Abstract
The study investigated the effects of Sr-doping on BaTiO₃ regarding the mean square displacement, diffusion coefficient, polarization-strain response, dielectric constant, and dielectric loss. Initially, increasing strontium doping up to 6% enhanced the mean square displacement (from 0.211 to 0.218 Å2) and the diffusion coefficient (from 0.361 to 0.380 nm2/ns) due to improved atomic mobility and lattice dynamics. However, further doping to 8% caused a decrease in both the mean square displacement and the diffusion coefficient, due to structural disorder and defects that impeded atomic movement. Similarly, the piezoelectric coefficient increased from 273.07 pC/N to 321.15 pC/N with 6% doping but declined to 308.65 pC/N at 8%. This indicated enhanced polarization at moderate doping levels but reduced performance due to excess impurities. The dielectric constant arose from 76 to 85 with 6% strontium but decreased to 80 at 8%, reflecting increased ferroelectric properties followed by structural degradation. Lastly, dielectric loss increased from 0.06 to 0.08 with 6% doping and then decreased to 0.073 at 8%. This suggested that while ionic mobility initially improved energy loss, excessive doping led to inefficiencies. Overall, optimal strontium doping enhanced the material’s ferroelectric and piezoelectric properties, while excessive doping introduced detrimental effects.
Introduction
Barium titanate (BaTiO3) is a prominent ferroelectric ceramic material known for its remarkable electrical properties, including high dielectric constant, piezoelectricity, and pyroelectricity1,2. This inorganic compound exhibits a perovskite crystal structure, which allows it to undergo phase transitions that enhance its ferroelectric characteristics3,4. BaTiO₃ is widely utilized in various applications, particularly in multilayer ceramic capacitors, making its efficient charge storage capacity crucial5. Additionally, its piezoelectric properties make it suitable for use in sensors and actuators, enabling the conversion of mechanical energy into electrical energy and vice versa6,7. Beyond electronics, BaTiO3 was garnered attention in the biomedical field for applications, such as drug delivery and cancer therapy due to its biocompatibility and therapeutic potential8,9. The doping of BaTiO3 with strontium (Sr) is a significant area of research that aims to enhance its piezoelectric and dielectric properties10. Sr doping modifies the lattice structure of BaTiO3, leading to improved stability and performance under various conditions11. This alteration can enhance the material’s dielectric constant and reduce dielectric loss, making Sr-doped BaTiO₃ particularly valuable for high-frequency applications. The introduction of Sr also influences the phase transition temperatures (Ts) of BaTiO₃, optimizing its performance in devices across various T ranges. Consequently, understanding Sr doping effects is vital for tailoring BaTiO₃ ceramics to meet specific performance requirements in advanced electronic applications.
Chen et al.12 examined the phase characteristics and electrical properties of lead-free BaTiO3 piezoelectric ceramics. Their findings indicated that an excessive amount of Sr contamination negatively impacted the piezoelectric performance of BaTiO3-based ceramics. Salem et al.13 focused on the structure and dielectric properties of doped BaTiO₃, revealing that increased zirconium contamination resulted in a decrease in both piezoelectric coefficient and grain size. However, they noted that remnant polarization rose with increased zirconium doping, indicating potential in permanent memory devices. Gang et al.14 explored three-dimensional polydimethylsiloxane/BaTiO3 elastomer networks for piezoelectric energy harvesting, finding that nanoparticle addition enhanced piezoelectric response. Yu et al.15 investigated the piezoelectric properties and biocompatibility of BaTiO3 produced via digital light processing. Their results demonstrated that ceramic scaffolds subjected to piezoelectric stimulation significantly promoted cell proliferation and growth compared to unstimulated scaffolds. Xiang et al.16 studied the self-assembly of BaTiO₃ nanoparticles for piezoelectric catalysis in tumor treatment. They found that these assembled nanoparticles not only caused mechanical damage to tumor cells but also improved piezoelectric catalytic efficiency. This resulted in generating more reactive oxygen species under ultrasonic stimulation compared to monodispersed counterparts, attributed to the confined space during collection. Rehman et al.17 examined the influence of silica nanoparticles on the dielectric properties and energy storage capabilities of BaTiO₃, reporting that adding 1% silica by weight enhanced the dielectric properties, achieving a peak relative permittivity of insert specific value or context if available. Arshad et al.18 synthesized Sr-doped BaTiO3 ceramics and found that increasing Sr content reduced Curie T, dielectric constant, and structural disorder up to x = 0.3. Oxygen