C-ACT: Applied Chemical Technology

In order to get the entire velocity profile, we need to use realistic values of the density and diffusion constant as functions of pressure and temperature, and then solve the above equations numerically for the velocity in the two different regions and match them at the edge of the boundary layer. We have given an example of such a calculation in Figures 2 and 3. These two figures are calculations done at 300 bars and 400 bars respectively. What these figures show is basically that there is a greater sensitivity to the stream velocity rather than temperature. We also learn from these figures that for velocities on the order of a few cm/s, the thickness of the boundary layer is about 100 microns.

Figure 2. Velocity profiles for turbulent and laminar flow as a function of distance from the wall.

Figure 3. Velocity profiles for turbulent and laminar flow as a function of distance from the wall.

One might be tempted to conjecture that this layer being turbulent (albeit less turbulent than the regions further away from the plate) will have an eddy swooping down once in a while to somehow "pick" a particle off the surface. We believe this is not very likely, since the sublayer is on the order of a few microns for the velocities under consideration, while the particles we would most like to remove are around less than a micron in radius. In fact, one may use standard estimates for turbulence variables such as the dissipation (loss of kinetic energy per unit time), e ~ u_c³/d (where u_c is a measure of the typical velocity in that region and d is the thickness of the viscous sublayer), and the turbulent kinetic energy (mean square velocity fluctuations), k ~ (ed)^2/3, to obtain a kinetic "pressure" due to turbulence rk ~ 10^-3 dynes/cm², r being the characteristic density. Multiplying this pressure by the cross-sectional area of a micron-sized particle yields a force of ~ 3.14 x 10^-13 dynes. This force due to turbulence may be compared to the force of adhesion (see Eq. (3.2) in the next section), which is ~ 0.0052 dynes for a micron sized particle. It is therefore clear that turbulence cannot scoop up a particle and move it along the surface.

On the other hand, it is in the viscous sublayer that the velocity changes are largest, so that the shear is greatest here. It may be possible for the shear stress in this region will be able to move a particle. We shall develop this idea in the next section.

ADHESIVE FORCES

As mentioned in the introduction, in addition to assuming a spherical particle and a smooth surface, we shall assume that electric charges are not present in the problem, and the effects of gravity will be ignored as well.

The adhesive force between a neutral particulate contaminant and the wafer is expected to be due to the attractive Van der Waal’s interaction between molecules (11). This is a macroscopic force found by averaging over the force between all the molecules of a particle and the neighboring surface. For a spherical particle sitting on a flat wafer, it is known that surface roughness will cause the mean distance of separation between the particle and the wafer to be nonzero. The attractive force between these two entities acts along the normal between the sphere and the wafer, and is given by:

Eq. (3.1)

where A is Hamaker’s constant (~ 10^-19 J), r is the radius of the spherical particle, and z is the distance of separation. Experimentally, it is found that h ~ 10 eV. In our theoretical study in this paper, A or h need to be considered as empirical parameters which will change depending on the nature of the particle and the surface upon which it is placed. The dielectric properties of the intervening fluid viz., supercritical carbon dioxide will change the strength of the interaction between the particle and the surface. Empirically, the separation distance for a smooth surface is found to be ~ 0.4 nm, because at distances smaller than this the interaction becomes repulsive. Using these numbers, we get:

Eq. (3.2)

where the adhesive force F_a is in dynes, r is in cm, and the negative sign indicates an attractive force. For r = 1 micron, the adhesive force is 0.00521 dynes.

The force due to the shear in the viscous sublayer on a particle lying on a surface, is analogous to the formula for Stoke’s drag (12):

Eq. (3.3 )

One sees immediately from this equation that when U ~ 1000 cm/s, the shear force will exceed the adhesive force for r = 1 micron. This puts movement of the particle due to the shear force in the realm of reality.

In general, for the particles to move, F_s must be greater than the force of friction F_f which is given by the following equation:

Eq. (3.4)

where µ is the coefficient of friction. The shear force is proportional to the square of the radius of the particle while the force of friction grows only linearly with the radius. Thus, there will be a radius below which the shear force can no longer overcome the force of friction. The radius of the smallest particle which can be moved is therefore given by:

Eq. (3.5)

The idea of utilizing a coefficient of friction is not new, having been proposed and tested for the cleaning of flat plates with air at least two decades ago by Zimon (5). Similar ideas have been proposed for the cleaning of wafers/discs by spinning them (13). If the particles that need to be removed are charged, the force of adhesion may be electrostatic in origin (14).

