Insights into the Impact of Yttrium Doping at the Ba and Ti Sites of BaTiO3 on the Electronic Structures and Optical Properties: A First-Principles Study.

We reported a systematic study of the effects of Y doping BaTiO3 at Ba and Ti sites. We assessed the structural, electronic, and optical properties by employing first-principles calculations within the Tran-Blaha-modified Becke-Johnson (TB-mBJ) potential and generalized gradient approximation + U approaches. We calculated the lattice constants and bond lengths for pure and Y-doped BaTiO3. We explored the consequences of electronic structure and optical property modification because of Y doping in BaTiO3. Indeed, Y doping has led to various modifications in the electronic structures of BaTiO3 by inducing a shift of the conduction band through lower energies for the Ba site and higher energies for the Ti site. It was found that Y doping, either at Ba or at Ti sites, strongly enhanced the BaTiO3 dielectric constant properties. The transformation in bonding was explored via the charge density contours and Born effective charges. We used the state of art of polarization theory based on finite difference and Berry-phase approaches to investigate piezoelectricity. Y doping has increased the dielectric constants but canceled the piezoelectricity as they changed to metallic nature. We could look into the future for potential doping, preserving the semiconductor nature of BaTiO3 and increasing the permittivity with large dielectric loss.

Introduction

Barium titanate materials have received much attention because of the scientific interest related to exceptional and multiple meaningful properties such as high dielectric permittivity, positive temperature coefficient of resistivity, high-voltage tunability, ferroelectricity, pyroelectricity, and piezoelectricity.[1−6] These characteristics have contributed the use of barium titanate ceramics in various applications including multilayer ceramic capacitors, piezoelectric and ultrasonic actuators, pyroelectric detectors, temperature sensors and controllers, and microwave devices for telecommunications.[7−13] Barium titanate (BaTiO3) has its structure based on ABO3 general formula in which A is the larger cation located in eight corners and six oxygen atoms located in the middle of surfaces, while B is the smaller cation located in the center of the unit cells.[5−8] BaTiO3 is composed of perovskite-based metal oxide materials that can be crystallized in cubic (Pm3̅m) and tetragonal (P4mm) crystal structures at room temperature,[14] which are the most investigated structures.[15] However, the tetragonal structure shows the most stable structure at ambient conditions. Despite their wide potential of applications, BaTiO3-based materials have been faced with certain practical limitations because of the narrow range of tetragonal stability and lower piezoelectric coefficient[16]. To further improve the ferroelectric and dielectric properties of BaTiO3, the current research strategy is driven by the prospects of synthesizing new systems, by means of replacing Ba2+ or Ti4+ by other ions of comparable ionic sizes. In fact, the perovskite structures show high flexibility and intrinsic capacity to host ions of different sizes, and a large number of different dopants can be incorporated at the Ba sites or at the Ti sites in perovskite-based metal oxide materials.[17−20] In particular, trivalent rare-earth dopants have attracted wide attention because of their ability to act as a donor or an acceptor. It is reported that doping a small amount of rare-earth dopants into BaTiO3 systems can greatly improve the microstructure of ceramic materials, phase composition, sintering properties, mechanical properties, physical and chemical properties, and dielectric properties.[6]

In consideration of their obvious importance, many experimental and theoretical works based on BaTiO3 materials with tailoring the physical–chemical properties and electrical properties through different dopants have been achieved. On the experimental side, Ren et al.[21] have synthesized Ba1–YTiO3 solubility associating with the electron compensation prepared by using a solid-state reaction method. They reported that doping Y at A site can lead to high conductivity, appended by a giant permittivity and large dielectric loss. Also, the effects of rare-earth doping in BaTiO3 were experimentally prepared using a conventional powder sintering technique.[17] Hence, it was observed that the different rare-earth element possessed a critical effect on dielectric properties, where Y-doping showed the largest leakage current compared with that of other rare-earth elements. Recently, Liu et al.[22] have investigated the experimental synthesis of Y-doped BaTiO3 ceramics by using the sol–gel technique. They proposed a new method to evaluate the depletion layer width of Y-doped BaTiO3 grains. On the theoretical side, it has been suggested from interatomic potential-based defect calculations that defect clustering in rare-earth-doped BaTiO3 can introduce a significant strain on the lattice, resulting in positive binding energy.[18] So far, to the best of our knowledge, there is no theoretical investigation on optoelectronics properties of rare-earth-doped BaTiO3.

