Usage of P–V–L bond theory in studying the structural/property regulation of microwave dielectric ceramics: a review

Abstract

Structure/property regulation is widely investigated for providing theoretical foundations and guidance for designing microwave dielectric ceramics. P–V–L bond theory targets the provision of detailed structural parameters of crystals and fundamental chemical bond characteristics, such as bond ionicity, bond covalency and bond susceptibility. In this review, the concept of P–V–L bond theory, the binary bonding formula theory of complex polycrystalline and the usage of P–V–L bond theory regarding the structural effects on dielectric properties in various crystal structures are introduced. And the bond traits provided via P–V–L bond theory in several ion-doping modification systems are generalized with structural evolution and dielectric properties. Furthermore, conjectures about the development and application directions of P–V–L bond theory in the study of structure/properties of microwave dielectric ceramics are put forward.

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Hongyu Yang

Hongyu Yang is a PhD candidate in the School of Electronic Science and Engineering, University of Electronic Science and Technology of China, Chengdu. He obtained his Bachelor's Degree in 2015 from the Southwest University of Science and Technology, China. His research interests include microwave dielectric materials and low temperature co-fired ceramics.

Shuren Zhang

Shuren Zhang is a professor in the School of Electronic Science and Engineering, University of Electronic Science and Technology of China, Chengdu. His research interests include electronic materials and devices, electronic materials and engineering, electronic thin films and integrated devices.

Hongcheng Yang

Hongcheng Yang is a PhD candidate in the School of Electronic Science and Engineering, University of Electronic Science and Technology of China, Chengdu. She obtained her Bachelor's Degree in 2016 from the Southwest University of Science and Technology, China. Her research interests include microwave dielectric materials and low temperature co-fired ceramics.

Enzhu Li

Enzhu Li is a professor in the School of Electronic Science and Engineering, University of Electronic Science and Technology of China, Chengdu. She received her PhD degree from Tokyo Institute of Technology in 2008. Her research interests include microwave dielectric materials and low temperature co-fired ceramics.

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1. Introduction

Microwave dielectric ceramics (MWDCs) are widely used in microwave frequency (300 MHz–300 GHz) circuits as dielectric materials to supply one or more functions. MWDCs have become promising in electronic communication technology for their widespread applications in filters, resonators, RF/antennae, frequency discriminators in electronic countermeasures, navigation, radar, home satellite live television receivers and hand-held mobile phones.^1,2 For their practical application, three key indicators are required: a moderate dielectric constant ε_r value to make the size of the device controllable, a high Q × f value (Q = 1/tan δ, f is the resonant frequency) to reduce the insertion loss of the application, and a near zero τ_f value to ensure the carrier signal of the application does not drift with the working environment of the communication devices.^3,4 Specifically, ceramics with various ε_r values show different application fields: e.g. ε_r < 20: Al₂O₃ (ε_r = 10, Q × f = 500 000 GHz)⁵ for electronic circuit packages and dielectric resonance antennae, Zn₂SiO₄ (ε_r = 6.6, Q × f = 219 000 GHz, τ_f = −61 ppm per °C)⁶ and (Mg_1−xZn_x)Al₂O₄ (ε_r = 7.9–8.5, Q × f = 82 000–106 000 GHz, τ_f = −73 ppm per °C)⁷ for microwave substrates; 20 < ε_r < 70: Ba(B_1/3B′_1/3)O₃ (B = Zn/Mg/Co/Ni, B′ = Ta/Nb) series of complex perovskite structures^1,8–11 (BaZn_1/3Ta_2/3O₃: ε_r = 29, Q × f = 150 000 GHz, τ_f = 0 ppm per °C), BaTi₄O₉,¹² Ba₂Ti₉O₂₀¹³ and (Zr, Sn)TiO₄^14,15 for microwave dielectric resonance devices; ε_r > 70: BaO–Ln₂O₃–TiO₂ (Ln = Sm, Nd)^16–18 and CaO–Li₂O–Ln₂O₃–TiO₂^19–21 systems. Up to now, several MWDC materials with ε_r values of 5–100 have been commercialized, possessing excellent high Q × f values and zero τ_f values:¹e.g. BaB_1/3Ta_2/3O₃ (B = Mg, Zn), Ba(Co,Zn)_1/3Nb_2/3O₃, SrTiO₃–LaAlO₃, and ZrTiO₄–ZnNb₂O₆. Driven by the rapid development of the internet of things (IoT) and 5th generation wireless systems (5G), much effort has been devoted to the exploration of novel MWDC materials with excellent dielectric performances.

Considering that crystal structures of different microwave dielectric ceramics are quite different, it is of great significance to study the influence of crystal structure and the origin of dielectric properties of ceramics in the microwave region for improvements in design and performance.^22,23 The properties of microwave dielectric ceramics are affected mainly by the crystal structure and fabrication process. The fabrication process, such as raw materials, preparation technology, phase structure, numbers of grain boundaries per volume, grain growth, porosity and sintering aids, inevitably results in extrinsic loss of microwave dielectric properties. For instance, the choice of boric acid as a raw material instead of boron oxide is attributed to its better solubility and dispersion in deionized water when preparing Li₂O–B₂O₃–SiO₂ glass frit.²⁴ A sol–gel method realizes better uniformity at the molecular level than a solid-state process, which is beneficial for improving the Q × f value.²⁵ A high sintering temperature positively affects the cation ordering of Ba(Mg_1/3Ta_2/3)O₃ ceramics and then improves the Q × f value.²⁶ Ceramic samples with compact microstructures and uniform distributions of average grain size can also reduce the extrinsic dielectric loss.²⁷ When controlling external factors, such as preparation/fabrication technology, the influence of crystal structure as an internal factor has a crucial impact on microwave dielectric performance. Up to now, several theories have been developed to study the regulation between crystal structure and dielectric properties. For example, THz time-domain spectroscopy and far-infrared reflectance spectroscopy can be used to analyse the intrinsic properties of the lattice vibration of material systems;^28–31 the Clausius–Mossotti equation predicts the macroscopic dielectric constant of an ionic crystal by using ionic polarizabilities;^32,33 bond-valence theory gives the actual valence state of ions in a crystal structure and provides the basis for the state of the ions.^34–37 These concepts help researchers gain better insights into the relationship between crystal structure and dielectric properties.

Nowadays, the chemical bond theory named the P–V–L bond theory, established by Phillips, Van Vechten and Levine,^38–43 which provides fundamental bond characteristics, has been paid increasing attention and been applied to various dielectric ceramic systems to discuss the influence of crystal structure on the variation in chemical bond parameters and to establish the relationship between chemical bond characteristics and dielectric properties.

Chemical bond characteristics of crystals are crucial for analysing and studying the chemical and solid-state physical aspects of crystal structures. Several crystal features can be explained qualitatively using these concepts: e.g. the greater the electronegativity difference between A and B atoms in binary compounds, the more likely it is that a rock salt structure can be formed, otherwise an open covalent structure is preferred.⁴¹ During the early 1970s, the chemical bond theory first proposed by Phillips and Van Vechten (abbreviated as P–V) was applied to characterize crystal features quantitatively. This theory relies on the dielectric definition of crystals and defines the calibration of chemical bond characteristics and the relationship between various micro-parameters in crystals. It has been used effectively to calculate many aspects of crystals, such as bond charge,^44,45 elastic and piezoelectric constants,^46,47 electronic structure,⁴⁸ thermodynamic properties,⁴⁹ cohesive energies and heats of formation,⁵⁰ nonlinear optical susceptibilities,^51–53 effective lattice charges,⁵⁴ valence band parameters,⁵⁵ band structures,⁵⁶ impurity binding energies,⁵⁷ photoelastic constants,^58,59 and electro-optic and elasto-optic phenomena.⁶⁰ It should be noted that the P–V theory applies only to eight-electron A^NB^8−N type crystals with one type of chemical bond: e.g. zinc blende, wurtzite, and rock salt. Following that, Levine then expanded the application of P–V bond theory to A_mB_n (NaCl, A^IVB^VI, A^NB^16−N), A^IB^IIIC^VI₂, A^IIB^IVC^V₂ and A^IIIB^VC^VI₄ systems by dividing the crystal properties into parameters related to a single bond, such as polarizability, ionic properties and the average number of electrons per bond.⁴² However, it should be realized that there are limitations since two types of bonds, with the same coordination number and nearly the same nearest-neighbour distances should be satisfied simultaneously. Even so, a normalization idea attributing the crystal properties to individual chemical bonds was developed.

In order to make P–V–L bond theory more applicable to extensive and complex multi-component crystals (A_aB_bC_cD_d), Zhang put forward the idea of decomposing complex polycrystals into binary crystals, where the crystal can be regarded as an aggregation of chemical bonds between ions.^61,62 Suppose that two ions that make up the chemical bond have the same symmetrical positions in the crystal, and bonds with the same length belong to one type of bond. Each kind of chemical bond with a binary expression has its own specific bond parameters, spatial configuration and proportion of elements. If the element proportional relationship of this kind of chemical bond is called a binary bond formula, then each kind of chemical bond in a crystal corresponds to a unique binary bond formula. The complex crystal is the aggregation of all the kinds of chemical bonds, and the molecular formula of the crystal is the sum of all the kinds of binary bond formulae. Then P–V–L bond theory can be applied to calculate the binary bond formula. Finally, once the relationship between the binary bond formula and complex crystals has been established, the purpose of studying the chemical bond properties of multiple complex crystals can be understood, and the relationship between microstructural parameters and macroperformance can be constructed. With these considerations in mind, P–V–L bond theory has been applied to several complex compounds. For instance, in a high-temperature oxide superconductor structure, the relationships between critical transition temperature and covalency have been established in REBa₂Cu₃O₇ (RE = Ln)^63,64 and HgBa₂Ca_n−1Cu_nO_2n+2+δ systems.⁶⁵ The calculation of chemical bond parameters of a hexagonal ferrite (such as M-type BaFe₁₂O₁₉ or W-type BaCo₂Fe₁₆O₂₇) crystal solved the calculation of isomeric displacement of Mössbauer spectra;⁶⁶ the stability of crystal structures was predicted via the calculations of lattice energy and thermal expansion of solid materials,^67,68etc. Therefore, the physical and chemical properties of complex crystals have been explained at the scale of chemical bonds. Similarly, with the help of the binary bond splitting theory of polycrystalline materials, the P–V–L bond theory is expected to be applicable in MWDC systems.

In this contribution, we review the bond theory established by Phillips, Van Vechten and Levine and the binary bonding formula theory of complex polycrystals proposed by Zhang, and we specifically precis current studies focusing on structure–properties relationships discussed using P–V–L bond theory, and summarizing how P–V–L bond theory correlates with microwave dielectric properties.

2. P–V–L bond theory

The chemical bond theory established by Phillips, Van Vechten and Levine (P–V–L) is a dielectric description of the chemical bonds in a crystal structure, which correlates the structure and dielectric properties of a crystal. Firstly, in the theoretical calculation of a binary crystal A^NB^8−N, Phillips chose to build a unit cell with two atoms, where a symmetrical coordinate system is based on the middle position of the atoms.⁴¹ Thus, the periodic potential of a crystal is expanded by Fourier transformation:

V_G = [V_A(G) + V_B(G)]·cos(G·r) + i[V_A (G) − V_B (G)]·sin(G·r) (2)

where G refers to the reciprocal lattice vector, V_A and V_B are the pseudopotentials of atoms A and B, and V_A + V_B is the homopolar potential, and V_A − V_B is the ionic potential. Therefore, eqn (2) between real basis functions is:

That is, the average energy gap between bonding and anti-bonding molecular orbitals E_g is divided into two parts: the average covalent energy gap E_h caused by homopolar potential energy and the average ionic energy gap C caused by ionic potential energy. The following relationship was put forward:

E_g² = E_h² + C² (4)

The expressed form of the complex dielectric function is given as ε(ω) = ε₁(ω) + iε₂(ω), which is correlated with the Kramers–Kronig relationship:

By using a nearly free electron model isotropically applied to three dimensions, the low frequency limit ε₁(0) was given as follows:⁶⁹

where ℏω_p refers to the Planck constant, E_F is the Fermi energy, E_g is the energy gap, N equals 4/atomic volume, and ω_p is the free electron plasma frequency.

Following that, Van Vechten provided calculation methods for the electronic dielectric constants of 68 crystals.^39,40 The homopolar part E_h is in reverse proportion to the nearest-neighbour distance, and the expression of the heteropolar part C is given as:

E_h = 39.74·r^−2.48 eV (9)

k_s = (4k_F/πa₀)^1/2 (12)

k_F = (3π²N_e)^1/3 (13)

where r refers to the ionic radius of atom A/B, Z is the number of valence electrons on atom A/B, e^−k′′sR is the Thomas–Fermi screening factor, b is the value of order of unity obtained from ref. 39, which lies in the range 1.4–1.6, a₀ is the Bohr radius, and N_e refers to eight electrons per diatomic volume. N_e is calculated as:for a crystal with one type of bond, v_b = 1/N_b (N_b refers to the bond density, which is also the number of bonds per unit volume), Z_A is the valence electron of atom A, and N_CA is the coordination number of atom A.

The effect of the d state in atomic cores should be considered; therefore, ε₀ is adjusted as:

D = Δ_αΔ_β − (δ_αδ_β)·(Z_α − Z_β)² (16)

Δ and δ are parameters for the rows of the periodic table corresponding to α and β, which are chosen from Table 1. Therefore, the dielectric constants of crystals, such as diamond, zinc blende, wurtzite and rock-salt type crystals can be predicted via P–V bond theory.

Table 1 Δ and δ values of elements from the periodic table

row	Δ	δ
1	1.0	1.0
2	1.0	1.0
3	1.0	1.0
4	1.12	1.0025
5	1.21	1.0050
6	1.31	1.0075

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As is known, chemical bonds are strong interaction forces between two or more atoms/cations in a molecule or crystal. Generally, the bonds formed by electrostatic attraction and electron transfer between atoms are called ionic bonds, in which ions can be single or atomic groups, such as Na⁺, Cl⁻ or NO₃⁻. While bonds formed between atoms by sharing pairs of electrons (overlapping clouds of electrons) are covalent bonds: e.g. HCl: H–Cl polar bond, and Cl₂: Cl–Cl nonpolar bond. In crystals, chemical bonds are beneficial for discussing several aspects of chemical and solid-state physical properties: for instance, the heat absorption and exothermic effects of a chemical reaction, the stability of a crystal structure, the melting/boiling point and the hardness of a crystal. Therefore, it is of great importance to study the chemical bonds at great length to achieve a better understanding of a crystal structure and its chemical–physical properties.

People find it appealing to determine the ionic and covalent contribution of chemical bonds since they are not in a fully ionic or covalent state. In an A^NB^8−N crystal, Pauling defined the ionicity from a thermochemical point of view, where the ionicity originates from extra-ionic energy Δ_AB, which is proportional to the absolute value of the electronegativity difference between A and B atoms (M refers to the number of resonating bonds):⁴¹

To give a spectroscopic definition of ionicity and covalency, Phillips defined the polar coordinates as in Fig. 1:

Fig. 1 A sketch showing how (E_g, φ) polar coordinates correspond to the covalent and ionic energies represented as Cartesian coordinates (E_h, C). The polar angle φ is called the ionicity phase angle (reprinted with permission from ref. 41. 2020 Copyright American Physical Society) (http://dx.doi.org/10.1103/RevModPhys.42.317).

Therefore, the ionicity or covalency of an A^NB^8−N crystal can be quantitatively calculated. By using the concept of ionicity, the coordination environments of 68 A^NB^8−N crystals, such as diamond, zinc blende, wurtzite and rock-salt type, were predicted precisely, as shown in Fig. 2, where crystals tend to be four-fold coordinated if f_i < 0.785 ± 0.010; otherwise, they are six-fold coordinated.

Fig. 2 The separation of four-fold and six-fold coordinated structures, using the spectroscopically defined covalent and ionic energy gaps E_h and C, respectively (reprinted with permission from ref. 41. Copyright 2020 American Physical Society) (http://dx.doi.org/10.1103/RevModPhys.42.317).

