Mechanism of ionic polarizability, bond valence, and crystal structure on the microwave dielectric properties of disordered Li₁₀MTi₁₃O₃₂ (M = Zn, Mg) spinels

Abstract

The influence mechanism of the microwave dielectric properties of disordered spinels (Fd3̄m) Li₁₀MTi₁₃O₃₂ (M = Zn, Mg) was investigated via Rietveld structural refinement, ionic polarizability, bond valence, P–V–L theory, Raman spectroscopy, and DC conductivity. For Li₁₀ZnTi₁₃O₃₂ and Li₁₀MgTi₁₃O₃₂, promising microwave dielectric properties of ε_r = 28.23 ± 0.3 and 29.23 ± 0.3, Q × f = 35 800 ± 500 GHz and 32 100 ± 500 GHz (at ∼7.5 GHz), and τ_f = −17.06 ± 2.0 ppm per °C and −11.05 ± 2.0 ppm per °C, respectively, were obtained at 980 °C. Bond valences reveal that almost all cations are rattling, weakening the bond strengths and widening the molecular dielectric polarizability. The expansion structures also result in τ_f values closer to zero and lower Q × f values. The bond ionicity and lattice energy of the Ti–O bonds are much greater than those of other bonds, indicating that the Ti–O bond is a major contributor to ε_r and Q × f. Moreover, the DC conductivity clarified that the Q × f values of the disordered spinels Li₁₀ZnTi₁₃O₃₂ and Li₁₀MgTi₁₃O₃₂ are not as high as other ordered spinels because of the transport of Li⁺ ions in their structures.

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1. Introduction

From 1G to 5G (the fifth generation mobile network), mobile communications are constantly expanding towards higher frequency bands (microwave, millimeter wave and centimeter wave) to increase speed and capacity, which is inseparable from the support of microwave dielectric ceramic materials. Microwave dielectric ceramics for 5G applications need to have a low relative permittivity (ε_r) for modulating signal delay, a high quality factor (Q × f) to suppress signal attenuation, and a near-zero temperature coefficient of resonance frequency (τ_f) to ensure device stability in high-temperature environments.^1–6

Microwave dielectric ceramics with a spinel structure (AB₂O₄), such as MgAl₂O₄,^7–9 and Mg₂TiO₄,¹⁰ have been reported to possess a relatively low ε_r and a high Q × f. In order to overcome the high sintering temperature, τ_f is not close to zero for traditional spinels, so a large number of Li-containing spinels have been reported, such as Li₂ATi₃O₈ (A = Zn, Mg),^11,12 LiGa₅O₈,^13,14 Li₂Zn₃Ti₄O₁₂,¹⁵ and Li₄Ti₅O₁₂.^16–18 Concerning the Li₂O–AO–TiO₂ (A = Zn, Mg) systems, recent studies have focused on the characterization and microwave dielectric properties of two ordered ceramics of Li₂ZnTi₃O₈ (ε_r = 25.6–26.2, Q × f = 62 000–72 000 GHz, and τ_f = −15 to −11.2 ppm per °C) and Li₂MgTi₃O₈ (ε_r = 20.2, Q × f = 42 000 GHz, and τ_f = +3.2 ppm per °C), but the microwave dielectric properties of disordered spinel ceramics have rarely been reported. For ordered Li-containing spinels, Li_4b and Ti_12d at the octahedral sites were arranged spontaneously into 1 : 3 cation ordering (i.e., [Li_0.5Zn_0.5]^tet[Li_0.5Ti_1.5]^octO₄ and [Li_0.55Mg_0.45]^tet[(Li_0.45Mg_0.05)Ti_1.5]^octO₄) with a space group of P4₃32. The spinel-structured Li₂MM′₃O₈ (M = Mg, Co, Ni, Zn; M′ = Ti, Ge) ceramics with an ordered spinel structure (P4₃32) have been reported to possess a relative ε_r of 10.5–28.9, a Q × f of 47 400–160 000 GHz, and a τ_f of −63.9 to +7.4 ppm per °C.^19–25

Recently, we found an interesting phenomenon in the ordered spinels Li_1.33xA_2−2xTi_1+0.67xO₄ (x = 0.5625; A = Zn, Mg),²⁶ namely, a large positive deviation [Δε_r = (ε_r(corr)–ε_r(C–M))/ε_r(C–M) = 16–91%] between the porosity-corrected permittivity ε_r(corr) and the calculated ε_r(C–M) using the Clausius–Mossotti (C–M) equation, which might be related to the combined effect of the rattling and the compressed cations at the A and B sites, which was mutually verified in garnet and zircon systems,^27–30 thus establishing the interaction mechanism of bond length, polarization, permittivity, and τ_f.

Hernandez et al.³¹ discovered that Li_1.33xZn_2−2xTi_1+0.67xO₄ spinels undergo two order–disorder transitions to the interval of x = [0–1], where x ≤ 0.5 is the disordered face-centered cubic phase (Fd3̄m), 0.5 < x < 0.9 is the ordered primitive cubic phase (P4₃32), and when x is beyond 0.9, it becomes face-centered cubic again. And a similar phenomenon occurs in magnesium analogs.³² The different cation occupation is the main reason for the order and disorder changes in the spinel structure. In the disordered structure, there is only one kind of regular octahedron (16d) and one tetrahedron (8a), compared with the two octahedra (4b, 12d) and one tetrahedron (8c) in the ordered structure. The disordered structure is simpler and more conducive to studying the influence of the bond length, bond valence, and polarization on the microwave dielectric properties.

