1. Introduction

Metal oxide nanoparticles attract considerable interest due to their potential applications in many important fields of science and technology such as gas sensors,magnetic phase transitions,catalysts and superconductors^{[1, 2]}. CuO has attracted much interest because of being the basis for several high temperature superconductors. It has a monoclinic crystal structure and presents p-type semiconductor behaviour with an indirect band gap of 1.21-1.51 eV. Copper oxide nanomaterials have the advantage of a lower surface potential barrier than that of the metals,which affects electron field emission properties. It also plays an important role in optoelectronics and solar cell devices^[3]. In recent years,there has been increasing interest in developing nanostructured metal oxides with p-type semiconductivity. Copper oxide,as one of the relatively few metal oxides that tend to be p-type,has been widely exploited for diverse applications such as heterogeneous catalysts,lithium ion electrode materials,and field emission (FE) emitters^[4].

Electrical properties of nanomaterials differ from those of bulk materials due to an increased number of interfacial atoms and a large number of defects at the grain boundaries. The ac conductivity measurements have been widely used to investigate the nature of defect centers in different materials. The defect centers are responsible for conduction in amorphous materials^{[5, 6]}. The surface of nanomaterials is similar to amorphous materials with lots of defects. These defects in nanomaterials have a major role in ac conduction. Various theoretical models such as quantum mechanical tunneling (QMT)^[7],correlated barrier hopping (CBH)^[8] and overlapping large polaron tunneling^[9] are available for explaining the conduction mechanism in amorphous semiconductors. Many works are available for the ac response studies on the basis of the above models^[10]. In the CBH model the conduction process is described by the hopping of charge carriers between defect centers over the potential barrier separating them. The height of the barrier is correlated with separation $R$ of the defect centers. Pike introduced this model for the dielectric studies of scandium oxide films^[11]. Later the CBH Model was extended by Elliot to explain conduction in chalcogenide glasses. According to Elliot,the correlated barrier hopping of bipolarons occurs between the D$^{-}$ and D$^{+}$ centers at low temperatures. The D° centers are formed at the expense of D$^{+}$ and D$^{-}$ centers at high temperatures causing single polaron hopping. In CuO nanoparticles oxygen vacancies act as defect centers. The ac conductivity of CuO nanoparticles is influenced by the volume fraction,thickness,and the heterogeneity of grain boundaries. Very few researchers have studied the ac conductivity of metal oxide such as ZnO and CuO^{[12, 13, 14]} even though a methodical investigation of the ac conduction mechanism in CuO nanoparticles is still lacking. We present here the temperature and frequency dependence of ac conductivity of CuO nanoparticles and the mechanism involved in the conduction process. The analysis of the ac conductivity measurements reveals that the CBH is the dominant conduction mechanism in CuO nanoparticles. The PL spectrum of CuO nanoparticles confirms the presence of defect centres,which are responsible for the conduction in CuO nanoparticles.

2. Experimental

Highly purified copper acetate and glacial acetic acid were used for the preparation of CuO nanoparticles. The synthesis was started with the mixing of 300 mL of 0.02 M copper acetate aqueous solution with 1 mL glacial acetic acid in a 500 mL beaker. Heating of the resulting solution followed by the rapid addition of about 0.8 gm of NaOH until the pH value of the mixture reached 6-7. A black precipitate is formed as a result of the reaction. After being cooled to room temperature,the precipitate was centrifuged and washed well with distilled water and then with ethanol^[15]. The resulting product was then dried in air at room temperature and crushed well before being used for the analysis. X-ray diffraction studies were carried out by a Bruker AXS D8 advance X-ray diffractometer employing Cu-K$\alpha$ radiation. The morphology of the product was examined by a Jeol/JEM 2100 transmission electron microscope. The obtained powder was consolidated into the form of cylindrical pellets each of diameter 10 mm and a thickness of 1 mm by applying a pressure of 4.5 tons for 3 min using a hydraulic press. Silver pellets were used as electrodes. The ac conductivity measurement was carried out using a HIOKI 3532-50 LCR meter in the temperature range of 373-573 K and within the frequency range of 50 Hz-1 MHz. The PL-spectrum was recorded using an LS 45,Perkin Elmer fluorescence spectrometer with a Xenon lamp as the excitation source.

3. Results and discussion

The morphology and size of the nanoparticles were analyzed by the TEM images presented in Figures 1(a) and 1(b). Figure1(a) shows that the product consists of nanoparticles with a spherical shape. It is found that the average diameter of CuO nanoparticles is 10 nm. The SAED patterns are indexed as shown in Figure1(b) confirming the monoclinic structure of the samples.

The XRD pattern of CuO nanoparticles is shown in Figure2,which reveals the formation of single phase CuO nanoparticles with a monoclinic structure and the diffraction data are in good agreement with the JCPDS card of CuO (JCPDS 80-1268). The broadening of the peak indicates the small size of the CuO nanoparticles. The average crystallite size of the copper oxide nanoparticles is calculated using Scherrer's equation^[16] $L=\frac{0.9\lambda }{\beta \cos \theta}$ where $L$ is the average crystallite size,$\lambda$ is the X-ray wavelength (1.5405 Å),and $\beta$ is full width at half maximum in radian. The average crystallite size of CuO nanoparticles is 8 nm,which strongly agrees with that obtained from TEM observations.

