Stoney Equation
The fundamental equation relating film stress to substrate curvature:
σ = (E_s / (1 - ν_s)) × (t_s² / 6t_f) × (1/R)
where σ = film stress, E_s = substrate Young's modulus, ν_s = substrate Poisson's ratio,
t_s = substrate thickness, t_f = film thickness, R = radius of curvature
Thermal Stress
σ_thermal = (E_f / (1 - ν_f)) × (α_s - α_f) × ΔT
where α = thermal expansion coefficient, ΔT = temperature change
Multi-Layer Stack
1/R = (6 / E_s × t_s²) × Σ(σ_i × t_i)
Sum over all layers i
Stress Types
- Tensile stress (positive): Film wants to contract, can cause cracking
- Compressive stress (negative): Film wants to expand, can cause buckling/delamination
- Intrinsic stress: From growth process (grain boundaries, defects, impurities)
- Thermal stress: From CTE mismatch during cooling