Transport Mechanisms and Dielectric Features of Mg-Doped ZnO Nanocrystals for Device Applications

Magnesium-doped zinc oxide “ZnO:Mg” nanocrystals (NCs) were fabricated using a sol gel method. The Mg concentration impact on the structural, morphological, electrical, and dielectric characteristics of ZnO:Mg NCs were inspected. X-ray diffraction (XRD) patterns display the hexagonal wurtzite structure without any additional phase. TEM images revealed the nanometric size of the particles with a spherical-like shape. The electrical conductivity of the ZnO NCs, thermally activated, was found to be dependent on the Mg content. The impedance spectra were represented via a corresponding circuit formed by a resistor and constant phase element (CPE). A non-Debye type relaxation was located through the analyses of the complex impedance. The conductivity diminished with the incorporation of the Mg element. The AC conductivity is reduced by raising the temperature. Its plot obeys the Arrhenius law demonstrating a single activation energy during the conduction process. The complex impedance highlighted the existence of a Debye-type dielectric dispersion. The various ZnO:Mg samples demonstrate high values of dielectric constant with small dielectric losses for both medium and high-frequency regions. Interestingly, the Mg doping with 3% content exhibits colossal dielectric constant (more than 2 × 104) over wide temperature and frequency ranges, with Debye-like relaxation. The study of the electrical modulus versus the frequency and at different temperatures confirms the non-Debye relaxation. The obtained results reveal the importance of the ZnO:Mg NCs for device applications. This encourages their application in energy storage.


Introduction
Zinc oxide (ZnO) has been the subject of rapid development in the past decades in order to meet different research requirements. Essentially, its low-cost eco-friendly nature is behind its success. This explains the significant demand of ZnO material in several technological fields such as: solar cells [1], gas sensing [2], and light-emitting devices [3]. Indeed, ZnO material owns a wide band gap energy (≈3.37 eV) as well as a broad extinction binding energy (≈60 meV) at room temperature [4,5]. It is also distinguished with its notable dielectric characteristics, particularly, an elevated dielectric constant and a minor dielectric loss [6].
In this regard, the dielectric properties of ZnO are extremely attractive to researchers. Previous studies reveal the importance of developing materials with a high dielectric constant that ensures better energy storage in devices. Such materials have been widely applied to modern wireless communication technology [7] and electronic components such as capacitors. Similarly, fabricating material with low dielectric loss guarantees

Synthesis of ZnO:Mg NCs
The ZnO:Mg NCs preparation technique was provided in our earlier study [23]. Briefly, zinc acetate dehydrate "Zn(CH 3 COO) 2 . 2H 2 O; 0.02M" and magnesium acetate tetrahydrate "Mg (CH 3 COO) 2 . 4H 2 O", with a purity of 99.99%, were used as starting materials. Citric acid (C 6 H 8 O 7 ) was used as a stabilizer and deionized water was used as a solvent. The synthesis process could be resumed as follows: 400 mL of the solvent was used to dissolve the precursors. The mixture was stirred for 2 h at 50 • C and pre-heated at 80 • C until the solvent was completely removed. Then, we proceeded to the second thermal treatment at 350 • C for 3 h. The obtained powders were crushed and annealed at 500 • C for another 3 h to achieve the crystallized phase. The thermal process was performed in two steps in order to ensure well-organized crystalline phases and reduce the amount of impurity in each sample. The prepared powders are denominated as ZnO:Mg x% where x = 0, 1, 2, 3 and 5, representing the percentage of [Mg]/ [Zn].The uniaxial press of 10 tons/cm 2 was used to press the crystallized powders in order to obtain disk pellets of 1 cm diameter and a thickness "e" of 2 mm. The fabricated pellets were investigated during impedance measurements.

Experimental Study
XRD measurements were accomplished with a Philips X'Pert system (Malvern, UK), using a copper X-ray tube (λ = 1.54056 A • ) at room temperature, at 40 kV and 100 mA. The obtained diffractometer in the angle domain was between 30 • to 70 • with a step size of 0.02 • . The transmission electron microscopy (TEM) images were performed with a Philips CM30 microscope (Malvern, UK). The pellets were calcined and put among two platinum electrodes in a furnace. We proceeded to take the measurements by varying the temperature for each sample with a step of 10 • C from 200 to 260 • C. The study of the dielectric properties at different temperatures was conducted by gathering complex impedance data in the frequency domain between 40 Hz to 5 MHz with an impedance analyzer (Agilent 4294A, USA). x = 0, 1, 2, 3 and 5, representing the percentage of [Mg]/[Zn].The uniaxial press of 10 tons/cm 2 was used to press the crystallized powders in order to obtain disk pellets of 1 cm diameter and a thickness "e" of 2 mm. The fabricated pellets were investigated during impedance measurements.

