Origin of Temperature Coefficient of Resonance Frequency in Rutile Ti_1−xZr_xO₂ Microwave Ceramics

Abstract

In this study, we report the effect of Zr⁴⁺ doping on the optical energy gap and microwave dielectric properties of rutile TiO₂. Rietveld analysis explicitly confirmed that Zr⁴⁺ occupies the octahedral site, forming a single-phase tetragonal structure below the solubility limit (x < 0.10). Notably, at x = 0.025, a significant enhancement in Q × f_o was observed. This enhancement was attributed to the reduction in dielectric loss, associated with a decrease in oxygen vacancies and a lower concentration of Ti³⁺ paramagnetic centers. This conclusion was supported by Raman and electron paramagnetic resonance spectroscopy, respectively. The origin of high τ_f in rutile Ti_1−xZr_xO₂ is explained on the basis of the octahedral distortion/tetragonality ratio, covalency, and bond strength.

1. Introduction

TiO₂ is one of the most studied metal oxides, used in many conventional applications (white pigment in paintings, paper, plastics, etc.) and advanced applications (microwave (MW) dielectric resonators, photocatalysts, resistive random-access memory (RRAM), and high-energy density capacitors) [1,2,3,4]. Naturally, TiO₂ exists in different polymorphs i.e., brookite, rutile, and anatase. Among these phases, the thermodynamically stable phase is the rutile, as brookite and anatase phase can transform irreversibly to the rutile phase in the temperature range of 700–920 °C [5]. In several studies, colossal permittivity was found for (Mg²⁺ + Ta⁵⁺) co-doped TiO₂ [6], (Tb^3+/4+ and Sb⁵⁺) co-doped TiO₂ [7], (Zr and Ta) co-doped TiO₂ [8], (Lu³⁺/Nb⁵⁺) co-doped TiO₂ [9], and (Nb and Mn) co-doped TiO₂ [10]. Cohn et al. [2] reported MW properties of TiO₂ for dielectric resonators, which gained significant attention [11]. TiO₂ possesses a high relative permittivity (ε_r) of ~100 and a suitable quality factor (Q × f_o) of ~10,000 GHz, but its poor temperature coefficient of resonant frequency (${\tau }_f$) of ~450 ppm/°C is not suitable for practical applications [12]. Several studies have proved that the dependency of the MW dielectric properties of rutile on the synthesis conditions, such as low oxygen partial pressure/high sintering temperature, results in a coring effect due to the reduction of Ti⁴⁺ to Ti³⁺ [13,14]. Noh et al. [15] sintered TiO₂ using spark plasma sintering and conventional sintering methods. The ɛ_r and Q × f_o of SPS-TiO₂ re-oxidized specimens were found to be 112.6 and 26,000, respectively. These properties were comparable to those samples sintered at 1300 °C using a solid-state route (ɛ_r = 101.3, Q × f_o = 41,600). Similarly, Weng et al. [16] sintered TiO₂ at 1040 °C/2 h and reported ɛ_r > 100, Q × f_o > 23,000 GHz, and ${\tau }_f$ = 200 ppm/°C. During sintering (in the air or under a low partial pressure of oxygen) at high temperatures, the generation of oxygen vacancies causes a coring effect, which significantly deteriorates the MW dielectric properties of TiO₂ [17]. Oxygen vacancies in dense rutile TiO₂ can be reduced by sintering it in an oxygen atmosphere or using a proper dopant. More recently, a tetravalent Hf⁴⁺ was doped in TiO₂ (Hf_xTi_1−xO₂), which exhibited ε_r = 17, Q × f_o = 84,020 GHz, and ${\tau }_f$ = −47 ppm/°C (at x = 0.05) and ε_r = 37, Q × f_o = 53,600 GHz, and ${\tau }_f$ = −28 ppm/°C (at x = 0.5) [18]. Mao et al. [19] studied the effect of Mn²⁺ and W⁶⁺ dopants in TiO₂, which improved the ε_r (~105) and Q × f_o (~39,000 GHz), but ${\tau }_f$ (~464.4 ppm/°C) further deteriorated. Similarly, other rutile TiO₂-based systems, such as Zn_0.15Nb_0.3Ti_0.55O₂ [20], Bi-doped TiO₂ [21], and Ni_0.5Ti_0.5TaO₄ [22] ceramics, exhibit a high ${\tau }_f$. Similarly, Souza et al. [23] synthesized TiO₂ − ZrO₂ (x = 0, 0.1, and 0.2) and reported ε_r = 88, Q × f_o = 16,285 GHz for the pure TiO₂, ε_r = 85.4, Q × f_o = 9765 GHz for x = 0.1, and ε_r = 79.2, Q × f_o = 8883 GHz for x = 0.2; however, they did not report the temperature coefficient of the resonance frequency of these ceramics. Therefore, the purpose of the present work was to study the effect of the Zr⁴⁺ substation on the MW dielectric properties of rutile TiO₂, which exhibited a high τ_f value; however, the quality factor was improved due to a reduction in the coring effect. Furthermore, a structure–property relationship was developed for the unaffected ${\tau }_f$ in Zr-doped TiO₂.

