1 Introduction

Perovskite oxides with general formula ABO3 have attracted much attention due to their excellent physical and chemical properties across a wide range of technological fields, including electronics, optics, optoelectronics, etc. Different A-site and B-site dopants (where A=Sr, Ba; B=Ti, Sn) are used to modify the electrical properties of perovskite [1,2,3,4,5,6,7,8,9,10,11,12]. A solid solution of both BaTiO3 and SrTiO3 produces the barium strontium titanate (BaxSr1−xTiO3, BST) [3, 13,14,15], or as mentioned by Yustanti et al. [16] and Pasha et al. [17] with the presence of Sr ions as a dopant in BaTiO3 formed BST crystal structure. BST is a promising material, due to its dielectric properties, adjusting the mole ratio of Ba/Sr to meet a wide variety of applications in electronics, such as microwave phase shifters, dielectric capacitors, dynamic random-access memories (DRAM), etc. Various preparation methods for BST have been investigated, as conventional solid-state reaction [15, 18,19,20], solvothermal [21], sol–gel [22, 23], coprecipitation [24], spray pyrolysis[25], organic precursor method [26] and hydrothermal methods [13, 27].

The pure compounds, BaTiO3 and SrTiO3, have the same cubic perovskite structure in the high-temperature phase. Barium titanate is a typical ferroelectric, which undergoes three consecutive phase transitions from a cubic Pm3m to a tetragonal P4mm phase at 403 K, then to an orthorhombic Amm2 phase at 278 K, and, finally, to a rhombohedral R3m phase at 183 K. Studies on the BaxSr1-xTiO3 report the dependence of the temperatures of such phase transitions versus Sr content (x) [16, 18, 28, 29].

The ideal perovskite has a cubic space group Pm-3 m that contains one unit formula per cell. In particular, the A-site cation is 12-fold coordinated, while the B-site cation is coordinated by six O anions in a body-face-centered cubic structure[30]. Furthermore, most of the perovskite materials can present a slightly distorted structure obtained from the cubic structure by different ways, such as the displacement of ions from the ideal position, tilting the BO6 octahedral, or defects in the structure that can break the cubic symmetry resulting in a “non-cubic” structure [10, 31].

Theoretical studies of perovskite compounds are of particular significance because they can be performed by different methods helping to explain the experimental results [32]. However, if a dopant is introduced in the compound, its properties are changed, and the ab initio simulations become far more complicated. In this case, the doping of titanate-based perovskites with several kinds of metals cations is widely performed because of the vast range of peculiar properties it allows to tune in fields as diverse as structural, electrical, and optical properties [33,34,35].

In this study, DFT calculations were applied to rationalize the effects of local structural changes induced by the introduction of Sn4+ in the crystal lattice and on the change in geometry of BST structure. We also demonstrate that the electronic properties are associated with the presence of different clusters with a unique bonding environment from electron density distribution. In particular, the formation of intermediate levels in the bandgap (Egap) region was discussed following the structural disorders. Also, the thermodynamic stability of the pristine BST (cubic and tetragonal; BSTc and BSTt) and Sn-doped (BSTSc and BSTSt) structures were analyzed to discuss the primary role of doping on the geometry stabilization. Notably, our models support the available experimental results, showing an excellent agreement. The manuscript is organized into three other sections in the following manner: (i) the next Sect. (2) describes in details the employed computational methodologies; (ii) Sect. 3 contains the results for structural, electronic, vibrational, and energetic properties; (iii) at last, the Sect. 4 contains the main conclusions of our work.

2 Computational details

The computational methods and theoretical procedures have been employed to study the bulk related to pristine cubic and tetragonal BST, as well as Sn-doped BST models with a doping amount of 50%, respectively. Calculations were carried out using the periodic ab initio CRYSTAL14 package [36] based on Density Functional Theory (DFT) using hybrid functional B3LYP [37, 38]. This computational technique has been successfully applied for the study of the electronic and structural properties of various materials, including perovskite and several other oxides[39,40,41,42,43,44,45,46,47,48,49]. In all calculations, the atomic centers were described by standard all-electron basis set 6–31G*, 976–41(d51)G, 9763–311(d631)G, 86–411(d31)G, and 9,763,111,631 for the O, Sr, Ba, Ti, and Sn atoms, respectively[40, 41, 50, 51].

