Hopping frequency and conductivity relaxation of promising chalcogenides: AC conductivity and dielectric relaxation approaches

X-ray diffractograms of as-prepared samples are illustrated in figure 1, which exhibit various peaks for [h k l] values for various nanophases [20]. Investigations on various peaks in figure 1 exhibit [1 0 0] and [2 0 1] for Te_0.5Se_0.5[JCPDS Card no. 01-078-1061], [2 2 0] for Te_0.5 Se_3.5[JCPDS Card no. 01–087–2414], [1 1 0] for Ag₂S (JCPDS Card no. 75–1061), [1 4 1] for S₃Se₅ (JCPDS Card no. 71–1118), [5 1 0] for Ag_4.53Te₃ (JCPDS Card no. 86–1953), [4 4 0] for Ag₅Te₃ (JCPDS Card no. 86–1168) and [4 2 2] for TeS (JCPDS Card no.78–1062) nanocrystallites. The different crystallites in figure 1 show the short range order of constituent atoms. Various peaks in figure 1 directly indicate their polycrystalline behaviour. To estimate crystallite size and corresponding lattice strain, developed in the present systems, various peak positions and nature of broadening of peaks have been exercised. Here, sulphur (S) nanoparticles are absent as (113), (222), (027), (046) and (318) peak positions [21] are not found in figure 1. This result may conclude that sulphur (S) in Ag₂S should take part in the bonding of the network structure.

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XRD data can be exploited to find out the broadening of peaks (β_T) due to combined effect of crystallite sizes (β_D) and corresponding micro strain (β_ε ) using the following relation [20]:

$$\begin{eqnarray}&&{\beta }_{{\rm{T}}}={\beta }_{D}+{\beta }_{\varepsilon }\end{eqnarray} \tag{ 1 }$$

Scherrer equation [22] can be employed to compute crystallite size in the form of :

$$\begin{eqnarray}&&D=\displaystyle \frac{0.89\lambda }{\left({\beta }_{{\rm{D}}}\,\cos \,\theta \right)}\end{eqnarray} \tag{ 2 }$$

Here, β_D (broadening of peak) is the measured full width at half maxima (FWHM) in radians, λ = 0.15406 nm (wavelength of the x-ray source), D is the crystallite size (in nm) and θ is the peak position in radians.

At the same time, XRD peak broadening due to micro strain [23] can be presented as:

$$\begin{eqnarray}&&{\beta }_{{\rm{T}}}=4\varepsilon \,\tan \,\theta \end{eqnarray} \tag{ 3 }$$

where, ε is the micro-strain developed in the samples under study.

Due to the course of formation of glass matrices, crystallographic irregularities or defects are developed in the form of dislocation, which are supposed to be highly responsible for the electrical transport via hopping of polarons. This dislocation density (δ) can be presented as [7]:

$$\begin{eqnarray}&&\delta =\displaystyle \frac{1}{\left({{\rm{D}}}^{2}\right)}\end{eqnarray} \tag{ 4 }$$

Estimated values crystallite sizes of different phases with their miller indices, ε and δ are included in table 1.

It is observed from table 1 that average crystallite size decreases with compositions. Micro-strain is found to decreases up to x = 0.1 and then to increase beyond it and dislocation density increases with compositions. This enhancement of dislocation/defect may be responsible to instigate the degree of structural alterations as well as grain boundary effect [7, 23]. So incorporation of more Ag₂S content in the present system suggests to enhance the dislocation and to decrease the crystallite sizes [7, 23]. Higher order defects may insist polaron conduction in a higher order via hopping of polarons in the present system [7, 23] up to a limit with x = 0.1. The gradual fall in the values of lattice strain (ε) up to x = 0.1 suggests to more stable network structure, which can be justified from the nature of diffractograms in figure 1. Sample with x = 0.2 shows higher micro-strain, which may be associated with different type of defects/dislocation. It is also noted from table 1 and figure 1 that the phases, formed with Se (i.e., Te_0.5Se_0.5, S₃Se₅ and Te_0.5 Se_3.5) are found to disappear with the increase of x. The crystallite sizes of Ag containing phases (i.e., Ag₂S, Ag_4.53Te₃ and Ag₅Te₃) are found to first increase and then decrease with x. It is clear from literature [24–27] that Se has a tendency to form long chain agglomerated structure. Hence, the nature of formation of nanocrystallites clearly indicates the reason for lowering of sizes of nanocrystallites, formed in the glassy matrices with Ag₂S content. Higher dislocation density for x = 0.2 may cause the possibility of higher collision of charge carriers/polarons which may decrease their mobility and hence the electrical conductivity. Present dislocation defect may be treated as Frenkel defect as here atom may be displaced from its lattice point to interstitial site in order to develop a vacancy at the lattice point.