2. Experimental Procedures

The Ti_1−xZr_xO₂ (x = 0–0.1) ceramics were prepared through a conventional solid-state route using reagent-grade TiO₂ (99.9%) and ZrO₂ (99.9%) precursors. The precursors were dried at 700 °C for 6 h to remove moisture (if any). The precursors were weighed and then mixed/milled using a mortar and pestle for 1 h in acetone. The mixed powders were calcined at 1000 °C (x = 0) and 1200 °C (x > 0) for 4 h at a heating/cooling rate of 5 °C/min in air. The calcined powders were re-milled and then pressed uniaxially into 10 mm-in-diameter cylindrical pellets. The green pellets were sintered in the temperature range of 1300−1400 °C for 4 h at a heating/cooling rate of 5 °C/min in air.

The phase analysis of the samples was carried out using an X-ray diffractometer (Bruker D8 Advance, Germany) with Cu-Kα radiation. For Rietveld analysis, the GSAS + EXPGUI package was used [24,25]. For background fitting, the Shifted Chebyshev polynomial was used with 12 terms. To increase the level of accuracy, the damping factor was kept at 9. To study the vibrational modes, the Raman spectra of all the samples were collected at room temperature using a Micro Ramboss spectrometer with a laser source of 532 nm and a power of 80 mW, with a frequency of 150–800 cm⁻¹. Electron paramagnetic resonance (EPR) spectroscopy was carried out at room temperature using a Bruker BioSpin GMBH apparatus, Germany, with an X-band frequency of 9.843 GHz. The gyromagnetic factor (g) value was calculated by using the Equation (1),

$$g=\frac{h{v}_{\circ }}{\beta H}$$

(1)

where h is the Planck constant ($h$ = 6.626 × 10⁻³⁴ Js), $v_{\circ }$ is the MW frequency, $\beta$ is the Bohr magneton ($\beta$ = 5.788 × 10⁻¹⁵ eVT⁻¹), and $H$ is the strength of the magnetic field. The microstructure and elemental distribution of the samples were examined via scanning electron microscopy with energy-dispersive X-ray analysis (SEM/EDX, Thermo Fisher Scientific, Waltham, MA, USA). The diffuse reflectance spectrum of the sample was obtained using a Jasco-V750 UV–vis spectrometer. The MW dielectric properties of the samples were measured using a network analyzer (Keysight, E5071C ENA, Santa Rosa, CA, USA) with the TE_01δ shielded cavity method [26]. The temperature coefficient of resonant frequency (${\tau }_f$) was calculated with the following formula:

$${\tau }_f=\frac{f_1-f_2}{f_1(T_1-T_2)}$$

(2)

where $f_1$ and $f_2$ are the resonant frequencies at temperature $T_1$ and $T_2$_, respectively.