Further, the electronic properties were analyzed in terms of the Density of States (DOS), Band Structure profiles, and the charge density map. Besides, Visualization for Electronic and Structural Analysis (VESTA) [52] and X-Window Crystalline Structures and Densities (XCrySDen) [53] software were used for visualization and representation of supercell models, X-ray diffraction (XRD) pattern calculation and the charge density maps construction. Electronic integration was performed using a dense 8 × 8 × 8 Monkhorst–Pack [54] k-mesh for the pristine and doped cells, containing 75 k-points for both BST (cubic and tetragonal) and Sn-doped BST models. The accuracy of the Coulomb and exchange integral calculations were controlled by five thresholds set to 8, 8, 8, 8, and 16. The converge criterion for mono- and bi-electronic integrals were set to 10–8 Ha, while the root-mean-square (RMS) gradient, RMS displacement, maximum gradient, and maximum displacement were set to 9.4 × 10–5, 7.3 × 10–4, 1.4 × 10–4, and 1.2 × 10–3 a.u. for BSTc, 1.9 × 10–4, 8.3 × 10–4, 3.4 × 10–4, and 1.7 × 10–3 a.u. for BSTSc, and 6.1 × 10–5, 4.7 × 10–4, 1.3 × 10–4, and 8.2 × 10–4 a.u., for BSTt respectively. In all cases, both lattice parameters and atomic positions were relaxed.

In this work, a cubic and tetragonal supercell model of 10 atoms, which corresponds to a 1 × 1 × 2 conventional cell, was used to simulate both pristine and Sn-doped BST structures. In particular, for the Sn-doped BST model, a 1 × 1 × 2 supercell (10 atoms) was considered, where one Ti4+ (ionic radius 0.605 Å) cations were replaced by Sn4+ (ionic radius 0.690 Å)[55] corresponding to a doping concentration of 50%.

3 Results and discussion

3.1 Structural properties

The initial lattice parameters and atomic position used in the optimization process were obtained from the BSTc, BSTt, BSTSc, and BSTSt, results of the Rietveld refinement, according to Souza et al. [56] and Chihaoui et al. [57]. The calculated values for the pristine barium strontium titanate (Ba,Sr)TiO3 (BST) (cubic and tetragonal; BSTc and BSTt) and Sn-doped (BSTSc and BSTSt) structures are collected in Table 1 and compared with experimental results.

Table 1 Theoretical and Experimental lattice parameters, unit cell volume, and percentual error for BSTc, BSTt, BSTSc, BSTSt materials
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As can be observed in Table 1, the crystal structure and the lattice parameters of theoretical results exhibit a small mean percentage error in comparison with the experimental results, evidencing that our calculations are in agreement with the experiments. Moreover, the unit-cell volume was underestimated by 1.41% and 3.01% for pristine BSTc, and BSTt, while the obtained results for BSTSc and BSTSt indicate an overestimation of 2.88% and 1.83%, respectively. Furthermore, comparing the crystalline parameters for both pristine and Sn-doped BST models, it was observed that for cubic polymorphs (BSTc and BSTSc) an expansion of cell parameters of ~ 2.13% was found after the Sn doping, as well as for BSTt and BSTSt models with a variation of 5.52%. This behavior can be explained by the differences between the ionic radius of Sn4+ (0.690 Å) and Ti4+ ions (0.605 Å).

