1 Introduction

Highly integrated and miniaturized microwave components that exhibit a minimal tangent loss and high dielectric constant, while still maintaining excellent mechanical properties, processing capability, and affordable cost, are highly significant for functioning optimally at certain microwave frequency bands [1,2,3]. Tangent loss ( ~ 10-2) and significant permittivity ( ~ 15) can minimize signal propagation time and eliminate cross-coupling with electrically conductive materials [4,5]. One of the most well-known wireless networking protocols for wideband mobile networks operating at microwave bands is the fifth generation (5 G). Reduced latency is one of the main characteristics of the 5 G system. Latency is entirely dependent on the delay time caused by signal transmission through dielectric material [6,7].

Reduced cost, small size, conducting efficiently, radiating optimally, broad bandwidth, and compatibility with various feed techniques are the most optimal features of dielectric resonator antennas (DRAs). DRAs can be characterized in a variety of sizes and forms [8,9]. They are widely utilized in conference video calls, mobile communication, direct digital transmission, and radar systems. Generally, dielectric materials can maintain the distribution of the electric field and reduce the heat-dissipated energy. Meanwhile, their performance is dependent on the dielectric loss; for instance, the more effective the dielectric material, the lower the losses [10].

Dielectric ceramic materials that are extremely polarized electrically have intrinsic physical characteristics that make them appropriate for use in a variety of electronic and industrial applications, including field-effect transistors [11], dynamic random access memories [12], high energy density storage [13], passive sensors [14,15], and commercial microwave elements, such as capacitors, antennas, microwave dielectric resonators, filters, and sensors [16,17,18]. Because of their unique characteristics and prospective use as active components in several fields of solid-state physics, highly efficient lead-free perovskite structures (ABO3) have garnered a lot of interest as dielectric ceramic materials. Barium titanate (BaTiO3), as a customized perovskite ceramic, is crucial for numerous types of electrical devices based on its ferroelectric and piezoelectric properties. For example, but not limited to, multilayer ceramic capacitors, lamp starters, thermistors, pulse-generating devices, infrared detectors, multilayer ceramic capacitors (MLCCs), piezoelectric and ultrasonic actuators, thermal sensors and controllers, microwave devices employing piezoelectric transducers, and energy storage devices [19,20,21,22].

It is possible to modify the properties of lead-free BaTiO3 (B-T) by doping or replacing host cations at the structure’s A-site and/or B-site [23,24,25,26,27,28]. B-T is supplemented with yttrium (Y) to increase its reliability. Due to its intermediate ionic radius between the two lattice ions, the Y3+ ion can occupy either the Ba2+ or Ti4+ site, functioning as either an acceptor or a donor ion [29,30,31]. The solubility of Y3+ ions is determined by their ability to fill the vacancies in the A-site or the B-site of the B-T lattice caused by the partial pressure of oxygen and sintering temperature, the Ba/Ti ratio, and the concentration of the dopant [32]. Y3+ ions’ solubility at the A-site when sintered at 1440–1470 °C is around 1.5% under ambient conditions, while it rises to 4% under reducing circumstances, as demonstrated by Wang et al. [33]. On the other hand, sintering in air at 1515 °C, the solubility of Y3+ ions at the B-site is increased to around 12.2%. Furthermore, structural alterations by the addition of Y3+ ions, such as the transition of phase from tetragonal to pseudo-cubic. Reaching the dopant’s solid solubility, as it completely occupies the empty B-sites in the B-T matrix, the dopant will be consumed for a charge-balancing mechanism and induce certain secondary phases like Ba6Ti17O40 and Y2Ti2O7, as reported by Belous et al. [34]. The industrial implementations of B-T could be significantly impacted by the induced Ba2TiSi2O8, Ba6Ti17O40, Y2TiO5, and R2Ti2O7 pyrochlore phases (where R is the rare-earth ions) as secondary phases, through influencing the dielectric and structural characteristics of the B-T matrix [35,36,37].

The physical properties of Y-doped barium titanate nanocomposite ceramics (Ba1-xYxTiO3 (BYT-x), x = 0–10 mol.%) could be altered by varying doping concentration up to reaching the solubility level and inducing some secondary phases, making it potentially suitable for advanced applications. There are a variety of techniques for fabricating BaTiO3, such as standard solid-state, co-precipitation, and hydrothermal [38,39,40,41]. While the sol-gel process can be carried out at reduced temperatures with minimal laboratory instruments and permits control over the material’s composition and homogeneity, it is especially relevant for creating novel structures [41,42,43,44,45,46,47,48].

This study reports on the synthesis of nano-structured Y-doped BaTiO3 via a rigorously controlled modified sol-gel route to achieve enhanced dielectric performance. The effect of Y incorporation on the structural and dielectric properties was investigated over a wide frequency range (42 Hz–1 GHz). Particular focus was placed on the high-frequency regime, where experimental data are limited, to elucidate the doping-induced development of self-resonance behavior and assess the material’s suitability for high-frequency applications. As summarized in Table 1, although numerous studies have investigated Y-doped and rare earth doped BaTiO3-based ceramics, most dielectric measurements were restricted to frequencies below 1 MHz, whereas investigations extending up to the GHz region remain scarce. Moreover, the evolution of the self-resonance frequency (SRF) has received limited attention in previous studies on Y-doped BaTiO3 ceramics.

