1 Introduction

Tin oxide is a functional material used in many applications, such as transparent conducting electrodes in solar cell, gas sensor, electrochromic devices, etc. Tin oxide is a crystalline solid with a tetragonal lattice structure similar to that of rutile. It is a n-type semiconductor material with direct wide band gap energy around 3.62 eV. Tin oxide has been doped with different chemical elements, such as F, In, Sb, Fe, and many others, in order to improve different physical properties [1, 2]. Fluorine doped tin oxide (FTO) is one of the most popular transparent conducting oxide (TCO) whose low resistivity strongly depends of the high carrier concentration produced by oxygen vacancy and the quantity of substitution fluorine. FTO thin film has been prepared by different experimental techniques, such as reactive sputtering, chemical vapor deposition (CVD), electrospun and spray pyrolysis (SP). The latter is a simple and cost effective technique applicable for large deposition area and with a good quality thin film as resulted [3,4,5,6,7].

In order to use FTO as a TCO semiconductor material for optoelectronic applications, accurate determination of several important parameters, such as complex permittivity, refraction index (n), extinction coefficient (k) and electrical resistivity is a fundamental requirement [8,9,10,11,12,13]. In the case of optical properties of semiconductors, important research works have been realized in order to find theoretical expressions [14]. Thus, the Drude model together with others optical models have been utilized to simulate experimental UV–Visible spectra of some TCO materials. For example, Drude and Lorentz models have been used to explain transmittance and reflectance spectra of CVD tin oxide (TO) and antimony-doped TO thin films [15]. In similar form, Drude and Kim oscillator models have been employed in the study of SP palladium-doped TO thin films [16]. Also, a combination of Drude and Forouhi–Bloomer models has been used to simulate the transmittance spectra of magnetron sputtered and electron beam evaporated indium tin oxide (ITO) thin films [17].

In the case of electrical response of TCO semiconductor materials, most of the research works invoke Bloch–Grüneisen (BG) law to explain the resistivity temperature dependence behavior [18]. Several mathematical treatments had been applied on BG resistivity equation in order to find semi-empirical relationship between integral constant and Debye temperature in noble materials as a function of temperature [19]. Thus, an electrical resistance study of thin ITO films has been described by the BG law with high two dimensional electron–electron interaction at very low temperature [20]. Also, CVD fluorine doped TO thin film has been found that its resistivity caused by the thermal phonons obeys BG model [21].

In this paper FTO thin film on glass substrate has been prepared by spray pyrolysis technique and discussed its optical and electrical properties, for first time, in terms of the Drude-Lorentz model, that to the best of our knowledge it is also the first time that both properties of a TCO are fitted simultaneously. While the FTO temperature dependence electrical behavior is analyzed in terms of Mathiessen’s rule [18, 22]. In addition, a first principle calculation has been realized in order to understand, from a theoretical point of view, the experimental results from Density Functional Theory (DFT) approach. Finally, the FTO film was evaluated as a transparent electrode in a dye-sensitized solar cell.

2 Theory and calculation

The Drude-Lorentz theory formalisms that have been found to describe TCO optical and electrical properties can be shown to yield the following relationship for the frequency dependence for the complex permittivity [15]:

$$\varepsilon \left( \omega \right)=~{\varepsilon _\infty } - \frac{{\omega _{p}^{2}}}{{{\omega ^2}+i\gamma \omega }}+\mathop \sum \limits_{k} \frac{{\omega _{{pk}}^{2}}}{{\omega _{{ok}}^{2} - {\omega ^2} - i{\gamma _k}\omega }}$$
(1)

where \({\varepsilon _\infty }~\)represents the high-frequency dielectric constant and the parameters \({\omega _{pk}},{\omega _{ok}}~\)and \({\gamma _k}\) are the plasma frequency, the transverse frequency and the damping constant respectively of the k-th Lorentz oscillator. In the Drude model \({\omega _p}\) and \(\gamma\) have similar meaning as those of Lorentz contribution. So that the complex permittivity can be written as [14]:

$$\varepsilon \left( \omega \right)=~{\varepsilon _1}\left( \omega \right)+i{\varepsilon _2}\left( \omega \right)$$

