TITLE: Dynamics of Particle Removal by

Supercritical Carbon Dioxide

AUTHOR(S): S. M Chitanvis

C. W. Patterson

W. D. Spall

K. Laintz

SUBMITTED TO: Cleaning with Supercritical Fluids

Ed. by S. Sawan and T. Mchurdy

**ABSTRACT**

We have investigated the possibility of cleaning semiconductor wafers
contaminated by particles by flowing supercritical carbon dioxide over
them. The hydrodynamics of turbulent flow of supercritical carbon
dioxide over a flat plate was calculated. Through these calculations it
is proposed that the drag on a particle sitting on a flat plate in the
viscous sublayer is capable of rolling submicron-sized spheres off the
wafer at sufficiently high velocities.

**INTRODUCTION**

Contamination of semiconductor wafers during manufacturing is a common
occurrence. Particle adhesion cannot always be prevented in the
manufacturing stage, making it necessary to clean the wafer in a
subsequent stage. As feature sizes on integrated circuits become
smaller, the need for ensuring particle removal increases. In the next
generation of semiconductor circuits, it is expected that feature sizes
of less than a micron will be attained. It will therefore become
imperative that all particles greater than about a tenth of a micron be
removed from semiconductor wafers.

Several schemes exist for cleaning wafers, such as an ultrasonic method
(1), high pressure jet scrubbing (2) and carbon dioxide snow-jet (3),
among others. The use of air to roll away particles has been
investigated previously by Bhattacharya and Mittal (4) and by Zimon (5).
The dynamics of liquid jets for cleaning has been studied recently by
Gim et al (6). Here we investigate the possibility of flowing
supercritical carbon dioxide over a dirty wafer in order to roll away
spherical particles. There are two advantages of using supercritical
carbon dioxide over air. The first advantage is that the density of
supercritical carbon dioxide is much larger than that of air, thereby
dramatically increasing the drag on the particle. The second is the
ability of supercritical carbon dioxide to dissolve organic material.
Since some of the contaminant particles are indeed organic in nature,
supercritical carbon dioxide will aid in loosening the adhesion between
such particles and the surface on which they are located. Furthermore, a
liquid such as water is not an option for cleaning semiconductor wafers,
as it would tend to hydroxylate the surface of the wafer and damage it
in other ways as well.

The basic idea behind using flow to move particles off surfaces is the
formation of a boundary layer near the surface. Within this boundary
layer, there is a shear in the velocity field, leading to a stress which
rolls particles away from any given position. As one might guess
intuitively, the wall shear in the case of turbulent flow is much
greater than in laminar flow. In laminar flow, the boundary layer is
relatively thicker, allowing the velocity to change gradually to its
stream value. In the case of turbulent flow, the viscous sublayer which
develops right next to the wall is much thinner, causing a much more
abrupt change in the velocity field, thereby setting up a larger wall
shear. The velocity profiles for turbulent and laminar flow are compared
in Figure 1 as a function of distance from the wall. It can be seen from
the figure that laminar flow shows a gradual velocity change while that
change is abrupt for turbulent flow. Therefore, turbulent shear is much
larger than that present in laminar flow. We must note however, that the
viscous sublayer displayed for the case of turbulent flow in Figure 1
must not be interpreted too literally as a laminar sublayer. While this
layer may be relatively quiescent compared to the turbulent core of the
flow, there is still turbulent motion within this sublayer.

In our treatment of particle removal we shall make a number of
simplifying assumptions. We shall assume that the force of adhesion is
only due to Van der Waals force of attraction. We shall not consider
other possible forces such as electrostatic interactions or gravity. We
shall also assume that the surface is smooth. In other words, the
roughness of the surface is due to randomness at the atomic level. This
means that over the width of the particle, the surface height varies by
a few angstroms at most. This assumption allows us to readily determine
the force of adhesion of glass particles on a silicon wafer. In this
paper, we have not investigated how the particles came to be attached to
the wafers (7).

We utilize the physics of rolling particles on a surface as developed by
Bhattacharya and Mittal (4). Our treatment differs from that of
Bhattacharya and Mittal (4) in that we provide a more detailed
description of the turbulent boundary layer which is formed in the
steady state when a fluid flows over a flat plate (5). The drag force we
use is consistent with the treatment of Gim et al (6).

