1 Introduction

The sol–gel procedure is a suitable synthetic path for creating materials with very different properties as compared to usual solid-state chemical techniques; on the other hand it offers a low cost approach for the production of electrochromic films [1]. It was found that the low temperature products have lower crystallinity and higher specific capacity than those synthesized by high temperature solid state method [2]. The wide variety of technological applications of V2O5·nH2O xerogel such as chemical sensors, optical memory switching and optical fibers give rise to its scientific interest [3–5]. Electronic transport is observed by electron hopping mechanism in materials containing transition metal ions such as Fe, Co, V, etc. if these ions are present in two distinct oxidation states via electrons hopping along e.g. V4+-O-V5+ paths [6]. The optical properties of highly oriented nanocrystalline hydrated vanadium pentoxide, V2O5·nH2O, prepared by dissolving V2O5 powder in hydrogen peroxide, H2O2 solution reveled that two different optical gaps are present, Eop1 = 0.37 eV and Eop2 = 2.2 eV, respectively [7]. Generally, the fundamental electronic structure of the present system is formed by the oxygen O-2p and vanadium V-3d orbitals [8]. The oxygen 2p states have the lowest energy (the highest binding energy). They form a band af 4.0 eV wide, which is fully occupied. The occupied part of the V-3d states is positioned about 3.0 eV above the top of the oxygen bands. Due to field splitting the V-3d manifold is spread over a range of at least 3 eV. The 3d xy level of the V 4+ is occupied with one electron. The unoccupied d xz and d yz levels are at least 1.0 eV higher. These in turn are located about 2 eV below \( d_{{x^{2} - y^{2} }} \) and the \( d_{{z^{2} }} \) levels. The relevance of the O-2p bands is that they provide a path for virtual hopping process between the V sites. The coupling between V sites is through virtual hopping via π-bonded O-2p y states [9]. These observations are consistent with the fact that these absorption bands are electronic d–d transitions associated with V 4+ centers in a low-symmetry crystal field environment with square pyramidal C 2v symmetry [10].

In the present study optical transmittance and reflectance were investigated for KxV2O5·nH2O films, where (0 ≤ x < 0.01). Investigation of the spectral distribution of the refractive index, n, the extinction coefficient, k, and hence the absorption coefficient, α, were made. An analysis of the absorption coefficient has been carried out to obtain the optical band gap, E Opt g , the band tail width, Ee, and the nature of transitions mechanism involved as well.

2 Experimental techniques

Five different compositions of KxV2O5·nH2O (where x = 0.00, 0.0017, 0.0049, 0.0064 and 0.0091 mol) have been prepared for the present investigation. Films of the investigated compounds were prepared by sol–gel technique (colloidal route) [11]. The powder of vanadium pentoxide (V2O5) (99.99%) and potassium hydroxide (KOH) (99.99%) in suitable proportions (mol %) were used as raw materials. A batch of (1.04 g) of V2O5 was dissolved in 80 mL of 17.5% hydrogen peroxide (H2O2) at room temperature [6]. The formed solution was stirred continuously using magnetic stirrer. The solution was then heated at 60 °C, with uninterrupted continuous stirring, while pH was 3–4. Water dissolved KOH (0.07 mol) was added to the V2O5 gel drop by drop using a Pruitt up to the desired concentration according to the following chemical reaction

$$ \begin{aligned} & {\text{KOH}} \to {\text{ K}}^{ + 1} + \left( {\text{OH}} \right)^{ - } \\ & {\text{V}}_{ 2} {\text{O}}_{ 5} \cdot {\text{nH}}_{ 2} {\text{O }} + {\text{ xK}}^{ + 1} \to {\text{K}}_{\text{x}} {\text{V}}_{ 2} {\text{O}}_{ 5} \cdot {\text{nH}}_{ 2} {\text{O}} \\ \end{aligned} $$

