Mixed Polaron and Bipolaron Transport in (xV₂O₅–(65–x) Sb₂O₃–35P₂O₅) Glasses

Abstract

This study presents the electrical and optical properties of 35P₂O₅–xV₂O₅–(65–x) Sb₂O₃ glasses for 0 ≤ x ≤ 65 mol%. The direct current (DC) resistivity was measured by the Van der Pauw method and optical absorption spectra were taken in the Ultraviolet–Visible-Near-Infrared (UV–VIS–NIR) range. Electrical transport is attributed to simultaneous hopping of small polarons (SPs) between V⁴⁺ and V⁵⁺ (vanadium ion) sites and small bipolarons (SBPs) between the Sb³⁺ and Sb⁵⁺ (antimony ion) sites. The resistivity exhibits a non-linear dependence on the ionic fraction of vanadium (n_v), whereas the resistivity exhibits a minimum in the composition range 0 ≤ n_V ≤ 0.3, and a resistivity maximum was observed in the range 0.3 ≤ n_V ≤ 0.5. On further increasing n_v, the resistivity exhibits a monotonic decline. In the composition range 0 ≤ n_V ≤ 0.3, where the hopping distance between V ions decreases, while that between the Sb ions increases, the resistivity minimum has been shown to be the consequence of decreasing tunneling distance of SPs between the V⁴⁺ and V⁵⁺ ion sites. In the composition range 0.3 ≤ n_V ≤ 0.5, the resistivity, activation energy for DC conduction, glass transition temperature, and density exhibit their respective maxima even though the separation between the V⁴⁺ and V⁵⁺ sites continues to decrease. This feature is explained by enhanced localization of electrons on account of increased disorder (entropy) among the SPs and SBPs, like that of Anderson localization. This argument is further supported by a shift in the polaronic optical absorption bands associated with the SPs and SBPs toward higher energies. The transport behavior of all the glasses except the x = 0 composition has been explained by adiabatic transport, principally, by the SPs on V ions while the Sb ions contribute little to the total transport process. The results provide a clear relation between composition, polaron/bipolaron contributions, and conduction in these glasses.

1. Introduction

In oxide glasses, the polarizable lattice almost always traps an electron or a hole (carriers) as a result of strong electron–phonon interaction (EPI) [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18], which produces local lattice distortions around the carrier. The ensemble comprising the carrier and the associated lattice distortion may be considered as a pseudo particle or polaron, the transportation of which is the origin of electronic conduction in glasses. Even in the absence of substantial EPI, the theoretical basis of the localization of electrons (Anderson localization or AL) was provided by Anderson [18]. Mott and others [8,9,10,11,12,13,14,15,16,17] have discussed the localization of charge carriers in oxide and chalcogenide glasses, which provide the perfect environment for AL. Such localization of charge carriers almost always renders glasses semiconducting. There is no example of inactivated metallic conduction in any oxide glass.

In glasses containing transition metal oxides, transition ions (TIs) often exist in multiple oxidation states. When an electric field is applied, electrons localized at the reduced ion sites (i.e., those with lower ionization energy) can hop to neighboring oxidized sites, contributing to electrical conduction. However, in oxide glasses, almost all of the electrons available for conduction are self-trapped [1,2,3,7] in the lattice by the EPI [1,2,3,4,5,7,8,9,10,11,12,13,14,15,16]. Such self-trapped electrons are known as polarons [1,2,3,4,5,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30]. Depending on the size of the lattice distortion, these polarons are termed either small or large polarons. In silicate, phosphate, and telluride glasses, the sizes of the lattice distortions around carriers are mostly a few atomic distances and are termed small polarons (SPs) [1,2,3,7].

Phosphate and telluride glasses incorporating mixtures of TIs have been shown to exhibit non-linear responses in several key aspects, including DC electrical resistivity (ρ), DC activation energy (W), and glass transition temperature (T_g) [1,2,3,4,5]. Keeping the total TI concentration constant, if the relative proportions of TIs are varied, ρ, W, and T_g exhibit pronounced maxima, similar to that of the well-known mixed alkali effect (MAE), observed in glasses containing two different alkali ion species [6]. The non-linear effect is commonly referred to as the mixed transition ion effect (MTE) [1,2,3,4,5] in the scientific literature. However, it is important to note that, whereas MAE is explained by a classical barrier-crossing mechanism of two competing alkali ions, MTE has its origin in AL (a quantum phenomenon). At a critical TI ratio, which is close to 0.5, the configurational entropy (disorder) among the TIs reaches a maximum, which coincides with the maximum Anderson disorder parameter [16,18,22]. Such disorder in the potential field impedes electron hopping on account of enhanced AL of the SPs [1], culminating in carriers trapped in deeper potential wells even as the inter-TI separation decreases. The corresponding resistivity maximum confirms the role of the configurational disorder (entropy) among the TIs in suppressing polaron transport and highlights the impact of MTE.