Table 1 Comparison of the frequency ranges employed for dielectric characterization of Y-doped and related BaTiO3-based ceramics reported in the literature and in the present work
Full size table

2 Theoretical and experimental methods

2.1 Sample preparation

Nano-structure pure barium titanate BaTiO3 (B-T), doped with different concentrations of yttrium (Y) coded as [BYT-x (x = 0–10)], has been synthesized using the sol-gel method, where x = 0, 1, 2, 5, 6, 7, 8, 9, and 10 mol.%. Barium Acetate Ba(CH3COO)2 (Aldrich 99%), Titanium (IV) Isopropoxide Ti(OCH(CH3)2)4 (Alfa Aesar 99 + %), Yttrium Nitrate Y(NO3)3.6H2O (ADVENT 98%), Acetylacetone C5H8O2 (Merck 99%), and Acetic Acid C2H4O2 are the chemical reagents used to prepare the nanostructure.

B-T is obtained after the stirring for about one hour of the following two solutions: titanium (IV) isopropoxide dissolved in acetyl acetone and barium acetate dissolved in acetic acid diluted with bidistilled water. In the next step, Y(NO3)3.6H2O was dissolved in bi-distilled water for 10 minutes and then added to the B-T solution and stirred for another 15 minutes, referred to as the [BYT-x (x = 0–10)] solution. [BYT-x (x = 0–10)] solutions were dried on a hot plate at 300 oC. Figure 1 (the successive pictures from a to f) shows an illustrative shrinkage example of some practical steps for preparing a B-T sample in the lab. First, the solution (Sol) is created (Picture a), then it hydrolysis and condenses into a porous linked structure (Pictures b and c). Next, it is aged (Picture d), and lastly, it is dried (Pictures e and f).

Fig. 1
Fig. 1
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Pictures of some steps for preparing one of the samples in the laboratory

The dried [BYT-x (x = 0–10)] samples were grinded for 25 min, and then the pure sample was calcined at 1000 °C for 4 h in air to get [BYT-x (x = 0–10)] nanoceramic material that will be grinded for 25 min. The muffle furnace type Carbolite CWF1200 was used for sample annealing. The successive processes are illustrated in Fig. 2.

Fig. 2
Fig. 2
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Preparation procedure for [BYT-x (x = 0–10)] nanostructure powder

2.2 Characterization

2.2.1 Structural analysis

Diffraction produced at room temperature using a Bruker D8 ADVANCE X-ray diffractometer equipped with monochromatized Cu Kα radiation (λ = 0.15406 nm) for phase identification of the prepared nanostructured samples. The diffraction data were collected in the Bragg-Brentano (θ–2θ) geometry over a 2θ range of 5°–80° at an operating voltage of 40 KV and a tube current of 40 mA, with a step size of 0.05° and a scanning rate of 2° min–1. Scherrer’s equation was used to estimate the crystallite size (C.S) from several representative diffraction peaks. The reported crystallite size for each sample corresponds to the arithmetic mean of the values calculated from the individual reflections [46]:

$${{\rm{C}}}.{{\rm{S}}}=\frac{{{\rm{K}}}.{{\rm{\lambda }}}}{{{\rm{\beta }}}.{{\rm{Cos}}}{{\rm{\theta }}}}$$
(1)

Where the wavelength (λ) is 0.15406 nm, the Scherrer constant (K) is 0.89, the (corrected) width of the intensity of the peak at half maximum (β) is in radians, and the diffraction angle (θ) is in degrees. The dimensionless Scherrer shape factor constant (K) depends on the crystallite size distribution, peak-width definition, and crystallite morphology. Based on high-resolution transmission electron microscope (HRTEM) observations, the particles exhibited an approximately equiaxed morphology. Since the crystallite size was estimated using the width of the intensity at half maximum (β) of the diffraction peaks, a Scherrer constant of K = 0.89 was adopted for the crystallite size calculations [49,50,51,52,53]. The experimentally measured width of the peak intensity at half maximum (FWHM) of each diffraction peak was corrected according to:\(\beta =\sqrt{{\beta }_{{{\rm{m}}}}^{2}-{\beta }_{{{\rm{i}}}}^{2}}\), where \(\beta \) is the corrected FWHM of the diffraction peak, \({\beta }_{m}\) is the experimentally measured FWHM of the sample, and \({\beta }_{i}\) is the instrumental broadening calculated using a standard corundum (α-AL2O3) reference specimen (Bruker AXS, Korundprobe 1976 C, A13-B77) [54]. The corundum reference was measured under the same instrumental and acquisition conditions used for the prepared samples. The instrumental contribution was determined from the measured broadening of the reference diffraction peak and was subsequently used as \({{{\rm{\beta }}}}_{i}\) to correct the FWHM of the corresponding sample peak. For each diffraction peak of the BYT-x (x = 0–10) samples, the corundum reflection closest in 2θ was used as the corresponding reference. The reference reflections used were located at approximately 25.5°, 35.1°, 37.7°, 43.3°, 52.4°, 57.5°, 66.5°, and 76.7°, and their FWHM was determined using the same peak-fitting procedure applied to the prepared samples diffraction peaks. These reference reflections cover the relevant diffraction-angle range of the selected BYT-x (x = 0–10) peaks. Twice the reciprocal power of the crystallite size was used to estimate the dislocation density [47]:

$$\delta =\,\frac{1}{{{{\rm{C}}}.{{\rm{S}}}}^{2}}$$
(2)

The Williamson-Hall relation was used to calculate crystallite size, confirming the calculated (C.S) by Scherrer’s equation. The relation between (\(\beta * \cos \theta \)) on the y-axis and (\(4* \sin \theta \)) on the x-axis was fitted as a straight line with a slope corresponding to the strain (ε), while the crystallite size is confirmed by the line intersection with the y-axis [47]:

$$\beta * \cos \theta =4\varepsilon * \sin \theta +\,\frac{K\lambda \,}{{CS}}$$
(3)

All numerical calculations, data processing, linear regression analysis, and graphical representation were performed using OriginPro 2019b (OriginLab Corporation, Northampton, MA, USA), a commercial scientific data analysis software package [55].

