Effect of (Ba_1/3Nb_2/3)⁴⁺ Substitution on Microstructure, Bonding Properties and Microwave Dielectric Properties of Ce₂Zr₃(MoO₄)₉ Ceramics

Abstract

In this study, Ce₂[Zr_1−x(Ba_1/3Nb_2/3)_x]₃(MoO₄)₉ (0.02 ≤ x ≤ 0.1, CZ_1−xN_x) ceramics were sintered at 600 °C and 700 °C using the traditional solid-state method. An analysis conducted through XRD and Rietveld refinement confirmed that all the CZ_1−xN_x ceramics displayed a single phase with a trigonal structure (space group R-3c). The observed increases in cell volume with increasing x values indicate the successful substitution of (Ba_1/3Nb_2/3)⁴⁺. The high densification of the synthesized phase was validated by the density and SEM results. Additionally, the P-V-L theory demonstrates a strong correlation between the Ce-O bond and ε_r, as well as τ_f, and between the Mo-O bond and Q×f. Notably, the CZ_0.98N_0.02 ceramics demonstrated superior performance at 675 °C, exhibiting ε_r = 10.41, Q×f = 53,296 GHz, and τ_f = −23.45 ppm/°C. Finally, leveraging CZ_0.98N_0.02 ceramics as substrate materials enabled the design of a patch antenna suitable for the 5G communication band, demonstrating its significant potential in this field.

1. Introduction

The advent of 5G has precipitated the rapid advancement of novel technologies such as autonomous driving, intelligent transportation, and smart manufacturing, marking the onset of an era characterized by the intelligent interconnection of all things. Microwave dielectric ceramics play a pivotal role in the evolution of 5G technology due to their substantial market size. Furthermore, 5G technology necessitates lower signal latency, increased information-carrying capacity, and broader coverage, imposing heightened requirements on related microwave components [1,2,3,4,5]. It is noteworthy that materials with low dielectric constants (ε_r) offer advantages such as enhanced signal transmission speed, reduced coupling loss between ceramic substrates and metal electrodes, and a high-quality factor (Q×f), rendering them the preferred choice for 5G technology. The nearly zero temperature coefficient of resonant frequency (τ_f) exhibited by ceramic substrates also constitutes an essential factor in maintaining signal stability within communication technology [6,7,8,9]. Furthermore, the utilization of low-temperature co-fired ceramics (LTCC) technology contributes to enhanced circuit density, facilitating device miniaturization and compliance with current environmental standards. This renders LTCC technology a highly promising multi-layer ceramic packaging solution. In LTCC technology, the firing temperature is typically maintained at a lower range of 800 °C to 900 °C to accommodate the sintering of ceramic materials with Au (melting temperature 1063 °C), Ag (melting temperature 961 °C), and Cu (melting temperature 1084 °C) electrode materials [10]. Notably, Mo-based ceramics have garnered attention from researchers due to their comparatively low sintering temperature and reduced losses [11,12].

In the double-molybdate system, Re₂Zr₃(MoO₄)₉ (Re = rare earth elements) has been reported to exhibit superior performance and can be synthesized at temperatures below 900 °C, showing great potential for development in LTCC applications [13,14,15]. Within this system, the Eu₂Zr₃(MoO₄)₉ ceramics demonstrate exceptional performance (ε_r = 10.75, Q×f = 74,900 GHz, τ_f = −8.88 ppm/°C), and they can be synthesized at 600 °C [15]. It is noteworthy that Tao et al. [12] and Shi et al. [16] have documented the low sintering temperature of Ce₂Zr₃(MoO₄)₉ (CZM) ceramics (575 °C and 750 °C respectively), along with their lowest τ_f value (−1.29 ppm/°C and −7.10 ppm/°C), indicating significant commercial application potential. However, the relatively low Q×f value (19,062 GHz and 24,720 GHz) of the CZM ceramics limits further scalability. Significant progress has been achieved in the doping and modification of CZM ceramics in recent years, representing an exciting development. Zheng et al. conducted studies on the substitution of Sn and Ti at the Zr site, resulting in Q×f value increases to 72,390 GHz and 84,200 GHz, respectively, while maintaining the τ_f value around −10 ppm/°C [17]. Furthermore, research on complex ion substitution at the Zr sites of CZM ceramics has expanded the range of substitution ions for performance enhancement. For instance, Xu et al. successfully elevated the Q×f value to 100,954 GHz at 750 °C by replacing the Zr site of CZM ceramics with (Zn_1/3Sb_2/3)⁴⁺. Xu et al. optimized the Q×f value to 59,381 GHz at 675 °C by substituting (Zn_1/3Nb_2/3)⁴⁺ for Zr in CZM ceramics [18]. Pan et al. also achieved an increased Q×f value of 82,696 GHz at 775 °C by replacing the Zr site with (Sr_1/3Sb_2/3)⁴⁺ [19]. It is evident that when using the pentavalent ion Nb for complex ion substitution in CZM ceramics, generally, a lower firing temperature is observed. Therefore, further research is warranted on (Ba_1/3Nb_2/3)⁴⁺ as a highly promising substitution ion in CZM ceramics.

