Quantum Mechanical Study of YTiO₃ to the Investigation of Piezoelectricity

Abstract

In previous articles we reported through theoretical studies the piezoelectric effect in BaTiO₃, SmTiO₃, and YFeO₃. In this paper, we used the Douglas-Kroll-Hess (DKH) second-order scalar relativistic method to investigate the piezoelectricity in YTiO₃. In the calculations we used the [6s4p] and [10s5p4d] Gaussian basis sets for the O (³P) and Ti (⁵S) atoms, respectively, from the literature in combination with the (30s21p16d)/[15s9p6d] basis set for the Y (³D) atom, obtained by generator coordinate Hartree-Fock (GCHF) method, and they had their quality evaluated using calculations of total energy and orbital energies (HOMO and HOMO-1) of the ²TiO⁺¹ and ¹YO⁺¹ fragments. The dipole moment, the total energy, and the total atomic charges in YTiO₃ in C_s space group were calculated. When we analyze those properties we verify that it is reasonable to believe that YTiO₃ does not present piezoelectric properties.

1. Introduction

Even today barium titanate (BaTiO₃), the first perovskite structure developed, is widely used in the industry. The polymorphic forms of BaTiO₃ have been likened by Kay and Vousden [1] to a displacement of the central Ti⁺⁴ ion within its oxygen octahedron towards one, two, and then three of the six adjacent oxygen ions as the temperature is lowered. This is a simplification of the actual atomic displacements, but it is a useful first approximation for understanding the structure. The piezoelectric properties in perovskite structure result from uncentersymmetric characteristics [2]. The structure is a network of corner-linked oxygen octahedral holes and large cation filling the dodecahedral holes [3]. For a review about the role of the perovskite structure in ceramic science and technology see Bhalla et al. [4].

In previous reports [5–7], we performed Hartree-Fock (HF) studies to investigate the piezoelectric effect in BaTiO₃, SmTiO₃ and YFeO₃. The obtained results offered evidences that the piezoelectricity can be caused by electrostatic interactions. The purpose of this paper is to present a quantum mechanical study of the YTiO₃ and offer some insight into the investigation of this property in perovskites not experimentally investigated yet.

In our study of YTiO₃ we used the 6s4p and [10s5p4d] Gaussian basis sets for O (³P) and Ti (⁵S) atoms, respectively, from the literature [5]. For Y (³D) atom, we utilized the Generator Coordinate HF method [8] to obtain the (30s21p16d) Gaussian basis set which was contracted to [15s9p6d]. The quality of the contracted basis sets [6s4p]/[10s5p4d]/[15s9p6d] was evaluated in the study of the total energy and orbital energies for the ²TiO⁺¹ and ¹YO⁺¹ fragments. Then it was done the addition of one d polarization function in the contracted basis set for the O atom and one diffuse function by symmetry for Ti and Y atoms. The [6s4p1d]/[11s6p5d]/[16s10p7d] molecular basis set was used to model the dipole moments, total energy, and total atomic charges in YTiO₃ in C_s space group. In these work, we intend to investigate the possible piezoelectricity property of YTiO₃ using theoretical calculations.

2. Computational Procedure

2.1. The Development of the (30s21p16d) Basis Set to Y (³D) Atom

By the GCHF approach, we choose the one-electron as the continuous superposition Ψ_𝑖=𝜑_𝑖(1,𝛼)𝑓_𝑖d𝛼,𝑖=1,2,…,𝑛, (1) where 𝜑_𝑖 are the generator functions (STFs and GTFs or other type of functions—in our case GTFs), the 𝑓_𝑖 are unknown weight functions, and 𝛼 is the generator coordinate. The application of (1) to calculate expectation values using a Slater determinant leads to the Griffin-Wheeler-HF (GWHF) equations, which underlie the CGHF method 𝐹𝛼,𝛽−𝜀_𝑖𝑆𝛼,𝛽𝑓_𝑖𝛽d𝛽=0,𝑖=1,2,…,𝑚, (2) where the 𝜀_𝑖 are the HF eigenvalues, and the Fock, 𝐹(𝛼,𝛽), and overlap 𝑆(𝛼,𝛽), kernels are defined in [8].