Rietveld refinement was performed using the FullProf Suite (Version 2020) based on the initial crystallographic information file (CIF, COD No. 15507756). Prior to refinement, the experimental XRD patterns were preprocessed using WinPLOTR with automatic background determination. The diffraction profiles were refined using the Thomson-Cox-Hastings pseudo-Voigt (TCH) profile function with axial-divergence asymmetry correction. The refinement procedure was carried out by sequentially refining the scale factor, lattice parameters, peak broadening (FWHM), and profile shape parameters, followed by simultaneous refinement of all variable parameters. The refinement quality was evaluated using the goodness-of-fit parameter (χ2), which ranged from 0.88 to 2.88 for all investigated compositions, indicating satisfactory refinement quality. Both tetragonal and cubic BaTiO3 structural models were examined during the refinement. The tetragonal structure (space group P4mm) was selected based on the lowest χ2 value and the overall refinement quality.

Energy dispersive X-ray spectroscopy (EDX) equipped with a silicon drift detector (SDD) was used to detect the elemental composition of the prepared samples. A field emission scanning electron microscope (FEI, model: Quanta 250 FEG) instrument provided with variable pressure was used to explore the morphology of the samples.

Microstructure examination and particle size measurement were performed using a high-resolution transmission electron microscope (HRTEM, JEOL JEM-2100, Tokyo, Japan) operated at an accelerating voltage of 200 kV. The microscope was equipped with a LaB6 thermionic electron gun and a Gatan ORIUS SC 1000 A CCD camera, providing a point-to-point resolution of 0.19 nm. Bright-field TEM images were collected to evaluate the particle morphology and determine the average particle size. Selected-area electron diffraction (SAED) patterns were recorded in ring mode using a camera length of 150 mm, whereas HRTEM images were collected to resolve the lattice fringes. Particle size analysis was performed using ImageJ software [56].

2.2.2 Dielectric measurements

The dielectric properties were evaluated through two precision impedance measuring instruments over a broad frequency range, from 42 Hz to 1 GHz, to ensure high reliability. The typical capacitance/impedance technique was employed, which entails sandwiching a thin sample sheet between two electrodes to form a parallel-plate capacitor. Measurements were carried out using a computerized HIOKI 3532-50 LCR Hi-TESTER over a low-frequency (LF) range from 42 Hz to 1 MHz, which provides good accuracy and a relatively easier measurement configuration for flat, thin samples. An ac signal was applied at selected test frequencies to probe the dielectric response, detecting the voltage and current vectors to determine the complex impedance (Z) and phase angle (θ). High-frequency dielectric measurements from 1 MHz to 1 GHz were performed using a network impedance analyzer (KEYSIGHT-E4991B). The E4991B-002 material measurement software included on the system enables advanced calibration procedures, including open, short, and load corrections, compensating for edge capacitance effects via built-in algorithms. These advanced correction capabilities significantly minimize systematic errors associated with parasitic contributions and measurement geometry, thereby ensuring high measurement accuracy and reliability across the entire frequency range [57]. The significance of the current studies is highlighted by the relatively limited availability of experimental data in this frequency range in the literature. Additionally, the use of such a well-calibrated measurement setup is necessary to extract the material’s intrinsic dielectric response free from extrinsic artifacts, which significantly improves the reported results’ reliability and scientific value.

The samples used in dielectric measurements were shaped under a pressure of 6 tons in the disk form, having a 10 mm diameter and 1.66 mm thickness. Silver paste was used to cover each surface of the tested samples to ensure good contact between the samples and the electrodes.

The dielectric properties were measured as follows [43,44]:

$${{\rm{Permittivity}}}\,{{\rm{\varepsilon }}}^{\prime} =\,\frac{{{{\rm{C}}}}_{S}}{{{{\rm{C}}}}_{0}}$$
(4)
$${{\rm{Dielectric\; loss}}}\,{{{\rm{\varepsilon }}}}^{{\prime} {\prime} }={{{\rm{\varepsilon }}}}^{{\prime} }\tan {{\rm{\delta }}}$$
(5)
$${{\rm{AC\; conductivity}}}\,{\sigma }_{{ac}}={{{\rm{\omega }}}{{\rm{\varepsilon }}}}_{{{\rm{o}}}}{{{\rm{\varepsilon }}}}^{{\prime} {\prime} }$$
(6)

where the sample electrical capacitance (\({C}_{S}\)) is in Farads, the air electrical capacitance (\({C}_{0}\)) is \({C}_{0}=\frac{{{{\rm{\varepsilon }}}}_{0}{A}_{s}}{t}\), free space permittivity εo = 8.854 × 10–12 in F/m, \(\omega =\,2\pi f\) is the applied electric field angular frequency, and the thickness (\(t\)) and surface area (\({A}_{s}\)) of the sample are in m and m2, respectively.

Jonscher’s power law was employed to model the frequency-dependent ac conductivity [43,44]:

$${\sigma }_{{ac}}={\sigma }_{{dc}}\,+A{\omega }^{s}$$
(7)

Where σdc is the dc conductivity, A is a temperature-dependent pre-exponential factor, and s is the frequency exponent.

To perform quantitative analysis of the impedance characteristics, the experimental data were fitted using the Zview software (Version 3.4e, Scribner Associates, Inc., USA) according to the complex non-linear least-squares (CNLS) fitting algorithm [58].