There appears to be empirical evidence that the approach outlined above is applicable to the sliding of particles (5). For cases of interest to us, the particles appear to roll, rather than slide off the wafer (6). As far as we are aware, well-characterized experiments have not been performed for the case of supercritical carbon dioxide. We now turn to the theory of rolling particles off surfaces. We use the treatment of Bhattacharya and Mittal (4) to compute the radius of the smallest particle that can be moved due to the torque exerted on the particle due to the wall shear. As the particles sit on the surface, there will be a very small deformation x₀ caused by the adhesive force between the particle and the surface. This deformation is given in the following equation:

Eq. (3.6)

where E is the modulus of deformation ~ 8.0 1011 dynes/cm² for silica. When rolling occurs, we may equate the torques acting on the sphere due to the vertical force of adhesion and the horizontal force of drag to find

Eq. (3.7)

Using Eq. (3.1), Eq. (3.3) and Eq. (3.6), we obtain:

Eq. (3.8)

Thus rolling yields a U^-4/3 dependence for rmin This may be contrasted with the inverse quadratic dependence obtained from Eq. (3.5), for sliding particles. In fact, we suggest that experimental data on rmin could be used to differentiate between the two different mechanisms discussed here, and indeed to verify whether such predictions are correct.

In the next section, we shall specialize these formulas to the case of supercritical carbon dioxide.

SUPERCRITICAL CARBON DIOXIDE

We pointed out in the introduction that using supercritical CO₂ as a removal agent would be more advantageous than air, and this is borne out by Eq. (3.8). This equation shows that the radius of the smallest particle that will be removed, rmin, has a r^-2/3 dependence, implying that for a given flow velocity, supercritical CO₂ with a density of ~1 g/cc will remove much smaller particles than air. Of course, one could use air and increase its flow velocity appropriately in order to overcome the deficit caused by its low density. One would have to make certain that the air was free of contaminants which could damage the wafer being cleaned. On the other hand, supercritical CO₂ has the inherent advantage of being able to dissolve organic compounds, which should help loosen organic particle contaminants on the wafer surface. Thus, besides flow characteristics, one needs to consider that chemistry could make a difference in this process. The Hamaker constant A which appears in the adhesive force defined by Eq. (3.1) is expected to be smaller for the case of supercritical CO₂ flowing over particles containing organic material and placed on a wafer. Supercritical CO₂ is also a drying agent, and this may be of some use in the cleaning process as well.

The critical temperature of CO₂ is T_c = 31° C, and the critical pressure P_c = 73.8 bars. In Figure 4 we plot the viscosity of supercritical CO₂ as a function of temperature (15), varying pressure parametrically.

Figure 4. Viscosity of supercritical CO2 as a function of temperature.

In Figure 5, we plot the density of carbon dioxide over the same range of temperature and pressure (9). In order to maximize the force on particles it is desirable to increase the density (see Eq. (3.7)).

Figure 5. Density of carbon dioxide over same range of temp/press.

There is only a weak dependence (see Eq. (3.7)) of the particle removal efficiency on the viscosity, through the Fanning factor. The Fanning factor changes very little as the Reynolds number is changed. In any event, we can optimize the process by choosing to work in a regime where the viscosity is relatively low. Accordingly, we choose a temperature of 325°K, and a pressure of 300 bars based on Figures 4 and 5. Lowering the pressure would not only decrease the density but bring the operating point closer to the critical pressure, which is not desirable. Increasing the pressure would increase the viscosity. Raising the temperature decreases the density, while lowering it would not only increase the viscosity, but again bring us uncomfortably close to the critical temperature. This is due to dramatic density changes as a function of temperature and pressure near the critical point. We recognize in this manner that there is not a truly optimal operating point. What we obtain is a range of parameters defining an operating regime whose boundaries are defined by the proximity to T_c and P_c, and within which one can perform cleaning experiments.

In Figure 6 we show the thickness of the boundary layer as a function of the stream velocity of supercritical CO₂ for temperatures of 325°K (solid line) and 375°K (dashed line) at a fixed pressure of 300 bars. In this regime the density of CO₂ is ~ 0.94 g/cc and its viscosity is ~ 7.4x10^-4 g/cm^-s. The thickness of the laminar sublayer is seen to depend inversely on the stream velocity. This figure indicates that for a stream velocity of 200 cm/s, the boundary laver thickness is about 2 microns.

Figure 6. Thickness of the boundary layer as a function of the stream velocity of supercritical CO2 for various temperatures at fixed pressure.

In Figure 7, we plot the radius of the smallest particle that can be rolled as a function of the stream velocity for temperatures of 325°K (solid line) and 375°K (dashed line) at a fixed pressure of 300 bars. We notice first of all from this plot that a difference of 25°K in the operating temperature causes a change by almost a factor of two in the radius of the smallest particle which can be moved. It is clear that it is more efficient to work at the lower of the two temperatures that are displayed. Such plots allow us to design experiments with a goal of removing particles larger than some critical value. As pointed out in the introduction, the critical size of particle contaminants is expected to be of the order of 0.1 micron for the next generation of semiconductor wafers. Figure 7 shows that the stream velocity of supercritical CO₂ required to remove particles of this size is about 200 cm/s at 375° K, and the required stream velocity is about 100 cm/s at 325° K. Such velocities are attainable with currently available pumps.

Figure 7. Radius of smallest particle as a function of stream velocity.