However, it should be noted that P–V theory is restricted to A^NB^8−N crystals with only one type of chemical bond. To make it more applicable to other complex systems, Levine expanded the P–V theory to A_mB_n-type crystals and some systems with multiple bonds by using the idea of decomposing bulk properties into individual chemical bonds. Therefore, the macro-properties of crystal can be regarded as the sum of micro-contributions of chemical bonds. In his modification, one can expect the macro-susceptibility to be given by:⁴²

where F^μ represents the proportion of μ-type bonds among all chemical bonds of crystals, χ^μ is the susceptibility of μ-type bonds, N^μ_b is the bond density of μ-type bonds, that is, the number of μ-type chemical bonds per unit volume, and χ^μ_b is the susceptibility of an individual bond in the μ-type bonds. The dielectric constant and susceptibility of μ-type bonds can be expressed as:

ε^μ = 4πχ^μ + 1 (23)

where ε^μ is the dielectric constant of a μ bond, and Ω^μ_p is the plasma frequency of a μ bond, which is composed of:where N^μ_e is the number of valence electrons of a μ bond, A^μ and D^μ, similar to eqn (7) and (15) corresponding to a μ bond. The Fermi energy E^μ_F is calculated from the Fermi wave vector k^μ_F:

k^μ_F = (3π²·N^μ_e)^1/3 (26)

where N^μ_e is defined as:with n^μ_v the number of valence electrons per μ bond, and v^μ_b the bond volume.

A special note for Levine's theory is that the heteropolar part C in an A_mB_n crystal is different and is adjusted as:

where r^μ₀ = (r_A + r_B)/2, and the value of b^μ is correlated with the average coordination number N^μ_c:

b^μ = 0.089·(N^μ_c)² (33)

Therefore, Levine's modification expanded the application of P–V theory to A_mB_n binary systems. Although it is still not practical for more complex multi-systems, the idea of decomposing the macro-properties into individual bonds is beneficial for future work.

3. Binary bonding formula theory of complex polycrystals

In the process of materials research, compositions of materials are generally not confined to binary systems. For instance, high-entropy alloys, usually formed of five or more equal or approximately equal metals, have excellent properties in hardness, high temperature resistance, corrosion resistance, and wear resistance.⁷⁰ In the MWDC field, the ionic modification is usually based on a ternary system as well, such as Bi₂O₃–TiO₂–V₂O₅,⁷¹ ZnO–TiO₂–Nb₂O₅,⁷² or BaO–MgO–Nb₂O₅⁷³ systems. It is important to have a better understanding of the impact of chemical bond features on crystal structure. Therefore, P–V–L bond theory should be appropriately adjusted for a good fit with multi-systems.

The composition of a crystal can be expressed by the chemical molecular formula, which represents the atomic ratio among various chemical elements that constitute the crystal. However, the expression does not correspond to the spatial configuration in the specific crystal structure. On the other hand, the crystal can also be regarded as the aggregation of various ions and chemical bonds between ions. If the two ions that compose the chemical bond sharing the same symmetrical positions in a crystal and bond length are classified as the same type of bond, then this type of bond with the same specific properties, spatial configuration, and proportion of elements is called a binary bonding formula. Since more than one type of bond exists in a crystal, each kind of bond corresponds to a unique binary bonding formula. The crystal is the aggregation of various types of chemical bonds, so that the molecular formula of the crystal should also be the sum of the binary bonding formulas. Hence, one can deal with the binary bonding formula with P–V–L bond theory to obtain bond features. Once the proportional relationship between the binary bonding formula and complex multi-element crystal formula has been built, the purpose of dealing with the complex polycrystalline system is realized. Briefly speaking, the key idea is to convert the chemical formula of a complex polycrystal into a combination of binary bonding formulas.

Zhang developed the method for decomposing complex polycrystals into the binary bonding formula of a crystal when the crystal structure, coordination environment and element proportion in the crystal formula have been determined.

Suppose that the molecular formula of any complex crystal is:^62,74

where A and B refer to cations and anions, respectively, Aⁱ and B^j symbolise different elements or symmetry positions of equivalent elements, a_i and b_j indicate the number of elements. Correspondingly, the binary bonding formula of any type of A–B bond can be resolved under the conditions of given coordination numbers N_CAi and N_CBj:where N(B^j − Aⁱ) is the number of Aⁱ cations in the ligand of the B^j ion, and N(Aⁱ − B^j) is the number of B^j cations in the ligand of the Aⁱ ion. The binary bonding formula should be electrically neutral like the crystal, and the same ion shows different valence states in different bonding formulas. Therefore, it is crucial to determine the effective valence electron number of the ion in the specific bonding formula.

Suppose that A and B ions in any μ bond present valences of P^μ(A–B) and P^μ(B–A), the following equation is expected:

The effective charges q^μ_A and q^μ_B of A and B ions are calculated as:

Therefore, the valence electron numbers Z_A and Z_B values in P–V–L bond theory should be adjusted as effective valence electron numbers (Z_A)* and (Z_B)*:⁷⁵

(Z_A)* = Z_A·q^μ_A (41)

(Z_B)* = Z_B·q^μ_B (42)

Combining P–V–L bond theory and the binary bonding formula expression, crucial fundamental bond parameters could be achieved, and two key indicators of MWDC, dielectric polarization ε_r and Q × f value, could be discussed. Starting from the crystal structure configuration, one can calculate the detailed bond volume v^μ_b using crystal structure parameters from the ICSD database or Rietveld refinement results. Combined with the coordinate environment and valence electrons of atoms, the parameter for the number of valence electrons of a μ bond, N^μ_e, could be obtained, and finally the bond ionicity f_i, bond covalency f_c and bond susceptibility χ could be specially assigned, which is helpful in discussing dielectric polarization. Furthermore, lattice energy U presents the binding ability between ions, where a high U value stands for a stable structure, and the internal loss caused by anharmonic vibration would be reduced. For MWDC, it reflects the internal dielectric loss tan δ (Q = 1/tan δ, f is the resonant frequency), which is closely corelated with the Q × f value. As mentioned, there is no 100% ionic chemical bond in a crystal; therefore, the ionic part of a chemical bond's contribution to lattice energy U_bi and the covalent part of contribution U_bc should be considered. The lattice energy U is calculated as follows:⁶⁸

U_total = ∑U^μ_b (43)

U^μ_b = ∑(U^μ_bi + U^μ_bc) (44)

where Z^μ₊ and Z^μ₋ are the valence states of the cation and anion, d^μ is the bond length, and the m/n value is obtained from the binary bonding formula.

4. Usage of P–V–L bond theory in dielectric ceramics

As mentioned above, much attention has been paid to the application of P–V–L bond theory in the MWDC field in recent years because investigations of chemical bond parameters are instrumental in studying structural features, and then providing the basis for ionic doping modification and the design of novel materials, which is helpful for understanding microwave dielectric properties. Up to now, several ceramic systems with excellent dielectric performances have been developed, as listed in Table 2.^13,76–95 In this section, current studies of applying P–V–L bond theory in the MWDC field with regard to structural-property investigations are summarized.

Table 2 Representative microwave dielectric ceramic systems with different dielectric constants

Material systems	ST (°C)	ε r	Q × f (GHz)	τ f (ppm per °C)	Ref.
Mg2SiO4	1450	6.8	270 000	−60	76
Al2O3	1550	10.0	333 000	−60	77
0.75MgAl2O4–0.25TiO2	1400–1460	11.0	105 400	−12	78
Mg4(TaNb0.975V0.025)O9	1200	12.0	200 000	−73	79
MgWO4	1150	13.1	69 000	−58	80
Mg2TiO4	1500	14.2	160 000		81
NdNbO4	1250	19.6	33 000	−24	82
0.95MgTiO3–0.05CaTiO3	1400	20.0	56 000	0	83
(Zn0.7Mg0.3)TiO3	950	22.0	65 000	−80	84
Ba(Mg1/3Ta2/3)O3	1640	24.0	430 000	5	85
0.9LaAlO3–0.1SrTiO3	1680	26.7	139 000	−50	86
Zn1.01Nb2O6/TiO2/Zn1.01Nb2O6	1200	26.8	99 500	0.5	87
La(Mg1/2Ti1/2)O3	1650	29.0	114 000	−81	88
ZnTiNb2O8	1250	34.0	42 500	−52	89
BaTi4O9	1250	36.0	50 470	16	13
Ba2Ti9O20	1300	37.0	57 000	−6	13
Zr0.8Sn0.2TiO4	1600	38.9	51 500	0.7	90
Ba5Nb4O15	1380	39.0	23 700	78	91
CaLa4Ti4O15	1550	41.1	50 240	−25	92
0.76ZrTiO4–0.24ZnNb2O6	1260–1300	47.0	34 000	0	93
Li0.95Nb0.45Ti0.7O3	1100	56.2	8350	15	94
BaO–Sm2O3–TiO2	1300–1450	74.0	12 000	10	95

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4.1 Monoclinic clinopyroxene-type

Clinopyroxene-type ABC₂O₆ (A = Ca; B = Co, Mg, Zn, Fe; C=Si, Ge) materials are crystallized as a monoclinic structure with a C2/c space group.^96–98 Huber found out that the petedunnite CaZnSi₂O₆ phase stabilizes at P > 0.8 GPa; otherwise, it decomposes into Zn₂SiO₄, Ca₂ZnSi₂O₇ and SiO₂ if the pressure is low and temperature exceeds 650 °C.⁹⁹ Distortions of the polyhedra were quantitatively analysed by Heuer in a CaBSi₂O₆ system (B = Fe, Zn).⁹⁶ Recently, microwave dielectric properties of CaBSi₂O₆ (B = Co, Mg) were reported: ε_r = 6.04, Q × f = 12 457 GHz, and τ_f = −18.91 ppm per °C for B = Co,¹⁰⁰ sintered at 1175 °C; and ε_r = 7.46, Q × f = 59 638 GHz, and τ_f = −46 ppm per °C for B = Mg,¹⁰¹ sintered at 1290 °C. CaMgSi₂O₆ ceramics are suitable candidates for electronic circuit packaging and microwave substrates as a result of their low ε_r value and low dielectric loss. Therefore, a number of studies focus on improving its dielectric properties and reducing its sintering temperature to lower than 950 °C to make it applicable in low-temperature cofired ceramic (LTCC) technology: e.g. if Mn²⁺, Co²⁺, Zn²⁺, Cu²⁺, (Mn_1/2Zn_1/2)²⁺ cations are used to partially substitute for Mg sites,^102–105 improvements in Q × f value could be achieved: an ε_r of 8.01, a Q × f of about 83 469 GHz, and a τ_f of ca. −45.27 ppm per °C for CaMg_0.98Mn_0.02Si₂O₆. In the CaMg_0.96Cu_0.04Si₂O₆ system: ε_r = 7.41, Q × f = 160 100 GHz, and τ_f = −42 ppm per °C. Furthermore, sintering aids, such as Li₂O–MgO–ZnO–B₂O₃–SiO₂, LiF, and Li₂CO₃–V₂O₅, were used to decrease the sintering temperature of CaMgSi₂O₆ ceramics, where it was effectively reduced from 1300 °C to 900 °C using 15 wt% Li₂O–MgO–ZnO–B₂O₃–SiO₂ with great microwave dielectric performances: ε_r = 8.2, Q × f = 32 000 GHz, and τ_f = −48 ppm per °C.¹⁰⁶

The schematic monoclinic crystal structure of CaMgSi₂O₆ can be seen in Fig. 3(a) and (b). A phase transition takes place between C2/c and P2₁/c when the arrangements of tetrahedral [SiO₄] frameworks change, as indicated in Fig. 3(c).¹⁰⁴ Structural parameters of the monoclinic phase reported by Thompson (COD #1000007) are:¹⁰⁷ lattice parameters of a = 9.7397 Å, b = 8.9174 Å, c = 5.2503 Å, α = γ = 90°, β = 105.866°, and V_cell = 438.631 Å.³ In its structure, atoms of Mg and Ca occupy the 4e Wyckoff position while Si and O (three types of O: O1, O2 and O3) atoms are in the 8f site. The structure is built from three types of oxygen polyhedra, where the Ca cation is eight-fold coordinated, forming a [CaO₈] dodecahedron, the Mg cation is coordinated with six oxygen anions and forms an [MgO₆] octahedron, and the Si cation is surrounded by four oxygen anions and tetrahedral [SiO₄] is constructed. Specifically, Ca–O1 × 2, Ca–O2 × 2, Ca–O3(1) × 2 and Ca–O3(2) × 2 bonds appear in the [CaO₈] dodecahedron. While the Mg cation coordinates with O1 and O2 anions, in which four Mg–O1 bonds and two Mg–O2 bonds exist: Mg–O1(1) × 2, Mg–O1(1) × 2 and Mg–O1(1) × 2 bonds. Si–O1, Si–O2 and Si–O3(1) and Si–O3(2) exist in an [SiO₄] tetrahedron. Based on the crystal structure and coordination environments of ceramic CaMgSi₂O₆, the binary bonding formula was considered first:

Fig. 3 CaMgSi₂O₆ crystal structure with different phases and Ca-clusters: (a) the C2/c structure patterns, (b) a cluster of eight-fold coordinated Ca polyhedra, (c) the P2₁/c structure (reprinted with permission from ref. 104. Copyright 2019 Elsevier).

For cations, the effective valence electronic numbers are Z_Ca = 2, Z_Mg = 2, and Z_Si = 4, respectively, while the Z_o value of O²⁻ anion depends on the electric neutrality in any chemical bond, which differs along with the bonding formula. For instance, Z_o equals 3, 9/4 and 3 in Ca–O1, Ca–O2 and Ca–O3 bonds.

Ohsato reported that silicates presenting with low ε_r values, such as forsterite Mg₂SiO₄ (ε_r ∼ 6.8) and willemite Zn₂SiO₄ (ε_r ∼ 6.6), are caused by the large covalency of the [SiO₄] tetrahedral framework.¹⁰⁸ In a CaMgSi₂O₆ ceramic, average chemical bond covalency (Af_c) values quantitatively calculated by Li are sequenced in descending order:¹⁰²Af_c(Si–O) = 35.55% > Af_c(Mg–O) = 29.97% > Af_c(Ca–O) = 16.50%, supporting the Si–O bond having a large bond covalency. Meanwhile, one can compare the contributions of chemical bonds to bond susceptibility χ to judge the dominant source of dielectric polarization. The bond susceptibility χ value calculated via their chemical bond parameters, can be seen in Table 1S in ESI,† where the contribution from Si–O bonds reaches approximately 66%, while Ca–O and Mg–O bonds provide small contributions of about 14% and 20%. This conclusion indicates that Si–O bonds play leading roles in dielectric polarizability. The calculated lattice energy U_total of the CaMgSi₂O₆ system is as large as 39645 kJ mol⁻¹, where Si–O bonds contribute 85% (33 627 kJ mol⁻¹), which is 6 times the total lattice energy of Ca–O and Mg–O bonds. Reported with same crystal structure, a CaCoSi₂O₆ ceramic with a U_total value (19899 kJ mol⁻¹) shows a Q × f value of about 12 457 GHz, which is smaller than that of CaMgSi₂O₆ (59 638 GHz) and also indicates the importance of structural stability to dielectric loss.¹⁰⁰

To improve microwave dielectric performances, bivalent Mn²⁺ cations have been used to partially replace Mg sites in Ca(Mg_1−xMn_x)Si₂O₆, and a solid solution forms in the region of x = 0–0.08, as shown in Fig. 4.¹⁰² The introduction of the Mn element leads to an increase in the average bond covalency f_c value, where the f_c value for Si–O bonds is still as large as 32%, resulting in an increasing trend in ionic polarizability based on the Clausius–Mossotti equation. Thus, the ε_r value increases monotonously. The increase in U_total suggests an enhancement in structural stability. The reported Raman spectra in Fig. 5 also show the most intense peak around 1011 cm⁻¹, which is caused by O–Si–O stretching vibration, which dominates the Raman-active vibration. On the other hand, by using the far-infrared reflectivity spectrum and obtaining the complex dielectric function by the Kramers–Kronig relationship, the intrinsic dielectric properties were acquired via a three-parameter classical harmonic oscillator model. The results indicate that the Mn²⁺ cation increases the ε_r value and decreases the dielectric loss (x = 0.02, Q × f = 83 469 GHz), also reflects the improvement in structural stability of Ca(Mg_0.98Mn_0.02)Si₂O₆.

Fig. 4 The XRD patterns of CaMg_1−xMn_xSi₂O₆ (x = 0–0.08) ceramics sintered at 1300 °C (reprinted with permission from ref. 102. Copyright 2019 Elsevier).

Fig. 5 The Raman spectra of CaMg_1−xMn_xSi₂O₆ (x = 0–0.08) ceramics (reprinted with permission from ref. 102. Copyright 2019 Elsevier).

4.2 Tetragonal akermanite and monoclinic melilite-type

4.2.1 Tetragonal A₂BSi₂O₇ structure. Based on a calculation of monoclinic CaMgSi₂O₆ ceramic, it is determined that Si–O bonds have the largest bond covalencies and play leading roles in dielectric polarization and crystal structural stability. However, it is not appropriate to jump to the conclusion that the f_c value of the Si–O bond is the largest in all silicates according to P–V–L bond theory. Because for a specific crystal, such as a silicate structure, [MO₄] (such as M = Co, Mg, Zn, Mn) and [SiO₄] tetrahedral configurations coexist in tetragonal akermanite Sr₂MSi₂O₇ compared with the existence of [MgO₆] and [SiO₄] frameworks in CaMgSi₂O₆, where the coordinate environments obviously change.¹⁰⁹ Therefore, careful considerations should be taken into account.

