Phase composition, crystal structure, complex chemical bond theory and microwave dielectric properties of high-Q materials in a (Nd_1−xY_x)NbO₄ system

Abstract

In this paper, (Nd_1−xY_x)NbO₄ ceramics are prepared via a conventional solid-state reaction method and their microwave dielectric properties have been reported for the first time. The Rietveld refinement was used to investigate the crystal structure of (Nd_1−xY_x)NbO₄ ceramics. Based on the refined results, the NdNbO₄ ceramics have a monoclinic fergusonite structure (I2/a (15) space group, Z = 4). The XRD patterns present a single monoclinic phase of NdNbO₄ in the range of x = 0.02 to 0.1, with a further increase in the substitution content of Y³⁺ ions, few impurity phases are formed. In order to evaluate the correlations between complex chemical bond theory and microwave dielectric properties, the ionic polarization, lattice energy and bond energy were calculated using the refined lattice parameters and bond length. The effects of substituting Y³⁺ ions for Nd³⁺ ions on the microwave dielectric properties of the (Nd_1−xY_x)NbO₄ ceramics were also discussed. The increase in the dielectric constant ε_r is due to increasing the corrected theoretical dielectric constant ε_rc. For high relative density samples, the Q × f values and τ_f values are really dependent upon the calculated lattice energy and bond energy. High-quality factor microwave dielectric materials can be obtained with x = 0.08 in the (Nd_1−xY_x)NbO₄ system, and show excellent dielectric properties of ε_r = 19.87, Q × f = 81 100 GHz and τ_f = −18.84 ppm °C⁻¹.

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1. Introduction

Demands of the wireless industry with a wide range of applications from microwave communication to intelligent transport systems have led to the development of low loss, high relative permittivity and near zero temperature coefficients of resonant frequency ceramics receiving more attention than ever. Microwave dielectric ceramics with high performance and structure property have always attracted considerable attention. Therefore, a number of microwave dielectric materials have been developed to meet the requirements for small-sized and better selectivity GPS patch antennas in the last few years.^1–4

Recently, a number of studies have focused on microwave dielectric materials of ABO₄ compounds due to their flexibility in substituting different elements at the A- and B-site, which could lead to optimum microwave dielectric properties.^5–7 When the A- and B-site is substituted by lanthanoid and niobium elements, respectively, a new ABO₄ composition material system of a rare-earth orthoniobate material system RENbO₄ (RE = lanthanoid atoms, being La to Lu as well as Y) is developed. The RENbO₄ materials have similar fergusonite-type structures (monoclinic, C_2/C) and properties, which are firstly studied in regards to their luminescence characteristics, damping characteristics and phase transformation characteristics.^8–11

When RE = Nd, the NdNbO₄ composition system is developed. The microwave dielectric properties of NdNbO₄ ceramics with ε_r = 19.6, Q × f = 33 000 GHz and τ_f = −24 ppm °C⁻¹ were first reported by Kim, et al.¹² In recent years, a number of studies on microwave dielectric properties have been carried out.^13–20 For example, Zhang et al.¹³ studied the effects of CaF₂, CaTiO₃ and CuO additives on the microwave dielectric properties and sintering behavior of NdNbO₄ ceramics. The NdNbO₄ ceramics with 2.0 wt% CaF₂ sintered at 1225 °C for 4 h show excellent microwave dielectric properties, Q × f ∼ 75 000 GHz and τ_f ∼ −19 ppm °C⁻¹. With the increase of the CaTiO₃ content, the NdNbO₄–CaTiO₃ system has a trend of shifting toward zero, in the whole range, for the τ_f values.¹⁴ Moreover, a high Q × f value of 70 000 GHz with 0.6 wt% CaTiO₃ additive could be obtained and sintered at 1275 °C for 4 h. The effects of CuO addition on the sintering properties of NdNbO₄ were also studied, the sintering temperature of NdNbO₄ can be lowered to 975 °C when doped with 0.2 wt% CuO, and the microwave dielectric properties of NdNbO₄ were not affected apparently.¹⁵ Recently, Zhang et al. discovered that the microwave properties of NdNbO₄ ceramics could be optimized using bivalent ions by substituting Nd³⁺ ions owing to the formation of solid solutions and the phase composition would be changed when the Nd³⁺ ions were substituted by bivalent ions (Sr²⁺, Ca²⁺, Mn²⁺, Co²⁺).^16,17 The effects of Ta⁵⁺ and Sb⁵⁺ ion substitution on the NdNbO₄ ceramics were also investigated, a small level of Sb⁵⁺ substitution (x = 0.06) could greatly improve the Q × f values of the NdNbO₄ ceramics.^18–20 And the phase change was also investigated which proved to play an important role in NdNbO₄ ceramics. However, few studies reported the effects of trivalent ion (especially the lanthanoid ions) substitution of the Nd³⁺ ions on the microwave properties of the NdNbO₄ ceramics. And the high quality factor and the crystal structure of the NdNbO₄ ceramics using trivalent ion substitution have not been discussed. Moreover, the effects of the substitution of Y³⁺ ions for Nd³⁺ ions on the ionic polarization, oxygen octahedral distortion, lattice energy and bond energy in the NdNbO₄ system have also not been investigated.