Vanadium pentoxide (V2O5) and (V2O5-(x) KOH) gels can be synthesized via the condensation of aqueous solutions of (V2O5) and (V2O5-(x) KOH). The obtained gels without precipitation are limited to the range of x recorded is due to the KOH chemical alkalinity effect which increases the pH value above the suitable acidic requirements. No surfactant was used during preparation of the films which were obtained by the deep coating of Pyrex substrate (microscope glass slide). The films thickness were measured using optical interference technique using type ZYGO-Maxim Interferometer microscope, the obtained results are given in Table 1. Film’s properties are determined by calculating reflectance and transmittance as a function of wavelength based on the values of the thickness and the n and k parameters. To measure the transmission and the reflection of the films under test, JASCO V-570 spectrophotometer, SP, was used. The SP is a double beam instrument consisting of a double monochromator of grating type. The SP is linked to a PC for data acquisition. Slit width or signal amplification gain is also controlled automatically to adjust the base line before any series of measurements. Its spectral range extends from 190 nm in ultraviolet to 2,500 nm near infra red range. A clean Pyrex substrate of the type used to hold up the films was attached to the reference slit of the SP.

Table 1 Film thicknesses-for the system KxV2O5·nH2O
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3 Results and discussion

3.1 Dispersion curves of refractive index and extinction coefficient

Recently the nanocrystalline structure [11] as well as the transport properties [12] of a series of KxV2O5·nH2O films, where (0 ≤ x ≤ 0.0091) were investigated. Current results indicated that all films are composed of highly oriented nanocrystalline rod-like shape of size ranging from 9.7 to 14.7 nm as reported recently by us in Ref. [11]. On the other hand the electrical transport properties indicate a non-adiabatic small polaron hopping conduction mechanism this together with that the system is n-type semiconductor within the temperature range investigated [12].

The optical spectral distribution curves of transmittance, T(λ), and reflectance, R(λ), for KxV2O5·nH2O films with 0.0 ≤ x < 0.01 shown in Fig. 1 have been used to compute the refractive index, n, and extinction coefficient, k, according to the following equations [13]:

$$ R = \frac{{(n - 1)^{2} + k^{2} }}{{(n + 1)^{2} + k^{2} }} $$
(1)

If k ≪ n we may put;

$$ n = \frac{1 + \sqrt R }{1 - \sqrt R } $$
(2)
Fig. 1
figure 1

Spectral distribution of a transmittance (T) and b reflectance (R) for KxV2O5·nH2O films with 0.0 ≤ x ≤ 0.0091

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While the absorption coefficient α for the films under test was calculated using the following equation [13]:

$$ \alpha = \frac{1}{d}\ln \left( {\frac{100 - R}{T}} \right) $$
(3)

where d is the film thickness. And the extinction coefficient is obtained by;

$$ k = \frac{\alpha \lambda }{4\pi } $$
(4)

Figure 2a and b show the obtained results where there was no significant difference between the forms of the curves for these films. While, n, follows the usual trend of increasing as the wave length of the incident photons decreases [14].

Fig. 2
figure 2

Spectral dependence of a the refractive index (n) and b the extinction coefficient (k) for KxV2O5·nH2O films with 0.0 ≤ x ≤ 0.0091

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3.2 Spectral distribution of the absorption coefficient α

In many amorphous materials the absorption edge at higher energy range [α > 104 cm−1] obeys Tauc’s law [13–15]:

$$ \alpha h\nu = \beta \left( {h\nu - \text{E}_{\text{op}} } \right)^{r} $$
(5)

where β is constant called band tailing parameter, hν is the incident photon energy and Eopt is the optical band gap energy. The exponent r = 1/2 or 3/2 are for direct transition depends on whether the transition is allowed or forbidden, and r = 2 or 3 for allowed and forbidden indirect transitions, respectively. In order to distinguish between those different exponents one have to know first the value of Eop. This situation may be surmounted by the prior knowledge of the imaginary part of the dielectric permittivity, ε 2 , as a function of photon energy and which is recognized as the absorption spectrum [16].

