Complex dielectric, electric modulus, impedance, and optical conductivity of Sr_3−x Pb _x Fe₂TeO₉ (x = 1.50, 1.88 and 2.17)

3.1 Structural characterization

Figure 1 presents the powder XRD patterns of Sr_3−x Pb _x Fe₂TeO₉ perovskites with x = 1.50, 1.88 and 2.17. The inset in Figure 1 shows excellent agreement between the observed and calculated patterns. The crystal structures were resolved using the Rietveld refinement method; it revealed that x = 1.50 and 1.88 adopt a tetragonal phase of space group I4/mmm (#139) and it also revealed that x = 2.17 adopts a hexagonal phase of space group R 3 ̄ m (#166). For x = 1.50 and 1.88, the Sr²⁺/Pb²⁺ cations were located at the 4d (0,1/2,3/4) positions of D _2d symmetry, Fe(1)/Te(1) at 2a (0,0,0) and Fe(2)/Te(2) at 2b (0,0,1/2) sites of D _4h symmetry, and O²⁻ anions at the 4e (0,0,z) and 8h (x,x,1/2) positions (C _4v and C _2v symmetries, respectively). The (Fe/Te)_2a–O_(8h,4e)–(Fe/Te)_2b bond angles are constrained to be 180° by I4/mmm space group. For this reason, the tilting angle of the (Fe/Te)O₆ octahedra is considered to be null. For x = 2.17, the Sr²⁺/Pb²⁺ cations were situated at the 6c (0,0,z) positions of C _3v symmetry, Fe(1)/Te(1) at 3a (0,0,0) and Fe(2)/Te(2) at 3b (0,0,1/2) sites of D _3d symmetry, and O²⁻ anions at the 18h (x,−x,z) positions of C _s symmetry. A certain degree of anti-site disordering effect of Te⁶⁺ and Fe³⁺ cations on the B-sites was detected, indicating the presence of a partial amount of Te⁶⁺ at Fe³⁺ positions and vice-versa. Recently, the current Sr_3−x Pb _x Fe₂TeO₉ series in the composition range (0 ≤ x ≤ 2.25) [12] was reported by our research team exhibiting two phase transitions from a tetragonal space group I4/m (0 ≤ x ≤ 1) to another tetragonal space group I4/mmm (1.25 ≤ x ≤ 1.88) and finally to a hexagonal space group R 3 ̄ m (2.08 ≤ x ≤ 2.25). XRD patterns can be used to evaluate peak broadening with crystallite size, and lattice strain due to grain surface, domain boundaries, dislocations, non-uniform lattice distortions and other contributions. The crystallite size of the compositions x = 1.50, 1.88 and 2.17 was determined by the X-ray line broadening method using the Scherrer formula and the Williamson–Hall (W–H) equation. The Scherrer formula can be expressed as follows [13]:

where D is the crystallite size in nanometers, λ is the X-ray wavelength (Cu-K_α average = 1.54178 Å), K is a Scherrer constant equal to 0.94, θ is the Bragg angle of the most intense peak and β _hkl is the line broadening, i.e., full width at half maximum (FWHM) on the highest peak of the tetragonal (112) (200) planes and the hexagonal (104) (110) planes, which are located at around 2θ ≈ 32°.

Figure 1:

XRD patterns of Sr_3−x Pb _x Fe₂TeO₉ (x = 1.50, 1.88 and 2.17) perovskites collected at 298 K. The inset shows the final Rietveld refinement plot of x = 1.88.

In the Williamson–Hall (W–H) method [14], strain-induced broadening resulting from imperfection and distortions was related as follows:

where (ε = Δd/d) is an upper limit of lattice distortion. Assuming that the contributions of particle size and strain to the line broadening are independent of each other and both have a Cauchy-type profile, the observed linewidth can be expressed from Equations (1) and (2) as:

The unit-cell parameters, crystallite size and microstrain of the studied samples are computed and presented in Table 1. It can be noted that the crystallite size calculated using the Scherrer technique is smaller than that obtained using the W–H method.

