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Modeling of multi-temperature-cycle wafer curvature

A multi-temperature-cycle wafer curvature experiment was performed in order to study the temperature dependent material properties of copper films. The specimen consists of 20µm copper film electrochemically deposited on a silicon wafer. Focused Ion Beam and Electron Back-Scatter Diffraction were used to characterize the microstructure of copper films. Wafer curvature (wafer bow) measurements were performed using a multi-laser-beam wafer curvature system, in a well controlled thermal chamber and at high-speed data acquisition. A very pronounced hysteresis of the curvature-temperature curves was observed. An empirical model was developed which specifically describes the transition from elastic to plastic behavior at low strain. Following the approach of Voce and Kocks, the stress-strain curve of polycrystalline copper is modeled, where the saturation stress is considered as model parameter, and the temperature dependence of the plastic behavior is described by the activation free enthalpy of steady state creep. The model describes biaxial stress states as they occur in a wafer; the initial slope of the stress-strain curve is related to the elastic modulus of the material, and the model is applied to cyclic stress curves. It is demonstrated that in the temperature range from −50°C to 500°C the model can accurately describe the results of the multi-temperature-cycle (Multi-TC) wafer curvature experiment.