Scaling of the Electrical Conductivity Spectra Reveals Distinct Transport Responses in A2SmTaO6 [A = Ba, Sr, Ca]

TL;DR

This study investigates how microstructural inhomogeneities in A2SmTaO6 double perovskites influence electrical conduction and relaxation behaviors, revealing distinct transport responses through scaling of conductivity spectra.

Contribution

It demonstrates the impact of microstructural disorder on conduction mechanisms and relaxation timescales in A2SmTaO6 materials using scaling analysis of electrical spectra.

Findings

Scaling of conductivity spectra reveals correlation between conduction and relaxation.

Microstructural inhomogeneities restrict universality of conduction mechanisms.

Disorder influences charge carrier hopping and relaxation processes.

Abstract

Disorder plays an important role in materials science, influencing material behavior across different length scales. Imperfections like vacancies, atomic substitutions, lattice distortions, and microstructural inhomogeneities, disrupt ideal periodicity thereby altering physical properties. Analogous to spin-glass systems, electrical 'glassiness' arises when charge carriers confront disordered energy landscapes, leading to a broad range of relaxation times, especially in polycrystalline materials where dipoles experience competing exchange interactions. Complex impedance, permittivity, and electric modulus distill out separate resistive and capacitive effects, offering insights into how microstructural inhomogeneities affects conduction mechanism. In polycrystalline double perovskites A2SmTaO6 (A = Ba, Ca), with a power law driven ac conductivity, the hopping and relaxation of carriers is affected by both grains and grain boundaries. Scaling of ac conductivity and impedance response reveals correlation between conduction and relaxation timescales. The inhomogeneities in local energy landscape of 'frustrated' dipoles restrict the 'universality' of conduction mechanism across the bulk length scale.

Equations

$$\frac{{\sigma }_{ac}}{{\sigma }_{dc}}=F(\frac{\omega }{{\omega }_s})$$

$${\sigma }_{ac}={\sigma }_{dc}[1+(\frac{\omega }{{\omega }_H}{)}^n]$$

Full text

Keywords

Scaling; Electrical conductivity; Impedance spectroscopy; Frustration; Disorder

I Introduction

Double perovskite oxides have garnered decades of research attention due to their exceptional chemical flexibility and a wide spectrum of tunable physical properties, such as high k dielectric constants, spin glass behavior, solar energy harvesting, and supercapacitor applications  [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]. In most of these functionalities, electrical conduction plays a pivotal role in dictating the overall performance of the materials. Consequently, studying their electrical conductivity provides crucial insight into the charge transport processes operating within these structurally complex systems. The ac conductivity, shaped by both grain and grain boundaries, reveals how structural ordering, defect densities, and microstructural characteristics collectively influence transport behavior. Typically, frequency-dependent conductivity consists of two main components: (i) frequency-independent dc conductivity that follows a temperature-dependent power law and (ii) a frequency-dependent ac component that emerges above a characteristic frequency, known as the hopping frequency ($\omega_{H}$)  [13, 14]. Interestingly, a wide variety of disordered solids, such as ion-conducting glasses  [15], amorphous and polycrystalline semiconductors  [16, 17], conducting polymers  [18], transition metal oxides  [19], organic-inorganic hybrids  [20], and single crystals  [21], exhibit remarkably similar ac conductivity spectra. The common factor across these diverse systems is their intrinsic ’disorder’ of a specific form, which governs the universal features of their ac response. The ac conductivity spectra in all of these materials can often be collapsed onto a single master curve, underscoring the importance of scaling analysis in revealing key aspects of conduction processes in functional materials  [22].