3. Results and Discussions

Figure 1a shows the room-temperature XRD patterns of Ti_1−xZr_xO₂ (x = 0–0.1) ceramics sintered at their optimum sintering temperatures. All diffraction peaks of the samples matched JCPDS #89−4920 for rutile TiO₂, having a tetragonal crystal structure with a space group P4₂/mnm. Figure 1b shows the Rietveld fitted and experimental data of Ti_1−xZr_xO₂ (x = 0.025). Fitted plots for other samples are given in the Supplementary Information (Figure S1). The refined lattice parameters, cell volume, R factors (goodness of fitting or reliability factor), tetragonality (c/a), bond lengths (d), and theoretical density are given in

Table 1. Rietveld refined lattice parameters, atomic positions, and refinement factors of Ti_1–xZr_xO₂ (x = 0–0.1) ceramics.

x Values		0	0.025	0.05	0.075	0.1
a = b (Å)		4.5890 (7)	4.5923 (2)	4.6001 (5)	4.6061 (5)	4.6137 (9)
c (Å)		2.9579 (8)	2.9635 (7)	2.9732 (6)	2.9811 (1)	2.9882 (8)
α = β = γ (°)		90	90	90	90	90
c/a		0.6445	0.6453	0.6463	0.6471	0.6476
Vcell (Å)3		62.29	62.45	62.92	63.26	63.61
Rwp (%)		11.36	14.12	12.06	11.98	12.97
Rp (%)		8.82	10.91	9.17	9.21	10.09
χ2		1.49	2.14	1.66	1.71	1.96
Atomic positions (x, y, z)	Ti	(0, 0, 0)	(0, 0, 0)	(0, 0, 0)	(0, 0, 0)	(0, 0, 0)
	O	(0.3025, 0.3025, 0)	(0.3053, 0.3053, 0)	(0.3026, 0.3026, 0)	(0.3050, 0.3050, 0)	(0.2956, 0.2956, 0)
	Zr		(0, 0, 0)	(0, 0, 0)	(0, 0, 0)	(0, 0, 0)

. Due to the substitutionally doped Zr⁴⁺, the lattice parameters and cell volume have been increased and the diffraction peaks were shifted to the lower 2θ values with an increase in x [27]. The shift in the diffraction peaks is attributed to the larger ionic radius of Zr⁴⁺ (r_VI = 0.72 Å) than Ti⁴⁺ (r_VI = 0.605 Å) [28,29]. A secondary phase (ZrO₂) was observed at x = 0.1, which suggests that the solubility limit of Zr⁴⁺ in TiO₂ is less than 0.10 [30,31]. The tetragonality ratio has been increased from 0.6445 to 0.6476 with the increase in the Zr concentration.

Raman spectroscopy was carried out to confirm the rutile phase of Ti_1–xZr_xO₂ (x = 0–0.1) ceramic samples. Rutile has 15 optical modes (i.e., 1A_1g + 1A_2g + 1A_2u +1B_1g + 1B_2g + 2B_1u + 1E_g + 3E_u) in which the E_g, B_1g, B_2g, and A_1g are Raman active [32,33]. Figure 2 shows the bands that are indicative of the rutile symmetry, which were observed at about 144 cm⁻¹ (B_1g), 242 cm⁻¹ (two phonon process), 441 cm⁻¹ (E_g), and 606 cm⁻¹ (A_1g) for all the samples. Furthermore, when the Zr concentrations increased, the E_g peak moved to a lower wavenumber, showing that the doping Zr ions enter the host lattice, which expands [33]. The 144 cm⁻¹ (B_1g) peak is associated with the bending vibration of the O—Ti—O bond [27]. The peak at 242 cm⁻¹ is the complex vibration peak due to the multi-phonon scattering (second-order effect), associated with the O—Ti—O bond’s complex vibration on rutile TiO₂ [34]. Generally, the second-order effect was believed to be due to internal stress/strain or might be because of the Zr having a different ionic radius than that of the host [35]. The shift in the phonon peak towards the lower frequencies shows the reduction in the anharmonic behavior of the crystals, which shows an increase in crystal stability and enhancement of optical, dielectric, and thermal transport properties [36]. The E_g mode (441 cm⁻¹) represents the vibrational motion of oxygen atoms along the c-axis, whereas the A_1g mode (606 cm⁻¹) is related to the Ti−O stretching vibration [27]. When Ti⁴⁺ ions in the lattice were replaced by Zr⁴⁺, the O²⁻ ions could be more closely bound, because the Zr⁴⁺ ions have a higher electro-positivity than Ti⁴⁺. As a result of this, the vibrational motion of oxygen atoms along the c-axis is hindered, resulting in a decrease in the vibrational energy of oxygen atoms that corresponds to the E_g mode (441 cm⁻¹), as well as a decrease in frequency. On the other hand, the replacement of Zr⁴⁺ ions has no effect on the A_1g mode (606 cm⁻¹), matching the Ti−O stretching vibration energy. Therefore, the peak at 606 cm⁻¹ is not shifted with an increase in the Zr⁴⁺ concentration [27].