From a theoretical point of view, three models were constructed using a conventional 1 × 1x2 supercell containing 10 atoms. First, structural and electronic properties were calculated for a perfect bulk of BSTc, and BSTt lattice, and BSTSc, posteriorly. The representation of the BSTc, BSTt, and BSTSc bulk structures is shown in Fig. 1. This Fig. 1(a-c) illustrates the green, blue, gray, and red balls correspond to Ba, Sr, Ti, Sn, and O atoms, respectively. In this case, the Ti and Sn atoms are coordinated by six O atoms, producing octahedral [TiO6] and [SnO6] clusters. Correspondingly, the Ba and Sr atoms are coordinated to twelve O atoms, resulting in [BaO12] and [SrO12] clusters.

Fig. 1
figure 1

Conventional unit cell for pristine (Ba,Sr)TiO3 and Sn-doped structures a cubic, b tetragonal polymorphs, c [Ba,SrO12], [TiO6] and [SnO4] clusters

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Furthermore, in order to analyze the structural disorder caused by the change in geometry, as well as the role of the Sn-doping in BST matrix, the bonding environment of both [TiO6], [SnO6], [BaO12] and [SrO12] clusters was investigated in details. Table S1 (Supplementary Information) shows the calculated B3LYP values of bond distances (M–O) and the atomic coordinates in Cartesian coordinates (Å) along the 1 × 1 × 2 supercell used for BSTc, BSTc, BSTSc, BSTt, and BSTSt, respectively.

It is important to note that atomic coordinates and M–O bond distances are stretched and shortened in the cubic, tetragonal and doped structures, respectively (see Table S1). This effect can be related to the structural order–disorder effect associated with the change in geometry and the insertion of the Sn dopant. Besides, the bond distances (M–O) undergo a slight increase in comparing Sn4+ and Ti4+ centers. The increase in bond distances (M–O) can be due to three reasons: (i) the distortion of the environment of the A ion during to tilting, (ii) the Sr atom is partially substituted by an atom (Ba) of higher ionic radios and (iii) the Jahn–Teller effect in by the partial removal of the eg and t2g degeneracies from cubic to tetragonal models. In previous studies, the same behavior has been observed by Joseph et al. [58].

Additionally, the structural study of the pristine BSTc and BSTt, BSTSc, and BSTSt crystals was performed by X-ray diffraction (XRD) pattern using VESTA software. The XRD pattern for the BSTc, BSTSc, BSTt, and BSTSt structures is shown in Fig. 2(a-d), and indicates a single-phase cubic and tetragonal perovskite with the Pm-3 m and P4mm space group, respectively.

Fig. 2
figure 2

Simulated X-ray diffraction patterns of a BSTc, b BSTSc, c BSTt, and d BSTSt models by VESTA software

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The simulation of XRD patterns for the cubic BST and BSTS (Fig. 2a-b) is in good agreement with the crystallographic report JCPDS 39–1395, according to experimental data reported by Souza et al. [56]. In both crystals, the strongest peak around 2θ = 32°corresponds to the (110) crystalline plane; however, doping with Sn4+ cations induces a shift of the main reflections toward 2θ values with the formation of clusters of [SnO6], from 32.11 to 31.42(°), due to the different electronic density to Ti4+ ions, which can be seen from the strongest diffraction peak (Fig. 2a–b). The same was also observed for the following diffraction peaks changing from 39.60 to 38.87(°); 46.05 to 45.03(°), 57.24 to 55.96 (°), continuously for BSTc and BSTSc. Moreover, in the doped system, some crystallographic planes were barely noticeable due to the low intensity as (100), (300), (212), (026), and (304). This observation is a strong indication of the successful substitution of Ti4+ by Sn4+ in the B sites of the perovskite BSTc material. Our observation for the simulation of XRD patterns of BSTt (Fig. 2c–d) follows the previous study reported by Chihaoui et al. [57]. The tetragonal phase and space group P4mm is confirmed by the presence of peaks (001), (100), (101), (110), (012), (111), (004), (200), (002), (102), (210), (211), (112), (220), (202), (204), (300), (212), (103), (310), (032), (106), (016), ((132), (312), (116), (224) and (133). In this case, similar behavior was observed for the doped system, as mentioned in the cubic system. In pristine BSTc and BSTt crystals structures, the main observation was the appearance of crystallographic plans, as shown in Fig. 2a and c.