In this paper, (Nd_1−xY_x)NbO₄ (0.02 ≤ x ≤ 0.15) ceramics were prepared to study the influence of Y³⁺ ion substitution on the phase evolution, octahedral distortion, lattice energy, bond energy and microwave dielectric properties. The lattice energy and bond energy were calculated based on the complex chemical bond theory to develop a relationship between the theoretical calculations and the microwave dielectric properties. Moreover, an available method based on the Rietveld refinement of X-ray techniques was used to analyze the crystal structure.

2. Experimental procedure

The (Nd_1−xY_x)NbO₄ (x = 0.02–0.15) compounds were prepared by a conventional solid-state reaction method. Nd₂O₃, Y₂O₃, and Nb₂O₅ (High-Purity Chemicals 99.9%) were used as raw materials. Stoichiometric mixtures of the starting materials were ball-milled with distilled water for 6 h. All the slurries were dried, crushed and sieved with a 40 mesh screen. Then the sieved specimens were calcined at 900 °C for 4 h, the obtained powders were re-milled for 6 h. After drying, the crushed powders were sieved with a 40 mesh screen firstly, then granulated and doped with 6 wt% paraffin as a binder and sieved with an 80 mesh screen, the powders were then pressed into disk-type pellets with a 10 mm diameter and 5 mm thickness at 100 MPa. Then these pellets were sintered at temperatures of 1225–1275 °C for 4 h in air with a heating rate of 5 °C min⁻¹ based on the previous process conditions.^14–18

The crystal structures of the synthesized samples were identified using X-ray diffraction (XRD, Rigaku D/max 2550 PC, Tokyo, Japan) with Cu Kα radiation generated at 40 kV and 40 mA. The microstructures of the ceramic surfaces were obtained and analyzed using scanning electron microscopy (SEM, MERLIN Compact, Germany). The microwave dielectric properties were measured in the frequency range of 8–12 GHz using a HP8720ES network analyzer. The temperature coefficients of the resonant frequency (τ_f) were measured in the temperature range from 25 °C to 85 °C. τ_f (ppm °C⁻¹) was calculated by noting the change in the resonant frequency (Δf)

where f₂₅ is the resonant frequency at 25 °C and f₈₅ is the resonant frequency at 85 °C.

The apparent densities of the sintered pellets were measured using the Archimedes method (Mettler ToledoXS64). To study the relative density of the sample, the theoretical density was obtained from the crystal structure and atomic weight using eqn (2):

where V_C, N_A, Z, and A are the volume of the unit cell (cm³), Avogadro’s number (mol⁻¹), the number of atoms in a unit cell, and the atomic weight (g mol⁻¹), respectively. The relative density was obtained using eqn (3):