Experimental Study
XRD measurements were accomplished with a Philips X'Pert system (Malvern, UK), using a copper X-ray tube (λ = 1.54056 A°) at room temperature, at 40 kV and 100 mA. The obtained diffractometer in the angle domain was between 30° to 70° with a step size of 0.02°. The transmission electron microscopy (TEM) images were performed with a Philips CM30 microscope (Malvern, UK). The pellets were calcined and put among two platinum electrodes in a furnace. We proceeded to take the measurements by varying the temperature for each sample with a step of 10 °C from 200 to 260 °C. The study of the dielectric properties at different temperatures was conducted by gathering complex impedance data in the frequency domain between 40 Hz to 5 MHz with an impedance analyzer (Agilent 4294A, USA).  The ZnO structure remains unchanged after the Mg incorporation with low content. This affects the intensity and the wideness of the peaks. Indeed, the full width at half maximum of the peaks along the (101) plane decreased with Mg doping. The FWHM of the doped sample is reduced compared with that of a pure ZnO, highlighting the improvement of the crystalline structure. On other hand, the crystallite size increases with the incorporation of Mg, which was found to be greater than that of the undoped sample.   The ZnO structure remains unchanged after the Mg incorporation with low content. This affects the intensity and the wideness of the peaks. Indeed, the full width at half maximum of the peaks along the (101) plane decreased with Mg doping. The FWHM of the doped sample is reduced compared with that of a pure ZnO, highlighting the improvement of the crystalline structure. On other hand, the crystallite size increases with the incorporation of Mg, which was found to be greater than that of the undoped sample. Such an enhancement of the crystallite size is assigned to the Zn 2+ substitution with Mg 2+ . The obtained values are gathered in Table 1. The TEM images of non-doped ZnO and ZnO:Mg (3%) NCs are given in Figure 2. A homogenous structure with a spherical shape could be observed on the fabricated NCs. The mean particle size is about 30 nm, which is in good agreement with the XRD analysis. Moreover, it is clear that the nanocrystal shape remains the same after doping. However, it is difficult to judge if the size decreases with Mg doping. Such an enhancement of the crystallite size is assigned to the Zn 2+ substitution with Mg 2+ .

Structural and Morphological Studies
The obtained values are gathered in Table 1. The TEM images of non-doped ZnO and ZnO:Mg (3%) NCs are given in Figure 2. A homogenous structure with a spherical shape could be observed on the fabricated NCs. The mean particle size is about 30 nm, which is in good agreement with the XRD analysis. Moreover, it is clear that the nanocrystal shape remains the same after doping. However, it is difficult to judge if the size decreases with Mg doping.

Electrical Study
a. Impedance Spectra The complex impedance plots of undoped ZnO at various temperatures is given in Figure 3. We also studied the complex impedance changes at 200 °C for the various Mg percentages (see, Figure 4). Both figures illustrate the well-defined semicircles in the complex plane. This confirms the homogeneity and the single phase of the prepared doped samples. The centers of the semicircles are localized below the Z′ axis, which may be related to a non-Debye relaxation nature [30,31]. Compared with pure ZnO, the semicircles of Mg-doped ZnO NCs appeared at higher temperatures. Here, the increase in the Mg concentration enlarges the semicircle radius. However, the sample doped with 3% of Mg displays an opposite tendency. The diminution in the radius of the semicircle illustrates the enhanced of conductivity in the sample [32]. 3.2. Impedance Spectroscopy 3.2.1. Electrical Study a. Impedance Spectra The complex impedance plots of undoped ZnO at various temperatures is given in Figure 3. We also studied the complex impedance changes at 200 • C for the various Mg percentages (see, Figure 4). Both figures illustrate the well-defined semicircles in the complex plane. This confirms the homogeneity and the single phase of the prepared doped samples. The centers of the semicircles are localized below the Z axis, which may be related to a non-Debye relaxation nature [30,31]. Compared with pure ZnO, the semicircles of Mg-doped ZnO NCs appeared at higher temperatures. Here, the increase in the Mg concentration enlarges the semicircle radius. However, the sample doped with 3% of Mg displays an opposite tendency. The diminution in the radius of the semicircle illustrates the enhanced of conductivity in the sample [32].   The investigation of the impedance plot requires modeling of the electric impedance with a corresponding circuit including one block consisting of a parallel association of the bulk resistance R and a capacitor.