In this contribution, we intend to present a theoretical study of the structural, electronic, optical, and piezoelectric properties of rare-earth-doped BaTiO3. The first-principles methods we use for simulations are based on density functional theory accompanied by Tran–Blaha-modified Becke–Johnson (TB–mBJ) with generalized gradient approximation (GGA) + U approaches. To get a better understanding of the mechanism, the doping implications on the properties of BaTiO3, it is necessary to contrast the doping effects at both A and B sites of BaTiO3. Herein, Y is doped at both A and B sites of BaTiO3, and their effects on the crystal structures and the electronic and optical properties of BaTiO3 are compared. Their bonding characteristics are also shown. In addition, the Born effective charges and piezoelectric properties are investigated in detail.

Results and Discussion

The crystal structure of pristine BaTiO3 is tetragonal (the space group is P4mm) with Ti atoms positioned at the cube center, Ba atoms located at the cube corner, and O atoms located at the face centers. The optimized lattice parameter values and the average bond length between the neighboring elements for pristine BaTiO3 are gathered in Table as well as the experimental data[23] and theoretical works for comparison. Our calculated lattice parameters well reproduced the measured X-ray diffraction values.[23] In the optimized geometry of Y doping at the Ba site (Ti site) in BaTiO3, it is found that a- and c-lattice parameters change slightly with respect to those of pristine BaTiO3. This variation results from distortion of the tetragonal crystalline structure after Y doping. The optimized structures of pristine BaTiO3 and Y-doped BaTiO3 at both Ba and Ti sites are depicted in Figure .

Table 1

Lattice Parameters, a, c (Å), Volume, V (Å3), and Bond Length (Å) Results for Pure and Doped BaTiO3

	a (Å)	c (Å)	V (Å3)	Ti–O (Å)	Ba–O (Å)	Y–O (Å)
BaTiO3	3.999	4.208	67.30	2.072	2.914
	4.000[23]	4.024[23]	64.384[23]
	4.0257[25]	4.667[25]	65.3989[25]
	3.984[24]	4.066[24]	64.537[24]
Ba0.875Y0.125TiO3	3.967	4.063	63.94	2.017	2.845	2.526
BaTi0.875Y0.125O3	4.021	4.122	66.66	2.023	2.676	2.179

Figure 1

Optimized crystal structures of (a) pristine BaTiO3, (b) Ba0.875Y0.125TiO3, and (c) BaTi0.875Y0.125O3. Ba is shown in green, Ti in gray, O in red, and Y in cyan.

The calculated lattice constants of Y doping at both Ba and Ti sites are also listed in Table , where a fair agreement is observed from the experimental values published by Ren et al.[21] Our results show that the lattice parameters a, c and volume (V) of Ba0.875Y0.125TiO3 are slightly smaller by ∼4% than that of pure BaTiO3. For BaTi0.875Y0.125O3, the lattice parameter a is slightly larger, but c is rather smaller than that of the pristine perovskite. In a perovskite structure, the Ti–O bond lengths are composed of the octahedral site occupied by Ti atoms, whereas the Ba–O bond lengths are related to the dodecahedral site. The relaxed Ba–O and Ti–O bond lengths of pristine BaTiO3 are 2.914 and 2.072 Å, respectively, which are in excellent accord with the experimental data[23] and previous theoretical results.[24,25] For Y doping at both Ba and Ti sites, the process reduces the Ti–O and Ba–O bond lengths because of the lattice distortion. Our findings indicate that Y doping has long-range effect on the crystalline structure in which all interatomic bond lengths are modified.

