Ab Initio Studies On Phase Behavior of Barium Titanate

Using DFT methods we have studied structure, equation of state, and phase behavior of BaTiO3. We have identified the pressure induced phase transformations from the rhombohedral to orthorhombic structure at ca. 5 GPa and from tetragonal phase to cubic phase at ca. 7.5 GPa. INTRODUCTION BaTiO3 is an important example perovskite structured ceramic displaying ferroelectric property. At high temperature, BaTiO3 is stable in a paraelectric cubic structure with five atoms per unit cell (Pm3m). At lower temperatures, the crystal has three phase transitions from the paraelectric phase to ferroelectric phases. At 393 K, the symmetry changes to tetragonal (P4mm). The second phase transition, orthorhombic (Amm2) occurs at 278 K. Finally, the last phase transition, from orthorhombic to rhombohedral symmetry (R3m) takes place at 183 K. In these ferroelectric phases, the polar axis is aligned from tetragonal phase to rhombohedral phase along <100>, <110> and <111> directions. BaTiO3 has been widely investigated since the discovery of its ferroelectric properties in 1945. To explain its ferroelectric behavior, there exists ample experimental data. For a long time, the theoretical studies depend on the empirical approaches but during the last decade, with the application of first-principles calculations to ABO3 compounds, computers have provided a theoretical understanding of the nature of ferroelectric perovskites. In this work, the characterization of four phases of BaTiO3, from cubic to rhombohedral phase, has been studied. Thus, we present zero temperature full equation of state (EOS) of BaTiO3 which is obtained using the commercial ab-initio program CASTEP. First we describe the methods used in the calculations and the EOS of different phases. In the succeeding section we present how we optimized the structures. Furthermore, we discuss the EOS and optimization results. CALCULATIONAL DETAILS We present ab-initio Quantum Mechanical calculations of the EOS of different phases of BaTiO3 using DFT in the GGA approximation; our calculations correspond to zero Temperature except that the zero point energy of the crystal in not considered. We use DFT [1,2,3], using PW-91 GGA for the exchange and correlation energy functional [4]. The k points sampled using a 5X5X5 Monkhorst-Pack mesh. Ultrasoft pseudopotentials are used for both Ba,Ti and O [5]. Energy cut-off for these potentials was chosen as 700 eV. In Figure 1, the zero temperature EOS of the different crystalline phases of BaTiO3 is shown, namely cubic, tetragonal, orthorhombic and rhombohedral. In the plot, the energy-volume curves of phases are plotted. The EOS is calculated in a volume range of 10 % expansion and 15 % compression. The details of how the total energy is obtained will be discussed in the optimization section. Our main objective in calculating the EOS is to use it to develop a polarizable reactive force field for largescale atomistic MD and MC simulations [6]. The EOS parameters were obtained by fitting our DFT-GGA data to Rose's Universal binding curve [7]: EEOS(a) = Ecoh ( 1 + a* + k a* ) e where a* = (a a0)/a0λ, λ = sqrt(Ecoh /9V0 B), k = λ (B' -1) /2 1/3, B' = [dB/dP]P=0. Here, the constants: a0, Ecoh, V0, and B are the lattice parameter, cohesive energy, zero pressure volume, and bulk modulus, respectively. The parameter k depends on the pressure derivative of the bulk modulus evaluated at zero pressure. The EOS parameters were tabulated in Table 1. Figure 1. Energy as a function of volume for the cubic, tetragonal, orthorhombic and rhombohedral phases of BaTiO3 from ab initio calculations using DFT-GGA. Enthalpy H=E+PV as a function of pressure for the cubic, tetragonal, orthorhombic and rhombohedral phases of BaTiO3 from ab initio calculations using DFT-GGA Table 1. EOS parameters, cohesive energy, bulk modulus, B, and its pressure derivative B' for the cubic, tetragonal, orthorhombic and rhombohedral phases for BaTiO3. V0 (A3) ECoh (eV) B (GPa) B' Cubic 64.2754 -37.9181 167.64 4.45 Tetragonal 65.9530 -37.964 98.60 Orthorhombic 66.0171 -37.970 97.54 Rhombohedral 65.9906 -37.973 103.50 Structural Optimization: Structural optimization is performed using density functional theory. Lattice parameters and atomic positions were obtained from the optimization procedure. We employed the following procedure: (a) minimize the total energy, (b) obtain pressure from the stress tensor and (c) obtain vanishing forces on the atoms with respect to atomic displacements. Cubic Phase: No minimization was done in cubic phase. At the desired volume, total