This study presents a novel enhancement of the performance of CZM ceramics through the incorporation of (Ba_1/3Nb_2/3)⁴⁺ for the first time and, further, an examination of its impact on the microstructure and sintering characteristics of CZM ceramics. The Phillips-Van Vechten-Levine (P-V-L) valence bond theory is an analytical approach for investigating the correlation between properties and bonding characteristics. Therefore, the structure and properties of Ce₂[Zr_1−x(Ba_1/3Nb_2/3)_x]₃(MoO₄)₉ (0.02 ≤ x ≤ 0.1, CZ_1−xN_x) ceramics were elucidated using the P-V-L theory. Additionally, a patch antenna tailored for the 5G communication band was developed based on the CZ_0.98N_0.02 ceramic substrate.

2. Experimental Section

Following the stoichiometric ratio required for CZ_1−xN_x ceramics, high-purity CeO₂ (>99.9%, Macklin, Shanghai, China), ZrO₂ (>99.99%, Macklin, Shanghai, China), BaCO₃ (>99.99%, Macklin, Shanghai, China), Nb₂O₅ (>99.99%, Macklin, Shanghai, China), and MoO₃ (>99.5%, Macklin, Shanghai, China) powders were precisely weighed and synthesized via the conventional solid-phase reaction route. Initially, the weighed powders were a ball mill with zirconia balls and ethanol for 15 h of milling. Subsequently, the homogeneous slurry was extracted and subjected to drying at 80 °C in an oven. The resulting dried powders were then sieved through a 200-mesh screen and transferred to an alumina crucible. The precursor was formed by sintering the powder at 600 °C in a muffle furnace for 2 h. This precursor was then blended with 8 wt.% PVA before being pressed into cylindrical green bodies in a cylindrical mold under 200 MPa pressure. Finally, the cylindrical green bodies were placed in an alumina crucible and subjected to sintering at 600 to 700 °C in an air atmosphere for 4 h.

The phase of CZ_1−xN_x ceramics was determined using an X-ray diffractometer (XRD) model BRUKER D8 ADVANCE in ambient air at room temperature, and the data were refined employing the Rietveld method. The apparent density of CZ_1−xN_x ceramics was measured by the Archimedes displacement method at a test water temperature of 25.5 °C. The surface morphology of CZ_1−xN_x ceramics was characterized using a scanning electron microscope (SEM) model JEOL JSM-IT200. The ε_r and τ_f values of CZ_1−xN_x ceramics were measured utilizing a vector network analyzer, model N5234A, from Agilent for TE₀₁₁ mode. The τ_f value was determined by measuring the resonant frequency of the sample at 25 °C (f₂₅) and 85 °C (f₈₅) using a temperature chamber, and then calculated using Equation (1). The Q×f value of CZ_1−xN_x ceramics was obtained through the closed-cavity method in TE_01δ mode [9]. Furthermore, the patch antenna of CZ_1−xN_x ceramics was designed by initially calculating the values for radiation patch length (L_p), width (W_p), and feed position (X_f and Y_f) based on ε_r and the designed center frequency (f₀), followed by optimization using ANSYS HFSS software [20,21].

$${\tau }_f={\displaystyle \frac{f_{85}-f_{25}}{60f_{25}}}$$

(1)

$$W_p={\displaystyle \frac{c}{2f_0}}{\left({\displaystyle \frac{{\epsilon }_r+1}{2}}\right)}^{-0.5}$$

(2)

$$L_p={\displaystyle \frac{c}{2f_0\sqrt{{\epsilon }_e}}}-2\mathsf{\Delta }L$$

(3)

$$\mathsf{\Delta }L=0.412h{\displaystyle \frac{({\epsilon }_e+0.3)(\frac{W_p}{h}+0.264)}{({\epsilon }_e-0.258)(\frac{W_p}{h}+0.8)}}$$

(4)

$${\epsilon }_e={\displaystyle \frac{{\epsilon }_r+1}{2}}+{\displaystyle \frac{{\epsilon }_r-1}{2}}{\left(1+{\displaystyle \frac{12h}{W_p}}\right)}^{-0.5}$$

(5)

$${\xi }_{\mathrm{re}}(L)={\displaystyle \frac{{\epsilon }_r+1}{2}}+{\displaystyle \frac{{\epsilon }_r-1}{2}}{\left(1+{\displaystyle \frac{12h}{L_p}}\right)}^{-0.5}$$

(6)

$$X_f={\displaystyle \frac{L}{2\sqrt{{\xi }_{re}\left(L\right)}}}$$

(7)

$$Y_f=0$$

(8)

where c represents the speed of light, ΔL denotes the corrected length, ε_e signifies the effective dielectric constant, and h stands for the thickness of the CZ_1−xN_x ceramics substrate.

