Thickness Effects on the Plasticity of Gold Films

Abstract

An indentation size effect is a common occurrence during nanoindentation. Thin and thick gold films, deposited using sputter deposition and evaporation, illustrate this at depths less than 100 nm. The indentation size effect, however, has been observed to be independent of film thickness. It has been modeled using a combination of an indentation size effect model and a parabolic hardening model. At the near surface regime, the indentation size effect model is dominant, and at larger depths, the parabolic hardening model is dominant, taking into effect the film thickness. The described model, which is a combination of these two, fits the experimental data for the sputter-deposited films and the evaporated films.

Appendix

In order to calculate the composite modulus of the 100-nm sputtered gold film on sapphire and the 730-nm evaporated gold film on silicon, the model of Gao et al.[14] was used. The equation

$$ \left( {\frac{1-\nu }{\mu }} \right)_{\rm eff} =\frac{1-\nu_s -\left( {\nu_f -\nu_s } \right)I_1 \left( {h/a} \right)}{\mu _s +\left( {\mu_f -\mu_s } \right)I_0 \left( {h/a} \right)} $$

(A1)

was used to determine the ratio of the effective Poisson’s ratio to shear modulus for the film-substrate system. In Eq. [A1], μ is the shear modulus for the film (f) and substrate (s), ν is the Poisson’s ratio for the film (f) and substrate (s), and I ₁(h/a) and I ₀(h/a) are the numerical values taken from Gao et al.’s[14] Figure 3. These parameters depend on the film thickness, h, and contact radius, a, of the indent. The material constants μ and ν for the Al₂O₃, Au, and Si were 180 GPa and 0.22, 28.5 GPa and 0.42, 65.7 GPa and 0.218, respectively. Using the area function for the tip, the contact radii were calculated for both films through the equation

$$a=\left(\frac{A}{\pi}\right)^{1/2}$$

(A2)

where A is the calculated area from the tip area function. Several (h/a) values were calculated for displacements up to 500 nm. With the (h/a) values, the corresponding I ₁ and I ₀ were ascertained from Figure 3 of Gao et al.[14]

Values for the ${\left( {\frac{1-\nu }{\mu }} \right)_{\rm eff} }$ were calculated using Eqs. [A1] and [A2] and I ₁ and I ₀ from Figure 3 of Reference 14. The effective elastic modulus for the two film systems was determined by assuming an effective Poisson’s ratio of 0.3 for the film systems and using the relationship between the elastic modulus, the shear modulus, and the Poisson’s ratio

$$ \mu_{\rm eff} =\frac{E_{\rm eff} }{2\left( {1+\nu _{\rm eff} } \right)} $$

(A3)

Substitution and rearrangement provides an equation for the effective elastic modulus of the film-substrate system:

$$ E_{\rm eff} =1.82\left( {\frac{\mu }{1-\nu }} \right)_{\rm eff} $$

(A4)