The AR grade chemicals with 99% purity such as boric acid (H₃BO₃) from HI media, Ammonium dihydrogen ortho-phosphate (NH₄H₂PO₄) from Sd fine, Vanadium pentoxide (V₂O₅) from Sigma-Aldrich and Sodium carbonate (Na₂CO₃) from Merc various international chemical making firms were selected and procured on swift accessibility basis. The chemicals were mixed for every calculated weight proportion of each glass system and transferred into an agate mortar and manually grinded to obtained molecular dimensions powder. The crushed mixture was shifted into infusil make scientific company made silica crucibles. The crucible was employed into high temperature electrical furnace and heated to a very high temperature of the order of 1223 K and left over for 2 hours. The melted mixture of liquid was quenched by putting it on a fine-grooved stainless-steel plate in the right shape and then quickly closing another plate on top of it. The glass fragments in the shape of a disc were gathered. By transferring the glasses into a muffle furnace, they were annealed at 523 K for about six hours to eliminate thermal strains if they exist in the glass matrix. After annealing, the samples were reshaped into fine dimensions using sandpaper and pile. The thickness of the glasses was calculated using digital screw gauge with precision of 0.01 mm and cross-sectional area using graph sheet with accuracy of 1 mm.

On each of the produced glasses, X-ray diffraction analyses were carried, using Malvern Panalytical X-ray diffractometer X’PERT³ powder instrument operated at 40 kV and current of 30 mA with Cu-Kα wavelength at room temperature and 2θ ranging from 10˚ to 80˚.

The glasses were subjected room temperature (RT) density studies by taking toluene as an immersion liquid with a density of 0.8669 g/cc and applying the Archimedes principle, by using citizen make A digital, single-pan balance with a precision of 0.1 mg. The RT density of the glasses were calculated by using succeeding Equation (1).

D = ( M a M a − M L ) D L (1)

Here, M_a stands for the weight of the glass samples when they are suspended in air; M_L refers to the weight of the sample when it is suspended in immersion liquid; and, D_L represents the density of immersion liquid, like toluene. The molar volume V_M was estimated from determined densities of the glasses using equation.

V = x M D (2)

Here x is molar fraction, M is the molecular weight and D is density of the glasses [29] [30].

The dc conductivity measurement was performed by painting conductive silver paste on either side of the large surfaces and placing them in a two probe Keithley made instrument and applying constant voltage across the glasses over a wide temperature, from 303 K to 498 K. The current flowing through glass and voltage applied across the glass system was enumerated by using digital Pico ammeter and digital multi-meter with accuracy of ±10 mV. The Chromel-alumel thermocouples, with a precision of 1K, were used to measure the temperatures of the samples. The error on conductivity was determined using the relations and was in the range of 3% - 5% [14].

The x-ray diffraction pattern depicted in the subsequent Figure 1 showed the non-existence of sharp peaks and confiding to the fact that the glasses lack crystalline structure. A bulge like structure can be noticed in the range between 18˚ to 38˚ revealing the existence of short-range atomic order [31].

The density and molar volume at room temperature of present glasses versus mole fractions of V₂O₅ are depicted in Figure 2 The measured density at room temperature and calculated molar volume of glasses were observed to be in the range of 0.943 g/cm³ - 1.776 g/cm³ and 8.130 cm³/mol - 14.322 cm³/mol respectively as recorded in the Table 1. With increasing V₂O₅ content up to 0.15 mole fractions, the density of the current glasses dropped and hits minimum value and thereafter, it has been observed density rose nonlinearly with increasing V₂O₅ concentration up to a mole fraction of 0.4 before decreasing again at 0.45 mole fraction of V₂O₅ content. The glasses’ molar volume was varied appositively as density. Figure 2 shows how density and molar volume change with increasing V₂O₅ content. Density and molar volume estimates agreed well with various alkali and TM-doped glasses. [14] [32] [33].

Themeantransitionmetaliondistancewascalculatedusing, R = ( 1 N ) 1 / 3

here,N representstheconcentrationoftotalTMionthatwascalculatedusingtherelationship N = 2 ( D x / M ) × N A ,hereD is the density, x is the mole fraction of TM ion and M is the molecular wt. of the respective ions of the corresponding glass system, N_AisAvogadro number.

The density of the present glasses decreased and molar volume increased with increase in the V₂O₅ content up to x = 0.15 mol fraction of vanadium oxide concentration (TM ions). This demonstrates that the glass network structure was relaxed and a rise in the number of vanadium ions Up to x = 0.25 mole fractions of vanadium ion content, the density of glasses increased while molar volume declined, which reveals that in this region, the topology of the glasses was jittery. Further increase of V₂O₅ concentration the density decreased and molar volume increased until x = 0.4 mol fraction of vanadium ion and hits peak value. And again, increase of vanadium ion concentration the density decreased up to x = 0.45 mole fractions of V₂O₅ content. The molar volume behaves oppositely as density. Hence the topology of the present glasses was continuously varied nonlinearly with increase of V₂O₅ content which was incorporated into a glass structure. The glass network structure opened and jittered nonlinearly with mole fractions of V₂O₅ ranging from x = 0.05 to 0.45 at the expense of sodium ion concentration. In the current set of glasses, the alkali ion Na₂O concentration was replaced by Transition metal ion V₂O₅ from x = 0.05 to 0.45, from the analysis of density and molar volume indicating that the topology of glass network continuously alters nonlinearly and It indicates the presence of a single transition ion effect at x = 0.4 mole fraction of V₂O₅ content. This is the first-time boro phosphate glasses doped with Na₂O and V₂O₅ explored to density and molar volume studies and exhibited transition effect which was not observed in any earlier reports [14].