3.2 Electronic properties

In this section, we present the theoretical results for the electronic properties of the investigated perovskite compounds. The electron distribution plays a fundamental role in determining the band structure, the density of states (DOS), and the charge density. The band structures plotted along the path Γ (0,0,0), X (½,0,0), M (½,½,0), Z (0,0, ½), R (½,0, ½), and A (½,½,½), as well as the calculated DOS projected for the atoms of the BSTc, BSTt, BSTSc, and BSTSt crystals are displayed in Fig. 3a–b. These figures show that the shaded region in the band structure indicates the bandgap region, while pink bands indicate the valence band maximum (VBM) and conduction band minimum (CBM).

Fig. 3
figure 3

a Band structure profiles and b atom-resolved DOS profiles for BSTc, BSTt, BSTSc, and BSTSt

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An analysis of the band structure and projected DOS are presented in Fig. 3a–b. For BSTc it was observed that the valence band (VB) is observed between 0 and − 4 eV. The conduction band (CB) was evaluated between 3.73 eV and 12 eV. The calculated indirect band gap energy was 3.73 eV, being an indirect electronic excitation between M and Γ points. On the other hand, for the BSTt structure the VB was taken between 0 and − 4 eV, while CB is between 3.76 and 12 eV. Regarding the bandgap region, an indirect electronic transition between M- Γ points was calculated around 3.76 eV. From BSTSc illustrated in Fig. 3, the VB was evaluated between 0 and − 4 eV. The CB is between 3.77 and 18 eV. For BSTSt, the VB was evaluated between 0 and − 4 eV and CB is between 3.77 and 18 eV, where a bandgap energy of 3.77 eV was calculated between M-G.

The introduction of Sn4+ dopants causes slight changes in the bandgap energy (3.77 eV) of BSTSc, originating from the intermediate electronic level due to the appearance of the localized states into the CB region from 4d Sn atomic orbitals. The dopant forms clusters that influence the short, medium and long range, the electronic density of the crystal, due to symmetry breaking. In this way, the results show the role of the dopant by introducing new properties to the semiconductor. This analysis is valid for the other dopants. Moreover, the detailed analysis of the upper panel of Fig. 3 indicates that Sn-doping mechanism affects the band distribution, mainly in the VB. Comparing the bands distribution for BST and BSTS models at different crystalline structures, it was noted that occupied electronic levels for BSTS are more degenerated in comparison with BST, which can be associated with the bonding character of Sn–O and Ti–O chemical bonds.

Besides, comparing the CB for both models of BST and BSTS enables us to interpretate the charge carriers properties (electrons) from the curvature of Conduction Band Minimum (CBM). Indeed, for BST models the CBM located at G point is almost linear up to X, showing a broad feature, while the CBM for BSTS models at the same point exhibit a parabolic-like distribution with a well-located minimum point. Based on the relation between the effective mass of charge carriers with the band curvature, we can argue that electron–hole recombination rate is distinct between BST and BSTS models. In this case, a broader band can induce a higher effective mass for the excited electrons reducing its mobility, while a well-defined parabolic band can be associated with a reduced effective mass and higher electron mobility. Therefore, the obtained results for band structure profiles of BST and BSTS indicate that Sn-doping induces a higher electron mobility, making BSTS a good candidate for electro-optical applications.

As regard the obtained bandgap values, experimental and theoretical values were compared, as shown in Table 2, evidencing a good agreement. Indeed, the B3LYP hybrid functional showed a very close representation of the experimental bandgap energy, showing deviation ranging from 2.19 to 13.5%. Here, it is important to point out that bandgap description is a challenging topic for quantum mechanical calculations due to the treatment of exchange–correlation effects. However, the obtained results confirm the predictive power of hybrid B3LYP functional for perovskite materials.

