Massively Parallel Solution of the Inverse Scattering Problem for Integrated Circuit Quality Control

Department We developed and implemented a highly parallel computational algorithm for solution of the inverse scattering problem generated when an integrated circuit is illuminated by laser. The method was used as part of a system to measure diffraction grating line widths on specially fabricated test wafers and the results of the computational analysis were compared with more traditional line-width measurement techniques. We found we were able to measure the line width of singly periodic and doubley periodic diffraction gratings (i.e. 2D and 3D gratings respectively) with accuracy comparable to the best available experiment al techniques. We demonstrated that our parallel code is highly scalable, achieving a scaled parallel efficiency of 90% or more on typical problems running on 1024 processors. We also , made substantial improvements to the algorithmic and our original implementation of Rigorous Coupled Waveform Analysis, the underlying computational technique. These resulted in computational speed-ups of two orders of magnitude in some test problems. By combining these algorithmic improvements with parallelism we achieve speedups of between a few thousand and hundreds of thousands over the original engineering code. This made the laser diffraction measurement technique practical. 1 Parallel Computing Sciences Department, Mailstop 0441, (505) 845-8439, [email protected]. 2 Advanced Radiation Hardened CMOS Technologies Department, Mailstop 1074, (505) 844-7865, [email protected]. 1 1. Background. When an integrated circuit is illuminated by laser, an interference pattern is created in the diffracted light. Geometric properties of the chip surface can, in principle, be inferred from computational analysis of the scattered radiation; this information can then be used to monitor and improve process quality. In most conventional methods of monitoring the chip patterning processes, wafers are removed from the line following key steps and inspected using standard white light microscopy a technique that is straightforward but which yields only one-dimensional information. Other more rigorous inspection/measurement techniques used in the IC industry (e.g. scanning electron microscopy) can give 3D information but are very time consuming and destructive. With the new laser diffraction method described in this report, a chip can be tested rapidly, automatically and non-destructively at many stages of processing. When mature, this technology is expected to decrease costs, speed process flow and increase yield, and hence be of substantial economic significance. Prior to the work described in this report, The University of New Mexico (UNM) had developed a prototype of the necessary laser scattering hardware and had used this successfully to make ID measurements on chips etched with simple diffraction gratings [1, 6]. This work, however, relied for solution of the inverse problem on an analytical result which does not hold for measurement of 2D or 3D samples. In the higher dimensions the only known method of solving the inverse scattering problem was and is to solve the forward problem repeatedly and use each newly calculated field to improve the previous approximation to the true geometry. Simple estimates showed that this iterative method would clearly require supercomputing capacity, and Sandia’s experience at the Massively Parallel Computing Research Laboratory (MPCRL) indicated that use of a parallel supercomputer would likely result in a faster solution at much lower cost than use of a traditional vector supercomputer. Hence there was a natural way to cooperate on the solution. 2. Project Summary. When we began, we were primarily interested in showing that the inverse problem was tractable by any means. We believed that eventually we could produce a highly efficient parallel algorithm that would run on a supercomputer stationed at the fabrication facility. Accordingly, we were aiming at a “proof of concept” result, and our goals were to: