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Thin film stress and wafer bow calculator (Stoney equation)

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KEY POINTS

The Stoney equation σ = [Es/(1−νs)]·ts²/(6tf)·Δ(1/R) gives film stress from the change in wafer bow before and after deposition, without knowing the film's elastic modulus. Tensile stress bows the wafer concave and compressive stress convex. On a 300 mm wafer, a 500 nm film at 1 GPa produces roughly 300 µm of bow.

Film stress from measured bow (Stoney equation)

SubstrateSubstrate thickness ts (µm)Film thickness tf (nm)InputScan length (chord) (mm)Bow before deposition (center up = positive) (µm)Bow after deposition (center up = positive) (µm)

Film stress σf55.3MPa

Type of stressTensile (wafer bows concave)

Stress × thickness (film force)27.7N/m

Change in radius of curvature from deposition-653.3m

Substrate bow is evaluated as the difference before and after deposition (initial bow subtracted). Enter convex (center up) as positive. Stress is shown as positive for tensile and negative for compressive.

Thermal stress (thermal expansion mismatch when cooling from deposition temperature)

Film materialTemperature change ΔT (cooling is negative) (°C)

Thermal stress σth177.3MPa

Bow of a 300 mm wafer with 500 nm of film at this stress-55.2µm

Calculated with a film thermal expansion coefficient of 3.3 ppm/K, substrate 2.6 ppm/K and a film biaxial modulus of 325 GPa. Real film stress adds intrinsic stress, set by deposition conditions, to this thermal stress.

Stress and bow at a glance

Si(100), substrate 775 µm, film 500 nm. Bow is the sagitta across the full diameter (center up = positive).
Film stress	Bow of a 200 mm wafer	Bow of a 300 mm wafer
-1000 MPa	138.4 µm	311.3 µm
-500 MPa	69.2 µm	155.7 µm
-200 MPa	27.7 µm	62.3 µm
+200 MPa	-27.7 µm	-62.3 µm
+500 MPa	-69.2 µm	-155.7 µm
+1000 MPa	-138.4 µm	-311.3 µm

When film stress becomes a problem

Thin films retain intrinsic stress, set by how atoms pack during deposition, and thermal stress from the expansion mismatch as they cool from the deposition temperature. A film that wants to shrink is under tensile stress and bows the wafer concave; one that wants to expand is under compressive stress and bows it convex. The Stoney equation lets you find the stress from the change in bow without knowing the film's elastic modulus.

In devices that stack hundreds of layers, such as 3D NAND, and in steps with thick nitride or metal films, wafer bow causes focus errors in lithography, overlay errors and wafer handling errors. Countermeasures include depositing a film with opposite stress on the back side and using nitride films with tuned stress.

FAQ

When does the Stoney equation stop being accurate?

It assumes a film much thinner than the substrate. Errors grow once the film exceeds a few percent of the substrate thickness, and the tool warns above 5%.

Is the measured stress all thermal?

No. Real film stress is the sum of thermal stress from cooling after deposition and intrinsic stress set by deposition conditions.

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Last updated: October 23, 2026