3. Results and discussion

3.1 Multiphase refinement

Fig. 1 shows the XRD patterns of (Nd_1−xY_x)NbO₄ (x = 0.02–0.15) ceramics sintered at 1250 °C for 4 h. All of the peaks are clearly indexed as the monoclinic phase for NdNbO₄ (PDF. no. 32-0680) in the range of x = 0.02 to 0.10. As the x value increases to 0.15, some impurity phases can be observed. The formation of the impurity phases would be due to the reaction of excess Y³⁺ ions with Nb⁵⁺ ions which indicates that the solid solution of (Nd_1−xY_x)NbO₄ is lower than 0.15. In order to clarify the effects of the substitution of Y³⁺ ions for Nd³⁺ ions on the crystal structure of the (Nd_1−xY_x)NbO₄ ceramics, the refinements were performed using Full-prof software based on the X-ray diffraction data of the (Nd_1−xY_x)NbO₄ (x = 0.02–0.15) samples. And the refined lattice parameters, cell volume, reliability factors, bond length and atomic coordinate information are presented in Tables 1 and 2. In addition, the structural refinement patterns of the (Nd_0.92Y_0.08)NbO₄ ceramic are offered in Fig. 2. According to the Rietveld refinement results, the lattice parameters for the (Nd_0.92Y_0.08)NbO₄ ceramic are calculated as a = 5.434 Å, b = 11.229 Å, c = 5.134 Å, β = 94.40° and V = 312.37 ų, the Rietveld discrepancy factors R_p and R_wp are 12.00% and 12.90%. As Table 1 shows, the lattice parameters and cell volumes slightly decrease with increased Y³⁺ ion content in the range of x = 0.02–0.08, which is due to the incorporation of smaller Y³⁺ ions (1.019 Å, CN = 8) in place of Nd³⁺ ions (1.109 Å, CN = 8).²¹ Therefore, the substitution of Y³⁺ ions for Nd³⁺ ions could decrease the unit cell volume of NdNbO₄. When the x value increases to 0.10, an increase tendency in the lattice parameters is observed, which indicates that there could be an abnormal change in the crystal structure of NdNbO₄ and a second phase is formed.

Fig. 1 The XRD patterns of the (Nd_1−xY_x) NbO₄ ceramics with different x values sintered at 1250 °C for 4 h.

Table 1 Refined lattice parameters, cell volume (ų), reliability factors and the bond length of the (Nd_1−xY_x)NbO₄ ceramics when x = 0.02–0.15 sintered at 1250 °C for 4 h

x value	x = 0.02	x = 0.04	x = 0.06	x = 0.08	x = 0.10	x = 0.15
a	5.457	5.454	5.449	5.434	5.442	5.443
b	11.271	11.267	11.257	11.229	11.251	11.252
c	5.144	5.143	5.141	5.134	5.139	5.141
β	94.44	94.43	94.42	94.40	94.38	94.36
Vcell	315.45	315.11	314.42	312.37	313.75	313.97
Rp	0.0638	0.0644	0.1190	0.1200	0.1430	0.1430
Rwp	0.0911	0.0985	0.1260	0.1290	0.1560	0.1500
Nd/Y–O(1)^1 (Å) × 2	2.4848	2.4838	2.4818	2.4362	2.5402	2.5201
Nd/Y–O(1)^2 (Å) × 2	2.5380	2.5370	2.5352	2.5461	2.6782	2.7151
Nd/Y–O(2)^1 (Å) × 2	2.4290	2.4282	2.4266	2.4117	2.3628	2.3160
Nd/Y–O(2)^2 (Å) × 2	2.6172	2.6163	2.6143	2.6058	2.6572	2.7092
Nb–O(1)^1 (Å) × 2	1.9177	1.9169	1.9154	1.8558	1.8248	1.8259
Nb–O(1)^2 (Å) × 2	2.3056	2.3049	2.3034	2.3676	2.2620	2.2681
Nb–O(2) (Å) × 2	1.7377	1.7372	1.7360	1.7331	1.7122	1.6822

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Table 2 Refined atomic fractional coordinates from the XRD data for (Nd_0.92Y_0.08)NbO₄

Element	Wyckoff site	x	y	z	OCC	Biso.
Nd	4e	0.25000	0.12164	0.00000	0.46	−0.76351
Y	4e	0.25000	0.12164	0.00000	0.04	−0.76351
Nb	4e	0.25000	0.64525	0.00000	0.5	−0.45242
O1	8f	0.03430	0.71966	0.21131	1.0	−1.48755
O2	8f	0.91186	0.45173	0.21261	1.0	−1.98655

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Fig. 2 The structural refinement patterns of the (Nd_0.92Y_0.08)NbO₄ ceramic.

