Synthesis of Sm³⁺ and Gd³⁺ Ions Embedded in Nano-Structure Barium Titanate Prepared by Sol-Gel Technique: Terahertz, Dielectric and Up-Conversion Study

The investigated nano-structure barium titanate (BT) doped with 1 mole % of Sm³⁺ and Gd³⁺ (NBT1Sm & NBT1Gd) has been prepared by sol-gel technique in powder form by sintering at 850 °C for three hours. The starting materials used in this study are Barium acetate (Ba(Ac)₂) (99%, Sisco Research Laboratories Pvt. Ltd., India) and titanium butoxide (Ti(C₄H₉O)₄), (97%, Sigma–Aldrich, Germany) were used as the starting materials; acetyl acetone (AcAc, C₅H₈O₂), (98%, Fluka, Switzerland) acetic acid (HAc)-H₂O mixture (96%, Adwic, Egypt) were adopted as solvents of (Ti(C₄H₉O)₄), and Ba(Ac)₂, respectively. Sm (NO₃)₃-H₂O and Gd (NO₃)₃-H₂O (99.9%, Aldrich), solutions were added to the precursor with 1 mole%. The solutions were kept in air at room temperature, yielding transparent solutions. Dry gels obtained by baking the gel at 130 °C then were sintered at temperature 850 °C for 3 h in a muffle furnace type (Carbolite CWF 1200).

The examination of crystal structure of NBT1Sm & NBT1Gd nano-powder sintered at temperature 850 °C for 3 h were done by the means of X-ray diffraction (XRD, Philips X' pert MPD, Panalytical, Netherlands) using CuK_α radiation (= 1.54178 Å). The high-resolution transmission electron microscopy (HRTEM, JEM-2100F, JEOL, Japan) were used for the purpose of investigation of microstructure as well as particle size and selected area electron diffraction (SAED) patterns. The selected area electron diffraction and lattice fringes were analyzed via the software attached to the HRTEM. The compositions and planar images of the samples were pictured by Energy dispersive X-ray unit attached to Scanning electron microscope (FESEM, Quanta FEG 250, and FEI, USA). For photoluminescence (PL) measurement, the samples were excited at the 850 nm (laser diode) using spectrofluorometer (SF, Jasco, FP-6500, Japan) and the excitation slit bandwidth was 5 nm. The SDT Q600 V20.9 Build 20 from TA Instruments was used to estimate the TGA and DSC data of NBT1Sm & NBT1Gd nano-powder. The process was completed in a nitrogen atmosphere (flow rate 30.0 ml min⁻¹) and the temperature range 20 °C–1000 °C at a heating rate of 10 °C min⁻¹. A parallel plate capacitor configuration consisting of two gold-coated brass electrodes of 20 mm in diameter were used to achieve the dielectric measurements by employing a high-resolution Alpha-A Analyzer from novocontrol. The frequency range of 10⁻¹−10⁷ Hz and at temperatures ranging from 303 to 433 K were covered. The Quatro controller system using pure nitrogen as a heating agent was used to stabilize the temperature; it was kept to be better than ±0.1 K. All terahertz measurements in the frequency range of 0.01 and 3 THz were conducted using TPS spectra 3000 system, Teraview Ltd. England whose number of scans of 1800 scans s⁻¹ and spectral resolution 1.2 cm⁻¹. The measurements were carried out in nitrogen-dry environment to overcome of any artifacts due to water vapor or other atmospheric absorption lines. Usually, both amplitude and phase of signal are measured. The data were acquired in the transmission-mode, where a sweep of a polyethylene sample was always kept as a reference measurement. In order to increase the signal-to-noise ratio every sample data was the average of five individual measurements.