The electronic band structures of pristine BaTiO3 and Y-doped BaTiO3 at both Ba and Ti sites are calculated by using TB–mBJ and TB–mBJ with GGA + U approaches, as shown in Figure . The electronic band structure of pristine BaTiO3 determined with TB–mBJ exchange potentials exhibited the valence band maximum (VBM) simultaneously at R- and M-points and conduction band minimum (CBM) at Γ-point. Such arrangement of valence and conduction band edges makes it an indirect band gap semiconductor of magnitude 2.78 eV, which is more consistent with the experimental band gap value (3.2 eV)[26] and theoretical works.[24,25,27] The substitution of Y in BaTiO3 at the Ba and Ti sites has significantly restructured its electronic band structure. As shown in Figure , the VBM of Y doping at the Ba site has been driven to lower energies in the valence band, while the VBM of Y doping at the Ti site moves to the higher energy situation. In contrast to the band structure of pure BaTiO3, the VBM in the case of Y doping at both sites appeared at Γ-point. Similarly, the conduction band has been altered significantly and shifted to lower energies for the Ba site and higher energies for the Ti site. The CBM in Ba0.875Y0.125TiO3 and BaTi0.875Y0.125O3 appeared at Γ-point in the BZ. Moreover, Y-originated impurity bands have been seen over the Fermi level for Ba0.875Y0.125TiO3. A larger energy gap has been seen between the conduction band edges and the valence band edges. The energy separation between the VBM and CBM at Γ-point in BZ amounts to 2.91 and 2.85 eV for Ba0.875Y0.125TiO3 and BaTi0.875Y0.125O3, respectively. We further investigated the electronic band structures of these materials by applying the Hubbard U-parameter to the d-band of Y and Ti elements. This has shown important effects on the electronic band structures of these materials. As shown by green-colored bands in Figure , the electronic band structure obtained with GGA + U has qualitatively similar dispersion to that determined only with TB–mBJ; however, the separation between valence and conduction band edges has been significantly reduced. For pristine BaTiO3, the GGA + U approach leads to a band gap of 1.69 eV, which is much smaller than the band gap value obtained with TB–mBJ as well as the experimental value. We can conclude that TB–mBJ provides an efficient framework for band gap prediction. Similarly, the separation between the valence band and the conduction band in the band structure determined with GGA + U has been recorded as 1.40 and 2.2 eV for Y doping at the Ba and Ti sites, respectively.

Figure 2

Electronic band structures of (a) BaTiO3, (b) Ba0.875Y0.125TiO3, and (c) BaTi0.875Y0.125O3 calculated with TB–mBJ and GGA + U.

To obtain a deeper insight into the changes in the electronic structure, we analyzed the density of states of the pure and Y-doped perovskite, as shown in Figure . The total and partial DOSs showed that the upper region of the valence band is mainly due to Ti 3d states mixed with 2p states. The minimum of the conduction band is dominated by unoccupied Ti 3d transition-metal states. However, all occupied Ti 3d states are strongly hybridized with the nearby O 2p states. It can be observed that Ba does not contribute to VBM and CBM, although it does provide electrons to balance the system charge.[25] Therefore, the band gap value of BaTiO3 depends on the relative energy positions of Ti 3d and O 2p states. With regard to the Y doping at both sites, the main feature of the VBM and CBM is similar to that of the archetype BaTiO3, indicating no evident change of the distribution of the VBM and CBM except for the band gap. It can be seen for Y doping that the VBM is mainly occupied by the Ti 3d states and 2p states, as shown in Figure b,c. Then, the CBM principally abides of the Ti 3d states with the presence of Y 4d states. The appearance of impurity bands is believed to result in the charge spillage across the Fermi level, which is likely to cause metallization of the BaTiO3 by substituting Y over Ba sites. The charge compensation in BaTiO3 by doping Y has also been reported experimentally by Ren et al.[21]

Figure 3

Density of states of (a) pure BaTiO3, (b) Ba0.875Y0.125TiO3, and (c) BaTi0.875Y0.125O3 using the TB–mBJ approach.