The optical characterization of CuO nanoparticles was done by using PL spectra. The PL spectrum helps to understand the surface defects of CuO nanoparticles. Figure3 shows the PL spectrum of CuO nanoparticles using the Xenon lamp as the source of excitation at a wavelength of 400 nm. The emission peak at 535 nm is due to the band edge emission. The emission at 600 nm and 652 nm corresponds to the defect states in CuO nanoparticles. The peak at 600 nm is due to the oxygen vacancy and that at 652 nm is due to the multiple oxidation state of copper^{[17, 18, 19]}.

Figure4 shows the frequency dependence of the total conductivity of CuO nanoparticles at different temperatures. It is clear from Figure4 that the total conductivity increases with the increase of frequency and the temperature. The conductivity dispersion is very low in the high frequency region. The conduction process in a CuO nanoparticle is controlled by two types of hopping mechanisms^[20]. First is the hopping of charge carriers from one defect potential well to the adjacent defect potential well,which is inter well hopping. Second is the hopping between the holes/electrons within one defect potential well,which constitutes the intra well hopping.

The presence of ac signal in CuO nanoparticles,does have a finite probability for the occurrence of both types of conduction mechanisms. However,the relative probabilities depend on many factors such as the energy of charge carriers,the frequency of the applied ac signal,temperature,etc. Hence the measured conductivity includes a frequency dependent part,$\sigma (\omega )$ and a frequency independent part,$\sigma$(dc).

In general the frequency dependence of the total conductivity can be expressed as

where $\sigma$(total) is the total conductivity and $\sigma (\omega )$ is due to intra-well hopping and $\sigma$(dc) is due to inter well hopping. It is observed that a linear relationship exists between the ac conductivity and the angular frequency $\omega$,which obey a power law in the form

where $A$ is a constant,which depends on temperature and `$s$',the power law exponent,which depends on temperature and frequency and its value is generally less than or equal to unity. $s$ is defined by

The variation of total conductivity of CuO nanoparticles as a function of temperature at different frequencies is shown in Figure5(a). It is clear from the figure that the total conductivity gradually increases at lower temperatures and rapidly increases at higher temperatures. The reason behind this rapid increase is that the number of charge carriers increases at higher temperatures due to thermal agitation.

The activation energy ($E_{\rm A})$ for the total conduction can be calculated using the Arrhenius plot as shown in Figure5(b). $E_{\rm A}$ for $\sigma$(dc) is 0.25 eV,which is higher than that of $\sigma (\omega )$ in which the former is due to inter-well hopping. This is because,the charge carriers need more energy to hop between one defect potential well to the adjacent defect potential well. At higher frequencies the activation energy of $\sigma (\omega )$ is almost equal to that of the total conductivity while at lower frequencies,it is very close to that of $\sigma$(dc). The reason for this is that at higher frequency ranges $\sigma$(total) is mostly contributed by intra-well hopping of charge carriers,which is well explained by the CBH model in the later sections. Also at higher frequencies the DC contribution can be neglected. Hence the activation energy for the most contributing ac conduction is almost equal to that of the total conduction in the higher frequency region. At lower frequencies the major contribution is by the inter-well hopping of charge carriers and the DC conduction takes a major role.

Table1 shows the variation of activation energy at different frequencies for total conductivity and ac conductivity. The variation of ac conductivity $\sigma (\omega )$ as a function of frequency at different temperatures is shown in Figure6. The ac conductivity of CuO nanoparticles shows a similar behavior to that of the total conductivity. The pure ac component of the conductivity tends to dominate at higher frequencies and/or at lower temperatures. This is due to the fact that under these conditions more charge carriers hop between adjacent potential wells. The possibility of inter well hopping is very low in this region and also the temperature dependence of this type of conduction is greater than that of intra-well hopping. The frequency exponent $s$ is determined by fitting the experimental data using Equation (2) as shown in Figure6. The data reveals that the value of $s$ decreases as temperature increases. For explaining the conduction mechanism of CuO nanoparticles,it is very necessary to take advantage of various theoretical models. These distinctive theoretical models include QMT,hopping over barrier (HOB) and CBH. The QMT model points to the motion of the carriers,which occurs through tunneling between two localized states near the Fermi level. However,in the CBH model the charge carriers hop over the potential barrier between two charged defect states. Three types of carriers are distinguished in the QMT process such as electrons,small polarons,and large polarons.

According to the QMT model,the expression for the frequency exponent $s$ is given by

The QMT model predicts the temperature independence of $s$,which is not observed for CuO nanoparticles. Hence it cannot be used for the analysis. OLPT is another conduction mechanism in which `$s$' varies according to the following equation:

where $\beta =1/K_{\rm B}T$. Here $s$ should be both temperature and frequency dependent. The frequency exponent $s$ is predicted to decrease from unity with an increase in temperature. The obtained values of $s$ appear to be in disagreement with the OLPT model.

