StoneyGleichung

The Stoney Gleichung (Stoney equation) is a relation in thin-film mechanics that links the residual stress in a deposited film to the curvature of the substrate on which the film sits. Named for G. G. Stoney, who derived the relation in 1909, it provides a practical method for stress metrology in microelectronics and coatings.

In its standard form for a uniformly stressed thin film on a thick, elastic substrate, the equation is written as

sigma_f = (E_s h_s^2) / (6 t_f) · (1/R),

where sigma_f is the biaxial stress in the film, E_s is the Young’s modulus of the substrate, h_s is the substrate thickness, t_f is the film thickness, and R is the radius of curvature of the substrate after film deposition (with the curvature indicating the bending direction). The expression assumes t_f is much smaller than h_s and that deflections are small, the substrate is elastically isotropic, the film adheres uniformly, and the film stress is uniform across the area considered.

Applications of the Stoney equation include measuring residual stresses during film deposition, annealing, or processing by monitoring wafer curvature with optical or laser-based methods. It is widely used in semiconductor fabrication, coating technology, and MEMS to assess mechanical integrity and performance.

Limitations and generalizations: the original form is less accurate for thick films, anisotropic substrates, or nonuniform stresses. For multilayer or complex structures, modified or generalized versions of the equation are employed, sometimes incorporating Poisson’s ratio, multilayer geometry, or nonuniform curvature.