3 Results and discussion

3.1 X-ray diffraction

Figure 3a shows the X-ray diffraction graph of [BYT-x (x = 0–10)] samples calcined at 1000 ⁰C for 4 h in air. All the observed peaks in the XRD graph of the pure B-T belong to tetragonal BaTiO3 as confirmed by ICDD number [01-074-1957]. For the doped samples, the observed XRD peaks in the graph belong to both BaTiO3 and Ba0.966Y0.004TiO3 tetragonal perovskite structures, as confirmed by ICDD numbers [01-074-1957] and [01-078-3299], respectively. There were no additional XRD peaks detected for the Y presence, as Ba ions were substituted by Y ions, where they were completely embedded in the B-T lattice structure for BYT-1 and BYT-2, producing a decrease in tetragonality. At higher Y concentrations, some additional secondary XRD peaks belonging to Y2Ti2O7 (Yttrium Titanium Oxide) and BaY2O4 (Barium Yttrium Oxide) were detected and confirmed by ICDD numbers [01-085-1584] and [01-078-5032], respectively, suggesting partial solubility limits and the onset of impurity phase formation. Definite sharp peaks of the powder diffraction patterns demonstrate accurate crystallinity for the prepared samples. The Y doping affects the main peak position, as illustrated in Fig. 3b. Up to 6 mol.% Y doping, the main peak position at 2θ ≈ 31° shifts slightly to a higher 2θ, indicating lattice contraction due to substitution of Ba²⁺ ( ~ 1.61 Å) by smaller Y³⁺ ( ~ 1.019 Å) ions [50, 59]. At higher doping levels, the shift changes its direction to lower 2θ, which may be attributed to a transition from ionic-size controlled contraction to defect and strain induced lattice relaxation effects [28].

Fig. 3
Fig. 3
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a XRD patterns of BYT-x (x = 0–10) respectively, b magnified XRD patterns from 2θ = 30.5° up to 31.5°, and c magnified XRD patterns from 44° up to 46°

Figure 3c shows the XRD patterns ranging from 2θ = 44° up to 46⁰ for all samples, exploring B-T tetragonal features as the peak indexed around 2θ = 45° was completely split. For lower Y concentrations, the peaks broaden and partially merge, indicating a reduction in the apparent tetragonality. This effect may be attributed to the possible substitution of Ba2+ ions with smaller ionic radius Y3+ ions and the crystallite size reduction accompanied by peak broadening, which makes it more difficult to distinguish between closely spaced reflections. At higher Y concentrations, peak broadening increases and a partial recovery of tetragonality is observed, reflecting defect-driven lattice distortions and charge compensation mechanisms (e.g., cation vacancy formation) that modify the local symmetry and stabilize the tetragonal distortion [29,30,31,32,33]. To confirm the tetragonality behavior, a numerical first-derivative analysis of the X-ray diffraction intensity profile was applied in the vicinity of the 2θ = 45°, as shown in Fig. 4. This method mathematically enhances local intensity fluctuations by illustrating inflection points and small curvature changes in the diffraction envelope.

Fig. 4
Fig. 4
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Intensity derivative of the XRD peaks for the prepared samples around 2θ = 45°

Table 2 shows the prepared sample codes and the corresponding chemical formula; moreover, all the assigned planes are used to calculate the lattice parameters (a & c), the tetragonal factor (c/a), the volume of the unit cell (V), and the structure of each sample. Up to 6 mol% Y doping, both lattice parameters and unit cell volume decrease, which may be attributed to the possible occupying the A site by smaller Y3+ replacing Ba2+, revealing oxygen vacancy formation suppressing the tetragonality by reducing the Ti4+ displacement. Further Y addition causes the lattice parameters and unit cell volume to increase. Grain growth, oxygen vacancy formation, and grain boundary diffusion are all influenced by the calcination temperature, impurities, and crystal orientation. Table 3 displays the strain (ε), the density of dislocation (δ), and the average nanocrystallite size (C.S) estimated from Scherrer’s equation and Williamson-Hall plot, confirming the nanocrystalline nature of the prepared samples. Figure 5 displays the linear fitting of Williamson-Hall parameters, while Fig. 6 illustrates the systematic variation of the lattice parameters, the tetragonality, the unit cell volume, and the crystallite size as a function of Y concentration.

Fig. 5
Fig. 5
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Crystallite size of BYT-x (x = 0–10), respectively, determined by a Williamson-Hall plot using linear fitting parameters

Fig. 6
Fig. 6
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Effect of Y-doping ratio on lattice parameters, tetragonal factor, volume, and crystallite size of BYT-x (x = 0–10), respectively

Table 2 The codes, the XRD-calculated lattice parameters, and the structure of BYT-x (x = 0–10), respectively
Full size table
Table 3 The crystallite size (C.S), micro-strain (ε), the density of dislocations (δ), and particle size (P.S) as estimated for BYT-x (x = 0–10), respectively
Full size table

The Rietveld refinement profiles of the XRD patterns for the prepared BYT-x (x = 0–10) samples using the tetragonal structure model (space group P4mm) are presented in Fig. 7. The excellent agreement between the observed and calculated diffraction patterns, together with the absence of significant systematic misfits, confirms the high quality of the refinement. This suggests that the tetragonal perovskite structure remains the dominant crystalline phase throughout the investigated composition range. Slight lattice distortions induced by Y incorporation are reflected in the calculated lattice parameters. Y3+ may substitute for either Ba2+ or Ti4+ in the BaTiO3 lattice and can behave as a donor or an acceptor depending on the processing conditions and defect chemistry. The structural evolution observed in the present work is consistent with the proposed substitution of Y3+ at the Ba2+ site, which is thermodynamically favored at relatively low processing temperatures due to the preferential formation of cation vacancies on the Ba sublattice [32,60]. Since the prepared samples were calcined at 1000 °C, the observed structural evolution agrees well with the substitution mechanism reported in the literature. This interpretation is further supported by the systematic decrease in the calculated lattice parameters and unit-cell volume, together with the gradual shift of the major diffraction peaks toward higher 2θ values up to 6 mol% Y. At higher Y concentrations, the appearance of secondary phases, mainly Y₂Ti₂O₇ with a minor amount of BaY₂O₄, indicates that the solid solubility limit has been exceeded. Consequently, excess Y³⁺ is accommodated through secondary-phase formation rather than further incorporation into the BaTiO₃ lattice, thereby limiting additional structural distortion of the perovskite lattice.

