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Comparative Measurements and Analysis of the Electrical Properties of Nanocomposites Ti_{x}Zr_{1−x}C+α-Cy (0.0 ≤ x ≤ 1.0)

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## Abstract

**:**

_{x}Zr

_{1−}

_{x}C+α-C

_{y}(0.0 ≤ x ≤ 1.0) nanocomposites produced by dual-source magnetron sputtering was determined. The films produced are biphasic layers with an excess of amorphous carbon relative to the stoichiometric composition of Ti

_{x}Zr

_{1−x}C. The matrix was amorphous carbon, and the dispersed phase was carbide nanoparticles. AC measurements were performed in the frequency range of 50 Hz–5 MHz at temperatures from 20 K to 373 K. It was found that both conductivity and permittivity relationships are determined by three tunneling mechanisms, differing in relaxation times. The maxima in the low- and high-frequency regions decrease with increasing temperature. The maximum in the mid-frequency region increases with increasing temperature. The low-frequency maximum is due to electron tunneling between the carbon films on the surface of the carbide nanoshells. The mid-frequency maximum is due to electron transitions between the nano size grains. The high-frequency maximum is associated with tunneling between the nano-grains and the carbon shells. It has been established that dipole relaxation occurs in the nanocomposites according to the Cole-Cole mechanism. The increase in static dielectric permittivity with increasing measurement temperature is indicative of a step polarisation mechanism. In the frequency region above 1 MHz, anomalous dispersion—an increase in permittivity with increasing frequency—was observed for all nanocomposite contents.

## 1. Introduction

_{x}Zr

_{1−}

_{x}C+α-C

_{y}nanocomposites in the concentration range (0.0 ≤ x ≤ 1.0) were produced using specially selected parameters of two-source magnetron sputtering. The transition metal carbide nanoparticles had average dimensions of 8 nm to 20 nm, depending on the composition [15], and are the dispersed phase. The second phase constituting the matrix was amorphous carbon. This structure of the fabricated nanocomposites clearly demonstrates that these materials are models for the study of percolation phenomena and hopping conductivity, based on electron tunneling quantum phenomena [41]. The tunneling phenomenon, as it is known, occurs when the dimensions of the potential wells from which the electrons tunnel, and the thicknesses of the barriers separating the wells, are nanometres [42,43,44,45]. In many papers, step conductance studies are performed using direct current—see, for example, [46,47,48,49,50]. The use of AC voltage for testing extends, significantly, the range of information about tunneling processes occurring in nanocomposites [51,52,53,54]. Such studies allow the determination of conduction mechanisms, dielectric relaxation processes, permittivity and polarisation mechanisms in nanocomposites. Investigations of the frequency-temperature electrical properties of nc-Ti

_{x}Zr

_{1−}

_{x}C+α-C

_{y}(0.0 ≤ x ≤ 1.0) nanocomposites may contribute to expanding their range of applications in micro and medium-power electrical devices and power electronics.

_{x}Zr

_{1−}

_{x}C+α-C

_{y}nanocomposites in the concentration range (0.0 ≤ x ≤ 1.0) at temperatures of 20 K-373 K to determine the mechanisms of conductivity and polarisation and to benchmark the structure and electrical parameters and, on this basis, determine the mechanisms of electron tunneling between the individual structural components of the nanocomposites.

## 2. Materials and Methods

#### 2.1. Materials

_{x}Zr

_{1−}

_{x}C+α-C

_{y}(0.0 ≤ x ≤ 1.0) nanocomposites were obtained by dual magnetron sputtering [15]. The layers were applied to the planar surface of silicon wafers designed for the manufacture of integrated circuits. The orientation of the wafers was (100) and their thickness was 0.3 mm. First source used five alloy targets with the following compositions: 1—Ti, 2—Ti