Crystalline volume fraction X_XRD of the present glassy system has been determined by equation [28]:

$$\begin{eqnarray}&&{{\rm{X}}}_{{\rm{XRD}}}=\displaystyle \frac{{{\rm{I}}}_{hkl}}{{I}_{a}+{K}_{hkl}\times {I}_{hkl}}\end{eqnarray} \tag{ 5 }$$

where, I_hkl is the area of a diffraction peak for a phase of a particular [hkl] value, I_a is the maximum area of peak of the considered XRD diffraction pattern and K_hkl is the calibration constant value, which can be computed from the relation [28]:

$$\begin{eqnarray}&&{{\rm{K}}}_{{\rm{hkl}}}=\displaystyle \frac{{{\rm{I}}}_{hkl}}{{I}_{a}}\end{eqnarray} \tag{ 6 }$$

The estimated average values of crystalline volume fractions of the present glassy system with x = 0.05, 0.1 and 0.2 are presented table 2. On increasing the value of x from 0.05 to 0.1 in the glass composition, the crystalline volume fraction percentage is found to decrease from 39.4% to 24.7% and then increase to 27.3% for x = 0.2 as shown in table 2. Higher value of crystalline volume fraction percentage for x = 0.2 is expected to be associated with different type of defects/ dislocation as discussed earlier.

The frequency dependent dielectric constant ε^/ (real) and dielectric loss ε^// (imaginary) in a wide frequency range (42Hz—5MHz) at various temperatures (343K—423K) have been studied for understanding the nature and origin of losses occurring in these materials [29, 32, 49]. Moreover, the incorporation of Ag₂S content in the chalcogenides may instigate to enrich its conductivity level, which may in turn improve its dielectric properties in terms of substantial decrease in activation energy for hopping of polaron [29, 32, 49]. This feature makes them capable of being used for integrated circuits (ICs) device applications [29]. ε^/ and ε^// spectra are presented in figures 8(a) and (b) respectively for x = 0.2. ε^/ and ε^// spectra of other samples under study have shown similar nature.

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Dielectric constant (ε^/) of the present system decreases on increasing frequency in the region of lower frequencies and it becomes independent on frequency in higher frequency region for the entire range of temperature as shown in figure 8(a). It is also noted in figure 8(a) that ε^/ increases with temperature, which shows it's thermally activated nature. Higher values of dielectric constant (ε^/) at low frequency and high temperature is because of the diverse constituents such as electronic, ionic, orientation and interfacial polarization [32, 49]. Electronic polarization does not exist here as it takes place at ∼10¹⁶ Hz. Displacement of positive and negative ions may be responsible for ionic polarisation, which may exists at ∼10¹³ Hz. So, it should not be present here due to the above-mentioned facts. Permanent dipole moments should instigate dipolar polarization (orientational polarization), which is associated with the rotation to the dipoles at ∼10¹⁰ Hz frequencies. Movement of charge of carriers by interfaces is the possible reason for space charge polarization, which exists at frequencies ranging between 1 and 10³ Hz. As the applied field is increased, the dipoles exhibit slow rotation, which may take more time compared to electronic and ionic polarizations. In the higher frequency region, the permanent dipolewill not be capable of being following the orientational polarization. So, a competition may take place between orientational polarization and space charge polarization in the higher frequency, where the former tries to go down and the later should have a tendency to go up. As a consequent, dielectric constant(ε^/) becomes a constant value at a higher frequency. The nature of temperature dependent dielectric loss (ε^//) of a system has been explained by Stevels [50] to interpret it as an outcome of conduction losses, dipole relaxation losses and ion vibrational losses. This process [32, 49, 50] depends on the migration of charge carriers (polarons) over long distances via loss of thermal energy in the lattice.