The scaling behavior of conductivity spectra has been widely investigated in glasses and amorphous systems  [15, 16]; however, studies on polycrystalline perovskite oxides remain relatively scarce. In polycrsytalline perovskites, the conduction and relaxation processes are influenced by a variety of factors, including structural order, defect density, and microstructural characteristics. In a previous study, Halder et al. demonstrated the applicability of the time–temperature superposition principle (TTSP) by scaling the ac conductivity spectra in double perovskite A2HoRuO6 (A = Ba, Sr, Ca)  [23]. The dc conductivity ($\sigma_{dc}$), hopping frequency ($\omega_{H}$), and the relaxation frequency ($\omega_{m}$) were found to be closely aligned, indicating that the onset of ac conduction coincides with the frequency of the relaxation maxima. The mathematical expression used for the conductivity scaling was given by;

[TABLE]

where the scaling factor F is independent of temperature and $\omega_{s}$ is the scaling parameter. The conductivity spectra were successfully scaled using $\omega_{s}$ = $\sigma_{dc}$T  [24] and $\omega_{s}$ = $\omega_{H}$  [25, 26]. Although the scaled spectra for the grain boundary region collapsed onto a single master curve, those corresponding to the grain interior did not exhibit universal scaling. This discrepancy arises from the distinct thermal activation behaviors of charge carriers in the grain and grain boundary domains. Furthermore, to understand the effect of structural disorder on carrier concentration, a composition-based investigation of TTSP was carried out in the complex perovskite Ba2HoRu${}_{1-\textit{x}}$Sb${}_{\textit{x}}$O6, where it was shown that the conductivity spectra of all individual materials collapsed onto a single master curve when the composition was added to the scaling function F($\frac{\omega\textit{x}}{\omega_{s}}$)  [27]. In polycrystalline perovskite oxides, the distribution of grains and grain boundaries are not uniform but follow a distribution law. Therefore, relaxation behavior in these materials does not have a single relaxation time, but a distribution of relaxation times that accounts for the difference in polarization of the dipoles within these different microstructures  [28]. Therefore, to critically analyze the effect of disorder on the conduction mechanism, one must consider the scaling behavior in each of these thermally activated regions separately.

This article presents a comprehensive analysis of the dynamics of charge carriers in the 1:1 structurally ordered double perovskite oxides A2SmTaO6 (AST; A = Ba, Sr, Ca) at selected temperatures, using ac conductivity and electrical impedance formalisms. In our earlier work, we explored the vibrational, dielectric, and electronic properties of AST compounds in detail  [29]. Ba2SmTaO6 (BST) was found to crystallize in a cubic Fm3m structure, featuring a 1:1 ordering of Sm and Ta atoms at the B site, as depicted in Figure 1(a) [29]. This atomic ordering arises primarily because of differences in the ionic radii and electronegativities of the B-site cations. In contrast, Sr2SmTaO6 (SST) and Ca2SmTaO6 (CST) adopt a monoclinic structure with P21/n symmetry (Figure 1(b) and  1(c) respectively) [29]. Like BST, SST and CST also exhibit 1:1 ordering of the SmO6 and TaO6 octahedra; however, octahedral tilting becomes evident due to the smaller ionic radii of Sr and Ca relative to Ba. This distortion increases from Sr to Ca. The electrical behavior of AST is strongly influenced by changes in crystal symmetry and the extent of hybridization between the Ta 5d and O 2p orbitals, which also leads to a decrease in the band gap, as seen in their electronic structure  [29, 30]. Unlike A2HoRuO6, the conductivity spectra of AST are predominantly governed by contributions from the grain boundary, especially at elevated temperatures. This makes them ideal candidates for investigating the scaling behavior of ac conductivity within the grain boundary regime. The observed Cole-Cole plots exhibit skewed semicircles, indicative of a distribution in relaxation times, often linked to nonuniform polarization processes within microstructural domains. Consequently, AST double perovskites offer a promising platform to investigate the effects of structural disorder on intrinsic charge carrier dynamics in polycrystalline materials, where conduction is dominated by a single microstructural component. Understanding and controlling electrical disorder is vital for the design and optimization of polycrystalline oxides for a broad range of technological applications.