TiO₂ belongs to a class of reducible oxides as it readily loses oxygen upon sintering at high temperatures in air with the formation of excess electrons and oxygen vacancies. The reduction in TiO₂ can be expressed using Kroger–Vink notations, in terms of either the formation of Ti⁴⁺ interstitials, oxygen vacancies, Ti³⁺ interstitials, or Ti³⁺ species and oxygen vacancies in octahedral sites given by the Equations (3)–(6), respectively [14].

$$T{i}_{Ti}^{\times }+2O_o^{\times }\iff T{i}_i^{••••}+4e^{\prime }+O_2$$

(3)

$$2O_o^{\times }\iff 2V_o^{••}+4e^{\prime }+O_2$$

(4)

$$T{i}_{Ti}^{\times }+2O_o^{\times }\iff T{i}_i^{•••}+3e^{\prime }+O_2$$

(5)

$$2T{i}_{Ti}^{\times }+O_o^{\times }\iff 2T{i}_{Ti}^{{ }^{\prime }}+V_o^{••}+\frac{1}{2}{O}_2$$

(6)

The excess electrons formed due to oxygen vacancies are usually stabilized at cations by forming Ti³⁺ as illustrated in Equation (5). This fact has been firmly confirmed, based on accurate theoretical calculations, that the electrons generated as a result of oxygen vacancies are preferentially stabilized on d orbitals of Ti [37] and were also shown by Chester [38] and Zimmermann [39]. For the detection of paramagnetic species, we have used EPR spectroscopy to investigate the bulk and surface defects in the rutile TiO₂. Ti_1−xZr_xO₂ (x = 0–0.1) samples were examined via conventional continuous wave-EPR (CW-EPR) by homogeneously changing the magnetic field in a specific range until a resonance condition was achieved, which entails the transition between spin states. Figure 3 displays the room temperature EPR spectra of rutile Ti_1−xZr_xO₂ (x = 0–0.1), where the signal belongs to the paramagnetic center Ti³⁺ formed due the generation of excess electrons due to oxygen vacancies, consistent with the previous result [40]. The line of the EPR signal for rutile TiO₂ recorded at room temperature tends to broaden and eventually disappear. This is because of the relaxation time of Ti³⁺, which causes the weakening of the EPR signal’s intensity at a temperature higher than 77 K [40]. With an increase in the Zr⁴⁺ concentration, the intensity of the signal decreases, which means the concentration of the paramagnetic center, implying that Ti³⁺ has been reduced in the sample. The corresponding g-value (1.97) belongs to the inner Ti³⁺ paramagnetic centers [41,42,43].

The relative density of rutile Ti_1−xZr_xO₂ (x = 0–0.1) with an increasing Zr concentration was measured. At an optimum sintering temperature, a relative density of ~93% was achieved. Figure 4a–e shows the SEM micrographs of sintered, thermally etched, and polished rutile Ti_1−xZr_xO₂ (x = 0–0.1) samples, illustrating their morphologies and grain-size distributions. The SEM images show a well-developed grain morphology and an almost dense microstructure with some pores or voids. The grain size increased from 5 to 19 µm as the Zr concentration increased from 0 to 0.1 (Figure 4f). This is because Zr promotes the grain-growth [44]. The composition (x = 0, sintered at 1300 °C) exhibited a uniform distribution of grain size, while the doped samples sintered at high temperatures (1375−1400 °C) displayed a bimodal distribution, which is an indication of the Ostwald-ripening mechanism for the grain-growth [45]. During sintering, grain growth is dominated by the grain boundary and volume diffusion, which results in larger grains [11]. The pores/voids of the doped samples have been decreased as compared to the pure rutile TiO_2, but residual porosity has been observed in all the compositions. To further confirm the homogeneity in the samples, elemental mappings for Ti_1−xZr_xO₂ (x = 0.025) are shown in Figure 4g–j, which are also supported by the EDX spectra, shown in Figure S2. At x = 0.10, the contrast of some small grains was different, consistent with the elemental mapping (Figure 5), which is in agreement with the XRD data (Figure 1), showing the formation of a secondary phase (ZrO₂) [30,31]. This confirmed the presence of Ti and O in the TiO₂ sample, while for the doped samples, Ti, O, and Zr were homogeneously distributed. However, for samples with x = 0.10, there are Zr-rich regions that show the formation of the secondary phase (Figure 5), consistent with the XRD and SEM.