In practical application the GWHF equations are solved through numerical integration. This is accomplished by discretization preserving the character of the CGHF method. This integration is implemented with a relabeling of the 𝛼 space Ω=ln𝛼𝐴,𝐴>1, (3) where 𝐴 is a scaling factor determined numerically.

The Ω space is discretized in an equally spaced mesh formed by Ω values so that Ω_𝑚=Ω_min+(𝑚−1)ΔΩ,𝑚=1,2,…,𝑁. (4) In (4), 𝑁 corresponds to the number of discretization points defining the basis set size, Ω_min is the initial point, and ΔΩ the increment.

In the solution of the discretized form of (2), the (30s21p16d) basis set was used as defined by the mesh of (3). The discretization parameters (which define the exponents) for the basis set built to Y (³D) are Ω_min(s)=−0.670, ΔΩ_(s)=0.123, and 𝑁_(s)=30; Ω_min(p)=−0.377, ΔΩ_(p)=0.109, and 𝑁_(p)=21; Ω_min(d)=−0.498, ΔΩ_(d)=0.115, and 𝑁_(d)=16.

2.2. Building the [15s9p6d] Contracted Basis Set Y (³D) Atom and Evaluating the Quality of the [6s4p]/[10s5p4d]/[15s9p6d] Contracted Basis Set in Polyatomic Calculations

The contraction of the (30s21p16d) basis sets was implemented by employing the segmented contraction scheme proposed by Dunning and Hey [9]. The (30s21p16d) basis set was contracted to 15s9p6d through the following scheme: (14,2,1,1,1,1,1,1,1,1,1,1,2,1,1/9,3,1,1,1,1,3,1,1/8,1,1,1,4,1).

The (30s21p16d) and [15s9p6d] basis sets were obtained through a slightly modified version of the ATOMSCF program of Pavani and Clementi [10].

In order to evaluate the quality of the [6s4p], [10s5p4d], and [15s9p6d] basis sets in polyatomic study, we performed calculations of total energies, highest occupied molecular orbital (HOMO) energy, and the energy of one level below the highest occupied molecular orbital (HOMO-1) for the ²TiO⁺¹ and ¹YO⁺¹ fragments. The results are compared with those obtained from the (24s14p), (30s19p14d), and (30s21p16d) basis sets.

Table 1 shows the HF total and orbital energies for ²TiO⁺¹ and ¹YO⁺¹ fragments obtained with the uncontracted and contracted basis sets. From this table, one can see that the total energy obtained with [10s5p4d]/[6s4p] and [15s9p6d]/[6s4p] basis sets are close to the corresponding values of (30s19p14d)/(24s14p) and (30s21p16d)/(24s14p) basis sets. The total energies differ by 0.885 and 0.172 hartree for ²TiO⁺¹ and ¹YO⁺¹ fragments, respectively. The HOMO energies obtained from contracted basis sets are closer to those of the uncontracted basis sets and differ by 0.00841 and 0.0138 hartree for the ²TiO⁺¹ and ¹YO⁺¹ fragments, respectively. The HOMO-1 energies obtained from contracted basis sets are also close to the values of uncontracted basis sets and present deviations of 0.0128 and 0.00577 hartree.

Table 1 

Total and HOMO and HOMO-1 energies (hartree).

To better describe the properties of the YTiO₃ in the implementation of the calculations it was necessary to include polarization function for the O (³P) atom. The polarization function value is 𝛼_d=0.622507 [5].

In ab initio calculations of the transition metal compounds, the role of the basis set is a crucial point, since the description of the configuration of the metal in the compound differs from the neutral state. The adequate diffuse functions were obtained through the total energy optimization of the ground state anion Ti⁻¹ (⁴F) and Y⁻¹ (³F) by HF method. The following diffuse functions for the [10s5p4d] and [15s9p6d] basis sets were determined for the Ti (⁵S) and Y (³D), respectively, using this strategy: 𝛼_s=0.00177827; 𝛼_p=0.0537032; 𝛼_d=0.0263027 and 𝛼_s=0.0081283; 𝛼_p=0.0489779; 𝛼_d=0.0323594.