vacancies were identified as main contributors to conduction, and stable dielectric properties were observed across a wide frequency range. Yadav et al.19 investigated Sr-doped BaTiO3 and observed enhanced ferroelectric properties compared to pristine BaTiO3. Sr doping lowered structural transition Ts and increased remnant polarization, reaching ∼8 µC/cm2 at phase transitions. Tian et al.20 synthesized Sr-doped BaTiO3 ceramics and studied their phase, dielectric, ferroelectric, and piezoelectric properties. They found that Sr promoted phase transition and improved properties, with optimal performance observed at 0.12 Sr content, whereas excessive Sr resulted in decreased piezoelectricity. Kumari et al.21 synthesized Ba(1−x)SrxTiO3 nanoparticles, finding that Sr doping (x > 0.3) shifted the crystal structure from tetragonal to cubic. Ba₀.₇Sr₀.₃TiO₃ exhibited a high dielectric constant of 4915 at 1 Hz with low loss, making it particularly suitable for electronic applications. Esmaeili et al.22 used molecular dynamics (MD) simulations to study the effect of T on the piezoelectric properties of tetragonal BaTiO3. They found that increasing the T from 300 to 400 K reduced the piezoelectric coefficient, saturation polarization, coercivity field, and residual polarization, while the dielectric coefficients varied between 64 and 73 depending on the mode.
Investigating the impact of Sr doping on the piezoelectric properties of lead-free BaTiO3 ceramics through MD simulations is essential for enhancing the performance of these materials. This study examined various parameters, including stress-strain (SS) curve, mean square displacement (MSD), diffusion coefficient, strain-electric field (S-E) response, hysteresis loops (butterfly and ferroelectric), dielectric constant, and dielectric loss at different levels of Sr doping. The significance of this research lies in its potential to optimize the piezoelectric features of BaTiO3 ceramics, making them more suitable for applications in sensors, actuators, and energy harvesting devices. By elucidating how Sr doping influenced these properties, the study aimed to contribute to the development of advanced lead-free piezoelectric materials that comply with environmental regulations while maintaining high performance. The findings could facilitate the design of next-generation electronic components that were efficient and sustainable, addressing growing demand for eco-friendly technologies across various industrial sectors.
Molecular dynamics simulation
Present simulation details
This research investigated how Sr doping influenced the piezoelectric characteristics of BaTiO3 ceramics using MD simulations. The first step involved creating a structural model with Avogadro software23 with the dimensions of 40 × 40 × 40 Å3. Next, Sr with 2, 4, 6, and 8% were replaced instead of Ba particles. To maintain a consistent simulation environment, periodic boundary conditions were implemented in all three spatial directions. After constructing the model, the simulation unfolded in two main stages. The initial stage aimed to stabilize the simulated structure using NVT ensemble, keeping T constant at 300 K. Nose-Hoover thermostat was used for effective T control, and fluctuations in T and potential energy (PE) were monitored for 10 ns to evaluate the equilibration process. Upon reaching equilibrium, the simulation shifted to the NVE ensemble, enabling an investigation into the piezoelectric properties of materials. Sr doping in BaTiO3 significantly enhanced its piezoelectric and dielectric properties. By replacing Ba ions with Sr, the stability of perovskite structure improves, which was critical for maintaining the material’s ferroelectric properties. The replacement of Ba ions was carried out randomly, with specific ions selected for substitution determined by a random distribution algorithm. This approach ensured that Sr ions were uniformly distributed throughout BaTiO3 lattice, mimicking realistic doping conditions. For various doping concentrations, the corresponding number of Ba ions were randomly replaced by Sr ions within the simulation box. This random replacement method maintained the overall structure of perovskite lattice while introducing the desired levels of Sr doping. The choice of substituted atoms can significantly affect the results obtained from simulations. In the case of Sr doping replacing Ba in BaTiO3, subtle changes in lattice parameters and structure symmetry may occur, impacting the stability of perovskite structure and the material’s ferroelectric properties. Due to their similar size to Ba ions, Sr ions may induce slight distortions in the lattice, affecting the dielectric and piezoelectric responses of the material. Furthermore, chemical interactions between Sr ions and TiO3 may differ from those between Ba ions and TiO3, leading to changes in the piezoelectric, dielectric, and ferroelectric properties of material. Figure 1 shows a view of the structure of BaTiO3 doped with Sr with different atomic ratios using OVITO software24.