3. Results and Discussion

The XRD patterns of CZ_1−xN_x ceramics sintered at the optimum temperature are depicted in Figure 1. All diffraction peaks of the CZ_1−xN_x ceramics exhibit identical positions to those of Pr₂Zr₃(MoO₄)₉ (PDF# 51-1851), with no additional diffraction peaks observed [17,18]. This observation suggests that (Ba_1/3Nb_2/3)⁴⁺ was successfully incorporated into the crystal lattice of the CZM ceramics, forming a trigonal structure (space group R-3c) within the CZ_1−xN_x solid solution. Furthermore, as the value of x increases, all diffraction peaks of CZ_1−xN_x ceramics display a leftward shift, which is attributable to the larger ion radius of (Ba_1/3Nb_2/3)⁴⁺ (1.41 Å, CN = 6) compared to that of Zr⁴⁺ (0.72 Å, CN = 6) [22]. This finding further validates the successful substitution by (Ba_1/3Nb_2/3)⁴⁺ at Zr sites in the CZM ceramics. To investigate the influence of the (Ba_1/3Nb_2/3)⁴⁺ on the crystal structure of the CZM ceramics, the diffraction peaks of the CZ_1−xN_x ceramics were refined using the Rietveld method, as illustrated in Figure 2 and

Table 1. The lattice parameters and reliability factors of CZ_1−xN_x ceramics sintered at the optimal sintering temperatures.

x	Lattice Parameter					Reliability Factors
	a = b (Å)	c (Å)	α = β (°)	γ (°)	Vm (Å3)	Rp (%)	Rwp (%)	χ2
0.02	9.8405	58.9013	90	120	4939.56	6.17	8.07	2.52
0.04	9.8356	58.8950	90	120	4934.08	5.25	6.65	1.70
0.06	9.8337	58.8883	90	120	4931.71	5.60	7.06	1.85
0.08	9.8404	58.9261	90	120	4941.58	4.96	6.31	1.49
0.10	9.8445	58.9552	90	120	4948.08	4.76	6.08	1.49

. Figure 2 demonstrates a good match between the measured and fitted diffraction peaks. Furthermore, the R_p, R_wp, and χ² values obtained after the refinement are all within 10, indicating the reliability of the refined structure [23]. The crystal lattice parameters of the CZ_1−xN_x ceramics are depicted in Figure 3. Owing to the disparity in ionic radii, the collective values of a, b, c, and V_m exhibit an increasing trend with higher substitution levels, corresponding to the shift of the peaks observed in the XRD pattern.

The crystal structure of the CZ_1−xN_x ceramics is illustrated in Figure 4. The CZ_1−xN_x ceramics consist of [CeO₉] polyhedra, [ZrO₆] polyhedra, and [MoO₄] tetrahedra interconnected with each other. Ce atoms are coordinated with 9 × O atoms (3 × O(1), 3 × O(2), and 3 × O(6)) to form [CeO₉] polyhedra. Both Zr(1) and Zr(2) atoms are coordinated with 6 × O atoms (Zr(1): 6 × O(4), Zr(2): 3 × O(1) and 3 × O(5)) to form [ZrO6] polyhedra. Similarly, both Mo(1) and Mo(2) atoms are coordinated with four O atoms (Mo(1): O(1), O(2), O(3), O(4), Mo(2): 2 × O(5) and 2 × O(6)) to form [MoO₄] tetrahedra. In the CZ_1−xN_x ceramics, the [CeO₉] polyhedra and [ZrO₆] polyhedra are not directly connected; they are linked through [MoO₄] tetrahedra serving as a bridge [18,24].

The performance of ceramics is influenced by both intrinsic and extrinsic factors. Extrinsic factors, such as second phases and density, play a significant role. The XRD results confirmed the purity of the CZ_1−xN_x ceramics phases, thus necessitating an investigation into the relationship between sintering behavior and the performance of CZ_1−xN_x ceramics. Figure 5 illustrates the diameter shrinkage behavior and density changes of the CZ_1−xN_x ceramics with temperature. With the increase in the sintering temperature, the shrinkage rate of the CZ_1−xN_x ceramics with varying x values exhibited a pattern of initial augmentation followed by diminishment. This phenomenon is attributable to grain growth and pore elimination during sintering. However, excessive temperatures can induce secondary crystallization in the grains, resulting in reduced shrinkage rates. The trends observed in the apparent density align closely with those of the shrinkage rate. Furthermore, at the optimal sintering temperature, the relative density of the CZ_1−xN_x ceramics consistently exceeded 90%, underscoring their favorable densification.

Figure 6 illustrates the microstructural images of the CZ_1−xN_x ceramics at the optimal sintering temperature for each dopant content. The presence of atomic arrangement disorder and vacancies at grain boundaries has a detrimental effect on performance. Moreover, the well-developed regular grain morphology indicates favorable grain growth, contributing to performance enhancement [25,26]. Consequently, the CZ_0.98N_0.02 ceramics exhibit the largest grain size and most regular grain shape, positively impacting performance improvement. It is noteworthy that all the CZ_1−xN_x ceramics demonstrated high densities without discernible pores, which is consistent with the relative density results.