In this case, the capacitor could be substituted by a constant phase element (CPE) ZCPE because of the semicircles depression [33]. ZCPE is given by:    The investigation of the impedance plot requires modeling of the electric impedance with a corresponding circuit including one block consisting of a parallel association of the bulk resistance R and a capacitor.
In this case, the capacitor could be substituted by a constant phase element (CPE) ZCPE because of the semicircles depression [33]. ZCPE is given by:  The investigation of the impedance plot requires modeling of the electric impedance with a corresponding circuit including one block consisting of a parallel association of the bulk resistance R and a capacitor.
In this case, the capacitor could be substituted by a constant phase element (CPE) Z CPE because of the semicircles depression [33]. Z CPE is given by: Here, j denotes the imaginary unit, ω corresponds to the angular frequency, A 0 is the capacitance value of Z CPE , and n is non-dimensional parameter (0 ≤ n ≤ 1) reflecting the trend to an exact semicircle [34].
The total impedance Z * is described as follows: where The suggested model is used to fit the Nyquist plots, proving the viability between the equivalent circuit and the existing system (see Figures 3 and 4). Table 1 lists the extract parameters. The capacitances, A g , is sized up at 10 −10 F cm −2 S n−1 , demonstrating that the grain boundary contribution is formed as a general response of the system [35,36]. The slight difference in ionic radius between Zn and Mg can generate a small displacement in the ZnO structure, since the Mg substitutes the interstitial Zn sites [23]. The n g exponent values remain unaffected with increasing the Mg content oppositely to the resistance R g that raises in this case (see, Table 2). This may be explained by the structural defects related to the doping. A similar behavior was observed for ZnO-doped Sb [37], where the enhancement of the resistance is coming from the substitution of the Zn 2+ ions with Sb 3+ . Previous work by M. Ben Ali et al. [38] examining the Ni-doping influence on the electrical properties of ZTO assigned the increase in resistivity rate to the placement of the excess of the Ni-doping element on the grain boundaries. This demonstrates that Ni atoms are non-localized in the ZTO matrix, where the Ni excess leads to the formation of a Schottky barrier blocking the carrier's transport. Based on these results, we can conclude that for high-doping concentrations, Mg atoms prefer to sit in the grain boundaries and form a barrier to hinder the carrier's transport.
where e is the sample thickness, S represents the surface region, and Z 0 characterizes the resistance found based on the interception of the semicircles with the real axis. The reciprocal temperature dependence to the DC conductivity is drawn in straight lines in Figure 5. Here, the Arrhenius law could be used: where, K B designed the Boltzmann constant and E a is the activation energy corresponding to the energy difference between the conduction levels and the donor. The E a values are deduced from the linear fit slope of the logarithmic DC conductivity (σ dc .T) versus 1000 T . We found low activation energy values of doped samples in comparison with unmodified ZnO (see Table 3). The activation energy is estimated at 0.25 eV for the 1% Mg-doping concentration, which is ascribed to Zn 2+ [39]. Besides, such an activation energy is found in the range (0.32 to 0.38 eV) for higher Mg percentages. These values originate from the oxygen vacancies V O [40]. Chaari et al. [19] reported comparable results for ZnO ceramics doped with high concentrations of Sn 2 O 3 . The lower activation energy values reveal the source of the conduction phenomenon process in ZnO:Mg NCs, which is associated with the polaron hopping. to the energy difference between the conduction levels and the donor. The values are deduced from the linear fit slope of the logarithmic DC conductivity ( . ) versus . We found low activation energy values of doped samples in comparison with unmodified ZnO (see Table 3). The activation energy is estimated at 0.25 eV for the 1% Mg-doping concentration, which is ascribed to Zn 2+ [39]. Besides, such an activation energy is found in the range (0.32 to 0.38 eV) for higher Mg percentages. These values originate from the oxygen vacancies VO [40]. Chaari et al. [19] reported comparable results for ZnO ceramics doped with high concentrations of Sn2O3. The lower activation energy values reveal the source of the conduction phenomenon process in ZnO:Mg NCs, which is associated with the polaron hopping.  The DC conductivity decreases significantly with increasing the Mg doping amount due to the diminishing of the free electrons density. In fact, the imperfections act as trapping/scattering centers to reduce the number of free electrons. Further, the Mg atoms located at the grain boundaries behave as an electrical barrier that raise the carrier scattering, therefore, decreasing the conductivity [19,41].