Another important aspect of electronic transport, comprising small bipolarons, or SBPs, has also been discussed in the literature [2,3,9,20]. In oxide glasses, group V ions such as As and Sb typically exist in As³⁺/As⁵⁺ and Sb³⁺/Sb⁵⁺ states. Two electrons may be localized in a deep trap, forming what is known as an SBP. Hopping of such bipolarons between the reduced and oxidized states has been reported to result in electronic conduction in glasses [15,16,17,18,19,20]. Kumar et al. [20] have demonstrated the formation of SBPs, which are bound pairs of small polarons in vanadium phosphate glasses containing antimony oxide. While the reduced TI, V⁴⁺, with a single electron forms a SP for the conduction process, the Sb³⁺ ion can provide two electrons, both of which can be trapped in the same potential well to form an SBP, as described in the theoretical formalism set out by Emin [9]. Previous work has attributed electrical conductivity in Sb–phosphate glasses to the transport of SBPs [21] between Sb³⁺ and Sb⁵⁺ sites. SBPs on the reduced (Sb³⁺) sites may hop over to the oxidized (Sb⁵⁺) sites under the influence of an external electromotive force (EMF) during electrical conduction. It is to be noted that such conductivity is lower than conduction on account of SP transport in the same glass if SPs are present in the same glass on account of the presence of TIs such as vanadium or iron ions. In addition, the activation energy attributed to SP conduction will, accordingly, be lower than that for SBP conduction. Moreover, an SBP optical absorption band is expected to be detected at a shorter wavelength (greater energy) than an SP optical band in the same glass.

Theoretical Background

The mechanism of electronic transport of a self-trapped carrier in TI-containing glasses, when subjected to an external EMF, is described by the quantum mechanical process by which an electron tunnels between TI sites. Such a transport mechanism may be visualized as the “hopping” of the SPs between TI sites and is commonly referred to as SP hopping (SPH) conduction. SPH conductivity, $\sigma$, is given by Equation (1) [1,2,3,4,5,6,8,9,16]:

$$\sigma ={\sigma }_0 exp exp (-{\displaystyle \frac{W}{kT}})$$

(1)

where the pre-exponential coefficient, ${\sigma }_0$, is given by

$${\sigma }_0=\left({\displaystyle \frac{\nu e^2}{kTR}}\right)C\left(1-C\right)exp exp (-2\alpha R)$$

(2)

Here, α is the wavefunction localization parameter, ν is the frequency of longitudinal optical phonons, R is the average distance between ions [8], and C is the fractional site occupancy [8]. The activation energy W comprises two components: W = W_H W_D/2, with W_H being the hopping activation energy and W_D being the site energy difference due to lattice inhomogeneities. Since in oxide glass containing TIs, W_D ≪ W_H at room temperature or higher [1,2,3,4,5,8], W_H is typically used as the activation energy for SPH conduction.

At temperatures above (θ_D/2), where θ_D is the Debye temperature, SPH conduction involves multiple unsuccessful and successful hopping attempts from a reduced to an oxidized TI site by tunneling. Such a process is facilitated by the optical phonon frequency, ν (Equation (2)), typical of the glass continuous random network (CRN) structure and the distance between the tunneling sites. Close proximities between such sites will culminate in a favorable α, the wavefunction localization parameter, enhancing the overlap between the carrier wavefunctions. Depending on the probability that the carrier will successfully hop between TI sites, polaron conduction may occur via adiabatic (high probability) or non-adiabatic (low probability) mechanisms [8,9,17,18,19,20,21,22,23,24,25,26]. In the adiabatic regime, the term exp(−2αR) in Equation (2) approaches unity and may be neglected [8,17]. Conversely, in the non-adiabatic regime, this term becomes significant, resulting in reduced hopping probability.

According to Emin′s model [9], the polaron formation energy, Wp, relates to the W as [1,2,3,4,5,8,16,17]

$$W={\displaystyle \frac{Wp}{2}}$$

(3)

Equation (3) provides the relationship between the activation energy, W, for DC conduction and the SP or SBP formation energy, Wp. Moreover, as Emin has demonstrated, the Wp of SBP is approximately double that of SPs in the same glass. It follows that the polaron formation energy is the average depth of the potential wells that the carriers are trapped in. The CRN structure of glasses renders such trap “depths” smeared over a range of values, determined by the Anderson randomness parameter [18], as described by Mott [19].