Fig. 7
Fig. 7
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Rietveld refinement of the XRD patterns for the prepared BYT-x (x = 0–10) samples

3.2 Field emission scanning electron microscope (FESEM)

Micrographs for BYT-0, BYT-1, BYT-5, and BYT-10 are collected by FESEM, as displayed in Fig. 8, respectively. It is apparent from the micrographs that the pure sample has irregular morphological shapes containing some larger grains, and the granules are

Fig. 8
Fig. 8
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FESEM images and FESEM-derived grain size distribution for (a) BYT-0, b BYT-1, c BYT-5, and d BYT-10, respectively

roughly tetragonal and morphologically compacted. Yttrium doping at the A site of B-T causes noticeable alterations in the grain’s texture and a change in the morphology as the grain size becomes more uniform, and it inhibits grain growth, leading to dense and more uniform size distribution with a nearly spherical shape. In general, Yttrium doping significantly reduces the average grain size. Quantitative FESEM grain-size analysis histograms provide additional evidence for the progressive reduction in average grain size and the increased uniformity of the microstructure with increasing Y3+ concentration is displayed in Fig. 8. However, FESEM micrographs reveal substantial particle agglomeration, and the boundaries between neighboring nanoscale particles are not consistently resolved. Consequently, the measurable features in the FESEM images may represent agglomerates or particle clusters rather than individual nanoparticles. The EDX shows the elemental composition of the synthesized samples, confirming the absence of impurities and detecting only the constituent elements (Ba, Y, Ti, and O). The detected XRD secondary phases Y2Ti2O7 and BaY2O4 at higher Y3+ concentration are composed of the same constituent elements of the main composition Ba1-xYxTiO3 [BYT-x (x = 0–10)], and therefore it cannot be distinguished as impurities by EDX, that provide elemental rather than phase information. Therefore, no foreign elemental impurities were observed. High-intensity peaks of O, Ba, and Ti elements for the BYT-0 sample, approving the absence of any impurities. Similarly, for the doped samples, the same peaks appeared, while lower-intensity peaks corresponding to the Y element were detected. Atomic proportions of the prepared samples at various lattice sites are illustrated in Fig. 9.

Fig. 9
Fig. 9
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EDX spectra and atomic proportion (at. %) of (a) BYT-0, b BYT-1, c BYT-5, and d BYT-10, respectively

3.3 High-resolution transmission electron microscopy (HRTEM) analysis

The Bright-field HRTEM micrographs, particle size distribution histograms, the selected-area electron diffraction (SAED) patterns, and HRTEM images with the corresponding fast Fourier transform (FFT) insets for the BYT-0 and BYT-7 samples are presented in Fig. 10. The HRTEM images reveal that the nanoparticles are agglomerated into clusters composed of fine, nearly equiaxed particles, as shown in Fig. 10a, b. Particle size analysis was performed using ImageJ software by measuring more than 150 particles from the bright-field images. The resulting particle size distributions were fitted

Fig. 10
Fig. 10
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a, b Bright-field HRTEM micrographs; c, d particle size distribution histograms fitted by Gaussian function; e, f selected-area electron diffraction (SAED) patterns; and g, h high-resolution transmission electron microscopy (HRTEM) images showing the lattice fringes of the BYT-0 and BYT-7 samples, respectively. The insets in (e, f) show the original SAED patterns, whereas those in (g, h) show the corresponding Fast Fourier transform (FFT) images

using a Gaussian function, yielding average particle size of 38.24 ± 12.92 nm for BYT-0 sample and 24.59 ± 8.06 nm for BYT-7 sample, as illustrated in the histograms presented in Fig. 10c, d. These values are in good agreement with the average crystallite sizes estimated from the XRD analysis. The nanostructured polycrystalline nature of the prepared samples is obviously shown as a series of well-defined concentric diffraction rings composed of discrete diffraction spots through patterns of SAED as shown in Fig. 10e, f. The diffraction rings were indexed using the ratio-method pattern indexing, and the corresponding lattice planes were assigned to each diffraction ring [61]. The radius of each diffraction ring (R) was measured from the center of the transmitted beam to the corresponding ring using ImageJ after calibration with the reciprocal-space scale of the SAED pattern. Following the ratio-method indexing approach (\({R}_{i+1}/{R}_{i}={d}_{i}/{d}_{i+1}\)), the ratios of successive ring radii were compared with the inverse ratios of the d-spacings independently determined from XRD. The best agreement was used to identify the diffraction rings and assign the corresponding (hkl) planes based on the d-spacing values of the prepared samples. Moreover, the assigned lattice planes of the diffraction rings was verified independently with the previously reported SAED patterns for tetragonal Sm3+ doped BaTiO3 [47]. The HRTEM images in Fig. 10g, h clearly reveal well-resolved lattice fringes, with measured interplanar spacing of approximately 0.20 nm corresponding to the (002)/(200) crystallographic planes of tetragonal BaTiO3 for BYT-0 sample and 0.21 nm and 0.23 nm corresponding to the (111) and (002)/(200) planes, respectively, for BYT-7 sample. These values are in excellent agreement with the XRD results. The corresponding FFT exhibit well-resolved diffraction spots, further confirming the high crystalline nature and excellent crystallographic ordering of the selected nanoparticle region as shown in the insets of Fig. 10g, h.