In order to get insight into the chemical bonding nature, the contour plots for the electron charge density of BaTiO3, Ba0.875Y0.125TiO3, and BaTi0.875Y0.125O3 are calculated and reported in Figure a–c. Analysis of Figure a–c indicates that the bonding between Ti and O atoms is characterized by a weak covalent character as a result of hybridization between the O 2p and the Ti 3d states, while the Ba–O bonding exhibits an ionic nature. In the case of Y doping at the Ba and Ti sites, the plot of Figure b reveals that a similar chemical bonding characteristic is observed between Y and O atoms. It can be shown that mixed covalent and ionic bonding is revealed for all materials.

Figure 4

Charge density maps for (a) pure BaTiO3, (b) Ba0.875Y0.125TiO3, and (c) BaTi0.875Y0.125O3.

Next, we present the optical spectra of pure and Y-doped BaTiO3 determined with the TB–mBJ exchange potential in the following. Figure represents the real and dielectric functions determined along the x- and z-axes. Because these materials have different lattice parameters along the x- and z-directions, the dielectric functions and the subsequent optical coefficient showed a considerable degree of anisotropy along the x- and z-directions. On the other hand, the optical spectra for these materials have been found isotropic because of similar lattice parameters along the x- and y-axes. Therefore, the optical spectra along the x- and z-directions have been shown in the present work.

Figure 5

Real (a,c,e) and imaginary (b,d,f) parts of dielectric functions determined for BaTiO3 (top), Ba0.875Y0.125TiO3 (middle), and BaTi0.875Y0.125O3 (bottom).

The static dielectric constants (ε0) obtained from the real part of the dielectric function of pure BaTiO3 shown in Figure a have been found to be ε0 ≈ 4.67 and ε0 ≈ 4.34 along the x- and z-axes, respectively. Similarly, the static dielectric constants obtained for Ba0.875Y0.125TiO3 and BaTi0.875Y0.125O3 are found at the x-axis of ε0 ≈ 14.01 and 9.78 at the z-axis of 6.04 and 9.66 at the x-axis, respectively. In this context, the incorporation of Y into BaTiO3 has been demonstrated to increase the static dielectric constants, which exhibited higher dielectric properties compared to that of pristine BaTiO3. Note that ε0 characterizes the electronic polarization features of an optical material. The large values ε0 determined for BaTiO3 and Y-doped BaTiO3 demonstrate the large electronic polarizability features of these materials. Moreover, Ba0.875Y0.125TiO3 shows higher dielectric constant than BaTi0.875Y0.125O3. εR achieves maximum values in the lower part of ultraviolet (UV) range of light at 4.22, 5.18 eV, and 4.75 for BaTiO3, Ba0.875Y0.125TiO3, and BaTi0.875Y0.125O3, respectively. With further increase in the photon energies, dispersion of εR goes through an abrupt decrease and achieve negative energies at 11.50 and 12.60 eV for BaTiO3, Ba0.875Y0.125TiO3, and BaTi0.875Y0.125O3, respectively. The positive and negative values of εR represent the dielectric and metallic behavior of these materials.

The imaginary part of dielectric function (εI(ω)) for pure and Y-doped BaTiO3 is displayed in Figure b,d,f. The peaks in the dispersion of εI represent the optical transition taking place, inter- and intra-band optical transition. Hence, the optical transition taking place between the occupied states in the valence band to the unoccupied states in the conduction band is encountered by peaks in the dispersion of εI. The sharp peaks in the x- and z-components of εI for BaTiO3 shown in Figure c are therefore believed to be originated from the optical transition taking place from the occupied states in the VBM located simultaneously at the R- and M-points in the BZ to CBM appeared at the G-point. Several peaks of low intensity have also been seen in the dispersion of εI determined for BaTiO3 at high photon energies. They are expected to take place between the occupied states in the deep valence band to the empty states in the conduction band. Moreover, the x-component of εI determined for Y doping at Ba and Ti sites exhibited a sharp peak at low photon energy of magnitude 0.21 and 0.15 eV, respectively. The observed peak is likely caused by the optical transition between the impurity band appeared in the vicinity of the Fermi level seen in Figure d,f.