When a glass containing polarons (SPs/SBPs) is subjected to electromagnetic radiation in the near-infrared region, the carriers in the traps or potential wells typically exhibit broad, noisy, and low intensity absorption bands [2,3,9]. The width of such absorption bands is indicative of the energy spread of the “depths” of the potential wells. The energy corresponding to the center (peak point) of the absorption band is approximately equal to twice the activation energy for DC conduction (W).

The polaron (SP/BSP) radius, r_p, is an important parameter that provides insight into the extent of localization. The polaron radius can be estimated under the assumption of a homogeneous atomic distribution as [15,19]

$$r_P={\displaystyle \frac{1}{2}}{\left({\displaystyle \frac{\pi }{6N}}\right)}^{1/3}$$

(4)

where N is the density of hopping sites. The average inter-site distance denoted by R can be found using [2,3,8]

$$R=N^{-1/3}$$

(5)

When donor cations are separated by only a few multiples of their polaron radius, adiabatic transport is likely. Conversely, when the separation between ions is large, non-adiabatic transport becomes dominant. In this regime, conduction is reduced because charge carriers encounter fewer opportunities to hop, as these hopping events are primarily facilitated by lattice vibrations [8,17].

The present investigation explores the combined effect of SPs and SBPs by systematically determining the electrical and optical properties as a function of the relative proportions of SPs and SBPs in the glasses with the following general composition: 35P₂O₅–xV₂O₅–(65–x) Sb₂O₃. In this study, the relationship between resistivity and optical absorption data has been attempted. Optical bands for SPs and SBPs are anticipated at frequencies that correspond to roughly 2W [9], facilitating the identification of SP/SBP transport through the correlation of W with the energy linked to the peak wavelengths (λ_SP/SBP) of the SP/SBP optical bands [2,3,9].

2. Experimental Section

2.1. Glass Synthesis

Glasses in the 35P₂O₅–x V₂O₅–(65–x) Sb₂O₃ composition systems were prepared employing the conventional melting and quenching method [27,31], where x was varied between 0 and 65. Another series of binary vanadium phosphate glasses with the general composition (100–x) P₂O₅–x V₂O₅ were also synthesized with x = 45, 55, 65, 75 by following the same procedure.

P₂O_5, V₂O_5, and Sb₂O₃ with 99.99% purity were obtained from Alfa-Aesar, Ward Hill, MA, USA. The batches comprising appropriate weights of the oxides were mixed, and the mixture was melted in fireclay crucibles at 1100–1150 °C for 30 min in the furnace (Deltech Inc., Denver, CO, USA, (DT29-BL56-E2404)). The melts were quenched between two ½ thick copper plates to obtain the glass. The binary antimony phosphate and vanadium phosphate were named PS and PV, respectively, in this work, while the ternary glasses containing P₂O₅, V₂O_5, and Sb₂O₃ were named PVS glasses. The optical images of glass pieces obtained after quenching are shown in Figure 1. The compositions of the glasses were confirmed by X-ray fluorescence (XRF) spectroscopy (PANalytical Wavelength Dispersive Axios max advanced system with SuperQ4 analysis software) [PANalytical, SuperQ4 Software, Almelo, The Netherlands: PANalytical B.V., 2012] and are presented in

Table 1. (Binary) PV glass XRF-normalized chemical composition and glass transition temperature.

Sample Name	[V/(V + P)]	P2O5 (mol%)		V2O5 (mol%)		Glass Transition Temperature (Tg) (°C)
		Nominal	XRF-Normalized	Nominal	XRF-Normalized
PV-1	0.45	55	48.95	45	51.05	482
PV-2	0.55	45	41.42	55	58.58	488
PV-3	0.65	35	30.32	65	69.68	498
PV-4	0.75	25	23.49	75	76.51	534

and

Table 2. (Ternary) PVS glass chemical composition (nominal and XRF-normalized) and glass transition temperature.

Sample Name	[V/(V + Sb)]	P2O5 (mol%)		V2O5 (mol%)		Sb2O3 (mol%)		Glass Transition Temperature (Tg) (°C)
		Nominal	XRF-Normalized	Nominal	XRF-Normalized	Nominal	XRF-Normalized
PS	0.00	35	35.20	0	0.00	65	64.80	469
PVS-1	0.15	35	46.6	10	13.3	55	40.03	476
PVS-2	0.31	35	32.55	20	27.93	45	39.52	475
PVS-3	0.38	35	38.55	25	29.04	40	32.41	512
PVS-4	0.46	35	34.11	30	31.54	35	34.36	521
PVS-5	0.62	35	38.28	40	43.33	25	18.39	464
PVS-6	0.77	35	38.84	50	51.23	15	9.93	447
PV	1.00	35	30.32	65	69.68	0.00	0.00	498

which lists the nominal as well as the XRF-determined actual molar compositions. We calculated V/(V + P) and V/(V + Sb) with molar concentration based on the nominal composition of the glasses (Table 1, Table 2,

Table 3. Comparison of the physical parameters of P₂O₅–V₂O₅ glasses.