c. AC conductivity Figure 6a represents the frequency-temperature dependency of the AC conductivity. This is demonstrated by the Jonsher's relationship [38,42]: where, is the DC conductivity associated to the flat region, is the frequency reliant expression that characterizes the dispersion phenomenon, A a constant associated with the strength of polarizability [42], and s is a factor verifying 0 ≤ ≤ 1 [43] which describes the interaction level between the mobile ions and the matrix. All these parameters depend on temperature [44].  The DC conductivity decreases significantly with increasing the Mg doping amount due to the diminishing of the free electrons density. In fact, the imperfections act as trapping/scattering centers to reduce the number of free electrons. Further, the Mg atoms located at the grain boundaries behave as an electrical barrier that raise the carrier scattering, therefore, decreasing the conductivity [19,41].
c. AC conductivity Figure 6a represents the frequency-temperature dependency of the AC conductivity. This is demonstrated by the Jonsher's relationship [38,42]: where, σ dc is the DC conductivity associated to the flat region, Aω s is the frequency reliant expression that characterizes the dispersion phenomenon, A a constant associated with the strength of polarizability [42], and s is a factor verifying 0 ≤ s ≤ 1 [43] which describes the interaction level between the mobile ions and the matrix. All these parameters depend on temperature [44]. As we can see from Figure 6a, all the curves depend on temperature. The power law is obeyed for the different ZnO:Mg samples since the fit matches well with the experimental data in Figure 6a-c. At low-frequency range, the plots show an unaffected flat region corresponding to DC conductivity, whereas the dispersion at high-frequency range is related to AC conductivity. Ionic conductors are characterized with such a behavior [45]. For the lower frequencies, the conductivity is reduced with further Mg addition due to the placement of the doping element on the interstitial position in the ZnO lattice that enhances structural defects concentration (Figure 6b). At the high-frequency region, the conductivity continues to decrease from 1 to 3% of Mg content while it shows an increase for 5% of Mg doping. These latest samples display a high conductivity due to the increase in oxygen vacancies that lead to augmentation of the hopping charge carrier's concentration [46].
The theoretical fits give σ and s values which can explain the mechanism related to the conduction process. The s parameter is dependent on the ZnO and ZnO:Mg (1%) temperature, shown in Figure 7. It slightly increases from 0.42 to 0.56 with temperature. This increase could be attributed to the quantum mechanical tunneling (QMT) conduction model [47]. Here, s is expressed as [48]: where, is the Boltzmann constant, is the energy of the polaron hopping, and is a typical relaxation time. When >> , the expression can be simplified as follows: As we can see from Figure 6a, all the curves depend on temperature. The power law is obeyed for the different ZnO:Mg samples since the fit matches well with the experimental data in Figure 6a-c. At low-frequency range, the plots show an unaffected flat region corresponding to DC conductivity, whereas the dispersion at high-frequency range is related to AC conductivity. Ionic conductors are characterized with such a behavior [45]. For the lower frequencies, the conductivity is reduced with further Mg addition due to the placement of the doping element on the interstitial position in the ZnO lattice that enhances structural defects concentration (Figure 6b). At the high-frequency region, the conductivity continues to decrease from 1 to 3% of Mg content while it shows an increase for 5% of Mg doping. These latest samples display a high conductivity due to the increase in oxygen vacancies that lead to augmentation of the hopping charge carrier's concentration [46].