A tetragonal akermanite-type A₂BSi₂O₇ structure with a space group of P4̄21m(113) usually exists in A₂BSi₂O₇ (A = Sr, Ca; B = Mg, Zn, Co, Mn) systems. It has attracted public attention for it broad application prospects, such as lighting, display, photoelectric conversion, optoelectronic information, and biomolecular identification. Moreover, its excellent dielectric properties¹⁰⁹ (Sr₂MgSi₂O₇: ε_r = 8.3, Q × f = 55 000 GHz, and τ_f = −47.5 ppm per °C; Sr₂MnSi₂O₇: ε_r = 8.8, Q × f = 32 000 GHz, and τ_f = −58.5 ppm per °C; Ca₂ZnSi₂O₇: ε_r = 11.0, Q × f = 13 500 GHz, and τ_f = −64.3 ppm per °C) also indicate their huge potential in the microwave communication field. Recently, Xiao investigated the structural effects of a tetragonal Sr₂MgSi₂O₇ microwave dielectric ceramic via P–V–L bond theory, providing an insight into the structural properties of tetragonal silicates.¹¹⁰ According to the tetragonal structure of Sr₂MgSi₂O₇ reported by Endo (COD #4317123),¹¹¹ as can be seen from Fig. 6, the Sr²⁺ cation is surrounded by eight O²⁻ anions, and Mg/Zn²⁺ and Si⁴⁺ cations are four-fold coordinated. An [MgO₄] tetrahedron connects with an [SiO₄] tetrahedron sharing a vertex (O3), while the [SiO₄] framework interconnects through shared vertices (O1). The structural detail is listed in Table 3. Therefore, the binary bonding formula is expressed as:

Fig. 6 Diagrammatic sketch of a tetragonal Sr₂MgSi₂O₇ crystal structure.

Table 3 Structural parameters of tetragonal Sr₂MgSi₂O₇ with a space group of P4̄21m, where the lattice parameters are: a = b = 8.011 Å, c = 5.164 Å, α = β = γ = 90° and V_cell = 331.355 ų

Atom	Position	Fractional coordinates			Occ
		x	y	z
Sr	4e	0.3341	0.1659	0.5074	1.0
Mg	2a	0.0000	0.0000	0.0000	1.0
Si	4e	0.1353	0.3647	0.9458	1.0
O1	2c	0.5000	0.0000	0.1580	1.0
O2	4e	0.1384	0.3616	0.2506	1.0
O3	8f	0.0788	0.1947	0.8111	1.0
Bond length (Å)	[SrO8]: Sr–O1 × 1: 2.6050; Sr–O2(1) × 2: 2.7480; Sr–O2(2) × 1: 2.5830; Sr–O3(1) × 2: 2.7920; Sr–O3(2) × 2: 2.5870
	[MgO4]: Mg–O3 × 4: 1.9450
	[SiO4]: Si–O1 × 1: 1.6240; Si–O2 × 1: 1.5740; Si–O3 × 2: 1.5950

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To make comparisons of bond features in tetragonal phases, a representative A₂BSi₂O₇ (A = Sr, Ca; B = Mg, Mn) with both reported structural details^111,112 and microwave dielectric performances (a Ca₂MgSi₂O₇ ceramic is not used as a comparison here due to the fact that secondary phase SiO₂ exists along with the Ca₂MgSi₂O₇ phase¹¹³) is chosen. Detailed comparisons of bond parameters can be found in Table 2S in ESI.†

Table 4 shows the average bond ionicity, lattice energy and proportion of bond susceptibility of T-type A₂BSi₂O₇ (A = Sr, Ca; B = Mg, Mn) structures. The variations in bond covalencies f_c are not similar to those in CaMgSi₂O₆, where the largest average bond covalencies are revealed in B–O bonds (46%–48%), followed by those in Si–O bonds and then A–O bonds. This different situation is possibly caused by the coordination environments of the B cation. However, based on the U_total and χ values, specifically, the U value of Si–O bonds drops from 34 891 to 34347.kJ mol⁻¹ (the U_total for all bonds still decreases), which illustrates that Sr₂MgSi₂O₇ is intrinsically more stable than the Ca₂ZnSi₂O₇ structure. Although the actual Q × f value is also affected by external loss, such as densification, grain growth, and sintering process. The higher lattice stability explains the lower intrinsic dielectric loss. The proportion of Si–O bond susceptibility shows an increasing trend from 47.26% to 51.72%, corresponding to the increase in ε_r value from 8.3 to 11.0, suggesting the importance of Si–O bonds in dielectric polarizability.

Table 4 Average bond ionicity Af_i value, total lattice energy U_total and proportion of bond susceptibility χ/∑χ of T-type A₂BSi₂O₇ (A = Sr, Ca; B = Mg, Mn, Zn) structures

Structure	Bond type	Af i (%)	U total (kJ mol^−1)	χ/∑χ (%)
Sr2MgSi2O7	Sr–O	85.75	5042	22.42
	Mg–O	47.99	4058	30.32
	Si–O	65.44	34 891	47.26
Sr2MnSi2O7	Sr–O	85.81	5025	22.39
	Mn–O	48.69	3910	30.33
	Si–O	65.55	34 859	47.27
Ca2ZnSi2O7	Ca–O	85.57	5166	22.40
	Zn–O	47.00	4271	25.88
	Si–O	65.66	34 347	51.72

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Hanuza reported the Raman and IR spectra of A₂BSi₂O₇ (A = Sr, Ca; B = Mg, Zn) and calculated vibrational modes through lattice dynamic calculation,¹¹⁴ where the most intense and sharp Raman peaks around 901 cm⁻¹, 650 cm⁻¹ and 615 cm⁻¹, are attributed to the symmetric stretching modes of the SiO₃ groups, symmetric stretching modes of the Si^OSi (where O refers to the bridging oxygen), and bending modes of SiO₃ groups. IR spectra of Ca₂ZnSi₂O₇ show that the largest oscillator strength Δε (1.497) originates from the E mode, which is determined by the bending mode of δ(Si^OSi), the bending mode of the SiO₃ group δ(SiO₃) and the symmetric stretching mode of [CaO8] T′(Ca²⁺). This result also show that Si–O bonds are important for intrinsic dielectric properties.

In A₂BSi₂O₇ structures, it is found that although Si–O bonds do not show the largest bond covalency value, they still play an indispensable role in structural stability and dielectric polarization.

4.2.2 Monoclinic A₂BSi₂O₇ and AB₂Si₂O₇ structures. Monoclinic melilite-type structure silicate compounds can be found in A₂BSi₂O₇ (A = Ba; B = Co, Zn, Cu, Mg) and AB₂Si₂O₇ (A = Ba; B = Co, Zn) systems. However, structural diversities appear in Ba₂ZnSi₂O₇ and BaCo₂Si₂O₇ crystal structures. This phenomenon is caused by the different arrangements of the tetrahedral framework, as sketched in Fig. 7,¹¹⁵ [SiO₄] interconnects with [ZnO₄] in the a–c plane in Ba₂ZnSi₂O₇, while [CoO₄] connects with [SiO₄] in the a–b plane. The structural details and bond parameters of Ba₂ZnSi₂O₇ can be obtained using COD #8100589 reported by Kaiser.¹¹⁶ Song investigated the bond features in the BaCo₂Si₂O₇ system through P–V–L bond theory.¹¹⁵ Due to the different coordination environments in Ba₂ZnSi₂O₇ and BaCo₂Si₂O₇ systems, as can be seen in Tables 3S and 4S in ESI,† the binary bonding formulas show large differences:

Fig. 7 Crystal structure of Ba₂ZnSi₂O₇ and BaCo₂Si₂O₇.

After comparing the bond parameters of Ba₂ZnSi₂O₇ from Table 5S in ESI,† and BaCo₂Si₂O₇ from ref. 115 in Table 5, similar conclusions can be drawn as in the tetragonal akermanite structure that although the Si–O bond does not have the largest bond covalency, its contribution to lattice energy and bond susceptibility still determines its importance in dielectric polarization and lattice stability.

Table 5 Average bond ionicity Af_i value, total lattice energy U_total and microwave properties of M-type Ba₂ZnSi₂O₇ and BaCo₂Si₂O₇

Structure	Bond type	Af i (%)	U total (kJ mol^−1)	Properties
Ba2ZnSi2O7	Ba–O	85.35	4851	ε r = 8.8
	Zn–O	46.90	4033	Q × f = 34 000 GHz
	Si–O	64.55	34 326
BaCo2Si2O7	Ba–O	83.65	2331	ε r = 9.3
	Co–O	44.94	7644	Q × f = 31 000 GHz
	Si–O	59.75	33 088

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4.3 Rock salt structure

Driven by the requirements for high reliability and high-frequency applications, the pursuit of novel material systems with low ε_r value and ultra-high Q × f (Q× f ≥100 000 GHz) value has been underway. Li₂O–AO–BO₂ based systems can be regarded as the substitution of the A site in AO (A = Mg, Zn, Ni) with Li₂BO₃ (B = Ti, Sn, Zr). For example, Li₂TiO₃ and MgO are subordinated to the same rock-salt-type structure, and the ionic radii of cations are close to each other (IR_Li,Ti = 0.708 Å, IR_Mg = 0.72 Å),¹¹⁷ which is beneficial for forming a solid solution. Therefore, material systems with the chemical formula of Li_2xA_1−xB_xO_1+2x (A = Mg, Zn, Ni; B = Ti, Sn, Zr)^118,119 show large diversities as x varies. For instance, x = 1/5: Li₂Mg₄TiO₇,¹²⁰x = 1/4: Li₂Mg₃TiO₆,¹²¹x = 2/5: Li₄Mg₃Ti₂O₉¹²² and x = 1/2: Li₂MgTiO₄.¹²³ In 2016, novel rock-salt ceramics with the chemical formula of Li₂O–3MgO–TiO₂ (B = Ti, Sn, Zr) ceramics possess excellent microwave dielectric performances: ε_r = 8.8–15.2, Q × f = 86 000–152 000 GHz (f = 8.3–10.7 GHz).¹²⁴ Besides, the raw materials are cheap and easy to obtain, attracting a lot of public attention. However, it is noticed that the sintering characteristics are not satisfactory. For instance, the secondary-phase Mg₂SnO₄ exists along with the formation of the Li₂Mg₃SnO₆ phase,¹²⁵ ZrO₂ usually accompanies the existence of the Li₂Mg₃ZrO₆ phase.¹²⁴ Most importantly, a large number of micropores formed by the volatilization of lithium at high temperature lead to cracks or fractures in the microstructure, which is detrimental to the development of properties. Therefore, it is crucial to solve these fundamental and nonnegligible issues.

The volatilization of lithium can be compensated by burying the Li₂Mg₃BO₆ (B = Ti, Sn, Zr) green pellets in sacrificial powders of the same composition, which improves the sintering densification and acquiring the pure phase.¹²⁶ A non-stoichiometric ratio of the lithium of Li_2(1+x)Mg₃ZrO₆ partially inhibits ZrO₂ and impurities, achieving a Q × f value as high as 307319 GHz.¹²⁷ By using a novel atmosphere-controlled sintering process, where ZrO₂ powder provides a protective atmosphere, and Li₂CO₃ powder provides an Li-rich sintering atmosphere, Li_2(1+x)Mg₃SnO₆ ceramics with very dense microstructures were achieved.^125,128 Therefore, it is reasonable to discuss the structural–property relationship when the external loss caused by the fabrication process and sintering characteristics is controlled.

4.3.1 Variation of ionic occupancy in cubic Li_2xMg_1−xTi_xO_1+2x (x = 1/4, 2/5 and 1/2) systems. Li_2xMg_1−xTi_xO_1+2x (x = 1/4, 2/5 and 1/2) systems belong to a rock-salt cubic structure with an Fm3m(225) space group. A schematic diagram of the crystal structure is presented in Fig. 8, where the Li/Mg/Ti atoms are randomly distributed in the 4a Wyckoff position, and one type of O atom is in the 4b site. All the cations are six coordinated, forming an [MO₆] octahedron (M = Li/Mg/Ti).

Fig. 8 Crystal structure of a cubic Li_2xMg_1−xTi_xO_1+2x (x = 1/4, 2/5 and 1/2) system.

It is interesting to note that the above-mentioned systems share the same crystal structure, spatial configuration, coordinate environments and distribution of valence states. While the only difference lies in ionic occupancy, which directly leads to variations in N(B^j − Aⁱ) and A_mB_n expression. Therefore, it is necessary to discover the impact of ionic occupancy on bond parameters and dielectric properties. The synthesis and structural properties of Li_2xMg_1−xTi_xO_1+2x (x = 1/4, 2/5 and 1/2) ceramics were discussed by several researchers using P–V–L bond theory,^122,123,126 where the binary bonding formulas are shown as:

(x = 1/4)Li₂Mg₃TiO₆ = Li₂O1₂ + Mg₃O1₃ + TiO₁

(x = 2/5)Li₄Mg₃Ti₂O₉ = Li₄O1₄ + Mg₃O1₃ + Ti₂O1₂

(x = 1/2)Li₂MgTiO₄ = Li₂O1₂ + MgO

where all the O²⁻ anions in Li–O, Mg–O and Ti–O bonds have the same effective valence electronic numbers (Z_o) of 3, 6, 12, while for the cations they are Z_Li = 1, Z_Mg = 2, and Z_Ti = 4, respectively. The relative bond features of Li_2xMg_1−xTi_xO_1+2x (x = 1/4, 2/5 and 1/2) systems are listed in Table 6.

Table 6 Bond parameters, bond ionicity and bond susceptibilities of Li_2xMg_1−xTi_xO_1+2x (x = 1/4, 2/5 and 1/2) systems

System	Bond type	E h (eV)	E g (eV)	d (Å)	f i (%)	U (kJ mol^−1)	χ ^μ	F ^μ	χ	χ/∑χ (%)
x = 1/4	Li–O	6.379	11.561	2.09065	69.537	2102	1.8019	0.3333	0.60	31.89
	Mg–O	6.379	16.029	2.09065	84.148	11 711	1.9310	0.5000	0.97	51.26
	Ti–O	6.379	22.611	2.09065	92.033	15 402	1.9054	0.1667	0.32	16.86
x = 2/5	Li–O	6.451	11.676	2.08160	69.500	4204	1.7871	0.4444	0.79	42.90
	Mg–O	6.451	16.191	2.08160	84.143	11 710	1.9138	0.3333	0.64	34.46
	Ti–O	6.451	22.852	2.08160	92.041	30 801	1.8861	0.2222	0.42	22.64
x = 1/2	Li–O	6.470	11.709	2.07910	69.467	2111	1.7865	0.5000	0.89	48.47
	Mg–O	6.470	16.239	2.07910	84.126	3920	1.9131	0.2500	0.48	25.95
	Ti–O	6.470	22.923	2.07910	92.034	15 467	1.8850	0.2500	0.47	25.57

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From Table 6, it is straightforward to find that variation in ionic occupancy leads to a difference in bond proportion F^μ, where the F^μ values of Li–O and Ti–O bonds increase along with the x value, while for the Mg–O bond, it decreases. Average bond ionicity (f_i) value shows a monotonous downward trend from 81.906% to 81.876%, where the f_i value is descending as f_i(Ti–O) > f_i(Mg–O) > f_i(Li–O). The lattice energy of the Ti–O bond U_Ti–O accounts for the majority contribution in the three types of systems. The susceptibility χ^μ of the Mg–O bond is the largest in the three types of systems; however, as indicated, the proportion of Mg–O bonds decreases, leading to different situations for the contribution to bond susceptibility, where the contributions of Li–O and Ti–O increase with the x value. The above-mentioned systems have the same structure and configuration; however, the primary relationship between the contribution of chemical bonds to bond susceptibility obviously changes along with the ionic occupancy, and then different bond features are displayed. This conclusion shows that it is necessary to consider variation in ionic occupancy when calculating bond properties of the same structure. Also, one can expect from Table 6 that the total lattice energy U in an x = 2/5 system has the largest value of 46715 kJ mol⁻¹ and the smallest value of 21499 kJ mol⁻¹ in an x = 1/2 system, which suggests superior structural stability in the Li₄Mg₃Ti₂O₉ system. And total bond susceptibility ∑χ shows a slight decline from 1.88 to 1.84 as x increases, indicating a gradually weakening dielectric polarization due to the χ value being proportional to bond susceptibility. Combined with the reported microwave dielectric performances: Li₂Mg₃TiO₆ ceramic well-sintered at 1600 °C with an ε_rp (porosity corrected) of about 16.0, a tan δ of about 7.3 × 10⁻⁵, and τ_f of ca. −13 ppm per °C;¹²⁹ for Li₄Mg₃Ti₂O₉ ceramic sintered at 1450 °C: ε_rp = 15.9, tan δ = 6.0 × 10⁻⁵, and τ_f = −7 ppm per °C;¹²² for Li₂MgTiO₄ ceramic sintered at 1350 °C: ε_rp = 15.1, tan δ = 8.4 × 10⁻⁵, and τ_f = −4 ppm per °C,¹²³ it is shown that experimental results provide evidence for the calculations. It is notable that the Q × f value is not only affected by intrinsic structural effects, but is very sensitive to extrinsic parts, such as sintering temperature, which promotes the growth of grain and alters the amount of grain boundaries. Thus, another nonnegligible contribution to actual dielectric loss comes from extrinsic loss.