_{0.75}+Zr

_{0.25}at. %, 3—Ti

_{0.5}+Zr

_{0.5}at. %, 4—Ti

_{0.25}+Zr

_{0.75}at. %, 5—Zr. The second sputtered target was made of carbon manufactured by Nanoshel (Punjab, India) with a purity of 99.99 %. The operating parameters of both sources were chosen in such a way that the carbon content of the layer was redundant compared to the stoichiometric composition of Ti

_{x}Zr

_{1−}

_{x}C. The structure and composition of the films produced were investigated by XRD, SIMS and energy-dispersive X-ray spectroscopy [15]. It was determined that the fabricated films are nano-grained nanocomposites with average carbide grain sizes varying from 8 nm to 20 nm, depending on the Ti content. The results of the investigations on the structure and chemical composition of the layers are shown in Table 1. Comparing the composition of the targets and the composition of the fabricated carbides, it can be seen that titanium is more easily atomised than zirconium. From the table, it can be seen that the thicknesses of the produced layers are similar for the different compositions and are approximately 1 µm. Each layer contains surplus carbon. Given the surplus carbon and its density of 2.25 g·cm

^{−3}and also that the densities of TiC and ZrC are much higher than those of carbon at 4.91 g·cm

^{−3}and 6.73 g·cm

^{−3}, respectively [55], it should be assumed that the amorphous carbon α-C phase acts as the matrix in the nanocomposites studied. This means that the nanocomposite layers produced are biphasic. In addition, there are further nanometre-sized and even smaller elements on the surface of the nano-grains. These are near-monoatomic carbon films. Their properties differ fundamentally from those of the amorphous carbon matrix [15].

#### 2.2. Methods

_{x}Zr

_{1−}

_{x}C+α-C

_{y}nanocomposites has been done using a measuring stand developed at the Department of Electrical Devices and High Voltage Technology at the Faculty of Electrical Engineering and Computer Science at the Lublin University of Technology. The configuration and functional principle of the stand was described in publications [56,57,58]. Measurements were carried out as a function of frequency in the range from 50 Hz to 5 MHz and temperature range from 20 K to 373 K. Time needed for a single sample measurement was approximately 5 h. In order to improve the efficiency of the measuring stand, two impedance meters measure two samples simultaneously. The stand is fully automated. Frequency and temperature settings, as well as saving of measurement results, are made by a dedicated computer programme.

_{0}—dielectric permittivity of a vacuum.

## 3. Hopping Conductivity of Nanocomposites Considering Quantum Mechanical Electron Tunneling Phenomenon

_{0}—electric field strength amplitude, σ

_{0}—conductivity, ω—angular frequency, t—time.

_{m}—standard deviation; τ

_{m}—expected value.

_{r}in accordance with Equation (15). The calculations were performed for probability values p ranging from 10

^{−6}to 0.5 with a small step. Figure 3a presents an example of a relationship j

_{r}(f · τ

_{m}).

_{max}increases, and its position on the x axis shifts to the lower frequencies area. The parameters that fully characterise the frequency dependence of conductivity for electron tunneling are the jump probability p, the expectation value of the relaxation time τ

_{m}o and the value of the conductivity in the high frequency area σ

_{0}. From these, using Equations (15) and (7), the waveforms σ(f) and α(f) can be calculated. There is more than one tunneling mechanism in the nanocomposite, which differs in p, τ

_{m}and σ

_{0}values, and a corresponding number of maxima appear on the α(f) relationship [15]. Experimental verification of the model was been carried out in many studies; see, e.g., [66,67].

## 4. Results and Discussion

#### 4.1. Frequency-Temperature Dependence of the Conductivity of nc-Ti_{x}Zr_{1−x}C+α-C_{y} Nanocomposites

_{y}nanocomposite for selected measurement temperatures. We will now compare the experimental results, shown in Figure 4, with the results of the electron tunneling conductivity model—Figure 3. From the figures, it can be seen that in the low-frequency area, the conductivity hardly depends on the frequency. It is a direct current conductivity. An increase in temperature causes an increase in conductivity. This means that there is a dielectric-type conductivity in the nanocomposite. As the frequency increases above 10

^{3}Hz, the conductivity starts to increase. Conductivity waveforms with increasing temperature shift to the area of higher frequencies. This is related to the reduction of relaxation times with increasing temperature. The σ(f) waveforms show slight inflection in the frequency area of about 10

^{5}Hz.

