1 Introduction

Correlated electrons in vanadium oxides are responsible for their unique structural, electrical, optical and magnetic properties. Their electronic band structures are affected by crystallographic structure, crystal field splitting and hybridization between O2p and V3d bands. There have been many experimental and theoretical studies of the band structure of the main vanadium oxides—V2O3, VO2 and V2O5. The first experimental studies on band structure were based on optical spectroscopy that utilized absorption and reflection of light. Ceramic materials, single crystals and thin films were the subjects of these studies. Lately, new methods such as photoemission spectroscopy (PES), X-ray absorption or emission spectroscopy (XAS, XES) [1], X-ray reflectivity (XRR), X-ray fluorescence (XRF) [2], photoluminescence (PL) [3, 4], Raman scattering and scanning tunnelling microscopy (STM) [5] have been used.

Theoretical calculations used several quantum mechanics models such as the Hartree–Fock self-consistent field method based on one-electron approximation, the Hubbard–Mott model [6] introducing the effects of electron correlations on the Hamiltonian, Peierls mechanism [7, 8] involving electron–phonon interactions, or the density function theory (DFT) [9].

One of the most important parameters with regard to the properties of materials is the bandgap energy (Eg). Generally, the Eg of a semiconductor or an insulator has been found to decrease with increasing temperature. The variation of the fundamental Eg with temperature is very important for both basic science and technological applications.

2 Electronic structure

The theoretical basis of optical properties results from Maxwell’s equations. From optical spectra, the complex dielectric function ε(ω) is derived [10]:

$$\varepsilon \left( \omega \right) = \varepsilon_{1} - i\varepsilon_{2}$$
(1)

where ω is angular frequency of light (ω =2πν), ε1 and ε2 represent real and complex parts of ε, and i is the imaginary unit.

$$\varepsilon _{1} = ~n^{2} - \kappa ^{2}$$
(2)

and

$$\varepsilon _{2} = 2n\kappa$$
(3)

where n and κ are real and imaginary parts of the refractive index.

Measurement of light absorption is one of the most important techniques used to determine the optical properties of solids. In absorption measurements, the intensity of light (I(d)) after it has travelled through a certain thickness of a material is compared with the incident intensity (Io), thereby defining the absorption coefficient (α):

$$I\left( d \right) = I_{o} { \exp }\left[ { - \alpha \left( \omega \right)d} \right]$$
(4)

The dependence of the absorption coefficient on frequency is shown in Fig. 1. Since I(d) depends on the square of the field variables, it immediately follows that

Fig. 1
figure 1

Frequency dependence of the absorption coefficient (α) near the threshold for interband transition

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$$\alpha \left( \omega \right) = 2\frac{{\omega ~\kappa }}{c} = 4~\frac{{\pi \kappa }}{\lambda }$$
(5)

where the factor of 2 results from the definition of α(ω) in terms of light intensity, which is proportional to the square of the optical fields. This expression entails that the absorption coefficient is proportional to ~ к(ω), the imaginary part of the complex index of refraction (extinction coefficient), so that к is usually associated with power loss.

It can be concluded from Eqs. (2), (3) and (5) that either set—(ε1, ε2) or (n, α)—represents the wavelength-dependent constants characterizing the optical properties of a studied solid. Many efforts have been devoted to the determination of the refractive index (n), but there is so far no universal approach. Several methods of calculating the refractive index of vanadium oxide [11,12,13,14,15,16,17,18] and other materials [19,20,21,22,23] have been proposed. Various assumptions were used in these methods, and the results are often subjective/debatable and ambiguous.

2.1 Optical properties V2O5-literature survey

The electronic structure of vanadium pentoxide has been the subject of intensive studies [24,25,26,27,28,29,30,31,32,33,34]. Various theoretical calculations of V2O5 band structure include both semi-empirical and ab initio techniques.

Lambrecht et al. [24, 25] presented a calculation of the energy band structure using a tight-binding model in which the oxygen p-bands and vanadium d-bands were decoupled. They applied a perturbation approximation in order to obtain an effective Hamiltonian for the valence and conduction bands separately. The theoretically determined dispersion of the energy bands was verified by applying electrical transport properties. The valence band density of states was compared with XPS (X-ray photoelectron spectroscopy) and SXS (soft X-ray spectroscopy) data. The valence-to-conduction band transitions were compared with optical and electron energy loss (ELS) data. A satisfactory agreement between theoretical and experimental data was found [25].

Kempf et al. [26] reported on pseudo potential periodic Hartree–Fock calculations on a V2O5 crystal. The determined V-O bond lengths and stretching force constants were found to be in good agreement with experimental data. The estimated band structure and density of states remain in contrast with tight-binding calculations. There is no gap between the conductor and valence bands. According to the authors, vanadium pentoxide is partially ionic.