In this paper, we focus on the disordered compositions Li₁₀MTi₁₃O₃₂ (M = Zn, Mg) with x = 0.9375 in Li_1.33xM_2−2xTi_1+0.67xO₄ spinels. It is a Z = 1 structure located between the disorder x = 0.9 and x = 1 (Li₄Ti₅O₁₂, ε_r = 30.1, Q × f = 29 530 GHz, τ_f = −15 ppm per °C).¹³ The main aim of this work is to investigate the effects of the bond length, bond valence, ionic dielectric polarizability, structural stability, and rattling and compressed cations on the microwave dielectric properties (ε_r, τ_f, and Q × f) both theoretically and experimentally.

2. Experimental procedure

Li₁₀MTi₁₃O₃₂ (M = Zn, Mg) ceramics were prepared via a solid-state reaction using high-purity (>99.99%) Li₂CO₃, ZnO, MgO, and TiO₂ (rutile) powders as raw materials. Before weighing, the MgO, ZnO, and TiO₂ powders were calcined to remove any moisture (900 °C/2 h). The raw materials were ground using a ball mill for 6 h, the powder was calcined at 800 °C for 4 h, and then it was ball milled once with a 5 wt% PVA binder. The obtained powder was uniaxially pressed into cylinders with a diameter of 10 mm and a height of 5 mm at a pressure of 150 MPa. Subsequently, the samples were heated from room temperature to 550 °C for 4 h to burn off the organic binder and then sintered at 940–1020 °C for 4 h with a temperature-ramping rate of 5 °C min⁻¹.

The phase purity and crystal structure of the Li₁₀MTi₁₃O₃₂ (M = Zn, Mg) ceramics were identified via a powder X-ray diffractometer (X’Pert PRO). Scanning electron microscopy (SEM; S4800, Hitachi) was used to observe the grain growth of the Li₁₀MTi₁₃O₃₂ (M = Zn, Mg) ceramics after polishing and thermal etching (below the sintering temperature of 50 °C for 30 min). The average grain size of the Li₁₀MTi₁₃O₃₂ (M = Zn, Mg) ceramics was evaluated using Nano Measurer software. The densities of the sintered ceramics were determined using the Archimedes principle. The Raman spectra of the ceramics were collected using a Raman spectrometer (Thermo Fisher Scientific) with a 532 nm laser. Impedance spectroscopy was carried out using an impedance analyzer (4291A, Agilent, America), and DC conductivity testing was performed using a high-temperature resistivity test system (RMS-1000I, Partulab, China) at selected temperatures. The ε_r, dielectric loss (tan δ = 1/Q), and resonant frequency (f) were determined using the Hakki–Coleman method using a network analyzer (N5230A, Agilent, America) (ESI†). The drift of the f and ε_r (i.e., τ_f and τ_ε) were calculated as follows:

where f₁ and f₂ represent the resonant frequency at T₁ and T₂, respectively.

3. Results and discussion

Fig. 1a shows the XRD patterns of the Li₁₀ZnTi₁₃O₃₂ and Li₁₀MgTi₁₃O₃₂ ceramics sintered at 980 °C, respectively. The diffraction peaks of Li₁₀MTi₁₃O₃₂ (M = Zn, Mg) were well indexed with the disordered spinel Li₄Ti₅O₁₂ phase (PDF#49-0207). Then, Rietveld structural refinements were carried out on Li₁₀MTi₁₃O₃₂ (M = Zn, Mg) established on the reported cubic Li₄Ti₅O₁₂ disordered structure model (ICSD#160655) with the space group of Fd3̄m. The refined plots for Li₁₀ZnTi₁₃O₃₂ and Li₁₀MgTi₁₃O₃₂ are displayed in Fig. 1b and c, respectively, and the refined structural parameters are listed in Table 1. It is notable that the refined lattice parameters of a = 8.3636 Å and V = 585.0211 ų for Li₁₀ZnTi₁₃O₃₂ are smaller than a = 8.3649 Å and V = 585.2944 ų for Li₁₀MgTi₁₃O₃₂, which is opposite to the ionic radii of Zn²⁺ (0.60 Å) and Mg²⁺ (0.57 Å, CN = 4), which might be related to the rattling effect of cations in Li₁₀MgTi₁₃O₃₂. The crystal structure and cation occupancy information of the Li₁₀MTi₁₃O₃₂ ceramics are presented in Fig. 1d and Table 1, respectively. It is observed that Li and M share the tetrahedral sites (8a), and Li and Ti share the octahedral sites (16d). The octahedra and tetrahedra are connected at their common vertices, and the octahedra are connected at their common edges to form the structural skeleton. The tetrahedra are isolated from each other, indicating that the influence of the octahedron on the structure is dominant. Moreover, Fig. S1 (ESI†) depicts the bond lengths in Li₁₀MTi₁₃O₃₂ (M = Zn, Mg) from the Rietveld refinements.

Fig. 1 (a) XRD patterns of the Li₁₀MTi₁₃O₃₂ (M = Zn, Mg) ceramics sintered at the optimum temperature. Rietveld refinement patterns of (b) Li₁₀ZnTi₁₃O₃₂ and (c) Li₁₀MgTi₁₃O₃₂. (d) Schematic diagram of the crystal structure of spinel-type Li₁₀MTi₁₃O₃₂ (M = Zn, Mg).