The XRD spectra of the of NBT1Sm and NBT1Gd nano powder obtained after being sintered at 850 °C for 4 h are exhibited in Fig. 1a. The obtained diffraction lines were compared with the standard data of BaTiO₃ (JCPDS file No. 81-2201), as can be seen in Table I. It has been found that all the peaks were nearly consistent with the standard data with preferred orientation along the (110) direction. Thus, investigated nano powders formed in distinct tetragonal perovskite structure (ferroelectric phases). It is conspicuous from the Table I that the diffraction lines slightly shift towards high Bragg angles compared to ICDD data, which is a strong indication of improved tetragonality of the samples when doped with Sm³⁺ and Gd³⁺ ions. ¹ The improved tetragonality arise from the variation of d-spacing followed by change of lattice parameters, c and a, and hence improve the c/a ratio to be close to unity as could be seen in Table II. Such shift may be explained by the incorporation of Sm³⁺ and Gd³⁺ ions into the nanostructure BTiO₃, which directly affected the volume of the unit cell and hence varied the d-spacing, which in turn is responsible for shift to high Bragg angles. ³⁹ The position of reflections of NBT1Sm and NBT1Gd is completely identical which could be accounted by the nearly atomic radii of Sm and Gd. Figure 1b presents the enlargement of area under rectangular magenta, displayed a distinct splitting of (200)/(002) peaks at around 45° and 45.7° asserting the enhancement of tetragonal distortion phase, which is a good indication with many previous works. ^40,41 It is known from our prior work ⁴ that pure BaTiO₃ sintered between 800 °C and 1000 °C frequently demonstrates the coexistence of majority tetragonal and minority cubic phases. Then, the cubic phase disappears at higher calcination temperature of about 1050 °C, where all peaks are attributed to the perovskite tetragonal phase BaTiO₃ with an unwanted residual amount of BaCO_3.

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However, the sintering has been carried on for 4 h at 850 °C, the XRD diffractogram had a trace of BaCO₃ labeled by asterisks on the Fig. 1a. As a consequence of the presence of some impurity phases, the perovskite phase percentage was estimated by the highest intense peak according to the below formula. ⁴²

$$\begin{eqnarray} &&\% \,{\rm{Perovskite}}\,{\rm{phase}}=\left(\displaystyle \frac{{{\rm{I}}}_{{\rm{perov}}}}{{{\rm{I}}}_{{\rm{perov}}}+{{\rm{I}}}_{{\rm{BaCO}}3}}\right)\,\times \,100\end{eqnarray} \tag{ 1 }$$

where I_perov, I _BaCO3 refer to intensity of (hkl) lines for BaTiO₃ and BaCO₃, respectively. The percentage of the perovskite phase found to be 97.26% and 96.41% for NBT1Sm and NBT1Gd, respectively. This means that the BaCO₃ phase represented around 4%, and hence could be negligible. The lattice parameters of studied doped powders were calculated by means of miller indices (hkl) and d-spacing using the below equation ^43,44 :

$$\begin{eqnarray}&&\displaystyle \frac{1}{{d}^{2}}=\displaystyle \frac{{h}^{2}+{k}^{2}}{{a}^{2}}+\displaystyle \frac{{l}^{2}}{{c}^{2}}\end{eqnarray} \tag{ 2 }$$

The computed estimated lattice constants for powders are a = b = 3.99 nm and c = 4.14 Å. It is obvious that the lattice constant a is the equal to standard file, whereas the lattice constant c has bigger values, i. e., the tetragonal unit cell has been stretching along the c-axis due to the incorporation of Sm³⁺ and Gd³⁺ ions. This distortion of unit cell is responsible for the shifting 2θ angles to higher values. Our results are in a good agreement with previous works. ^9,45

The average crystallite size (D) and microstrain (ε) of Sm³⁺ and Gd³⁺ doped BTiO₃ were estimated by the mean of Williamson-Hall (W-H) relations. ⁴⁶

$$\begin{eqnarray}&&\beta \,\cos \,\theta =\displaystyle \frac{k\lambda }{D}+4\varepsilon \,\sin \,\theta \end{eqnarray} \tag{ 3 }$$

where β refer to the full width at the half maximum, k implies the shape factor ≈ 0.94, λ denotes to the wavelength of the incident X-rays (λ = 0.15406 nm). Figure 1c represented the variation of FWHM of each plane with the 4sin(θ) as well as the linear fit of the plotted points. The values of the goodness of fit (R²) of a regression of NBT1Sm and NBT1Gd are 0.94 and 0.93, which very close to unity expressing the regression line highly fitted the data. From Eq. 2, the crystallite size and strain are computed from the slope and intercept of the fitted line and the date is listed in Table II. The crystallite sizes of NBT1Sm and NBT1Gd are 38.36 and 33.26 nm, whereas the strains are 5.77 and 1.61, respectively.

