Free energy of formation of defects in polar solids

Abstract

A more exact method than hitherto available, based on lattice statics and quasi-harmonic lattice dynamics, is presented for the direct minimisation of the free energies of periodic solids with very large unit cells. This is achieved via the calculation of analytic derivatives of the vibrational frequencies with respect to all external and internal variables. The method, together with large defective supercells, is used to calculate the free energies of defects in MgO as a function of temperature. A major advantage of the supercell approach is that constant-volume and constant-pressure quantities are calculated independently. This allows a critical appraisal of the common approximations used for many years: (i) to convert constant-volume defect parameters to constant-pressure and (ii) to justify the use of static calculations at constant volume in the interpretation of experimental data obtained at constant pressure and at high temperatures. Defect enthalpies show only a small variation with temperature and differ by ca. 2% from the internal energy change in the static limit. An assessment is also made of the commonly used ZSISA approximation, in which the free energy at each temperature is minimised with respect to external strains only, simultaneously determining the internal strains by minimising the static lattice energy.