Ab initio study of the elastic properties of single and polycrystal TiO2, ZrO2 and HfO2 in the cotunnite structure

Abstract

In this work, we study theoretically the elastic properties of the orthorhombic (Pnma) high-pressure phase of IV-B group oxides: titania, zirconia and hafnia. By means of the self-consistent SIESTA code, pseudopotentials, density functional theory in the LDA and GGA approximations, the total energies, hydrostatic pressures and stress tensor components are calculated. From the stress–strain relationships, in the linear regime, the elastic constants Ci j are determined. Derived elastic constants, such as bulk, Young’s and shear modulus, Poisson coefficient and brittle/ductile behavior are estimated with the polycrystalline approach, using Voigt–Reuss–Hill theories. We have found that C11, C22 and C33 elastic constants of hafnia and zirconia show increased strength with respect to the experimental values of the normal phase, P21/c. A similar situation applies to titania if these constants are compared with its normal phase, rutile. However, shear elastic constants C44, C55 and C66 are similar to the values found in the normal phase. This fact increases the compound anisotropy as well as its ductile behavior. The dependence of unit-cell volumes under hydrostatic pressures is also analyzed. P–V data, fitted to third-order Birch–Murnaghan equations of state, provide the bulk modulus B0 and its pressure derivatives B 0. In this case, LDA estimations show good agreement with respect to recent measured bulk moduli of ZrO2 and HfO2. Thermo-acoustic properties, e.g. the propagation speed of transverse, longitudinal elastic waves together with associated Debye temperatures, are also estimated.