3.2 Crystal structure analysis

The schematic crystal structures of the monoclinic fergusonite structure of the (Nd_0.92Y_0.08)NbO₄ ceramic are shown in Fig. 3. There are four NdNbO₄ molecules per primitive cell, and the Nb⁵⁺ ions are connected with six oxygen atoms forming a distorted NbO₆ octahedron. Three Nb-site octahedra, by edge sharing, form a “V” type arrangement. In the monoclinic fergusonite structure, Nd³⁺ and Nb⁵⁺ occupy 4e Wyckoff positions whereas the two distinguishable oxygen atoms occupy the same positions, 8f, named O1 and O2. As Fig. 3 shows, O1 atoms are connected with two Nd atoms and two Nb atoms, and the O2 atoms are connected with two Nd atoms and one Nb atom. In this paper, with the increasing content of Y³⁺ ions, the atomic interactions of the NdNbO₄ ceramics can be changed, which could result in oxygen octahedron distortion. The change in oxygen octahedron distortion has a close connection with the lattice energy and bond energy, which as intrinsic factors affect the microwave dielectric properties. Fig. 4 presents the variation tendency of Nb-site octahedron distortion. In order to investigate the influence of Y³⁺ substitution on the interaction of the oxygen octahedron distortion accuracy, a calculation for the bond strength is need; therefore the bond strength is calculated from the following formula:²²where R is the refined bond length as shown in Table 1, and R₁ = 2.137, N = 6.5 for a Nd-site, and R₁ = 1.907, N = 5 for a Nb-site are the universal bond-strength-bond-length parameters from ref. 22. The details of the bond strengths of the (Nd_1−xY_x)NbO₄ (x = 0.02–0.15) ceramics are given in Table 3. Notice that the Nb-site octahedron has a higher bond strength than that of the Nd-site, which suggests that the Nb-site octahedron distortion has a greater contribution to the microwave dielectric properties of the NdNbO₄ ceramic. The variation of the Nb-site bond strength with different substitution ions is given in Fig. 5. Obvious conclusions can be observed from Fig. 5, with the x values increasing from 0.02 to 0.15, the Nb-site bond strength maintains an increasing tendency which indicates a decrease in the Nb-site octahedron distortion. A stable Nb-site oxygen octahedron system could be obtained for the (Nd_0.85Y_0.15)NbO₄ ceramic.

Fig. 3 The crystal structure pattern (1 × 1 × 1) supercell of the monoclinic fergusonite structured (Nd_0.92Y_0.08)NbO₄.

Fig. 4 Distortion in BO₆ for (Nd_1−xY_x)NbO₄ when (a) x = 0.02; (b) x = 0.08; (c) x = 0.15.

Table 3 The bond strength for the (Nd_1−xY_x)NbO₄ (0.02 ≤ x ≤ 0.15) ceramics sintered at 1250 °C for 4 h

Bond type	Bond strength
	x = 0.02	x = 0.04	x = 0.06	x = 0.08	x = 0.10	x = 0.15
Nd/Y–O(1)^1 × 2	0.3753	0.3762	0.3782	0.4267	0.2305	0.2109
Nd/Y–O(1)^2 × 2	0.3270	0.3278	0.3293	0.3203	0.3252	0.3424
Nd/Y–O(2)^1 × 2	0.4350	0.4359	0.4378	0.4556	0.2426	0.2139
Nd/Y–O(2)^2 × 2	0.2678	0.2684	0.2697	0.2755	0.5205	0.5928
Nb–O(1)^1 × 2	0.9724	0.9744	0.9783	1.1458	1.2465	0.2427
Nb–O(1)^2 × 2	0.3871	0.3877	0.3890	0.3390	0.4259	0.4202
Nb–O(2) × 2	1.5918	1.5941	1.5996	1.6130	1.7139	1.8723
Total Nb–O	2.9513	2.9562	2.9669	3.0978	3.3863	3.5352

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Fig. 5 The Nb-site bond strength for (Nd_1−xY_x)NbO₄ as a function of x values.