Bullett [27] determined the electronic structure of vanadium pentoxide using direct and non-empirical atomic orbital techniques. He postulated an indirect semiconducting energy gap of 2.6 eV.

Eyert and Höck [28] computed the band structure of bulk vanadium pentoxide using the density-functional theory (DFT) and local-density approximations (LDA). Its electronic properties were modified via strong hybridization between O 2p and crystal-field-split V3d orbitals. A strong deviation of VO6 octahedra from the cubic coordination led to a narrow split-off conduction band.

The electronic structure of vanadium pentoxide is strongly connected with its anisotropy, which in turn is associated with its crystal structure. The atoms form double chains within planes that are separated by a van der Waals gap.

Kenny et al. [29] studied the optical absorption coefficients of V2O5 single crystals using incident polarized light with wavelengths in the range of 0.47–1.8 μm and unpolarized light with wavelengths from 1.5 to 7.5 μm. Fundamental absorption was observed at incident photon energies of 2.15, 2.22 and 2.17 eV for E║a, E║b and E║c, respectively. Some evidence for a direct forbidden transition mechanism with bandgaps of 2.36 and 2.34 eV was observed for E║a and E║c, respectively. The most notable property of V2O5 is its ability to produce monolayers (or materials only several layers thick) [30]. Vanadium pentoxide is the second material known to exhibit such a property. The first one was graphite forming a single monolayer, known as graphene. Chakarbarti al. [31] determined the V2O5 monolayer band structure using ab initio density-functional theory (DFT). The obtained results are in excellent agreement with experimental crystallographic data as well as with other experimentally determined surface properties [32, 33].

Tolhurst et al. [34] studied a double-layered polymorph of V2O5 (named ε′-V2O5) using soft X-ray spectroscopic measurements and density-functional theory calculations. This polymorph has increased interlayer separation, which leads to a dramatic increase in the bandgap. Table 1 summarizes the representative literature data [28, 34,35,36,37] of the band gap and its temperature dependence.

Table 1 Bandgap (Eg) of V2O5 and its temperature dependence—dEg/dT
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Figure 2 shows the refractive index for vanadium pentoxide thin films as a function of wavelength [13, 38]. Parameter n decreases with wavelength. This dependence may be verified using the theoretical equation proposed by Cauchy [39]:

Fig. 2
figure 2

Refractive index (n) determined for V2O5 thin films [13, 38]

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$$n\left( {{\lambda }} \right) = A + \frac{B}{{\lambda^{2} }}$$
(6)

where A and B are independent of λ. According to Fig. 3, the experimental points are consistent with the theory postulated by Cauchy.

Fig. 3
figure 3

Cauchy’s plot of the refractive index (n) for V2O5 thin films [13, 16, 38]

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The analysis of the dependence of the absorption coefficient on light frequency is very significant from the viewpoint of the semiconducting properties of vanadium oxide and its subsequent areas of application. Generally, the frequency dependence of the absorption coefficient (α(ω)) is rather different for various physical processes which occur during the interaction of light with the solid. In particular, the following cases can be observed [10]:

  1. 1.

    Free carrier absorption

  2. (a)

    typical semiconductor

$$\alpha \left( \omega \right) \sim \omega^{ - 2}$$
(7)
  1. (b)

    metals at low frequencies

$$\alpha \left( \omega \right) \sim \omega^{{\frac{1}{2 }}}$$
(8)
  1. 2.

    Direct interband transition (conservation of crystal momentum)

  2. (a)

    allowed transition

$$\alpha \left( \omega \right) \sim \frac{{(\hbar \omega - E_{g} )^{{\frac{1}{2}}} }}{\hbar \omega }$$
(9)
  1. (b)

    forbidden transition

$$\alpha \left( \omega \right) \sim \frac{{(\hbar \omega - E_{g} )^{{\frac{3}{2}}} }}{\hbar \omega }$$
(10)
  1. 3.

    Indirect interband transition (change in crystal momentum)

  2. (a)

    allowed transition

$$\alpha \left( \omega \right) \sim \frac{{(\hbar \omega - E_{g} \pm \hbar \omega_{\text{phonon}} )^{2} }}{\hbar \omega }$$
(11)
  1. (b)

    forbidden transition

$$\alpha \left( \omega \right) \sim \frac{{(\hbar \omega - E_{g} \pm \hbar \omega_{\text{phonon}} )^{3} }}{\hbar \omega }$$
(12)

The ħωphonon factor is generally omitted in Eqs. (11) and (12) because of the fact that phonon energy is several times lower than the energy of electron transition. Equations (9)–(12), known as Tauc equations, are applied to determine the bandgap (Eg) of semiconductors.