Table 1 Refined structural parameters, thermal parameters, and reliability factors of the Li₁₀MTi₁₃O₃₂ (M = Zn, Mg) ceramics

Compound	Atom	Site	Occ.	x, y, z	B (Å^2)	a (Å)	V (Å^3)	R p (%)	R wp (%)	χ ^2
Li10ZnTi13O32	Zn(1)	8a	0.1250	0.1250	0.3866	8.3636	585.0211	2.91	3.91	3.96
	Li(1)	8a	0.8750	0.1250	0.3866
	Li(2)	16d	0.1875	0.5000	0.5396
	Ti(1)	16d	0.8125	0.5000	0.5396
	O(1)	32e	1.0000	0.2621	0.4660
Li10MgTi13O32	Mg(1)	8a	0.1250	0.1250	0.5066	8.3649	585.2944	2.84	3.80	3.92
	Li(1)	8a	0.8750	0.1250	0.5066
	Li(2)	16d	0.1875	0.5000	0.8772
	Ti(1)	16d	0.8125	0.5000	0.8772
	O(1)	32e	1.0000	0.2622	0.7794

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Raman spectra can reflect the vibrations of chemical bonds and their structural characteristics. The Raman spectra of the Li₁₀ZnTi₁₃O₃₂ and Li₁₀MgTi₁₃O₃₂ ceramics are shown in Fig. 2. Six modes at 228, 258, 342, 417, 512, and 675 cm⁻¹ for Li₁₀ZnTi₁₃O₃₂ and 231, 260, 340, 418, 514, and 676 cm⁻¹ for Li₁₀MgTi₁₃O₃₂ were observed, which are very close to the previously reported values for disordered Li₄Ti₅O₁₂ (246, 274, 360, 429, and 680 cm⁻¹; A_1g + E_g + 3F_2u).^33,34 However, the Raman spectra of Li₁₀ZnTi₁₃O₃₂ and Li₁₀MgTi₁₃O₃₂ show a new mode at 512 cm⁻¹ and 514 cm⁻¹, respectively, which may be ascribable to Zn–O and Mg–O stretches in the [ZnO₄] and [MgO₄] tetrahedra.^35,36 For Li₁₀ZnTi₁₃O₃₂, the Raman band at 676 cm⁻¹ is considered to be the A_1g mode associated with the stretching vibration of [TiO₆]. The 417 cm⁻¹ band is roughly the E_g mode associated with the [LiO₄] stretching vibration. The mode at 340 cm⁻¹ is due to the bending vibrations at the O–Li–O and O–Ti–O bonds. The same is true for the Li₁₀MgTi₁₃O₃₂ ceramic, and the Raman vibration modes are listed in Table 2. In addition, most of the Raman shifts of Li₁₀MgTi₁₃O₃₂ were found to be lower than those of Li₁₀ZnTi₁₃O₃₂, which may be related to the expansion of the cationic-oxygen bond and its larger unit cell parameter (a). In general, an increase in bond expansion is accompanied by a weakening of the bond, which leads to a decrease in the frequency.³⁷

Fig. 2 Raman spectra of the Li₁₀MTi₁₃O₃₂ (M = Zn, Mg) ceramics studied at different sintering temperatures.

Table 2 Raman vibration modes the Li₁₀MTi₁₃O₃₂ (M = Zn, Mg) ceramics

Li10ZnTi13O32	Li10MgTi13O32
Raman mode (cm^−1)	Raman mode (cm^−1)	Symmetry mode	Assignment
231	228	F2u	O–Li–O, O–Ti–Obending
258	260	F2u	O–Li–O, O–Ti–Obending
340	342	F2u	O–Li–O, O–Ti–Obending
418	417	Eg	[LiO4]stretching
514	512	Eg	[Zn/MgO4]stretching
676	675	A1g	[TiO6]stretching

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In the sintering behavior of ceramics, grain growth is a manifestation of the densification process. SEM was carried out to observe the polished and thermally etched grains. As shown in Fig. 3a–c, the average grain size (D̄) of Li₁₀ZnTi₁₃O₃₂ increased from 3.81 μm to 5.06 μm as the sintering temperature was increased. When sintered at 940 °C, the grain size was not uniform, and there was some slight porosity. As the temperature was increased to 980 °C, the grain size increased with the disappearance of the pores, and the microstructure became uniform and compact. When the temperature was raised further, the size of some grains increased abnormally. As shown in Fig. 3d–f, the grain growth process of the Li₁₀MgTi₁₃O₃₂ ceramic (the average grain size increased from 2.66 μm to 4.85 μm) is similar to that of Li₁₀ZnTi₁₃O₃₂, and the densest microstructure is obtained at 980 °C.

Fig. 3 Polished and thermally etched SEM images of the Li₁₀ZnTi₁₃O₃₂ ceramic sintered at (a) 940 °C, (b) 980 °C, and (c) 1020 °C, and the Li₁₀MgTi₁₃O₃₂ ceramic sintered at (d) 940 °C, (e) 980 °C, and (f) 1020 °C.

Fig. 4a presents the variation of the bulk density and relative density of the Li₁₀MTi₁₃O₃₂ (M = Zn, Mg) ceramics at different sintering temperatures. The theoretical density is calculated based on structural refinement (3.6009 g cm⁻³ for Li₁₀ZnTi₁₃O₃₂ and 3.4827 g cm⁻³ for Li₁₀MgTi₁₃O₃₂). Both Li₁₀ZnTi₁₃O₃₂ and Li₁₀MgTi₁₃O₃₂ show high relative densities (>95%) in the sintering range. The saturation relative density of Li₁₀ZnTi₁₃O₃₂ was 98.19% at 980 °C, and the maximum relative density of Li₁₀MgTi₁₃O₃₂ was 98.25%.