energy and macroscopic strains (diagonal hydrostatic stress tensor) was obtained. For only cubic phase, we have done calculations up to the volume of 7 times of the equilibrium volume in order to obtain the cohesive energy from EOS. We obtained the result of cohesive energy as -37.9181 eV/cell which is Ecoh =-38.23 eV/cell in good agreement with a previous LDA calculation [8] at the experimental volume. The experimental value is -31.57 eV/cell. Ferroelectric Phases: In the ferroelectric phases, the optimization procedure seems to be complicated because when we go to the lower symmetry structures, number of degree of freedoms increases. In our density functional theory calculations, the tetragonal, orthorhombic and rhombohedral structures contain respectively 4, 6 and 5 atomic degrees of freedom. In tetragonal structure, total energy structure calculations were performed with respect to volume. By keeping the volume fixed, at different c/a ratios, the structure was optimized by letting the atomic fractional coordinates change in the way symmetry allows. Thus, c/a ratio, ∆T-Ti, ∆T-O1 and ∆T-O2 are the quantities that changed during the optimization. We used the equivalent pseudo-monoclinic cell for orthorhombic phase. In this phase, lattice parameters a (a=b), cm and the angle γ, and ∆O-Ti, ∆O-O1, ∆O-O2 and ∆O-O3 are the varied quantities. Optimization procedure for this structure is as follows: During the optimization procedure macroscopic stress was kept constant. First, by keeping the cell lengths and the angle constant, fractional atomic coordinates were optimized in order to obtain zero forces on each atom in the cell. Then keeping these new fractional atomic coordinates fixed, we let cell parameters change. This self-consistent optimization procedure was repeated until we got the desired stress and zero force in the system. For rhombohedral phase, the same optimization procedure used in orthorhombic phase was used. Lattice parameter a (a = b = c), the angle α (α = β = γ), ∆R-Ti , ∆R-O1 and ∆R-O2 were the optimized quantities. Table 2 Notation of atomic positions (in reduced coordinates) in the cubic, tetragonal, orthorhombic and rhombohedral phases of BaTiO3. The Ba position is (0, 0, 0) Cubic Atom Position Tetragonal Atom Position Ti (0.5, 0.5, 0.5) Ti (0.5, 0.5, 0.5 + ∆T-Ti ) O1 (0.5, 0.5, 0.0) O1 (0.5, 0.5, 0.0 + ∆T-O1) O2 (0.5, 0.0, 0.5) O2 (0.5, 0.0, 0.5 + ∆T-O2) O3 (0.0, 0.5, 0.5) O3 (0.0, 0.5, 0.5 + ∆T-O3) Orthorhombic Rhombohedral Ti (0.5, 0.5 + ∆O-Ti}, 0.5 + ∆O-Ti) Ti (0.5 + ∆R-Ti, 0.5 + ∆R-Ti, 0.5 + ∆R-Ti) O1 (0.5, 0.5 + ∆O-O1, 0.0 + ∆O-O2) O1 (0.5 + ∆R-O1, 0.5 + ∆R-O1, 0.0 + ∆R-O2) O2 (0.5, 0.0 + ∆O-O2, 0.5 + ∆O-O1) O2 (0.5 + ∆R-O1, 0.0 + ∆R-O2, 0.5 + ∆R-O1) O3 (0.0, 0.5 + ∆O-O3, 0.5 + ∆O-O3) O3 (0.0 + ∆R-O2, 0.5 + ∆ R-O1, 0.5 + ∆RO1) During the calculations, Ba atom was chosen as the reference atom and was kept fixed at (0,0,0) position. Fractional atomic positions of paraelectric and ferroelectric phases were given in Table 2. In the tetragonal structure, O2 and O3 and in the rhombohedral structure O1 and O2 atoms are equivalent because of the symmetry. The optimization results are shown in Tables 3, 4, and 5. Table 3. Lattice parameters and atomic displacements of BaTiO3 in the tetragonal phase. A (A) c(A) ∆T-Ti ∆T-O_1 ∆T-O2 Reference 3.99095 4.0352 0.0224 -0.0244 -0.0105 [9](Exp. 300 K) 3.994 4.036 0.0143 -0.0307 -0.0186 [8] (LDA) 3.986 4.026 0.015 -0.023 -0.014 [10] 4.000 4.000 0.0138 -0.0253 -0.0143 [11] 3.9759 4.1722 0.0188 -0.0473 -0.0266 (P = 0.0 GPa) 3.99095 4.0352 0.0165 -0.0272 -0.0156 (Exp. Volume) 3.9417 3.9905 0.0139 -0.0204 -0.0129 (P = 7.8 GPa) In Table 3, lattice parameters and atomic displacements are reported. The results of Ghosez et. al. [8] were density functional theory results using LDA at the experimental volume at 290 K of Kwei et. al. [9]. As reported in LDA calculations (underestimation of lattice constant at zero stress), our results of GGA underestimate a (0.5 %) and overestimate c (3.5 %). The distortions of oxygen atoms are 2-2.5 times of the experimental result. But by doing calculations at the experimental volume, the distortions of the oxygen atoms become close to experimental data. Because of the importance of the macroscopic strain in the stabilization of the tetragonal phase, we checked the results at the stress of 7.8 GPa in our pressure induced phase transition calculations. It is also important to note that the mechanisms of the temperature induced phase transitions are different than the mechanisms of the pressure induced ones. Table 4. Lattice parameters and atomic displacements of BaTiO3 in orthorhombic phase. 