On the Methodology of Numerical Etching
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Mat. Res. Soc. Symp. Proc. Vol. 563 © 1999 Materials Research Society
discontinuous deformation analysis [8-10], is implemented to simulate a dynamic etching process and record all the changes in the mechanical fields of the Ti pattern, the insulating materials, and Si substrate. An iterative process is followed until the computed and measured deformation of the final Si diaphragm is reasonably matched. THEORY Background The numerical etching algorithm is designed to simulate the random processes of dry and wet etching. It is developed on the basis of discontinuous deformation analysis (DDA) [8], which is a powerful numerical model used for analyzing a system containing many solid bodies. The technique uses the variation principle in solid mechanics.
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thin silicon substrate
optimal thickness\ (membrane + plate)
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/ C
Figure I. Energy loss for different patterning
thick silicon substrate'--"---_J __ _po sition. ............ .o.. ........ ......... ........ ........................................... ...... . Figure 2. Deformation profile at different thickness
Successive Etching Model Etching is an irreversible process. The process of mass or energy loss from the system cannot be reversed. For a specified patterning, energy loss will be different for various methods used in the etching procedure. If we etch portions a and b instantaneously in case I and sequentially in case II as shown in Fig. 1, the energy loss in case II is approximately five times of that in case I. During the process of substrate thinning, the substrate varies from a thick plate to a thin membrane, thus, the modeling problem involves a solid/plate problem, and through combined plate plus membrane (M+P) transition stage, to a membrane problem. For a very thin diaphragm that is traditionally considered as a membrane problem, combined M+P effects still need to be considered due to the fixed-end boundary condition near the corners of the substrate window. Plate and membrane have totally different mechanisms. Bending and shearing are considered in a plate problem to support the vertical load. On the other hand, when the membrane effect governs, in-plane stress produces vertical forces due to local curvature to balance the vertical load. The plate effect is neglected in a membrane case, and a plate problem does not include the membrane effect. One may think that with a patterned thin film on top of the substrate, the thinner the substrate, the larger the substrate deformation caused by the residual stress in the thin film. However, calculations show that there exists an optimal thickness such that the diaphragm deflection reaches its maximum. This creates the highest measurable deflection profile under the fixed TGLI resolution. This optimal thickness lies in the mechanical transition zone of M+P behavior.
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Figure 3. Computed interferometric pattern for Si thickness of (a) 0.5 pin, (b) 3 pm (optimal thickness), and (c) 10 prn With the complexity of mechanical behaviors involved during the substrate etching process
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