Figure summarizes the calculated absorption coefficients of pure and Y-doped BaTiO3 along the x- and z-axes. As seen, these materials showed an exceptionally larger absorption of incident light. They showed moderate absorption in the lower part of UV range (below 15 eV). The absorption spectra of the investigated materials have experienced an abrupt increase with increase in photon energy beyond 15 eV. All materials showed slightly different optical absorption along the x- and z-directions. The maximum optical absorption for BaTiO3 along the x- and z-directions have been recorded as 3.63 × 106 cm–1 at 19.63 eV and 3.48 × 106 cm–1 at 19.81 eV, respectively. For Ba0.875Y0.125TiO3, the optical absorption approached as large as 2.71 × 106 cm–1 at 20.13 eV and 2.85 × 106 cm–1 at 19.79 eV along the x- and z-axes. In the case of BaTi0.875Y0.125O3, the optical absorption is found to be 3.39 × 106 cm–1 at 19.73 eV and 3.375× 106 cm–1 at 19.9 eV along the x- and z-axes.

Figure 6

Absorption spectra of (a) pure BaTiO3, (b) Ba0.875Y0.125TiO3, and (c) BaTi0.875Y0.125O3.

Figure illustrates the reflectivity spectra against the photon of pure and Y-doped BaTiO3. These materials have shown a moderate reflection in the visible and low UV ranges, and large reflection has been seen in the UV range beyond 20 eV. Figure shows that doping of Y into BaTiO3 has resulted in the reduction of its reflectivity features. These materials exhibited a slightly different reflectivity along the x- and z-directions. It has been found slightly larger along the x-direction for the photon energies below 10 eV; however, the reflectivity along the z-direction is found larger than the x-direction for the photon energies above 10 eV. For BaTiO3, the maximum reflection along the x- and y-directions has been recorded as 0.56 at 24.00 eV and 0.57 at 24.97 eV, respectively. Similarly, the maximum reflection along the x- and z-axes for Ba0.875Y0.125TiO3 is evaluated as 0.47 at 26.14 eV and 0.45 at 25.76 eV, respectively. For BaTi0.875Y0.125O3, the maximum reflection along the x- and y-axes is found to be 0.49 at 22.55 eV and 0.5 at 22.53 eV along the x- and z-axes, respectively.

Figure 7

Reflectivity spectra of (a) pure BaTiO3, (b) Ba0.875Y0.125TiO3, and (c) BaTi0.875Y0.125O3.

To investigate the optical transparency of the considered materials, we determined their refractive indices n(ω) as shown in Figure . As seen, the refraction spectra of these materials exhibited the refractive indices larger than unity in the infrared, visible, and in a larger part of UV range. The optical refraction reduced to less than unity for photon energies beyond 12.5 eV for investigated materials. Materials exhibiting refractive indices equal to or larger than unity are considered as transparent for the incident light. Therefore, it is reasonable to claim the pure and Y-doped BaTiO3 as transparent for the incident light below 12 eV and as opaque beyond 12 eV. The maximum values of refractive indices are estimated to be 3.08 at 4.41 eV and 2.96 at 4.65 eV along the x- and z-axes, respectively, for BaTiO3. Thus, we obtained that the maximum values for optical refraction are 3.84 at 0.01 eV and 2.99 at 5.44 eV in the case of Ba0.875Y0.125TiO3 and 2.98 at 0.01 eV and 2.98 at 4.04 eV in the case of BaTi0.875Y0.125O3 along the x- and z-axes, respectively.

Figure 8

Refraction spectra of (a) pure BaTiO3, (b) Ba0.875Y0.125TiO3, and (c) BaTi0.875Y0.125O3.

