Non-axisymmetric models for light scattering from a particle on or near a plane surface
Introduction
The foregoing miniaturization of microchips come to be a real factor of a technical progress in semiconductor manufacturing. Being a first-step chain for the chip manufacturing, silicon wafers demand an extra care of their purity, which is determined by certain defects of real wafers: contaminating micro-particles, pits, subsurface bulks (crystal originated particles), scratches, etc. Therefore, the control of silicon wafer surface defects is of prime significance in semiconductor manufacturing. The purity monitoring is being carried out through optical surface scanners. The design of the surface scanner includes a laser source of light and collector's system to capture the intensity scattered by a micro-particle. Problems of surface scanner's operating include both false counts of non-existing contaminants detected, and missing counts of real defects not detected by the instrument. As the minimum line width of wafer structures continues to decrease, the importance of proper detection and characterization of scattering features grows. One response to this problem is the use of increasingly sophisticated models to predict light scatter signals using mathematical modeling and computer simulation analysis.
Several studies have addressed the scattering problem by spherical micro-particles 1, 2, 3, 4. In 5, 6, 7, 8the scattering by axisymmetric particles on a silicon substrate has been investigated in the framework of the discrete sources method and the T-matrix method. The analysis was restricted to an axisymmetric model by assuming that the particle symmetry axis coincides with the normal direction to the plane interface. Unfortunately, real contaminating particles are not axisymmetric. In this context the elaboration of mathematical models for such kind of applications appears to be of great interest in the design of surface scanners.
In the present communication, we extend our previous models [8]to a non-axisymmetric structure consisting in a nonspherical particle deposited on a plane surface. The aim of our analysis is to investigate the influence of the micro-particle shape on the scattering characteristics.
Section snippets
Discrete sources method
The geometry of the scattering problem is shown in Fig. 1. An arbitrarily shaped particle with a smooth boundary S and interior Di is situated on a plane surface Σ. The upper half-space corresponding to the ambient medium is denoted by D0, while the lower half-space corresponding to the substrate is denoted by D1. Let us introduce a rectangular coordinate system Oxyz by choosing the origin O at the tangent point between the particle and the substrate. The z-axis coincides with the normal
Numerical simulations
In this section we investigate from a computational point of view the influence of particle shape on the response of a surface scanner. Usually, the diameter of the contaminating particles is much smaller than the diameter of a laser beam. In this context the assumption that the exciting field is a polarized plane wave seems to be realistic. In addition, since the diameter of the particle is smaller than the exciting wavelength, we restrict our analysis to the scattering by Rayleigh particles.
Conclusion
The discrete sources method and the T-matrix method have been extended to analyze the scattering characteristics of a non-axisymmetric structure consisting in a non-spherical particle and a plane surface. The computer results have shown a visible dependence of the scattering characteristics on the particle shape especially for particles with high refractive index. These results needs to be accounted in the design of surface scanners.
Acknowledgements
This research was supported by DAAD (ref. 325, N A/99/09581).
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