The theoretical model of fcc ultrathin film

A three-dimensional lattice model is constructed to theoretically study the size effects on the elastic properties of ultrathin films with face-center-cubic crystal structure. The lattice model directly takes the discrete nature in the thickness direction into account and treats the deformations along the film plane with continuum mechanics. Only the interactions between the nearest and second nearest atoms are considered in this model and represented as harmonic springs. The constitutive relation of the ultrathin film is then derived using the energy approach and the analytical expressions of the elastic moduli of ultrathin films, including in-plane, out-plane Young's modulus and Poisson's ratio, are obtained. Moreover, the analytical expressions of ultrathin films with different crystal orientations are also formulated. It is shown that the ultrathin film along in-plane directions may be stiffer or softer than its bulk counterpart, but it is always softer along the out-plane direction.