On the effective coefficient of thermal expansion (CTE) of bilayer/trilayer in semiconductor package substrates

The coefficient of thermal expansion (CTE) is a material parameter used to measure the degree of expansion divided by the change in temperature. For single layer films, there exists an explicit definition and well-established technique (Thermomechanical Analysis) to measure the CTE. However, there is no explicit definition of effective CTE for multilayer films/composites composed of multiple materials with different individual CTE's. Definition and calculation of effective CTE of multilayer films/composites are still unexplored fundamental problems. Bilayer and trilayer films are major components in the semiconductor package substrate. Three different effective CTE's located at the top surface, bottom surface and mid-plane of bottom film of bilayers were derived explicitly and compared to finite element modeling. Furthermore, the effective CTE of symmetrical trilayer films was derived explicitly and compared to finite element modeling. It shows that not only the individual CTE's of constituent materials but also the ratios of thicknesses and moduli contribute to the effective CTE behavior. The theoretical and numerical study of effective CTE of bilayer and trilayer fills our knowledge gap of effective CTE of multilayer films/composites and provides better understanding of the thermal-mechanical behavior of semiconductor package substrate. The principles also provide solutions and insights to solve critical issues (i.e. warpage) induced by thermal mismatch strains.