Sample Name	[V/(V + P)] Nominal	Density (g/cm3)	Activation Energy (W) (eV)	Resistivity (ρ) at 220 °C (Ωcm)	Concentration of Vanadium Ions (cm)−3 (NV)
PV-1	0.45	2.04	0.60	4.90 × 103	3.45 × 1021
PV-2	0.55	2.06	0.59	1.83 × 103	4.16 × 1021
PV-3	0.65	2.53	0.42	8.37 × 102	5.92 × 1021
PV-4	0.75	2.63	0.40	2.164 × 102	6.90 × 1021

,

Table 4. Comparison and transport data of V₂O₅–Sb₂O₃–P₂O₅ glasses.

Sample Name	[V/(V + Sb)]	Activation Energy (W, eV)	Resistivity (ρ) at 220 °C (Ωcm)	Pre-Exponential ${\mathit{\sigma }}_0{\left(\mathit{\Omega }\mathbf{c}\mathbf{m}\right)}^{-1}$	Wavelength * (nm)
PS	0.00	0.73	4.50 × 106	10.96	849
PVS-1	0.15	0.79	4.92 × 105	190.55	774
PVS-2	0.31	0.67	3.00 × 104	269.15	898
PVS-3	0.38	0.67	2.43 × 105	11.48	984
PVS-4	0.46	0.71	1.79 × 106	12.59	849
PVS-5	0.62	0.65	3.20 × 104	76.61	911
PVS-6	0.77	0.63	7.93 × 103	275.42	984
PV	1.00	0.42	8.37 × 102	20.42	1430

* wavelength of the optical absorption band peak.

and

Table 5. Comparison of the physical parameters of V₂O₅–Sb₂O₃–P₂O₅ glasses.

Sample Name	[V/(V + Sb)]	Density (g/cm3)	N Concentration of Ions (cm)−3		VV Distance R (nm)	rp (nm)
			NV	NSb
PS	0.00	3.74	0.00 × 1000	6.14 × 1021
PVS-1	0.15	3.53	9.35 × 1020	5.13 × 1021	1.02	0.41
PVS-2	0.31	3.44	1.91 × 1021	4.29 × 1021	0.81	0.32
PVS-3	0.38	3.38	2.41 × 1021	3.86 × 1021	0.75	0.30
PVS-4	0.46	3.36	2.95 × 1021	3.44 × 1021	0.70	0.28
PVS-5	0.62	3.18	3.93 × 1021	2.45 × 1021	0.63	0.26
PVS-6	0.77	3.11	5.09 × 1021	1.53 × 1021	0.58	0.23
PV	1.00	2.53	5.93 × 1021	0.00 × 1000	0.55	0.22

).

X-ray diffraction (XRD) and Differential Thermal Analysis (DTA) were employed to determine the amorphicity and glass transition temperature (Tg) of each sample. Small amounts from each batch were crushed and ground for XRD, XRF, and DTA measurements. The rest of the glasses were successively polished with 400, 600, 1200, and 2000 grit abrasives to obtain polished co-planar,~2 mm thick samples of arbitrary shapes to be utilized for electrical conductivity and optical (UV-VIS-IR) spectroscopy measurements. For XRD measurements, a Rigaku Smart Lab-II diffractometer (scanning range: 10° < 2θ < 80°) was used to verify the amorphous nature of the glass samples (Figure 2). In XRF analysis, minor impurities of up to 0.5–1 wt% were identified, including multiple metal oxides. Some of the reported compositions were renormalized, which ignores the existence of such impurities (Table 1 and Table 2). For some glasses, the actual composition of the glasses differed significantly from the nominal composition owing to the volatile nature of P₂O₅.

2.2. Electrical Resistivity Measurements

The Van der Pauw four-probe method [7] was utilized to measure DC resistivity (ρ) with a computerized setup (H-50, MMR Technologies) functioning within the temperature range of −191 to 427 °C. The Van der Pauw method is particularly suited for measuring resistivity in irregularly shaped glass samples, as it allows accurate determination without requiring a specific geometry. The average of 24 measurements will be undertaken at each temperature to generate an average value of resistivity by incorporating the dimensions of the sample.