3.2 Optical parameters

The optical properties of Sr_3−x Pb _x Fe₂TeO₉ (x = 1.50, 1.88 and 2.17) ceramics have been investigated using UV/Vis/NIR spectrophotometer measurements. Absorbance and transmittance plots of the present compositions in the wavelength range 250–800 nm are shown in Figure 2a and b, respectively. By decreasing the wavelength from 800 to 500 nm, the absorbance spectra show a significant increase, while the transmittance spectra show a substantial decrease.

Figure 2:

(a) Absorbance, and (b) transmittance spectra of Sr_3−x Pb _x Fe₂TeO₉ (x = 1.50, 1.88 and 2.17) ceramics as a function of wavelength, (c) absorption coefficient versus photon energy, (d) Tauc plots of (αhυ)² against hυ.

Figure 2c exhibits the absorption coefficient (α) as a function of photon energy (hʋ) which can be computed from the transmittance data using the following Equation [15]:

where T is the optical transmittance and d is the thickness of the material.

The relation between the absorption coefficient (α) and the incident photon energy (hʋ) can be expressed using Tauc’s relationship in the high absorption region of semiconductor as follows [16]:

where B is a constant, hʋ is the photon energy (h = 6.626 × 10⁻³⁴ J is Planck’s constant, ʋ = c/λ is the light frequency s⁻¹) and m represents the nature of the material transition (m = 2 for direct transition, m = 1/2 for indirect transition).

The optical bandgap energy of the compositions x = 1.50, 1.88 and 2.17 was determined using the expression of a direct bandgap semiconductor, m = 2, close to the band edge, with hʋ greater than E _g of semiconductors, i.e., (αhʋ)² ≈ B(hʋ − E _g). Plots of (αhʋ)² versus photon energy hʋ (known as Tauc plots [16]) for some selected compositions are displayed in Figure 2d. The values of the direct bandgap energy were estimated by extrapolating the linear region of the Tauc plots to the energy axis with a condition of (αhʋ)² = 0 (as seen in Figure 2d). Therefore, E _g values are estimated to be ∼1.89 eV for (x = 1.50), ∼1.87 eV for (x = 1.88) and ∼1.74 eV for (x = 2.17). The bandgap energy (E _g) reduces with increasing Pb concentration in the system, indicating that the hybridization between the metal and oxygen orbital is enhanced. It can be confirmed that the observed values of E _g obtained by multiplying the absorption coefficient by hʋ (Equation (5)) are slightly lower than those obtained in Reference [12] using the (F(R) × hυ)² Kubelka–Munk (K–M) equation [17] for the same compositions. The absorption and refraction of a medium can be described by a single quantity called the complex refractive index which is written as follows [18]:

where the real part n is called the refractive index, and the imaginary part k is called the extinction coefficient. The refractive index (n) is an important parameter for the application of optical materials in optics-based devices because of its direct relationship to energy dispersion. The extinction coefficient (k) vanishes for lossless materials. The both parameters (n and k) are collectively called the optical constants of the material, were calculated using the following expressions [19, 20]:

where n is computed from the reflectance and extinction coefficient data. The photon energy hʋ dependence of refractive index (n) and extinction coefficient (k) of ceramics Sr_3−x Pb _x Fe₂TeO₉ (x = 1.50, 1.88 and 2.17) is depicted in Figure 3a and b, respectively. It was found that the refractive index (n) of the studied samples strictly decreases when the photon energy rises from 1.60 to 2.50 eV; then it becomes saturated in the high photon energy region.

Figure 3:

(a) Refractive index, and (b) extinction coefficient dependence on hυ for the Sr_3−x Pb _x Fe₂TeO₉ (x = 1.50, 1.88 and 2.17) perovskites, (c) real, and (d) imaginary parts of the dielectric permittivity as a function of photon energy.

However, the behavior of the extinction coefficient (k) differs from the refractive index (n); where k strictly increases when the photon energy rises from 1.70 to 2.43 eV to reach the maximum at 0.1709 for (x = 1.50), at 0.1789 for (x = 1.88), and at 0.1826 for (x = 2.17); then a considerable decrease was observed above k _max with increasing photon energy.

