1 Introduction

Glassy materials have some exclusive features that are not found in other engineering materials. Some of these unique features are high hardness and transparency at room temperature, along with sufficient strength and excellent corrosion resistance. Due to the potential applications in various engineering and technological fields, the study of the properties of glasses is of great significance. Among the glassy materials, those containing transition metal oxides (TMO) have been extensively studied due to their interesting physical properties [1–29], and also technological advantages in electrochemical devices such as solid-state batteries [30, 31], optoelectronic devices such as fiber Raman amplifiers [32, 33], precursors for infrared fibers [34–36], and waveguide devices [37]. Also, glassy materials containing V2O5 have been proposed as cathode materials for lithium solid-state batteries [31]. Tellurite glasses are very promising materials for laser and nonlinear applications in optics, due to their important characteristic features, such as high refractive index, low phonon maxima, and low melting point [38].

Electrical properties of glasses containing TMO have been studied extensively for a number of years due to their potential use in solid-state devices. The addition of transition metal oxide makes them electronic or mixed electronic–ionic conductors, which are of potential interest as cathode materials for solid-state battery [39]. In previous works, physical, electrical, optical, and thermal properties of the different glasses like MoO3–TeO2–V2O5 [10] Sb2O3–TeO2–V2O5 [4] Sb–TeO2–V2O5 [2, 9, 11] have been studied. Tellurate glasses have been also studied for their structural and vibrational properties [40]. In the present work, tellurate–vanadate glasses containing silver oxide have been selected, and temperature dependence of their conductivity (above and below the Debye temperature) has been investigated. The addition of third ionic component, Ag2O, has been chosen for samples having higher conductivity. Also, the transport mechanism of xAg2O–40TeO2–(60 − x)V2O5 glasses has been studied over the temperature range of 150–380 K. Besides, charge carrier density and mobility, density of states near the Fermi level, and other hopping parameters have been determined.

2 Experimental details

In the present study, the glass systems of compositions xAg2O–40TeO2–(60 − x)V2O5 with 0 ≤ x ≤ 50 mol%, hereafter termed as TVAgx, were prepared by melting tellurite dioxide TeO2 (Alderige, 99.995 % pure), vanadium pentoxide V2O5 (Merck, extra pure), and silver oxide Ag2O (Alderige 99 % pure) as starting materials with prescribed compositions. The melt was poured on to a steel block, kept at room temperature, and immediately pressed by another steel block. Prepared samples were brown in color and are transparent especially in the film form. The obtained bulk samples were annealed at 80 °C for 2 h. More details of glass preparation process, structural characterization, XRD patterns, glass density, and some physical properties were precisely described in our recent work [8]. Some of the physical and polaronic properties of the TVAgx glasses such as density (determined by Archimedes method), phonon optical frequency, vanadium ion density, silver ion density, Mean V-ion spacing, polaron radius [r p = 1/2(π/6 N)1/2], reduced fraction of vanadium ions [determined by titration method by using KMnO4 and FeSo4(NH4)2So4 solutions], and oxygen molar volume were determined; these parameters have been listed in Table 1. Glassy nature of the samples was confirmed by XRD studies using a using a diffractometer (Unisantis-XMD-300, CuKα, India). Density, ρ, at room temperature was measured by following Archimedes’ principle using sensitive single pan balance. The para-xylene (C8H10, ρ = 0.861 g cm−3) was used as buoyant.

Table 1 Data of T R as the temperature at which the slope change occurred, W as activation energy for electrical conduction, ρ as glass density, N i as transition metal ion density, R V as mean V-ion spacing, Θ D as Debye temperature, C V as the ratio of reduced vanadium ions, V O as oxygen molar volume and υ o as optical phonon frequency
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The total vanadium ion concentration and vanadium ion reduction ratio (C V = [V4+]/[Vtot]) for the samples were determined by titration method using KMnO4 and FeSO4 (NH4)2SO4 solutions. The obtained values of the ratio of reduced vanadium ions, C V, are listed in Table 1.

The annealed slabs were carefully polished for electrical measurement. The dc electrical conductivity of slabs of thickness 1 mm and radius 1 cm was measured by using the Keithley electrometers (model 617) within the temperature range of 150–380 K using four-point method.

3 Results and discussion

XRD patterns of TVAgx samples have been reported in our recent work [8], confirming their non-crystalline nature except for TVAg50.

Figure 1 shows the temperature dependence of dc electrical conductivity for the present samples in the temperature range of 150–380 K.