Fig. 4 (a) Density, (b) ε_r, (c) τ_f, and (d) Q × f of the Li₁₀MTi₁₃O₃₂ (M = Zn, Mg) samples as a function of the sintering temperature, the inset in (d) shows the contribution of each bond to the lattice energy.

As shown in Fig. 4b, the ε_r of the Li₁₀ZnTi₁₃O₃₂ ceramic increases from 27.42 at 940 °C to 28.23 at 980 °C, after which it decreases slightly with increasing temperature, and the change in the ε_r of the Li₁₀MgTi₁₃O₃₂ ceramic is consistent with that of Li₁₀ZnTi₁₃O₃₂, reaching a maximum of 29.23 at 980 °C. In general, the relative permittivity of the ceramics is correlated with the porosity, which can be corrected using the Bosman–Havinga equation:³⁸

ε_r(corr) = ε_r(1 + 1.5P) (3)

where P denotes the fractional porosity, and ρ_th and ρ_mea are the theoretical and measured density values, respectively. The corrected permittivity ε_r(corr) at the optimum sintering temperature is 28.99 for Li₁₀ZnTi₁₃O₃₂ and 30.00 for the Li₁₀MgTi₁₃O₃₂ ceramic. The evaluated permittivity ε_r(C–M) is 16.05 for Li₁₀ZnTi₁₃O₃₂ and 15.47 for Li₁₀MgTi₁₃O₃₂, which are calculated using the Clausius–Mossotti equation:³⁹

α_D = 10α(Li⁺) + α(Zn/Mg²⁺) + 13α(Ti⁴⁺) + 32α(O²⁻) (6)

Here, α_D is the molecular dielectric polarizability, which is the sum of the dielectric polarizabilities of the individual ions: Li⁺ (1.20 ų), Zn²⁺ (2.04 ų), Mg²⁺ (1.32 ų), Ti⁴⁺ (2.93 ų) and O²⁻ (2.01 ų). It is clear that ε_r(C–M) is rather low compared with ε_r(corr), as listed in Table 3. In addition, the ε_r(corr) of Li₁₀MgTi₁₃O₃₂ is larger than Li₁₀ZnTi₁₃O₃₂, an opposite trend to that of ε_r(C–M). The discrepancy between ε_r(corr) and ε_r(C–M) can be due to the “rattling” or “compressed” cations. When the motion of cations and/or anions is restricted by crystal symmetry, larger than normal bond distances and/or thermal motions may occur, resulting in abnormally large polarizabilities.^40,41 Macroscopically, the variation in the bond length is manifested as the change in the bond valence (BV; V_i):

Where R_ij is the bond valence parameter, d_ij is the bond length and b is a constant (0.37 Å).⁴²Table 4 presents the BV values of the Li₁₀ZnTi₁₃O₃₂ and Li₁₀MgTi₁₃O₃₂ ceramics. The average bond valence (1.0921 v.u.) of the A-site (8a) in Li₁₀ZnTi₁₃O₃₂ is slightly lower than the normal value (1.1250 v.u.), indicating that the A-site is in an expanded state and that both Li(1) and Zn are rattling cations. The B-site (16d) also shows an expanded state, but only Ti is a rattling cation, and Li(2) is tightly bound in its octahedral site and is not a rattling cation. The same is true for each ion in Li₁₀MgTi₁₃O₃₂. The weighted average discrepancy factor (<d>) can give the average deviation from the normal bond valence sums in a compound:where W is the Wyckoff site multiplicity, D_i is the discrepancy factor for each of the n ions on a distinct crystallographic site, and N is the number of atoms in the unit cell. As listed in Table 4, the negative 〈d〉 values of the Li₁₀MTi₁₃O₃₂ (M = Zn, Mg) ceramics indicate that the ions are, on average, under-bonded and the overall structure of the material is in a slightly expanded state. The ε_r(corr) is higher than ε_r(C–M) because longer and weaker bonds are more likely to cause polarization. Furthermore, the |<d>| value of Li₁₀MgTi₁₃O₃₂ (0.0670 v.u.) is slightly higher than that of Li₁₀ZnTi₁₃O₃₂ (0.0601 v.u.), which explains why, although Li₁₀MgTi₁₃O₃₂ has a lower α_D value than Li₁₀ZnTi₁₃O₃₂, its ε_r(corr) value is slightly higher.

Table 3 Permittivity and τ_f values of the Li₁₀MTi₁₃O₃₂ (M = Zn, Mg) ceramics at the optimal sintering temperature

Compounds	S.T. (°C)	ε r	ε r(corr)	ε r(C–M)	Δεr (%)	τ f (ppm per °C)
Li10ZnTi13O32	980	28.23 ± 0.3	28.99	16.05	80.62	−17.06 ± 2.0
Li10MgTi13O32	980	29.23 ± 0.3	30.00	15.47	93.92	−11.05 ± 2.0

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Table 4 Bond ionicity and bond valences of the Li₁₀MTi₁₃O₃₂ (M = Zn, Mg) ceramics

Compound	Atom	f ^μi (%)	Site	BV (v.u.)	Average BV (v.u.)	Normal BV (v.u.)	D i (v.u.)	<d> (v.u.)
[Zn0.125Li0.875]^tet[Li0.1875Ti0.8125]^octO4	Zn(1)	43.2111	8a	1.8671	1.0921	1.1250	−0.1329	−0.0601
	Li(1)	65.7780		0.9813			−0.0187
	Li(2)	66.9681	16d	1.4366	3.2672	3.4375	−0.5634
	Ti(1)	92.8605		3.6897			−0.3103
	O(1)		32e	2.0819	2.0819	2.0000	0.0819
[Mg0.125Li0.875]^tet[Li0.1875Ti0.8125]^octO4	Mg(1)	43.2442	8a	1.8027	1.0794	1.1250	−0.1973	−0.0670
	Li(1)	65.8038		0.9760			−0.0240
	Li(2)	66.9655	16d	1.4390	3.2725	3.4375	−0.5610
	Ti(1)	92.8611		3.6957			−0.3043
	O(1)		32e	2.0703	2.0703	2.0000	0.0703