In order to detail particle size, shape and microstructure of studied powders, NBT1Sm and NBT1Gd were analyzed by HRTEM. Figures 2a, 2b displayed the low magnified image of NBT1Sm and NBT1Gd, respectively. clearly, the HRTEM micrograph indicated highly compact, organized grains of definite shape that seems to be a tetragonal. The prepared sample had high degree of crystallinity with homogeneous distribution of shape and particle size due to the annealing temperature. The particles tend to aggregate in cluster due to its small size. Clearly, the grain size of NBT1Sm is bigger than that of NBT1Gd. Figures 2c, 2d illustrates the histogram of the estimated grain size and their Gaussian fit which asserted that the grain size of NBT1Sm and NBT1Gd are 43.66 and 36.38 nm, respectively confirming presence of the nano-structure phase. It is worth mentioning here that the grain size estimated by HRTEM have the same trend like XRD, but the values of grain size has a little different. This deviation could be ascribed to the fact that XRD refers to coherent domains and the HRTEM images may be polydomain in nature. ⁴⁷ Figure 2e represents the high magnification image of NBT1Sm including the line profile on the bottom as well as Fourier transformer analysis in the left for accurate measurement of d-spacing. The existence of lattice fringes indicates structural uniformity polycrystalline nature of the studied powder. The inset line profile in Fig. 2e shows that the lattice distance is 2.65 nm which is almost near to (112) reflection. selected area electron diffraction (SAED) patterns are clearly pictured in Fig. 1f indicating the polymorphic rings consisting of spots indicating the nanostructure nature of prepared powder alongside polycrystalline nature. The values of measured interplanar spacing are indexed and labelled using the HRTEM software. It is visible that the estimated d-spacing from the HRTEM are nearly agree with that obtained from XRD measurement.

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The elemental composition of NBT1Sm and NBT1Gd sintered at 850 °C is investigated by energy dispersive X-ray analysis, as seen in Figs. 3a, 3b, respectively. the presence of both Sm³⁺ and Gd³⁺ have been asserted and their values were listed in Table III. After close inspection of weight percentages, it was found that BaTiO₃ + 1 wt.% (Sm³⁺ and Gd³⁺) and the ratio of Ba:Ti:O are almost 1:1:3, which indicating that the Sintered BaTiO₃ are near in stoichiometry. The weight percentages of the elements are consistent with previous work. ⁸ The data of EDX alongside XRD and HRTEM revealed the successful preparation of NBT1Sm and NBT1Gd. Figures 3c, 3d, respectively, displayed the SEM planar micrograph of NBT1Sm and NBT1Gd sintered at 850 °C. It is apparent that the granules present in an agglomerated form and showed a compact morphology. It well-known that the agglomeration is distinct feature in nanopowder due to large specific area. ^6,48 Because of the many agglomerated particles, we find a difficulty in estimation of grain size where there is a broad particle size distribution which make the estimation is not accurate.

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Under the excitation of the NBT1Sm powder sample sintered at 800 °C for 4 h by laser diode at 850 nm, the green and red emissions characteristic of trivalent Sm³⁺ ions were readily observed in Fig. 4a. When laser diode is tuned to a strong Sm³⁺ ions absorption (⁶H_5/2 → ⁶F_11/2) at wavelength of 850 nm, respectively two green and red photoluminescence group emissions assigned to (⁴F_3/2 → ⁶H_5/2) (528 nm) and (⁴G_5/2 → ⁶H_5/2) (567 and 597 nm) were detected. The green emission attributed to the intra 4-F transitions of Sm³⁺ ions, whereas the red emission is attributed to another intra 4-F transition of Sm³⁺ ions and assigned to (⁶F_11/2 → ⁶H_5/2) (604, 611, 632, 643, 660, 672, 688, 716 and 732 nm), respectively. After the laser diode light brings the Sm³⁺ ion into ⁶F_11/2 level, the Sm³⁺ ion will decay non-radiatively from the ⁶F_11/2 state to the ⁶F_9/2, ⁶F_7/2, ⁶F_5/2, ⁶F_3/2, ⁶H_15/2, ⁶H_13/2, ⁶H_11/2, ⁶H_9/2, ⁶H_7/2 and subsequently into the ⁶H_5/2 metastable levels state following the initial energy transfer from the Sm³⁺ ion. The energy transfer processes bring the Sm³⁺ ion into (⁴G_7/2) state from which (⁴F_3/2) is populated through the non-radiative relaxation giving rise to the green up-conversion mechanism as shown in the schematically energy level diagram of Sm³⁺ ion levels shown in Fig. 4b. The second mechanism is the red up-conversion luminescence, in which, the laser beam brings the Sm³⁺ ion to the excited state ⁶F_11/2 level, which then decays through a non-radiative process into ⁶F_9/2, ⁶F_5/2, ⁶H_11/2, ⁶H_9/2 and subsequently into the ⁶H_5/2 metastable levels as ca be seen in Fig. 4b. Energy transfer processes bring the Sm³⁺ ion into (⁴F_3/2) state from which (⁴G_5/2) is populated through the non-radiative relaxation process. The spin-allowed radiative transition from the preceding mentioned levels to the ground state gives rise to visible (green and red) emissions. The obtained results are in compatible with the previously reported works. ⁴⁹ It worth to be mentioned that the NBT1Gd sample has not shown any upconversion on the visible region in our measurements.