II Experimental Details

A detailed description of the synthesis of AST is provided in our previous communication  [29]. The initial reactants BaCO3, SrCO3, CaCO3, Sm2O3, and Ta2O5 are taken in stoichiometric ratios and mixed homogeneously by grinding in acetone medium for 10 h followed by calcination at 1370°C for 20 h. The powders were then allowed to cool to room temperature at a rate of 50°C/h. The sintering of the disk-shaped pellets (8 mm diameter and 1.25 mm thick) was carried out at 1390°C for 10 h followed by slow cooling to room temperature (40°C/h). The crystalline phase and lattice constants of the samples were determined by powder XRD (Rigaku MiniflexII diffractometer, Tokyo, Japan) using CuK$\alpha$ radiation. The structural and refinement parameters are provided in our previous communication  [29]. The sintered pellets were polished, and electroded by high purity ultrafine silver paste for electrical characterization. The impedance (Z), phase angle ($\phi$) and conductance (G) of the samples were measured in a frequency window of 42 Hz to 5 MHz and at selected temperatures between 303 and 673 K using a computer-controlled LCR-meter (HIOKI-3552, Nagano,Japan).

III Results and Discussion

Dynamical processes in materials are probed using a range of spectroscopic techniques, among which electrical relaxation measurements, typically frequency dependent, are widely employed to study charge dynamics. With advances in ac conductivity measurement methods, impedance spectroscopy has emerged as a powerful and reliable tool for analyzing experimental data. Figure 2(a)–(c) present the real (Z*′) and imaginary components (Z′′) of the complex impedance for BST, SST, and CST at selected temperatures. The frequency-dependent behavior of the impedance components (Z′* and Z*′′) reveals a shift in the relaxation response with increasing frequency, eventually reaching a plateau. This saturation at higher frequencies arises from the reduced polarization of the dipoles in response to the alternating electric field. A distinct peak in the imaginary part of the impedance Z′′, centered around the relaxation region of Z′, signifies the maxima of the relaxation process. As the temperature increases, this peak shifts toward higher frequencies, indicating a thermally activated relaxation phenomenon. In general, impedance spectra primarily capture contributions from the most resistive components in the system, such as grain boundaries. In contrast, grain contributions, which are more capacitive in nature, are better represented using the electric modulus formalism  [31]. As the frequency increases toward the peak, enhanced charge carrier mobility indicates long-range conduction. Beyond the peak, conduction is dominated by localized hopping, where carriers are trapped and move through short-range forward–backward hopping due to reduced mobility. Thus, $\omega_{m}$ represents the frequency limit for the transition from dc to dispersive conduction. The rise in temperature improves the dynamics of the charge carriers, decreasing the relaxation time ($\tau_{m}$; $\omega\tau$ = 1) and causing a corresponding shift of the relaxation frequency ($\omega_{m}$) to higher values. Interestingly, as we progress from BST to SST and CST, $\omega_{m}$ shifts toward lower frequencies, highlighting the increasing influence of high-resistance grain boundaries and slower relaxation dynamics. This trend is further illustrated in the comparative Cole-Cole plots in Figure 3(a) at 513 K, where BST exhibits a lower intercept on the Z′* axis than SST and CST. The grain boundaries are more resistive in SST and CST compared to those of BST. The scaling behavior of the imaginary part of the impedance (Z*′′*) has been examined to gain deeper insight into the relaxation dynamics in AST compounds. Figures 3(b)–(d) display normalized spectra, where both axes are scaled by the respective peak values corresponding to the relaxation maxima at each temperature. For BST and SST, the impedance spectra at various temperatures collapse onto a single master curve, indicating a consistent relaxation dynamics in the grain boundary region. In contrast, CST shows a noticeable deviation in the high-frequency region, attributed to the significant difference in the local activation energy barriers of the grain boundaries.