3.4 Dielectric properties

Measuring dielectric properties was accomplished over two frequency ranges, from 42 Hz up to 1 MHz for the former and from 1 MHz up to 1 GHz for the later. Figures 11, 13, and 14 illustrate the variation of \({{\rm{\varepsilon }}}^{\prime} \) representing the permittivity, \({{{\rm{\varepsilon }}}}^{{\prime} {\prime} }\) representing the dielectric loss, and \({\sigma }_{{ac}}\) representing the ac conductivity for the prepared samples as a function of frequency at room temperature (RT). The frequency at which the sample exhibits a minimum impedance value is called the self-resonance frequency (SRF); it depends on the physical size and microstructure of the sample. It can be realized from Fig. 11a that \({{\rm{\varepsilon }}}^{\prime} \) declined by growing the frequency for all the samples up to approximately 104 Hz, and then it was nearly constant, roughly up to the SRF, at which \({{\rm{\varepsilon }}}^{\prime} \) exhibited a sharp peak due to the resonance effect. The cumulative contributions of all polarization components, including space charge, orientation (dipolar), ionic, and electronic polarizations, at low frequencies result in an increase in ε′ [62]. As the electric field varies rapidly with the frequency, the slower polarization components (space charge and orientation polarization) are no longer able to follow the rapidly alternating field, which causes the ε′ values to be reduced. This behavior is commonly observed in dielectric materials [63]. Orientation polarization results from dipolar molecules flipping or rotating, or from ions shifting positions under an external electric field, while electronic polarization is induced by an electron displacement with respect to the nucleus and persists even in the high-frequency region. The permittivity variation as a function of composition is displayed in Fig. 12, showing size-dependent dielectric properties for all the prepared samples, except for the BYT-7 sample, which may be attributed to the increased internal stresses. The prepared samples demonstrate a notably high and frequency-independent permittivity (ε′ ≈ 770, for BYT-0) throughout a broad frequency range up to 100 MHz, as shown in Fig. 11b. The weak dispersion observed in this window indicates that the frequency response is significantly affected by rapid frequency components (ionic and electronic); in contrast, slower components are greatly diminished with this frequency window. Moreover, the current system shows nearly an eightfold increase in ε′ as compared with the previously reported pure sample (ε′ ≈ 100) [45], even though it is sintered at a lower temperature (1000 °C versus 1050 °C). This improvement highlights the importance of processing optimization for better dielectric performance, along with reduced thermal budget, improved microstructural homogeneity, and reduced defect density likely arising from the optimized synthesis and thermal treatment conditions, as shown in Fig. 1.

Fig. 11
Fig. 11
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The permittivity \({\varepsilon }^{{\prime} }\) as a function of frequency F for BYT-x (x = 0–10) a at the low frequency and b at the high frequency, respectively

Fig. 12
Fig. 12
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Variation of permittivity \({\varepsilon }^{{\prime} }\) as a function of sample compositions at different frequencies

Following Y3+ incorporation, the \({{\rm{\varepsilon }}}^{\prime} \) decreases markedly and is governed by the combined effects of defect chemistry, structural evolution, grain refinement, and secondary-phase formation rather than by a single mechanism. At the relatively low calcination temperature employed in the present work (1000 °C), Y3+ ions more likely substitute for Ba2+ sites, where they predominantly behave as donor dopants. Charge compensation under these conditions suppresses oxygen-vacancy-related defects and promotes cation-vacancy formation, thereby reducing defect-assisted polarization and charge-carrier hopping, which is consistent with the simultaneous decrease in \({{\rm{\varepsilon }}}^{\prime} \) and \({\sigma }_{{ac}}\) observed experimentally [60,64]. The reduction in \({{\rm{\varepsilon }}}^{\prime} \) is further enhanced by the suppression of grain growth induced by Y3+ incorporation, as confirmed by FESEM and HRTEM analyses. The resulting increase in the grain-boundary volume fraction introduces a larger fraction of low-permittivity interfacial regions that restrict domain-wall motion and consequently reduce the effective dielectric response [65,66]. At higher Y3+ concentrations, the formation of secondary phases, predominantly Y2Ti2O7 with a minor amount of BaY2O4, further decreases the effective volume fraction of the ferroelectric BaTiO3 phase, thereby limiting the overall polarization response. Nevertheless, the partial recovery of \({{\rm{\varepsilon }}}^{\prime} \) observed at high Y3+ concentrations may be attributed to local lattice distortion and changes in the defect-compensation mechanism as the Y3+ content approaches the solid-solubility limit [64].

The frequency dependence of dielectric loss \({{{\rm{\varepsilon }}}}^{{\prime} {\prime}} \) for the prepared samples is viewed in Fig. 13. At low frequency, the dielectric loss at the nanoscale decreases with frequency without obvious relaxation peaks as the conduction loss dominates over the dipolar

Fig. 13
Fig. 13
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The dielectric loss \({\varepsilon }^{{\prime} }{\prime} \) as a function of frequency F for BYT-x (x = 0–10) a at the low frequency and b at the high frequency, respectively

relaxation, which may be due to the increased grain boundary density, while it begins to be nearly constant from 105 Hz up to 1 MHz. In case of using the high-frequency bridge, the dielectric loss seems to be nearly constant up to 107 Hz, and then it increases with frequency, reaching its maximum at the SRF point.