The theoretical fits give σ dc and s values which can explain the mechanism related to the conduction process. The s parameter is dependent on the ZnO and ZnO:Mg (1%) temperature, shown in Figure 7. It slightly increases from 0.42 to 0.56 with temperature. This increase could be attributed to the quantum mechanical tunneling (QMT) conduction model [47]. Here, s is expressed as [48]: where, K B is the Boltzmann constant, W H is the energy of the polaron hopping, and τ 0 is a typical relaxation time. When W H >> K B T, the expression can be simplified as follows: = 1 + 4 (9) For the samples ZnM2 and ZnMg3, the exponent s was found to vary from 0.50 to 0.42 with increasing temperature from 200 to 260 °C. However, its value is more important for the ZnO:Mg (5%) sample, around 0.7, and decreases slightly with temperature. The diminution of the s parameter leads to the principal transfer route which is correlated with the barrier hopping (CBH) model [49][50][51], where charge carriers skip among sites over the potential barrier parting them instead of tunneling via the barrier [52]. The parameter s obeys the following equation [53]: is the upper barrier height at an infinite separation. This is also identified as the energy relative to polaron binding in its localized sites.  For the samples ZnM2 and ZnMg3, the exponent s was found to vary from 0.50 to 0.42 with increasing temperature from 200 to 260 • C. However, its value is more important for the ZnO:Mg (5%) sample, around 0.7, and decreases slightly with temperature. The diminution of the s parameter leads to the principal transfer route which is correlated with the barrier hopping (CBH) model [49][50][51], where charge carriers skip among sites over the potential barrier parting them instead of tunneling via the barrier [52]. The parameter s obeys the following equation [53]: W M is the upper barrier height at an infinite separation. This is also identified as the energy relative to polaron binding in its localized sites.

Dielectric Study a. Permittivity and loss studies
The complex dielectric constants describe the dielectric properties of materials. They are determined from Z′ and Z″ (Equations (3) and (4), respectively) and given by the following formula [38]: where the real ε′ and the imaginary ε″ parts illustrate the quantities of energy accumulated and dispersed in the dielectric owing to the applied electric field, correspondingly. Their expressions are:

Dielectric Study a. Permittivity and loss studies
The complex dielectric constants describe the dielectric properties of materials. They are determined from Z and Z" (Equations (3) and (4), respectively) and given by the following formula [38]: where the real ε and the imaginary ε" parts illustrate the quantities of energy accumulated and dispersed in the dielectric owing to the applied electric field, correspondingly. Their expressions are: Here, C 0 is geometrical capacitance of samples (C 0 = ε 0 S e , ε 0 is the permittivity of the vacuum, S is the cross-sectional area of the flat surface of the pellet, and e is the thickness). Z and Z" are the real and the imaginary parts of impedance. The dielectric loss (tan δ) is calculated from the values of the dielectric constants and could be written as follows [54]: Figure 9a depicts the frequency dispersion ε for all the ZnO:Mg samples. It shows an increase of ε with the Mg content. For each sample, the dielectric constant gradually reduces as the frequency augments, and reaches an almost constant value (relatively high) in the high frequencies range. Here, is geometrical capacitance of samples ( = , ε0 is the permittivity of the vacuum, is the cross-sectional area of the flat surface of the pellet, and is the thickness). Z′ and Z″ are the real and the imaginary parts of impedance. The dielectric loss ( tan ) is calculated from the values of the dielectric constants and could be written as follows [54]: Figure 9a depicts the frequency dispersion ε′ for all the ZnO:Mg samples. It shows an increase of ε′ with the Mg content. For each sample, the dielectric constant gradually reduces as the frequency augments, and reaches an almost constant value (relatively high) in the high frequencies range.  Figure 9b depicts the frequency dispersion ε′ versus the temperature of the ZnO:Mg (3%) sample. As shown, ε′ is gradually reduced through the increasing of the frequency to reach almost a constant value in the high frequencies range. This way points to a Debyetype dielectric dispersion [55,56]. According to the Maxwell-Wagner model [57], the large values of ε′ is related to the large polarization effect produced at the grain boundaries due to the charge carrier's migration between grain boundaries under the effect of the applied external field [58]. At high frequencies range, the decrease in ε′ values could be explained by polarizability loss, since the dipoles are enabled to rotate quickly which leads to a delay among frequencies of the oscillating dipole and applied fields [59]. It can be pointed out that the reached ε′ value (at medium and high frequencies) is considered relatively high, making the ZnO:Mg samples very promising for energy storage applications.