The complex dielectric function (ε*) is usually defined in terms of its real and imaginary parts using the following Equation [18]:

The real (ε ₁) and imaginary (ε ₂) parts of the dielectric permittivity were evaluated using the relations derived from the complex refractive index as follows:

The variation of ε ₁ and ε ₂ as a function of photon energy hʋ of the samples (x = 1.50, 1.88 and 2.17) is presented in Figure 3c and d, respectively. It can be noted that the real part (ε ₁) of the dielectric function exhibits the same behavior as the refractive index (n), where it strictly decreases when the photon energy rises from 1.60 to 2.50 eV; and then becomes saturated in the high photon energy region (>2.50 eV). In contrast, the imaginary part (ε ₂) shows a sharp decrease with increasing photon energy from 2.50 to 4.94 eV.

An obvious consistency between the spectral plots of Figure 3a and c was observed, because there is a relationship between the dielectric permittivity (ε ₁, ε ₂) and the optical constants (n, k):

It can be seen that the real part (ε ₁) shows the same trend as the refractive index (n) when the photon energy hʋ increases from 1.54 to 4.94 eV, and the values of the real part are larger than those of the imaginary part.

The optical conductivity is one of the important quantities that describe the optical properties of solids, and it is mainly used to detect any further allowed interband optical transition of a material. The complex optical conductivity (σ* = σ ₁ + iσ ₂) is related to the real and imaginary parts (ε ₁, ε ₂) of the complex dielectric function (ε*) by the following expressions [21]:

where ω (=2πν) is the angular frequency and ε ₀ (=8.854 × 10⁻¹² F m⁻¹) is the free space dielectric constant. The photon energy (hυ) dependence of the real and imaginary parts (σ ₁, σ ₂) of the complex optical conductivity (σ*) of ceramics Sr_3−x Pb _x Fe₂TeO₉ (x = 1.50, 1.88 and 2.17) is presented in Figure 4a and b, respectively.

Figure 4:

(a) Real, and (b) imaginary parts of the complex optical conductivity versus photon energy for the Sr_3−x Pb _x Fe₂TeO₉ (x = 1.50, 1.88 and 2.17) ceramics, (c) optical conductivity σ _opt, and (d) dissipation factor as a function of hυ.

It can be seen that the real part (σ ₁) progressively rises with increasing photon energy, while the imaginary part (σ ₂) decreases in the low photon energy region to reach the minimum at ∼2.50 eV for (x = 1.50), at ∼2.37 eV for (x = 1.88), and at ∼2.32 eV for (x = 2.17); then it shows a significant increase above its minimum in the high photon energy region. The observed values of the real part (σ ₁) are much smaller than those observed in the imaginary part (σ ₂).

To evaluate the optical conductivity of the present compositions x = 1.50, 1.88 and 2.17, we used the refractive index and its relationship with the absorption coefficient as follows [22]:

Figure 4c shows the variation of optical conductivity as a function of photon energy. It can be observed that the plots of σ _opt as a function of hυ exhibit the same behavior as that observed in the plots of the real part (σ ₁) of the complex optical conductivity versus hυ.

In physics, the dissipation factor (or dielectric loss), tanδ, is a measure of the rate of energy loss from a mechanical or electrical oscillation mode in a dissipative system. For example, electrical energy in all dielectric materials is generally dissipated in the form of heat. The dissipation factor (tanδ) can be written as follows [23]:

The variation of tanδ as a function of photon energy hυ for the samples x = 1.50, 1.88 and 2.17 is displayed in Figure 4d.

The dispersion of the refractive index depends on the frequency of the light beam. As part of the energy evaluation that characterizes our compounds, we also investigated the dependence of the photon energy on the refractive index using the single effective oscillator model proposed by Wemple—DiDomenico [24] as:

where E ₀ and E _d are single oscillator constants. E ₀ is the single oscillator energy, which is considered to be the average excitation energy of the electronic transitions; it is proportional to the bandgap, while E _d is the dispersion energy, which is considered as a direct measure of the average strength of the optical transitions between the bands.