Glass samples, characterized by uniform thickness (800–1200 μm) and various shapes, were polished and fitted with silver electrodes using silver paste (PELCO^® High Performance Silver Paste). Measurements were executed under ohmic conditions, with each sample affixed on top of an aluminum nitride ceramic heater, utilizing thermal grease to ensure effective thermal contact. Current and voltage probes were configured as described above, with the current ranging between 10⁻⁶ and 10⁻¹¹ A, chosen according to the measurement temperature. All measurements were taken in a dark, evacuated chamber (8–10 mTorr). The estimated uncertainties were ±7.5% for resistivity and ±0.02 eV for activation energy. Conductivity values were derived from the inverse of the measured resistivity. The electrical resistivity data are presented in Table 3 and Table 4 and also presented diagrammatically in Figure 3a,c, Figure 4 and Figure 5.

2.3. Optical Absorption Spectroscopy

Optical absorption spectra were recorded using a Perkin-Elmer Model 330 dual-beam spectrophotometer, scanning in the 250–3000 nm range at a rate of 120 nm/min. This instrument was chosen for its broad spectral coverage, allowing observation of both UV-edge transitions and polaronic absorption in the near-infrared (NIR) region. The glasses were polished using 600- and 1200-grit silicon carbide abrasive papers to obtain smooth optical surfaces with a final thickness of approximately 500–1000 μm. Careful preparation ensured minimized surface scattering and consistent sample geometry, which is critical for reliable absorption measurements. Spectra were recorded in transmission mode at room temperature. Figure 6 shows the absorption behavior of selected P₂O₅–V₂O₅–Sb₂O₃ glasses, exhibiting broad absorption bands in the visible–NIR region.

2.4. Density Measurement

Density measurements were conducted using standard pycnometric techniques with deionized water as the immersion medium. Glass samples were precisely shaped, cleaned, and dried prior to measurement to ensure accuracy. Temperature corrections were applied to account for variations in fluid density. The estimated uncertainty in these measurements is ±0.01 g/cm³. The density data are listed in Table 3 and Table 5.

2.5. Differential Thermal Analysis

The glass transition temperature (Tg) was measured using a PerkinElmer DTA 7 system, (PerkinElmer, 710 Bridgeport Ave, Shelton, WA, USA, CT06484). operated at a heating rate of 10 °C/min across a temperature range of 200–1000 °C. The system was calibrated with DTA standards of alumina, quartz, and gold. The associated uncertainty in the T_g values, provided in Table 1 and Table 2 is estimated to be ±5 °C.

3. Results and Discussion

3.1. Electrical Resistivity

The conductivity of the PV glasses (Figure 3a,b) increases with increasing V concentration in the glass composition (Table 1 and Table 3), indicating that V ions are primarily responsible for electrical transport in this glass system. The activation energy (W) for dc conduction for all glasses, which corresponds to the slope of the log ($\rho$) vs. 1000/T plot, is found to increase with increasing temperature—indicating a temperature-dependent transport mechanism. For clarity, the log ($\rho$) vs. 1000/T plot of the PV-1 is presented in Figure 3b. In the low-temperature region (73–170 °C), the activation energy W_low = (0.49 ± 0.02)  eV, while in the high-temperature range (190–220 °C), W_high = (0.64 ± 0.02) eV.

For all binary and ternary glasses, activation energies (W) were determined from linear segments of the plots in the temperature range of 73–220 °C. The conductivity (σ) measurements for the binary PS glass were conducted in the 147–220 °C region on account of its exceptionally high resistivity at lower temperatures. Figure 3c presents the variation in conductivity of the PS and PVS glasses, which follows the general trend of increasing σ with temperature. The pre-exponential factor (${\sigma }_0$) was found from the intercepts of the Arrhenius plots of Figure 3c.

There is an extensive literature on the hopping conduction of SPs and SBPs prevalent in oxide glasses containing TIs [1,2,3,4,5,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30]. The characteristics of hopping can be distinguished from the conduction by holes and electrons [28]. Whereas the activation energy of transport in a crystalline semiconductor is constant as a function of temperature, that of a glassy semiconductor varies with temperature, as discussed earlier. In glassy semiconductors, conduction may take place by two distinct mechanisms: (i) small polaron hopping between localized states in the bandgap via tunneling; (ii) direct transfer of carriers from the localized states to the extended states (conduction/valence bands). Typically, the latter takes place at higher temperatures [19] because SPs/SBPs are trapped quite deep inside the forbidden band. It is established that SPH is the mode of conduction in glasses containing V ions [1,2,3,4,5,7,15,19], and small bipolaron hopping (SBPH) takes place in glasses containing Sb ions [20,21].