The imaginary part ε″ also decreases with the frequency but it varies slightly versus the temperature. This demonstrates the loss of polarization with the frequency by the disappearance of dipoles or to their inability to rotate properly at high frequencies [60].
We located a considerable reduction of the imaginary part ε″ with the rate of the Mg doping which confirms the rule of the space charge [61] (see Figure 10a). The same performance was also monitored by Hafef et al. [24] with high MgO-doping rates (10% and 20% of MgO). This also was explained by the separation of the MgO phase in the ZnO host matrix. In this work, the Mg is well incorporated into the ZnO lattice and hence, it induces less crystallinity, as was previously shown in the XRD analysis. Thus, this behavior could be related to the increased defects density induced by further Mg incorporation, which leads to less dielectric polarization. Moreover, ″ increases with the temperature (see Figure 10b) where no peaks in the loss tangent curves are observed ( Figure 11). This demonstrates the slightly high conduction losses [62]. Finally, the present material could be an encouraging candidate for device applications acting in the high-frequency range.  Figure 9b depicts the frequency dispersion ε versus the temperature of the ZnO:Mg (3%) sample. As shown, ε is gradually reduced through the increasing of the frequency to reach almost a constant value in the high frequencies range. This way points to a Debyetype dielectric dispersion [55,56]. According to the Maxwell-Wagner model [57], the large values of ε is related to the large polarization effect produced at the grain boundaries due to the charge carrier's migration between grain boundaries under the effect of the applied external field [58]. At high frequencies range, the decrease in ε values could be explained by polarizability loss, since the dipoles are enabled to rotate quickly which leads to a delay among frequencies of the oscillating dipole and applied fields [59]. It can be pointed out that the reached ε value (at medium and high frequencies) is considered relatively high, making the ZnO:Mg samples very promising for energy storage applications.
The imaginary part ε also decreases with the frequency but it varies slightly versus the temperature. This demonstrates the loss of polarization with the frequency by the disappearance of dipoles or to their inability to rotate properly at high frequencies [60].
We located a considerable reduction of the imaginary part ε with the rate of the Mg doping which confirms the rule of the space charge [61] (see Figure 10a). The same performance was also monitored by Hafef et al. [24] with high MgO-doping rates (10% and 20% of MgO). This also was explained by the separation of the MgO phase in the ZnO host matrix. In this work, the Mg is well incorporated into the ZnO lattice and hence, it induces less crystallinity, as was previously shown in the XRD analysis. Thus, this behavior could be related to the increased defects density induced by further Mg incorporation, which leads to less dielectric polarization. Moreover, ε increases with the temperature (see Figure 10b) where no peaks in the loss tangent curves are observed ( Figure 11). This demonstrates the slightly high conduction losses [62]. Finally, the present material could be an encouraging candidate for device applications acting in the high-frequency range.  b. Modulus analysis The electric modulus is generally used to examine the electrical characteristics of bulk materials. At low frequencies, the conductivity phenomenon usually hides the interfacial polarization generally present in such materials [63]. To surmount this difficulty, modulus is mostly appropriate to obtain phenomena such as the conductivity relaxation times and the electrode polarization. The complex electric modulus is written as [64]: where, = ′′ and = ′ are the real and imaginary parts of the complex electric modulus, respectively. Figure 12a represents the variations of M′ within the frequency at different temperatures. It raises with increasing frequency to reach a flat zone to the highest frequency values corresponding to the limiting value of M′. This behavior illustrates the weak contribution of the electrode polarization that could be neglected for Mg-doped ZnO NCs [65]. Such behavior was also detected for Na-doped ZnO NCs [22].  b. Modulus analysis The electric modulus is generally used to examine the electrical characteristics of bulk materials. At low frequencies, the conductivity phenomenon usually hides the interfacial polarization generally present in such materials [63]. To surmount this difficulty, modulus is mostly appropriate to obtain phenomena such as the conductivity relaxation times and the electrode polarization. The complex electric modulus is written as [64]: where, = ′′ and = ′ are the real and imaginary parts of the complex electric modulus, respectively. Figure 12a represents the variations of M′ within the frequency at different temperatures. It raises with increasing frequency to reach a flat zone to the highest frequency values corresponding to the limiting value of M′. This behavior illustrates the weak contribution of the electrode polarization that could be neglected for Mg-doped ZnO NCs [65]. Such behavior was also detected for Na-doped ZnO NCs [22]. b. Modulus analysis The electric modulus is generally used to examine the electrical characteristics of bulk materials. At low frequencies, the conductivity phenomenon usually hides the interfacial polarization generally present in such materials [63]. To surmount this difficulty, modulus is mostly appropriate to obtain phenomena such as the conductivity relaxation times and the electrode polarization. The complex electric modulus is written as [64]: where, M = ωC 0 Z and M = ωC 0 Z are the real and imaginary parts of the complex electric modulus, respectively. Figure 12a represents the variations of M within the frequency at different temperatures. It raises with increasing frequency to reach a flat zone to the highest frequency values corresponding to the limiting value of M . This behavior illustrates the weak contribution of the electrode polarization that could be neglected for Mg-doped ZnO NCs [65]. Such behavior was also detected for Na-doped ZnO NCs [22]. The frequency reliance of M″ with the temperature for ZnO:Mg (3%) is represented in Figure 12b. The M" position shifts near the high-frequency region when the temperature increases, and the shift of the M" position is usually related to the conductivity relaxation. An asymmetric peak could be observed for each investigated temperature, which exhibited the non-Debye kind of behavior in the relaxation of the Mg-doped ZnO NCs [66].