In section 2 we discuss the properties of turbulent flow and, in
particular, the flow velocity as a function of distance from the wall in
the turbulent regime-the so-called law of the wall. In this respect we
believe our treatment of particle removal is unique. We discuss the
magnitude of adhesive forces in section 3. In section 4 we apply our
treatment of boundary layer flow to supercritical carbon dioxide and we
calculate the requirements for spherical glass particles to be removed
from a smooth silicon surface using supercritical carbon dioxide.

**THE BOUNDARY LAYER IN TURBULENT FLOW OVER A PLATE**

Turbulent flow over a flat plate is characterized by three regions (8):
(a) a viscous sublayer often called the laminar sublayer, which exists
right next to the plate, (b) an adjacent turbulent boundary layer, and
(c) the turbulent core. Viscous forces dominate inertial forces in the
viscous sublayer, which is relatively quiescent compared to the other
regions and is therefore also called the laminar sublayer. This is a bit
of a misnomer, since it is not really laminar. It is in this viscous
sublayer that the velocity changes are the greatest, so chat the shear
is largest. Viscous forces become less dominant in the turbulent
boundary layer. These forces are not controlling factors in the
turbulent core.

In the supercritical region, the kinematic viscosity of carbon dioxide
is D-10-3 cm2 s-l, (9) so that even for a low flow speed u of 100 cm/s,
the Reynolds number Re = u s/D is \~105, s being the typical length
scale chosen here to be \~1 cm. At such Reynolds numbers we expect the
flow of supercritical carbon dioxide over a flat plate to be turbulent
at least in the vicinity of the plate. We use standard empirical laws of
turbulent flow to get the velocity distribution both near and far from
the wall. The velocity in the laminar boundary layer grows linearly and
connects in a continuous fashion with a logarithmic velocity profile in
the turbulent boundary layer (10):

Eq. (2.1)

where t is the wall shear, r is the fluid density at the wall, D is
diffusion constant, u\* is a velocity called the frictional velocity,
and k, c and B are empirically determined dimensionless constants.

The transition between the laminar region and the turbulent region is
made by demanding continuity, which yields a self-consistent equation to
be solved:

Eq. (2.2)

where Rc = ycuc/D. The solution is Rc = 114. We can use this value to
estimate the stream velocity required to develop a boundary layer of
thickness yc = d. For d = 100 microns, we get uc \~ 5 cm/s, using D\~
5.4 10-4 cm2/s at T=325°K. From Eq. (2.1), u\* = uc/(Rc)1/2 \~ 0.5 cm/s.
Now, there exists a relation between the frictional velocity and the
stream velocity U viz.,

Eq. (2.3)

where f is Fanning's friction factor \~ 0.04 which finally yields U \~
13 cm/s. When we examined these values with detailed numerical
calculations, it turns out that the estimates of the velocity required
to achieve a certain thickness of the viscous sublayer is quite
accurate, as long as realistic estimates of the density and diffusion
constant are used.

We notice from Eq. (2.1) that the wall shear stress t is constant within
the laminar boundary layer, and for this case is \~ 0.25dyne/cm2. As far
as calculation of the wall shear stress for a known stream velocity is
concerned, we point out that the use of Fanning's factor alone suffices
to determine its value, and we do not need all the formulas stated
above.

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.

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 \~ uc3/d (where uc 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 r k \~ 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 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 Fa 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, Fs must be greater than the force
of friction Ff 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 Tc = 31° C, and the critical pressure
Pc = 73.8 bars. In Figure 4 we plot the viscosity of supercritical CO2
as a function of temperature (15), varying pressure parametrically. 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)). 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 Tc and Pc, 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.

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.

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.

**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**

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(4) Zimon, A., Adhesion of Dust and Powder, p. 252, Plenum Press, N.Y.,
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(5) Gim, R, Lesniewski, T., and Middleman, S., Particles on surfaces,
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(6) Vatistas, N., "Particle Adhesion to Surface under turbulent flow
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(7) Stanisic, M.M., The Mathematical Theory of Turbulence, p. 63,
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(9) Amsden, A.A., O'Rourke, P.J. and Butler, T.D., KIVA-II: A program
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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.