Table 5 presents the calculated average distance between V ions (V-V distance) denoted by R, and the SP radius, rₚ, determined [2,3,15,19] by using Equations (4) and (5) [15,19]. When electrons are able to respond quickly to lattice vibrations, as in systems where donor cations are spaced by only a few multiples of their ionic radii, the conduction mechanism is considered adiabatic. In such cases, the charge carriers can efficiently utilize available lattice vibrations to hop between localized states. Conversely, in the non-adiabatic regime, where the coupling among charge carriers and the lattice is weaker or slower, the conduction is significantly reduced due to the limited number of effective hopping events [17,28]. In all the glasses of this investigation (Table 5), R is determined to be within 2.5 times that of its corresponding rₚ. Such a condition is considered to be conducive to the adiabatic mechanism of nearest-neighbor SPH.

For V-rich glasses within the composition range 65 $>$ [V/(V + Sb)] > 0, the pre-exponential factor σ₀ varies between (10.96 ± 0.82) and (275.42 ± 20.6) S·cm⁻¹, which are values with orders of magnitude lower than those observed in typical metals [29]. This confirms that the electrical transport mechanism is governed by small polaron hopping (SPH). In the binary PV glass, ${\sigma }_0$ is found to be (20.42 ± 1.53) Scm⁻¹, which is in close agreement with the results reported by Dutta et al. for the mixed transition metal glass system P₂O₅–V₂O₅–MnO [1]. Figure 4 shows the relationship between the ln ${\sigma }_{0}R$ and V-V distance (R).

Doweidar et al. [30], Gohar et al. [32], and Dutta et al. [2,3] demonstrated SPH conduction in a variety of glasses (borate, phosphate, vanadate, telluride, etc.) containing V and Fe ions. The general consensus of all previous investigations of SPH conduction is that the nature of the hopping process becomes increasingly adiabatic as the distance between the TI sites decreases. Another important conclusion is that α of trapped electron/s for SP/SBP remains largely unchanged, regardless of the TI concentration. If the optical phonon frequency (ν) and C (redox ratio of the TIs) are assumed constant, the term exp(−2αR), the wavefunction decay parameter, becomes important.

For polaronic conduction in general, a plot of ln(${\sigma }_{0}R$) vs. R gives a straight line, the slope of which gives −2α, if C, the redox ratio of V or Sb, does not vary among the glasses [30,31]. However, this is true for glasses containing only SPs or SBPs, not a combination of SPs and SBPs. The ln(${\sigma }_{0}R$) vs. R plot provided in Figure 4 is not a perfect straight line but exhibits a maximum where V/(V + Sb) is ≈ 0.5 (Table 5). If, however, the ln(${\sigma }_{0}R$) data corresponding to R_VV values between 0.65 and 0.75 are ignored, a straight line is obtained as shown by the dotted line as a guide for the eye. In other words, some other factor is active around V/(V + Sb) ≈ 0.5, which will be treated next.

The probability of two electrons from two V⁴⁺ sites to simultaneously hop into one Sb⁵⁺ site and contribute to electrical transport would, indeed, be vanishingly small. Even in such an unlikely event, both electrons would be buried in a deeper potential well of an SBP. Hence, the active role of Sb ions in the conductivity of ternary glasses is extremely limited, as demonstrated by the drastic rise in conductivity even with modest additions of V₂O₅ to the binary antimony phosphate (PS) glass composition, which leads to the conclusion that the SPs on V sites are the principal carriers in the ternary PVS glasses. Hence, the ln(${\sigma }_{0}R$) vs. R plot provided in Figure 4 may be considered to be predominantly a reflection of SPH transport, contributed, principally, by the V ions, influenced by the potential field of the Sb ions.

In the composition range 0 ≤ V/(V + Sb) ≤ 0.31, the log(ρ) values decrease sharply. This behavior can be attributed to the increase in the number of available sites for SPH, resulting in increased conductivity. In polaronic systems, the conductivity (σ) depends exponentially on the hopping probability, which itself is highly sensitive to carrier concentration. Therefore, the sharp decrease in resistivity in this regime takes place on account of increasing the relative proportion of V.