The detected peak aids in determining the relaxation time τ using the following relationship [58]: where designates the relaxation frequency. The change of M" with temperature images the enhancement of the dielectric relaxation time τ with temperature [37]. Such a change affirms the thermal stimulation of the dielectric relaxation [67]. As shown in Figure 13, varies linearly with the inverse of the absolute temperature T, following the equation: where, τ0 is the pre-exponential factor. From a linear fit of the plot, the value of the activation energy is estimated to 0.73 eV, which is quite different to that determined while considering the plot of Ln( dcT). Hence, it provides information about the non-statistic distribution of the Mg 2+ ions in the ZnO matrix and suggests an arbitrary conductivity. Therefore, the relaxation of dipoles manifests itself arbitrarily [37,68]. The frequency reliance of M with the temperature for ZnO:Mg (3%) is represented in Figure 12b. The M max position shifts near the high-frequency region when the temperature increases, and the shift of the M max position is usually related to the conductivity relaxation. An asymmetric peak could be observed for each investigated temperature, which exhibited the non-Debye kind of behavior in the relaxation of the Mg-doped ZnO NCs [66].
The detected peak aids in determining the relaxation time τ using the following relationship [58]: where f max designates the relaxation frequency. The change of M max with temperature images the enhancement of the dielectric relaxation time τ with temperature [37]. Such a change affirms the thermal stimulation of the dielectric relaxation [67]. As shown in Figure 13, Lnτ varies linearly with the inverse of the absolute temperature T, following the equation: where, τ 0 is the pre-exponential factor. From a linear fit of the plot, the value of the activation energy E a is estimated to 0.73 eV, which is quite different to that determined while considering the plot of Ln(σ dc T). Hence, it provides information about the non-statistic distribution of the Mg 2+ ions in the ZnO matrix and suggests an arbitrary conductivity. Therefore, the relaxation of dipoles manifests itself arbitrarily [37,68].

Conclusions
Mg-doped ZnO NCs were synthesized via sol-gel method. The electrical and dielectric characteristics were investigated as function of temperature and frequency, using impedance spectroscopy. The conductivity and the activation energy are influenced by Mg concentration. From temperature reliance, we reveal that the conduction mechanism is managed from the model of the hopping correlated barrier (CBH). Dielectric features are

Conclusions
Mg-doped ZnO NCs were synthesized via sol-gel method. The electrical and dielectric characteristics were investigated as function of temperature and frequency, using impedance spectroscopy. The conductivity and the activation energy are influenced by Mg concentration. From temperature reliance, we reveal that the conduction mechanism is managed from the model of the hopping correlated barrier (CBH). Dielectric features are largely affected by Mg doping due to less dielectric polarization. The low dielectric losses at high frequencies and the shift toward high frequencies of M with temperature, make the Mg-doped ZnO NCs a suitable material for application in non-linear optics. The analysis of the electrical modulus illustrates an extension in the relaxation time with further Mg incorporation. All the dielectric properties relative to the prepared ZnO:Mg NCs are of interest to bring a considerable influence to several technological applications, such as microwave devices.