Figure 7a illustrates the temperature-dependent variation in the ε_r for the CZ_1−xN_x ceramics. The consistency between the ε_r and apparent density suggests that density is the primary factor influencing the dielectric constant of the ceramic. As evidenced by the presence of pores in the ceramic, where air has an ε_r value of 1, it was observed that the ε_r value of the CZ_1−xN_x ceramics significantly exceeded this baseline, indicating that increased porosity leads to a reduction in dielectric constant [19]. Consequently, considering the presence of pores, Equation (9) was employed to correct for the ε_r (ε_corr.) of the CZ_1−xN_x ceramics, as depicted in Figure 7c, where P (P = 1 − ρ_relative) is the porosity. Moreover, the relevant numerical values of the CZM ceramics matrix were derived from the reported findings of Tao et al. and Shi et al. [12,16]. The values of ε_r and ε_corr. increased proportionally with x, a trend explicable by Equation (10) [27]. As per the Clausius–Mossotti equation, it is evident that ε_r exhibits a positive correlation with the polarizability (α). Additionally, based on the α values reported by Shannon et al. [28], the theoretical polarizability (α_theo.) of the CZ_1−xN_x ceramics was calculated using Equation (11).

$${\epsilon }_{corr.}={\epsilon }_r\left(1+1.5P\right)$$

(9)

$${\displaystyle \frac{{\epsilon }_r-1}{{\epsilon }_r+2}}={\displaystyle \frac{4\pi \alpha }{3V_m}}$$

(10)

α_theo.(CZ_1−xN_x) = 2α(Ce³⁺) + 3(1−x)α(Zr⁴⁺) + xα(Ba²⁺) + 2xα(Nb⁵⁺) + 9α(Mo⁶⁺) + 36α(O²) = 2 × 6.15 + 3(1−x) × 3.25 + x × 6.04 + 2x × 3.97 + 9 × 3.28 + 36 × 2.01

(11)

Figure 7. CZ_1−xN_x ceramics (a) ε_r at a sintering temperature of 600 ℃ to 700 ℃, (b) ε_r, (c) ε_corr., (d) α_theo., and (e) f_iave.(Ce-O) at the optimal sintering temperature as a function of x values.

Figure 7. CZ_1−xN_x ceramics (a) ε_r at a sintering temperature of 600 ℃ to 700 ℃, (b) ε_r, (c) ε_corr., (d) α_theo., and (e) f_iave.(Ce-O) at the optimal sintering temperature as a function of x values.

Figure 7d illustrates the continuous increase in the α_theo. values of the CZ_1−xN_x ceramics as the x value increases, aligning with the observed trend in the ε_r values. This observation suggests that α plays a significant role as an internal factor contributing to the enhancement of ε_r values. Furthermore, the P-V-L theory serves as a widely adopted approach for investigating the correlation between material structure and performance [29,30]. Within CZ_1−xN_x ceramics, an analysis of the interna influence of (Ba_1/3Nb_2/3)⁴⁺ on the CZM ceramics was conducted through calculations of chemical bond variations using the P-V-L theory. Initially, complex chemical formulas were decomposed into binary bond formulas, as depicted in Equation (12).

$$\begin{gathered} {\mathrm{Ce}}_2\left[\right({\mathrm{Zr}}_{1-x}\left({\mathrm{Ba}}_{1/3}{\mathrm{Nb}}_{2/3}\right)x{]}_3({\mathrm{MoO}}_4{)}_9 \\ {\mathrm{Ce}}_{2/3}\mathrm{O}{\left(1\right)}_3+{\mathrm{Ce}}_{2/3}\mathrm{O}{\left(2\right)}_3+{\mathrm{Ce}}_{2/3}\mathrm{O}{\left(6\right)}_3+\mathrm{Zr}/({\mathrm{Ba}}_{1/3}{\mathrm{Nb}}_{2/3}){\left(1\right)\mathrm{O}\left(4\right)}_3+\mathrm{Zr}/({\mathrm{Ba}}_{1/3}{\mathrm{Nb}}_{2/3}){\left(2\right)\mathrm{O}\left(3\right)}_3 \\ +\mathrm{Zr}/({\mathrm{Ba}}_{1/3}{\mathrm{Nb}}_{2/3}){\left(2\right)\mathrm{O}\left(5\right)}_3+\mathrm{Mo}{\left(1\right)}_{3/2}{\mathrm{O}\left(1\right)}_3+{\mathrm{Mo}\left(1\right)}_{3/2}{\mathrm{O}\left(2\right)}_3+{\mathrm{Mo}\left(1\right)}_{3/2}{\mathrm{O}\left(3\right)}_3+{\mathrm{Mo}\left(1\right)}_{3/2}{\mathrm{O}\left(4\right)}_3 \\ +{\mathrm{Mo}\left(2\right)}_{3/2}{\mathrm{O}\left(5\right)}_3+{\mathrm{Mo}\left(2\right)}_{3/2}{\mathrm{O}\left(6\right)}_3 \end{gathered}$$