Calculation of Born effective charge (Z*) tensors has been considered as useful for the quantification of the charges associated with atoms/ions. For the present BaTiO3 and doped structure with yttrium, the Z* values, calculated with density functional theory by respecting charge neutrality sum rule, are gathered in Table . For the pure BaTiO3, by considering the ferroelectric direction set as z-axis, only the diagonal charge tensors are nonzero. In Table , we listed, however, the trace of the diagonal charges. Furthermore, oxygen sites show two equivalent directions producing two nonequivalent directions with a view to Ti–O bond, named O1, O2, and O3. We performed the present calculation using the optimized lattice parameters. For the pure BaTiO3, we reproduced the earlier results performed by Ghosez et al.[27] Obviously, the atypical larger (Z) values of Ti and O when compared to their nominal charges are associated with the aforementioned complex ionic and covalent character of bonding, showing a charge transfer between Ba 5p and O 2p as well as the hybridization between Ti 3d and O 2p orbitals (observed in partial densities of states). As the second step, we performed both structures doping Y at both A and B sites in BaTiO3. Our first observation was the difference in the Born charge tensors when compared to the pure one as the effective charges were neither diagonal nor symmetrical because of the structural anisotropy induced by the Y doping in different substitution sites: Ba and Ti. In order to compare these three different situations, we have calculated the eigenvalue of the tensors and reported the Born effective charges in the doped structures. It is important to note that the absolute values of the effective charges become slightly larger than those of the nominal (formal) charges (Y = +3; O = −2). When we analyze carefully the values of effective charges in both new doped structures, we might observe that substitution introduces anisotropy in Ba, Ti, and O charge tensors, while tensors for Y and Ba remain diagonal. Also, O atoms show an increase in dynamical charge by +10, +1.5, and 2% for the oxygen atoms along the polarization direction. In parallel to this, there is a pronounced decrease of the charge in the case of Ba atoms (51%) and Ti (29%) along the direction of polarization. In the case of Y doping at the Ti site, the trend in the effective charges is almost the same, but with the strong increase of O2 and O3 charges, less loss in effective charge in Ba compared to the former structure. Titanium has lost 25% of its effective charge, while Y increases by 14% compared to its charge when it replaces a Ba atom. The results indicate that the doped BaTiO3 with Y increases the anisotropy of Ti environments and at the same time a slight decrease in Ti–O and Ba–O bond lengths (see Table ), which leads to the appearance of anisotropy but more stabilized structure. The same behavior was observed in the recent work on Zr/Ca doping BaTiO3, where Born effective charges calculated have shown that doping BaTiO3 with Ca or Zr increases the dynamical charges on Ti as well as on O and decreases the dynamical charge on Ba.[16] Furthermore, it is well known that oxide perovskites produce anomalous Born effective charges values because of the Coulomb interactions and its destabilization that produces ferroelectricity. The pure BaTiO3 well reproduces this anomaly when doped with Y.

Table 2

Calculated Born Effective Charge Tensors (in e) (Trace) and Piezoelectric Constants (in C/m2) of Pure and Doped BaTiO3

ion/compound	BaTiO3	Ba0.875Y0.125TiO3	BaTi0.875Y0.125O3
Ba	2.76	1.35 (−51%)	2.56 (−9%)
Ti	6.28	4.40 (−29%)	4.70 (−25%)
O1	–2.63	–2.367 (+10%)	–2.0 (−23%)
O2	–3.20	–3.25 (+1.5%)	–4.5 (+40%)
O3	–3.20	–3.4 (+2%)	–4.48 (+40%)
Y		3.18	3.70
e33 (C/m2)	4.75
	4.78[30,31]