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Figures 5a, 5b shows TGA-DSC curves of the synthesized NBT1Sm and NBT1Gd, respectively, obtained after calcination at 800 °C for 4 h. The analysis was done at a heating rate of 10 °C min⁻¹ in nitrogen atmosphere. The TGA curves depicts only about 6% weight losses below 1000 °C for both NBT1Sm and NBT1Gd. This weight loss could be attributed to the vaporization of residual or non-structural water and volatile organic solvents in 18 °C–300 °C temperature range, ^50,51 and due to the decomposition of a small amount of an impurity phases (such as BaCO₃ as confirmed by XRD analysis) in higher temperature ranges. ⁵² Generally, there is no distinct peak in the DSC curve of the obtained powder, which reflects the purity and thermal stability of the synthesized material. ⁵² A small endothermic peak situated at ∼360 °C (for both NBT1Sm and NBT1Gd) denotes the decomposition of organic compounds. ⁵³ The other small exothermic beak at ∼815 °C (for both NBT1Sm and NBT1Gd) points out the transformation of amorphous decomposition products in BaTiO_3. ⁵³ According to the previous works, ^54–59 crystallization of barium titanate takes place at 700 °C. Furthermore, the weight loss is minimal and the weight remains unaffected after that. It was stated that Ba²⁺ ions in the A site are mainly replaced by rare earth elements. ⁶⁰ Sm³⁺ (0.136 nm) and Gd³⁺ (0.107 nm) ions are most probably replaced Ba²⁺ (0.135 nm) cations rather than Ti⁴⁺ (0.065 nm). ⁵³

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The thermal kinetic parameters, i.e. thermal activation energy (E_a), Arrhenius constant (A), activation entropy ΔS, The activation enthalpy ΔH and Gibbs free energy change ΔG can be calculated by application of Broido method ⁶¹ on TGA data. In this method, for 1st order transition:

$$\begin{eqnarray}&&\mathrm{ln}\,\mathrm{ln}\left(1/y\right)=-\left({E}_{Ta}+1/R\right)+constant\end{eqnarray} \tag{ 4 }$$

where y is the fraction of undecomposed substance, E_Ta is the thermal activation energy, and R is the universal gas constant. Figure 5c Shows the relation of ln (1/y) with 1000/T of the observed main decomposition stage, i.e. 760 °C–860 °C.

The other thermal kinetic parameters are calculated as follows:

$$\begin{eqnarray}&&{\rm{\Delta }}S=R\,\mathrm{ln}\left(Ah/k{T}_{m}\right)\,\end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray}&&{\rm{\Delta }}H=E-RT\end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray}&&{\rm{\Delta }}G={\rm{\Delta }}H-T{\rm{\Delta }}S\end{eqnarray} \tag{ 7 }$$

Where ΔS is activation entropy, A is Arrhenius constant, k is the Boltzmann constant, h is the Planck constant, Tm is the maximum rate reaction temperature, ΔH is activation enthalpy and ΔG is Gibbs free energy change. The values of thermal kinetic parameters, A, Tm, Ea, ΔS, ΔH, and ΔG of the three thermal decomposition stages are summarized in Table IV. The high E_a values are indication of high thermal stability of NBT1Sm and NBT1Gd. The exothermic nature of thermal degradation is indicated by the negative value of enthalpy ΔH, as indicated from DSC result at this temperature range. ⁶² The less ordered activated state is indicated from the positive values of entropy ΔS and the spontaneity of thermal degradation is indicated by negative ΔG values. ⁶²