Fig. 1
figure 1
Simulated BaTiO3 ceramic with (a) 2, (b) 4, and (c) 6, and (d) 8% Sr-doping.
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Equilibration process
In this first part of equilibration, T changes in the atomic sample were done using NVT ensemble. Figure 2 shows T changes as a function of time for different Sr concentration. According to the results presented in (Fig. 2), as simulation time increased, the T of atomic samples approached the target value and ultimately converged at 300 K. Furthermore, investigations in this section indicated that adding atomic doping to the simulated samples did not cause any thermodynamic disturbances. Consequently, the structures reached the desired final T without any atomic divergence.
Fig. 2
figure 2
T variations in BaTiO3 structure according to different amounts of Sr doping.
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On the other hand, the variations in PE of studied samples were further investigated to understand the equilibration process in the simulated atomic samples. Figure 3 illustrates PE variations as a function of time for different Sr concentrations. According to the graphs presented in (Fig. 3), after 10 ns, PE values converged to −84482.1, −84486.4, −84644.2, and −84587.8 eV. The convergence of these PE values indicated that the system reached a stable state. Additionally, this convergence demonstrated that chosen force field was suitable for accurately modeling interactions within the material, effectively capturing the energy landscape of system.
Fig. 3
figure 3
PE variations in BaTiO3 structure based on different amounts of Sr doping.
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Validation
Figure 4 illustrates the changes in SS diagram due to increasing Sr doping levels. The findings show that the magnitude of stress in the structure increased with Sr doping up to 6%, after which it decreased with further doping up to 8%. This behavior can be explained by the interaction of structural and electronic changes in the material. Initially, as Sr doping increased to 6%, material may experience enhanced bonding and structural integrity, leading to improved stiffness and stress resistance. This enhancement was often attributed to better lattice interactions and increased domain alignment in ferroelectric materials, making material more robust against deformation. However, at higher doping levels, such as 8%, excessive doping can significantly disrupt crystal lattice. This disorder may weaken effective bonding and create defects that hinder the movement of domain walls, ultimately reducing the performance and ferroelectric properties of structure. Moreover, SS curves were in good agreement with Lematre et al.25, which validated the present research.
Fig. 4
figure 4
The variations in SS diagram in the structure of BaTiO3 due to the increase in Sr doping.
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Results and discussion
In our previous study, calculations for undoped BaTiO3 were conducted. Specifically, at 300 K, MSD was found to be 0.195 Å2, and the diffusion coefficient was determined to be 0.42 nm2/ns22. With Sr doping, these values increased due to the creation of additional lattice sites and the reduction of energy barriers for atomic motion, which facilitated enhanced atomic mobility within the material. Figure 5 illustrates the changes in the mean square displacement (MSD) and diffusion coefficient as a function of time for different Sr concentrations. The numerical results showed that as Sr doping increased up to 6%, MSD increased from 0.211 to 0.218 Å2, while the diffusion coefficient increased from 0.361 to 0.380 nm2/ns. However, with further doping up to 8%, MSD and diffusion coefficient decreased to 0.205 Å2 and 0.361 nm2/ns, respectively (see Table 1). The behavior of MSD and diffusion coefficients in relation to Sr doping can be understood through the effects of atomic interactions and structural changes in the material. Initially, as Sr doping increased to 6%, MSD reflected enhanced atomic mobility within the material. Sr substitution created additional lattice sites and potentially lowered energy barriers for atomic motion. This increased mobility allowed atoms to explore more positions in the lattice, resulting in higher MSD values and an increased diffusion coefficient. Improved atomic interactions contributed to a more dynamic system where atoms can move more freely, enhancing diffusion. However, when doping exceeded 6%, such as at 8%, the MSD decreased. At higher Sr concentrations, lattice structure may become increasingly disturbed due to the presence of additional impurities. This disorder can introduce defects and strain that hindered atomic movement. Strong interactions between Sr atoms and the host lattice may also reduce overall mobility within the structure. As a result, the increase in defects and structural instability disrupted the ability of atoms to diffuse, leading to a decrease in both MSD and diffusion coefficients.