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The bond ionicity (f_i) can be calculated using Phillips–Van Vechten–Levine (P–V–L) theory and structural refinement of the bond length (Table S1, ESI†), which is usually related to the variation of the measured ε_r.^43,44 The relationship between the bond ionicity (f_i) and ε_r can be described using the following formula:⁴⁵

where n is the refractive index. The f_i of the μ bond is obtained using the following formula:⁴⁶where E^μ_g and C^μ are the average energy interval and heteropolar part, respectively (Table S2, ESI†). As listed in Table 4, the smaller bond ionicity of all the bonds in Li₁₀ZnTi₁₃O₃₂ is observed, suggesting a lower ε_r for Li₁₀ZnTi₁₃O₃₂ for than Li₁₀MgTi₁₃O₃₂. Besides, f^μ_i(Ti–O) possesses the maximum value in the Li₁₀MTi₁₃O₃₂ (M = Zn, Mg) ceramics, indicating that the Ti–O bond contributes the most to ε_r.

As shown in Fig. 4c, the τ_f values of the Li₁₀MTi₁₃O₃₂ (M = Zn, Mg) ceramics sintered at various temperatures show a weak temperature dependence and remained stable at around −17.06 ppm per °C (Li₁₀ZnTi₁₃O₃₂) and −11.05 ppm per °C (Li₁₀MgTi₁₃O₃₂). Generally, τ_f is strongly determined by the coefficient of thermal expansion (α_L) and the temperature coefficient of the dielectric constant (τ_ε) [i.e., τ_f = −(τ_ε/2 + α_L)].⁴⁷ The measured τ_ε values are +2.61 ppm per °C for Li₁₀MgTi₁₃O₃₂ and +15.77 ppm per °C for Li₁₀ZnTi₁₃O₃₂ at microwave frequencies between 25 °C and 85 °C. The α_L values of the Li₁₀MgTi₁₃O₃₂ and Li₁₀ZnTi₁₃O₃₂ ceramics are calculated as 9.74 ppm per °C and 9.18 ppm per °C, respectively. Clearly, τ_ε is larger than α_L and thus plays a dominant role in τ_f. And more deeply, τ_ε is related to the temperature coefficient of ionic polarizability (τ_αm):⁴⁸

The τ_αm values are 29.16 ppm per °C for Li₁₀ZnTi₁₃O₃₂ and 29.48 ppm per °C for Li₁₀MgTi₁₃O₃₂. Therefore, the negative τ_f values for Li₁₀MTi₁₃O₃₂ (M = Zn, Mg) originate from the positive τ_αm value being slightly higher than 3α_L, which makes the τ_f values close to zero.

As shown in Fig. 4d, the maximum Q × f values of 35 800 and 32 100 GHz are obtained, respectively, for Li₁₀ZnTi₁₃O₃₂ and Li₁₀MgTi₁₃O₃₂ sintered at 980 °C. The Q × f of Li₁₀MgTi₁₃O₃₂ is slightly lower than that of Li₁₀ZnTi₁₃O₃₂, which may be due to the increase in intrinsic dielectric loss caused by weakly bonded cations in the extended coordination environment. In a classical sense, the more under-bonded cations contribute to the dielectric loss because the cations “rattle” more freely in the polyhedral cage and may interact more strongly with the anharmonic phonons.⁴⁹ For intrinsic losses, the lattice energy (U) based on P–V–L theory can be evaluated as follows:^50–52

where Z^μ₊ and Z^μ₋ are the valence states of the cation and anion, d^μ is the bond length, and U^μ_bi and U^μ_bc are the ionic part and the covalent part of the μ bond, respectively. As listed in Table 5, the total lattice energy (U_total) of the Li₁₀ZnTi₁₃O₃₂ ceramic is greater than that of the Li₁₀MgTi₁₃O₃₂ ceramic, and the larger U_total corresponds to the higher Q × f value. This is because the Q × f value is mainly determined by the lattice anharmonicity.⁵³ As the lattice energy increases, the lattice anharmonicity will also increase, which reduces the inherent dielectric loss and increases the Q × f value. Moreover, the lattice energy of the Ti–O bond is much greater than that of other bonds, accounting for 76.12% of the total lattice energy for Li₁₀ZnTi₁₃O₃₂ and 76.13% for Li₁₀MgTi₁₃O₃₂, indicating that the contribution of the Ti–O bond to the Q × f value is dominant. In addition, the Q × f value is strongly dependent on the packing fraction, which is defined by summing the volume of packed ions over the unit cell volume:⁵⁴

Table 5 Bond length (d), lattice energy (U), Q × f, and packing fraction of the Li₁₀MTi₁₃O₃₂ (M = Zn, Mg) ceramics

Ceramic	Bond type	d (Å)	U (kJ mol^−1)	Q × f (GHz)	Packing fraction (%)
Li10ZnTi13O32	Li(1)–O × 4	1.9859	913.4093	35 800 ± 500	68.42
	Zn–O × 4	1.9859	419.3192
	Li(2)–O × 6	1.9949	314.7417
	Ti–O × 6	1.9949	5250.0245
	U total = 6897.4947
Li10MgTi13O32	Li(1)–O × 4	1.9879	912.5777	32 100 ± 500	68.36
	Mg–O × 4	1.9879	418.9638
	Li(2)–O × 6	1.9943	314.8187
	Ti–O × 6	1.9943	5250.1225
	U total = 6897.4827