The curves of refractive index factor (n ² − 1)⁻¹ versus (hυ)² of ceramics Sr_3−x Pb _x Fe₂TeO₉ with x = 1.50, 1.88 and 2.17 are shown in Figure 5a. The plots of Equation (15) allow us to determine the single oscillator constants (E ₀ and E _d) directly from the slope 1/(E ₀ E _d) and intercept (E ₀/E _d) of the linear fit. The estimated values of E ₀ and E _d are listed in Table 2.

Figure 5:

(a) Linear fitting curves of (n ² − 1)⁻¹ versus (hυ)², (b) linear fitting curves of ln(α) against hυ, (c) linear, and (d) nonlinear optical susceptibilities as a function of (hυ)².

The results indicate that the values of E ₀ follow the same variation of E _g and we can observe that E ₀ ≈ E _g, while the values of E _d change in the opposite direction with increasing x. The dispersion energy E _d may depend on the charge distribution within a unit cell, which is closely related to chemical bonding.

The values of the static refractive index (n ₀) are calculated by extrapolating the Wemple–DiDomenico dispersion equation to (hυ → 0), which is given by:

The values of n ₀ are calculated to be ∼1.79 for (x = 1.50), ∼1.62 for (x = 1.88) and ∼1.95 for (x = 2.17). The high frequency dielectric constant ε _∞ = (n ₀)² is 3.204 for (x = 1.50), 2.624 for (x = 1.88) and 3.803 for (x = 2.17). According to the model mentioned above, the single oscillator parameters E ₀ and E _d are correlated to the imaginary part (ε ₂) of the complex dielectric function and the −1 and −3 moments of the ε ₂ optical spectrum can be derived from the following expressions [25]:

The values of the M ₋₁ and M ₋₃ moments are summarized in Table 2, where M ₋₁ is dimensionless and M ₋₃ is in (eV)⁻².

The Urbach energy is generally defined as the width of the tail of localized defect states in the bandgap. In fact, the disorder is associated with structural defects in a material, leading to an elongation of the density of states in the band tails. The impurities can cause band defects, which increase the intermediate levels within the bandgap region.

The Urbach energy is taken into account as the width of the exponential absorption edge and can be determined by the following relationship [26]:

where α ₀ is a constant and E _U is the Urbach energy. Figure 5b shows the plots of ln(α) as a function of photon energy hυ for ceramics Sr_3−x Pb _x Fe₂TeO₉ with x = 1.50, 1.88 and 2.17. The E _U values were derived from the slope (1/E _U) of the linear fit of the ln(α) versus hυ plots, as shown in Figure 5b. Hence, the E _U values are estimated to be ∼1.07 eV for (x = 1.50), ∼1.05 eV for (x = 1.88) and ∼0.99 eV for (x = 2.17).

Nonlinear optics deals with the interaction between light and matter. In general, the optical response of a solid varies linearly with the magnitude of the electric field. Otherwise, at high powers, the properties of the solid can change more rapidly, resulting in nonlinear effects. The linear and nonlinear susceptibilities are correlated to the induced polarization as a power series of the electric field strength of light, which can be described by the following expression [27]:

where χ ⁽¹⁾ is the linear optical susceptibility, χ ⁽²⁾ and χ ⁽³⁾ are second- and third-order nonlinear optical susceptibilities, respectively. However, Lines et al. [28] found that χ ⁽²⁾ equals zero for optically isotropic glasses and for centro-symmetric crystals.

The linear optical susceptibility χ ⁽¹⁾ in an isotropic medium can be expressed from the linear refractive index by:

Figure 5c displays the linear optical susceptibility, χ ⁽¹⁾, as a function of the square of the photon energy. It was found that χ ⁽¹⁾ strictly decreases with increasing the squared photon energy in the low region and becomes saturated above 6 (eV)².