The resulting samples’ ac conductivity (σac) increased with increasing frequency, as illustrated in Fig. 14. σac appears to be constant and denoted as σdc in the low-frequency range. However, the σac obeys the universal power law and increases significantly in the high-frequency zone due to free charge hopping in infinite clusters [67,68]. Jonscher’s power law was employed to model the frequency-dependent σac over the frequency range from 42 Hz up to 1 MHz, illustrating the conduction mechanism through the value of s. The frequency exponent (s) ranges from 0 to 1; it denotes the charge-lattice interaction parameter, providing the information about the degree of interaction between the mobile ions and the neighboring rigid lattice. The material’s conduction mechanism can be clarified through the value of s. The values of s obtained are in the range from 0.59 to 0.97, which lie within the characteristic interval 0 < s < 1, indicating that AC conduction is controlled by a hopping mechanism of localized charge carriers rather than band conduction [28]. These values are commonly attributed to translational hopping motion (THM) in which charge carriers hop between energetically favorable localized states under the influence of the applied alternating electric field. The variation in the s values of the investigated compositions indicates that the introduction of Y3+ modifies the local defect environment and the interaction between localized charge carriers and the surrounding lattice, thus affecting the hopping probability and the overall charge transport process, as seen in the frequency-dependent conductivity behavior [28,69].

Fig. 14
Fig. 14
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The \({\sigma }_{{ac}}\) as a function of frequency for BYT-x (x = 0–10) a at the low frequency and b at the high frequency, respectively

From the mentioned figure, the conductivity of the produced samples exhibits semiconducting characteristics. Since nanocrystalline materials are compact polycrystalline, the creation of ionic space charge layers can often increase conductivity [45]. Space charge layers are created, as the nanocrystalline grain boundaries separate defects and charge carriers. This modifies the charge carrier concentration and reduces the effective barrier height for charge transfer across the interfacial zones, improving the overall conductivity.

Numerous models of conduction mechanisms are available, including Correlated Barrier Hopping (CBH), Non-overlapping Small Polaron Tunneling (NSPT), Quantum Mechanical Tunneling model (QMT), and Overlapping Large Polaron Tunneling (OLPT) model [70]. Table 4 indicates the parameters of Jonscher’s power law, which was used to accurately fit the experimental data for the prepared samples, as seen in Figs. 15 and  16. The inter-grain hopping or long-range translational motion of ions, which is caused by the ions successfully hopping to the closest possible site because of the availability of long time, is responsible for the dominance of the σdc at low frequencies. However, intra-grain hopping or short-range hopping is the cause of the high-frequency σac spectrum [71].

Fig. 15
Fig. 15
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Jonscher’s fitted ac conductivity for BYT-x (x = 0–10)

Fig. 16
Fig. 16
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The goodness of fitting using Jonscher’s power law data for BYT-x (x = 0–10) samples, respectively

Table 4 Jonscher’s power law fitted parameters for the prepared samples BYT-x (x = 0–10)
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SRF is the frequency at which the sample’s frequency response has a minimum impedance point, as illustrated in Fig. 17, where the capacitive and inductive reactance values are equivalent (\(\frac{1}{\omega C}=\omega {L}_{s}\)). Consequently, the phase angle is 0°, and the impedance is resistive. The phase angle shifts to a positive value; consequently, the inductive reactance owing to the parasitic inductance is dominant after SRF. The samples have an inductive behavior at frequencies above the SRF and, therefore, cannot be used as a capacitor. Parasitic inductance (\({L}_{s}\)) can be ignored at a very low frequency range compared with SRF, as well as the sample exhibits a high reactance (\(\frac{1}{\omega C}\)), and parallel resistance (\({R}_{p}\)) dominates compared with series resistance (\({R}_{s}\)). Consequently, the complex circuit model can be regarded as a parallel equivalent circuit consisting of C and \({{R}}_{p}\). As the frequency rises, the reactance decreases, and the series resistance (\({R}_{s}\)) dominates. Accordingly, the series equivalent circuit is the most appropriate [57]. As discussed previously, Y3+ incorporation leads to a progressive reduction in the dielectric permittivity of the BYT-x (x = 0–10) ceramics. At the SRF, the inductive reactance equals the capacitive reactance, satisfying the resonance condition (\(\frac{1}{\omega C}=\omega {L}_{s}\)), where\(\,\omega =2\pi f\).

Fig. 17
Fig. 17
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The impedance magnitude │Z│ as a function of frequency F for BYT-x (x = 0–10) samples in the high-frequency region

Accordingly, the SRF is given by

$${f}_{{SRF}}=\frac{1}{2\pi \sqrt{{L}_{s}C}}$$

Under identical sample geometry and measurement conditions, the effective inductance remains essentially unchanged, whereas the capacitance decreases with decreasing dielectric permittivity. Consequently, the reduction in capacitance shifts the resonance condition toward higher frequencies, leading to an increase in the SRF. Therefore, the introduction of yttrium results in a shift of the SRF to higher frequencies, indicating modification in the effective electrical microstructure and charge-carrier dynamics. As a result, the operating frequency range is broadened, demonstrating that doping facilitates the precise adjustment of both permittivity and frequency responsiveness. This behavior is consistent with the reported inverse relationship between dielectric permittivity and resonance frequency in dielectric resonator systems [72]. The measured SRF values of the prepared BYT-x (x = 0–10) samples are summarized in Table 5. These features are particularly favorable for high frequency applications, such as microwave dielectric resonator applications, microwave filters, and frequency stabilizers. The BYT-9 sample exhibits the highest SRF besides the lowest \({{{\rm{\varepsilon }}}}^{{\prime} }\) and\(\,{{{\rm{\varepsilon }}}}^{{\prime} {\prime}} \).

