A compact analytic model of the strain field induced by through silicon vias
Journal
IEEE Transactions on Electron Devices
Journal Volume
59
Journal Issue
3
Pages
777-782
Date Issued
2012
Author(s)
Jan, S.-R.
Chou, T.-P.
Yeh, C.-Y.
Liu, C.W.
Goldstein, R.V.
Gorodtsov, V.A.
Shushpannikov, P.S.
Abstract
The thermoelastic strains are induced by through silicon vias due to the difference of thermal expansion coefficients between the copper ( ~ 18 ppm / °C) and silicon ( ~ 2.8 ppm /°C) when the structures are exposed to a thermal ramp in the process flow. A compact analytic model (Bessel function) of the strain field is obtained using Kane-Mindlin theory, and has a good agreement with the finite-element simulations. The elastic strains in the silicon in the radial direction and angular direction are tensile and compressive, respectively. The linear superposition of the analytic model of a single via can be used in the multi-via configuration. Due to the interaction of vias, the slightly larger errors of strain occur between the two close vias when the linear superposition is used.
Type
journal article