(12)

In the P-V-L theory, the correlation between ionicity (f_i) and ε_r was established, and the impact of (Ba_1/3Nb_2/3)⁴⁺ on the ε_r of the CZM ceramics was determined by evaluating the change in f_i for each chemical bond. As depicted in Equation (13), there was a positive correlation between f_i and ε_r. Furthermore, Equations (14)–(17) were utilized to compute the f_i of each chemical bond in the CZ_1−xN_x ceramics [31,32].

$${\epsilon }_{\mathrm{r}}={\displaystyle \frac{n_0^2-1}{1-f_i}}+1$$

(13)

$$f_i^{\mu }={\displaystyle \frac{{\left(C^{\mu }\right)}^2}{{\left(E_g^{\mu }\right)}^2}}$$

(14)

$${\left(E_g^{\mu }\right)}^2={\left(E_h^{\mu }\right)}^2+{\left(C^{\mu }\right)}^2$$

(15)

$${\left(E_h^{\mu }\right)}^2={\displaystyle \frac{39.74}{{\left(d^{\mu }\right)}^{2.48}}}$$

(16)

$${\left(E_h^{\mu }\right)}^2={\displaystyle \frac{39.74}{{\left(d^{\mu }\right)}^{2.48}}}{C}^{\mu }=14.4b^{\mu }\exp \left(-k_s^{\mu }{r}_0^{\mu }\right)\left[{\left(Z_A^{\mu }\right)}^{*}-{\displaystyle \frac{n}{m}}{\left(Z_B^{\mu }\right)}^{*}\right]/r_0^{\mu }$$

(17)

where n₀, d^μ, and b^μ denote the refractive index, bond length, and periodic correction factor, respectively. ($Z_A^{\mu }$)* and ($Z_B^{\mu }$)* represent the effective valence electron numbers of the cation and anion, respectively. Additionally, exp(−$k_s^{\mu }{r}_0^{\mu }$) represents the Thomas-Fermi factor. The calculated f_i values for each chemical bond in the CZ_1−xN_x ceramics are summarized in

Table 2. The f_i of the CZ_1−xN_x ceramics sintered at optimal temperature.

Bond Type	x = 0.02	x = 0.04	x = 0.06	x = 0.08	x = 0.10
Ce-O(1) a	0.8393	0.8490	0.8476	0.8506	0.8517
Ce-O(1) b	0.8393	0.8490	0.8476	0.8506	0.8518
Ce-O(1) c	0.8393	0.8490	0.8476	0.8507	0.8518
Ce-O(2) a	0.8382	0.8509	0.8485	0.8493	0.8533
Ce-O(2) b	0.8382	0.8509	0.8485	0.8493	0.8533
Ce-O(2) c	0.8382	0.8510	0.8485	0.8494	0.8533
Ce-O(6) a	0.8638	0.8561	0.8511	0.8534	0.8538
Ce-O(6) b	0.8638	0.8561	0.8511	0.8534	0.8538
Ce-O(6) c	0.8639	0.8561	0.8511	0.8534	0.8539
Zr(BaNb)1-O(4) × 6	0.7865	0.7913	0.7952	0.7923	0.7913
Zr(BaNb)2-O(3) a	0.7666	0.7817	0.7781	0.7821	0.7835
Zr(BaNb)2-O(3) b	0.7666	0.7817	0.7781	0.7821	0.7836
Zr(BaNb)2-O(3) c	0.7666	0.7817	0.7782	0.7820	0.7836
Zr(BaNb)2-O(5) a	0.7643	0.7876	0.7762	0.7803	0.7836
Zr(BaNb)2-O(5) b	0.7644	0.7877	0.7762	0.7803	0.7836
Zr(BaNb)2-O(5) c	0.7644	0.7877	0.7763	0.7803	0.7837
Mo1-O(1)	0.7021	0.7170	0.7046	0.7103	0.7117
Mo1-O(2)	0.7095	0.7191	0.7062	0.7179	0.7198
Mo1-O(3)	0.7161	0.7370	0.7289	0.7312	0.7390
Mo1-O(4)	0.7048	0.7315	0.7222	0.7267	0.7280
Mo2-O(5) × 2	0.7104	0.7251	0.7268	0.7307	0.7317
Mo2-O(6) × 2	0.6281	0.7037	0.7006	0.7030	0.7104

^a–c Three different bonds.

. Notably, the f_i value of the Ce-O bond surpasses that of other chemical bonds. Moreover, the average f_i value (f_iave.(Ce-O)) for this bond was computed, as shown in Figure 7e. It was observed that the ε_r exhibited a similar upward trend to the f_iave.(Ce-O), suggesting that the Ce-O bond exerts a dominant influence on the ε_r of CZ_1−xN_x ceramics.