3.3 Microstructure analysis

Fig. 6 presents the SEM images of the compounds using the (Nd_1−xY_x)NbO₄ (x = 0.02–0.15) ceramics sintered at 1250 °C for 4 h. The results indicate that well-developed microstructures and nearly full densification of the (Nd_1−xY_x)NbO₄ ceramics can be achieved at a suitable x value. The increase of sintering temperature helped to promote grain growth, and a relative increase in the grain size was achieved for the specimen shown in Fig. 6(a)–(d). The optimum microstructure could be achieved at x = 0.08 (Fig. 6(d)) which has homogeneous grains and a smooth surface. Moreover, all the estimated mean particle sizes were in the range of 3–5 μm. However, a further increase in the x values from 0.10 to 0.15 would result in a decrease in the grain size and the number of pores, which causes the low relative densities of the specimens, as illustrated in Fig. 6(e) and (f).

Fig. 6 The SEM micrographs for the (Nd_1−xY_x)NbO₄ ceramics sintered at 1250 °C for 4 h when (a) x = 0.02, (b) x = 0.04, (c) x = 0.06, (d) x = 0.08, (e) x = 0.10 and (f) x = 0.15.

3.4 Microwave dielectric properties analysis

Fig. 7 shows the relative density, dielectric constant and quality factor of the (Nd_1−xY_x)NbO₄ (x = 0.02–0.15) ceramics with different sintering temperature. As sintering temperature varied from 1225 °C to 1275 °C, both the relative density and dielectric constant increase and reach a saturated value, which is due to that a high relative density means a low number of pores (ε_r = 1). And a higher dielectric constant with a high relative density for the (Nd_1−xY_x)NbO₄ (x = 0.02–0.15) ceramics could be obtained when sintered at 1250 °C for 4 h. The quality factor also presents a close relationship with the sintering temperature because the extrinsic dielectric loss of the microwave dielectric ceramics are determined by the universal defects caused by the porosity, grain boundary, etc. The optimized sintering process, especially the sintering temperature could help to decrease the extrinsic dielectric loss and enhance the Q × f values. In this paper, the sintering time and heating rate based on our previous work are not under consideration in the investigation of the influencing factors on the Q × f values. We observed that the Q × f values of the sintered specimens increased firstly with sintering temperature and then decreased, which indicated that an appropriate sintering temperature can help to enhance the Q × f values.

Fig. 7 (a) Relative densities and ε_r values of the (Nd_1−xY_x)NbO₄ ceramics when x = 0.02–0.15 at different sintering temperatures; (b) Q × f values of the (Nd_1−xY_x)NbO₄ ceramics when x = 0.02–0.15 at different sintering temperatures.

The microwave dielectric properties of the (Nd_1−xY_x)NbO₄ (x = 0.02–0.15) ceramics sintered at the optimal sintering temperature are presented in Fig. 8. Our recent work demonstrated that the microwave dielectric properties of sintered ceramics possessing a high relative density have no obvious relationship with the extrinsic factors like the sintering temperature, grain boundary and pores.^23–26 In this paper, the dielectric constant of the (Nd_1−xY_x)NbO₄ (x = 0.02–0.15) ceramics at the optimal sintering temperature is dependent on the dielectric polarizabilities, and has a vital effect on the theoretical dielectric constant. In order to clarify the effects of the substitution of Y³⁺ for Nd³⁺ on the dielectric constant, the theoretical dielectric constant ε_tr was calculated based on the Clausius–Mosotti equation²⁷ using the ionic polarizabilities, which is described as:

Fig. 8 (a) The corrected theoretical dielectric constants ε_rc and experimental dielectric constants ε_r of the (Nd_1−xY_x)NbO₄ ceramics when x = 0.02–0.15; (b) the Q × f values of the (Nd_1−xY_x)NbO₄ ceramics when x = 0.02–0.15 as a function of the Nd-site lattice energy; (c) the τ_f values of the (Nd_1−xY_x)NbO₄ ceramics when x = 0.02–0.08 as a function of the bond energy.