4.3.2 Ionic modification of the Mg/Ti site in cubic Li₂Mg₃TiO₆ systems. As important as the A_mB_n binary bonding formula expression, the discrepancy in elementary substance also affects the bond parameters because of their differences in ionic radii and electronegativity. To improve the microwave dielectric properties of Li₂Mg₃TiO₆ ceramics, ionic substitutions of the Mg/Ti sites were extensively studied: e.g. divalent cations M²⁺ (M = Zn, Co, Cu) for the Mg site and tetravalent cations M⁴⁺ (M = Zr, Sn, Mg/Sb, M/Ta) for the Ti site. The relationship between bond parameters and microwave dielectric properties were correlated using P–V–L bond theory.

By using Cu²⁺ cations to partially replace Mg sites, it is shown that a cubic solid solution is formed within x = 0.00–0.20 in Li₂Mg_3−xCu_xTiO₆, as shown in Fig. 9.¹³⁰ Lattice parameters a, b, c, and V_cell monotonously increase due to the larger ionic radii of Cu²⁺ than Mg²⁺ at CN = 6 (0.73 Å > 0.72 Å). With the increase in Cu²⁺ content, the ionicity f_i values of all types of bonds present increasing trends with the ε_r value, indicating an enhancement in dielectric polarization. This conclusion is confirmed by the increases in observed and theoretical dielectric polarizability using the addition rule of dielectric polarizability, as shown in Fig. 10. The lattice energy U value decreases along with the x value, reflecting that the stability of crystal structure is gradually weakening and then the intrinsic loss would be reduced. Variations in the Q × f value shown in Fig. 11 provide evidence for the drop in dielectric loss. On the other hand, Kim defined the evaluating indicator of packing fraction (PF) in a scheelite compound as the proportion of volume of packed ions to the volume of a primitive unit cell, which is directly proportional to the Q × f value.¹³¹ In a cubic rock salt Li₂Mg₃TiO₆ structure, a similar relationship between PF and the Q × f value can be revealed as well.

Fig. 9 The XRD patterns of Li₂Mg_3−xCu_xTiO₆ (x = 0.00–0.20) ceramics obtained at optimum temperatures: (a) x = 0, 1310 °C, (b) x = 0.04, 1280 °C, (c) x = 0.08, 1250 °C, (d) x = 0.12, 1250 °C, (e) x = 0.16, 1220 °C, (f) x = 0.20, 1220 °C (reprinted with permission from ref. 130. Copyright 2019 Elsevier).

Fig. 10 The theoretical polarizability and ε_r of Li₂Mg_3−xCu_xTiO₆ (x = 0.00–0.20) ceramics obtained at optimal densification temperatures (reprinted with permission from ref. 130. Copyright 2019 Elsevier).

Fig. 11 The packing fraction, total lattice energy and Q × f of Li₂Mg_3−xCu_xTiO₆ (x = 0.00–0.20) ceramics obtained at optimal densification temperatures (reprinted with permission from ref. 130. Copyright 2019 Elsevier).

In an Li₂(Mg_1−xCo_x)₃TiO₆ (0 ≤ x ≤ 0.4) system doped with Co²⁺ cations, the chemical bond length increases from 2.091 to 2.095 Å, which leads to an increasing trend for all types of bond ionicity f_i values.¹²⁹ The ε_r value increases with x value. Here, an increase in bond ionicity indicates increased dielectric polarization, which can be confirmed by dielectric polarizability using the Clausius–Mossotti equation. Similarly, the introduction of Co²⁺ cations decreases the crystal structural stability; as a result, the intrinsic loss would be increased. The decline in PF value also serves as a verification of this conclusion. Furthermore, ZnO is usually used as a substitution source for Mg sites. Yang reported the Zn²⁺ ionic doping of an Li₂(Mg_1−xZn_x)₃TiO₆ (x = 0–0.2) system. In the region of x = 0–0.15, a cubic solid solution is formed, the secondary Zn₂TiO₄ phase appears when the solid solution exceeds the limit of x = 0.15. Similarly, the crystal structural parameters increase for the larger ionic radius of the Zn²⁺ cation than Mg²⁺ (0.74 Å > 0.72 Å) cation.¹³² Based on P–V–L bond theory calculation, it is also found that ionicity for all types of bonds increases with x value, corresponding to an increase in dielectric polarization and ε_r value. The decline in lattice energy also shows a similar variation trend to PF and Q × f value. The Mg site was also reported with Ca²⁺ ionic replacement; however, due to the large discrepancy between ionic radii (39% > 15%), it is unlikely to form a solid solution even over a small concentration range (x = 0.03) in Li₂Mg_3−xCa_xTiO₆.¹³³

From the three cases, one can find a common interesting phenomenon that the dielectric polarization increases, and the structural stability drops when the Mg site is replaced by a larger radius of cations. According to the definition of bond ionicity:

The homopolar potential energy E_h can only be smaller as bond length increases, while heteropolar energy E_g decreases. Thus, the variations in f_i value should depend on the changing ratio of homopolar energy to heteropolar energy. In a cubic structure, since cations randomly distribute at the same site, and one type of M–O1 bond exists in an [MO₆] (M = Li/Mg/Ti) octahedron, decreasing ratios of E_h in all type of bonds are larger than the decreasing ratios of E_g. Thus, this phenomenon causes an increase in the bond ionicity f_i value.

For substitution of Ti sites, single cations Zr⁴⁺, Sn⁴⁺, Ge⁴⁺ and complex cations (Mg_1/3Sb_2/3)⁴⁺, (Mg_1/3Nb_2/3)⁴⁺ and (Mg_1/3Ta_2/3)⁴⁺ were chosen as sources for substitution. When a Ti site was replaced by an Sn⁴⁺ cation in an Li₂Mg₃Sn_1−xTi_xO₆ system, a solid solution could be formed over the entire range of x = 0–1, as shown in Fig. 12.¹³⁴ Due to the larger radius of the Sn⁴⁺ (0.69 Å) cation than Ti⁴⁺ (0.605 Å), the introduction of SnO₂ leads to an increase in the crystal structural parameters a, b, c, and V_cell (from x = 1 to 0) and bond length. Based on reported bond features, it is found that the bond covalency f_c value decreases with the increase in SnO₂ content, which is in accordance with the variation in the ε_r value. This different result indicates that dielectric polarizability decreases as the content of SnO₂ increases, which is confirmed by the drop in ionic polarizability. Such a situation exists in other systems as well, such as the Ca(Mg_1−xMn_x)Si₂O₆ system.¹⁰² Also, increasing content of SnO₂ leads to a declining trend in structural stability and then the intrinsic loss would be increased, as shown in Fig. 13. To determine whether this phenomenon is an exception or exists in most Ti site doping cases for Li₂Mg₃TiO₆ ceramics, ZrO₂ was chosen in Li₂Mg₃Zr_1−xTi_xO₆ (0 ≤ x ≤ 1).¹³⁵ Similarly, a solid solution was formed over the whole range of x = 0–1. An increase in Zr⁴⁺ cations (ionic radius: 0.72 Å) leads to an expansion of cell volume and an increase in bond length. The covalency f_c value of all bonds decreases, accompanied by a decrease in ε_r value. Structural stability also shows a similar tendency to PF and Q × f values.

Fig. 12 The XRD patterns of Li₂Mg₃Sn_1−xTi_xO₆ (x = 0.0–1.0) ceramics obtained at optimal temperatures: (a) x = 0.0, 925 °C, (b) x = 0.2, 925 °C, (c) x = 0.4, 925 °C, (d) x = 0.6, 900 °C, (e) x = 0.8, 900 °C, (f) x = 1.0, 900 °C (reprinted with permission from ref. 134. Copyright 2020 Elsevier).

Fig. 13 The total lattice energy U_total, packing fraction and Q × f of the Li₂Mg₃Sn_1−xTi_xO₆ (x = 0.0–1.0) samples obtained at optimal temperatures (reprinted with permission from ref. 134. Copyright 2020 Elsevier).

By substituting Ti sites with (Mg_1/3Ta_2/3)⁴⁺ complex cations (ionic radius: 0.72 Å) in Li₂Mg₃Ti_1−x(Mg_1/3Ta_2/3)_xO₆ (0 ≤ x ≤ 0.2) system,¹³⁶ the cell volume and bond length increase, and based on P–V–L bond theory, the decreasing trend in bond covalency f_c value is also consistent with the ε_r value. From another point of view, Pan confirmed this decrease in ε_r value using far-infrared reflectivity. By transforming infrared spectra into a complex dielectric function with the Kramers–Kronig relationship and fitting it with a classical Lorentz three-parameter model, the real part and the imaginary part of the dielectric constant are shown in Fig. 14, and the intrinsic ε_r value is also confirmed as decreasing from 13.68 to 12.57 along with the increase in x value.

Fig. 14 Real and imaginary parts of complex permittivity for Li₂Mg₃Ti_1−x(Mg_1/3Ta_2/3)_xO₆ (0 ≤ x ≤ 0.2) ceramics sintered at the optimum temperatures (reprinted with permission from ref. 136. Copyright 2018 Elsevier).

To make comparisons between the B site in cubic Li₂Mg₃BO₆ (B = Ti, Sn, Zr) systems, the bond parameters are listed in Table 7. From Table 7,^126,128,138 a similar conclusion could be jumped to that an increase in bond length from Ti to Zr leads to a similar decrease in the bond covalency f_c value, which corresponds to a decline in the ε_r value. Also, the higher structural stability in the B = Ti system shows its smaller intrinsic loss, and the Q × f value presents the largest value among Li₂Mg₃BO₆ (B = Ti, Sn, Zr) systems. Also, in (Mg_1/3Sb_2/3)⁴⁺ and (Mg_1/3Nb_2/3)⁴⁺ complex ionic substitution, the cell volume increases, and the ε_r value presents a decreasing trend. At this point, one may guess that it is more convenient to predict the variation in dielectric polarization through the addition rule of ionic polarizability (α_theo). However, this guess does not fit with the substitution in the Ti site with Zr⁴⁺, (Mg_1/3Sb_2/3)⁴⁺, (Mg_1/3Nb_2/3)⁴⁺ or (Mg_1/3Ta_2/3)⁴⁺ cations^136–138 for their large α_theo values of about 3.25, 3.29, 3.09 and 3.59 compared with the 2.93 of the Ti cation. Here, we summarize modified Li₂Mg₃BO₆ (B = Ti, Sn, Zr) ceramics with bond ionicities and bond covalency values and lattice energies in Table 8.

Table 7 Comparisons between bond ionicity, lattice energy and microwave dielectric properties of B site difference in cubic Li₂Mg₃BO₆ (B = Ti, Sn, Zr) systems

B	Bond type	Bond length	f i (%)	U (kJ mol^−1)	ε r	Q × f (GHz)
Ti	Li–O	2.0900	69.54	2102	14.8	175 000
	Mg–O	2.0900	84.15	11711
	Ti–O	2.0900	92.03	15402
	Af i = 81.91%, Utotal = 29215 kJ mol^−1
Sn	Li–O	2.1216	69.78	2078	12.4	168 330
	Mg–O	2.1216	84.25	11510
	Sn–O	2.1216	92.06	15231
	Af i = 82.03%, Utotal = 28819 kJ mol^−1
Zr	Li–O	2.1365	69.89	2066	12.2	113 000
	Mg–O	2.1365	84.29	11519
	Zr–O	2.1365	92.07	15150
	Af i = 82.08%, Utotal = 28735 kJ mol^−1

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Table 8 Modified Li₂Mg₃BO₆ (B = Ti, Sn, Zr) ceramics with bond ionicity, bond covalency value, lattice energy and microwave dielectric properties

System	V cell	f i/fc	ε r	U	Q × f
Li2Mg3−xCuxTiO6	↗	f i ↗	↗	↘	↘
Li2(Mg1−xCox)3TiO6	↗	f i ↗	↗	↘	↘
Li2(Mg1−xZnx)3TiO6	↗	f i ↗	↗	↘	↘
Li2Mg3Sn1−xTixO6	↘	f c ↗	↗	↗	↗
Li2Mg3Zr1−xTixO6	↘	f c ↗	↗	↗	↗
Li2Mg3Ti1−x(Mg1/3Ta2/3)xO6	↗	f c ↘	↘	↘	↘

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Therefore, we may find that the expansion or shrinkage in cell volume relies on the radius of the donor ion, while Mg site substitution with a large size of ion leads to an increase in bond ionicity f_i value and ε_r value. While in B site substitution, the variation in ε_r value is proportional to the bond covalency f_c value. Based on the bond features and microwave dielectric properties, we may put forward a conjecture that one could improve the Q × f value with a smaller size ionic substitution. Xiang reported Ti sites partially replaced with smaller sized Ge⁴⁺ cations (0.535 Å), and the result also showed an improvement from 1.1 × 10⁵ at x = 0.04 to 1.35 × 10⁵ at x = 0.1.¹³⁹ However, whether this conjecture is ubiquitous in cubic Li₂Mg₃BO₆ systems still requires considerable future experiments.

4.3.3 Bond features in orthorhombic Li₃Mg₂NbO₆ systems. Li₃Mg₂NbO₆ is another typical rock salt structure studied worldwide because of its excellent microwave dielectric properties: ε_r = 14.9, Q × f = 100 965 GHz, τ_f = 21.96 ppm per °C when sintered at 1225 °C for 4 h.¹⁴⁰ It is crystallized as an orthorhombic structure with an Fddd space group. The XRD patterns of Li₃Mg₂NbO₆ ceramics sintered at 1175–1300 °C are shown in Fig. 15.¹⁴⁰ A diagrammatic sketch of a crystal structure is presented in Fig. 16, where in its structure, three types of Li and Mg atoms and two types of O atoms are shown, where Li1Mg1 and Li2Mg2 atoms occupy the 16 g Wyckoff position, while Li3Mg3 atoms are in the 8b site and Nb is in the 8a site. 16f and 32 h sites are distributed with O(1) and O(2) atoms. All the cations are six-fold coordinated, forming an oxygen octahedron. Detailed structural parameters can be traced in ref. 140. Recently, the relationships between bond characteristics and microwave dielectric properties were studied by several researchers based on the actual crystal structure, coordinate environments, and ionic occupancy. First and foremost, the binary bonding formula expression was considered by Wu:¹⁴⁰

Fig. 15 X-ray diffraction patterns of Li₃Mg₂NbO₆ ceramics sintered at different sintering temperature from 1175 °C to 1300 °C for 4 h (reprinted with permission from ref. 140. Copyright 2016 Elsevier).

Fig. 16 Diagrammatic sketch of Li₃Mg₂NbO₆ structure.