Fig. 1
figure 1

Inverse temperature dependence of ln(σ dc) for different TVAgx glasses. Inset shows the slope change of the plots changes at the special temperature (T R ) for TVAg50

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In Fig. 1, the slope of the plots changes at the special temperature (T R ), which is shown for TVAg50 as an inset of Fig. 1. The conduction mechanism in temperatures higher than T R is different from that at lower temperatures. The DC electrical mechanism at T > T R is small polaron hopping (SPH) mechanism, which is discussed below.

The theoretical consideration of SPH is based on the molecular crystal model introduced by Holstein [41]. The semiconducting behavior of the transition metal oxide glasses is believed to arise from the hopping of small polarons from the ions of low-valence state to the ions of high-valence state. Generally, in such glasses electron–phonon interaction is responsible for the formation of small polarons at high temperature (T > Θ D/2), where Θ D is the Debye temperature given by hν 0/k with h as Planck’s constant, ν 0 as optical phonon frequency, and k as Boltzmann constant. Thus, it is expected that the SPH model dominates charge transport in transition metal oxide glasses for T > Θ D/2 [41–45].

On the other hand, the conduction mechanism at T < T R is variable-range hopping (VRH) mechanism. In the VRH model, the charge carrier transport should be an acoustic phonon-assisted hopping process. In addition, the polaron-binding energy is small, and the disorder energy (i.e., W D) plays a dominant role. Mott [46] has proposed that polarons may hop preferentially beyond nearest neighbors. Greaves [47] has suggested that the VRH dominates even at intermediate temperature. Triberis and Friedman [48] and Triberis [49] have studied dc electrical hopping conduction in disordered systems at low and high temperatures using percolation considerations. In this case, the electronic transport is described as a single phonon-induced tunneling of electrons between localized states which are randomly distributed in energy and position [39].

Thus, generally, the change in slope in the conductivity is due to the change from SPH to VRH mechanism. In this section, we analyze our data in terms of SPH model at high temperatures. SPH regime is observed at high temperatures (T > Θ D/2) given as [39].

$$ \ln (\sigma T) = \ln (\sigma_{0} ) - W/kT $$
(1a)

with the pre-exponential parameter (in non-adiabatic regime) as:

$$ \sigma_{0} = \frac{{\nu_{0} Ne^{2} R^{2} C(1 - C)}}{k}\exp ( - 2\alpha R) $$
(1b)
$$ {\text{and}}\;\left\{ {\begin{array}{*{20}l} {W = W_{\text{H}} + W_{\text{D}} /2} \hfill & {({\text{for}}\quad T > \varTheta_{\text{D}} /2)} \hfill \\ {W = W_{\text{D}} } \hfill & {({\text{for}}\quad T < \varTheta_{\text{D}} /4)} \hfill \\ \end{array} } \right. $$
(1c)

where σ 0, ν 0 , C V (=V4+/Vtot), N, α, k, W, W H, W D are pre-exponential factor as temperature-independent parameter, optical phonon frequency, the ratio of the fraction of sites occupied by an electron (or polaron) namely the ratio of transition metal ions in the low-valence state to the total transition metal ion content, the transition metal ion density, the tunneling factor (the ratio of the wave function decay), the Boltzmann constant, the activation energy for hopping, the hopping energy, and the disorder energy (the difference of electronic energies between two hopping sites), respectively [39].

Thus, W and ln(σ 0) can be calculated from the slope of ln(σT) versus T −1 and its intercept at T −1 = 0. Figure 1 shows the ln(σT) versus T −1 plots for all of the present samples.

Figure 2 shows the mean V-ion spacing dependence of the activation energy, showing a change at x = 20 and x = 40. For 0 ≤ x ≤ 20 mol%, it seems that the conduction is due to polaron hopping, and for 20 ≤ x ≤ 40 the mixed polaronic–ionic mechanism is governed. Such behavior change can be seen from Figs. 3, 4, 5 and 6. For the understudied glasses, the dc electrical conductivity ranges from 7.54 × 10−7 to 2.01 × 10−9 S cm−1 at 305 K (as listed in Table 2), indicating that the conductivity decreases with increasing Ag2O content up to 40 mol%, but in the range of x = 40–50, it shows an increasing trend. Such behavior can be seen in conductivity at other temperatures. From Figs. 2 and 5, such behavior change is observable in activation energy W and also for carrier mobility. This behavior can be attributed to a change in conduction regime from polaronic to mixed polaronic–ionic mechanism, which arises with addition of Ag2O for 20 ≤ x ≤ 50 and especially for x ≥ 40 (Table 2).