As it is known, this expression can be related to different optical parameters, such as the complex refraction index and the complex optical conductivity. Also, several dc electrical parameters can be related to the Drude’s dielectric function contribution. Thus, the dc resistivity can be calculated with the expression [16]:

$$\rho =\frac{\gamma }{{2\pi c{\varepsilon _o}\omega _{p}^{2}}}$$

where c is the light speed and \({\varepsilon _{\text{o}}}\) is the free space permittivity. Also, the carrier concentration (n) and the mobility (µ) can be obtained by the following relations [16]:

$$n=\frac{{4{\pi ^2}{c^2}{\varepsilon _o}{m^*}}}{{{e^2}}}\omega _{p}^{2}$$

and

$$\mu =\frac{e}{{2\pi {m^*}c\gamma }}$$

where e is electron charge and m* = 0.28me (me the rest electron mass) is the effective mass reported for FTO [23]. Thus, the electron mean free path (\(\ell\)) can be calculated using n and µ.

In addition, the film degeneracy can be evaluated through n by calculating the Fermi energy position relative to the conduction band minimum. It is called the Burstein–Moss shift \(\Delta {E_{BM}}\) [24].

3 Experimental details

The precursor materials used to prepare fluorine doped tin oxide were dehydrate stannous chloride (SnCl2‧2H2O) and ammonium fluoride (NH4F). A starting solution was obtained by mixing the precursors with distilled water and hydrochloric acid (HCl) of the following form: 6.0 g of SnCl2‧2H2O was dissolved in 90 ml distillated water. After, 12 ml of HCl and 10 wt% NH4F are added to the solution and it is stirred for 1 h. The final result is a solution completely homogeneous and transparent.

This solution was sprayed onto a heat glass plate using a semi-automated homemade spray pyrolysis system, see Fig. 1. The deposition temperature was 500 °C for 10 min and a source-substrate distance of 30.0 cm. After deposition, the film was allowed to cool to room temperature.

Fig. 1
figure 1

Semi-automated home-made spray pyrolysis system

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The surface morphologies of the film were analyzed at room temperature using a Scanning Electron Microscope (SEM) Zeiss Evo 40 vp. For optical characterization, an UV–Visible spectrometer model V-670 from Jasco was utilized in wavelength range of 200–1100 nm. The transmittance spectrum was fitted using ReFfit Program code [25, 26].

The sheet resistance measurements were performed by four-point Van der Pauw method. Silver epoxy paste was employed to ensure good electrical contacts. A temperature dependence study of the electrical response was realized using a Peltier cell in temperature range from 278 to 403 K.

4 Results and discussion

Figure 2a presents the SEM micrograph of the FTO thin film. It is compact and adherent in nature. Homogeneous and well define grains can be observed, which indicates the polycrystalline nature of the sample. The inset figure shows the grain size histogram obtained through a standard image analysis process. Thus, a grain size distribution from 16 to 380 nm was found, with an average value of 155 nm. The X-ray diffraction pattern observed to the FTO film (Fig. 2b) presents bands associated to planes whose Miller´s indices are (110), (101), (200), etc. They are all characteristics of the well known tetragonal-rutile structure. The obtained rutile phase is comparable with the standard SnO2 (TO) material, Card No. 88-0287 from JCPDS data. It corresponds to a material structure with P42/mnm space group. But, it is expected that the presence of fluorine atoms must break the structural symmetry. However, a similar XRD diagram is found between TO and FTO materials. It is due to the very close ionic radii between oxygen (1.36 Å) and fluorine (1.30 Å) with an atom coordination number of three.