Figures 6a, 6b show the dielectric real part (storage) εʹ, while parts c and d show the dielectric imaginary part (loss) εʹʹ of both NBT1Sm and NBT1Gd, respectively. Obviously, the low frequency relaxation, in particular in the εʹ loss, in both cases (NBT1Sm and NBT1Gd) deviates from the expected contribution of the conductivity εʹʹ(ν) ∝ ν⁻¹. Thus, a remnant of a relaxation process is affecting the low frequency part of the loss as well as the storage, which describes the relaxation of free charges at the electrode/crystal interfaces activated process. ^63,64 In addition, another relaxation process is affecting the high frequency side and becomes active upon cooling (cf. the loss data at T = 303 K). This process has been discussed in details in the work of Artemenko et al., ⁶⁵ where he attributed this relaxation peak to the presence of polaronic states arising from charged defects. Additionally, it has been addressed to this relaxation is thermally activated. ^63,64 These processes are clearly observed in the modulus representation as shown in Figs. 7a, 7b, which is obtained according to M* = 1/ε*. In both cases (NBT1Sm and NBT1Gd), the spectral features are similar_. Yet, the relaxation processes in the case of NBT1Gd are slightly shifted to high frequencies, and enhanced in the amplitude. Figures 7c, 7d illustrates the frequency dependence of the real part of the conductivity for NBT1Sm and NBT1Gd, respectively. Clearly, the σʹ (ν) changes from power-law regime to the other. Such power-law dependence may be correlated by power-law of Jonscher, ⁶⁶

$$\begin{eqnarray}&&{\sigma }_{{\rm{Ac}}}\left(\omega \right)={\sigma }_{{\rm{Dc}}}+{\rm{A}}{\omega }^{{\rm{n}}}\,\end{eqnarray} \tag{ 8 }$$

with ${\sigma }_{{\rm{Dc}}}$ represents the DC conductivity, A is the pre-exponential factor, which depends on temperature and material, and n is the exponent of frequency. The latter nominates the degree of interaction between mobile ions with the lattices. ⁶⁶ At lower frequencies the conductivity remains almost constant, which corresponds to the DC conductivity (σ_DC). According to the double layer model of Maxwell–Wanger for dielectrics, the resistive grain boundaries at low frequency is significant. consequently, the transportation occurs by penetration process. ⁶⁷ At high frequencies, the conductivity in contrast increases suddenly defining the σ_AC part. A power-law exponent n = 0.9 is found, which manifests the hopping by enhanced number of charge carriers. ^68,69 Moreover, the frequency-temperature superposition FTS was tested where a value σ_DC is used along vertical axes and a value ν_c is used for horizontal axes. The latter represents the onset of the plateau, and usually interpreted as the rate with which the ions escape from their local entrapment (a Coulombic 'cage). ^33,70 The data are collapsed in a good master curve, ⁷¹ i.e., the FTS holds well (cf. inset in Fig. 7c). Doing so, the values of the σ_DC (T) as well as τ_c = 1/2πν_c are obtained; all are included in the relaxation map shown in Fig. 8a. Also, the relaxations times obtained from the modulus M' data, using the "peak picking" where the equation ωτ_M = 2πν_M τ = 1 holds, are included in Fig. 8a.

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Further, the temperature dependence of the dielectric permittivity εʹ is evaluated at ν =1 kHz for both BT1Sm and BT1Gd samples and displayed in Fig. 8b. In contrast to the reported giant dielectric constant values of bulk BaTiO₃, our investigated samples (BT1Sm and BT1Gd) have less values that lies in the order of tens. Yet, such giant values were discussed as the grain size, particle size, and film thickness dependencies; the presence of thin low-permittivity layers at the grain boundaries was taken as a point. ^72–74 Here, filling the BaTiO₃ ceramics with Sm³⁺ reduces the εʹ compared to the case filled with Gd. Also, from Fig. 8b, one may estimate the ferroelectric phase transition temperature T_c in both cases is T_c > 423 K, which is higher than that reported of the pure nanocrystalline BaTiO₃. ⁷⁵ Taken the grain size as an effect and in accordance with that previously proposed by Zhang et al. ⁷⁶ and Lee and Auh ⁷⁷ ; the T_c decreases with decreasing of grain size. Thus, the doping shows an opposite effect to reducing the grain size.

In Figs. 9a, 9b display the evolution of the electric field in the time domain and frequency domain of THz signals passed through the NBT1Sm and NBT1Gd with respect to that passes through the polyethylene discs as a reference. Comparing to the reference, one observes that the signal amplitude is attenuated by ∼26% in the case of NBT1Sm, while it is lowered by ≈40% in the case of NBT1Gd. In both cases the signal slightly shifts to a higher position.