Figure 8a illustrates the variation in the Q×f values for the CZ_1−xN_x ceramics at different temperatures. The trend in the Q×f values mirrors that of the apparent density, suggesting that density is the primary external factor influencing Q×f values. Lattice energy (U) is commonly linked to the Q×f values of ceramics. The magnitude of the U value reflects the stability of the ceramic, while higher U values indicate greater compound stability, thereby reducing nonharmonic vibration-induced losses and, consequently, enhancing the Q×f value of the ceramic. According to the P-V-L theory, the U values for the chemical bonds in the CZ_1−xN_x ceramics were computed using Equations (18)–(21) [33].

$$U_{cal}=\sum _{\mu }{U}_b^{\mu }$$

(18)

$$U_b^{\mu }=U_{bc}^{\mu }+U_{bi}^{\mu }$$

(19)

$$U_{bc}^{\mu }=2100m{\displaystyle \frac{P_{A-B}^{1.64}}{d^{0.75}}}{f}_c^{\mu }$$

(20)

$$U_{bi}^{\mu }=1270{\displaystyle \frac{(m+n)P_{A-B}{P}_{B-A}}{d}}(1-{\displaystyle \frac{0.4}{d}})f_{\mathrm{i}}^{\mu }$$

(21)

where $U_{bc}^{\mu }$ and $U_{bi}^{\mu }$ represent the covalent and ionic bond energies of the material, respectively. $f_c^{\mu }$ and $f_{\mathrm{i}}^{\mu }$ represent the bond covalency and ionicity, respectively. $P_{A-B}$ and $P_{B-A}$ represent the valence states of the cation and anion, respectively.

Table 3. The U (kJ/mol) of the CZ_1−xN_x ceramics sintered at optimal temperature.

Bond Type	x = 0.02	x = 0.04	x = 0.06	x = 0.08	x = 0.10
Ce-O(1) a	1095	1100	1080	1076	1085
Ce-O(1) b	1095	1100	1080	1075	1085
Ce-O(1) c	1095	1100	1080	1075	1085
Ce-O(2) a	1103	1086	1074	1084	1074
Ce-O(2) b	1102	1085	1074	1084	1074
Ce-O(2) c	1102	1085	1074	1084	1074
Ce-O(6) a	906	1048	1055	1056	1070
Ce-O(6) b	906	1048	1055	1056	1070
Ce-O(6) c	906	1048	1055	1056	1070
Zr(BaNb)1-O(4) × 6	10,161	10,653	10,145	10,476	10,702
Zr(BaNb)2-O(3) a	3726	3711	3676	3665	3696
Zr(BaNb)2-O(3) b	3726	3709	3676	3665	3696
Zr(BaNb)2-O(3) c	3726	3709	3675	3666	3695
Zr(BaNb)2-O(5) a	3761	3613	3707	3694	3695
Zr(BaNb)2-O(5) b	3761	3612	3707	3694	3695
Zr(BaNb)2-O(5) c	3760	3611	3706	3694	3694
Mo1-O(1)	43,587	44,245	45,021	44,810	45,295
Mo1-O(2)	42,314	43,919	44,773	43,585	44,009
Mo1-O(3)	41,121	40,777	40,864	41,257	40,646
Mo1-O(4)	43,133	41,786	42,088	42,075	42,641
Mo2-O(5) × 2	42,157	42,900	41,254	41,343	41,993
Mo2-O(6) × 2	53,631	46,310	45,640	45,941	45,491

^a–c Three different bonds.

presents the specific U values for each chemical bond. The Mo-O bond exhibits a notably higher U value compared to the other two chemical bonds, indicating its significant influence on the Q×f values of the ceramic. Additionally, Figure 8b,c illustrates the Q×f and average U (U_ave.(Mo-O)) of the CZ_1−xN_x ceramics at optimal temperature. The sample with x = 0.02 demonstrates the highest CZ_1−xN_x value (53,296 GHz), significantly surpassing that of the CZM ceramics, suggesting that (Ba_1/3Nb_2/3)⁴⁺ serves as an effective substituted complex ion. Furthermore, U_ave.(Mo-O) displays a similar trend to the Q×f value, underscoring the major influence of the Mo-O bond on the Q×f in the CZ_1−xN_x ceramics.

The τ_f of a ceramic material characterizes its stability in diverse environments, and the closer τ_f is to 0, the greater the stability of the ceramic. As per Equation (22), an inverse relationship exists between the thermal expansion coefficient (α) and the τ_f value of the ceramics. The α value of the CZ_1−xN_x ceramics was determined using Equations (23)–(26) derived from the P-V-L theory [34,35].

$${\tau }_f=-\left({\displaystyle \frac{{\tau }_{\epsilon }}{2}}+\alpha \right)$$

(22)

$$\alpha =\sum _{\mu }{F}_{mn}^{\mu }{\alpha }_{mn}^{\mu }$$

(23)

$${\alpha }_{\mathrm{mn}}^{\mathsf{\mu }}=-3.1685+0.8376{\gamma }_{mn}$$

(24)

$${\gamma }_{mn}={\displaystyle \frac{k{Z}_A^{\mu }{N}_{CA}^{\mu }}{U_b^{\mu }{△}_A}}{\beta }_{mn}$$

(25)

$${\beta }_{mn}={\displaystyle \frac{m\left(m+n\right)}{2n}}$$

(26)

where τ_ε represents the temperature coefficient of the dielectric constant, $N_{CA}^{\mu }$ denotes the coordination number of the cation, $Z_A^{\mu }$ indicates the valence state of the cation, and k and Δ_A stand for the Boltzmann constant and the ionic periodicity constant, respectively. The α values of the chemical bonds in the CZ_1−xN_x ceramics are detailed in

Table 4. The α (10⁻⁶/K) of the CZ_1−xN_x ceramics sintered at optimal temperature.