Fig. 2
figure 2

a SEM micrograph of FTO thin film deposited on glass substrate. b XRD diffraction pattern of FTO thin film deposited on glass substrate

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Figure 3 shows the optical transmittance spectrum of FTO on glass substrate. The well-developed fringes pattern in transmittance indicates that the film has a smooth surface with well define air-FTO and FTO-glass substrate interfaces [27]. In addition, the good interference behavior shows that the film is morphologically homogeneous, and this is in agreement with the scanning electron microscope picture obtained of the sample. As shown in Fig. 3, the sample exhibit sharp absorption edges at 290–350 nm, indicating the high quality of the crystallites. The average transmission in the visible region has been found to be 81.2%.

Fig. 3
figure 3

Measured and simulated FTO transmittance spectra. The reflectance spectrum is obtained using the dielectric function fitting parameters

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This FTO/glass spectrum is fitted using Drude-Lorentz model, which is a multilayer’s classical dispersion theory. Thus, Fig. 3 shows the fitted transmittance spectrum for the FTO sample; where the Van der Pauw conductivity measurement has been considered a condition in the fitting process, see Fig. 4. A very good result was obtained and the Table 1 presents the fitted spectrum parameters. It can be observed, in Fig. 4, that the real part optical conductivity presents four (4) bands with maximum at 0 (0 eV), 24,061 cm−1 (3.0 eV), 29,214 cm−1 (3.6 eV) and 32,315 cm−1 (4.01 eV). These features also appear, as expected, in others optical parameters and they will be discussed in the next paragraph. In addition, Fig. 3 shows the theoretical FTO reflectance spectrum obtained using the fitted parameters, which it is in perfect agreement with the experimental spectrum reported in the literature [3, 28]. So, the frequency dependence of the dielectric function and many others optical properties can be determined completely in the UV–Visible range.

Fig. 4
figure 4

Electrical conductivity of FTO thin film. The open dot point in zero (0) wavenumber corresponds to DC conductivity value measured by Van der Pauw technique, and it is a fitting condition. The solid line is the conductivity frequency dependency obtained by the dielectric function parameters

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Table 1 Fitting parameters of FTO thin film using Drude-Lorentz model. It includes the film thickness d
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Figure 5 shows the real and imaginary parts of the dielectric function for FTO layer. Three main features are observed in this figure. (1) At low energy, around 0–2.0 eV, the sharp increase and decrease for the real and imaginary parts respectively is due to the intra-band transitions of electrons in the conduction band and close to the Fermi level, corresponding to a metal-like behavior and described by the Drude model. (2) The increase at high energies, around 4.01 eV is related to the optical band gap transitions [15], and (3) The two soft increases at 3.0 and 3.6 eV are described by Lorentz oscillators and they seem to be related to a trap created inside of the band gap and close to the conduction band, as a consequence of oxygen vacancies [29]. In order to prove this, without the presence of fluorine, a TO thin film sample was prepared with the same FTO spray pyrolysis condition, where a high oxygen vacancy concentration is known to take place when TO is prepared as a TCO material. Table 1 shows the Drude-Lorentz fitting parameters of TO thin film, where two soft features lower to the optical band gap, around 18,461 cm−1 (2.29 eV) and 24,817 cm−1 (3.07 eV), can be observed. Thus, the 3.0 eV found in FTO thin film clearly corresponds to electron transition from the valence band to the defect level (trap) inside of the band gap due to oxygen vacancies. If this transition involves the VBM, then the defect level is found around to 0.62 eV under the conduction band minimum. In this approach, the 3.6 eV band can also be explained as a transition from deeper valence band (0.6 eV under VBM) to the defect level.