Fig. 5
figure 5
The variations in (a) MSD and (b) diffusion coefficient in BaTiO3 structure with increasing Sr doping.
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The polarization-strain curve of Sr-doped BaTiO3 was essential for understanding its ferroelectric properties. BaTiO3 was recognized for its strong piezoelectric and ferroelectric characteristics, which were significantly influenced by its crystal structure and domain configurations. The piezoelectric coefficient of undoped BaTiO3 was measured to be 228.59 pC/N. When Sr was introduced as a dopant, it altered lattice parameters and phase stability of BaTiO3, thereby affecting the relationship between applied strain and induced polarization. Figure 6 illustrates the changes in polarization as a function of strain for different Sr concentrations. The numerical results indicate that piezoelectric coefficient increased from 273.07 pC/N to 321.15 pC/N with Sr doping up to 6%. However, with further doping up to 8%, the piezoelectric coefficient decreased to 308.65 pC/N (see Table 1). Increasing Sr doping enhanced the polarization-strain response of BaTiO3. The presence of Sr ions can modify the material’s symmetry, influencing the polarization response in polarization-strain curve. A higher polarization coefficient indicated that a greater polarization was achieved for a given amount of strain, which was advantageous for applications in sensors and actuators that required strong piezoelectric responses. As Sr doping increased to 6%, the dipole balance improved, leading to stronger bonding and an enhanced ability to polarize the material. This doping contributed to increased domain stability, a stronger piezoelectric response, and higher polarization coefficients. However, excessive Sr doping can negatively affect the polarization coefficient and alter the polarization-strain behavior. High levels of impurities may disrupt dipole alignment and create localized strains that hindered polarization. Consequently, structural instability can lead to a decrease in the polarization coefficient. In summary, polarization-strain curve and polarization coefficient of Sr-doped BaTiO3 demonstrated a complex interplay between doping concentration, lattice structure, and ferroelectric properties. Moderate doping enhanced piezoelectric response by promoting better domain alignment and increased polarization, while excessive doping can have detrimental effects, reducing the overall ferroelectric performance of the material. The negative polarization values in the diagram indicate instances where the polarization was reversed due to using strain in the opposite direction. This phenomenon can occur under specific conditions in ferroelectric materials26. This reversal of polarization can be attributed to changes in domain configurations and the internal electric field induced by strain27. The asymmetry in the polarization values around P = 0 indicated the nonlinear behavior of material response to strain. This polarization response was influenced not only by doping concentration but also by the interplay between strain and the material’s inherent properties. In some cases, dopants like Sr may alter phase transitions or domain stability, resulting in a response that was not perfectly symmetric. This asymmetry reflected the complex behavior of ferroelectric materials under strain, where domain switching and polarization reversal did not occur symmetrically. Understanding these relationships was essential to optimize BaTiO3-based materials for various electronic and electromechanical applications. Jian et al.28 reported a piezoelectric constant of 293 pC/N for Ba1 − xSrxTiO3 in their experimental study.
Table 1 Calculated parameters for BaTiO3 nanostructure doped with increasing Sr doping.
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Fig. 6
figure 6
Polarization-strain diagram changes in BaTiO3 structure due to increasing Sr doping.