According to P–V–L bond theory calculation, it is found that Nb–O bonds have the largest average ionicity f_i value of about 98.72%, followed by that of Li–O bonds (95.72%) and Mg–O bonds (93.62%). As the following relationship indicates,¹⁴¹ the ε^μ of a μ bond is proportional to the f_i value; it is then supposed that Nb–O bonds make the predominant contribution to dielectric polarization. Meanwhile, due to the larger total lattice energy of Nb–O bonds (25382 kJ mol⁻¹) than Mg–O (9378 kJ mol⁻¹) or Li–O (5308 kJ mol⁻¹) bonds, the importance of Nb–O bonds on structural stability is noneligible, affecting the dielectric loss:

where n is the refractive index. Up to now, several attempts have been made to improve the microwave dielectric properties of Li₃Mg₂NbO₆ ceramics at Li, Mg and Nb sites. For instance, Li⁺/Ti⁴⁺ ions were chosen as a substitution source in an Li_3+xMg₂Nb_1−xTi_xO₆ (x = 0.02–0.08) system, where the cell volume drops along with the x value.¹⁴² In the region of x = 0.02–0.06, the ε_r value and Q × f value show an increasing tendency. The Mg site can usually be replaced with Mn²⁺, Ca²⁺, Zn²⁺, Ni²⁺, and Co²⁺ ions.^143–147 Tetravalent Ti⁴⁺ and pentavalent V⁵⁺, Sb⁵⁺, Ta⁵⁺, and hexavalent W⁶⁺ ions are reported to substitute for Nb sites.^148–154 For instance, a Ta⁵⁺ ion replaces an Nb site in Li₃Mg₂Nb_1−xTa_xO₆ (x = 0–0.04) system leading to a decrease in cell volume, and the ε_r value increases along with the x value owing to the enhancement in dielectric polarizability. The Q × f value consistent with PF and relative density show an increasing trend with the x value. The Nb site also can be replaced with an Ti⁴⁺ ion because of the smaller discrepancy in ionic radius, resulting in the formation of a solid solution. Similarly, the ε_r value and Q × f value both increase from x = 0 to 0.04, improving the Q × f value from 1 × 10⁵ to 1.31 × 10⁵ GHz. An Sb⁵⁺ cation was used to substitute for the Nb site in an Li₃Mg₂(Nb_1−xSb_x)O₆ system (x = 0.02–0.08). An orthorhombic solid solution was formed and the cell volume declined as x increased. It is also found that Sb₂O₅ leads to an increase in total bond ionicity f_i value, corresponding to an increase in ε_r value. By controlling the extrinsic loss, such as densified temperature, the extrinsic contribution to dielectric properties is minimized. The total lattice energy in accordance with the variation in Q × f value increases in the range x = 0.02–0.06, illustrating that the dielectric loss is highly correlated with structural stability in an Li₃Mg₂NbO₆ system. Furthermore, for reported Li bond parameters in an Zn²⁺ doped Li₃Mg₂SbO₆ system, a proportional relationship between bond ionicity f_i value and ε_r value can still be found, whereas the gradually decreasing total lattice energy reflects the decline in Q × f values.¹⁵⁵

Just as section 4.1 shows, however, even Ca–O bonds obtain a larger bond ionicity (86.267%) value than Si–O bonds (67.983%) in a CaMgSi₂O₆ system, and the results show that the majority contribution to dielectric polarization originates from Si–O bonds because of their largest proportion of bond susceptibility value. Besides, it is also shown from the lattice dynamic calculation that the largest oscillator strength Δε (1.497) in the IR spectra of Ca₂ZnSi₂O₇ shows that Si–O bonds are important for the intrinsic dielectric properties. A possible reason is that the refractive index differs with an A_mB_n crystal; therefore, it would be more convincing to compare bond susceptibility in an Li₃Mg₂NbO₆ system to predict the variation in dielectric polarization.

4.4 Tetragonal scheelite CaWO₄ structure

As a tetragonal scheelite structure, alcium tungstate (CaWO₄) is broadly applied in laser, phosphor, and scintillation counters due to the merits of outstanding luminescence characters. The microwave dielectric properties of scheelite-type A²⁺B⁶⁺O₄ (A²⁺: Ca, Pb, Ba; B⁶⁺: Mo, W) ceramics were first reported, where CaWO₄ ceramics sintered at 1100 °C show excellent dielectric properties: ε_r = 10.4, Q × f = 76 550 GHz, τ_f = −24.4 ppm per °C.¹³¹

Based on the structural detail reported by Rodríguez–Hernández (COD #1546034),¹⁵⁶ lattice parameters of a = b = 5.205 Å, c = 11.275 Å, and V_cell = 305.46 ų with a space group of I4₁/a are shown. In its crystal structure (Z = 4), as shown in Fig. 17, the W⁵⁺ cation is four-fold coordinated, while the Ca²⁺ cation is surrounded by eight oxygen anions. The O²⁻ anion, however, is nearly close to two Ca²⁺ ions and one W⁵⁺ ion, showing that its coordination number CN = 3. Thus, according to the actual structural configuration, its binary bonding formula is expressed as:

CaWO₄ = CaO_8/3 + WO_4/3

Fig. 17 Crystal structure of CaWO₄: (a) the coordination number of ions in CaWO₄, (b) the unit cell of CaWO₄ (reprinted with permission from ref. 159. Copyright 2018 Elsevier).

The effective valence electronic numbers of O²⁻ anions in a Ca–O bond and a W–O bond are 2.25 and 13.5, respectively. The ionicity f_i value for Ca–O(2) bond is largest (75.87%), followed by that for the Ca–O(1) bond of about 75.80%, and the smallest f_i value is observed in the W–O bond (68.86%).

Shao investigated the electronic structure of the CaWO₄ crystal by choosing the molecular-cluster model with the framework of the fully relativistic self-consistent Dirac–Slater theory equipped with the DV-Xα method, and reported the total/partial densities of states of a perfect CaWO₄ crystal in Fig. 18.¹⁵⁷ It is proved that O 2p and W 5d states both provide contributions to the top of the valence bands and the bottom of the conduction bands, which illustrates that the W–O bond has a much stronger covalent property than other bonds in the CaWO₄ crystal. This finding provides evidence for P–V–L bond theory calculation. Moreover, the lattice energy U_W–O (29564 kJ mol⁻¹) is about ten times larger than that in Ca–O(1) (2722 kJ mol⁻¹) or Ca–O(2) (2720 kJ mol⁻¹) bonds, emphasizing the significance of W–O bonds in structural stability and intrinsic loss. It is widely acknowledged that the Raman peak with the highest intensity dominates the Raman-active vibrations, and most researchers found that the half width (FWHM) value of the most intense Raman peak is closely related with dielectric loss because the half width of the Raman scattering peak is related to the damping coefficient of structural vibration. The larger the damping coefficient of Raman scattering, the wider the half width of the peak. The damping coefficient of structure vibration is a parameter directly related to the intrinsic loss of microwave dielectric ceramics. The larger the damping coefficient of structural vibration is, the larger the intrinsic loss of microwave dielectric ceramic material would be; as a result, a lower Q × f value is presented. Gracia reported the Raman spectrum of the CaWO₄ crystal, as shown in Fig. 19.¹⁵⁸ It is shown that the most intense peak around 912 cm⁻¹ is attributed to the A_g mode of the stretching vibrational behaviour of the WO₄ unit. This conclusion also supports that W–O bonds provide the major contributions to the intrinsic loss. Moreover, due to the close ionic radius between Ca²⁺ (IR = 1.12 Å at CN = 8) and Sr²⁺ (IR = 1.26 Å at CN = 8) ions, Xiao prepared a series of tetragonal Ca_1−xSr_xWO₄ (x = 0–0.1) solid solutions, where the total bond covalency f_c shows a decreasing trend, corresponding to a decline in dielectric polarization, while the introduction of an Sr²⁺ cation is beneficial for improving the structural stability, which demonstrates an improvement in Q × f value.¹⁵⁹

Fig. 18 Total densities of states and partial densities of states for a perfect CaWO₄ crystal (reprinted with permission from ref. 157. Copyright 2007 John Wiley and Sons).

Fig. 19 FT-Raman spectra in the range from 50 to 1000 cm⁻¹ of CaWO₄ crystals processed at 140 °C for different times in the MH system (reprinted with permission from ref. 158. Copyright 2011 AIP Publishing).

4.5 Glass-free systems for LTCC application

As a key technology of RF passive integration, several merits of low temperature cofired ceramic (LTCC) have attracted a lot of public attention: e.g. adjustability of dielectric properties of LTCC material based on different ingredients, allowing circuit design flexibility; adaptability to the requirements of high current and high temperature resistance, high reliability for application to harsh environments; circuit boards with multiple layers and passive components can be integrated, avoiding the costs of packaging, realizing the integration of passive and active components, and improving the assembly density of circuits; and excellent compatibility with other multi-layer cabling technologies.¹⁶⁰

However, the excellent microwave dielectric properties of material candidates sintered at a temperature lower than 950 °C and chemical compatibility with Ag and Cu electrodes are required. Until now, two mainstream methods to prepare LTCC materials have been broadly acknowledged, where one type is to add low melting point glass frits into a ceramic matrix. The sample can be well-sintered at low temperatures with the help of a liquid phase formed by glass, such as Li₂O–B₂O₃–SiO₂, BaO–ZnO–B₂O₃, ZnO–B₂O₃–SiO₂, and CuO–B₂O₃.^161–163 However, dielectric performances have deteriorated due to the large dielectric loss of glass. Another type of ceramic has the feature of being “glass-free”, where the ceramic contains low-melting oxides on their own: for instance, BiVO₄, Bi₂O₃–MoO₃, Li₂MoO₄, NaAgMoO₄ and (Ca,Bi)(Mo,V)O₄.^164–168 They can be sintered at low temperature without additives.

4.5.1 Orthorhombic olivine structure. LiMPO₄ (M = Mg, Mn, Co, Ni, Zn) compounds make themselves promising in LTCC because of the existence of low melting phosphates. Not only the magnetoelectric and electrochemical aspects, but the dielectric properties in the microwave region were explored: e.g. a low ε_r value of 6.6, a Q × f of about 79 100 GHz, and τ_f of ca. −55 ppm per °C for M = Mg when sintered at 950 °C.¹⁶⁹ For M = Ni sintered at 825 °C: ε_r = 7.18, Q × f = 27 754 GHz, τ_f = −67.7 ppm per °C.¹⁷⁰ The chemical compatibility of LiMnPO₄ with Ag presents excellent properties: ε_r = 8.1, Q × f = 44 224 GHz, τ_f = −90 ppm per °C.¹⁷¹ Due to the excellent microwave properties of LiMPO₄ ceramics at low temperatures, several efforts have been made to analyse the structure–property relationship.

Based on the crystal structure reported by Abrahams (COD #2001689), as presented in Fig. 20,¹⁷² LiMPO₄ crystallizes as an orthorhombic structure with Pnma space group, where the lattice parameters are: a = 10.0317 Å, b = 5.8539 Å, c = 4.6748 Å, and V_cell = 274.643 ų. In this structure, the Li atom is in a 4a Wyckoff position, while Ni and P are in 4c sites. Besides, three types of O atoms are shown where O(1) and O(2) are in 4c sites, and O(3) is in a 8d site. Li⁺ and Ni²⁺ cations are surrounded by six O²⁻ anions, forming [LiO₆] and [NiO₆] octahedra, while the P⁵⁺ cation is four-fold coordinated and presents a [PO₄] tetrahedron. The [LiO₆] octahedron interconnects with itself along the b-axis direction through sharing an edge (O(1)–O(2)). The [PO₄] tetrahedron links with the [LiO₆] octahedron by sharing a corner (O(1)) and an edge (O(3)–O(3)). The Li⁺ cation appears in the top corner, face center of the ac plane and body center, thus, multi-layer [LiO₆] units appear, where [NiO₆] octahedra interpenetrate between layers and connect with the [LiO₆] octahedron by sharing edge (O(1)–O(3)) and corner (O(3)). By decomposing LiMPO₄ into binary compounds, the binary bonding formula is obtained as:¹⁷⁰

Fig. 20 Crystal structure of the orthorhombic LiNiPO₄ system.

The coordination environment and charge distribution can be briefly explained in Fig. 21.¹⁷⁰ Citing O(3) as an instance, it connects with Li⁺, Ni²⁺ and P⁵⁺ ions, where the effective valence electron numbers (Z) for the cations are 1, 2 and 5, respectively. Combined with the coordination number, an electron number of 1/6 from the Li⁺ cation is given to the Li–O(3) bond. The Z values for O in Li–O, Ni–O and P–O bonds are 2, 4 and 15, respectively. Based on these considerations, the bond parameters can be calculated via P–V–L bond theory.

Fig. 21 The coordination number and charge distribution of ions in LiNiPO₄ ceramics (reprinted with permission from ref. 170. Copyright 2018 Elsevier).

The average ionicity Af_i values are sequenced as: Af_i(Ni–O) = 76.4% > Af_i(P–O) = 74.2% > Af_i(Li–O) = 57.1%, emphasizing the indispensable role of Ni–O and P–O bonds in the ε_r value. While the lattice energy of P–O bonds provides about an 86.5% contribution (28055 kJ mol⁻¹), followed by that of Ni–O (3379 kJ mol⁻¹) and Li–O (986 kJ mol⁻¹) bonds, suggesting that P–O bonds are also important in determining the structural stability. Furthermore, as just mentioned, it would be more convincing to compare contributions to bond susceptibility to predict the ε_r value since the refractive index in eqn (48) varies with the type of binary crystal. By applying the reported bond parameters, bond susceptibility values are listed in Table 9.

Table 9 Susceptibility of chemical bonds in an LiNiPO₄ system

Bond type	N e ^μ (Å^−3)	E g (eV)	χ ^μ	F ^μ	χ	χ/∑χ (%)
Li–O(1)	0.123	9.162	1.549	0.125	0.194	8.16
Li–O(2)	0.144	10.317	1.430	0.125	0.179	7.54
Li–O(3)	0.134	9.788	1.482	0.125	0.185	7.81
Ni–O(1)	0.265	13.074	1.900	0.063	0.119	5.01
Ni–O(2)	0.277	13.486	1.861	0.063	0.116	4.90
Ni–O(3)^1	0.285	13.765	1.836	0.125	0.229	9.67
Ni–O(3)^2	0.236	11.975	2.017	0.125	0.252	10.63
P–O(1)	2.746	28.722	4.148	0.063	0.259	10.93
P–O(2)	2.256	24.722	4.610	0.063	0.288	12.15
P–O(3)	2.454	26.365	4.405	0.125	0.551	23.21

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As Table 9 shows, the contributions of P–O bonds reach about 46.3%, while the secondary contribution comes from Ni–O bonds (30.2%), and after that Li–O bonds account for 23.5%. These results suggest that P–O bonds may possibly be the dominant factor for dielectric polarization. Xiao reported the vibrational spectra of LiMPO₄ ceramics to study the Raman and IR active vibrations. Specifically, the Raman scatting spectra are shown in Fig. 22.¹⁷³ The Raman-active vibrational behaviours are divided into three regions, where in the frequency of 50–400 cm⁻¹, Raman peaks are related to lattice vibration. Internal modes associated with P–O–P bending vibrational modes are detected in 400–700 cm⁻¹, while Raman peaks in 930–1100 cm⁻¹ are stretching vibrations of the [PO₄] tetrahedral unit. As can clearly be seen, the most intense peaks located at 940 cm⁻¹ and 1010 cm⁻¹ are assigned to the [PO₄] tetrahedron, suggesting its dominant role in Raman-active vibration. It is also found the peak position of the mode at 940 cm⁻¹ is contrary to the change in ε_r value. This phenomenon is explained by the fact that when the Raman peak blue-shifts, the V_cell and electron mobility space are increased, causing the crystal to be polarized and improving the ε_r value. That is to say, the [PO₄] tetrahedron is significant for dielectric polarization. If we look further, the damping value (FWHM) of both this mode (940 cm⁻¹, the symmetric stretching of the [PO₄] tetrahedron) is in inverse proportion to the Q × f value, suggesting the relationship between damping coefficient of structure vibration and dielectric loss. Again, the importance of P–O bonds is emphasized.

Fig. 22 (A) The Raman spectra of LNP ceramics; (B) the modes in 50–400 cm⁻¹ representing the lattice modes; (C) the modes in 400–900 cm⁻¹ representing mainly symmetric and antisymmetric bending modes of O–P–O and Ni–O; (D) the modes in 900–1200 cm⁻¹ representing the symmetric and asymmetric stretching mode vibrations of the [PO₄]³⁻ anion (reprinted with permission from ref. 173. Copyright 2019 John Wiley and Sons).

Furthermore, far-infrared reflectivity spectra of LiNiPO₄ were also recorded in Fig. 23¹⁷³ and analysed using a four-parameter semi-quantum (FPSQ) model to obtain intrinsic dielectric properties, as shown in Fig. 24.¹⁷³ Based on the fitted phonon parameters, it is suggested that [LiO₆] and [NiO₆] octahedra account for 50.86% of the ε_r value. This finding is close to P–V–L bond theory calculation, where 53.71% contributions to bond susceptibility are indicated for [LiO₆] and [NiO₆] octahedra. However, infrared spectra analysis shows the majority contribution to dielectric loss (85.79%) originates from [LiO₆] and [NiO₆] octahedra, while bond theory shows 86.54% contributions come from P–O bonds. One possible reason for this discrepancy is that there is no specific assignment of infrared modes via first principle calculations, and only general vibrations of polyhedra are classified. Thus, it is reasonable to use first-principles calculation to find out the reason for this difference.

Fig. 23 Measured and fitted infrared reflectivity spectrum of LNP ceramics, together with the real parts (Re(ε)) and the imaginary parts (Im(ε)) of the complex dielectric function, as well as the reciprocal (Im(ε⁻¹)), obtained from K–K analysis and the 4-P model as indicated (reprinted with permission from ref. 173. Copyright 2019 John Wiley and Sons).