The [6s5p1d]/[11s6p5d]/[16s10p7d] basis set was used in the modelling of dipole moment, total energy, and total atomic charges from [YTiO₃]₂ fragment.

2.3. The Representation of the Crystalline 3D Periodic YTiO₃ System

In previous papers [5–7], the fragment model shown in Figure 1 was used to simulate the necessary conditions to the existence of piezoelectricity in BaTiO₃ as full solid. In this paper, we used the same model to represent the crystalline 3D periodic YTiO₃ system. The Ti atom is located in the center of the octahedron, being wrapped up by six O atoms, disposed in the reticular plane (200), and two Y atoms arranged in the reticular plane (100).

Figure 1 

The octahedral [YTiO3]2 fragment. (Ia): (a) represents the [YTiO3]2 fragment with the Ti atoms fixed in the space; (b) represents the [YTiO3]2 being moved +0.005 Å in the symmetry x axis, while the Y and O atoms are maintained fixed; (c) it presents the [YTiO3]2 fragment in which the Ti atoms are moved −0.005 Å symmetry x axis, and Y and the O atoms are maintained fixed; (Ib), represents the [YTiO3]2 fragment with the bond lengths Ti1–O3, Ti1–O4, and Ti2–O4 shortened 0.005 Å.

To study the crystalline 3D periodic YTiO₃ system it is necessary to choose a fragment (or a molecular model), which represents adequately a physical property of the crystalline system as a whole. The fragment [YTiO₃]₂ was chosen because, after its optimization, the obtained structural parameters (interatomic distances) were closer to experimental values with good precision. The study was developed according to the following strategy: (i) first, it was carried out the geometry optimization of the [YTiO₃]₂ fragment in the C_s symmetry and ¹A’ electronic state. (ii) Finally, calculations of single points were performed with the geometry optimized according to the descriptions shown in Figure 1.

In Figure 1 (I), (a) represents the [YTiO₃]₂ fragment with the Ti atoms fixed in the space. (b) represents the [YTiO₃]₂ fragment with the Ti atoms being moved +0.005 Å in the x axis, while the Y and O atoms are maintained fixed. (c) represents the [YTiO₃]₂ fragment in which the Ti atoms are moved −0.005 Å in the x axis, and Y and the O atoms are maintained fixed.

Figure 1 (II) represents the [YTiO₃]₂ fragment with the bond lengths Ti₁–O₃, Ti₁–O₄, and Ti₂–O₄ shortened 0.005 Å.

All molecular calculations in this work were carried out by Hartree-Fock method with inclusion of the DKH second-order scalar relativistic calculations [11–13] as implemented in the Gaussian 03 program [14].

3. Results and Discussion

Aiming at the verification of possible piezoelectric properties of YTiO₃, the Ti⁺³ central ion, should be centersymmetric and the [YTiO₃]₂ fragment polarises when submitted to mechanical stress. Firstly, we would like to call the attention for the comparison among the calculated bond lengths and the experimental values [15]. For Y–O and Ti–O bond lengths, our results are 2.009 and 2.421 Å, while the experimental values are 2.077 and 2.508 Å, respectively. The differences between the theoretical and experimental values are 0.0682 and 0.087 Å. This shows a very good performance of our 6s5p1d/11s6p5d/16s10p7d basis set to model the geometry of the fragment studied.