Between V/(V + Sb) ≈ 0.30 and ≈ 0.50, the resistivity, however, starts to increase with increasing V, which contradicts the prevalent theory of SPH conduction, which mandates that the resistivity should decrease when the hopping distance becomes shorter. Such a behavior can only be explained by the mixed transition ion effect (MTE), mediated by Anderson localization [18], as demonstrated earlier by Dutta et al. in phosphate and telluride glasses [2,4,5]. As discussed earlier, the electrons on V⁴⁺ are localized by the disorder of the potential field offered by the cations comprising the glass matrix to start with. The maximum disorder or configurational entropy or maximum AL is reached when V/(V + Sb) approaches a value of 0.5. Accordingly, the glass composition with V/(V + Sb) = 0.46 (Table 4, Figure 5) exhibits the highest resistivity among all the glasses in this investigation. This also provides the reason why the ln(${\sigma }_0\mathrm{R}) \mathrm{v}\mathrm{s}. \mathrm{R}$ plot in Figure 4 deviates from linearity. As discussed earlier, the AL is accentuated by increasing the configurational entropy by virtue of the mixing of two different ions, reaching a maximum at V/(V + Sb) ≈ 0.50. For (V/V + Sb) > 0.5, the configurational disorder among the V and Sb ions continues to decrease monotonically as the V ion concentration increases until it reaches the binary PV composition. Concomitantly, the AL component decreases monotonically, resulting in a monotonic decrease in ρ and W (Figure 5).

The fact that Sb ions play a negative role in the conduction process can be demonstrated by comparison of the resistivity (ρ) and activation energy (W) of selected ternary PVS glasses with those of binary PV glasses containing similar concentrations of V₂O₅, as shown in

Table 6. Comparison of ternary PVS-6 and binary PV-1, PV, and PS glasses with comparable vanadium concentrations.

Sample Name	PVS-6	PV-1	PV	PS
V2O5	51.23	51.05	69.68	0.00
P2O5	38.84	48.95	30.32	35.20
Sb2O3	9.39	0.31	0.00	64.80
Resistivity (ρ) at 220 °C (Ω cm)	7.93 × 103	4.90 × 103	8.37 × 102	4.50 × 106
Activation energy (W) (eV)	0.63	0.60	0.42	0.73

. For example, when comparing PVS-6 and PV-1, which contain 51.05 mol% and 51.23 mol% V₂O₅, respectively, notable differences are observed. The ρ and W of the two glasses differ by factors of approximately 1.6 and 2, respectively. It is important to note that PVS-6 also contains a significant amount of Sb₂O₃ (9.93 mol%). Furthermore, among the binary glasses, the PS composition shows the highest resistivity, while the PV glass exhibits the lowest. The ρ of the PS glass surpasses that of the PV glass by over 3.5 orders of magnitude (refer to Table 6 and Figure 4). These findings strongly suggest that under the conditions of our experiment, Sb ions actively influence the conduction mechanism in these glasses and play a significant role in suppressing the transport of SPs on V ions.

3.2. Optical Absorption Spectrum

The optical absorption spectra of two PVS glasses in the composition range 0.00 ≤ n_V ≤ 1), as well as the PS and one PV glass, are presented in Figure 6. The wavelengths corresponding to the optical absorption peaks are listed in Table 4. The small polaron and small bipolaron absorption bands, characteristically broad and noisy, have been smoothed for better display along with the matching glass compositions. With decreasing n_V, it is noticed that the absorption band at 1300 nm (for the PVS-2 glass) shifts to higher wavelengths. The absorption band at 1430 nm for PV glass (n_V = 1) is absent in the PS-1 glass (n_V = 0). This validates the assignment of the 1430 nm peak to V ions. The spectrum of the binary PV glass (Figure 6) indeed exhibits a diffused absorption band, which seems close to the characteristic absorption band of the PS glass, but it is clearly shifted towards lower wavelengths for the latter. This background absorption (in the PV glass) is attributed to the continuous random network of the glass former, P₂O₅. The transformation of the diffused (background absorption) to a pronounced and well-formed band at a lower wavelength is ascribed to the SBPs in the binary antimony phosphate glass.