After considering the influence of porosity, the theoretical dielectric constant ε_tr was corrected as follows:²⁸

where ε_tr is the theoretical dielectric constant; ε_rc is the corrected theoretical dielectric constant; V_m is the molar volume; α_D is the theoretical dielectric polarizability; b has the value of 4π/3; P is the fractional porosity. Based on the additivity rule of molecular polarizabilities, the molecular polarizability of a complex substance (like (Nd_1−xY_x)NbO₄) can be broken up into the molecular polarizabilities of simper substance states:

α_D((Nd_1−xY_x)NbO₄) = (1 − x)α_D(Nd³⁺) + xα_D(Y³⁺) + α_D(Nb⁵⁺) + 4α_D(O²⁻) (8)

Table 4 shows the change of the corrected theoretical dielectric constant (ε_rc) and the molecular polarizability (α_D) of the (Nd_1−xY_x)NbO₄ compounds at optimal sintering temperature. With the increase of the x values, both of the ε_rc and α_D have an increasing tendency, which indicates that the substitution of Y³⁺ ions for Nd³⁺ ions could enhance the polarizability of the NdNbO₄ ceramics. According to the calculation results, the variation of the experimental dielectric constant ε_r as a function of the corrected theoretical dielectric constant ε_rc is presented in Fig. 8(a). Notice that the ε_r and ε_rc present a similar variation tendency, which suggests that the ε_rc could predict the variation of the experimental dielectric constant ε_r when the specimens possess a high relative density.

Table 4 Theoretical dielectric constant (ε_tr), corrected theoretical dielectric constant (ε_rc), the molecular polarizability (α_D) and coordination numbers (Z) for the (Nd_1−xY_x)NbO₄ (0.02 ≤ x ≤ 0.15) ceramics sintered at 1250 °C for 4 h

x value	x = 0.02	x = 0.04	x = 0.06	x = 0.08	x = 0.10	x = 0.15
Z	4	4	4	4	4	4
αD	16.996	16.972	16.948	16.924	16.900	16.840
εtr	28.847	28.751	28.974	30.532	28.772	27.608
εrc	26.740	26.799	27.245	28.867	26.909	25.561

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Fig. 8(b) shows the Q × f values for (Nd_1−xY_x)NbO₄ with different substitution content sintered at the optimal sintering temperature. As we all know, the Q × f values were affected by many factors, and these can be divided into two fields, the intrinsic loss and extrinsic loss. In this paper, the effect of extrinsic loss like the second phase, oxygen vacancies, grain boundaries, and densification or porosity is minimal to the Q × f values due to the highly densified compounds. In previous work, the Q × f values with an oxygen octahedron structure are observed to relate to the lattice energy.^19,29,30 Based on the generalized P-V-L theory,³¹ the lattice energy for a single-bond crystal consists of ionic and covalent parts. The ionic part mainly results from electrostatic interactions and repulsive interactions of the ion pairs, and the covalent part arises from the overlap of electron clouds. With an increase in the lattice energy, the Q × f values would increase. And the lattice energy U_cal of a complex crystal can be calculated as follows:

For any binary crystal A_mB_n type compound, the lattice energy U_b is described as:

U_b = U_bc + U_bi (10)

where U_bc is the covalent part and U_bi is the ionic part of bond type A–B; d is the distance between cation A and anion B; m and n are the numbers of cation A and anion B; Z₊ and Z₋ are the valence states of the cation and anion which constitute the bond type A–B; f_i and f_c are the bond ionicity and bond covalence, which can be obtained from ref. 19. The calculated lattice energy for the (Nd_1−xY_x)NbO₄ (x = 0.02–0.15) ceramics sintered at 1250 °C is illustrated in Table 5. Fig. 8(b) presents the variation of the Q × f values as a function of the Nd-site lattice energy. When the Y³⁺ ion content increased, the lattice energy increased which led to an increase in the Q × f values. This phenomenon is because the Q × f values are mainly decided by the lattice anharmonicity, and when the lattice energy increased, the lattice anharmonicity would be enhanced which would decrease the intrinsic loss, therefore, the Q × f value would increase.