Figure 4 shows the mechanisms of electron interband transition for direct and indirect semiconductors. A direct transition corresponds to the photon–electron interaction process in which the k-vector does not change. The crystal momentum of electrons and holes is the same in both the conduction band and the valence band. In an indirect transition photon, electron and phonon of the lattice take part. This process is accompanied by a change in the k-vector. The allowed transitions remain in agreement with particular selection rules, assuming a dipole model. On the other hand, if this model is not valid, the transition is called forbidden. More complex models can then be taken into account (involving, for instance, a magnetic dipole, electric quadrupole, etc.).

Fig. 4
figure 4

Direct and indirect mechanisms of electron interband transition

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According to [29, 37], the edge is direct and forbidden. Diffuse reflectance spectra [40] give an Eg of 2.31 eV at room temperature, but the band edge has been determined to be direct and Mousavi et al. [41] observed that for V2O5 films prepared by means of spray pyrolysis Eg changes with the substrate temperature (Tsub). When Tsub increases, the Eg decreases gradually from 2.46 to 2.22 eV.

Kang et al. [5] studied the interband transition in a V2O5 film deposited via RF magnetron sputtering using absorption and photoluminescence spectral measurements. Transmission measurements indicate two distinct interband transitions, implying indirect and direct transitions.

2.2 Optical properties of V2O5 thin films: experimental results

Vanadium pentoxide thin films were deposited by means of reactive radio frequency sputtering. Deposition conditions of the thin films and their properties such as structure, morphology were described in detail elsewhere [42]. Table 2 summarizes characterization of the films used in the studies.

Table 2 Deposition conditions for a series of vanadium oxide thin films and their properties
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Optical transmittance and reflectance spectra were measured over a wide wavelength range from 180 to 3200 nm with a Lambda 19 Perkin-Elmer double beam spectrophotometer equipped with a 150 mm integrating sphere. Thin films from Series 1, 2 and 3 were the subject of spectrophotometric studies. Each of Figs. 5, 6 and 7 shows the reflectance (R), transmittance (T) and absorbance (A) spectra recorded for one of the thin films. The transparency region of vanadium pentoxide is limited by the fundamental absorption edge at ca. 500 nm. The reflectance values vary in the range of 10–20% for the Series 1 sample (mostly amorphous), in the range of ca. 0–10% for the Series 2 and 3 samples (crystalline). The observed non-monotonic plots of R, T and A may result from additional absorption bands due to the departure from stoichiometry [11, 12].

Fig. 5
figure 5

Reflectance (R), transmittance (T) and absorbance (A) spectra recorded for a Series 1 V2O5 thin film

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Fig. 6
figure 6

Reflectance (R), transmittance (T) and absorbance (A) spectra recorded for a Series 2 V2O5 thin film

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Fig. 7
figure 7

Reflectance (R), transmittance (T) and absorbance (A) spectra recorded for a Series 3 V2O5 thin film

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The absorption coefficient (α) and photon energy were determined from Figs. 5, 6 and 7 using the following equations:

$$\alpha = \frac{1}{d}\ln \frac{1 - R}{T}$$
(13)
$$E_{\text{photon}} \left[ {eV} \right] = \hbar \omega = \frac{1240}{{\lambda \left[ {nm} \right]}}$$
(14)

where d represents film thickness.

One of the crucial parameters used to evaluate a semiconductor’s properties is the bandgap (Eg). The bandgap of a semiconductor can be determined from experimentally measured transmittance T and reflectance R within the range of fundamental absorption using the following Tauc equation:

$$(\hbar \omega \alpha )^{{\frac{1}{n}}} = A\left( {\hbar \omega - E_{g} } \right)$$
(15)

where the A coefficient is constant and n, according to Eqs. (9)–(12), assumes values ½, 3/2, 2 and 3 for direct allowed (DA), direct forbidden, indirect allowed and indirect forbidden transitions, respectively.

Figures 8, 9, 10 and 11 illustrate the absorption coefficient data experimentally determined for the Series 2 thin film in the coordinate system of (αħω)1/n versus ħω for n = ½, 3/2, 2 and 3, respectively.

Fig. 8
figure 8

Tauc plot (n = 2 − corresponding to a direct allowed transition) for a Series 2 V2O5 thin film

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Fig. 9
figure 9

Tauc plot (n = 3/2 − corresponding to a direct forbidden transition) for a Series 2 V2O5 thin film

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Fig. 10
figure 10

Tauc plot (n = 2 − corresponding to an indirect allowed transition) for a Series 2 V2O5 thin film

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Fig. 11
figure 11

Tauc plot (n = 3 − corresponding to an indirect forbidden transition) for a Series 2 V2O5 thin film

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The bandgap was determined by extrapolating the linear part of the best fit of (αħω)1/n vs. ħω to αħω = 0. The values of Eg were obtained from the ħω axis intercepts. Similar plots were computed for the other two V2O5 samples: Series 1 and Series 3. The results are presented in Table 3.

