When dealing with stress in thin film / substrate combinations, it is usual to assume that the film is elastically isotropic and that the substrate on which it lies is also elastically isotropic. The classic Stoney formula can then be used to determine the magnitude of the stress in the thin film. Silicon wafers are widely used in single crystal form as substrates. These can be purchased in a variety of substrate (hkl) surface orientations. Silicon is noticeably anisotropic elastically. Therefore, the stiffness of such substrates is orientation-dependent within the plane of the substrate, unless the wafer has a (001) or (111) surface orientation.

In this work, formulae for the biaxial elastic moduli along the directions of principal stress for general (hkl) interfaces of cubic materials are derived for situations in which there is equi-biaxial strain within the plane, such as when isotropic thin films are deposited on (hkl) silicon single crystal substrates. Within a particular (hkl), the directions defining these principal biaxial moduli are those along which there are the extreme values of both the shear modulus and Poisson’s ratio. Conditions for stationary values of the biaxial moduli have also been derived, from which the conditions for the global extrema of the biaxial moduli in substrates of cubic materials have been established.

Future work will consider the effect of anisotropy for other readily available single crystal substrates of arbitrary surface orientation, such as alumina (trigonal), rutile (tetragonal) and zinc oxide (hexagonal).

Figure: Equi-biaxial elastic strains within isotropic thin film / single crystal substrate combinations introduced during deposition and/or by a change in temperature cause curvature associated with the two principal biaxial elastic moduli in the plane of the substrate. In general, these biaxial elastic moduli are different, so that the curvature induced in a thin film / single crystal substrate combination is not radially symmetric.