To validate dielectric loss data, dielectric loss tangent (tanδ = ε^///ε^/) has been estimated for all samples under study and one such set of results for x = 0.2 is presented in figure 9(a) at various temperatures. The results in figure 9(a) show that tanδ increases as the frequency decreases and increases sharply at a particular frequency. It is also observed that tanδ increases with temperature, indicating thermally activated nature. The rate of loss of thermal energy [32, 49] is proportional to σ/ω (σ is the conductivity and ω is the angular frequency). Conduction loss [32, 49] is also found to increase with temperature and plays a dominant role. At higher temperature, all types of losses contribute to the dielectric loss of the present system. For this reason, tanδ exhibits a sharply pronounced valueat a particular frequency (call it critical frequency).But the values of tanδ at high frequency region may strongly depend on ion vibrations [32, 49, 50] in the lattice of the glassy matrices under study.

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Critical frequency (ω_c) has been estimated from the point of intersection between drawn straight lines and frequency axis in figure 9(a) at various temperatures for x = 0.2. Similarly, ω_c has been computed for x = 0.05 and 0.1 respectively at various temperatures. The reason for dielectric loss has already been discussed. In search of other possible reasons for dielectric loss, contribution of hopping frequency (ω_H) may be considered as a matter of fact that discussion of hopping frequency (ω_H) and corresponding activation energy (E_H) in figure 6 should impel us to think over in that direction. To investigate its contribution in dielectric loss process, hopping frequency (ω_H) has been projected in figure 9(b) with respect to critical frequency (ω_c) for all samples under study at various temperatures. It is observed from figure 9(b) that hopping frequency (ω_H) does not show any considerable changes with respect to critical frequency (ω_c) for all samples at various temperatures. It may be directly concluded from this result that hopping frequency (ω_H) does not contribute any role in the dielectric loss process. It depends on conduction loss [32, 49, 50] and ion vibrations [32, 49, 50] as discussed earlier.

Electrode polarization effects [29, 32] in dielectric spectra of disordered solids may cause a difficulty in the computation of various dielectric parameters, particularly at low frequencies.Efforts [38, 51] have been made in the past to correct the electrode polarization effect. Electric modulus formalism [38, 51] can be used for investigating complex electrical response of a system by nullifying the electrode polarization effect as the reciprocal of complex permittivity is taken into account. The complex electrical modulus can be presented as [51]:

$$\begin{eqnarray}&&{{\rm{M}}}^{* }=\displaystyle \frac{1}{{{\rm{\varepsilon }}}^{* }}={{\rm{M}}}^{/}+{\rm{j}}\,{{\rm{M}}}^{//}=\displaystyle \frac{{{\rm{\varepsilon }}}^{/}}{{\left({{\rm{\varepsilon }}}^{/}\right)}^{2}+{\left({{\rm{\varepsilon }}}^{//}\right)}^{2}}+{\rm{j}}\displaystyle \frac{{{\rm{\varepsilon }}}^{//}}{{\left({{\rm{\varepsilon }}}^{/}\right)}^{2}+{\left({{\rm{\varepsilon }}}^{//}\right)}^{2}}\end{eqnarray} \tag{ 13 }$$

where, M* is the complex modulus, ε*is the complex dielectric permittivity, M^/ is the real and M^// is the imaginary parts of the electric modulus.

The assimilation of the electric modulus spectra of the present system in a wide frequency and temperature range may compel us to explore complete dielectric properties. Figure 10(a) displays the imaginary part of the electric modulus M^//spectra for x = 0.2. It can be seen from figure 10(a) that at low frequency region, M^//approaches to zero, which implies the suppression of electrode polarization effect. At higher frequencies,impact of relaxation process makes M^//maximum [29, 52, 53]. As the frequency increases, M^//is found to decrease, which may suggest the short-range mobility of charge carriers in the system.

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It is evident from figure 10(a) that M^// peaks have been shifted towards higher frequency region as the temperature increases. This is an indication of thermally activated relaxation process in the system. Similar nature of electrical relaxation is also observed for other samples under study. M^// peak positions in 10(a) corresponds topeak relaxation frequency (ω_max). In this approximation [29, 52, 53], ω_max indicates the limiting frequency up to which the charge carriers (polarons) may hop over long distances. Beyond ω_max region, charge carriers are supposed to be confined to potential wells, where they are freely accessible [29, 52, 53]. It may be concluded from this discussion, that the transition of mobility of charge carriers from long range to short range order takes place at ω_max.