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In our previous study, the saturation polarization for undoped sample was found to be 0.37 C/m222. With Sr doping, this value initially decreased at lower concentrations due to the disruption of the lattice structure, which hindered the alignment of dipoles. Figure 7 illustrates the changes in polarization as a function of electric field for different Sr concentrations. The saturation polarization increased from 0.34 to 0.37 C/m2 with Sr doping up to 6%. This increase can be attributed to enhanced ferroelectric properties of material. When Sr ions were incorporated into the BaTiO₃ lattice, they improved dipole alignment and raised overall dielectric constant. This resulted in a greater ability to polarize in an electric field, resulting in higher saturation polarization values. However, while saturation polarization and piezoelectric coefficient initially increased with Sr doping, they decreased after reaching optimal level at 6%. This behavior indicated that moderate Sr doping enhanced domain stability and polarization change. Beyond this level, defects and structural disorders may disrupt the efficiency of piezoelectric effect. The negative polarization at -1 MV/m for Sr = 2% may result from the orientation of dipoles in the material, particularly at low doping concentrations. At these levels, dipoles may not align effectively under the applied electric field, causing polarization to reverse when the field is negative. This reversal can lead to negative polarization values at specific fields, reflecting material’s response to an electric field that counteracted dipole alignment. In contrast, positive polarization observed at + 1 MV/m across all cases suggested that, at higher electric fields, applied field overcomes internal dipole misalignment or reverses polarization, resulting in a positive net polarization. The electric field at + 1 MV/m may be strong enough to realign the dipoles or switch domains, inducing polarization in the direction of the field. In the work of S. L. Miller et al.29, the key mechanism driving the polarization-electric field hysteresis loop was domain switching. Their model explained how the alignment and reorientation of ferroelectric domains under an external electric field contributed to hysteresis behavior. It obtained insight into how polarization changed in response to applied electric field through the realignment of domains within the material. On the other hand, J. Yu et al.30 developed a compact theoretical model based on dipole switching to describe polarization-electric field hysteresis. Their simulation results aligned well with experimental observations of hysteresis loops, and the model can be easily integrated into electronic design automation tools for circuit simulations. This capability made it valuable for circuit design and enhanced the understanding of ferroelectric material behavior. The hysteresis loop results of BaTiO3 at room T showed strong agreement with the ferroelectric hysteresis behavior of BaTiO3 reported in the studies by Cheng et al.31 and Tan et al.32.
Fig. 7
figure 7
The variations in the polarization-electric field diagram in BaTiO3 structure due to the increase of Sr doping.
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Figure 8 shows the changes in strain as a function of electric field for different Sr concentrations. The numerical results indicated that strain increased with Sr doping up to 6% in modeled structure. However, with further doping up to 8%, the strain began to decrease. The increase in strain with Sr concentration up to 6% can be attributed to several factors. First, the substitution of Sr ions for other ions in the crystal lattice enhanced the material’s polarizability, leading to increased strain. This optimal concentration may also facilitate better alignment of dipole moments, increased polarization, and an enhanced piezoelectric response. Additionally, Sr substitution may induce beneficial phase transitions that improved the ferroelectric properties of material, resulting in a more pronounced strain response under an electric field. Beyond a concentration of 6%, the strain started to decrease due to structural distortion and the formation of defects within the network. As more Sr was added, this distortion can disrupt dipole alignment and reduce the material’s ability to respond effectively to an electric field. Increased ionic concentration may also lead to electrostatic repulsion among positively charged Sr ions, counteracting the mechanisms that previously facilitated strain. Ultimately, this resulted in a saturation effect, where further increased in Sr concentration did not increase and may even decrease the strain response.
Fig. 8
figure 8
The variations in strain-electric field diagram in BaTiO3 structure due to increasing Sr doping.
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The dielectric constant of undoped BaTiO3 was measured to be 7322. With Sr doping, this value increased due to enhanced ferroelectric properties and improved dipole alignment within the material. Figure 9 indicates the variations in the dielectric constant over time for different Sr concentrations. The numerical findings indicated that the dielectric constant increased from 76 to 85 with Sr doping up to 6%. However, with further doping up to 8%, the dielectric constant peaked at 80. The increase in dielectric constant with Sr doping in BaTiO₃ up to 6% can be attributed to enhanced ferroelectric properties and improved dipole alignment within the material. Sr ions effectively replaced barium, resulting in a more stable crystal structure and an increased polarization response under an electric field. This led to a higher dielectric constant, indicating that the material can store more electrical energy. However, as Sr doping continued up to 8%, dielectric constant began to decrease. This reduction can be explained by the introduction of structural defects and disorder in the crystal lattice that occurred at higher doping levels. These defects can impair the mobility of charge carriers and diminished material’s ability to polarize effectively, resulting in decreased dielectric performance. This behavior emphasized the importance of optimizing doping levels to balance enhanced dielectric properties with potential negative effects from excessive doping. The results obtained in this study were in great agreement with the findings of Yussof et al.33. Their study showed that the dielectric coefficient in BaTiO3 at 300 K was in the range of 70. In their studied sample, the structure of BaTiO3 is the hexagonal. Also, Wu et al.34 indicates that decreasing particle size and tetragonal-phase content can reduce the dielectric constant, with values approaching the desired range for certain conditions. So, current research predicted the Sr doping caused the structural evolution occur inside pristine sample and type/size of them changed by doping ratio variation. This structural process caused the low dielectric constant and high piezoelectric constant occur inside computational box. Previous papers for adding validation are summarized in (Table 2).