Diffuse reflectance spectroscopy (DRS) studies were performed in the region 200–800 nm to evaluate the optical bandgap and the effect of Zr⁴⁺ doping on the bandgap of rutile TiO₂. Besides studying the possible transitions between conduction and the valence band, we can also find any other transition due to impurities with DRS as well. Figure 6a shows the DRS spectrum of Ti_1−xZr_xO₂ (x = 0–0.1) samples, indicating a sharp decrease in reflectance at about 310 nm for the un-doped rutile sample, caused by strong absorption. After increasing the concentration of Zr⁴⁺, the absorption edge shifted to a higher wavelength for x ≤ 0.075, showing a decrease in the optical band gap energy. This absorption may be attributed to the charge transfer between O_2p and M_3d (M = Ti, Zr) [46]. The optical band gap ($E_g$) was calculated by solving the Kubelka-Munk equation (Equation (7)).

$$[F(R_{\infty })h\upsilon ]=\alpha {(h\upsilon -E_g)}^n$$

(7)

where $F(R_{\infty })$ is the Kubelka–Munk function, hv is energy, $E_g$ is band gap energy, and $\alpha$ is the proportionality constant. The power $n$ describes the type of electronic transition and has a constant value, i.e., $n$ = 0.5 for direct allowed and $n$ = 2 for indirect allowed $E_g$. Furthermore, $n$ = 3/2 and $n$ = 3 are used for direct and indirect forbidden energy gaps, respectively [47,48]. The key parameters affecting the bandgap of materials are the crystallinity, processing technique, cell parameters, and average particle size [49]. The $E_g$ values were calculated from the Tauc plot (Figure 6b). $E_g$ for the x = 0 sample was estimated to be 2.92 eV, which decreased to 2.62 eV with the Zr concentration from x = 0 to 0.075 because Zr doping will modify the band structure and hence band gap. It has been speculated that the position of the conduction band has decreased while the position of the valence band has remained unchanged [46,50,51]. With a further increase in the Zr concentration (x = 0.1), the bandgap increased to 2.66 eV due to the presence of the ZrO₂ phase, which has a higher band gap value, i.e., 4.6−5.25 eV [52,53].

Table 2. MW dielectric properties and optical bandgap of Ti_1−xZr_xO₂.

x	ST (°C)	Relative Density (%)	εr	αtheo (Å3)	Q × fo (GHz)	τf (ppm/°C)	Packing Fraction (%)	Bandgap Energy (eV)
0	1300	93	96	6.95	9500	417	70.6	2.92
0.025	1375	91	99	6.81	32,360	424	70.4	2.84
0.05	1375	89	101	6.96	29,710	432	70.0	2.81
0.075	1400	89	103	6.97	21,050	429	69.6	2.62
0.1	1400	--	99	6.98	18,800	453	69.3	2.68

shows the MW dielectric properties of the rutile Ti_1–xZr_xO₂ (x = 0−0.1) ceramics. A high ε_r for rutile Ti_1–xZr_xO₂ (96–103) was, observed which may be due to their characteristic extended apical bonds as compared to the equatorial bonds causing the rattling of cations in a rutile structure [20]. ε_r increased from 96 (x = 0) to 103 (x = 0.075) with an increase in the dopant (Zr) concentration and then decreased to 99 for sample x = 0.10. The enhancement of the ε_r could be related to the Zr substitution at the Ti lattice site [54]. This increase may also be attributed to the complex process of grain growth and densification of the ceramic [55,56]. Another crucial factor contributing to the enhancement of ε_r is the ionic polarizability. The ionic polarizability of Zr⁴⁺ (3.25 ų) is higher than that of Ti⁴⁺ (2.93 ų), which may be the phenomenon behind the increase in the ε_r [57]. Therefore, theoretical dielectric polarizability and ionic polarizabilities (${\alpha }_{\mathrm{theo}}$) were calculated using Shannon’s additivity rule [57] as follows:

$${\alpha }_{\mathrm{theo}}=(1-x){\alpha }_{T{i}^{4+}}+x{\alpha }_{Z{r}^{4+}}+2{\alpha }_{O^{2-}}$$