230 K [9] LDA [8] Present Work a(A) 3.984 3.984 3.97031 cm(A) 4.0184 4.0184 4.0791 α (deg) 89.824 89.824 89.605 ∆O-Ti 0.0079 0.0127 0.0169 ∆O-O1 -0.0233 -0.0230 -0.0197 ∆O-O2 -0.0146 -0.0162 -0.0153 ∆O-O3 -0.0145 -0.0144 -0.0314 Except the tetragonal phase, all of the theoretical calculations, including our calculation also, overestimate the distortion in the position of the Ti atom in orthorhombic and rhombohedral structures (Tables 4 5). As seen from these tables, our results are not better than the other theoretical works but it must be kept in mind that we are presenting the results at zero stress or at a stress where the structure in question is stable. Our main aim here is to get volume-energy dependence of the phases and get the ferroelectric transition path in the right manner. The details of the transition path will be discussed in the result section. During the optimization procedure, we relaxed the atomic positions until the force on each atom is less than 0.003 eV/A. Table 5. Lattice parameters and atomic displacements of BaTiO3 in rhombohedral phase. a(A) α(deg) ∆R-Ti ∆R-O1 ∆R-O2 Reference 4.003 89.84 -0.013 0.011 0.0191 [9] 4.001 89.87 -0.011 0.013 0.0192 [8] 4.000 90.00 -0.012 0.010 0.0195 [11] RESULTS The goal of this study is to get energy-volume behavior of paraelectric and ferroelectric phases of BaTiO3 and check whether we can get the thermodynamic phase transition path in the correct order from our pressure induced first-principles phase transition calculations. In Fig. 1A energy volume behavior of BaTiO3 from EOS is plotted. From the plot it can be easily deduced that the minimum energy structure is rhombohedral. To get the phase transitions, we calculated enthalpy (H=E+PV) of the system (Fig. 1B). Thus, we have observed the rhombohedral-orthorhombic, orthorhombic -tetragonal and tetragonal-cubic phase transitions respectively at 5.0, 6.0 and 7.5 GPa. These transition pressures are completely consistent with the dielectric measurements of Ishidate et.al. [12]. They had determined the temperature-pressure phase diagram of BaTiO3. Their phase transition pressures are 5.4, 6.0 and 6.5 respectively. The consistency with the experiment is remarkable. Figure 2. Variation of distorsions as a function of pressure. Rhombohedral distortions, ∆R-Ti, ∆RO1, and ∆R-O2 as a function of pressure. Orthorhombic distortions, ∆O-Ti , ∆O-O1, ∆O-O2, and ∆O-O3 as function of pressure. The tetragonal distortions, ∆T-Ti , ∆T-O1 and ∆T-O2 as a function of pressure show similar behavior, not shown. Taking into consideration of high pressure up to 25 GPa on all phases, all the ferroelectric phases structurally become equal to the cubic structure (Fig. 2). We see that the atomic distortions of the ferroelectric phases become equal to zero as the pressure reaches to 30 GPa. Besides at around the same pressure, the cell angle α of the rhombohedral and orthorhombic structures become equal to 90 degrees (Fig. 3A). From Fig. 3B, the lattice constants of all the ferroelectric phases converged to the cubic structure lattice constant as the pressure increased over 30 GPa. From all these figures, it can be deduced that all the ferroelectric structures become equal to the cubic structure around 30 GPa pressure and lose their ferroelectric property and becomes paraelectric [13]. Figure 3. A) Rhombohedral and orthorhombic structure (pseudo monoclinic cell structure) cell angle α as a function of pressure. B) Lattice constants of cubic, tetragonal, orthorhombic and rhombohedral structures as a function of pressure. CONCLUDING REMARKSIt this paper, we have obtained the energy-volume behavior of paraelectric and ferroelectricphases of BaTiO3 to derive force field from ab-initio results. So it is important to obtain pressureinduced phase transitions in the correct order and at right pressures where phase transitionsexperimentally occur. In that manner we have achieved the goal of the work. During theoptimization procedure, the lattice constants and atomic position distortions were relaxed forferroelectric phases. Except the tetragonal phase, we see the ratio for the zero stress latticeconstants with the experimental ones were less than 1 %. But in the tetragonal phase the latticeparameter c was 3 % higher than the experimental result. This result is mainly because of theexchange-correlation energy calculation within GGA (PW-91). REFERENCES[1] P. Hohenberg and W. Kohn, Phys Rev. 136, 864A (1964)[2] W. Kohn, and L.J.Sham, Phys. Rev. 140, 1133B (1965)[3] M.C. Payne, M.P. Teter, D.C. Allan, T. A. 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