1. Introduction

Zinc oxide (ZnO) is a very important material for many applications like microelectronics, short-wavelength light-emitting devices, lasers, field emission devices, solar cells, and gas sensors^{[1, 2]}. It is a near-stoichiometric n-type semiconductor with low resistivity values of ZnO films that may be adjusted between 10$^{-3}$ and 10$^{-4}$ $\Omega$$\cdot$cm by changing the annealing conditions and doping^[3]. ZnO is a Ⅱ-Ⅵ compound semiconductor; most of the group Ⅱ-Ⅵ binary compound semiconductors crystallize in either cubic zinc blende or hexagonal wurtzite (Wz) structure. It has a large exciton binding energy about 60 meV at room temperature and has a wide band gap energy 3.37 eV^[4-7].

ZnO thin films can be doped with a variety of semiconductors to meet the demands of several application fields. Stoichiometric ZnO films are highly resistive. Conducting films can be made either by creating oxygen vacancies, which act as donors or by doping with various dopants such as Ga$^{3+}$, Mn$^{4+}$, Al$^{3+}$, In$^{3+}$, Co$^{2+}$, N$^{1+}$ and Fe$^{3+}$^[8-13]. Many attempts were reported about doped ZnO films, but most of them were related to Al doping. Now the use of dopants such as Co, In or Al can enhance the optical and electrical conductivity of ZnO. However, these dopants were studied in our published papers, usually the experimental results^{[1, 30-34]} to evaluate a correlation for the optical gap energy and electrical conductivity. Moreover, the doped thin films have been used in various applications such as transparent electronics, ferromagnetism, semiconductors, piezoelectric devices, gas sensors and transparent electrode window layer of thin film solar cells^[8-15]. The films (ZnO:Al) are considered to be a material of utmost importance due to its high conductivity, good transparency and lower cost.

The aim of this work is to study the possibility of the correlation between the optical and electrical properties of ZnO thin films with precursor molarity and doping level. Wojtczak et al.^[16] presented a new approach to the description of the density of states for thin films which allows us to take into account the broadening of spectral lines, and were the first to conclude that the formulation of the CPA approximation for thin films allows us to include the boundary conditions in the natural way and consider them without any additional approximations; secondary investigation is described for the same boundary conditions given by various crystallographic orientations of the copper surfaces, the Friedel oscillations of the electronic density appear. Tudose et al.^[17] studied the correlation of ZnO thin film surface properties with conductivity. There is a limited amount of literature dedicated to systematic and detailed studies on the surface evolution with growth parameters and their effect on the ZnO surface conductivity under reduction/oxidation. However, Benramache et al.^{[18, 19]} studied the correlation for crystallite size in undoped ZnO thin film with the bandgap energy--precursor molarity--substrate temperature; we found that the correlation between the structural and optical properties suggests that the crystallite sizes of the films are predominantly influenced by the band gap energy of the thin films. The above discussion has been about ZnO, but the same works investigated the dependence of physical properties of ZnO thin film as a function of parameter conditions such as temperature, thickness, oxidizing conditions, nitrogen addition and doping for characterizing the thin films^[20-29].

In this paper, we aim to present a new approach to calculate the electrical conductivity by varying the optical gap energy, precursor molarity and doping level of doped ZnO thin films. Detailed calculations are developed from doped ZnO thin films with Al, Co and In.

2. Methods and model

The ZnO, ZnO:Al, ZnO:Co and ZnO:In samples were deposited on glass substrates by ultrasonic spray technique at a substrate temperature of 350 ℃. The optical gap energy and electrical conductivity of the films were measured with the doping level in our published papers, where we have studied the effect of various parameters on the ZnO thin films such as doping level, growth times, substrate temperature and annealing temperature^{[1, 30-34]} (see Table 1).

First, the correlation between the electrical and optical properties of undoped ZnO thin films was studied with precursor molarity as expressed in our published paper^[35]; second, we aim in this study to investigate a correlation with the doping concentration in doped films, in which the precursor molarity of ZnO is fixed at 0.1 M.

The correlation between the electrical and optical properties was studied for the electrical conductivity ($\sigma )$ as a function of the band gap energy ($E_{\rm g})$, precursor molarity $M$ and doping level $X_{0}$ of doped ZnO thin films resulting from the following equation:

$\begin{equation} \label{eq1} \begin{cases} \sigma _{(\ast )} =\dfrac{\sigma _{\rm (e)} }{\sigma _{\rm (e)Max} }, \\[4mm] E_{\rm g(\ast )} =\dfrac{E_{\rm g(e)} }{E_{\rm g(e)Max}}, \\[4mm] M_{(\ast )} =\dfrac{M_{\rm (e)} }{M_{\rm (e)Max} }, \\[4mm] X_{0(\ast )} =\dfrac{X_{\rm 0(e)} }{X_{\rm 0(e)Max} }, \\ \end{cases} \end{equation}$

(1)

where $\sigma _{\rm (e)}$, $E_{\rm g(e)}$, $M_{\rm (e)}$ and $X_{\rm 0(e)}$ are the experimental data; $\sigma _{\rm (e)Max}$, $E_{\rm g(e)Max}$, $M_{\rm (e)Max}$ and $X_{\rm 0(e)Max}$ are maximal experimental values and $\sigma _{(\ast )}$, $E_{\rm g(\ast )}$, $M_{(\ast )}$ and $X_{0(\ast )}$ are the first values that have been calculated in the correlation relationships. This model was used to delete the form units of the electrical conductivity, optical gap energy and Al concentration. The model proposed of pure and doped ZnO thin film was discussed.