(8)

where ${\alpha }_{T{i}^{4+}}$, ${\alpha }_{Z{r}^{4+}}$, and ${\alpha }_{O^{2-}}$ are the ionic polarizabilities of Ti, Zr, and O, respectively. The increasing trend of the ${\alpha }_{\mathrm{theo}}$ is in good agreement with the increasing trend of ε_r; therefore, it can be concluded that in rutile Ti_1−xZr_xO₂, the change in ε_r arises from the ionic polarizability difference, as shown in Table 2. The non-linear behavior of ε_r for sample x = 0.1 is because of the presence of ZrO₂ as a secondary phase, which has an ε_r value of 15–40, which depends upon the frequency and crystal structure [58], leading to the decrease in the ε_r of the x = 0.1 sample. In addition, Ravichandran et al. [59] established a relationship between the ε_r and optical bandgap energy, i.e., the ε_r is inversely proportional to the $E_g$ [60], which is consistent with our study. With a decrease in the $E_g$ value, the ε_r values increased.

Generally, the microstructures, densities, grain size, crystal lattice, packing fraction, secondary phases, and valency of Ti in the Ti-based ceramics affect the Q × f_o [61,62,63]. The Q × f_o initially increased from 9500 (x = 0) to 32,366 GHz (at x = 0.025) and then decreased to 18,798 GHz (at x = 0.1). The relative density of the samples showed a decreasing trend with an increase in the Zr concentration, which is consistent with the previous study [23]. At 1300 °C, TiO₂ has a density of ~93% but shows oxygen deficiency due to a low diffusion coefficient, making it unable to allow the oxygenation of the ceramic. However, oxygen diffused only to the surface of the ceramic because the core appears dark due to the presence of the reduced Ti³⁺ species, i.e., the bulk non-stoichiometry of oxygen is less, but it may be sufficiently large in the core to provoke the formation of condensation/shear structures of oxygen vacancies, just like that of Magnelli phases [17]. The reduction in TiO₂ increases the dielectric loss, leading to the deterioration of the Q × f_o. In comparison, the doping of Zr⁴⁺ in TiO₂ has overcome the problem and enhanced the Q × f_o value. This phenomenon can be explained using the shift in the E_g band towards the lower wavenumber (Figure 2) with an increasing Zr concentration, which is attributed to the decrease in the oxygen vacancies in the ceramics [32]. Similarly, this fact is also confirmed using EPR spectroscopy, which showed a decrease in the concentration of the Ti³⁺ paramagnetic center. The Q × f_o of ceramics, having a rutile structure, also strongly depends on the c/a ratio [19,64], which agrees with the present findings (Table 1). The slight decrease in the Q × f_o from 32,366 (x = 0.025) to 18,798 GHz (x = 0.1) may be due to an increase in the number of voids (Figure 4). The Q × f_o and lattice vibrations of the rutile TiO₂ can also be influenced by the packing fraction (PF). The PF shows the density of atomic stacking and can be calculated using Equation (9) [65]:

$$\%PF=\left[\frac{volume of the packed ions}{volume of the unit cell}\right]\times Z$$

(9)

where $Z$ shows the formula units per unit cell. As the Q × f_o and PF are directly proportional, the decrease in Q × f_o may also be attributed to the decrease in PF with an increase in the Zr concentration, as given in Table 2. At x = 0.10, the presence of the ZrO₂ phase may also be responsible for the decrease in the Q × f_o [23].