To investigate the nature of M^// _, the following expression [52–54] can be considered:

$$\begin{eqnarray}&&{{\rm{M}}}^{//}=\displaystyle \frac{{{\rm{M}}}_{{\rm{\max }}}^{//}}{\left(1-\beta \right)+\tfrac{\beta }{1+\beta }\left[\beta \left({\omega }_{{\rm{\max }}}/\omega \right)+\left(\omega /{\omega }_{{\rm{\max }}}\right){.}^{\beta }\right]}\end{eqnarray} \tag{ 14 }$$

where, 0 < β <1, M^// _max is the peak or maximum value of M^//, ω_max is angular frequency corresponding to M^// _max and β stands for Kohlrausch-Williams-Watts (KWW) stretched coefficient [54].The experimental data in figure 9(b) have been analysed by best fitted curves using equation (14). The values of M^// _max, ω_max and β have been estimated from this fitting.The activation energy (E_τ ) related to the relaxation time τ was calculated by using the equation [29, 53]:

To analyse the nature conduction mechanism, it is approximated that the present systems undergo Debye type relaxation [29, 32], which requires the essential condition: ω_max x τ = 1, where, τ is the relaxation times. The estimated relaxation time τ with reciprocal temperature is projected in figure 10(b). Figure 10(b) exhibits thermally activated nature of τ. As the temperature increases, τ is found to decrease, which shows its semiconducting behaviour. It should also be pointed out that τ decreases with composition up to x = 0.1 and then increases, which can be validated from the nature of AC conductivity and hopping frequency as presented in figures 5 and 6 respectively.

To investigate thermally activated relaxation times in figure 10(b), theoretical expression of relaxation time [29, 49, 53] can be used in its form as:

$$\begin{eqnarray}&&\tau ={\tau }_{0}\exp \left(-\displaystyle \frac{{{\rm{E}}}_{\tau }}{{{\rm{K}}}_{B\,}{\rm{T}}}\right)\end{eqnarray} \tag{ 15 }$$

where, τ₀ is pre-exponential factor, K_B is the Boltzmann constant and T is the absolute temperature. Experimental data in figure 10(b) have been best fitted by linear fits using equation (15). Activation energy (E_τ ) corresponding to relaxation time has been estimated from the slopes of the best fitted solid straight lines and presented in table 5. E_τ is found to increase and becomes maximum for x = 0.1. After x = 0.1, E_τ is found to decrease, indicating higher values of relaxation times. Smaller values of relaxation times and higher values of corresponding activation energy are the indication of higher conductivity due to higher rate polaron hopping in the system. Short time relaxation process may be considered to be trivially associated with motion of polaron. A detailed understanding of these relaxation processes may be related to the least vibration of dipoles and conducting species at lower frequencies. At higher frequencies, the dipoles may vibrate rapidly and hence it requires more activation to attain relaxed state. It is essential to mention that relaxation times have also computed earlier FT-IR spectra. These values of relaxation times (table 3) are found to be much higher than those obtained from dielectric relaxation process. The reason for such discrepancy is that computation of relaxation times from FT-IR study entirely depends on the vibrations due to only optical phonon frequency (ν₀). But in the dielectric relaxation process, all types of lattice vibrations are counted and the results are supposed to be more accurate.

Like AC conductivity scaling data, attempts have been made to scale the conductivity relaxation spectra (modulus spectra) at various temperatures [29, 49, 53]. In this scaling process, M^// values have been divided by M^// _max and ω values have been divided by ω_max to acquire modulus scaling. One such scaling spectra for x = 0.05 has been illustrated in figure 11(a) at various temperatures. All other samples show similar nature of temperature scaling. Perfect overlapping of all the scaled data in figure 11(a) explores temperature independent relaxation process [29, 52, 53] in the present system. M^// scaling spectra at a fixed temperature (343K) for all the samples under study are projected in figure 11(b), which indicates imperfect overlapping nature of M^// scaling spectra.The imperfect overlapping nature of the plots in figure 11(b) clearly indicates composition dependent relaxation process in the present system. So, it can be concluded from the discussion that conductivity relaxation process in the present system is independent of temperature, but it depends on compositions. It is also noteworthy that the nature of conductivity scaling process is same as that of AC conductivity scaling. In this regard, point should be noted that two methods (AC conductivity approach and Modulus approach) are independently employed here to explore conduction process. The outcomes of these two distinguished processes are of similar interpretations of the experimental results.

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