving, except for the dependence on z.

As a consequence, we can take a vertical slice at any radial position and examine transport due to diffusion and convection, where in each case we only include transport along the vertical (z) axis. Recall that (in this approximation) the vertical velocity is just linear in position. (In the derivation below we’ll ignore thermal expansion to keep the expressions simpler; for most processes that’s at most a factor-of-two error.)

The change in concentration at any location is due to stuff carried in by the vertical (downward) velocity, and is thus proportional to the gradient: if the stuff above us has the same concentration as where we are, convection doesn’t change anything. The net change due to diffusion is the difference in the flux in and the flux out, both of which are proportional to the gradient in concentration, so the change is proportional to the second derivative of concentration. The net result in steady state is:

which has solutions:

where the characteristic length is:

The characteristic length d turns out to be very useful in categorizing the solutions. When d >> Hc, the solution just looks like a line. When d <<Hc, we can divide the solution into two nearly-linear parts, for regions z<d and z>d:

This is great, but to use the solutions we need to know what the concentration at the top (z=Hc) is. To figure this out, we conserve species. The stuff that goes in has to either end up on the wafer, or be carried out into the exhaust:

The average concentration at the exit can be conveniently obtained for the piecewise-linear approximation:

The approximate values are very accurate for large and small values of d, and off by less than 20% for d=Hc:

Using these expressions, we can find the concentration at the top of the chamber, and in consequence the deposition rate and efficiency of use (assuming only one active precursor) as a function of inlet velocity:

Here’s a plot for a ceiling height of 1 cm, total pressure of 10 Torr, and binary diffusion coefficient of 15 cm2/s (a moderately heavy precursor molecule at modest temperature), for a chamber radius of 20 cm. We can see that at low flows we get great efficiency but low deposition rate; at high flows the deposition rate goes up, but at the cost of wasting most of what we put in. These extremes correspond to the cases of the starved reactor and differential reactor we examined in our 0-dimensional treatment.

Generally speaking, semiconductor processes take place in moderately to strongly diffusion-controlled conditions, with the transfer length d larger than the ceiling height.

Details Part 3: 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 [2] :

where ν is the kinematic viscosity. Note that this has almost the same form as the scaling length d for concentration defined above: the boundary layer is just the diffusion length for momentum, as we’ve noted before. 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 this chamber at this flow. 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!

Howling and colleagues [1] have provided an explicit expression for the x- and z-velocities in a linear showerhead for the case of creeping flow (small Reynolds number):

A plot of this expression shows that it is only moderately different from the linear approximation we have used in section (2) above:

However, the radial velocity is parabolic rather than constant with z. As long as species concentrations don’t vary much with radius we can get away with ignoring this fact, but it becomes important if we need to account for the origin of streamlines within the reactor.

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References:
1: ”Plasma Deposition in an Ideal Showerhead reactor: A two-dimensional analytical solution“, A. Howling, B. Legradic, M. Chesaux and Ch. Hollenstein, Plasma Sources Sci. Technol v. 21 015005 (2012), DOI 10.1088/0963-0252/21/1/015005
2: Thin Film Deposition, D. Smith, p. 324