Thin Film Stress Measurement
Calculate film stress from substrate curvature using the Stoney equation. Includes thermal stress decomposition and stored elastic energy.
Substrate
Film
Measurement
Deflection is the bow of the center relative to the edge of the measured radius; positive means the film side is concave, which a tensile film produces.
Film stress
119.7 MPa
tensile
Thermal mismatch stress
-125.5 MPa
compressive, at 25 °C
Intrinsic stress
245.1 MPa
tensile
Curvature change Δκ
3.200e-3 m⁻¹
R = 312.5 m
Force per width (σ·t)
11.97 N/m
Film strain
1419 µε
Stored energy 16.98 mJ/m²
Film Stress vs Temperature
The intrinsic stress is assumed constant; the thermal part changes linearly with temperature (constant α, E, ν).
Model
σf = Es ts² Δκ / (6 (1 − νs) tf), κ = 2h / (a² + h²)
σth = Ef/(1 − νf) · (αs − αf)(T − Tdep), σintrinsic = σf − σth
ε = σf(1 − νf)/Ef, U = σf² tf (1 − νf)/Ef
Stoney's equation with the substrate's biaxial modulus (Stoney 1909; Janssen et al., Thin Solid Films 517, 1858, 2009), using the change of curvature between the two deflection measurements (spherical cap). Thermal mismatch for a film on a much thicker substrate (Freund & Suresh, Thin Film Materials, 2003).
Textbook approximation: tf ≪ ts, deflection small compared with ts, uniform equibiaxial stress, isotropic substrate. For Si(001) wafers, E = 130 GPa and ν = 0.28 give the biaxial modulus E/(1 − ν) ≈ 180 GPa.
Model and references
Textbook approximation. A standard approximation from the literature (for example paraxial, plane-wave or Gaussian-beam), valid within the range the page states. About the tiers
Stoney's equation with the substrate's biaxial modulus, for a thin film, small deflections, uniform equibiaxial stress and an isotropic substrate. The thermal-mismatch stress assumes a much thicker substrate.
- Stoney G. G. (1909). The tension of metallic films deposited by electrolysis. Proc. R. Soc. Lond. A 82, 172–175. doi:10.1098/rspa.1909.0021
- Janssen G. C. A. M. et al. (2009). Celebrating the 100th anniversary of the Stoney equation for film stress: Developments from polycrystalline steel strips to single crystal silicon wafers. Thin Solid Films 517, 1858–1867. doi:10.1016/j.tsf.2008.07.014
- Freund L. B., Suresh S. (2004). Thin Film Materials: Stress, Defect Formation and Surface Evolution, ch. 2. Cambridge University Press. doi:10.1017/CBO9780511754715