2.3 Cantilever Bending Method for Measurement of Changes in Surface …
49
Fig. 2.8 Scheme for the bending of a rectangular substrate (the both edges are not clamped) with
a thickness of d s when a thin film with a thickness of d f is uniformly deposited on the top side of
the substrate: a the substrate is convexly bended in the case where the internal compressive stress
(σ f < 0) is developed within the film, and b the substrate is concavely bended in the case where
the internal tensile stress (σ f > 0) is developed within the film. The curvature of the substrate κ is
reciprocal of the curvature radius R
and
M = ∫ σ zdA = 0,
(2.11)
where A is the sectional area (z–y plane in Fig. 2.8), and z is the distance from
the neutral plane (where an elastic strain becomes zero) in the substrate. Elasticity
theory was first applied by Stoney [20] to derive the relationship between surface
stress and curvature of cantilever. Stoney considered a “thin steel rule” with a thin
nickel layer with a thickness d f deposited on steel. Assuming that the thickness of
the rule d s ( d f ) is negligibly small as compared to the radius of curvature R, the
following relationship was derived [1, 20]:
g = σ f d f =
E s κd
2
s
6
=
E s d
2
s
6R
,
(2.12)
where σ f is the stress in the film, κ is the curvature (equal to reciprocal of R), and E s
is Young’s modulus of the substrate. The unit of σ f d f in Eq. (2.12) is J m
−2 , which
corresponds to the unit of surface stress g. Equation (2.12) is also valid in the case
where the uppermost surface (free from a thin film) on the top side of the substrate is
49
Fig. 2.8 Scheme for the bending of a rectangular substrate (the both edges are not clamped) with
a thickness of d s when a thin film with a thickness of d f is uniformly deposited on the top side of
the substrate: a the substrate is convexly bended in the case where the internal compressive stress
(σ f < 0) is developed within the film, and b the substrate is concavely bended in the case where
the internal tensile stress (σ f > 0) is developed within the film. The curvature of the substrate κ is
reciprocal of the curvature radius R
and
M = ∫ σ zdA = 0,
(2.11)
where A is the sectional area (z–y plane in Fig. 2.8), and z is the distance from
the neutral plane (where an elastic strain becomes zero) in the substrate. Elasticity
theory was first applied by Stoney [20] to derive the relationship between surface
stress and curvature of cantilever. Stoney considered a “thin steel rule” with a thin
nickel layer with a thickness d f deposited on steel. Assuming that the thickness of
the rule d s ( d f ) is negligibly small as compared to the radius of curvature R, the
following relationship was derived [1, 20]:
g = σ f d f =
E s κd
2
s
6
=
E s d
2
s
6R
,
(2.12)
where σ f is the stress in the film, κ is the curvature (equal to reciprocal of R), and E s
is Young’s modulus of the substrate. The unit of σ f d f in Eq. (2.12) is J m
−2 , which
corresponds to the unit of surface stress g. Equation (2.12) is also valid in the case
where the uppermost surface (free from a thin film) on the top side of the substrate is