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As seen in Table 5, the packing fraction of Li₁₀ZnTi₁₃O₃₂ (68.42%) is higher than that of Li₁₀MgTi₁₃O₃₂ (68.36%). The increase in packing fraction can reduce the vibration of the crystal lattice and thus enhance the quality factor. Raman spectroscopy is also an effective tool for studying lattice vibration information and dielectric loss.⁵⁵Table 6 shows the FWHM values of the A_1g mode at different sintering temperatures for the Li₁₀MTi₁₃O₃₂ (M = Zn, Mg) ceramics, and the trend of Q × f is opposite to the FWHM. The lower the FWHM, the smaller the space for lattice vibration and the non-harmonic vibration, resulting in a decrease in the inherent dielectric loss or an increase in Q × f.⁵⁶

Table 6 Variation of the FWHM value of the Raman spectrum with the sintering temperature for the Li₁₀MTi₁₃O₃₂ (M = Zn, Mg) ceramics

Compound	Sintering temperature (°C)	FWHM (cm^−1)
Li10ZnTi13O32	940	79.03337
	960	78.40893
	980	78.31708
	1000	78.53711
	1020	79.51513
Li10MgTi13O32	940	75.12137
	960	74.89138
	980	74.47194
	1000	75.18652
	1020	76.04403

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The Q × f values of the disordered spinel (Fd3̄m) microwave dielectric ceramics Li₁₀ZnTi₁₃O₃₂ and Li₁₀MgTi₁₃O₃₂ are not as high as those of ordered spinels, which may be related to the transport of Li⁺ ions in the structure. Complex impedance analysis was employed to explain the effects of Li⁺ ions on the structure and Q × f. Fig. 5a and b show the complex impedance plane plots of the Li₁₀MTi₁₃O₃₂ (M = Zn, Mg) ceramics at 633–573K. The Li₁₀MTi₁₃O₃₂ (M = Zn, Mg) ceramics show high resistivity and good insulation. The relationship between the DC (direct current) conductivity (σ_dc) and the temperature can be expressed via the following Arrhenius equation:

where σ₀ is the pre-constant, k_B is the Boltzmann constant (8.617 × 10⁻⁵ eV K⁻¹), T is the temperature (in Kelvin), and E_dc is the activation energy from DC conductivity. The E_dc values for Li₁₀MTi₁₃O₃₂ (M = Zn, Mg) can be obtained according to the slope of the fitted straight line for each ceramic: E_dc = 0.4230 eV for Li₁₀ZnTi₁₃O₃₂, and E_dc = 0.5531 eV for Li₁₀MgTi₁₃O₃₂ (Fig. 5c). The DC conductivity activation energy can reflect the sum of the free energies of the carriers to produce long jumps, and the small E_dc values in the Li₁₀MTi₁₃O₃₂ (M = Zn, Mg) ceramics might come from the embedding and detaching of Li⁺ in the spinel structure. The DC conductivities of the Li₁₀MTi₁₃O₃₂ (M = Zn, Mg) ceramics as a function of temperature were also measured in Fig. 5d. When the temperature is increased from 303 K to 683 K, the conductivity of Li₁₀ZnTi₁₃O₃₂ increases from 5.67 × 10⁻¹⁰ S cm⁻¹ to 2.21 × 10⁻⁵ S cm⁻¹, and the conductivity of Li₁₀MgTi₁₃O₃₂ increases from 6.11 × 10⁻⁹ S cm⁻¹ to 9.27 × 10⁻⁵ S cm⁻¹, where the conductivity of Li₁₀ZnTi₁₃O₃₂ is less than that of Li₁₀MgTi₁₃O₃₂. Li-rich spinel materials are also candidates for lithium-ion conductors, such as Li₄Ti₅O₁₂, which brings about the higher dielectric loss of this material relative to other spinels.

Fig. 5 Complex impedance plane plots of (a) Li₁₀ZnTi₁₃O₃₂ and (b) Li₁₀MgTi₁₃O₃₂ at 633–573 K. (The inset shows a magnification of the impedance). (c) Arrhenius plots of DC conductivity (σ_dc) for the Li₁₀MTi₁₃O₃₂ (M = Zn, Mg) ceramics. (d) DC conductivity of the Li₁₀MTi₁₃O₃₂ (M = Zn, Mg) ceramics as a function of temperature.