The actual value of the optical gap for the investigated films can be established by studying the absorption band profile. The absorption is best given by a plot of the complex dielectric permittivity, ε* = ε 1 –iε 2 as a function of the incident energy, hν, where generally, ε 1  = n 2 − k 2 and ε 2  = 2nk. When the value of ε 2 is plotted as a function of energy in eV, the fundamental absorption spectrum is obtained [16]. When the onset of the absorption spectrum (ε 2 vs. hν) is determined then, the value of the optical gap is obtained accordingly. Figure 3 shows the calculated values of ε 2 as a function of, E = hν, in eV for KxV2O5·nH2O (with 0.00 ≤ x ≤ 0.0091) system. In Fig. 3 it is observed that there are two independent absorption edges, indicating two different optical gaps [14]. The first edge if extrapolated to intersect with the E-axes produces the actual value of the optical gap, Eop1, while the second edge if extrapolated as well will bring the second actual optical gap, Eop2, as shown in Fig. 3. Accordingly the two different absorption edges are at (Eop1 = 0.37–0.41 eV) and (Eop2 = 2.02–2.23 eV) depending on different concentration of K-ions, respectively.

Fig. 3
figure 3

The imaginary part of the dielectric constant ε2 as a function of photon energy (hν) for KxV2O5·nH2O for K different content

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As the optical gaps were determined the usual procedure for determining the value of, r, is by applying Eq. 5 which involves plotting a graph of (αhν) 1/r against hν as shown in Fig. 4a and b [13–15]. The optical band gap can be determined by extrapolation of the linear parts of the obtained curves as shown in Fig. 4a and b, while β can be calculated on the other hand from the slope of the linear parts of the curves [13–15]. It is clearly found that the obtained optical gap changes with changing the value of the exponent, r, as presented for example in Tables 2 and 3. The obtained linear regression coefficient doesn’t as well facilitate a good indication of the value of r, which may be considered. To avoid this uncertainty one should apply the values of the optical gaps introduced from the absorption spectrum, (ε 2 vs. hν), of Fig. 3 explained above. This indicates that KxV2O5·nH2O films have more than one type of conduction mechanism. The linear fitting to the data at the band edge and intersecting with the hν axis at the same energy which is coincide with those obtained from the absorption spectrum, will give the value of r as well as the optical gap as shown in Fig. 4a and b.

Fig. 4
figure 4

Optical gaps as produced by extrapolating the linear parts of the absorption edge according to Eq. 5. a EOpt2 with the exponent r = 3/2, b EOpt1 with the exponent r = 1/2 for KxV2O5·nH2O films with different K content

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Table 2 Representation of the obtained values of the optical gap EOpt1 at different values of the exponent r for KxV2O5·nH2O system with 0 ≤ x < 0.01
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Table 3 Representation of the obtained values of the optical gap EOpt2 at different values of the exponent r for KxV2O5·nH2O system with 0 ≤ x < 0.01
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Applying the results obtained from the absorption band (ε 2 vs. E of Fig. 3), it is noted that the first high energy band gives, EOpt2, is in the range 2.02–2.27 eV as given in Table 4. This gap is characterized by a direct forbidden transition with r = 3/2, see Fig. 4a. The result for V2O5·nH2O sample is in good agreement with results that obtained in [17]. These results can be compared with those obtained for sputtered V2O5 thin films in [18] and other works [19–22]. Furthermore, the second region for hν < 0.6 eV is best presented by an absorption band with a gap of EOpt1 around 0.4 eV and r = 1/2, see Fig. 4b. This low energy absorption is associated with the intra band transitions involving deep impurity states within the energy gap and/or to a second gap in the phase space of lower energy and is described by a direct allowed transition. Figure 5 shows the obtained results for EOpt1 and EOpt2 as a function of composition of KxV2O5·nH2O with 0.00 ≤ x < 0.01.

