Quantitative Modelling of Deposition of Airborne Molecular Contamination
Authors: Mertens Paul W. 
Published: February 2020

During the fabrication, micro-chips circuits are very sensitive to all kinds of contamination present in the environments fluid in contact with the wafers [1].
Therefore the fabrication is done in clean rooms and makes use of clean mini-environments (e.g. FOUPs). Historically such clean environments contain an extremely low concentration of particulate contamination, obtained by intensive filtering through different types of particle filters including HEPA filters.
In contrast, the air is not purified in terms of molecular, volatile contaminants, except locally inside some specific process equipment.
Measurement in state-of-art clean rooms reveal the presence of many molecular contaminants coming from different sources (environmental pollution in the intake air, but also from components and materials in the clean room construction or equipment and from process steps that make use of volatile chemicals).
The dominant species are organic compounds comprising a benzene ring with various ligands.
The relative molecular concentration is on the order of 1 ppb, corresponding to a molecule volume concentration, C, = 1x1010 molec./cm3.
A question that typically arises in this context is that of how clean the air or gas ambient needs to be.
This study addresses this issue by presenting a simple but quantitative model, built from first principles, to establish the relation between the amount of contamination in the air and the surface concentration of the contaminants that deposits on the wafer as a function of the exposure time and other relevant parameters.
The transfer of contaminants from the air ambient to the surface occurs in 2 steps.

Surface reaction: at the surface adhesion of molecules takes place

Bulk transport in the air: the contaminants that have deposited are replenished from the volume of the air towards the surface

At the surface, the initial adsorption flux, ja, is given by the the thermal bombardment flux (= arrival flux) multiplied with the sticking probability (hi) of species, I, as shown in equation (1) (blue dash-dot line).
Where:

v
th= average thermal velocity (air: 470 m/s)

Ci
= volumetric density of species i in air ambient adjacent to the surface.

The hi = sticking coefficient of species i on the surface, ranges typically from 10-6 to 1 [2,3].
When the resulting surface concentration of species i, si , increases, the reverse process, desorption, will occur simultaneously.
A simple mass balance shows that for a case with a sticking coefficient approaching 1 and with a molecule volume concentration of 1x1010 molec./cm3 , (i.e. 1 ppb) a surface concentration of 1x1012 molec./cm2 on an infinitely large flat surface, would require all the contaminants from a volume contained in a layer with a thickness of 1 m adjacent to the surface. This simple assessment illustrates that the supply and transport from the ambient is an important limiting phenomenon.
Below the focus is on estimating the potential maximum contamination achieved, assuming a sticking probability = 1. This provides an estimation for the upperlimit for the case with a lower sticking probability.
The transport of contaminants in the air occurs by convection and by diffusion.
In clean room ambient, laminar flow patterns are quite typical. One can approximate the transport by defining a static boundary layer adjacent to the wafer surface (with a thickness L
BL) in which the (lateral) velocity is approximated as 0. Outside the boundary layer the flow is assumed to be intense enough to maintain a constant contamination concentration C
i,
∞.
During an initial transient phase contaminants diffuse from within the static boundary layer to the surface from a layer proportional to the diffusion length SQRT(Dt) (black dashed line).
As soon as the diffusion length reaches the edge of the static boundary layer the supply flux of contaminants reaches a constant value and is determined by diffusion over the static boundary layer thickness (red solid lines).
Combining the initial transient and the steady state supply yields equation (2).
The values thus obtained for C = 1x1010 molec/cm3 are shown in figure 1 for 3 different values of the boundary layer thickness (LBL).
Fig 1 Black and red lines represent the surface contamination as a function of time for a sticking coefficient of 1, C= 1x1010 mol/cm3 and for a static boundary layer thickness of 0.1, 1 or 10 cm.

References

Saga and T. Hattori, J. Electrochem. Soc 144, p. L253 (1997).

Hendriks, Microelectronic Engineering, 25, p. 185 (1994).

Den et al., J. Electrochem. Soc 153, G149 (2006).

Figure 1