To compare the cleaning characteristics of supercritical carbon dioxide with that of air, we have plotted in Figure 8 the radius of the smallest particle (in the range of a few microns) that can be removed with air at 1 atmosphere and a temperature of 300 K. It shows that even for a relatively high velocity of 1000 cm/s, the minimum radius is ~ 0.2 microns. This may be compared to 0.005 microns for supercritical CO₂. Requirements of such fast flow for air will probably force the use of jets. This raises an additional complication of having to install a mechanism to "sweep" the wafer clean. The sweeping motion will most certainly lead to additional contamination from lubricated joints, etc. Such complications do not arise for the case of supercritical C0₂, given the relatively low velocities needed.

Figure 8. Radius of smallest particle that can be removed with air at 1 atmosphere and 300 K.

CONCLUSIONS

We have shown that particles can be theoretically rolled off semiconductor wafers using a turbulent flow of supercritical CO₂ over them. Flow speeds of a few hundred cm/s will be required to remove particles less than a tenth of a micron in radius. The relative merits of using supercritical carbon dioxide over air include the use of lower flow velocities with supercritical CO₂, it is pointed as well as an advantage in terms of chemistry over air. Supercritical CO₂ dissolves organic molecules, implying that it can loosen the adhesion between organic contaminants (particles) and the semiconductor wafer.

REFERENCES

1. Mayer, A. and Shwartzman, S., J. Electronic Mater., 8, 855 (1979).
2. Musselman, R.P. and Yarbrough, T.W., J. Environmental Sci., 51, (1987).
3. Archibald, L.C. and Lloyd, D., Particles on surfaces, Vol. 3, Ed., K. L. Mittal, Plenum Press, N.Y., (1988).
4. Bhattacharya, S. and Mittal, K.L., Surface Technology, 7, 413, (1978).
5. Zimon, A., Adhesion of Dust and Powder, p. 252, Plenum Press, N.Y., (1966).
6. Gim, R, Lesniewski, T., and Middleman, S., Particles on surfaces, Detection, Adhesion and Removal, ed. K.L Mittal (1995).
7. Vatistas, N., "Particle Adhesion to Surface under turbulent flow conditions," in Particles on Surfaces, Vol.3, Ed., K. L Mittal, Plenum Press, N.Y., (1988).
8. Stanisic, M.M., The Mathematical Theory of Turbulence, p. 63, Springer-Verlag, N.Y. (1988).
9. Ely, J.F., NIST CO2PAC, Proceedings of the 65th Annual Convention of the Gas Processors Association, San Antonio, TX, (1986).
10. Amsden, A.A., O’Rourke, P.J. and Butler, T.D., KIVA-II: A program for chemically reactive flows with sprays, Los Alamos National Laboratory Report LA-11560-MS, Appendix B, (1989).
11. Israelichvili, J.N., Intermolecular and Surface Forces, p. 128 Academic Press, (1991).
12. Goldman, A.J., Cox, RG. and Brenner, II., Chem. Eng Sci., 22, 637 (1967).
13. Reed, J., "The Adhesion of small particles to a surface," Particles on Surfaces, Ed. by K.L Mittal, vol., 7, p 3 Plenum Press, N.Y. (1988).
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FIGURE CAPTIONS

Figure 1. Velocity past a flat plate as a function of distance from the surface of the plate under turbulent and laminar flow conditions using a flow (stream) velocity of 11 cm/s. Notice the gradual change in the laminar flow (dashed line), compared to the turbulent case (solid line). It follows that the shear in the turbulent case, near the plate is much larger than in the laminar case.

Figure 2. Velocity profiles for supercritical CO₂ calculated at 300 bars of pressure. The solid line corresponds to 325°K and a viscous sublayer thickness of 100 microns, the dashed line corresponds to 375°K and a viscous sublayer thickness of 100 microns, and finally the dot-dashed line corresponds to 325°K, but a viscous sublayer thickness 200 microns.

Figure 3. This figure is similar to Figure 2, except that the pressure used is 400 bars. Notice the correspondingly higher velocities required to attain the same viscous sublayer thicknesses (100 microns and 200 microns).

Figure 4. Viscosity as a function of temperature for supercritical CO₂ for three different pressures.

Figure 5. Density as a function of temperature for supercritical CO₂ for three different pressures.

Figure 6. The relation between the stream velocity and the thickness of the viscous sublayer for 325°K (solid line) and 375°K (dashed line) at a pressure of 300 bars. The thickness ranges from about 1 micron for 350 cm/s to 10 microns for 50 cm/s.

Figure 7. The theory of rolling developed for turbulent flow conditions can used to calculate (at 300 bars) the radius of the smallest particle that can be moved as a function of the stream velocity for 325°K (solid line) and 375°K (dashed line). Notice the sensitivity to temperature. Velocities of 100 cm/s to 200 cm/s are needed to move particles about a tenth of a micron.

Figure 8. In order to compare the cleaning efficiencies of air and supercritical CO₂, we have displayed in this figure the radius of the smallest particle that can be moved with air at one atmosphere and 300°K, for velocities comparable to those used in the previous figure. Notice that the velocities required to move the particles with air are much higher than in the case of supercritical CO₂.