Table 2 Calculated and experimental values of the bandgap energy of BSTc, BSTt, BSTSc, and BSTSt
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Additionally, Fig. 3b summarizes the analysis atomic contribution for both VB and CB, showing a well-defined pattern that is directly associated with the local clusters centered on Ba, Sr, Sn, Ti, and O atoms. The main contribution to the VB region is due to the 2p (px, py, pz) orbitals of the oxygen anions with a small content of Ba and Sr orbitals for both BST and BSTS models. In contrast, the CB was mostly based on empty valence (3dxz, 3dxy, 3dyz, 3dz2, 3dx2-y2) orbitals from Ti atom and a small content of Sn cations hybridized with oxygen atomic orbitals, revealing the role of [TiO6] and [SnO6] clusters. These results confirm the role of Sn-doping mechanism in the control of CBM distribution, which can be associated with the electron mobility along the electronic structure, as previously discussed.

Electronic density maps of the BSTc, BSTt, BSTSc, and BSTSt, were obtained from the optimized wavefunction, where the electronic density matrix was resolved as isolines that describe the density in an area, as an exhibit in Fig. 4(a-d). These electronic density maps were described along the Ba–O, Sr–O, Ti–O, and Sn–O bonds direction of the materials, which corresponds more specifically to the diagonal (110) plane for all models of cubic and tetragonal symmetries (Fig. 4a–d).

Fig. 4
figure 4

Electron density maps in the diagonal plane (110) for a BSTc, b BSTSc, c BSTt and d BSTSt crystals structures

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Figure 4(a–d) illustrates the strongly covalent character in the interaction of the Ti and Sn atoms with the oxygen atoms on the analyzed (110) plane, which is represented by homogenous distributions of the contour lines. The observed behavior takes place because of the hybridization between the oxygen 2p atomic orbitals and the Ti (3d) and Sn (4d) atomic orbitals. The density distribution remained the same for Fig. 4(a–c) illustrating the inhomogeneous distribution of the contour lines that represent the ionic character of the Ba–O and Sr–O bonds.

Furthermore, the significant difference between the contour plots reported in Fig. 4(a–d) is associated with the charge corridor along the z-axis for BSTc, BSTt, BSTSc, and BSTSt models. In particular, for both cubic models (BSTc and BSTSc), the charge corridor exhibits a homogenous distribution for electron density along the Ti–O–Ti or Sn–O–Ti bond path. On the other hand, for BSTt and BSTSt, the intermetallic Ti–O–Ti showed a charge depletion area for a bridge-like oxygen atom, confirming the existence of spontaneous polarization along the [001] direction due to the off-centering atomic displacement for Ti atoms in tetragonal symmetry [3].

3.3 Vibrational properties

According to group theory, cubic perovskite with space group, \({O}_{h}^{1}\), Pm-3 m (221), has no Raman-active phonon modes. In contrast, tetragonal perovskite with space group,\({C}_{4v}^{1}\), P4mm (99) has the following phonon modes at the Γ point:

$$ \Gamma_{{{\text{optical}}}} = \, 3A_{1} + \, 4E \, + \, B_{1} $$

In the cubic phase (m-3 m), the zone-center optical phonons belong to 3F1u + F2u irreducible representations. Each of the Fu modes is triply degenerate, and all of them are of odd symmetry concerning the inversion, therefore, Raman inactive. The F1u modes are infrared active, while the F2u modes are silent. Upon transition to the tetragonal phase (4 mm), the F1u modes split into A1 and E modes, and the F2u phonon gives rise to B1 and E modes.

The A1 and E modes are both Raman and infrared active, while the B1 mode is only Raman active. The four optical E modes for the tetragonal phase are doubly degenerate and polarized along the x and y axes, and the three optical A1 modes are polarized along the z-axis. The E and A1 modes are split into longitudinal (LO) and transverse (TO) components as a result of long-range electrostatic forces associated with lattice ionicity [17, 61, 62]. Table 3 summarizes the experimental Raman active modes (cm−1) for tetragonal or pseudo-cubic perovskite BST.