In Table 2 are shown the dipole moment (in Debye) and the total energy (in hartree) from [YTiO₃]₂ fragment. In this table, one can see that the fragment with Ti⁺³ at (b) is 0.0111 hartree more stable when compared at (a); this evidence is confirmed by the decrease of the dipole moment. Also, when the central ion is moved to the position c, it can be verified that the fragment is 0.00813 hartree more stable than at (a). This indicates that the Ti⁺³ central ion is not centersymmetric. Still in Table 2 one can see that reducing the Ti₁–O₃, Ti₂–O₄, and Ti₂–O₄ bond lengths, the calculations in the atomic position d did not show either a deviation of atomic charges among the atoms of the [YTiO₃]₂ fragment, besides the decreasing of the dipole moment of the fragment. These results suggest that there is not a probable polarization of YTiO₃ crystal when submitted to mechanical stress. Configurations b, c, and d show lower total energy compared to configuration a, as well as, lower values of total dipole moment μ and its individual components μ_x, μ_y, and μ_z. We suggest in this work that symmetric distribution of atomic charges provides a more stability of studied molecular fragment.

Table 2 

Dipole moment (Debye) and total energy (hartree) of the [YTiO₃]₂ fragment.

Table 3 shows the values of the total charges of the atoms in the [YTiO₃]₂ fragment with Ti⁺³ at a, b, and c positions and the Ti–O bond length reduced 0.005 Å. According to the table, when the Ti⁺³ central ion is deviated from the position a to b and c positions, it occurs the rearrangement of the electronic density, without being clearly established, however, a tendency in the migration of the atomic charges. On the other hand, in Table 3 one can see that when the Ti₁–O₃, Ti₁–O₄, and Ti₂–O₄ bond lengths are shortened, at atomic position (d), by a mechanical stress, it also occurs the rearrangement of the electronic density, without again verifying the clear tendency in the migration of the atomic charges in the material.

Table 3 

Total atomic charges from the [YTiO₃]₂ fragment.

For the fragment a, which represents the probable configuration of this work, the HOMO and LUMO orbitals are HOMO = − 0.14 4s Ti₁ − 0.27 2pz O₁ − 0.48 2pz O₄ + 0.82 2px O₄ + 0.11 2s O₆ – 0.27 2pz O₆ + 0.12 4dxz Y₁; LUMO = + 0.17 4s Ti₁ + 0.16 3px Ti₁ + 0.14 2s O₂ − 0.21 2py O₂ + 0.16 2px O₂ − 0.24 2px O₃ + 0.24 4s Ti₂ + 0.38 3px Ti₂ + 0.43 3py Ti₂ + 0.33 3dx²-y² Ti₂ + 0.10 3dxy Ti₂ + 0.10 3dz² Ti₂ − 0.20 5s Y₂ + 0.16 4px Y₂ + 0.274py Y₂.

Thus, we suggest that the metal-O distance in the [YTiO₃]₂ fragment is formed by π-symmetry covalent bonds between 3d and 4d orbitals of the Ti (⁵S) and Y (³D) atoms, respectively, to 2p orbital of the O (³P) atom in opposition to BaTiO₃ where it was verified that those bonds were ionic. The ionic characteristic between metal-O bonds is a fundamental condition to occurrence of piezoelectricity in our opinion, which is not observed in YTiO₃. Therefore, previous theoretical studies of BaTiO₃ [5] have suggested that known piezoelectric properties are caused by electrostatic effects, and for SmTiO₃ [6] they have suggested the possible piezoelectric effect as well as electrostatic effects in its chemical bonds. On the other hand, for YFeO₃ [7] and YTiO₃ (this work), the theoretical calculations suggest absence of piezoelectric effect with covalent bonds between their atoms. For us, it seems that ionic bonds contribute to the occurrence of piezoelectric effect.

4. Conclusions

In the theoretical modeling of YTiO₃ in the C_s symmetry and ¹A’ electronic state, the calculated results of the bond lengths showed a very good concordance with the experimental values from the literature. Obtained values of the moment of dipole, the total energy, and the total atomic charges showed that it is reasonable to believe that YTiO₃ does not present behavior of piezoelectric material.

Acknowledgments

The authors gratefully acknowledge the financial support from the Brazilian agencies CNPq and CAPES. They employed computing facilities at the LQTC-UFPA and CENAPAD-UNICAMP.