Fig. 2
figure 2

Variation of activation energy with mean V-ion spacing for different TVAg glasses

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Fig. 3
figure 3

Variation of dc electrical conductivity as a function of Ag2O (mol%) at 230, 270, 305 K for different glasses

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Fig. 4
figure 4

Plot of ln(σ) versus l/T 1/4 for different TVAgx glasses

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Fig. 5
figure 5

Variation of carrier mobility as a function of V2O5 (mol%) at 225, 260, 305 K for different TVAgx glasses

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Fig. 6
figure 6

Variation of carrier density as a function of V2O5 (mol%) at 225, 260, 305 K for different glasses

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Table 2 DC electrical conductivity at 225, 260, 305, and 380 K and pre-factor of Mott–Austin formula (σ 0)
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The expression for the conduction by the VRH model, according to Mott [50], is based on a single optical phonon approach and is given by:

$$ \sigma = A\exp \left( { - BT^{ - 1/4} } \right) $$
(2)

Here A and B are constants given by

$$ A = \left[ {e^{2} /2(8\pi )^{1/2} } \right]\nu_{0} \left[ {N(E_{\text{F}} )/\alpha kT} \right]^{1/2} $$
(3)
$$ B = 2.1\left[ {\alpha^{3} /kN(E_{\text{F}} )} \right]^{1/4} $$
(4)

where N(E F) is the density of states at the Fermi level, α describes the decay of the localized state wave function and ν 0 is the optical phonon frequency (≈1013 s−1 [50, 51]). So the VRH model predicts a T −1/4 dependence of the logarithm of conductivity at low temperatures as shown in Fig. 4. The values of slopes A and B from the best fits in Fig. 6 are shown in Table 3. The parameters α and N(E F), the hopping distance (R VRH), and hopping energy (W D) as calculated from Eqs. (3)–(6) are also given [50, 51]:

$$ R_{\text{VRH}} = \left[ {(9/8)\pi \alpha N(E_{\text{F}} )kT} \right]^{1/4} $$
(5)
$$ W_{\text{D}} = 3/\left[ {4\pi R_{\text{VRN}}^{3} N(E_{\text{F}} )} \right] $$
(6)
Table 3 VRH-Mott parameters for TVAgx glasses
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At low temperatures, the polaron-binding energy (W 0) becomes lower than the disorder energy, W D; it is, therefore, reasonable to assume that W 0 = W D. On the basis of this assumption, one can calculate the Mott variables as listed in Tables 3 and 4 [11]. It should be mentioned that, by using the data of R V and α (see Tables 1 and 3), exp(−2αR V) ≠ 1 and then the conduction occurs in non-adiabatic regime.

Table 4 Continuity of Mott parameters for TVAgx glasses
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Small polaron hopping variables (at temperatures above Θ D/2) are discussed in terms of the polaron bandwidth, J; for adiabatic hopping conduction, the hopping energy W H is given as

$$ W - W_{\text{H}} \simeq W_{\text{D}} /2 = W_{P} /2 = W_{{P^{\prime } }} /2 - J $$
(7)

where W P and W P′ are the polaron-binding energy and the maximum polaron-binding energy, respectively. By contrast, for the non-adiabatic region W H is given by

$$ W - W_{\text{D}} /2 \simeq W_{\text{H}} = W_{{P^{\prime } }} /2 $$
(8)

To calculate the W H values from Eq. (8), we evaluate the disorder energy, W D, from the low temperature conductivity, as discussed in this section. For these glasses, W H is nearly equal to W P′ /2. By using the maximum value of the polaron-binding energy (W P′ ), it is possible to determine the J value from Eq. (7). For our samples, J values are in the range 0.002–1.092 eV (Table 4). It is customary to consider the maximum hopping energy as W P′ /2 [11]. Therefore, we consider the J value equal to zero for sample TVAg40, because this sample has the maximum hopping energy value for all samples. The value of the polaron-coupling constant, γ p, which is a measure of the electron–phonon interaction, is evaluated using the formula γ p = 2W H/hν 0. The effective dielectric constant, ɛ p, is calculated from the relation \( \varepsilon_{\text{p}} = (\frac{{e^{2} }}{{4W_{\text{H}} }})[\frac{1}{{r_{\text{p}} }} - \frac{1}{{R_{V} }}] \). Also, the small polaron-coupling constant (γ p), polaron bandwidth, J, carrier mobility, μ, and charge carrier density, N c, are calculated for TVAgx samples. The values of parameters are listed in Tables 3 and 4.