The nonlinear optical susceptibility χ ⁽³⁾ and the linear optical susceptibility χ ⁽¹⁾ are directly correlated according to Miller’s rule [20, 29]:

where A ≈ 1.7 × 10⁻¹⁰ is a constant (for χ ⁽³⁾ in esu with 1 esu equal to 1.4 × 10⁻⁸ m² V⁻²). Figure 5d shows the plots of χ ⁽³⁾ versus (hυ)² for the studied compositions x = 1.50, 1.88 and 2.17. It can be seen that χ ⁽³⁾ shows the same behavior as χ ⁽¹⁾. Moreover, with a tendency towards the long-wavelength edge (hυ → 0), Equation (21) becomes:

The nonlinear refractive index (n ₂) can be determined from the nonlinear susceptibility χ ⁽³⁾ using the following relation:

The calculated values of χ ⁽³⁾ and n ₂ are given in Table 2. For comparison, we have also provided the values of the optical parameters of some perovskite compounds (see: Table 2) [30, 31].

The complex electric modulus is a very important and convenient parameter to analyze the electric transport phenomenon and the relaxation mechanism in the materials [32], it has been defined as being proportional to the reciprocal of the dielectric permittivity of the materials (M* = 1/ε*). The complex electrical modulus (M*) and complex impedance (Z*) are related to the real (ε ₁) and imaginary (ε ₂) parts of the complex dielectric constant by the following expressions [30, 33]:

where M ₁ and M ₂ are the real and imaginary parts of the complex electrical modulus, respectively, Z ₁ and Z ₂ are the real and imaginary parts of the complex impedance, ω (= 2πν) is the angular frequency, C ₀ = (A/e)ε ₀ denotes the vacuum capacitance of the cell (where ε ₀ is the permittivity of free space, A = area and e = thickness of the material).

For example, the diameter of the pellets is 1.30 cm (= 1.30 × 10⁻² m) and their thickness (e) = 250 × 10⁻⁷ cm (= 250 × 10⁻⁹ m). The area (A) of the samples = 1.32732 × 10⁻⁴ m², and therefore, the C ₀ = (1.32732 × 10⁻⁴ m²/250 × 10⁻⁹ m) × (8.854 × 10⁻¹² F m⁻¹) = 4.70084 × 10⁻⁹ F which is equal to 4.70084 nF. Figure 6a and b illustrates the variation of the real (M ₁) and imaginary (M ₂) parts of the complex electric modulus as a function of the photon energy hυ.

Figure 6:

Variation of (a) real, and (b) imaginary parts of the complex electric modulus, (c) real, and (d) imaginary parts of the complex impedance as a function of photon energy.

The M ₁ curves show the dispersion in the high photon energy region, which may have arisen due to the conductivity relaxation. A continuous sigmoidal increase was observed for the real part M ₁ with increasing photon energy hυ, which may be due to the short-range mobility of charge carriers. In addition, M ₁ exhibits a saturation tendency at maximum asymptotic values in the high energy region, which is likely related to a lack of restoring force governing the mobility of charge carriers under the influence of an induced electric field. This behavior originates from the long-range mobility of charge carriers that cause electrode polarization.

On the other hand, the variation of the imaginary part M ₂ as a function of photon energy gives important information about charge transport processes such as the electrical transport mechanism, conductivity relaxation and ion dynamics. The region in which the peaks occur indicates the transition from long-range to short-range mobility of the charge carriers with increasing photon energy hυ. The low-energy region of the M ₂ peaks represents the range in which the charge carriers (ions) are mobile over long distances, i.e., ions can perform successful hopping from one site to the neighboring site. Whereas the high-energy region represents the range in which the charge carriers are confined to the potential wells, i.e., they are mobile over short distances and the ions can perform localized motion within the well.

Figure 6c and d presents the variation of the real (Z ₁) and imaginary (Z ₂) parts of the complex impedance versus photon energy hυ. It can be noted that the plots of Z ₁ and Z ₂ as a function of hυ for the present compositions (x = 1.50, 1.88 and 2.17) show the same behavior with broad peaks. This behavior indicates the occurrence of relaxation processes in the studied samples. All these results indicate that the difference in electronegativity between Sr and Pb plays a fundamental role in the variation of the bandgap and the optical properties.