168
6 Stresses of Anodic Oxide Films Grown on Metal Electrode
compared to that for the anodic oxide film with a large relative dielectric constant
such as TiO 2 film (ε f = 60). A small value of −20 MPa [19] obtained by the cathodic
potential scan (1 mV s
−1 ) after the potentiostatic anodic oxidation of Ti for 1 h in pH
8.4 borate solution may result from the compositional alteration of the anodic oxide
film such as hydration due to cathodic polarization for a long time.
6.7 Residual Stress of Substrate Metal
The thickness of substrate metal decreases due to growth of the anodic oxide film
or anodic dissolution during anodization of the metal. When the substrate metal is
subjected to residual stress (compressive or tensile), the stress variations (σ · d)
measured during anodic oxidation are influenced by the decrease in thickness of the
substrate metal layer as represented by
(σ · d) = σ m d m + σ f d f ,
(6.32)
where σ m is the residual stress of the substrate metal, d m (< 0) the decrease in
thickness of the substrate metal layer due to consumption, σ f the growth stress of
the anodic oxide film, and d f ( > 0) is the increase in film thickness due to film
growth. In Eq. (6.32), it is assumed that σ m and σ f are uniform in the whole depth
of the substrate metal layer and of the anodic oxide film, respectively. Moreover,
the contribution of compressive stress component due to electrostriction is removed
from Eq. (6.32). Equation (6.32) can be transformed to [20]:
(σ ·d) =
σ m + σ f
d f
d m
d m = (σ m − η f α PB σ f )d m ,
(6.33)
where α PB is the Pilling–Bedworth ratio, and η f is the formation efficiency of the
anodic oxide film. The efficiency of the anodic dissolution of the substrate metal into
solution through the anodic oxide film is given by (1 − η f ).
It has been reported that the residual stresses of a sputter-deposited Zr film on Si
(100) wafer are −80 MPa for a thickness of 235 nm and −40 MPa for a film thickness
of 440 nm, respectively [24]. The compressive residual stress for the sputter-deposited
film results from an atomic peening due to bombardment of target atoms and ions
or inert gas atoms with a kinetic energy of 5–10 eV [44]. In contrast, the residual
stresses of evaporated Fe and Cr films with a thickness of 100 nm on MgF 2 at 27 °C
are 1.35 GPa and 1.45 GPa (i.e. tensile), respectively [45]. Figure 6.7 shows the stress
variations due to the consumption of Fe when a Fe thin film evaporated on a glass
plate is subjected to anodic dissolution in pH 8.4 borate solution [46]. Assuming that
the anodic dissolution of Fe proceeds with a current efficiency of 100%, i.e., η f = 0
in the active dissolution region of Fe, the residual stress σ Fe of the Fe thin film is
derived from Eq. (6.33):
6 Stresses of Anodic Oxide Films Grown on Metal Electrode
compared to that for the anodic oxide film with a large relative dielectric constant
such as TiO 2 film (ε f = 60). A small value of −20 MPa [19] obtained by the cathodic
potential scan (1 mV s
−1 ) after the potentiostatic anodic oxidation of Ti for 1 h in pH
8.4 borate solution may result from the compositional alteration of the anodic oxide
film such as hydration due to cathodic polarization for a long time.
6.7 Residual Stress of Substrate Metal
The thickness of substrate metal decreases due to growth of the anodic oxide film
or anodic dissolution during anodization of the metal. When the substrate metal is
subjected to residual stress (compressive or tensile), the stress variations (σ · d)
measured during anodic oxidation are influenced by the decrease in thickness of the
substrate metal layer as represented by
(σ · d) = σ m d m + σ f d f ,
(6.32)
where σ m is the residual stress of the substrate metal, d m (< 0) the decrease in
thickness of the substrate metal layer due to consumption, σ f the growth stress of
the anodic oxide film, and d f ( > 0) is the increase in film thickness due to film
growth. In Eq. (6.32), it is assumed that σ m and σ f are uniform in the whole depth
of the substrate metal layer and of the anodic oxide film, respectively. Moreover,
the contribution of compressive stress component due to electrostriction is removed
from Eq. (6.32). Equation (6.32) can be transformed to [20]:
(σ ·d) =
σ m + σ f
d f
d m
d m = (σ m − η f α PB σ f )d m ,
(6.33)
where α PB is the Pilling–Bedworth ratio, and η f is the formation efficiency of the
anodic oxide film. The efficiency of the anodic dissolution of the substrate metal into
solution through the anodic oxide film is given by (1 − η f ).
It has been reported that the residual stresses of a sputter-deposited Zr film on Si
(100) wafer are −80 MPa for a thickness of 235 nm and −40 MPa for a film thickness
of 440 nm, respectively [24]. The compressive residual stress for the sputter-deposited
film results from an atomic peening due to bombardment of target atoms and ions
or inert gas atoms with a kinetic energy of 5–10 eV [44]. In contrast, the residual
stresses of evaporated Fe and Cr films with a thickness of 100 nm on MgF 2 at 27 °C
are 1.35 GPa and 1.45 GPa (i.e. tensile), respectively [45]. Figure 6.7 shows the stress
variations due to the consumption of Fe when a Fe thin film evaporated on a glass
plate is subjected to anodic dissolution in pH 8.4 borate solution [46]. Assuming that
the anodic dissolution of Fe proceeds with a current efficiency of 100%, i.e., η f = 0
in the active dissolution region of Fe, the residual stress σ Fe of the Fe thin film is
derived from Eq. (6.33):