Fig. 5
figure 5

Real and imaginary dielectric functions of FTO thin film

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Table 2 shows the calculated resistivity, carrier concentration, mobility, mean free path, optical band gap and relative Fermi level using the fitted dielectric function parameters. A resistivity value of 1.15 × 10−3 Ω-cm is obtained, which it is in good agreement with 1.14 × 10−3 Ω-cm found experimentally. A carrier concentration of 1.70 × 1020 cm−3 and a mobility of 32.05 cm2/Vs are obtained and they are similar to those with good electrical properties reported in the literature [30]. The FTO film shows a mean free path of 36.20 Å. It is considerably shorter than the 155 nm average grain size measured by SEM. This indicates that the mobility is limited by scattering of the ionized fluorine atoms and others bulk structural defects rather than scattering of the grain boundary. The calculated relative Fermi energy (Burnstein–Moss effect) is 0.40 eV, which is higher than the room temperature thermal energy (KT = 0.025 eV). It evidences the degenerate nature of the FTO film. Thus, the 4.01 eV optical band gap corresponds to electron transition from valence band to lowest unoccupied conduction band. This optical transition energy is given by the following expression [24]:

Table 2 FTO electrical and optical parameters calculated using Drude model
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$$E_{g}^{o}=~{E_g}+~\Delta {E_{BM}} - \Delta {E^{BGN}}$$

where Eg corresponds to the intrinsic TO band gap, with experimental value reported in the literature around 3.62 eV. While ΔEBGN is the band gap shrinkage usually results from scattering against ionized impurities and the correlated motion of charged carriers. The results seem to indicate that the ΔEBGN term is very small (around 0.01 eV). So that, the well documented renormalization of the host band gap in degenerate doping semiconductor seems not to take place practically. It suggests that most of the scattering process is due to bulk structural defect, such as vacancy and atoms in non-equilibrium position, than due only to fluorine impurity. However, it seems contradictory with the amount of fluorine used in the starting solution. So that, it probably indicates that the presence of fluorine in the FTO thin film is much less than that in the precursor solution, as it has also been observed by other researchers [4, 31, 32].

Figure 6 shows the electrical resistivity behavior of the FTO film from 278 to 403K. According to Mathiessen’s rule the electrical resistivity in many materials can be expressed as a sum of the residual and the temperature-dependent resistivity contributions.

Fig. 6
figure 6

Temperature dependence of dc electrical resistivity in FTO thin film. The dash line is the fitting using BG equation. The solid line is the fitting using the polynomial ρ(T) = ρ(0) + αT2 + βT5

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$$\rho \left( T \right)=~\rho \left( 0 \right)+~{\rho _{el - ph}}\left( T \right)+{\rho _{el - el}}\left( T \right)+ \ldots ~$$
(2)

where ρ(0) corresponds to the residual resistivity, ρel−el(T) is the electron–electron interaction, and ρel−ph(T) is the electron–phonon interaction given by BG model [18, 22].

$$~{\rho _{el - ph}}\left( T \right)=\beta T{\left( {\frac{T}{{{\theta _D}}}} \right)^4}\mathop \smallint \limits_{0}^{{\frac{{{\theta _D}}}{T}}} \frac{{{x^5}dx}}{{({e^x} - 1)(1 - {e^{ - x}})}}$$
(3)

where \(x=\hbar \omega /{K_B}T\), β is a constant and θD is the Debye temperature.

It can be observed that no good fitting is obtained using Eq. (2), with the two first terms, and Eq. (3). So that, the electrical responses do not obeys the BG model on this temperature range as it has been found in others metal oxide materials. However, a good result is found with the expression ρ(T) = ρ(0) + αT2 + βT5, where α and β are constants. The second term can be associated with electron–electron interaction, while the third term is the electron–phonon interaction for temperature higher than Debye temperature. Thus, the little variation of the electric resistivity with the temperature indicates that the residual contribution, which is temperature independent, is the most important term. It corresponds to an electron–defect interaction. This result is in perfect agreement with the small electron mean free path value obtained before.