Table 5 Lattice energy for the (Nd_1−xY_x)NbO₄ (0.02 ≤ x ≤ 0.15) ceramics sintered at 1250 °C for 4 h

Bond type	Lattice energy U (kJ mol^−1)
	x = 0.02	x = 0.04	x = 0.06	x = 0.08	x = 0.10	x = 0.15
Nd/Y–O(1)^1 × 2	1463	1464	1465	1487	1437	1447
Nd/Y–O(1)^2 × 2	1438	1438	1439	1435	1377	1361
Nd/Y–O(2)^1 × 2	1406	1407	1408	1415	1439	1462
Nd/Y–O(2)^2 × 2	1324	1325	1326	1329	1310	1289
Nb–O(1)^1 × 2	7135	7136	7140	7306	7393	7389
Nb–O(1)^2 × 2	6203	6204	6207	6074	6293	6279
Nb–O(2) × 2	6912	6913	6916	6924	6981	7068
Total UNd–O	5631	5634	5638	5666	5563	5559

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As we all know, usually a shorter bond length correlates with higher bond energy, and higher bond energy suggests a more stable system. Our recent work suggests that the bond energy can affect the temperature coefficient of the resonant frequency τ_f values, and a higher bond energy correlates to a smaller |τ_f| value.^20,29 The bond energy E of a complex crystal could be written as:

where E^μ_b is the bond energy for a μ type bond, which is composed of nonpolar covalence energy E^μ_c and complete ionicity energy E^μ_i parts as follows:

E^μ_b = t_cE^μ_c + t_iE^μ_i (14)

The energy of the ionic form E^μ_i is the unit charge product divided by the bond length d^μ, adjusted to kcal mol⁻¹ by a factor of 33 200 when the bond length is pm.

For any binary crystal A_mB_n type compounds, the nonpolar covalence energy E^μ_c parts could be calculated as follows:

where r_cA and r_cB are the covalent radii, E_A–A and E_B–B are the homonuclear bond energy, which can be obtained from the handbook of bond energies.³² In this paper, E_Nd–Nd = 82.8 kJ mol⁻¹, E_Y–Y = 270 kJ mol⁻¹, E_Nb–Nb = 513 kJ mol⁻¹, and E_O–O = 498.36 kJ mol⁻¹; r_cNd = 174 pm, r_cY = 163 pm, r_cNb = 147 pm and r_cO = 63 pm.

For eqn (14), t_c and t_i are the covalent and ionic blending coefficients, respectively. The relationship of t_c and t_i can be described by the following formula:

t_c + t_i = 1 (17)

The ionic blending coefficient t_i is defined as:

where S_A and S_B are the electronegativities of A and B ions. ΔS_B is the change for complete loss of an electron. In this paper, S_Nd = 1.14, S_Y = 1.22, S_Nb = 1.59, S_O = 3.44 and ΔS_B = ΔS_O = 3. The details of the bond energies are given in Tables 6 and 7. Fig. 8(c) shows the τ_f values of the (Nd_1−xY_x)NbO₄ ceramics (x = 0.02 to 0.08) as a function of the bond energy. The variation of the τ_f values is consistent with the bond energy. With the increase of the bond energy, the |τ_f| values decreased. That’s because with the increase of the bond energy the distortion of the oxygen octahedron would be recovered, and the τ_f values have a close relationship with the distortion of the oxygen octahedron. Therefore, a higher bond energy would recover the distortion of the oxygen octahedron which correlates to a smaller |τ_f| value. As the x value increases to 0.10, the second phase is formed, which has a vital effect on the τ_f values. Therefore, the increasing tendency of the |τ_f| value in Table 8 could be attributed to the formation of the second phase.

Table 6 The nonpolar covalence energy E^μ_c (kJ mol⁻¹), complete ionicity energy E^μ_i (kJ mol⁻¹), covalent blending coefficients t_c and ionic blending coefficients t_i for the (Nd_0.92Y_0.08)NbO₄ ceramics sintered at 1250 °C for 4 h