Fig. 24 (A) Fitted and measured infrared reflectivity spectra of an LNP ceramic; (B) the reciprocal of the imaginary part of permittivity obtained from K–K; (C) the real part of permittivity obtained from K–K and 4-P; and (D) the imaginary part of permittivity obtained from K–K and 4-P (reprinted with permission from ref. 173. Copyright 2019 John Wiley and Sons).

4.5.2 Double trigonal molybdate structure. Mo-rich compounds, such as (Na_0.5Bi_0.5)MoO₄, Bi₂Mo₂O₉, Li₂MoO₄, (Na_xAg_2−x)MoO₄, and Bi(Sc_1/3Mo_2/3)O₄ ceramic systems,^{164,174–177} have the feature of low-temperature sintering because of the low melting point of MoO₃ of about 795 °C without using additives. Thus, the preparation technology and microwave dielectric properties are much easier to control since there is no concern about the effects of additives on phase formation, microstructure or crystal structure. Their excellent microwave dielectric performances, for instance: (Na_1.2Ag_0.8)MoO₄: ε_r = 8.1, Q × f = 44 800 GHz, τ_f = −82 ppm per °C when sintered at 410 °C, Li₂MoO₄: ε_r = 5.5, Q × f = 46 000 GHz, τ_f = −160 ppm per °C when sintered at 540 °C, show prospects in LTCC usage. Recently, a series of molybdate-based trigonal structures with the chemical formulas of Ln₂Zr₃(MoO₄)₉ (Ln = La, Ce, Eu, Gd) have been reported with excellent properties when sintered at a low temperature (Ln = Gd: ST = 725 °C, ε_r = 11.2, Q × f = 57 460 GHz, τ_f = −31.9 ppm per °C).¹⁷⁸ By applying P–V–L bond theory, it is easier to understand the structure–property relationship.

Ln₂Zr₃(MoO₄)₉ shows a trigonal structure with a space group of R3̄c(167). Its crystal structure is sketched in Fig. 25 (Ln = Ce is a case in point),¹⁷⁹ where two types of Zr (Zr(1) and Zr(2)) and Mo (Mo(1) and Mo(2)) atoms are presented, and six different O atoms exist. Ce and Zr2 atoms occupy 12c Wyckoff sites, and Zr1 is in a 6b site, while Mo1 and Mo2 atoms distribute in 36f and 18e sites, respectively. For O atoms, they all occupy 36f sites. The structure consists of five types of polyhedra because of different coordination environments of cations. For instance, the Ce³⁺ ion is surrounded by nine O²⁺ anion, where the bond types are classified as Ce–O1 × 3, Ce–O2 × 3 and Ce–O6 × 3. Both Zr(1)⁴⁺ and Zr(2)⁴⁺ ions are six-fold coordinated, while the Zr(1)⁴⁺ ion is only in the center of an O4²⁺ anion, forming a [Zr(1)O₆] octahedron, and O3²⁻ and O5²⁻ anions are around the Zr(2)⁴⁺ ion: Zr(2)–O3 × 3 and Zr(2)–O5 × 3. For the [MoO₄] tetrahedron, Mo(1)–O1 × 1, Mo(1)–O2 × 1, Mo(1)–O3 × 1 and Mo(1)–O4 × 1 bonds are shown in [Mo(1)O₄], while Mo(2)–O5 × 2 and Mo(2)–O5 × 2 bonds are presented in [Mo(2)O₄].

Fig. 25 Schematic diagram of the crystal structure for Ce₂Zr₃(MoO₄)₉ ceramics (reprinted with permission from ref. 179. Copyright 2019 Elsevier).

Firstly, the binary bonding formula is decomposed into:

The effective valence electron numbers of O anions in Ce–O, Zr–O and Mo–O bonds are Z_o = 2, 4 and 9, respectively. The coordination environment and charge distribution of the Ln = Zr system are sketched in Fig. 26.¹⁸⁰ Just as shown in section 4.3.2, discrepancies in Mg sites and Ti sites lead to different variations between ionicity and ε_r values in the Li₂Mg₃TiO₆ system. To figure out whether this situation occurs in trigonal Ln₂Zr₃(MoO₄)₉ structures, comparisons of bond ionicity, lattice energy and microwave dielectric properties among Ln₂Zr₃(MoO₄)₉ (Ln = La, Ce, Eu, Gd) systems are first listed in Table 10.^179–183

Fig. 26 The coordination number and charge distribution of ions in La₂(Zr_1−xTi_x)₃(MoO₄)₉ ceramics (reprinted with permission from ref. 180. Copyright 2019 John Wiley and Sons).

Table 10 Comparisons between average bond ionicity, lattice energy and microwave dielectric properties of the Ln₂Zr₃(MoO₄)₉ (Ln = La, Ce, Eu, Gd) system

Average bond ionicity Afi value (%)
Ln site	Ln–O	Zr–O	Mo–O	Af itotal	ε r
La	85.146	79.306	72.801	79.324	10.17
Ce	85.178	78.975	73.250	79.321	10.69
Eu	84.967	79.100	72.575	79.121	10.75
Gd	85.000	78.825	72.863	79.097	11.17

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Total lattice energy Utotal (kJ mol^−1)
Ln site	Ln–O	Zr–O	Mo–O	U total	Q × f
La	3302	10 752	43 253	364 555	51 578
Ce	3317	10 933	42 856	363 358	24 720
Eu	3331	10 835	43 559	367 241	74 900
Gd	3336	10 988	43 180	366 144	57 460

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In Table 10, it is shown that Ln–O bonds present the largest bond ionicity value, and the Af_itotal values show a decreasing trend as the Ln site ionic radius declines (IR values for La, Ce, Eu, Gd at CN = 9 are 1.216 Å, 1.196 Å, 1.120 Å, and 1.107 Å, respectively), while the increase in ε_r value indicates an enhancement in dielectric polarization. Considering that the densified temperatures are different, the ε_rc value after porosity is corrected also increases from 11.31 to 12.01. This phenomenon is similar to the ionic modification of the Ti site in Li₂Mg₃TiO₆ ceramics, suggesting that the bond covalency value of the Ln-site in Ln₂Zr₃(MoO₄)₉ (Ln = La, Ce, Eu, Gd) systems may possibly be closely related to the dielectric polarization. For lattice energy, one can still find that the largest intrinsic structural stability exists in Ln = Eu, followed by that in Gd, La, and Ce systems, which corresponds to the variations in Q × f values, where the Ln = Eu system obtains the largest value of about 74 900 GHz. This conclusion emphasizes the proportional relationship between structural stability and intrinsic dielectric loss in Ln₂Zr₃(MoO₄)₉ (Ln = La, Ce, Eu, Gd) systems. More importantly, by applying far-infrared reflectivity spectra, taking Ln = La as an example, as can be seen in Fig. 27,¹⁸⁴ the intrinsic dielectric loss is fitted and sequenced as: Eu (1.3 × 10⁻⁴) < Gd (2.4 × 10⁻⁴) < La (2.9 × 10⁻⁴) < Ce (2.3 × 10⁻³). This finding is also solid evidence to support the relationship between structural stability and dielectric loss. However, due to different sinterabilities, for instance, the densified temperatures of Ln = La, Ce, Eu, Gd are 650 °C, 575 °C, 600 °C, and 725 °C, the actual growth of grains and amounts of grain boundaries are different, which also negatively affects the dielectric loss.

Fig. 27 (a) Measured (black line) and fitted (red line) IR reflectivity spectrum of La₂Zr₃(MoO₄)₉ ceramic prepared by a reaction-sintering process sintered at 650 °C for 4 h; (b) real and imaginary parts of complex permittivity for an La₂Zr₃(MoO₄)₉ ceramic prepared by a reaction-sintering process sintered at 650 °C for 4 h (points are measured values in the microwave region) (reprinted with permission from ref. 184. Copyright 2019 Elsevier).

TiO₂ is usually chosen as the Zr site substitution source. In La₂(Zr_1−xTi_x)₃(MoO₄)₉ ceramics, solid solutions are formed in the range x = 0–0.1.¹⁸⁰ The introduction of a Ti⁴⁺ cation leads to an expansion in cell volume and bond length. Based on P–V–L theory calculation, decreasing bond ionicity is consistent with the ε_r value, which is confirmed by the observed and theoretical dielectric polarizability. It should also be mentioned that the Ti site situation can be also referred to the Li₂Mg₃TiO₆ system (Mg site substitution). Also, when the Zr site is partially replaced by an Sn⁴⁺ cation in Ce₂(Zr_0.94Sn_0.06)₃(MoO₄)₉, one can still find a declining trend between bond ionicity and ε_r values in Ce₂Zr₃(MoO₄)₉ and Ce₂(Zr_0.94Sn_0.06)₃(MoO₄)₉ systems, where the Af_total (ε_r) values are 79.32% (10.69) and 74.00% (10.35), respectively.¹⁸⁵ This conclusion indicates that in Zr site substitution, the bond ionicity value may possibly be proportional to dielectric polarization.

4.6 Cubic spinel structures

MgO–TiO₂ systems are appealing to the public for their richness in raw materials, low cost and excellent microwave dielectric properties. For instance, three main crystalline phases are reported in the binary MgO–TiO₂ phase diagram: spinel Mg₂TiO₄, ilmenite MgTiO₃ and pseudobrookite MgTi₂O₅, where Mg₂TiO₄ and MgTiO₃ exhibit low ε_r values and ultra-low dielectric loss (Mg₂TiO₄: ST = 1480 °C, ε_r = 14.51, Q × f = 161 570 GHz, and τ_f = −49.3 ppm per °C; MgTiO₃: ST = 1400 °C, ε_r = 17.75, Q × f = 250 000 GHz, and τ_f = −42.5 ppm per °C).^186,187

Li reported the chemical bond characteristics of the Mg₂TiO₄ system by using P–V–L bond theory. The Mg₂TiO₄ system has a cubic structure with a space group of Fd3m(227) and lattice parameters of a = b = c = 8.443 Å and V_cell = 601.84 ų. Structural details, including fractional coordinate, atomic position and occupancy are listed in Table 11.¹⁸⁶

Table 11 Crystal structural detail, fractional coordinate, atomic position and occupancy of cubic spinel Mg₂TiO₄ structures

Atom	Position	Fractional coordinate			Occ
		x	y	z
Mg1	8a	0.1250	0.1250	0.1250	0.964
Mg2	16d	0.5000	0.5000	0.5000	0.518
Ti1	8a	0.1250	0.1250	0.1250	0.036
Ti2	16d	0.5000	0.5000	0.5000	0.482
O	32e	0.2616	0.2616	0.2616	1.000

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The crystal structure is presented in Fig. 28, where the M1 site (8a) is occupied by Mg1 and Ti1, while Mg2 and Ti2 are in the M2 site (16d). A cation in the M1 site is four-fold coordinated, forming an [M1O₄] tetrahedron, and an octahedral ligand [M2O₈] forms in the M2 site. The O²⁻ anion is surrounded by four cations. The binary bonding formula is then expressed as:¹⁸⁸

Fig. 28 Crystal structure of a cubic spinel Mg₂TiO₄ crystal.

Thus, Ti–O bonds show larger bond ionicity (88.34%) values than Mg–O bonds (76.62%), and the lattice energy of Ti–O bonds (13290 kJ mol⁻¹) is two times larger than that of Mg–O bonds (7473 kJ mol⁻¹). By substituting the Ti site with an Sn⁴⁺ cation in Mg₂(Ti_1−xSn_x)O₄, a cubic solid solution was formed over the whole range of x = 0–1. An increase in Sn⁴⁺ content leads to an enhancement in bond lengths. Based on P–V–L bond theory calculation, it is found that the decrease in average bond covalency is responsible for the drop in ε_r value, as shown in Fig. 29.¹⁸⁶ The structural stability reflected by the U_total value gradually weakens as x increases. To verify the structural changes, Raman scatting spectra are presented in Fig. 30.¹⁸⁶ Raman-active vibrational modes are assigned into three parts, where the Raman modes located in the high-frequency region (600–750 cm⁻¹) are attributed to the symmetric stretching (A_1g) and asymmetric stretching behaviours (T¹_2g) of the [M1O₄] tetrahedron. Modes in (350–600 cm⁻¹) are classified as symmetric (T²_2g) and asymmetric (E_g) bending behaviours of the [M1O₄] tetrahedron. The mode at 323 cm⁻¹ implies a local lattice in the octahedral sub-lattice. The most intense Raman peak (A_1g) usually has a dominant effect on structural variation. It is shown that the A_1g mode redshifts as x increases, which verifies that the increase in cell volume and bond length leads to a decrease in vibrational energy. Thus, the decrease in the ε_r value is verified by the shift in A_1g Raman peak position.

Fig. 29 The dielectric constant ε_r and the average bond covalency Af_c variation with x varying from 0 to 1 (reprinted with permission from ref. 186. Copyright 2019 Elsevier).

Fig. 30 (a) The Raman spectra of Mg₂(Ti_1−xSn_x)O₄ compounds in the range 200 cm⁻¹–850 cm⁻¹; (b) the variation in Raman shift in the high frequency region (600 cm⁻¹–730 cm⁻¹); (c) the variation in Raman shift in the range 280 cm⁻¹–630 cm⁻¹; (d) the relationship between dielectric constant and A_1g Raman shift (reprinted with permission from ref. 186. Copyright 2019 Elsevier).

4.7 Orthorhombic europium titanium structure

Ln₂TiO₅ ceramics have potential in the usage of forms of nuclear waste because of their excellent chemical durability and tolerance to high-energy particles. Recently, Zheng reported a novel orthorhombic europium titanium type Eu₂TiO₅ microwave dielectric system and analysed the bond parameters using P–V–L bond theory for the first time.¹⁸⁹

Unlike reported crystal structural types, Eu₂TiO₅ is an orthorhombic structure with a Pnam(62) space group and lattice parameters of a = 10.5342 Å, b = 11.2957 Å, c = 3.7785 Å and V_cell = 449.61 ų, where cations are mixed with seven and five-fold coordination, as shown in Fig. 31. Two types of Eu atoms, and six different types of O atoms are shown. Specifically, both Eu1³⁺ and Eu2³⁺ cations are seven-fold coordinated, which is regarded as an irregular octahedron with the addition of a seventh atom. It is also reported that based on the polyhedral configuration of seven-fold coordination, the coordination of the Ln cation (for instance Ln = Y) can be regarded as a cube missing one oxygen atom, and a distorted trigonal prism with a seventh oxygen atom connected via a face.¹⁹⁰ The Ti⁴⁺ cation is surrounded by five O²⁻ anions. Eu1 is the center cation connecting with three O1, two O2, one O3 and one O4 anions in an Eu1 polyhedron; however, Eu2 connects with one O1, one O2, one O3, two O4 and two O5 anions in an Eu2 polyhedron. The Ti⁴⁺ cation links with one O2, two O3, one O4 and one O5 anion. The binary bonding formula is written as:¹⁸⁹

Fig. 31 Crystal structure of an orthorhombic Eu₂TiO₅ crystal.

It is important to note that even for the same cation, taking Eu(2) as an example, the effective valence electron numbers of the O anion (Z_o) in Eu–O bonds are different: Z_o = 36/7 in an Eu(2)–O(4) bond, and Z_o = 27/7 in an Eu(2)–O(5) bond. Those values in Ti–O bonds also vary as the m and n values in A_mB_n change.

After taking these factors into consideration, the average bond ionicity f_i value is arranged as follows: f_i(Eu1−O) = 88.17% > f_i(Eu2−O) = 85.99% > f_i(Ti–O) = 77.04%, where the chemical bonds tend to be more ionic, and the largest f_i value is found in the Eu1–O bond. For the lattice energy, Ti–O bonds have the largest average lattice energy AU value of about 3626 kJ mol⁻¹, followed by that of Eu1–O (1251 kJ mol⁻¹) and then Eu2–O (1241 kJ mol⁻¹) bonds, which shows that Ti–O bonds play an indispensable role in determining the structural stability of an orthorhombic Eu₂TiO₅ structure.

Niu reported the electron localization function (ELF) based on first principles calculation to give an insightful estimate of the ionicity or covalency aspects of bonds in orthorhombic Ln₂TiO₅ systems (Ln = La, Sm, Gd, Dy), as presented in Fig. 32.¹⁹¹ As a dimensionless, empirical function, the ELF that generated comparatively large values (>0.5) is ascribed to the bonding/nonbonding localized electrons and smaller values (<0.5) for delocalized electrons. It is found that in both Ln/Ti–O bonds, the bond ionicity should be the essential factor, and Ti–O bonds are more covalent than Ln–O bonds for the σ-bonding Ti (3d)–O(2p) orbital interaction. This theoretical calculation also provides evidence for the bond ionicity results that Ti–O bonds have the largest bond covalency value of about 22.96% compared to that of Eu–O bonds of 14.01%.