Table 3 Tauc plot results
Full size table

The analysis of spectrometric results presented in Figs. 8, 9, 10 and 11 as well as in Table 3 suggests that V2O5 thin films undergo both direct and indirect transitions. However, it is not possible to decide which type of the electron interband transition is predominant in this case. Based on the following evidence,

  • agreement with theoretical band calculations [24,25,26, 30],

  • agreement with recent reports on single crystals [26, 43],

  • agreement with recent experimental studies based not only on

    spectrophotometric measurements such as photoluminescence

    and ellipsometry [17],

the direct allowed (DA) transition can be considered the most probable. The available literature on the bandgap of V2O5 thin film is vast. The impact of the following factors affecting the bandgap of films based on vanadium pentoxide has been studied:

  • thin film deposition technique [44,45,46] (Table 3),

  • anisotropy [28, 29, 31, 47]

  • film thickness [34]

  • substrate type [47]

  • non-stoichiometry [13, 48, 49]

  • UV irradiation [48]

  • chemical composition [50]

  • temperature [17, 43, 44, 47]

  • morphology [13, 30, 34, 45]

The results are presented in Table 4.

Table 4 Summary of findings concerning the bandgap in vanadium pentoxide where DA, DF, IA and IF represent direct allowed, direct forbidden, indirect allowed and indirect forbidden transitions, respectively
Full size table

Figure 12 illustrates the Tauc plots corresponding to ħω energy between 2.97 and 3.23 eV. The best agreement with the theoretically predicted dependence is observed for the direct allowed (DA) transition. The results obtained by applying the least squares method are listed in Table 5.

Fig. 12
figure 12

Tauc plots for a Series 2 V2O5 thin film and the ħω range of 2.97–3.23 eV

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Table 5 Results of calculations of the Tauc plot for the Series 2 V2O5 thin film and the ħω range of 2.97–3.23 eV, obtained using the least squares method
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2.3 Optical properties of V2O3

Based on papers [52,53,54,55], the most significant calculations were reported [56, 57]. The main controversy has been over the ordering of the components of the trigonally split t2g band. Several band schemes have been suggested [58,59,60]. The V3+ ions in V2O3 have a 3d2 electronic configuration. These vanadium ions in the metallic phase (corundum) occupy two-thirds of the octahedral sites formed by oxygen anions. A trigonal distortion causes the splitting of the t2g orbital into a non-degenerate a1g and a doubly degenerate \({e^{\text{II}}}_{\text{g}}\) orbital [61]. Castellani et al. [57] suggested the formation of a molecular bond between the a1g orbitals of V–V pairs spread into a band. This model is consistent with many experimental results [62,63,64,65]. However, taking into account the fact that the c/a lattice parameter ratio is too high for a1g to couple between two V cations, Ivanov [66] and Ezhov et al. [58] contradicted the postulated model of the molecular orbital. Shinna et al. [67] assumed a strong hybridization of the V pair.

The electronic properties of V2O3 are strongly dependent on oxygen stoichiometry.

A slight variation in oxygen concentration changes the effective mass [68, 69].

V2O3 is treated as the model system used to study the MIT in a correlated electron system (TMIT= 160 K).

2.4 Optical properties of VO2

The calculation of the electronic structure of VO2 has been the subject of intensive research involving many models such as the cluster type [70, 71], tight-binding type [72,73,74], or augmented-plane-wave (APW) [75, 76], as well as energy band studies using Bloch functions in a linear combination of atomic orbitals [77].

Gavini et al. [78] determined the real (n) and imaginary (κ) parts of the refractive index. Studies of the absorption coefficient performed by Gavini et al. and Merenda et al. [79] revealed that the electronic structure at E < 1.8 eV can be attributed to d-d transitions with a threshold at 0.6 eV. At 1.82 eV, and the threshold for O2p–V3d transitions is observed, with peaks at 2.64 eV and 3.56 eV.

The temperature of MIT for bulk single-crystal VO2 is 541 K [80]. Below TMIT, VO2 exhibits a monoclinic structure with the P21/c space group in which the partially filled d-band is split into an unoccupied part pushed past the π* band and the filled part of the d-band. Above the TMIT, VO2 transforms to a tetragonal (rutile) phase with the partially filled d-band located at the Fermi level and the material is metallic [81]. Jiang et al. [82] studied the optical properties of vanadium dioxide thin films deposited under different oxygen partial pressures via reactive magnetron sputtering. The bandgap decreased from 339.6 to 319.4 K. The near-infrared extinction coefficient (k) and optical conductivity increased with decreasing oxygen partial pressure.

3 Conclusions

The electronic structure of the three main vanadium oxides (V2O3, VO2 and V2O5) was reviewed. The optical properties of vanadium pentoxide thin films were determined. It was found that the direct allowed (DA) transition is the most probable type observed in the case of the studied films.