Modeling and controlling of warpages and layout-dependent local-deformations are challenges to overcome to realize 3D stacking of dies with through-silicon vias and micro-bumps. Dies larger than about 500 mm2 are now being used for high performance computing, and large cylindrical warpage of the die and local die surface deformations can greatly affect the yield and reliability of the stacked dies. We have analyzed and modeled large cylindrical warpage with layout-dependent local-deformations by using a combination of an analytical buckling model, curvature measurements of dies with simple layout patterns, and finite element analysis (FEA) simulation. The effects of warpage on the buckling behavior were roughly estimated by using an analytical model based on non-linear plate theory and were precisely calculated using FEA simulation. The model was applied to actual dies by measuring the curvature of simple structures and by homogenizing complicated device structures into simple elastic films. Finally, die warpage with layout-dependent local deformations was modeled using a patchwork of elastic film on a Si substrate. This approach was tested on dies with a redistribution layer (RDL) structure and a 50-μm-thick Si substrate. The simplified FEA simulation results and experimentally measured deformations matched well. The RDL consisted of elastic materials (polymer) and plastic materials (metal interconnections) so this method can be applied to other structures such as multilayer interconnects with LSI or bump structures. It can be used to calculate the deformation of a large die during the design phase, which will enable die warpage and local deformations to be managed by optimizing the device layout. This will improve yield and reliability in 3D fabrication.
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