_{y}nanocomposite. The figure shows that in the frequency range above 10

^{5}Hz there are two maxima very close to each other. In the low frequency area, around 200 Hz, at low temperatures, there is a third, much weaker maximum. The similarity of the waveforms of both conductivity and frequency factor, shown in Figure 3 and Figure 4, implies that the observed relationships are consistent with the hopping mechanism of charge transfer, considering the quantum phenomenon of electron tunneling described in Section 3. This indicates the presence of a hopping conduction mechanism in the nanocomposite. The occurrence of three maxima means that there are at least three types of tunneling in the material, differing in the values of the relaxation times.

_{0.86}Zr

_{0.14}C+α-C

_{y}nanocomposite. The introduction of zirconium into the nanocomposite resulted in bends becoming apparent in the conductivity waveforms (Figure 5a). The frequency dependence of the α(f) coefficient (Figure 5b) shows two distinct maxima in the investigated frequency range. Their position and maximum values depend on the measurement temperature. The value of the maximum in the mid-frequency area increases with increasing temperature, and its position shifts to the higher-frequency area. The high-frequency maximum, on the other hand, decreases with increasing temperature and shifts to the higher-frequency area much less than the low-frequency maximum.

_{x}Zr

_{1−}

_{x}C+α-C

_{y}nanocomposites in the entire range of titanium concentration changes of 0.0 ≤ x ≤ 1.0 exhibit dielectric-type conductivity.

_{x}Zr

_{1−x}C+α-C

_{y}nanocomposites. This means that in the investigated frequency range, three tunneling mechanisms are observed, differing in the values of relaxation times and the effect of temperature on them. Low and high frequency peaks decrease in value with increasing temperature. This is due to an increase in the electron tunneling probability p from the middle well to the right well (compare Figure 3 and Figure 4, Figure 5, Figure 6, Figure 7 and Figure 8). On the other hand, the increase in the maximum value of the coefficient α(f) for the maximum in the medium frequencies area is caused by the reduction of the probability of tunneling p with increasing temperature.

_{1}and P

_{2}. Then, the following tunneling mechanisms will be possible: P

_{1}↔ P

_{1}, P

_{2}↔ P

_{2}, P

_{1}↔ P

_{2}. One type of well is metal carbide nanograin. It was established in [15] that on the surface of nanograins there is a transition layer consisting of carbon with a structure significantly different from that of the amorphous carbon matrix. Since the matrix is amorphous carbon, it can be assumed that the second type of potential well is the carbon coatings on the surface of carbide nanograins. Due to the fact that the volume of such coatings is smaller than that of metal carbide grains, the coatings are probably responsible for a weak peak in the low frequency area (P

_{2}↔ P

_{2}). Comparing the temperature dependence of the values of the maxima, it should be assumed that the maximum in the area of medium frequencies is caused by tunneling between the nanograins (P

_{1}↔ P

_{1}). The transitions between nanograins and coatings (P

_{1}↔ P

_{2}) are responsible for the high-frequency maximum.

#### 4.2. Frequency-Temperature Dependencies of the Permittivity of Ti_{x}Zr_{1−x}C+α-C_{y} Nanocomposites

_{0.86}Zr

_{0.14}C+α-C

_{y}and nc-Ti

_{0.39}Zr

_{0.61}C+α-C

_{y}).

^{4}Hz.

_{S}(static permittivity) to the value of ε

_{S}/2 [68]:

_{1/2}—frequency at which permittivity decreases twice.