4. Conclusions

Li₁₀MTi₁₃O₃₂ (M = Zn, Mg) ceramics with a disordered spinel structure (Fd3̄m) were synthesized using a solid-state reaction method. Li₁₀ZnTi₁₃O₃₂ and Li₁₀MgTi₁₃O₃₂ with relative permittivities of 28.23 ± 0.3 and 29.23 ± 0.3, Q × f values of 35 800 ± 500 GHz and 32 100 ± 500 GHz (at ∼7.5 GHz), and τ_f values of −17.06 ± 2.0 ppm per °C and −11.05 ± 2.0 ppm per °C, respectively, were obtained at 980 °C. The permittivities ε_r(C–M) calculated using the Clausius–Mossotti (C–M) equation are significantly lower than the permittivities ε_r(corr) corrected by the porosity (ε_r(C–M) = 16.05, ε_r(corr) = 28.99 for Li₁₀ZnTi₁₃O₃₂; ε_r(C–M) = 15.47, ε_r(corr) = 30.00 for Li₁₀MgTi₁₃O₃₂). The main reason for this phenomenon comes from the longer and weaker bonds in the Li₁₀MTi₁₃O₃₂ (M = Zn, Mg) ceramics that are more facile to polarization. The larger negative weighted average discrepancy factor (<d>) of Li₁₀MgTi₁₃O₃₂ (−0.0670 v.u.) than that of Li₁₀ZnTi₁₃O₃₂ (−0.0601 v.u.) is a good explanation for the higher ε_r(corr) value of Li₁₀MgTi₁₃O₃₂ (α_D = 115.73 A³), although its dielectric polarizability (α_D) is lower than that of Li₁₀ZnTi₁₃O₃₂ (α_D = 116.45 A³). This expansion structure also results in a closer value of τ_f to zero and a lower Q × f. Values of the temperature coefficient of ionic polarizability (τ_αm) are 29.22 ppm per °C for Li₁₀ZnTi₁₃O₃₂ and 29.84 ppm per °C for Li₁₀MgTi₁₃O₃₂, and the small negative τ_f values of Li₁₀MTi₁₃O₃₂ (M = Zn, Mg) are derived from the proximity of τ_αm to 3α_L. Calculations using P–V–L theory also confirmed that the bond ionicity (f_i) of all the bonds in the prepared Li₁₀MgTi₁₃O₃₂ structure was higher than that of Li₁₀ZnTi₁₃O₃₂, and the Ti–O bond contributed the most to ε_r. The lattice energy of the Ti–O bonds is much greater than that of other bonds, indicating that it is also a major contributor to Q × f in the Li₁₀MTi₁₃O₃₂ (M = Zn, Mg) ceramics. In addition, a study of the DC conductivity clarified that the Q × f values of the disordered spinels Li₁₀ZnTi₁₃O₃₂ and Li₁₀MgTi₁₃O₃₂ are not as high as other ordered spinels, which may be due to the transport of Li⁺ ions in their structures.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

This work was supported by the National Natural Science Foundation of China (No. 22105048), Guangxi Science and Technology Plan Project (No. AD20297035), the Natural Science Foundation of Guangxi Zhuang Autonomous Region (No. 2022GXNSFBA035602), the Guilin University of Technology Research Startup Project (No. GUTQDJJ2021073), and the Guangxi BaGui Scholars Special Funding.

