Twinning in thin films—II. Equilibrium microstructures

Abstract

We consider the energetics of twinning in misfitting films. While the analysis is general, we consider the case of a misfitting coherent thin film with a cubic crystal structure which transforms to a tetragonal crystal structure on an initially lattice matched cubic substrate; a commonly occurring phase transformation in ferroelectric films. The strain energy change upon twinning, estimated in Paper 1, along with the interfacial energy contributions are incorporated into a thermodynamic analysis of twinning and the equilibrium twinned microstructure is determined as a function of the film thickness, lattice misfit, elastic properties of the film—substrate system, the film surface energy, the film—substrate interface energy and the domain (twin) boundary energy. We develop stability diagrams that examine the role of these parameters in determining the equilibrium microstructure. Our results show that for like variants, a periodic array of domains is always energetically favorable compared to a monovariant film and therefore there is no critical thickness below which the monovariant film is stable. However, the equilibrium wavelength of the twinning microstructure depends on the normalized film thickness as well as the orientation of the domain boundaries with respect to the cubic substrate. Examination of the elastic energy as a function of domain boundary orientation shows that boundaries oriented parallel to the (110) plane of the substrate are preferred. However, (100) oriented boundaries are likely favorable from an interfacial energy viewpoint. The equilibrium wavelength of the microstructure scales as the exponential of the inverse film thickness for small thicknesses, and is either independent of film thickness or proportional to the square root of film thickness at large film thicknesses for the (100) and (110) oriented boundaries, respectively. For the unlike variants case, there is a critical thickness below which the monovariant film is energetically favored. This critical thickness depends on the relative misfit strains, the normalized domain (twin) boundary energy, and the normalized film—substrate interface energy and the normalized surface energies. Finally, we modify this thermodynamic analysis to account for externally applied stress fields and examine their effect on the equilibrium domain structure. Using these results, we derive an expression for the applied stress required to inhibit twinning in these films.