For our samples, values of ɛ p are in the range 5.64–6.96, which is higher than those for TeO2–Sb–V2O5 glasses [11], and is of the same order as values obtained for other glasses.

The values of γ p are 12.79–73.01, and therefore we conclude that a strong electron–phonon interaction occurs in these glasses similar to those in TeO2–Sb2O3–V2O5 glasses [52]. The values of the N c are also evaluated for these glasses by use of the formula \( N_{\text{c}} = \frac{\sigma }{e\mu } \) [27, 39]. This value at four different temperatures is calculated. The results are listed in Table 5. Here μ is carrier mobility in non-adiabatic regime expressed by Mott and Davis [39] and Mott [53] as

$$ \mu = \frac{{eR_{\text{V}}^{2} }}{kT\hbar }\left( {\frac{\pi }{{4W_{\text{H}} kT}}} \right)J^{2} \exp \left( {{\raise0.7ex\hbox{${ - W}$} \!\mathord{\left/ {\vphantom {{ - W} {kT}}}\right.\kern-0pt} \!\lower0.7ex\hbox{${kT}$}}} \right) $$
(9)
Table 5 Carrier mobility at 225, 260, 305, and 380 K for different TVAgx glasses upon Mott’s model
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Values of μ are estimated using the experimental results for R V, W H, and W for different concentrations of V2O5. Table 5 gives the calculated values of μ at 225, 260, 305, and 380 K (also see Fig. 5). Note that, μ decreases with decreasing V2O5 content for non-adiabatic hopping conduction. Furthermore, electrons are localized at vanadium ion sites in our glasses as small polarons, since μ ≤ 10−2 cm2 V−1 s−1 [54]. So conduction in the present glasses is due to polaron hopping. These results also suggest that with increasing vanadium concentration, R V decreases, which lowers W for TVAg10–TVAg40 (Fig. 2) and increases hopping probability. Thus, μ becomes larger with increasing V2O5 (see Fig. 5). Knowing the value of mobility, it is straightforward to calculate the carrier density, N c, by the relation \( N_{\text{c}} = \frac{\sigma }{e\mu } \). Values of N C at 225, 260, 305, and 380 K are given in Table 6, which shows that the carrier density for electronic conduction increases with decreasing V2O5 mol% for TVAg10–TVAg40 (see Fig. 6).

Table 6 Carrier density at 225, 260, 305, and 380 K for different TVAgx glasses upon Mott’s model
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The expression for the conduction by the VRH model, according to Greave’s, is based on a single optical phonon approach and conductivity is given by

$$ \sigma T^{1/2} = A\exp \left( { - BT^{ - 1/4} } \right) $$
(10)

Here A and B are constants given by

$$ A = \left[ {e^{2} /2(8\pi )^{1/2} } \right]\nu_{0} \left[ {N(E_{\text{F}} )/\alpha kT} \right]^{1/2} $$
(11)
$$ B = 2.1\left[ {\alpha^{3} /kN(E_{\text{F}} )} \right]^{1/4} $$
(12)

So the Greave’s VRH model predicts a T −1/4 dependence of the ln(σT 1/2) at intermediate and low temperatures [39] as shown in Fig. 7.

Fig. 7
figure 7

Plot of ln(σT 1/2) versus l/T 1/4 for different TVAgx glasses

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Model parameters are obtained by fitting measured conductivity data to the prediction of the model and are listed in Tables 7, 8, 9 and 10 (same as Mott’s VRH model).

Table 7 Greave’s parameters for different TVAgx glasses
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Table 8 Continuity of Greave’s parameters for different TVAgx glasses
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Table 9 Carrier mobility at 225, 260, 305, and 380 K for different TVAgx glasses upon Greave’s model
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Table 10 Carrier density at 225, 260, 305, and 380 K for different TVAgx glasses upon Greave’s model
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It is observed that the parameters obtained from Mott’s VRH model and Greave’s VRH models are in same order.