^{4}Hz, and for the high-frequency stage about 4 × 10

^{5}Hz. Based on Equation (16) the expected values of the relaxation times were calculated to be about 8 × 10

^{−6}s and about 4 × 10

^{−7}s, respectively. As the temperature decreases below 319 K, the relaxation times increase, while above the temperature of 319 K they decrease. Similar values of relaxation times and their temperature variation are found for the nanocomposite with higher zirconium content. It can be seen from Figure 9a and Figure 10a that the static permittivity values for both stages increase with increasing temperature. The observed increase in static dielectric permittivity due to increasing temperature is indicative of a hopping polarisation mechanism caused by tunneling. In this mechanism, the concentration of dipoles formed by tunneling increases with increasing temperature [64]. When solid dipoles are present in the materials (orientation polarisation), the permittivity is described by the Debye formula [68], and the value of the static permittivity decreases with increasing temperature as 1/T.

_{0}—dielectric permittivity of a vacuum.

_{m}. For this mechanism, the graph of ε″(ε′) assumes an arc shape. The value at the maximum of this relationship is smaller than the width of the arc on the x-axis. The Davidson-Cole mechanism is also applicable [71,72]. For this mechanism, an asymmetric ε″(ε′) dependence graph is characteristic. The asymmetric empirical model of Havriliak and Nagami [73] is also used for dielectric relaxation analysis. It can be seen from Figure 9b and Figure 10b that three symmetrical maxima are evident in the Cole-Cole diagrams over the studied temperature range. The maximum in the area of small values of the permittivity ε′ (Figure 10) has been truncated due to the end of the meter range. The values in the maxima are much smaller than their widths. The relationships seen in Figure 9b and Figure 10b are characteristic of the Cole-Cole dielectric relaxation mechanism. This means that there are probability distributions of relaxation times in the nanocomposites. This is consistent with the electron tunneling conductivity model presented in Section 3.

## 5. Conclusions

_{x}Zr

_{1−}

_{x}C+α-C

_{y}(0.0 ≤ x ≤ 1.0) nanocomposites produced by dual-source magnetron sputtering was determined. The samples contained an excess of amorphous carbon phase compared to the stoichiometric composition of Ti

_{x}Zr

_{1−}

_{x}C. Consequently, the films produced are biphasic layers. The matrix in the nanocomposites studied was amorphous carbon and the dispersed phase is metal carbide nanoparticles. Measurements were made in the frequency range 50 Hz–5 MHz at temperatures from 20 K to 373 K. It was found that the conductivity and permittivity of the nanocomposites are due to electron tunneling. Three maxima are observed on the frequency dependence of the conductivity coefficient. The low-frequency maximum is related to the tunneling of electrons between carbon coatings found on the surface of metal carbide nanoshells. The structure of such coatings differs from that of amorphous carbon. The mid-frequency maximum is due to electron transitions between metal carbide nanoshells. The high-frequency maximum is associated with tunneling between the nano-grains and the carbon coatings.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Cross-section of Ti

_{x}Zr

_{1−}

_{x}C+α-C

_{y}nanocomposite sample: 1—nanocomposite layer, 2—silicon substrate (bottom plate), 3—silver paste top plate, 4—silver paste electrode.

**Figure 2.**Potential wells: A—excitation of the electron from the highest occupied state to the lowest unoccupied state, 1—tunneling of the electron from the left well to the middle well, 2—tunneling from the middle well to the right well, 3—return tunneling from the middle well to the left well, ∆E—distance between occupied and unoccupied states or activation energy of tunnelling, r—distance between wells of potential.

**Figure 3.**Relationship of the actual component of current density j

_{r}—(

**a**) and frequency coefficient α(f)—(

**b**) as a function of the product of the frequency value f and the expected value of the relaxation time τ

_{m}. Computer simulation for: 1—p = 0.01; 2—p = 0.001; 3—p = 0.0001.

**Figure 4.**Frequency dependence of conductivity (

**a**) and frequency factor α(f) (

**b**) for selected measurement temperatures for the nc-TiC+α-C

_{y}nanocomposite.

**Figure 5.**Frequency dependence of conductivity (

**a**) and frequency factor α(f) (

**b**) for selected measurement temperatures for the nc-Ti

_{0.86}Zr

_{0.14}C+α-C

_{y}nanocomposite.