${\tau }_f$ slightly increased with an increase in the Zr concentration. The mechanism responsible for the increase in the ${\tau }_f$ is mostly the increase in ε_r because the materials with high ε_r have high ${\tau }_f$ and vice versa [66]. This may also be attributed to the substitution of Zr⁴⁺, a larger cation, for Ti⁴⁺, a smaller cation, which distorts the octahedra, increasing the ${\tau }_f$ of the samples. Figure 7b shows the dependence of the ${\tau }_f$ on the unit cell volume of Ti_1−xZr_xO₂ (x = 0–0.1) ceramics. ${\tau }_f$ is the combined effect of thermal expansion (${\alpha }_L$), as well as the temperature coefficient of permittivity (${\tau }_{\epsilon }$), as shown in the Equation (10) [67]:

$${\tau }_f=-\frac{1}{2}{\tau }_{\epsilon }-{\alpha }_L$$

(10)

The ${\alpha }_L$ value for ceramics is constant and in the range of 10 ppm/°C; therefore, the magnitude and sign of the ${\tau }_f$ depend on ${\tau }_{\epsilon }$. According to the Clausius–Mossotti relation, ${\tau }_{\epsilon }$ is given by Equations (11) and (12) [67].

$${\tau }_{\epsilon }=\frac{1}{\epsilon }\left(\frac{\partial \epsilon }{\partial T}\right)=\frac{(\epsilon -1)(\epsilon +2)}{\epsilon }(A+B+C)$$

(11)

$$\left(A=-\frac{1}{3V}{\left(\frac{\partial V}{\partial T}\right)}_p,B=\frac{1}{3{\alpha }_m}{\left(\frac{\partial {\alpha }_m}{\partial V}\right)}_T{\left(\frac{\partial V}{\partial T}\right)}_p,C=\frac{1}{3{\alpha }_m}{\left(\frac{\partial {\alpha }_m}{\partial T}\right)}_V\right)$$

(12)

Here, ${\alpha }_m$ represents polarizability, while Havinga and Bosman [67,68] relate the terms $A$ and $B$ to the expansion of volume, but their effects cancel out each other due to their equal magnitude and opposite sign. As shown by Lee [67], the term $C$ is the restoring force acting upon the ions and correlates with the potential well’s shape, which shows its high dependence on the lattice parameters and structure. The ${\tau }_{\epsilon }$ proportionally decreased with the increase in the volume of the unit cell, as with an increase in x, which in turn increased the ${\tau }_f$.

The ${\tau }_f$ is also related to the structural characteristics of the oxygen octahedra, such as the bond strength ($s$), bond length, and degree of covalency of constituents [65,69]. These characteristics can be calculated based on the degree of covalency, which is related to covalency ($f_c$) and $s$ using the following equations [70,71].

$$s={\left(\frac{R}{R_1}\right)}^{-N}$$

(13)

$$f_c=a{\left(\frac{R}{R_1}\right)}^{-NM}$$

(14)

$$degree of covalency=\frac{a{\left(\frac{R}{R_1}\right)}^{-NM}}{{\left(\frac{R}{R_1}\right)}^{-N}}\times 100=\frac{a{s}^M}{s}\times 100=\frac{f_c}{s}\times 100$$

(15)

where $R_1$, $N$, $a$, and $M$ are constants. The values of $R_1$ and $N$ depend on the cation site, while $a$ and $M$ depend on the number of electrons. The values of these constants are already reported in Refs. [70,71,72]. $R$ is the average bond length obtained from the Rietveld refinement results. The degree of the Ti/Zr−O bond covalency increases from 38.441 to 38.756% with an increase in the Zr concentration as given in

Table 3. Bond strength and covalency of Ti_1−xZr_xO₂ (x = 0–0.1).

x	Bond Type	R (Å)	R1	N	s	fc	Covalency (%)	Degree of Covalency (%)
0	Ti−O	1.9601	1.806	5.2	0.6532	0.2511	38.441	38.441
0.025	Ti−O	1.9633	1.806	5.2	0.6477	0.2477	38.255	38.496
	Zr−O		1.950	6	0.9600	0.4596	47.874
0.05	Ti−O	1.9664	1.806	5.2	0.6424	0.2446	38.077	38.554
	Zr−O		1.950	6	0.9509	0.4528	47.616
0.075	Ti−O	1.9699	1.806	5.2	0.6365	0.2411	37.876	38.585
	Zr−O		1.950	6	0.9408	0.4453	47.327
0.1	Ti−O	1.9710	1.806	5.2	0.6346	0.2400	37.814	38.756
	Zr−O		1.950	6	0.9377	0.4429	47.237

. Therefore, an increase in ${\tau }_f$ values may be associated with the decrease in s, which is in good agreement with Q × f_o, i.e., a decreasing trend from x = 0.025–0.1.