In order to understand from a theoretical point view the inclusion of F atoms in TO structure a DFT first-principles approach was implemented within the Quantum Espresso computer code [33]. The hybrid functional PBE0 is adopted for describing the exchange-correlation interactions [34]. Figure 7a shows the electronic density of state (DOS) of dielectric TO with well defined direct band gap (Eg = 3.62 eV), where the Fermi level is found in the valence band maximum (VBM) as expected. This means that the conduction bands are completely unoccupied and the material exhibits electrical insulation property. This DOS result is in perfect agreement with the experimental low energy high resolution XPS measurement [35], which indicates that our DFT theoretical approach is correct. Thus, Fig. 7b shows the DOS for a FTO sample with fluorine concentration of 8.33 at.%. It can be observed that the Fermi level is in the conduction band region with Burstein-Moss shift of 3.47 eV, band gap of 0.33 eV and optical band gap of 5.24 eV. So, the theoretical sample (TS) exhibits TCO’s properties. Also, inside the conduction bands a band gap like-behavior can be observed with energy value around 2.00 eV. A complete FTO theoretical study (not shown here) indicates that the band gap and the band gap like behavior decrease and increase with increasing F content, respectively. Thus, inter-band transitions from occupied to unoccupied conduction bands can take places. Therefore, transition bands with energy lower than the optical band gap are possible with introducing F content in TO structure. So, they can also contribute to the two lorentzian oscillators corresponding to transitions at 3.0 and 3.6 eV, obtained in UV–Visible experimental measurements and discussed above only in term of oxygen vacancies.

Fig. 7
figure 7

a Electronic density of state (DOS) of TO by DFT first principle calculation. b DOS of FTO structure with 8.33 at.% fluorine concentration. All the energy values are relatives to the Fermi level (EF). The positive DOS value (black) corresponds to spin up structure, while the negative DOS (red) corresponds to spin down structure. (Color figure online)

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Also, the FTO sample was utilized as an electrode in the preparation of a dye-sensitized solar cell (DSSCs). For fabrication of DSSCs, the FTO film was coated with a compact 800 nm-TiO2 film by spray pyrolysis method and after with a 4 µm-TiO2 nanoparticles layer using Doctor Blade technique. The electrode was immersed in the commonly used dye-N719 (0.2 mM) with soaking time of 24 h. The photovoltaic parameters indicate a short-circuit current density of Jsc = 12.3 mA/cm2, an open-circuit voltage of Voc = 0.60 V, a fill factor of ff = 63.5%, a power conversion efficiency of η = 4.7% and a quantum efficient QE = 43% at 535 nm (see Fig. 8).

Fig. 8
figure 8

Density of current–voltage curve for FTO film utilized as electrode in the dye-sensitized solar cells (DSSCs). Inset the quantum efficient spectrum

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5 Conclusions

Thin film of FTO was prepared by spray pyrolysis technique. The real and imaginary dielectric functions show an intra-band transition in the conduction band corresponding to free-like electron, which it is modeled with Drude theory. Thus, the calculated relative Fermi energy (Burnstein–Moss effect) reveals the degenerate nature of the FTO film. While the electron mean free path indicates that the electron mobility is limited by scattering of the ionized fluorine atoms and others bulk structural defects rather than scattering of the grain boundary.

Also, a 4.01 eV feature in the UV–Visible spectrum is related to optical band gap associated to direct inter-band transition, from the valence band to the lowest unoccupied conduction band, out of the Brillouin zone center. Additionally, two optical transitions at 3.0 and 3.6 eV associated to oxygen vacancies are obtained using Lorentz model. They seem to correspond to transitions from VBM and deeper valence band to a defect level close to the conduction band, respectively. The density functional theory shows that inter-band transitions from occupied conduction band to unoccupied conduction band are possible, and they can contribute to the two optical transitions mention before. In addition, a power conversion efficiency of η = 4.7% is found in a dye-sensitized solar cell when the FTO thin film is used as a transparent electrode.

It can be noted that no empirical process is utilized to analyze the optical and electrical measurements of FTO thin film at room temperature, as it is found in many others papers. An analytical classical dispersion theory has been used in this study for obtaining optical and electrical parameters simultaneously. After, a first principle calculation, through DFT method, is realized to SnO2 and FTO material in order to explain optical transitions with its respective electronic bands structure.