Table 5 Self-resonance frequency (SRF) of the prepared BYT-x (x = 0–10) samples
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The Nyquist plot for the prepared samples is explicitly shown in Fig. 18a. It provides important insights into a variety of relaxation processes, whether individual or collective, by plotting the imaginary part (Z′) on the vertical axis versus the real part (Z′) on the horizontal axis. Every point on the Nyquist plot represents a distinct frequency of the measurement’s AC signal. The conductivity in BYT-x (x = 0–10) samples clearly follows the Col-Col model rather than the Debye model due to this significant characteristic [73]. While exposed to a lower frequency range, the sample showed a semi-circular behavior; as frequency got higher, it showed some bending prior to change into a linear response. This suggests that the polarization response in the prepared samples is caused by the involvement of the grain and the grain boundary at upper and lower frequency ranges, respectively. Additionally, it indicates the existence of distributed relaxing periods instead of only one. An alternative technique for analyzing a material’s electrical properties, particularly its induced polarization response, is to determine its electrical modulus [74]. A divergence from perfect Debye-type relaxation is suggested by the Cole-Cole plot, which displays a deformed semicircular arc rather than an ideal one. This behavior is indicative of dispersed relaxation times, which can be explained by the reduction in grain size leading to a structural disorder and a localized charge carrier. The semicircle shape shown in Figs. 18b and  19 further supports the Cole-Cole concept of non-Debye relaxation. The presence of a single depressed semicircular arc suggests that the overall impedance response is dominated by the grain boundary contribution. Although both grains and grain boundaries contribute to the impedance response, only the grain boundary semicircle is observed within the investigated frequency range, indicating that the grain boundary resistance is significantly higher than the grain resistance [75].

Fig. 18
Fig. 18
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a Nyquist and b Cole-Cole plots for BYT-x (x = 0–10) samples at the low-frequency region

Fig. 19
Fig. 19
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The Cole-Cole plots for BYT-x (x = 0–10) samples at the high-frequency region

Impedance spectra of the prepared BYT-x (x = 0–10) samples were successfully fitted using the equivalent circuit shown in Fig. 20, which consists of a resistor (Rₚ) in parallel

Fig. 20
Fig. 20
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Equivalent electrical circuit (RP | | CPE) used to fit the impedance spectra of the BYT-x (x = 0–10) samples

with a constant phase element (CPE). Fitting parameters along with the fitting errors, χ2, and WSS values are presented in Table 6. Low fitting errors and χ2 values indicate that the chosen equivalent circuit provides an adequate description of the electrical behavior of the studied samples. Equivalent circuits are used in the analysis of impedance spectra to link the observed electrical behavior to the conduction and polarization phenomena in heterogeneous dielectric materials [76]. The deviation from the ideal capacitive response due to the grain size distribution, grain boundary effects, and defects is explained by the use of the CPE instead of the ideal capacitor. Here, the Rp value is the effective resistance for charge transfer, while the CPE-T and CPE-P terms are used to account for the pseudo-capacitance and the deviation from the Debye behavior, respectively. The obtained CPE-P values (0.87–0.97) indicate a near-ideal capacitive response with a slight deviation from Debye relaxation behavior, in agreement with the depressed semicircular arc observed in the Cole-Cole plots. This behavior reflects a distribution of relaxation times arising from microstructural heterogeneity, consistent with the dominant contribution of the grain boundaries to the overall impedance response [28,76]. The variations in Rp and CPE-T with increasing Y3+ concentration suggest that donor doping affects the electrical microstructure through alteration of grain boundary effects and polarization behavior [77].

Table 6 Optimized equivalent circuit parameters (RP, CPE-T and CPE-P), fitting errors χ2 and weighted sum of squares (WSS) as estimated for BYT-x (x = 0–10) samples, respectively
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4 Conclusion

The nanostructure pure BaTiO3 (B-T) and BaTiO3 doped with varying concentrations of yttrium ranging from 1 to 10 mol.%, coded as [BYT-x (x = 0–10)] and calcined at 1000 oC, were successfully prepared using a modified sol-gel process. Structural and frequency-dependent dielectric studies in a wide frequency range 42–109 Hz at room temperature were examined. By altering the tetragonality of the crystal lattice, the doped-produced samples have an impact on the crystal structure. The Scherer equation and Williamson-Hall plot, which were used to estimate the crystallite size (C.S), both verified that the C.S was reduced as a result of increasing the Y doping concentrations, and the nanostructure phase presence was confirmed by HRTEM. Tetragonal perovskite structures belonging to BaTiO3 and BYT-1 and BYT-2 emerged with no extra XRD peaks found for the Y, acknowledging that it was fully embedded in the B-T lattice structure for BYT-1 and BYT-2, resulting in a decrease in tetragonality. However, new secondary peaks corresponding to Y2Ti2O7 (Yttrium Titanium Oxide) and BaY2O4 (Barium Yttrium Oxide) were found at the higher Y concentrations.

The dependence of permittivity on the Y ions’ concentration was observed, and the highest value of ε′ is 770 for the pure sample BYT-0. The [BYT-x (x = 0, 5, 7, and 10)] had the highest value of conductivity. BYT-9 sample had the lowest dielectric loss value. The Cole-Cole plot displays a deformed semicircular arc rather than an ideal one, indicating dispersed relaxation times, which can be explained by the reduction in grain size leading to a structural disorder and a localized charge carrier. Y3+ incorporation modified the defect chemistry and microstructure of BaTiO3, resulting in grain refinement, reduced permittivity, and enhanced SRF. Impedance spectroscopy confirmed a grain boundary-dominated non-Debye relaxation characterized by distributed relaxation times. To further understand thermally induced polarization processes and the material’s stability, future research will focus on investigating the temperature-dependent dielectric properties to evaluate the potential suitability of the system for electrocaloric (ECE) applications.