The relationship among the relative permittivity, structure parameter, and ${\tau }_f$ value of rutile Ti_1−xZr_xO₂ (x = 0–0.1) is also important. ε_r and ${\tau }_f$, for rutile tetragonal ceramic, mainly dependent on the tetragonality (c/a ratio) [73] or octahedral distortion [74]. The distortion of each TiO₆ octahedra is affected by the c/a ratio and oxygen positional parameter ($\mu$). The relationship between $\mu$ and c/a ratio is in the given (Equation (16)) [73].

$$\mu =\frac{2-{\left(4-2\left(1-{\left(\frac{d_e}{d_a}\right)}^2\right)\left({\left(\frac{c}{a}\right)}^2+2\right)\right)}^{\frac{1}{2}}}{4\left(1-{\left(\frac{d_e}{d_a}\right)}^2\right)}$$

(16)

where $d_e$ and $d_a$ are the lengths of four equatorial and two apical Ti—O bonds, respectively, and can be calculated from $a$, $c$, and $\mu$ [64]. In special cases, when $\mu$ = $\mu$* =$\frac{1}{4}\left[1+\frac{1}{2}{\left(\frac{c}{a}\right)}^2\right]$, all the $d_e$ and $d_a$ bonds are equal. A perfect octahedron in a tetragonal structure would have an ideal c/a ≈ 0.586 and $\mu$_ideal = 1/2 (c/a)_ideal, giving all twelve O—O bonds equal lengths [64]. The degree of octahedral distortion can be indicated by the c/a ratio up to some extent [64]. Figure 7a shows the dependence of τ_f and ε_r on the c/a ratio of Ti_1−xZr_xO₂ (x = 0–0.1) ceramics. It is observed that the c/a ratio of Ti_1−xZr_xO₂ increased with an increase in the Zr concentration showing an increase in ε_r and τ_f. The c/a ratio for the Ti_1−xZr_xO₂ (x = 0–0.1) ceramic was higher than the ideal c/a ratio (~0.586) and increased with an increase in x, which resulted in a high distortion of the TiO₆ octahedra. When the temperature increased, the rise in thermal energy was supposed to be absorbed completely in recovering the octahedral distortion, rather than in restoring the direct dependence of the polarizability on temperature [74], which means that the tetragonality ratio would change towards the ideal c/a ratio [64]. When the c/a ratio is closer to the (c/a)_ideal (0.586), then there will be a slight change in c/a upon the increase in temperature from 25 to 85 °C. Therefore, an increase in the τ_f value for the Ti_1−xZr_xO₂ solid solution is due to an increase in the c/a ratio.

4. Summary

In this study, Ti_1−xZr_xO₂ ceramics were processed and their crystal structure, defect chemistry, microstructure, optical bandgap energy, and MW dielectric properties were investigated. Rutile TiO₂ crystallized into a tetragonal structure (P4₂/mnm). At x = 0.1, a secondary phase (ZrO₂) was observed, which shows that the solubility limit of Zr is below 0.10. The SEM micrographs showed an increase in the grain size from 5 µm to 19 µm with an increasing Zr concentration. The electron paramagnetic resonance spectroscopy showed the existence of Ti³⁺ species in the rutile sample, which reduces with the increase in the Zr⁴⁺ concentration. The Tauc plots showed a decreasing trend in the bandgap energies of Ti_1−xZr_xO₂ (x = 0–0.075), i.e., 2.92−2.62 eV. The relative permittivity slightly increased, due to the extended apical bonds, as compared to the equatorial bonds causing the rattling of cations in the rutile structure. The quality factor initially increased with an increase in the Zr concentration (i.e., x = 0.025), which was explained based on a decrease in the coring effect and hence a decrease in oxygen vacancies. Above x = 0.025, the quality factor started decreasing due to a decrease in the packing fraction. Similarly, the temperature coefficient of the resonance frequency increased with an increase in x, due to an increase in the tetragonality (c/a ratio), which is indirectly associated with the distortion of the octahedra. Similarly, the degree of covalency also decreased, which is also a reason for the increase in ${\tau }_f$.