Semi-analytical modeling and experimental measurement of elastic thermal expansion and aspect-ratio effects in Cu/SiO2 vias for 3D IC integration

Thermally induced protrusion of Cu pads embedded in SiO 2 vias is important for fine-pitch Cu/SiO 2 hybrid bonding, but its dependence on via geometry remains incompletely quantified. In this work, in situ heating atomic force microscopy (AFM) measurements at 25 and 200 °C are compared with a semi-analytical axisymmetric thermoelastic model obtained by free-energy minimization. The model is linear thermoelastic and does not include plasticity, creep, or grain-boundary-mediated deformation. For nominal 9-, 3-, and 1-μm-diameter pads with H = 1.5 μm, the measured central-area protrusions used as experimental U max were 8.4, 3.7, and 2.3 nm, whereas the calculated values were 8.6, 7.1, and 2.3 nm. The corresponding measured/calculated expansion volumes were 0.4883/0.4515, 0.0228/0.0315, and 0.0018/0.0009 μm 3 . Within the prescribed displacement field and idealized cylindrical, rigid-periphery assumptions, constant-volume calculations predict a broad maximum in U max near H/R ≈ 1.2 and a broad ΔV/S maximum near H/R ≈ 0.9. Across the three tested nominal aspect ratios, the model and measurements showed the same overall trend, although quantitative agreement depended on pad size and the selected metric. Finite-element calculations further benchmark the assumed displacement field, reproduce the overall aspect-ratio trends, and highlight the effects of SiO 2 compliance and surrounding constraints.