In the CBH model the frequency exponent $s$ can be evaluated as

Thus the CBH model predicts a temperature and frequency dependent exponent $s$,which increases towards unity as the temperature tends to zero. Here the overlapping of the Coulomb well of the neighboring sites separated by $R$,results in the lowering of the effective barrier height from $W_{\rm m}$ to a value $W$,which for the case of two electrons hopping is given by^{[20, 21, 22, 23, 24]}

where $W_{\rm m}$ is the maximum barrier height,$n$ is the number of electrons in the hopping process,$n =$ 1 for single polaron hopping and $n =$ 2 for bipolaron hopping,$\varepsilon^{\prime}$ the dielectric constant of the material,and $\varepsilon_0$ is that of free space.

In CuO nanoparticles the variation of $s$ with temperature shows the same variation as that of the CBH model. Moreover,Deepthi et al. in 2013 reported the conductivity mechanism of CuO nanostructures based on correlated barrier hopping^[23]. However,no one has studied the role of surface defects in the ac conduction of CuO nanoparticles in depth. In the present work,the correlated barrier hopping in CuO nanoparticles is verified and studied by analyzing the ac conductivity data thereby confirming the existence of defect states in them. According to this model the ac conductivity is

where $N$ is the concentration of a pair of sites,and $R_{\omega }$ is the hopping distance given by

The obtained $s$ value as a function of temperature is presented in Figure7. Theoretical fit to the experimental data using Equation (6) is also plotted. From the figure,it is clear that the value of $s$ decreases as the temperature increases. Also the fit appears to be reasonably good throughout the temperature range under consideration. $W_{\rm m}$ and $\tau_0$ are the fitting parameters and their values are obtained by fixing $\omega$ as 10$^{5}$ Hz. The value of $W_{\rm m}$ is 0.77 eV and that of $\tau_0$ is 5.07 $\times$ 10$^{-11}$ s. Using the best fitted value of $W_{\rm m}$ and $\tau_0$,the value for hopping distance $R_{\omega}$ is obtained as 2.3 Å. Bipolaron hopping is reported in many amorphous and doped semiconductors. In nanomaterials one would expect the interior of the grain to be perfectly crystalline and hence devoid of all defects. However,due to the large surface to volume ratio the surface of the nanoparticles have a high density of defects,vacancies,interstitials etc. The presence of Jahn-Teller effects in CuO proved the existence of bipolarons in CuO^{[26, 27]}. In CuO nanoparticles hopping between positive and negative defect centers (e.g. V$_{\rm O}$,V$_{\rm Cu})$ are responsible for the bipolaron conduction mechanism.

Figure8 shows the fitted curve of experimental data using Equation (8). It is clear from the figure that a reasonable fit for bipolaron hopping is obtained up to 463 K. When temperature increases beyond this temperature a deviation from bipolaron hopping occurs. At high temperature,the density of neutral defects increases due to the conversion of D$^{+}$ and D$^{-}$ centers in to D$^{0}$ centers,which in turn enhances single polaron hopping as predicted by Shimakawa^{[28, 29]}. The analysis of the experimental data is in good agreement with the bipolaron conduction up to 463 K and after this particular temperature single polaron hopping is the predominent conduction mechanism. The density of defect centers at various temperatures are calculated using Equation (9) and are shown in Table2.

The concentration of defect density $N$ of CuO nanoparticles decreases as the temperature increases. The large number of D$^{0}$ centers is responsible for the high conductivity in CuO nanoparticles at high temperature. More activation energy is needed for this process in this region and hence this process is activated only at high temperatures. In this region holes are hopping between D$^{0}$ and D$^{+}$ centers and electrons between D$^{0}$ and D$^{-}$ centers. Hence we conclude that,for CuO nanoparticles,bipolaron hopping is the predominant conduction mechanism up to 463 K and beyond this temperature single polaron hopping is dominant.

4. Conclusions

The ac conduction mechanism of CuO nanoparticles was successfully analyzed in the frequency range of 50 Hz $\times$ 1 MHz and in the temperature range of 373-573 K. The ac conductivity has been investigated in the light of the various theoretical models and it was found that the correlated barrier hopping model was the most appropriate conduction mechanism for explaining the temperature and frequency dependence of the ac conductivity in CuO nanoparticles. The activation energy for DC conductivity was found to be less than that of ac conductivity. The theoretical analysis using the CBH model gave reasonable values for maximum barrier height and characteristic relaxation time as 0.7998 eV and 5.60267 $\times$ 10$^{-11}$ s respectively. The PL spectrum of CuO nanoparticles supports the occurrence of the defect centers which contribute to the ac conduction process. The study also calculated the density of charged defect centers which showed a decrease with an increase in temperature. It was also found that the bipolaron conduction mechanism was prominent at lower temperatures and single polaron hopping was dominated at very high temperatures where the density of neutral defect centers was increased.

Acknowledgements

The authors gratefully acknowledge the financial support of Kerala State Council for Science,Technology and Environment (KSCSTE). The authors thank the Central Instrumentation Facility (CIF),St Berchmans College for the analysis carried out.