The thickness and cross-sectional area that have been measured, ranging from 2.01 mm to 2.90 mm and from 16 mm² to 51mm² respectively. The present glasses dc conductivity was estimated at 573 K and lies between 4.48 × 10⁻⁵ (Ω·m)⁻¹ to 7.94 × 10⁻³ (Ω·m)⁻¹ and hence which are semiconducting in nature [20] [32] [34].

In Figure 3, we see a graph of ln(T) vs (1/T) that was generated using Mott’s Small polaron hopping (SPH) model for current existing glasses [19] [35] [36] [37].

In the non-adiabatic conduction region, the linear lines were fits to the graph, above T = θ D 2 subsequently the corresponding activation energy W for all

glasses were estimated and were in the range of 0.232 eV to 0.696 eV. These values are comparable to many published values of Alkali and TM ion glass systems [25] [32] [38] [39]. For present series of glasses, the dc electrical conductivity at 468 K, as well as the calculated high temperature activation energies, were plotted and illustrated in Figure 4.

It is observed, at x = 0.15 mole fractions of V₂O₅, the high temperature activation energy reduces and the dc electrical conductivity rises, as seen in Figure 4, and further growth of V₂O₅ content, As the activation energy W rises, the conductivity σ falls, and further growth of vanadium content in the glass structure the conductivity increases and activation energy decreases until 0.4 mol fraction of V₂O₅ and further increase of vanadium the conductivity diminishes and activation energy raises further again. The incorporation of V₂O₅ content, high temperature DC activation energy and the DC conductivity varies oppositely and conductivity increases and hit peak value at 0.4 mol V₂O₅ and conductivity decrease from 0.4 mol to 0.45 mol of V₂O₅ the same values were tabulated in Table 2. It is clear in Figure 4 that from 0.05 to 0.2 mol of V₂O₅ content glasses reveals the ionic predominant region and from 0.2 to 0.3 mol fraction of V₂O₅ content shows mixed conduction region and form from 0.3 onwards up to 0.45 mol of V₂O₅ content reveals the electronic predominant region. Hence in the current series of BPVN glasses the varying conductivity and activation energy with V₂O₅ mole fraction exhibit the ‘single Transition Effect’ (STE) at x = 0.4 mole fraction of vanadium content in glass matrix. Because of the addition of V₂O₅ content at the cost of Na concentration, both sodium ions and polarons contribute to conductivity. Hence, the observed increase in conductivity at 0.15 mole fraction of V₂O₅ from 0.05 content demonstrates that alkali ions are responsible for the conductivity, indicating ionic predominant region and from 0.2 to around 0.25 mole fraction of V₂O₅ content shows mixed conductivity both due to alkali ions and transition metal ions. Further, from 0.3 to up to 0.45 mole fraction of V₂O₅ content is electronic predominant region and the conductivity is due to only polarons. Similar electronic predominant variations were found in several literatures [25] [40] [41] [42], which have been depicted in Figure 4

Conductivity in the non-adiabatic regime is governed by the Mott’s SPH model and is given by

σ T = σ o e − W k B T (3)

Here, W is the activation energy and σ o is the pre-exponential term, and it is denoted as

σ o = υ o N e 2 R 2 C ( 1 − C ) e − 2 α R k B (4)

Here, υ o = θ D k B / h is the optical phonon frequency, θ D is the Debye temperature, N, R have their regular meanings, as previously stated. α is the tunnelling factor and C is reduced fraction TM ions concentration [25].

It is taken into account for the strong electron-phonon interaction proposed by Austin and Mott [37],

W { = W H + W D / 2 T         > θ D / 2 ≅ W D T                                 < θ D / 4 (5)

Here, W H = W P / 2 is polaron hopping energy, W D e is disorder energy occurring due to a disparity in the energy levels of nearby neighbours between two hopping sites. The polaron energy W_H is estimated by Greaves model [33] given as

W H = W P 2 = e 2 4 ε p ( 1 r p − 1 R ) (6)

Here W P is the polaron binding energy and ε p is the effective dielectric constant and is given by

ε p = e 2 4 W r p (7)

Here r p is small polaron radius and given as

r p = 1 2 ( π 6 N ) 1 / 3 (8)

The calculated data for ε p and r p are tabulated in the Table 2 and they are similar to the reported data in the literature [25] [33] [39] [42] [43].