Table 4 Optical parameters as calculated, EOpt is the optical gap, Ee is band tail width
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Fig. 5
figure 5

The obtained results for EOpt1 and EOpt2 as a function of composition KxV2O5·nH2O with 0 ≤ x ≤ 0.0091

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Apparently, it should be noted that any variation in the band gap might be associated with the level of ordering in KxV2O5·nH2O films. Where when, r = 3/2, in a direct transition no phonons are created or adsorbed during the transition itself. Therefore, this may be true in amorphous or glassy materials where phonons concept is due to the ordered structure of crystalline solids. Additionally, if one compare the electrical conductivity results from Ref. [12] and the present optical gap, it is clear that EOpt1 = 2W dc ; here W dc is the activation energy of electronic conduction; indicating an intrinsic like semiconducting state. Moreover, from plotting ln α as a function of, hν, as shown in Fig. 6, values of, Ee; (known as the Urbach tail and which is the band tail width); as a function of composition can be calculated and given in Table 4, where generally [13, 15];

$$ \alpha = \alpha_{\text{o}} \exp (h\nu /{\text{E}}_{\text{e}} ) $$
(6)

Ee representing the exponential band tail which may take place from static spatial fluctuations in potential arising from the lack of long range order or charged defects centers [16].

Fig. 6
figure 6

Plots of ln α as a function of photon energy (hν) for films of different K content

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3.3 Determination of carrier concentration and the electron effective mass

The obtained data of refractive index n can be further analyzed to obtain the high frequency dielectric constant ε∞ according to the following procedure [7]. This procedure is describing the contribution of the free carriers and the lattice vibrationl modes of the dispersion. According to Pankove [13]:

$$ \varepsilon_{1} = \varepsilon_{\infty } - \frac{{e^{2} N_{t} }}{{\pi \in_{\text{o}} m^{*} c^{2} }}\lambda^{2} $$
(7)

where ε 1 is the real part of the dielectric constant, ε ∞ is the lattice dielectric constant, λ is the wavelength, e is the electronic charge, N t is the free-charge carrier concentration, ∈ o is the permittivity of the free space, m* is the effective mass of charge carriers, c is the velocity of light.

It is observed that the dependence of ε 1 on λ 2 is linear at longer wavelengths, as shown in Fig. 7. Extrapolating the linear part to intersect the Y axis will gives the values of ε ∞ and from the slopes of these lines we can calculate the values of N/m* for the investigated films. Values of ε ∞ and N/m* are given in Table 5. Applying the previously reported values of the xerogel films carrier concentration, N t , [12] it is possible to calculate the charge carriers effective mass relative to the free electron rest mass, m e , as presented in Table 5. The obtained values of the optical relative dielectric permittivity ε ∞  = n 2o are of the same order with those obtained by Chopra et al. [15] for V2O5-TeO2 films as well as for xerogel system of the composition LixV2O5·nH2O [23].

Fig. 7
figure 7

Real part of the dielectric constant as a function of λ2 for different compositions KxV2O5·nH2O with 0 ≤ x ≤ 0.0091

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Table 5 The obtained results of charge carrier concentration and effective mass of the present system
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4 Conclusions

In the present work, highly orientated nanocrystalline films with excellent properties can be prepared by sol–gel technique, where:

  1. (1)

    Optical measurements revealed that two optical gaps are present.

  2. (2)

    EOp1 with exponent r = 1/2 suggesting a direct allowed transition with an evaluated optical gap from 0.37 to 0.42 eV.

  3. (3)

    EOp2 with exponent r = 3/2 suggesting a direct forbidden transition with an evaluated optical gap from 2.02 to 2.23 eV.

  4. (4)

    The width of the localized states (band tail) Ee were estimated from the slopes of ln α & hν plots of the samples and have values in the range 0.45–0.80 eV.

  5. (5)

    Electronic effective masses have been calculated in the range 0.42–0.55 m e , indicating a relatively light Fermions.