The determination of film hardness from the composite response of film and substrate to nanometer scale indentations

Abstract

Two equations for determining the hardness of thin films from depth-sensing indentation data are examined. The first equation is based on an empirical fit of hardness versus indenter displacement data obtained from finite element calculations on a variety of hypothetical films. The second equation is based on a model which assumes that measured hardness is determined by the weighted average of the volume of plastically deformed material in the coating and that in the substrate. The equations are evaluated by fitting the predicted hardness versus contact depth to data obtained from titanium coatings on a sapphire substrate. Only the volume fractions model allows the data to be fitted with a single adjustable parameter, the film hardness; the finite element equation requires two thickness-dependent parameters to obtain acceptable fits. It is argued that the difficulty in applying the finite element model lies in the use of an unrealistic area function for the indenter. For real indenters, which have finite radii, the area function must appear explicitly in the final equation. This is difficult to do with the finite element approach, but is naturally incorporated into the volume fractions equation. Finally, using the volume fractions approach the hardnesses of the titanium films are found to be relatively insensitive to film thickness. Thus, the apparent increase in hardness with decreasing film thickness for the titanium films is most likely due to increased interactions between the film and substrate for the thinner films rather than to a change in the basic structure of the titanium films.