3. Results

We have estimated the relationships between the electrical conductivity and the band gap energy with the precursor molarity in undoped ZnO thin films, in our published paper^[35]. We have obtained the following empirical relationships:

$\begin{equation} \label{eq2} \sigma _{\rm (c)} =a b^{E_{\rm g(\ast )} } M_{(\ast )} ^{\rm c}, \end{equation}$

(2)

where $\sigma _{\rm (c)}$ is the correlated electrical conductivity; $a$, $b$ and $c$ are empirical constants as $a\approx$ 0.0003105, $b\approx$ 5697 and $c\approx$ 2.761.

In the investigation of the correlation with the doping concentration, the doped ZnO thin films were deposited at a precursor molarity of 0.1 M; on the other hand the correlation was based on Eq. (1) using the constant parameters $a, b$ and $c$ in all doped films. The correlation can be written in another form:

$\begin{equation} \begin{split} \label{eq3} \sigma ={}& d\left(3.105\, \times\, 10^{-4}\, \times\, 5697^{E_{\rm g(\ast )} }\, \times \, M_{(\ast )} ^{2.761}\right) \\[1mm]& \times\, [1\, +\, \ln (1\, -\, eX_0)], \end{split} \end{equation}$

(3)

where $d$ and $e$ are empirical constants, as well as measurement supposed these parameters were related by dopants and doping concentration. These parameters are collected in Table 2 and estimated as a function of dopant element. Table 3 presents the correlated values.

4. Discussion

In this study, we will show the evolution of the doping level and optical gap energy on the electrical conductivity, and we try to establish correlations for each model proposed. In our calculations the electrical conductivity of doped ZnO thin films are calculated from Eq. (3); the ZnO exhibiting single crystals are n-type semiconductors with a high transparent conducting.

As shown in Figs. 1, 2 and 3, significant correlation was found between the electrical conductivity and the optical gap values of the doped ZnO thin films as a function of Al, Co and In concentrations, respectively. The measurement in the electrical conductivity of doped films by Eq. (3) is agreed with the experimental data, the error of this correlation is smaller than 13%, which can be calculated from the relationship $\left| {(\sigma _{\rm Exp}\, -\, \sigma _{\rm Corr} )/\sigma _{\rm Exp} } \right|\times$ 100. The minimum error value was estimated in the cobalt-doped ZnO thin films (see Fig. 4). Thus the result indicates that such Co-doped ZnO thin films are chemically purer and have fewer defects and less disorder owing to an almost complete chemical decomposition and contained higher optical band gap energy.

The maximum enhancement of the electrical conductivity was found to be minimum error at doping level smaller than 3 wt.% (see Fig. 4). The amount of Al, Co and In doping contents achieved in doped ZnO film is smaller than 3 wt.%. Based on that, the experimental and correlation values for the electrical conductivity were developed, and good agreement was found between the calculated and experimental values.

In our calculations the electrical conductivity characterizes the doped ZnO thin films; stoichiometric-doped ZnO films are highly transparent, and have good optical band gap and a high electrical conductivity.

We have estimated the electrical conductivity of the undoped and doped ZnO thin films by varying the optical band gap with doping concentrations; it is predominantly influenced by the transition tail width of undoped and doped films. The correlation between the electrical conductivity and the band gap was investigated.

5. Conclusion

In summary, high-quality, transparent, pure and doped zinc oxide thin films with aluminum, cobalt and indium were deposited by ultrasonic spray technique on glass substrate at 350 ℃. The correlations between the electrical and optical properties with dopants' concentration of Al, Co and In were investigated. In this paper we have presented a new approach to the description of correlation between electrical conductivity and optical gap energy by varying the concentration of dopants of Al, Co and In. The electrical conductivity of the films is predominantly estimated by the band gap energy and the concentration of Al, Co and In. The measurement in the electrical conductivity of doped films with correlation is equal to the experimental value in the range between 0 and 3 wt.%, here the error is smaller than the region higher than 3 wt.%. The minimum error value was estimated in the cobalt-doped ZnO thin films. Thus the result indicates that such Co-doped ZnO thin films are chemically purer and have far fewer defects and less disorder owing to an almost complete chemical decomposition.