The expression for the conduction by the Triberis [49] and Friedman’s [48] model is based on the assumption that electron moves between localized states, which are randomly distributed in energy and position. The most interesting result of Triberis and Friedman’s model is a 1/T 1/4 dependence of log of conductivity for the few phonons at low temperature as well as for multi-phonon processes at high temperature:

$$ \sigma = \sigma_{0} \exp \left[ { - (T_{0} /T)^{1/4} } \right] $$
(13)

where the constant T 0 has different forms at high and low temperatures:

$$ T_{0} = \left\{ {\begin{array}{*{20}l} {12.5\alpha^{3} /kN_{0} } \hfill & {{\text{for}}\;{\text{high}}\;{\text{temperatures}}} \hfill \\ {17.8\alpha^{3} /kN_{0} } \hfill & {{\text{for}}\;{\text{low}}\;{\text{temperatures}}} \hfill \\ \end{array} } \right. $$
(14)

where N 0 (the density of states) is assumed to be constant.

After considering Mott’s and Greave’s model, the values of density of states at the Fermi level are too high at low temperatures, and this is opposite to SPH assumption in disorder systems. Thus, it assumes that Mott’s and Greave’s models are in qualitative agreement (Tables 9 and 10).

It is clear from Table 11, that by noting Triberis and Friedman’s model, the values of N 0 = N(E F) are calculated. Thus, this model provides good qualitative as well as quantitative fit to our dc conductivity data.

Table 11 Values of T 0 obtained from Eq. (14), and other parameters of this model (by assuming \( \alpha = 1\dot{A}^{ - 1} \))
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Survey of values of oxygen molar volume indicates the effect of Ag2O concentration change on thermal stability or fragility in samples. In other words, the oxygen molar volume is a too useful parameter for understanding the strength of glass structure. The values of oxygen molar volume for TVAgx glasses are obtained by using Eq. (15).

$$ V_{\text{O}} = \frac{{\left\{ {\left( {M_{{{\text{V}}_{2} {\text{O}}_{5} }} - 16C_{V} } \right)x_{{{\text{V}}_{2} {\text{O}}_{5} }} + M_{{{\text{TeO}}_{ 2} }} x_{{{\text{TeO}}_{2} }} + M_{\text{Ag}} x_{\text{Ag}} } \right\}}}{{\rho \left\{ {(5 - C_{V} )x_{{{\text{V}}_{2} {\text{O}}_{5} }} + 2x_{{TeO_{2} }} + x_{\text{Ag}} } \right\}}} $$
(15)

where M i is the sample molar weight of each component, x i mole fraction of the glass component, and ρ is the density.

Figure 8 shows the dependence between oxygen molar volume and Ag2O concentration in TVAgx samples. Decrease in oxygen molar volume is due to the decreasing of NBO’s. Thus, the thermal stability of glass increases with decrease in oxygen molar volume. In present samples, V O change in the range of 12.70–15.44 cm3 mol−1 for TVAgx samples. Results show an increasing trend for 0 ≤ x ≤ 40 except x = 20. Thus, between different compositions of present samples, TVAg20 glass has maximum thermal stability. In other word, this composition is considered to be strong glass and suitable choice for devices fabrication such as photodiodes, LEDs, solar cells, and laser diodes.

Fig. 8
figure 8

Variation of oxygen molar volume with Ag2O content, for TVAgx glasses

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4 Conclusions

The dc electrical conductivity of studied glasses is measured within the temperature range of 150–380 K. For the studied glasses, the dc electrical conductivity ranges from 7.54 × 10−7 to 2.01 × 10−9 S cm−1 at 305 K, indicating that the conductivity decreases with increasing Ag2O content up to 40 mol%; but in the range of x = 40–50, it shows an increasing trend. The changes in the electronic and ionic conduction in these glasses can be divided into three separate areas. The conductivity of the glasses in the range of x = 0–20 is electronic, and conductivity due to the presence of ions (ionic conductivity) has no role. But in the range of x = 20–40 due to the effective presence of silver ions in the glass composition, mixed of ionic conduction, and electronic conduction is proposed. In this study, the measured data of the dc electrical conductivity are investigated upon the different theories (conduction mechanism) of amorphous solids. The results show that at high temperatures, the dc conductivity is consistent with the predictions of the phonon-assisted SPH model of Mott. At low temperatures, the Mott’s VRH and Greaves’ hopping model require unusually large values of the density of states at the Fermi level for all compositions compared with those of other transition metal oxide glasses. The fitting of the Triberis and Friedman’s model for the present system gives reasonable values of the density of states (within the order of 1019 cm−3 eV−1) at the Fermi level. This model can interpret the electronic dc conductivity at high as well as at low temperatures. Also, according to the values of oxygen molar volume, among the fabricated glasses, TVAg20 glass is the best choice for electrical-to-optical or optical-to-electrical transducers or instruments that use such devices in their operation. Namely, this sample has the highest resistance against thermal shock compared with other samples.