Fig. 32 ELF at the (040) plane for La₂TiO₅, Sm₂TiO₅, Gd₂TiO₅ and Dy₂TiO₅. (reprinted with permission from ref. 191. Copyright 2011 Cambridge University Press).

4.8 Monoclinic fergusonite structure

As a typical case of fergusonite RENbO₄, NdNbO₄ is interesting to the public and is widely studied for its widespread application prospects. For instance, a phase transformation of RENbO₄ takes place between a tetragonal scheelite structure and a monoclinic ferroelastic structure at about 550 °C.¹⁹² It could be promising for the application of anode-supporting substrates, proton conducting cells and energy-absorption by virtue of its surface properties, chemical compatibility and the strain resulting from the phase transformation.^193–196 For MWDC applications, a RENbO₄ ceramic also shows excellence in dielectric performance because of the existence of an [NbO₆] framework and rare-earth elements. e.g. an ε_r of 19.6, a Q × f value of about 33 000 GHz and a τ_f of −24 ppm per °C are obtained for an NdNbO₄ ceramic,⁸² and for an LaNbO₄ ceramic sintered at 1275 °C: ε_r = 20.3, Q × f = 59 740 GHz, τ_f = 7.44 ppm per °C,¹⁹⁷ and a much lower dielectric loss is found in (La_0.92Sm_0.08)NbO₄ ceramics with an ε_r value about 19.37, a Q × f value of 62 203 GHz and a τ_f of ca. 2.57 ppm per °C.¹⁹⁸ Thus, much effort has been put into studying the structure–property relationship and improving the microwave dielectric properties.

Based on the reported monoclinic fergusonite crystal structure configuration of NdNbO₄ ceramic: a = 5.146 Å, b = 11.294 Å, c = 5.451 Å and V_cell = 317.25 ų with a space group of I2/a(15), as shown in Fig. 33,¹⁹⁹ the structure consists of a distorted [NdO₈] dodecahedron and [NbO₆] octahedron. [NbO₆] frames form a one-dimensional zigzag chain by sharing an edge, while the [NdO₈] units are set in the propagating chains of [NbO₆] sharing the oxygen. Both Nd and Nb atoms are in a 4e Wyckoff site, and both connect with two Nd³⁺ cations and two Nb⁵⁺ cations (CN = 4), while the O2 anion links with two Nd³⁺ cations and one Nb⁵⁺ cation (CN = 3). This discrepancy results in different effective valence electron numbers for the O anion (Z_o). Firstly, the binary bonding formula should be decomposed into:¹⁹⁹

Fig. 33 The crystal structure patterns of the (1 × 1 × 1) supercell of monoclinic fergusonite structured NdNbO₄ (reproduced from ref. 199 with permission from the PCCP Owner Societies).

Apparently, the Z_o values in Nd_1/4–O(1), Nd_1/4–O(2), Nb_1/3–O(1) and Nb_1/3–O(2) are 4.5, 3.375, 10 and 7.5, respectively. With the same crystal structure, bonding environment and binary bonding expressions of RENbO₄ (RE = Nd, La) ceramics, detailed comparisons between chemical bond features are listed in Table 6S.† Here, we summarize the bond ionicity, lattice energy and microwave dielectric properties in Table 12.^197,199

Table 12 Comparisons of average bond ionicity, total lattice energy and microwave dielectric properties in RENbO₄ (RE = Nd, La) ceramics

RE	Bond type	Af i (%)	U total (kJ mol^−1)	χ/∑χ (%)	ε r	Q × f (GHz)
Nd	Nd–O	85.55	5767	48.22	19.6	33 000
	Nb–O	85.56	19 530	51.78
	∑χ = 2.98, Afitotal = 85.55%, Utotal = 25297 kJ mol^−1
La	La–O	84.82	5412	53.87	20.3	59 740
	Nb–O	84.61	20 553	46.13
	∑χ = 3.62, Afitotal = 84.73%, Utotal = 25965 kJ mol^−1

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Combined with Table 6S in ESI† and Table 12, it can be found that all the ionicity f_i values of the chemical bonds in LaNbO₄ are smaller than those in the NdNbO₄ system. For the two structures, specifically, Af_i values of Nd–O bonds are very close to those of Nb–O bonds, and their susceptibility contribution ratios are also very similar. However, large contributions of lattice energy originate from Nb–O bonds (79%). This is to say, both Nd–O and Nb–O bonds are responsible for the dielectric polarization, but Nb–O bonds are the crucial factor for stabilizing the crystal structure. Moreover, it is also shown that the ε_r value of LaNbO₄, which is higher than that of NdNbO₄, corresponds to a lower bond ionicity value, suggesting that in the Nd–La transformation, an increase in bond covalency may possibly be closely related to dielectric polarization. This can be confirmed by the increase in bond susceptibility. High stability in LaNbO₄ ceramics also agrees with a higher Q × f value, which indicates that the structural stability in monoclinic fergusonite RENbO₄ (RE = Nd, La) is also an intrinsic factor determining intrinsic dielectric loss. Zhang reported the substitution of an La³⁺ ion on an Nd site in (Nd_1−xLa_x)NbO₄ ceramics,¹⁹⁹ and the results show that an increase in the ε_r value is consistent with average bond covalency, and an increasing content of La₂O₃ (x = 0–0.06) is beneficial to improving the structural stability and Q × f value. This finding supports the conclusion above. The Nd site also can be replaced by an Sm³⁺ cation in (Nd_1−xSm_x)NbO₄ ceramics. Interestingly, the bond ionicity decreases along with the ε_r value, and an increase in the Q × f value is also attributed to the enhancement in U_total value.²⁰⁰ There is a temperature-induced phase transition in SmNbO₄ ceramics from a monoclinic to a tetragonal structure when the temperature exceeds 800 °C but no relevant chemical bond parameters have been reported for SmNbO₄; therefore, it is difficult to compare detailed bond features in RE = Nd, Sm systems.²⁰¹

As noted, Nb–O bonds are the dominant influence on structural stability, apart from Nd site substitution. The Nb site is also substituted by pentavalent single ions and complex ions to improve the Q × f value: e.g. in the solid solution region of the (1 − x)LaNbO₄–xLaVO₄ system (x = 0–0.1), the drop in ε_r value is also ascribed to the decrease in bond ionicity, which can be confirmed by the drop in dielectric polarizability of V⁵⁺ (2.92 Å) compared to Nb⁵⁺ (3.97 Å) and an increase in U_total value confirms the great improvement in Q × f value from 6770 GHz to 62370 GHz.²⁰² Substitution of the Nb site using Ta₂O₅ as a source in LaNbO₄ ceramics presents a similar trend, where the ε_r value consistently decreases with bond ionicity, and the increase in structural stability induced by the Ta⁵⁺ cation improves the Q × f value from 47 000 GHz to 66000 GHz.²⁰³

Complex ions, such as (Al_1/3W_2/3)⁵⁺, (Mg_1/4W_3/4)⁵⁺, (Zr_1/2W_1/2)⁵⁺ and (AMo)⁵⁺ (A = Mg, Al, Si, Zr) were used to replace the Nb site. For instance, Yang took variations of (Al_1/3W_2/3)⁵⁺ and (Mg_1/4W_3/4)⁵⁺ ionic occupancies into consideration in modified NdNbO₄ systems,^204,205 and the results show that even the cell volume shows different changes. Both the average bond ionicity values are responsible for the ε_r value, and, for the Q × f value, variations in lattice energy and actual average size of grain together make contributions to the actual dielectric loss. They also found a close relationship between total lattice energy and cell volume in (AMo)⁵⁺ (A = Mg, Al, Si, Zr) substitution, as shown in Fig. 34,²⁰⁶ where shrinkage in cell volume improves the structural stability, consequently reducing the intrinsic loss. Based on their reports on microwave dielectric properties, an SiMo doping system also shows the largest Q × f value of about 49 466 GHz, which is higher than that of the undoped situation of 24 750 GHz.

Fig. 34 Correlation between the lattice energy and molecular volume of different NdNbO₄-based ceramics (reprinted with permission from ref. 206. Copyright 2019 Elsevier).

4.9 Complex perovskite structure

Complex perovskite Ba(M_1/3M′_2/3)O₃ (M = Mg, Ni, Zn; M′ = Nb, Ta) structures are well known for and have been broadly studied for their outstanding microwave dielectric properties For instance, Ba(Mg_1/3Ta_2/3)O₃ ceramic has a moderate ε_r value of about 24, an ultra-high Q × f value of 250 000 GHz and a near-zero τ_f value,¹ and it is mostly used in resonators and has achieved widespread commercial value and thus continuously attracts public attention. Until now, much effort has been put into exploring the intrinsic structural impacts on dielectric performances: e.g. a 1 : 2 ordered structure was clearly revealed using transmission electron microscopy in Ba((Co_0.5Zn_0.3Mg_0.2)_1/3Nb_2/3)O₃ ceramic along 〈110〉_c, 〈111〉_c and 〈112〉_c projections, where the ordering of M and M′ cations was considered to be a crucial factor for microwave dielectric properties.²⁰⁷

The Raman active and infrared active modes were assigned in detail by using first principles lattice dynamics calculation based on density functional theory, with the conclusion that the low dielectric loss found in this ordered Ba(Mg_1/3Ta_2/3)O₃ ceramic can be ascribed to the long phonon lifetime and weak coupling among phonons.²⁰⁸ The Ba(Mg_1/3Ta_2/3)O₃ system is crystallized as a hexagonal structure with a P3̄m1(164) space group, as shown in Fig. 35. Based on reported structural details (COD #4000400) of a = b = 5.77385 Å, c = 7.09376 Å, V_cell = 204.804 ų and P–V–L bond theory consideration, coordination environments for each cation are listed in Table 13. Also, the binary bonding formula is written as:

Fig. 35 Crystal structure of a hexagonal Ba(Mg_1/3Ta_2/3)O₃ system.

Table 13 Coordinate environments of the Ba(Mg_1/3Ta_2/3)O₃ system

Ba3MgTa2O9 = Ba11Ba21MgTa2O13O26
CN: coordination number, N(B^j–A^i) represents the number of A^i cations in the ligand of the B^j ion.
Ba1	Ba2	Mg	Ta
CN = 12	CN = 12	CN = 6	CN = 6
N(Ba1–O1) = 6	N(Ba1–O1) = 3	N(Mg–O2) = 6	N(Ta–O1) = 3
N(Ba1–O2) = 6	N(Ba1–O2) = 9		N(Ta–O2) = 3
O1	O2
CN = 6	CN = 6
N(O1–Ba1) = 2	N(O2–Ba1) = 1
N(O1–Ba2) = 2	N(O2–Ba2) = 3
N(O1–Ta) = 2	N(O2–Mg) = 1
	N(O2–Ta) = 1

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Detailed bond features of the Ba(Mg_1/3Ta_2/3)O₃ system can be found in Table 7S in ESI.† The average bond ionicity, effective valence electron number of oxygen anions, total lattice energy and bond susceptibility are summarized in Table 14.

Table 14 Average bond ionicity, effective valence electron number of oxygen anion, total lattice energy and bond susceptibility of the Ba(Mg_1/3Ta_2/3)O₃ system

Bond type	Afi (%)	Z o	U total (kJ mol^−1)	χ	χ/∑χ (%)
Ba1–O	95.61	3	2309	0.174	9.93
Ba2–O	95.61	3	3443	0.353	20.11
Mg–O	77.40	6	3864	0.564	32.14
Ta–O	89.78	15	49 016	0.664	37.82

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Based on the results from Table 14, it can be seen that Ba–O bonds have the largest bond ionicity f_i value; however, the majority contributions to bond susceptibility come from Ta–O bonds, followed by Mg–O and Ba–O bonds. Furthermore, Ta–O bonds contribute about 89.8% of total lattice energy. This conclusion illustrates that Ta–O bonds in [TaO₆] octahedra may be the dominant factor determining the dielectric properties.

By performing first principles calculations to obtain thorough vibrational modes and comparing them with an experimental far-infrared reflectivity spectrum, Wang assigned the specific vibrational behaviours in a Ba(Mg_1/3Ta_2/3)O₃ system.²⁰⁹ For instance, symmetry coordinates and polarization are illustrated in Fig. 36. According to their fitting results of the FIR reflection spectrum using a four-parameter semiquantum model, it is shown that the E_u⁽⁴⁾ mode at 220 cm⁻¹ and the E_u⁽⁵⁾ mode at 241 cm⁻¹ provide the largest theoretical contributions to dielectric properties. More importantly, the E_u⁽⁴⁾ and E_u⁽⁵⁾ modes are assigned as the vibrations of [TaO₆] octahedra (Ta motion versus O and vibration of O octahedra); that is to say, based on first principles calculation, it is confirmed that [TaO₆] octahedra should be the dominant reason for the dielectric properties, which is consistent with the results from P–V–L bond theory. Furthermore, the E_u⁽⁴⁾ and E_u⁽⁵⁾ modes in total contribute about 53.8% and 90.2% to the dielectric constant ε_r and dielectric loss tan δ, which are also close to the results from Table 14. Similarly, first principles calculation indicates that the E_u⁽¹⁾ mode at 149 cm⁻¹ and the A_2u⁽²⁾ mode at 212 cm⁻¹, where the vibrational behaviours closely related to [NbO₆] octahedra, dominate the dielectric properties in the Ba(Mg_1/3Nb_2/3)O₃ system.²¹⁰

Fig. 36 Symmetry coordinates and polarization of Ba(Mg_1/3Ta_2/3)O₃. The arrows () stand for the polarizing direction and the crosses (×) stand for no polarization (reprinted with permission from ref. 209. Copyright 2010 John Wiley and Sons).

At this point, we are driven to wonder whether the dominant factor in a complex perovskite structure is dependent on the heavier atom. Another typical complex perovskite can be found in cubic Ba(Mg_1/2W_1/2)O₃ ceramics (ε_r = 17, Q × f = 42 000 GHz, τ_f = −33.6 ppm per °C). Based on reported first principles calculations, it is shown that the F_1u⁽²⁾ mode at 277 cm⁻¹ makes about 47.5% and 80.5% contributions to ε_r and tan δ, while the mode is regarded as the stretching behaviours of [MgO₆] and [WO₆] octahedra.²¹¹ Ba(Mg_1/2W_1/2)O₃ is a cubic structure with a space group of Fm3m and lattice parameters of: a = b = c = 8.112 Å, and V_cell = 533.807 ų as reported by Filip'ev from COD #1537301.²¹² In the view of P–V–L bond theory, its coordination environment of cations is totally different from that in the Ba(Mg_1/3Ta_2/3)O₃ system, where the binary bonding formula is much simpler with only one type of bond shown in the oxygen polyhedron:

Ba₂MgWO₆ = Ba₂O₄ + MgO + WO

The bond parameters are listed in Table 15. Comparing Table 15 with Table 14, it can be seen that even the f_i values are close to each other; however, the contributions to bond susceptibility are quite different, where the Mg–O bond accounts for 58.36% of the total bond susceptibility in this situation, illustrating that the Mg–O bond is indispensable for dielectric polarization. First principles calculation also shows that the [MgO₆] octahedra make large contributions to dielectric loss, while P–V–L bond theory indicates that W–O bonds are indispensable in structural stability. This difference may possibly be related to the boundary condition constraints in the establishment of the model. Therefore, the theoretical conclusions must be verified through experiments on Mg site and W site substitutions to give more precise results.

Table 15 Bond ionicity, lattice energy, and bond susceptibility in Ba(Mg_1/2W_1/2)O₃ system

Bond type	f i (%)	U (kJ mol^−1)	χ ^μ	F ^μ	χ	χ/∑χ (%)
Ba–O	95.6	4627	0.71	0.67	0.47	30.12
Mg–O	77.6	3756	5.49	0.17	0.91	58.36
W–O	91.3	37 319	1.08	0.17	0.18	11.51

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The discrepancy between the two structures may possibly lie in structural differences and ionic occupancies. From this section, it can be concluded that although both belong to complex perovskites, differences in structures, coordination environment and effective valence electron number of ions together lead to different bond parameters and different contributing sources to bond susceptibility and lattice energy.

This conclusion is similar in cubic Li_2xMg_1−xTi_xO_1+2x (x = 1/4, 2/5 and 1/2) systems that although all of them are subject to a cubic structure and have a similar bonding environment, a discrepancy in ionic occupancy results in a different bond susceptibility. By comparing the bond susceptibility, lattice energy and dielectric properties of Ba(Mg_1/3Ta_2/3)O₃ and Ba(Mg_1/2W_1/2)O₃ systems in Table 16, one can clearly find from Table 16 that there is also a close relationship between bond susceptibility, lattice energy and microwave dielectric properties, suggesting that bond features may also be crucial for intrinsic dielectric properties.