The polaron bandwidth J S P H according SPH model [37] [44] is given by

J S P H { > ( 2 K T W H / π ) 1 4 ( h ν o / π ) 1 2         foradiabaticSPH < ( 2 K T W H / π ) 1 4 ( h ν o / π ) 1 2         fornon-adiabaticSPH

Using the relation J t h = J o e − 2 α R , were determined to fall within the acceptable range of 0.017 eV to 0.020 eV and, here J o = ( W H ) min 4 [18] varies from

0.032 eV to 0.104 eV and same recorded in Table 3 in that same table it’s found that the values of J w h fulfils both the Holstein condition for the creation of small polaron hopping and the non-adiabatic SPH conduction, which is given in the equation above. These conditions are necessary for the formation of small polaron hopping. [45] that were observed in the limit of 0.043 eV to 0.138 eV represented in the Table 3, the value of α = 20 ( nm ) − 1 in earlier equation is according to reports various TM ion doped glasses [44]. The relationship [45] was used to calculate the density of states at the fermi level N(E_F), N ( E F ) = 3 / 4 π R 3 W , and vary as to the order of 10²⁶ eV⁻¹m⁻³ - 10²⁷ eV⁻¹m⁻³ and shown in Table 3 and the reported values agree with various such glasses systems [14] [32] [46]. The small polaron hopping constant γ P , is essential in understanding electron-phonon interaction, hence estimated by the relation γ P = 2 W H / h ν o and is presented in Table 3. In accordance with Austin & Mott [37], when γ P > 4 suggests an interaction between electrons and phonons in glasses.

At temperatures lesser than the Debye temperature, the DC conductivity were observed to exhibiting temperature dependent nonlinear variations, and signifying the prominent role of disorder energy due to which hopping of electron may happen between the nearest neighbors [35] [45] is given by

σ = A e B / T − 1 / 4 (9)

Here, A = 4 [ 2 α 3 / 9 π k N ( E F ) ] 1 4

B = [ e 2 / 2 ( 8 π ) 1 2 ] υ 0 [ N ( E F ) / α k T ] 1 2

From slopes of graph ln σ _Vs T − 1 / 4 plotted in the Figure 5 the values of A, B are found.

The estimated values of N ( E F ) are in the range of 10³¹ eV⁻¹m⁻³ to 10⁴⁰ eV⁻¹m⁻³ as recorded in Table 4 and it is clear that values determined are very much higher and deviate from the values comprising oxide glasses.

In the same low temperature range, lesser than the Debye temperature, variable range hopping (VRH) has been analysed in the light of Greave’s VRH model and is given by

σ T 1 / 2 = A e − B / T 1 / 4 (10)

Here A, B are constants and are determined from the least square (LS) fits to data in the plot of ln σ T − 1 / 2 Vs T − 1 / 4 is shown in Figure 6

The density of states N ( E F ) was measured from the theoretical expression for constant B and given by

B = 2.1 [ α 3 / k B N ( E F ) ] 1 4

The values of N ( E F ) varies from 10²⁶ eV⁻¹m⁻³ to 10³⁰ eV⁻¹m⁻³ and both are presented in Table 4, found to be in agreement with the several determined values by Greave’s VRH model for alkali and TM ion doped oxide glasses [16] [25] [32] [40] [42] [47]. Thus, it may be concluded that the Greaves VRH model is sufficient to explain the electrical conductivity data in the temperature range lesser than the Debye temperature for the present glasses.

According to N.F. Mott and E.A. Davis [36], N.F. Mott and I. Austin [37], in the adiabatic and non-adiabatic hopping regime carrier mobility μ , which involves electron diffusion via polaron hopping furnished in following

μ = ( υ o e R 2 k T ) e − W k T For adiabatic (11)

μ = ( e R 2 k T ) ( 1 ℏ ) ( π 4 W H k T ) 1 2 J 2 e − W k T For non-adiabatic (12)

carrier mobility μ were calculated at temperature of 195˚C and found to vary from 9.710 × 10⁻⁹ to 1.091 × 10⁻⁶ [18], the hopping carrier concentration N_C estimated from σ = e N C μ [42], and it ranges from 5.719 × 10²⁶ eV⁻¹·m⁻³ to 3.086 × 10²⁷ eV⁻¹·m⁻³ and both are presented in Table 5.

The mobility and charge carrier density versus mole fractions of V₂O₅ content is shown in Figure 7. From the graph it is observed that up to 0.2 mole concentration of V₂O₅ the mobility decreases from starting 0.05 mol of V₂O₅ content, further increase in the V₂O₅ content the mobility is increases attains maximum value, then after, it keeps falling down continuously with growth of V₂O₅ content in the glass maximum until it reaches bare minimum value at x = 0.45 mole fraction, but At the same time, the largest concentration of charge carriers occurs at this location, suggesting Anderson’s localization [48] [49] of charge carriers taking place around vanadium ions where glass network becomes more closed the presence of the Single Transition Effect (STE) at 0.4 mol fraction of V₂O₅ content in the glass matrix. It’s very clear in similar Figure 7 that, with increasing V₂O₅ content, charge carrier density acts practically opposite to mobility [50].