Table 2 Summary of relevant studies on dielectric and piezoelectric properties of BaTiO3 and Sr-doped BaTiO3.
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Fig. 9
figure 9
The variations of dielectric constant diagram in the structure of BaTiO3 with increasing Sr doping.
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The dielectric loss for undoped BaTiO3 was calculated to be 0.0422. With Sr doping, the dielectric loss increased due to enhanced ionic mobility and polarization. Figure 10 indicates the variations in the dielectric loss over time for different Sr concentrations. The numerical findings indicated that dielectric loss increased from 0.06 to 0.08 with Sr doping up to 6%. However, with further doping up to 8%, the dielectric loss decreased to 0.073. The increase in dielectric loss from 0.06 to 0.08 with Sr replacement up to 6% suggested an increase in energy loss due to enhanced ionic mobility and polarization. However, beyond 6%, dielectric loss decreased to 0.073 at 8%, indicating that excess Sr led to structural disorders, which reduced material’s ability to store and dissipate energy effectively. For Sr-doped BaTiO3, dielectric loss was found to be approximately 0.06, which was in good agreement with our findings35.
Fig. 10
figure 10
The variations of dielectric loss diagram in the structure of BaTiO3 with increasing Sr doping.
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Conclusion
This research surveyed the impact of Sr-Doping on the piezoelectric and dielectric properties of lead-free BaTiO3 ceramic using MD simulation. The results reveal that:
The results indicate that increasing Sr content up to 6% enhanced materials’ bonding and structural integrity, leading to increased stiffness and resistance to stress. Additionally, with Sr doping up to 6%, MSD increased from 0.211 to 0.218 Å2, and the diffusion coefficient increased from 0.361 to 0.380 nm²/ns. However, with further doping up to 8%, MSD and diffusion coefficient decreased. The behavior of MSD and diffusion coefficients in relation to Sr contamination can be understood through the effects of atomic interactions and structural changes within the material.
Numerical results show that the piezoelectric coefficient increased from 273.07 pC/N to 321.15 pC/N with Sr doping up to 6%. However, with further increases in impurity up to 8%, the piezoelectric coefficient decreased to 308.65 pC/N. Increasing Sr impurity enhanced the strain polarization response of BaTiO₃.
Numerical results show that the dielectric constant and dielectric loss increased with the increase of Sr contamination up to 6%. With the further increase of Sr content up to 8%, these quantities decreased. The obtained results show that excess Sr led to structural disorders and reduced the material’s ability to store and dissipate energy effectively.
Data availability
The data and executable code related to this study are available upon request from the corresponding author. Please contact M. H. Ehsani at Ehsani@semnan.ac.ir for access to the data. The data will be made available in a public repository following publication.
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Acknowledgements
we’ll mention we have not received any external funding for the research underlying our paper, so there is no funding body we could acknowledge.
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Authors and Affiliations
Faculty of Physics, Semnan University, Semnan, P.O. Box: 35195-363, Iran
Shadi Esmaeili & M. H. Ehsani
Department of Mechanical Engineering, Khomeinishahr Branch, Islamic Azad University, Khomeinishahr, Iran
Davood Toghraie
New Technologies Research Center, Amirkabir University of Technology, Tehran, Iran
S Saber-Samandari
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Shadi Esmaeili
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2. M. H. Ehsani
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Contributions
M. H. Ehsani and Davood Toghraie. Conceived of the presented idea. Shadi Esmaeili. Developed the theory and performed the computations.M. H. Ehsani, Davood Toghraie., and S Saber-Samandari. Verified the analytical methods and supervised the findings of this work. All authors discussed the results and contributed to the final manuscript.
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Correspondence to M. H. Ehsani.
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Esmaeili, S., Ehsani, M.H., Toghraie, D. et al. Sr-doping effects on piezoelectric and dielectric properties of lead-free barium titanate via molecular dynamics approach. Sci Rep 15, 8404 (2025). https://doi.org/10.1038/s41598-025-92959-w
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Received:21 November 2024
Accepted:04 March 2025
Published:11 March 2025
DOI:https://doi.org/10.1038/s41598-025-92959-w
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Keywords
Sr doping
Piezoelectric features
Barium titanate
Molecular dynamics simulation