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The exploring of the refractive index in the terahertz region is very considerable for the many application like waveguide for terahertz radiation and terahertz sensor that involved in many biomedical instruments. Therefore, the refractive index (n) has been computed using the following equation ^71,78 :

$$\begin{eqnarray}&&n\left(\nu \right)=\displaystyle \frac{\varphi \left(\nu ,d+\delta d\right)-\varphi \left(\nu ,d\right)}{2\pi \nu \delta d}\left(C\right)+{n}_{o}=\displaystyle \frac{{\rm{\Delta }}\varphi \left(\nu \right)}{2\pi \nu \delta d}\left(C\right)+{n}_{o}\end{eqnarray} \tag{ 9 }$$

φ denotes to the phase of the THz pulse, ν is the frequency, d refers to the thickness of the cell and C equals to the velocity of light in vacuum. Figure 10a represents the refractive indices for frequencies between 0.5 to 3 THz. The refractive index follows a normal dispersion behavior with frequency where a monotonous decrease with frequency can be noted for both the index of refraction. All over the frequencies range, the index of reflection of NBT1Gd is larger than in NBT1Sm. Such reduction of the refractive index may be ascribed to grain size as well as roughness the surface. The values of refractive index at 1 THz is about 3.71 and 4.97 for NBT1Sm and NBT1Gd, respectively. it is worth to mention that our measured values of refractive indices have not match with the values of bulk BaTiO₃ which have higher values around 19 at 1 THz. ⁷⁹ Such incompatibility comes from the fact that our investigated films is prepared in nano structure which stand behind the low values of refractive index. Thus, the low values of refractive index are a further demonstration of formation of nano-structure NBT1Gd and NBT1Sm. The absorption coefficient α(ω) of the investigated samples has been calculated by the means of the below relation. ⁸⁰

$$\begin{eqnarray}&&\alpha \left(\omega \right)=\,\displaystyle \frac{1}{d}\,\mathrm{ln}\,\displaystyle \frac{\left|{E}_{s}\left(\omega \right)\right|}{\left|{E}_{r}\left(\omega \right)\right|}={t}_{12}{t}_{21}{e}^{id\left(k-{k}_{o}\right){e}^{\displaystyle \frac{-\alpha d}{2}}}\end{eqnarray} \tag{ 10 }$$

where d is the thickness of the cell, E_s is the transmitted pulse and E_r denotes to reference spectrum, where t₁₂ and t₂₁ are the frequency-dependent complex Fresnel transmission coefficients, α is the power absorption, k = 2πnr/λ and k₀ = 2π/λ are the propagation wave vectors. The THz absorption coefficient spectra of NBT1Gd and NBT1Sm in the terahertz range 0.03 to 3 THz are presented in the Fig. 10b. It is clear that the absorption coefficient improved upon the increasing of frequencies. Also, no apparent peaks appear all over the THz range, which is very similar to the trend of refractive index. The inset of Fig. 10b displayed the frequency-dependent extinction coefficient (k) which calculated from absorption coefficient by αλ/4πs. It is observed that the extinction coefficient increases monotonically with the increase of frequency. The plots have no scatterings or spikes reflecting the pure nature of prepared samples. the values the attenuation here is very small and reneged from 0.1 to 0.2 in sub-terahertz region (0.1 to 0.8 THz) and begin to increase to reach 1.5 when the frequencies increase to 3 THz. This means that the attention increases in the terahertz region compared to far and mid IR region due to the wide band gap of BaTiO₃.

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The complex-valued dielectric function of NBT1Sm and NBT1Gd, are calculated from the refractive index according to below equations ⁸¹ :

$$\begin{eqnarray}&&\varepsilon ^{\prime} \,=\,\,{n}^{2}-{k}^{2}\,{\rm{and}}\,\varepsilon ^{\prime\prime} \,=\,2nk\end{eqnarray} \tag{ 11 }$$

with k denoting the extinction coefficient. Both εʹ and εʹʹ are shown in Figs. 10c, 10d. In the case of NBT1Gd, two peaks locate around 0.05 and 1 THz can be recognized in the ε'' spectrum, which accompanied by downward changes in εʹ. These peaks are again observed in the case of NBT1Sm, which may describe the lattice vibration in crystalline NBT1Sm and/or NBT1Gd. Yet, the dielectric loss and storage are enhanced by factor about 4. The dielectric response in THz region of both cubic and tetragonal phases of BaTiO₃ exhibit the lattice vibrational modes defined as soft mode (SM) or central mode (CM). ^82–86