Bond Type	x = 0.02	x = 0.04	x = 0.06	x = 0.08	x = 0.10
Ce-O(1) a	10.2227	10.1619	10.4087	10.4592	10.3461
Ce-O(1) b	10.2227	10.1619	10.4087	10.4719	10.3461
Ce-O(1) c	10.2227	10.1619	10.4087	10.4719	10.3461
Ce-O(2) a	10.1256	10.3337	10.4846	10.3586	10.4846
Ce-O(2) b	10.1377	10.3461	10.4846	10.3586	10.4846
Ce-O(2) c	10.1377	10.3461	10.4846	10.3586	10.4846
Ce-O(6) a	13.0163	10.8233	10.7304	10.7173	10.5356
Ce-O(6) b	13.0163	10.8233	10.7304	10.7173	10.5356
Ce-O(6) c	13.0163	10.8233	10.7304	10.7173	10.5356
Zr(BaNb)1-O(4) × 6	3.8445	3.5103	3.8340	3.6024	3.4493
Zr(BaNb)2-O(3) a	3.2064	3.2224	3.2734	3.2828	3.2189
Zr(BaNb)2-O(3) b	3.2064	3.2258	3.2734	3.2828	3.2189
Zr(BaNb)2-O(3) c	3.2064	3.2258	3.2751	3.2811	3.2207
Zr(BaNb)2-O(5) a	3.1471	3.3957	3.2195	3.2322	3.2207
Zr(BaNb)2-O(5) b	3.1471	3.3976	3.2195	3.2322	3.2207
Zr(BaNb)2-O(5) c	3.1488	3.3994	3.2212	3.2322	3.2224
Mo1-O(1)	−0.4054	−0.4465	−0.4934	−0.4808	−0.5096
Mo1-O(2)	−0.3223	−0.4263	−0.4786	−0.4053	−0.4319
Mo1-O(3)	−0.2397	−0.2150	−0.2213	−0.2494	−0.2055
Mo1-O(4)	−0.3763	−0.2863	−0.3070	−0.3061	−0.3441
Mo2-O(5) × 2	−0.3117	−0.3612	−0.2492	−0.2554	−0.3005
Mo2-O(6) × 2	−0.9229	−0.5679	−0.5297	−0.5470	−0.5211

^a–c Three different bonds.

. Among these ceramics, it was observed that the Ce-O bond exhibited the highest α value, thus exerting a significant influence on the τ_f value. As depicted in Figure 9, the average α value (α_ave.(Ce-O)) of the Ce-O bond demonstrated a similar trend to the τ_f. This suggests a strong correlation between the Ce-O bond and the τ_f value in the CZ_1−xN_x ceramics.

It is noteworthy that bond energy (E) is frequently employed to characterize the stability of materials. A higher E value indicates a stronger connection between chemical bonds. The E value of the CZ_1−xN_x ceramics was determined using Equations (27)–(31) [36].

$$E^{\mu }=t_c{E}_c^{\mu }+t_i{E}_i^{\mu }$$

(27)

$$E_c^{\mu }={\displaystyle \frac{\left(r_{cA}+r_{cB}\right)}{d^{\mu }}}{\left(E_{A-A}{E}_{B-B}\right)}^{1/2}$$

(28)

$$E_i^{\mathsf{\mu }}={\displaystyle \frac{1389.088}{d^{\mu }}}$$

(29)

$$t_i=\left|{\displaystyle \frac{\left(S_A-S_B\right)}{6}}\right|$$

(30)

$$\left(t_c+t_i\right)=1$$

(31)

Specifically, E_A–A and E_B–B represent the bond energies of atoms A and B, respectively, while r_cA and r_cB denote the covalent radii of ions A and B. The calculation results are presented in Figure 9c and

Table 5. The E (kJ/mol) of the CZ_1−xN_x ceramics sintered at optimal temperature.

