An aluminum film (Young's modulus 70 GPa, Poisson's ratio 0.35, linear thermal expansion coefficient 23×10⁻⁶ K⁻¹) on a silicon substrate (linear thermal expansion coefficient 2.6×10⁻⁶ K⁻¹) is cooled from a stress-free 125 °C to 25 °C. Assuming the substrate is sufficiently thick, what are the sign and magnitude of the thermal stress in the film, σ = EΔαΔT/(1 − ν)?

Answer: C. Tensile, about 220 MPa

Δα = 23×10⁻⁶ − 2.6×10⁻⁶ = 20.4×10⁻⁶ K⁻¹ and ΔT = 100 K, so σ = 70×10⁹ × 20.4×10⁻⁶ × 100/(1 − 0.35) ≈ 2.2×10⁸ Pa, about 220 MPa. On cooling, aluminum tries to shrink more than silicon, but constrained by the substrate it cannot, so the film is pulled into tension. Calling it compressive gets the sign wrong. About 143 MPa forgets to divide by (1 − ν). The film is in a biaxial stress state, constrained simultaneously in both in-plane directions, so the biaxial modulus E/(1 − ν) must be used. Thermal stresses like this cause stress migration and voids in interconnects.