Table 3 Experimental Raman active modes (cm−1) for tetragonal or pseudo-cubic perovskite
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Thus, tetragonal BST has the Raman active crystal symmetry of \({C}_{4v}^{1}\), showing distinguishable Raman peaks at room temperature, in contrast to the cubic BST of the Raman inactive \({O}_{h}^{1}\) symmetry, for which the peaks disappear. This means that it is possible to discern crystal phases according to the Raman spectrum.

Figure 5 shows the calculated Raman spectra of the BSTt and BSTSt at the B3LYP level of theory in the frequency range 100–900 cm−1. On the Raman spectrum of the tetragonal BST crystal structure, we observe seven active vibration modes observed at 121, 178, 280, 298, 546, 579, and 801 cm−1. It is well known that DFT calculation at the B3LYP method tends to overestimate the values of the vibrational frequencies; therefore, a scaling factor of 0.94 is used [66].

Fig. 5
figure 5

Theoretical Raman spectra calculated for BSTt and BSTSt structures

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Analyzing Fig. 5, it was observed that theoretical results are in good agreement with experimental data reported by Chihaoui et al. and other papers presented in Table 3. The bands around 180, 280 and 579 cm−1 are assigned to the transverse optical (TO) mode of A1 symmetry, whereas the band around 298 and 801 cm−1, which are characteristic of the tetragonal phase, are attributed to the E (TO + LO) + B1 mode and A1(LO) + E(LO) [34]. The A1 longitudinal optic (LO) modes can be identified at 121 and 546 cm−1. Nine peaks ascribed to the BSTSt Raman spectrum are observed at 100–900 cm−1 and seven for BSTt. This behavior means that the substitution for Sr and Sn in the A and B sites of the perovskite structure introduces a significant disorder in the structure, favoring the relaxor character in the BSTS material.

3.4 Formation enthalpy and thermodynamic stability

The discussion involving the thermodynamic stability of perovskite polymorphs corresponds to an important topic due to the structural transformations associated with the symmetry order. In this context, the evaluation of Formation Enthalpy (ΔHf) from its forming binary oxides plays a fundamental role in the discussion of phase stability [67,68,69]. From a theoretical point of view, such a process can be interpreted as a function of the reaction involving the perovskite material (ABO3) and its component oxides (AO + BO2). In this case, the ΔHf can be calculated as the difference between the lattice energies, following the expression [70, 71]:

$$\Delta {H}_{f}={H}_{AB{O}_{3}}-\sum_{i}^{\mathrm{metal} \mathrm{oxides}}{H}_{i}$$

where Hi = ET + pV. Here, ET is the total energy the solid at P = 0, and the corresponding equilibrium structures were obtained by optimizing all of the geometric parameters. Additionally, the zero-point energy contribution was estimated from vibrational analyses for comparative purposes.

Herein, the overall phase stability was calculated for BSTc, BSTt, BSTSc, and BSTSt perovskites, respectively. Table 4 summarizes the obtained results for ΔHf as a function of the component oxides (BaO, SrO, TiO2 for BSTc and BSTt; BaO, SrO, TiO2, SnO2 for BSTSc and BSTSt, respectively).

Table 4 Calculated values of Formation Enthalpy (eV) before and after ZPE corrections for BSTc, BSTt, BSTSc, and BSTSt perovskites, respectively
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At first glance, it was observed that all investigated reaction channels indicate that BSTc, BSTt, BSTSc, and BSTSt perovskites are stable in comparison with its component oxides, once negative values were founded. Moreover, the contribution of ZPE was calculated to be 0.022, 0.036, 0.038, and 0.041 eV for BSTc, BSTt, BSTSc, and BSTSt perovskites, respectively. However, the stability order remains the same, even with the ZPE correction.

Further, comparing the ΔHf for different polymorphs of BST (BSTc and BSTt), the calculated values indicate that the cubic phase is more stable in comparison with tetragonal symmetry. However, the values are quite similar, indicating the existence of a few differences between the crystalline structures. Similarly, the obtained values for tetragonal and cubic polymorphs of BSTSt indicate that the high-symmetric cubic lattice is more stable than tetragonal lattice, notwithstanding the polymorphs are very close in energy. Therefore, it was possible to observe that BST and BSTS perovskites are stable in comparison with their component oxides, being the cubic polymorph the low-energy arrangement expected in the complex phase diagram with the tetragonal phase.