Bond Type	x = 0.02	x = 0.04	x = 0.06	x = 0.08	x = 0.10
Ce-O(1) a	406.8505	410.6544	401.4945	399.9166	404.4490
Ce-O(1) b	406.7539	410.5723	401.4004	399.8233	404.3694
Ce-O(1) c	406.6895	410.5067	401.3377	399.7611	404.3058
Ce-O(2) a	410.0645	404.6082	398.7994	403.8610	399.6523
Ce-O(2) b	410.0155	404.5445	398.7530	403.8133	399.6057
Ce-O(2) c	409.9500	404.4967	398.6911	403.7499	399.5435
Ce-O(6) a	326.0845	388.2832	390.5555	391.2096	397.8577
Ce-O(6) b	326.0742	388.2539	390.5109	391.1798	397.8269
Ce-O(6) c	326.0224	388.1952	390.4516	391.1053	397.7653
Zr(BaNb)1-O(4) × 6	459.2762	489.8926	461.1076	481.5940	496.2735
Zr(BaNb)2-O(3) a	518.9672	518.3938	512.9597	512.3371	519.4946
Zr(BaNb)2-O(3) b	518.8108	518.2381	512.9597	512.3623	519.3393
Zr(BaNb)2-O(3) c	518.7847	518.2122	512.7823	512.5390	519.3134
Zr(BaNb)2-O(5) a	525.3543	500.8407	518.6251	517.7693	519.3134
Zr(BaNb)2-O(5) b	525.2741	500.7680	518.5733	517.7177	519.2358
Zr(BaNb)2-O(5) c	525.1139	500.6227	518.4178	517.5632	519.1066
Mo1-O(1)	592.0846	602.8673	619.4640	614.6348	623.9548
Mo1-O(2)	567.3968	596.3924	614.4154	590.1899	598.0154
Mo1-O(3)	544.9440	536.5641	538.9721	545.7791	534.0395
Mo1-O(4)	583.1916	555.2862	561.8104	561.0780	571.3516
Mo2-O(5) × 2	564.4201	576.4860	546.1831	547.3699	559.0432
Mo2-O(6) × 2	820.3593	645.2379	632.2044	637.9052	627.9761

^a–c Three different bonds.

. It is noteworthy that despite not displaying the highest E value, the Ce-O bond was indicated by the α results as linked to the τ_f value. Moreover, the average E value (E_ave.(Mo-O)) for the Ce-O bond aligned with the trend of the τ_f, suggesting the significant role played by the Ce-O bond in ceramic stability, which is consistent with the α calculation outcomes.

Microstrip antennas, which are widely utilized in wireless communication and radar fields due to their simple design, ease of production, relatively low cost, and very narrow bandwidth, were the focus of this study. A rectangular microstrip antenna with a center frequency of 3.47 GHz was designed using a coaxial feeding method and CZ_0.98N_0.02 ceramics as the basic materials for the antenna. It is noteworthy that 3.47 GHz falls within the S-band of the microwave radio frequency commonly employed in wireless communication and radar fields. Furthermore, it is situated within the frequency range of the 5G communication band (3.3 GHz~4.2 GHz), specifically the n77 and n78 frequency bands [37,38,39].

Figure 10a illustrates the optimized antenna model and corresponding design dimensions centered at 3.47 GHz, with radiation patches and ground parts designed using silver (Ag). The return loss parameters (S11) based on the designed model are depicted in Figure 10b. It is evident from the figure that the antenna’s center frequency is 3.47 GHz, exhibiting a bandwidth of 75 MHz and an S11 value of −33.15 dB. Notably, when S11 falls below −10 dB, the minimal reflection impact on the transmission system allows for normal antenna operation. Furthermore, Figure 10c–e presents simulations of the antenna’s radiation in both three-dimensional and electromagnetic field views. The distribution of the main lobe and side lobe radiation from the antenna is clearly delineated, with a maximum radiation gain of 4.32 dB observed. Additionally, unidirectional and symmetrical maximum electric field and magnetic field gain directions indicate favorable radiation characteristics for this antenna design. Consequently, CZ_0.98N_0.02 ceramics exhibit significant potential for application in 5G communication.

4. Conclusions

In this work, CZ_1−xN_x (0.02 ≤ x ≤ 0.1) ceramics were synthesized via the conventional solid-state method. The XRD and Rietveld refinement analyses confirmed that all the CZ_1−xN_x ceramics exhibited a trigonal structure (space group R-3c) as a single phase. Furthermore, the increase in the unit cell volume with the increasing x value substantiated the successful substitution of (Ba_1/3Nb_2/3)⁴⁺. The density and SEM results validated the high density of the synthesized ceramics. Notably, the CZ_0.98N_0.02 ceramics exhibited the most uniform grain size, which enhances performance. Specifically, at 675 °C, the CZ_0.98N_0.02 ceramics demonstrated superior performance, with ε_r = 10.41, Q×f = 53,296 GHz, and τ_f = −23.45 ppm/°C. The P-V-L theory was employed to establish a correlation between structure and performance characteristics. The Ce-O bond predominantly influenced the ε_r and τ_f of the CZ_1−xN_x ceramics. Furthermore, the Mo-O bond was strongly correlated with Q×f. Finally, a patch antenna tailored for 5G communication bands (n77 and n78) was designed using the CZ_0.98N_0.02 ceramics as the substrate materials. This study represents an initial exploration of the practical applications of CZM ceramics.