Ferroelectric oxides have the ability to not only exhibit a switchable polarization but distinct interesting functional properties for technological applications: the large dielectric and piezoelectric constants which are directly related to their ferroelectric character. Furthermore, piezoelectric materials have the qualification to produce an electric polarization besides the application of external macroscopic strain. Materials with piezoelectric character can be used in a variety of microelectromechanical systems and capacitors, ferroelectric random-access memory with the help of polarization reversion. The modern theory of polarization[28] has helped to understand the dielectric response of ferroelectric materials such as oxide perovskites. Using Quantumwise code,[29] we computed the electronic and ionic contributions separately. We also listed in Table the piezoelectric constant e33 (C/m2) for the tetragonal BaTiO3. Technically, it was impossible to calculate the piezoelectricity for the doped structures Ba0.875Y0.125TiO3 and BaTi0.875Y0.125O3 as they turned to metallic systems.

Conclusions

In the current studies, we reported a first-principles study of the electronic structures and optical properties of pristine and Y doping BaTiO3 at Ba and Ti sites using the full potential linear augmented plane wave method, with the modified Becke–Johnson potential as well as U correction to describe the d states of the Ti and Y dopants. The calculated band structure and the density of states have confirmed the semiconductor nature of the pure perovskite, whereas the doped structures have shown a shift of the conduction band through lower energies for Ba site and higher energies for Ti site. The linear optical properties as real and imaginary parts of dielectric constant and related optical coefficients shed more light about the changes in the behavior of the considered compounds. Our findings show that the dielectric constants for different axes have been increased because of the introduction of the Y dopant. We extended our interest to calculate the Born effective charges and piezoelectric properties by using polarization theory, finite difference approach, and Berry-phase approach. Compared to the original pure BaTiO3, Y doping has destroyed the piezoelectricity because they changed to metallic systems for both considered cases.

Computational Methodology

Our computational calculations were done with the WIEN2K code,[32] which is based on the linearized augmented plane wave method.[33,34] Exchange and correlation were treated within the GGA in the form of Perdew–Burke–Ernzerhof to express the exchange–correlation energy.[35] The TB–mBJ[36] method was used for the treatment of exchange–correlation effects. GGA + U methods were applied as well to describe the d states of Ti and the Y dopant. Here, U (equal to what is often called Ueff = U – J = 7 (Ti) and 4 (Y) eV) is used as the on-site interaction term, as proposed in refs (37−41). Moreover, the all-electron calculations were performed by combining the TB–mBJ approach with the Hubbard U correction to investigate their suitability in describing electronic structures. The cutoff Rmt × Kmax was set to 7.0 as for basis set. The Monkhorst–Pack special k-point approach[42] has been endorsed for the integration of the Brillouin zone (BZ). The integrals over the special points in BZ were performed with 8 × 8 × 2 k-point grid to warranty an acceptable energy convergence. The total energy was converged up to 10–5 Ryd/unit cell in the present self-consistent computations for well-defined results.

For Y-doped BaTiO3, we carried out ab initio calculations by adopting a 2 × 2 × 2 supercell that contains eight BaTiO3 formula cells. The dopant concentration of 12.5% has been modeled by substituting one Ba (or Ti) atom with one rare-earth Y atom in the 40 atom BaTiO3 supercell. Correspondingly, the concentrations are Ba1–YTiO3 for A site doping and BaTi1–YO3 for B site doping.

The vibrational properties, mainly Born effective charges and dielectric tensors, were calculated using the DFPT,[43] while the piezoelectric constants were calculated using the metric tensor formulation of strain as described in ref (44). For the polar BaTiO3, the piezoelectric responses can be nonzero. We calculated the piezoelectric coefficients as the derivative of the polarization with respect to the strain at zero electric field. In fact, for any solid, the total macroscopic polarization P is the sum of the spontaneous polarization Peq (strain independent) of the equilibrium structure in the absence of external fields, which induces piezoelectricity by strain Pp named strain-dependent. The total polarization is then P = Peq + Pp. The piezoelectric tensor can be expressed as . For the present perovskite, the γδα parameters were calculated using the finite difference approach, whereas the polarization was extracted from the Berry-phase approach as implemented in Quantumwise (ATK).[29]