Bond type	E^μc	E^μi	tc	ti
Nd/Y–O(1)^1 × 2	213.95	568.91	0.6177	0.3823
Nd/Y–O(1)^2 × 2	204.71	545.31	0.6177	0.3823
Nd/Y–O(2)^1 × 2	216.12	575.70	0.6177	0.3823
Nd/Y–O(2)^2 × 2	200.02	532.82	0.6177	0.3823
Nb–O(1)^1 × 2	572.16	748.15	0.6917	0.3083
Nb–O(1)^2 × 2	448.48	586.43	0.6917	0.3083
Nb–O(2) × 2	612.67	801.12	0.6917	0.3083

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Table 7 The bond energies for the (Nd_1−xY_x)NbO₄ (0.02 ≤ x ≤ 0.15) ceramics sintered at 1250 °C for 4 h

Bond type	Bond energy EB (kJ mol^−1)
	x = 0.02	x = 0.04	x = 0.06	x = 0.08	x = 0.10	x = 0.15
Nd/Y–O(1)^1 × 2	336.14	338.67	341.30	350.02	320.47	321.06
Nd/Y–O(1)^2 × 2	329.09	331.57	334.11	334.91	337.88	345.91
Nd/Y–O(2)^1 × 2	343.86	346.43	349.06	353.58	363.25	376.39
Nd/Y–O(2)^2 × 2	319.13	321.52	324.00	327.24	323.01	321.76
Nb–O(1)^1 × 2	606.21	606.46	606.93	626.43	637.07	636.68
Nb–O(1)^2 × 2	504.22	504.37	504.70	491.01	513.93	512.55
Nb–O(2) × 2	669.00	669.19	669.65	670.78	678.96	691.07
EB	3107.65	3118.21	3129.75	3153.97	3174.57	3205.42

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Table 8 The relative density R.D. (%), dielectric constant ε_r, quality factor Q × f (GHz) and the temperature coefficient of the resonant frequency τ_f (ppm °C⁻¹) of the (Nd_1−xY_x)NbO₄ (0.02 ≤ x ≤ 0.15) ceramics sintered at 1250 °C for 4 h

x value	R.D.	εr	Q × f	τf
0.02	94.87	18.91	36 700	−26.29
0.04	95.23	19.34	38 700	−25.46
0.06	95.81	19.43	50 800	−20.07
0.08	96.18	19.87	81 100	−18.84
0.10	95.45	19.33	53 300	−17.48
0.15	94.78	19.16	36 600	−16.35

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Table 8 shows the relative density, ε_r values, Q × f values and τ_f values for the (Nd_1−xY_x)NbO₄ samples with different x values sintered at the optimal sintering temperature. When x = 0.08, excellent microwave dielectric properties with an ε_r value of 19.87, a very high Q × f value of 81 100 GHz and τ_f value of −18.84 ppm °C⁻¹ could be obtained.

4. Conclusions

The phase composition, lattice energy, bond energy and microwave dielectric properties of (Nd_1−xY_x)NbO₄ (x = 0.02–0.15) ceramics were investigated in the present study. The Rietveld refinement revealed that the NdNbO₄ ceramics showed a monoclinic fergusonite structure and possessed an Nb-site oxygen octahedron. With an increase of the x values, oxygen octahedron distortion, which c characterized by the Nb-site bond strength, would decrease. In order to demonstrate the intrinsic factors of the microwave dielectric properties, theoretical calculations based on the complex chemical bond theory were introduced in this paper. The calculated results indicate that the ε_r, Q × f value and τ_f value for the (Nd_1−xY_x)NbO₄ ceramics with high relative density are mainly dependent on the dielectric polarizabilities, lattice energy and bond energy, respectively. The change of the Y³⁺ ion content has an obvious effect on the microwave dielectric properties of the (Nd_1−xY_x)NbO₄ ceramics which is due to the differences of Y³⁺ and Nd³⁺ in their polarizabilities, electronegativities, and bond energies, etc. At 1250 °C, the (Nd_1−xY_x)NbO₄ ceramics when x = 0.08 possess excellent microwave dielectric properties with an ε_r value of 19.87, a high Q × f value of 81 100 GHz and τ_f value of −18.84 ppm °C⁻¹.

Acknowledgements

The authors gratefully acknowledged support from the Key Laboratory of Advanced Ceramics and Machining Technology, Ministry of Education (Tianjin University). The authors would like to thank Hai-Tao Wu for his help in the XRD experiments.

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