The binary PS glass lacks the distinctive absorption band (1430 nm) of the PV composition. However, the absorption band centered around 750 nm, which becomes much reduced in intensity, is assigned to the Sb ions. Most likely, it has a signature of the phosphate matrix of the glass. Further study is required to confirm this assignment. The broadness and asymmetry of the SP absorption band of PVS glasses are standard properties of small polaron bands [2,3,5], which are observed in the absorption spectra of the PV/PVS glasses (Figure 6). Absorption of a photon by the electron trapped in an SP comprises the change in the electronic energies between the V⁴⁺/V⁵⁺ and the Sb³⁺/Sb⁵⁺ states. The energy required is approximately double that of polaron production (W_P), which is connected to hopping conduction by Equation W = W_P/2_. According to Emin’s model [9], the absorption band centered around 850 nm corresponds well in energy with bipolaron transport in the PS glass, and the band at 1430 nm is caused by small polarons in the glass. According to Emin’s model, the DC activation energy, W, calculated from the optical data presented in Table 4, should be around (0.40 ± 0.02) eV for the PV-1 glass, which corresponds with the left shoulder of the optical absorption band (Figure 6). This is close to the experimentally found value of (0.42 ± 0.02) eV (Table 4) in the temperature range of 73–220 °C. Given the extremely wide and noisy character of SP absorption bands, the W − W_P association seems to be confirmed. Moreover, a shift in the left shoulders of the absorption bands to lower wavelengths (higher energy) is observed as the V/(V + Sb) ratio of the PV (1.0) and PVS-4 (0.46) glasses. In other words, the maximum activation energy and resistivity of the PVS-4 glass are verified by the optical absorption data.

3.3. Density

The effect of compositional variation on glass density was systematically analyzed for the binary PV glasses. As shown in Table 3 and Figure 6, the density increases with vanadium content. This trend can be attributed to the enhanced network connectivity introduced by V ions, which likely leads to a denser glass structure. In contrast, the density of ternary PVS glasses decreases with increasing vanadium, attributed to the lower mass of V⁴⁺ as compared to Sb³⁺ ions. The data obtained were used to calculate the concentration (N/cm³) of V and Sb ions in the glass, assuming homogeneous distribution. The concentration values were used to subsequently determine the SP/SBP radii (r_p) and R_V-V, the average separation of V ions, using Equations (4) and (5), respectively. Moreover, Figure 7 shows that the density increases with the V/(V + P) ratio, supporting the R_VV data presented in Table 5.

3.4. Glass Transition Temperature (T_g)

Figure 8a,b and Table 1 and Table 2 illustrate the variation in glass transition temperature (T_g) across both binary and ternary glass compositions. The minima and maxima observed in the log(σ) vs. 1/T curves correlate well with variations in the glass transition temperature (T_g). For the binary PV glasses, the T_g monotonically increases with V concentration, as the continuous random network structure is increasingly fortified with the increasing addition of V₂O₅ to the glass composition. This trend is attributed to the role of vanadium ions as a conditional network former, enhancing the degree of connectivity within the glass network structure. Accordingly, as the V concentration increases, the structural rigidity of the glass network improves, resulting in an elevated T_g [6,30]. For the PVS glasses, the maximum in T_g is observed at V/(V + Sb) = 0.46, exactly where the maximum resistivity is located (Table 4 and Figure 5). This feature was also observed earlier in mixed TI glasses [2,5].

4. Conclusions

In summary, the electrical and optical studies on 35P₂O₅–xV₂O₅–(65–x)Sb₂O₃ glasses show that charge transport occurs through the combined hopping of small polarons (SPs) between V⁴⁺ and V⁵⁺ sites and small bipolarons (SBPs) between Sb³⁺ and Sb⁵⁺ sites. Among these, SPs on vanadium sites are the primary carriers, while the contribution from Sb-related SBPs to total conductivity is minimal. Instead, Sb ions play a more prominent role in creating cationic potential disorder that suppresses SP transport, an effect attributed to Anderson localization superimposed on the inherent disorder of the glass matrix. All compositions except the Sb-only glass (x = 0) exhibit adiabatic SP hopping conduction. A distinct maximum in resistivity, activation energy for DC conduction, and glass transition temperature occurs near V/(V + Sb) ≈ 0.5, which corresponds to the highest level of cationic disorder. Optical absorption data agree with the electrical measurements, confirming the transport mechanism and the influence of composition on polaron/bipolaron dynamics.

These findings deepen the understanding of mixed transition ion effects in phosphate-based glasses and provide guidelines for tailoring conduction through compositional control. Future investigations should explore other transition metal combinations, examine temperature-dependent optical spectra in greater detail, and evaluate the long-term stability of glasses optimized for specific electrical properties. The conclusions that arrived can be further tested by the following:

i.

Incorporating Fe₂O₃ in the same glass system instead of V₂O₅. The former has been established to contribute mobile small polarons in phosphate glasses.

ii.

Using As₂O₃, known to contribute small bipolarons in glasses, instead of Sb₂O₃ in a similar glass system, with P₂O₅ being the principal glass former.

iii.

Using different glass formers such as SiO₂, GeO₂, B₂O_3, and Te₂O₂ in place of P₂O₅ to test the universality of Anderson localization in the transport process of glasses.