**Figure 6.**Frequency dependence of conductivity (

**a**) and frequency factor α(f) (

**b**) for selected measurement temperatures for the nc-Ti

_{0.75}Zr

_{0.25}C+α-C

_{y}nanocomposite.

**Figure 7.**Frequency dependence of conductivity (

**a**) and frequency factor α(f) (

**b**) for selected measurement temperatures for the nc-Ti

_{0.39}Zr

_{0.61}C+α-C

_{y}nanocomposite.

**Figure 8.**Frequency dependence of conductivity (

**a**) and frequency factor α(f) (

**b**) for selected measurement temperatures for the nc-ZrC+α-C

_{y}nanocomposite.

**Figure 9.**Frequency dependence of permittivity (

**a**) and Cole-Cole diagrams (

**b**) for selected measurement temperatures for the nc-Ti

_{0.86}Zr

_{0.14}C+α-C

_{y}nanocomposite.

**Figure 10.**Frequency dependence of permittivity (

**a**) and Cole-Cole diagrams (

**b**) for selected measurement temperatures for the nc-Ti

_{0.39}Zr

_{0.61}C+α-C

_{y}nanocomposite.

Sample Number | Target Composition | Real Sample Composition and Structure | Surplus C, a.u. | Thickness, µm | Grain Size L, nm | Thickness of the Carbon Coating on the Nanoparticles Surface, nm |
---|---|---|---|---|---|---|

1 | Ti; C | nc-TiC+α-C_{1.25} | 1.25 | 0.984 | 18 ± 2 | 0.3–0.5 |

2 | Ti_{0.75}+Zr_{0.25};C | nc-Ti_{0.86}Zr_{0.14}C+α-C_{2.62} | 2.62 | 1.165 | 9 ± 2 | |

3 | Ti_{0.5}+Zr_{0.5};C | nc-Ti_{0.75}Zr_{0.25}C+α-C_{0.95} | 0.95 | 1.097 | 11 ± 2 | |

4 | Ti_{0.25}+Zr_{0.75};C | nc-Ti_{0.39}Zr_{0.61}C+α-C_{0.21} | 0.21 | 1.132 | 15 ± 2 | |

5 | Zr; C | nc-ZrC+α-C_{0.20} | 0.20 | 1.015 | 20 ± 2 |

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**MDPI and ACS Style**

Żukowski, P.; Gałaszkiewicz, P.; Bondariev, V.; Okal, P.; Pogrebnjak, A.; Kupchishin, A.; Ruban, A.; Pogorielov, M.; Kołtunowicz, T.N.
Comparative Measurements and Analysis of the Electrical Properties of Nanocomposites Ti* _{x}*Zr

_{1−x}C+α-Cy (0.0 ≤

*x*≤ 1.0).

*Materials*

**2022**,

*15*, 7908. https://doi.org/10.3390/ma15227908

**AMA Style**

Żukowski P, Gałaszkiewicz P, Bondariev V, Okal P, Pogrebnjak A, Kupchishin A, Ruban A, Pogorielov M, Kołtunowicz TN.
Comparative Measurements and Analysis of the Electrical Properties of Nanocomposites Ti* _{x}*Zr

_{1−x}C+α-Cy (0.0 ≤

*x*≤ 1.0).

*Materials*. 2022; 15(22):7908. https://doi.org/10.3390/ma15227908

**Chicago/Turabian Style**

Żukowski, Paweł, Piotr Gałaszkiewicz, Vitali Bondariev, Paweł Okal, Alexander Pogrebnjak, Anatolyi Kupchishin, Anatolyi Ruban, Maksym Pogorielov, and Tomasz N. Kołtunowicz.
2022. "Comparative Measurements and Analysis of the Electrical Properties of Nanocomposites Ti* _{x}*Zr

_{1−x}C+α-Cy (0.0 ≤

*x*≤ 1.0)"

*Materials*15, no. 22: 7908. https://doi.org/10.3390/ma15227908