TWENTY-FOURTH MATHEMATICAL AND STATISTICAL MODELING WORKSHOP FOR GRADUATE STUDENTS

Thin films are widely used in various devices including micorelectronics and microelectromechanical systems. For this reason, it is important to accurately identify their properties. A Laser Deflectometer (LD) measures the warpage of thin films as temperature varies. To control the temperature, a thermal enclosure is needed. The top quartz plate of the thermal enclosure affects the path of the beam; hence, it distorts the warpage measurement. In this report, we provide two physical models: one relies on geometrical analysis, whereas the other is developed using operator-based modeling to correct the wafer warpage measurement. Also, an efficient statistical model based on experimental data predicts the true warpage given the LD measurement. We then analyze the statistical properties and validation using new datasets. In conclusion, we summarize our analysis and provide recommendations. 1 Problem Description Microfabricated thin films are key constituents of coatings, microelectronics, and microelectromechanical systems (MEMS). Thermomechanical behavior of thin films is critical to the functionality and reliability in these applications. Thin films are merely nanometers to micrometers thick and their thermomechanical material properties differ from those observed in bulk dimensions, which is sensitive to fabrication processes. Thin films are fabricated on thicker, functional substrates; removing the films from their substrates can destroy the films. This presents a challenge in characterizing the thin film thermomechanical properties, as isolating the thin films for testing may be impractical and adding fabrication steps to produce film samples free of the substrate may alter the film properties. Therefore, a method to identify thin film properties in situ is needed. The integrated circuit (IC) industry has deployed a non-contact, laser-based profilometer that is tailored to measuring warpage of thin films on standard substrates (wafers) at varying temperatures. The profilometer determines warpage by measuring how a laser beam impinging on the film surface reflects onto a position-sensitive photosensor as it scans across the film. These measurements are readily realized at ambient temperatures; however, to measure warpage at controlled, non-ambient temperatures, the wafer must be enclosed to maintain thermal stability. The laser and photosensor lie outside the thermal enclosure, so the laser light must enter the enclosure before reaching the film and exit the enclosure after reflecting off the film during the temperature-controlled program. The alteration of the beam path through a quartz window in the enclosure distorts film warpage data. Traditionally, the IC industry has disregarded distortion of warpage data by the enclosure, since it assumes that film stress, a primary quantity of interest derived from differential warpage data, is unaltered. Yet, highly desired thin film mechanical properties can be identified using the absolute warpage data, provided effects of the enclosure are accommodated. Modifying hardware or employing an alternate technology that could be adopted by the IC industry may be a long-term solution, but such changes are impractical. Instead, there is an immediate need to use existing profilometer functionality to ascertain non-distorted wafer warpage. This project aims to develop computational techniques that can effectively 1Department of Mathematics, University of Texas El Paso 2Department of Mathematics, Southern Methodist University 3Department of Mathematics, University of Central Florida 4Department of Mathematics, Virginia Tech 5Department of Mathematics, University of Michigan 6Applied Mathematics, North Carolina State University 7Sandia National Labs 8Department of Mathematics, North Carolina State University