Analysis of Barium titanate and Hafnium oxide using Phonon Calculation

Abstract

BaTiO3 has 4 structures. These are cubic (Pm3 ̄m), orthorhombic (Amm2), rhombohedral (R3m), and tetragonal (P4mm). These structures has a constant energy when the k point meshes are 888 (Gamma centered grid). These also has a constant energy when the energy cutoff is around 700 eV. Their the phonon band structures and Gibbs free energies were calcu- lated by DFT and phonon method. The phonon dispersions of Pm3 ̄m, Amm2, R3m, and P4mm are the unstable phases. On the other hand, their stabilities of the phases were measured by a experiment. Pm3 ̄m, Amm2, and P4mm were the unstable phases. R3m was the stable phase. Using the quasi harmonic approximation, I could compute their Gibbs free energies. From their Gibbs free energies, we could get a phase diagram. Temperature has a significant impact on the phase transition. Gibbs free energies of the phases were calculated under the tempera- tures 0 ~ 500 K at a pressure 0 GPa. The diagram was calculated as orthorhombic ! tetragonal ! cubic, and the corresponding transition temperatures are 242 and 391 K, respectively. The rhombohedral structure was computed that the structure did not transform to the orthorhombic structure. The experiment requires that the crystal structure transforms among from the rhom- bohedral (R3m) to the orthorhombic (Amm2) between 200 and 240 K, then to the tetragonal (P4mm) between 250 and 280 K, and to the cubic (Pm3 ̄m) between 300 and 320 K. The phase diagrams and phonon band structures of HfO2 was calculated for cubic (F m3 ̄ m), monoclinic (P21/c), tetragonal (P42/nmc), orthorhombic (Pca21, Pbca, and Pnma). Fm3 ̄m, P21/c, and P42/nmc belong to the nonpolar phases. These has a constant energy when the k point meshes are 666 (Monkhorst-pack). When the energy cutoff has around 550 eV, they have a constant energy. The orthorhombic phases (Pca21, Pbca, and Pnma) have a constant energy when the k point meshes are 444 (Monkhorst-pack). When the energy cutoff has around 550eV, the phases have a constant energy. From the phonon band structures of HfO2, Fm3 ̄m and P42/nmc are the unstable phases. P21/c, Pbca, Pnma, and Pca21 are the stable phases. I got the different result on the phonon dispersion of the orthorhombic phases (Pnma). The refer- ence argued that the orthorhombic phase (Pnma) is the unstable phase. The phase diagram was calculated under the temperatures from 0 to 3000 K and at the pressures between 0 and 30 GPa. The diagram is described the monoclinic (P21/c) is the most stable at the low tem- peratures and at the low pressures. When the pressure is from 9 to 15.8 GPa, the orthorhombic (Pbca) or the tetragonal (P42/nmc) are the most stable. When the pressure increases more than 15.8 GPa, the orthorhombic (Pnma) is the most stable. This results are similar to their the equilibrium of phase diagram for HfO2. In addition, when Fm3 ̄m and Pca21 consider into the phase diagram, I got the new phase diagram. The cubic phase transforms to the tetragonal at P = 10.1 GPa. The monoclinic phase (P21/c) is the most stable at the low temperatures and at the low pressures. When the temperature increases, the orthorhombic (Pca21) or the tetrag- onal (P42/nmc) are the most stable. Between the pressures 9 and 15.8 GPa, the orthorhombic (Pbca) is the most stable at the low temperatures. When the temperature increases, P42/nmc or Pca21 are the most stable. Also, when the pressure increases more than 15.8 GPa, the orthorhombic (Pnma) is the most stable.