References

1. C. L. Huang and T. H. Hsu, J. Eur. Ceram. Soc., 2022, 42, 3892–3897.
2. J. Li, J. Zhu and G. Wang, Ceram. Int., 2022, 48, 15261–15267.
3. S. Z. Hao, D. Zhou, L. X. Pang, M. Z. Dang, S. K. Sun, T. Zhou, S. Trukhanov, A. Trukhanov, A. S. B. Sombra, Q. Li, X. Q. Zhang, S. Xia and M. A. Darwish, J. Mater. Chem. C, 2022, 10, 2008–2016.
4. J. Bao, Y. Zhang, H. Kimura, H. Wu and Z. Yue, J. Adv. Ceram., 2023, 12, 82–92.
5. X. Zhou, L. Liu, J. Sun, N. Zhang, H. Sun, H. Wu and W. Tao, J. Adv. Ceram., 2021, 10, 778–789.
6. H. Tian, J. Zheng, L. Liu, H. Wu, H. Kimura, Y. Lu and Z. Yue, J. Mater. Sci. Technol., 2022, 116, 121–129.
7. T. Y. Qin, C. W. Zhong, Y. Shang, L. Cao, M. X. Wang, B. Tang and S. R. Zhang, J. Alloys Compd., 2021, 886, 161278.
8. K. P. Surendran, P. V. Bijumon, P. Mohanan and M. T. Sebastian, Appl. Phys. A: Mater. Sci. Process., 2005, 81, 823–826.
9. S. Takahashi, Y. Imai, A. Kan, Y. Hotta and H. Ogawa, Jap. J. Appl. Phys., 2015, 54, 10NE02.
10. A. Belous, O. Ovchar, D. Durilin, M. M. Krzmanc, M. Valant and D. Suvorov, J. Am. Ceram. Soc., 2006, 89, 3441–3445.
11. H. F. Zhou, X. L. Chen, L. Fang, D. J. Chu and H. Wang, J. Mater. Res., 2010, 25, 1235–1238.
12. S. George and M. T. Sebastian, J. Am. Ceram. Soc., 2010, 93, 2164–2166.
13. L. Y. Ao, Y. Tang, J. Li, W. S. Fang, L. Duan, C. X. Su, Y. H. Sun, L. J. Liu and L. Fang, J. Eur. Ceram. Soc., 2020, 40, 5498–5503.
14. S. Takahashi, A. Kan and H. Ogawa, Jap. J. Appl. Phys., 2016, 55, 10TE01.
15. H. Zhou, X. Liu, X. Chen, L. Fang and Y. Wang, J. Eur. Ceram. Soc., 2012, 32, 261–265.
16. H. F. Zhou, J. Z. Gong, N. Wang and X. L. Chen, Ceram. Int., 2016, 42, 8822–8825.
17. N. Wang, H. F. Zhou, J. Z. Gong, G. C. Fan and X. L. Chen, J. Electron. Mater., 2016, 45, 3157–3161.
18. J. Zhang, R. Z. Zuo, Y. Wang and S. S. Qi, Mater. Lett., 2016, 164, 353–355.
19. H. Xiang, L. Fang, W. Fang, Y. Tang and C. Li, J. Eur. Ceram. Soc., 2017, 37, 625–629.
20. K. Xiao, Y. Tang, Y. F. Tian, C. C. Li, L. Duan and L. Fang, J. Eur. Ceram. Soc., 2019, 39, 3064–3069.
21. W. S. Fang, Y. M. Sun, L. Fang, Y. Tang and C. C. Li, J. Alloys Compd., 2017, 722, 1002–1007.
22. S. K. Singh, S. R. Kiran and V. R. K. Murthy, Mater. Chem. Phys., 2013, 141, 822–827.
23. R. Z. Zuo and J. Zhang, J. Am. Ceram. Soc., 2016, 99, 3343–3349.
24. H. Luo, L. Fang, H. Xiang, Y. Tang and C. Li, Ceram. Int., 2017, 43, 1622–1627.
25. L. Fang, D. Chu, H. Zhou, X. Chen and Z. Yang, J. Alloys Compd., 2011, 509, 1880–1884.
26. M. Li, Y. Tang, H. Xiang, J. Li, D. Zhou and L. Fang, Ceram. Int., 2023 DOI:10.1016/j.ceramint.2022.11.021.
27. Y. Tang, H. Li, J. Li, W. Fang, Y. Yang, Z. Zhang and L. Fang, J. Eur. Ceram. Soc., 2021, 41, 7697–7702.
28. Y. Yang, Y. Tang, J. Li, L. Fang and H. C. Xiang, ACS Appl. Electron. Mater., 2022, 4, 3512–3519.
29. X. W. Hu, J. W. Chen, J. Li, H. C. Xiang, Y. Tang and L. Fang, J. Eur. Ceram. Soc., 2022, 42, 7461–7467.
30. Y. M. Dai, J. W. Chen, Y. Tang, H. C. Xiang, J. Li and L. Fang, Ceram. Int., 2023, 49, 875–881.
31. V. S. Hernandez, L. M. T. Martinez, G. C. Mather and A. R. West, J. Mater. Chem., 1996, 6, 1533–1536.
32. L. T. Nong, X. F. Cao, C. C. Li, L. J. Liu, L. Fang and J. Khaliq, J. Eur. Ceram. Soc., 2021, 41, 7683–7688.
33. H. D. Lutz, Z. Naturforsch., A: Phys. Sci., 1969, 24, 1417–1419.
34. D. Z. Liu, W. Hayes, M. Kurmoo, M. Dalton and C. Chen, Phys. C, 1994, 235, 1203–1204.
35. H. D. Lutz, B. Müller and H. J. Steiner, J. Solid State Chem., 1991, 90, 54–60.
36. L. Malavasi, P. Galinetto, M. C. Mozzati, C. B. Azzoni and G. Flor, Phys. Chem. Chem. Phys., 2002, 4, 3876–3880.
37. A. Magrez, M. Cochet, O. J oubert, G. Louarn, M. Ganne and O. Chauvet, Chem. Mater., 2001, 13, 3893–3898.
38. A. J. Bosman and E. E. Havinga, Phys. Rev., 1963, 129, 1593–1600.
39. R. D. Shannon, J. Appl. Phys., 1993, 73, 348–366.
40. H. S. Park, K. H. Yoon and E. S. Kim, Mater. Chem. Phys., 2003, 79, 181–183.
41. Y. S. Cho, K. H. Yoon, B. D. Lee, H. R. Lee and E. S. Kim, Ceram. Int., 2004, 30, 2247–2250.
42. N. E. Brese and M. O. Keeffe, Acta Crystallogr., 1991, 47, 192–197.
43. H. Y. Yang, S. R. Zhang, H. C. Yang, Y. Yuan and E. Z. Li, J. Am. Ceram. Soc., 2019, 102, 5365–5374.
44. H. Y. Yang, S. R. Zhang, H. C. Yang and E. Z. Li, Inorg. Chem. Front., 2020, 7, 4711–4753.
45. S. S. Batsanov, Russ. Chem. Rev., 1982, 51, 684–697.
46. B. F. Levine, J. Chem. Phys., 1973, 59, 1463–1486.
47. E. L. Colla, I. M. Reaney and N. Setter, J. Appl. Phys., 1993, 74, 3414–3425.
48. A. J. Bosman and E. E. Havinga, Phys. Rev., 1963, 129, 1593.
49. M. W. Lufaso, Chem. Mater., 2004, 16, 2148–2156.
50. H. Yang, S. Zhang, Y. Chen, H. Yang, Y. Yuan and E. Li, Inorg. Chem., 2019, 58, 968–976.
51. D. Liu, S. Zhang and Z. Wu, Inorg. Chem., 2003, 42, 2465–2469.
52. R. Zurmühlen, J. Petzelt, S. Kamba, V. V. Voitsekhovskii, E. Colla and N. Setter, J. Appl. Phys., 1995, 77, 5341–5350.
53. Y. G. Zhao and P. Zhang, RSC Adv., 2015, 5, 97746–97754.
54. B. D. Silverman, J. Phys. Rev., 1962, 125, 1921–1930.
55. S. Y. Wang, J. D. Chen, Y. J. Zhang and Y. C. Zhang, J. Alloys Compd., 2019, 805, 852–858.
56. Y. Tang, S. Y. Shen, J. Li, X. G. Zhao, H. C. Xiang, H. P. Su, D. Zhou and L. Fang, J. Eur. Ceram. Soc., 2022, 42, 4573–4579.

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Footnote

† Electronic supplementary information (ESI) available. See DOI: https://doi.org/10.1039/d2ma01074g

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