Figures 4(a), 5(a), and 6(a) show the ac conductivity plots for BST, SST and CST, respectively, at selected temperatures. The frequency-dependent conductivity can be interpreted in terms of the grain boundary conductivity and jump relaxation model  [13, 14].The conductivity spectra follow the power law behavior, typical of semiconductors, given by:

[TABLE]

where $\sigma$dc represents the dc conductivity which is the total conductivity of the sample at the limit $\omega$ $\rightarrow{0}$. $\sigma$dc results from the long-range translation of charge carriers between localized sites. $\omega$ is the angular frequency, n is the power law exponent that describes the electrical relaxation behavior of the material. $\omega$H represents the frequency of hopping of charge carriers, marking the crossover from the long-range translational to the dispersive conduction region at $\omega$ $\geq$ $\omega$H. The dispersion regime is caused by the correlated backward and forward hopping motion of the charge carriers. Alternatively, this can be visualized as a competition between two relaxation processes; successful and unsuccessful hopping of charge carriers. In a polycrystalline ceramic system, conductivity relaxation occurs as a result of the contributions of the grain, grain-boundary, and electrode-specimen interface. In the absence of electrode polarization, electro-ceramics show the presence of plateaus and dispersion regions in their conductivity spectra due to the contribution of the grains and grain boundaries  [29]. The spectroscopic plots for BST, SST, and CST show different regions corresponding to grains and grain boundaries. For BST (Figure 4a), a single plateau is observed in the low-frequency range due to $\sigma$dc followed by a high-frequency dispersion region. The dispersion region can be attributed to relaxation of the grain boundary of the sample. In the case of SST and CST (Figures 5a and  6a respectively), the presence of a second plateau is observed due to the contribution of the grains to the total conductivity. The relaxation frequency $\omega_{m}$, which provides the upper bound between the crossover between the two regions, is directly correlated to the hopping frequency obtained from the conductivity spectra. To examine the correlation between dc conductivity ($\sigma_{dc}$) and hopping frequency ($\omega_{H}$), logarithmic plots are drawn between these two parameters as shown in Figures 4(b),  5(b) and  6(b) for BST, SST, and CST respectively. The figures show a near linear nature with a slope of almost unity indicating a power law dependency of the form $\sigma_{dc}$ $\sim$ $\omega_{H}$n, where ‘n’ is the slope that proves our hypothesis that the relaxation and conduction mechanisms are strongly correlated in AST.

To gain a deeper understanding of the temperature dependence of the conduction mechanism and the distinct responses of grains and grain boundaries, the scaling of the frequency-dependent conductivity spectra has been analyzed. In this study, two established scaling approaches are used to scale the frequency axis: (i) $\omega_{s}=\sigma_{dc}T$  [24] and (ii) $\omega_{s}=\omega_{H}$  [25, 26]. The conductivity axis for BST, SST and CST has been normalized by the dc conductivity ($\sigma_{dc}$). For BST, as shown in Figures 4(c)–(d), both scaling formalisms result in a complete collapse of the spectra to a single master curve, indicating a consistent scaling behavior. SST also exhibits similar trends in the scaled conductivity spectra as seen in Figures 5(c)-(d), although scaling is more effective when the frequency axis is scaled with $\omega_{s}=\omega_{H}$ than $\omega_{s}=\sigma_{dc}T$. It is important to note that deviations from ideal scaling emerge when disorder occurs within a similar microstructural environment. Since the grain contribution in SST predominantly lies outside the frequency window, the inhomogeneity mainly arises from the unequal response of dipoles trapped in the grain boundaries with different activation energies. A prominent deviation in the scaling of the conductivity spectra is observed for CST. The scaled conductivity spectra for CST (Figures 6c-d) fail to collapse into a single master curve, particularly beyond $\omega_{H}$ in the grain boundary relaxation region. For CST, this frequency window above $\omega_{H}$ has contributions to the conductivity from two different sources: (i) grain boundary relaxation and (ii) long range translation of carriers in grains. The two different microstructural regions have different thermal motion of charge carriers. Figure 7(a)-(c) shows the temperature dependence of the dc conductivity ($\sigma_{dc}$) which follows the Arhennius exponential form. The activation energies (Ea) obtained from the fitting for BST, SST and CST are obtained to be 0.12 eV, 0.19 eV and 0.24 eV respectively, showing the highest energy barrier for CST.