3.5 Dielectric properties

Perovskite materials, such as BST, have been attracted an increased interest due to their inherent high dielectric constant [72,73,74]. In this context, the possibility to rationalize the dielectric properties through quantum–mechanical calculations correspond to an alternative tool to design new materials with potential technological applications. Herein, the static dielectric tensor (ε0) was evaluated by means of Couple-perturbed Kohn–Sham scheme (CPKS) combined with longitudinal-transverse optical (LO-TO) splitting, [75,76,77] as reported in Table 5.

Table 5 Independent components of static dielectric tensor (ε0) for BSTt and BSTSt perovskites, respectively
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Our theoretical results for both BSTt and BSTSt materials show that both oxides were described by two independent components of the static dielectric (ε0) tensor, where the \({{\varvec{\varepsilon}}}_{0}^{{\varvec{z}}{\varvec{z}}}\) is the dominant. Moreover, our purpose is to investigate the dielectric behavior of BSTSt materials in order to compare with BSTt. Therefore, it was observed that for BSTSt the calculated values of the static dielectric tensor are smaller than for BSTt for the dominant \({{\varvec{\varepsilon}}}_{0}^{{\varvec{z}}{\varvec{z}}}\) component. On the other hand, the \({{\varvec{\varepsilon}}}_{0}^{{\varvec{x}}{\varvec{x}}}\) component for BSTt becomes higher than for BSTSt. This effect may be related to a higher mobility of Sn atom, which induces a greater asymmetry in the electronic density of BO6 octahedral reducing the dielectric constant. In particular, the obtained results confirms the superior dielectric behavior of BSTt, but also shed a light on BSTSt perovskite materials as promising candidates for dielectric devices.

4 Conclusion

In summary, pristine cubic and tetragonal BST (BSTc and BSTt) and Sn-doped BST (BSTSc and BSTSt) were investigated from the Density Functional Theory calculations to gain an in-depth understanding of structural, electronic, vibrational and energetic properties. XRD patterns by VESTA software and DFT analysis confirmed the perovskite phases showing both Pm-3 m and P4mm space groups in pristine and Sn-doped BST structures. The building blocks of the BSTc, BSTt, BSTSc, and BSTSt crystals, i.e., a local coordination structure for both the Ba, Sr, Ti, and Sn atoms, were confirmed as deltahedral [Ba,SrO12], and octahedral [TiO6], and [SnO6] clusters. In the viewpoint of thermodynamic analysis, all models were calculated to be stable in comparison with their component oxides. The indirect bandgap energy was calculated as 3.73, 3.76, 3.77, and 3.77 eV for BSTc, BSTt, BSTSc, and BSTSt structures, respectively. Moreover, the detailed analysis of electronic structures for BST and BSTS models indicates that Sn-doping induces a higher electron mobility, being a potential candidate for electro-optical applications. Such results indicate that the change in geometry and insertion of the Sn atom generates an intrinsically order–disorder effect, which will promote a presence of intermediate levels, causing an increase in the gap energy. According to the charge density maps, the periodic models showed well-defined electron density distribution characterizing the symmetrical structure and confirmed the covalent bonds between transition Ti and Sn metals with oxygen atoms, while the Sr–O and Ba–O bonds were defined as ionic. Moreover, the vibrational analysis indicates the presence of singular disorders for BSTt and BSTSt associated with the doping process along with the A- and B-site of perovskite structure, resulting in a more pronounced relaxor character for BSTSt. For the dielectric applications, it was observed that dielectric tensor components for BSTSt were comparable with the calculated values for BSTt, suggesting a potential application in dielectric devices. These results were an essential tool for understanding the electronic and structural effects caused by a change in geometry and element doping on the perovskite BST structure.