Laser-induced stress wave thermometry applied to silicon wafer processing: Modeling and experimentation

Abstract

We present a methodology correlating the group velocity of guided plate waves to temperature in anisotropic silicon substrate. The model is developed through numerical solution and manipulation of the dispersion relations, while elastic constants and plate thickness are treated as functions of temperature. Analytical results demonstrate that adequate thermal resolution is provided by both the lowest-order antisymmetric and symmetric dispersive Lamb wave modes to serve as an effective diagnostic in a noncontact thermometry scheme applicable during rapid thermal processing of silicon waters. Validation is made through a combination of experimentation using laser-generated ultrasound in silicon wafers and analysis employing the Gabor wavelet transform to extract frequency- and temperature-dependent group velocities from the dispersive Lamb waves.