To understand the electrical ”glassiness” observed in AST, one can draw an analogy to the spin-glass behavior found in polycrystalline magnetic materials [32, 33]. A spin-glass state arises from competing magnetic interactions that prevent the establishment of a true magnetic order, causing spins to freeze in random orientations. The associated energy landscape is highly complex, characterized by numerous local minima separated by energy barriers. These minima represent metastable configurations that do not necessarily correspond to the global energy minimum of a true magnetic ground state. The height and distribution of these barriers determine the system’s ability to explore the energy landscape, leading to slow non-equilibrium polarization dynamics. This frustrated landscape, resulting from random and competing interactions, results in a broad distribution of relaxation times, which is both temperature dependent and lacks a single characteristic timescale [32, 34]. As temperature increases, the distribution of relaxation typically shifts towards shorter times (i.e. higher frequencies), reflecting faster dynamics. In ac susceptibility measurements, this behavior manifests itself as a frequency-dependent cusp in the real part of the susceptibility ($\chi^{\prime}$) at the spin-freezing temperature ($T_{f}$), while the imaginary part ($\chi^{\prime\prime}$) shows a peak at $T_{f}$ that shifts with frequency, a signature of spin-glass dynamics, distinguishing it from conventional magnetic ordering. A similar mechanism underlies the conduction and relaxation dynamics in perovskite oxides under an applied ac electric field. Here, charge carriers experience energy barriers associated with grain boundaries, which arise from a heterogeneous distribution of grains. As observed in Figure 2(a)-(c), the peak frequency ($\omega_{m}$) in the imaginary part of the impedance (Z*′′) shifts to lower frequencies from BST to CST, indicating slower relaxation. BST, with a lower activation energy (Ea* = 0.12 eV), corresponds to a smoother energy landscape and faster charge dynamics, while CST exhibits higher energy barriers (Ea = 0.24 eV) and consequently slower relaxation processes. Analogously to spin frustration caused by competing exchange interactions, electrical relaxation in these oxides is influenced by local electrostatic environments that introduce a ’dipolar frustration.’ This results in a wide range of relaxation times and sluggish charge dynamics. Furthermore, the trapping of charge carriers at the grain boundaries further contributes to the slow dynamics. The degree of this inhomogeneity and disorder in the energy landscape can be inferred from the scaling behavior of the conductivity and impedance spectra. A universal master curve typically indicates reduced disorder and more homogeneous dynamics, whereas deviations suggest that a higher degree of disorder impedes carrier transport.

IV Conclusions

A quantitative analysis of the scaling behavior of ac conductivity and complex impedance has been carried out for polycrystalline double perovskite oxides A2SmTaO$6$ (AST; A = Ba, Sr, Ca) over a frequency range of 42 Hz to 5 MHz at selected temperatures. AST shows a 1:1 structural ordering at the B site, therefore eliminating any structural disorder effects on the conduction process. The conductivity spectra primarily reflect contributions from grain boundary, with minor signatures from grains. A clear correlation is observed between the dc conductivity ($\sigma{dc}$), the hopping frequency ($\omega_{H}$), and the relaxation frequency ($\omega_{m}$) indicating that the onset of ac conduction is intrinsically related to the relaxation dynamics of the system. The extracted activation energies fall within the range of 0.12 to 0.24 eV highlighting polarons as the main charge carriers. Nonlinear behavior in the Arrhenius plots confirms the presence of distinct thermally activated polaronic hopping mechanisms at different temperature windows. The ac impedance spectra reveal a thermally activated relaxation mechanism characterized by a broad distribution of relaxation times. Among the studied samples, CST exhibits significantly slower relaxation dynamics compared to BST, reflecting higher levels of electrical ”frustration” due to competing energy barriers. Importantly, scaling analysis of the conductivity spectra reveals that the time–temperature superposition principle holds predominantly in the grain boundary regime, but even there, deviations from ideal behavior are evident due to local inhomogeneities in the energy landscape. These findings highlight that the ’universal’ nature of scaling behavior serves as a sensitive probe of the degree of disorder in the charge transport pathways. The extent to which conductivity spectra collapse onto a master curve reflects microscopic energetic inhomogeneity of the system, providing evidence for dipolar frustration and ’glassy’ charge carrier dynamics in these materials.

V Declaration of Competing Interest

The authors declare no competing financial conflicts of interest.