Table 16 Comparisons of bond susceptibility, lattice energy and microwave dielectric properties of complex perovskite Ba(Mg_1/3Ta_2/3)O₃ and Ba(Mg_1/2W_1/2)O₃ systems

System	χ	U total (kJ mol^−1)	ε r	Q × f (GHz)
Ba(Mg1/3Ta2/3)O3	1.755	58 632	24	250 000
Ba(Mg1/2W1/2)O3	1.566	45 702	17	42 000

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4.10 Monoclinic wolframite structure

Wolframite-type AZrB₂O₈ (A = Mn, Zn, Mg, Co, Ni; B = Nb, Ta) is crystallized as a monoclinic structure with a space group of P2/c (13).²¹³ As mentioned in section 4.8, the appearance of [NbO₆] units is beneficial for microwave dielectric properties, which is also the case for a wolframite structure. For instance, a ZnZrNb₂O₈ ceramic has a moderate ε_r value of 29.27, a high Q × f value of about 55 460 GHz, and a negative τ_f value of ca. −77.21 ppm per °C. Improvements in the Q × f value have been reported with ionic modification on the cation site: e.g. small amounts of Ni²⁺ cation (0.08 mol) on the Zn site greatly improve the Q × f value to 73224 GHz.²¹⁴ A Ti site replaced by a Ge⁴⁺ cation (0.2 mol) is beneficial for the development of the Q × f value to 80493 GHz.²¹⁵ An Nb site partially substituted by Ta⁵⁺ and Sb⁵⁺ cations also testified to positive effects on the Q × f value (Sb = 0.08 mol, Q × f = 89 400 GHz).²¹⁶ Therefore, structural impacts, especially the evolution of bond features in an AZrB₂O₈ monoclinic solid solution, may be crucial for discussing the structure–property relationship.

4.10.1 A site difference in AZrNb₂O₈ (A = Mn, Zn, Mg, Co). The monoclinic structure of an AZrNb₂O₈ system is shown in Fig. 37.²¹⁷ A and Zr atoms are in the 2f Wyckoff position while Nb is in a 2e site. All the cations are six-fold coordinated, forming structural basic units of [MO₆] octahedra (M = A, Zr, Nb). Based on the coordination environments, the binary bonding formula is expressed as:
where the effective valence electron numbers of O anions in A–O, Zr–O and Nb–O bonds are considered to be 3, 6 and 7.5, respectively. The coordination number and charge distribution can be generalized as in Fig. 38.²¹⁸

Fig. 37 Schematic representation of the Zn_1−xCo_xZrNb₂O₈ (2 × 2 × 2) supercell and the A-site (A = Zn²⁺/Co²⁺, Zr⁴⁺) and B-site (B = Nb⁵⁺) oxygen octahedrons of Zn_0.94Co_0.06ZrNb₂O₈ with their bond lengths (reproduced from ref. 217 with permission from the Royal Society of Chemistry).

Fig. 38 The coordination number and charge distribution of ions in AZrNb₂O₈ ceramics (here A = Co ions) (reprinted with permission from ref. 218. Copyright 2016 Elsevier).

Due to the difference in ionic radii of A-site cations (IR values for Mn, Zn, Mg, Co at CN = 6 are 0.83, 0.74, 0.72, 0.65 Å), the V_cell and chemical bond lengths after Rietveld refinement present a declining trend from A = Mn to Co elements. Comparisons between bond ionicity, lattice energy, and microwave dielectric properties are listed in Table 17.²¹⁸ The largest bond ionicity is shown in Nb–O bonds, followed by that in Zr–O and A–O bonds. The average f_i for all types of bonds present a declining trend from 76.212% to 76.146%, corresponding to a decrease in dielectric polarization. As a result, the ε_r value declines from A = Mn to Co. This result indicates that variation in bond ionicity in different A site situations possibly affects the ε_r value. The largest total lattice energy value is found in A = Mg (52335 kJ mol⁻¹), and then A = Zn (52297 kJ mol⁻¹), while the smallest value is shown in A = Mn (51936 kJ mol⁻¹), suggesting that the internal losses in A = Zn, Mg may be smaller than those in A = Mn, Co. By using the far-infrared reflectivity spectra, it is also found that the fitting of intrinsic loss by a classical three-parameter model of A = Zn, Mg systems is much smaller than those in A = Mn, Co systems.

Table 17 Comparisons between bond ionicity, lattice energy, and microwave dielectric properties of AZrNb₂O₈ ceramics (A = Mn, Zn, Mg, Co)

A-site	Bond type	Af i (%)	U (kJ mol^−1)	ε r	Q × f (GHz)
Mn	Mn–O	64.11	3212	27.56	27 400
	Zr–O	80.51	11 673
	Nb–O	84.02	37 051
Zn	Zn–O	64.02	3233	27.08	58 000
	Zr–O	80.48	11 744
	Nb–O	83.99	37 320
Mg	Mg–O	63.92	3255	26.55	68 000
	Zr–O	80.47	11 824
	Nb–O	84.05	37 256
Co	Co–O	64.00	3233	25.59	47 100
	Zr–O	80.47	11 744
	Nb–O	83.98	37 320

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NiO (0–0.08 mol) was used to partially replace the Zn sites in ZnZrNb₂O₈ ceramics, and the results also provide evidence that the decrease in f_iNb–O value is responsible for the drop in the ε_r value, and the lattice energy consistently increases along with structural stability.²¹⁴

4.10.2 Ionic doping on Zr/Nb sites in AZrNb₂O₈ (A = Zn, Mg). Due to the excellence of the microwave dielectric properties in AZrNb₂O₈ (A = Zn, Mg) systems, a large number of studies have developed around the goal of further improving the Q × f value. Just as mentioned in section 4.10.1, the improvement in structural stability may possibly be positive for the development of the Q × f value. Thus, many studies around the MgZrNb₂O₈ system seek whether a similar relationship exists and what types of substitution sources are suitable.

For instance, taking the Zr site as an example, the cell volume is obviously decreasing when the Zr site is partially substituted by a Ge⁴⁺ cation in an Mg(Zr_1−xGe_x)Nb₂O₈ (x = 0–0.4) system, which is caused by its smaller ionic radius of 0.53 than the 0.72 of a Zr⁴⁺ cation. Consequently, the Q × f value is greatly enhanced from 64 784 GHz to 80493 GHz.²¹⁵ The Sn⁴⁺ cation (IR = 0.69 at CN = 6) replacing a Zr site in an MgZr_0.85Sn_0.15Nb₂O₈ ceramic also shows that lattice energy is advantageous to the Q × f value, where the largest U_total corresponds to the highest Q × f value of 94014 GHz.²¹⁹

For Nb site substitution, Ta⁵⁺, Sb⁵⁺, (Sn_1/2W_1/2)⁵⁺ cations are used to study the structure–property relationship based on P–V–L bond theory: e.g. in ZnZr(Nb_1−xA_x)₂O₈ (A = Ta, Sb) ceramics,²¹⁶ an increasing content of donor reduces the cell volume, and the results show that a decrease in the Af_i value for all types of bonds in a Ta⁵⁺ doping system (from 76.19% to 76.03%) should be responsible for the decline in ε_r value (from 26.7 to 25.7). Also, the increase in lattice energy improves the Q × f value (from 65 100 GHz to 89 400 GHz). In an MgZrNb₂O₈ system, the Ta⁵⁺ cation is also proved to be an effective source to improve the lattice energy and Q × f value, and the average bond ionicity Af_i value decreases similarly with the ε_r value.²²⁰ Complex (Sn_1/2W_1/2)⁵⁺ cation replacement for the Nb site in MgZrNb₂O₈ system (0–0.06 mol) is also confirmed by the similar variation tendency.²²¹

From the above-mentioned reports, it can be concluded that the variation in bond ionicity may be proportional to the dielectric polarization, and an improvement in lattice energy actually enhances the structural stability and further improves the Q × f value.

4.11 ZnO–TiO₂–Nb₂O₅ ternary systems

The ternary system of ZnO–TiO₂–Nb₂O₅ was firstly introduced by Baumgarte and Blachnik in 1992.²²² It is proposed that in TiO₂–Zn_1/3Nb_2/3O₂, the rutile TiO₂ phase appears when the molar content of Zn_1/3Nb_2/3O₂ is lower than 58%, ixiolite ZnTiNb₂O₈ appears when the content is in the range of 69% to 85%, and columbite ZnNb₂O₆ is formed when the content is higher than 93%. In 2008, Kim first reported the formation and microwave dielectric properties of the Zn_0.15Nb_0.3Ti_0.55O₂ phase with a rutile structure.²²³ It should be noted that all the above systems contain Zn, Ti, and Nb elements and the molar ratio of Zn : Nb remains unchanged at about 1 : 2. Further in-depth study shows that the above materials can be understood as the coordinated substitution of Ti⁴⁺ in TiO₂ by (Zn_1/3Nb_2/3)⁴⁺ ions: (Zn_1/3Nb_2/3)_xTi_1−xO₂ (x = 0.45, 0.75, 1). Phase structures vary with the increase in x value, as indicated in the ternary phase diagram of Fig. 39.

Fig. 39 Ternary ZnO–TiO₂–Nb₂O₅ systems.

This ZnO–TiO₂–Nb₂O₅ system has attracted much attention because of the excellence of its dielectric properties. For instance, columbite ZnNb₂O₆ has a considerable Q × f value of about 83 700 GHz with a moderate ε_r value of about 25.⁸⁹ Both ixiolite ZnTiNb₂O₈ and rutile Zn_0.15Nb_0.3Ti_0.55O₂ ceramics can be well-sintered at relatively low temperatures with various great dielectric performances (ixiolite: ST = 1100 °C, ε_r = 37.21, Q × f = 41 231 GHz; rutile: ST = 1050 °C, ε_r = 94.35, Q × f = 10 889 GHz).²²⁴ Thus, many investigations have been undertaken to not only further improve the dielectric properties but also to decrease the sintering temperature, exploring applications in LTCC applications.

From the point of view of crystal structure and P–V–L bond theory, an attractive feature of this system is that three distinctive crystal structural types appear with the variations in mole contents of the (Zn_1/3Nb_2/3)⁴⁺ ions. That is to say, the ionic occupancy is presented with large differences, which causes the different bond parameters. And taking this effect into consideration, one may have a better understanding of the structure–property relationship in the ZnO–TiO₂–Nb₂O₅ system.

4.11.1 Variations in ionic occupancy in (Zn_1/3Nb_2/3)_xTi_1−xO₂ (x = 0.45, 0.75, 1). Just as indicated, variations in complex ionic occupancy lead to different phase compositions, which directly causes various bond parameters. For x = 1, a columbite orthorhombic ZnNb₂O₆ structure with a space group of Pbcn(60) is formed, and its crystal structural parameters are reported as: a = 14.213 Å, b = 5.732 Å, c = 5.040 Å, V_cell = 410.60 ų. The crystal structure is shown in Fig. 40,²²⁵ where Zn and Nb atoms occupy 4c and 8d Wyckoff positions, and the structure consists of [ZnO₆] and [NbO₆] octahedral units, and it is arranged as Zn–Nb–Nb–Zn–Nb–Nb⋯ along the a-axis orientation. A [ZnO₆] octahedron links with an [NbO₆] octahedron through sharing a vertex, while both of them interconnect with each other via a shared edge when the x value changes to x = 0.75, and an ixiolite orthorhombic ZnTiNb₂O₈ phase exists. This phase can also be treated as the formation of an equimolar solid solution between ZnNb₂O₆ and TiO₂. However, it should be noted that the disappearance of a superlattice diffraction peak (040) indicates the complete disorder of the solution structure, where Zn/Ti/Nb atoms are randomly distributed in 4c sites, coordinating with six oxygens to form [MO₆] (M = Zn/Ti/Nb) octahedra, as shown in Fig. 41. Furthermore, when the x value reaches 0.45, the phase structure changes into rutile tetragonal Zn_0.15Nb_0.3Ti_0.55O₂, where the cations are also six-fold coordinated and in a 2a site.

Fig. 40 Crystal structure of columbite ZnNb₂O₆ (reprinted with permission from ref. 225. Copyright 2019 John Wiley and Sons).

Fig. 41 Crystal structure of ixiolite ZnTiNb₂O₈.

It is necessary to investigate the structural impacts via P–V–L bond theory due to the three types of crystal structures showing large diversities in microwave dielectric properties. Based on a consideration of bonding environments and charge distributions, the binary bonding formulas for the three structures can be summarized as:^224,225

Detailed comparisons between bond ionicity and lattice energy can be found in Table 8S in ESI.† Here we summarize the average bond ionicity, total lattice energy and microwave dielectric properties of (Zn_1/3Nb_2/3)_xTi_1−xO₂ (x = 0.45, 0.75, 1) systems in Table 18. From Table 8S† and Table 18, it can be seen that the columbite structure is presented with a much lower average bond ionicity Af_i value, while the largest Af_i is shown in the rutile structure. More specifically, f_iNb–O bonds in the three types of structure are largest, followed by Ti–O bonds and Zn–O bonds. The lattice energy U value for each bond also shows similar variation. The largest U_total in the columbite ZnNb₂O₆ system shows excellent structural stability; correspondingly, its intrinsic loss would be a minimum. It is proved that Nb–O bonds make about a 68% contribution to dielectric polarization in the ZnNb₂O₆ system, emphasizing the importance of Nb–O bonds.

Table 18 Comparisons between average bond ionicity, total lattice energy and microwave dielectric properties in (Zn_1/3Nb_2/3)_xTi_1−xO₂ (x = 0.45, 0.75, 1) systems

x (mol)	Structure	Af i (%)	U total (kJ mol^−1)	ε r	Q × f (GHz)
C, I and R refer to columbite, ixiolite and rutile structures. The U value is normalized based on the formula (Zn1/3Nb2/3)xTi1−xO2.
1	C	74.04	13 532	23.1	72 885
0.75	I	76.06	13 145	37.2	41 231
0.45	R	76.52	12 872	94.4	10 889

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Furthermore, far-infrared reflectivity spectra were applied to further testify to the intrinsic dielectric properties in (Zn_1/3Nb_2/3)_xTi_1−xO₂ (x = 0.45, 0.75, 1) systems; taking x = 1 as an example, the infrared spectrum is presented in Fig. 42.²²⁵ The real part and imaginary part of the complex dielectric constant fitted using a classical three-parameter model (LTPC) and a four-parameter semiquantum model (FPSQ) are shown in Fig. 43.²²⁵ Both these fitting processes indicate an intrinsic low ε_r value of about 23.3 and an ultra-high Q × f value of 122 807 GHz. Similarly, intrinsic dielectric properties for ixiolite ZnTiNb₂O₈ (ε_r = 32.16, Q × f = 40 151 GHz) and rutile Zn_0.15Nb_0.3Ti_0.55O₂ (ε_r = 97.63, Q × f = 10 443 GHz) are also close to experimental values, indicating that the intrinsic factor may have a dominant effect on microwave dielectric performance.²²⁴

Fig. 42 Experimental and fitted far-infrared reflectivity spectra of ZnNb₂O₆ ceramic (reprinted with permission from ref. 225. Copyright 2019 John Wiley and Sons).

Fig. 43 Real ε′(ω) and imaginary parts ε′′(ω) of relative permittivity after (a) LTPC and (b) FPSQ mode fitting. FPSQ, four parameter semiquantum model; LTPC, Lorentz three parameter semiquantum model (reprinted with permission from ref. 225. Copyright 2019 John Wiley and Sons).

4.11.2 Ionic modification in (Zn_1/3Nb_2/3)_xTi_1−xO₂ (x = 0.45, 0.75, 1). In the three typical structures, columbite ZnNb₂O₆ and ixiolite ZnTiNb₂O₈ ceramics are mostly studied for their considerable dielectric properties with moderate ε_r values. However, a number of investigations have been undertaken to improve the negative τ_f values (−52 ppm per °C for ixiolite). For instance, in ZnTiNb₂O₈ ceramics, Mg²⁺, Co²⁺, Ni²⁺, Ca²⁺/Sn⁴⁺ and Ta⁵⁺ cations were used to substitute the cation site, and most studies have shown that the ceramic system with ionic doping presents an increase in ε_r value and the τ_f value could be adjusted to the positive side. Interestingly, based on previous reports, it is the formation of secondary rutile Zn_0.15Nb_0.3Ti_0.55O₂ that regulates the dielectric properties.^226–229

In our previous study, combined with the phase diagram, Zn_0.15+0.35xTi_0.55−0.05xNb_0.3+0.7xO_2+2x ceramics (shortened to xZTN-(1-x)ZNT) were planned through directly introducing Zn_0.15Nb_0.3Ti_0.55O₂ into Zn_0.5Ti_0.5NbO₄ ceramics.²²⁴ The purpose of this design is to understand the phase evolution and variations in bond features in both orthorhombic and tetragonal solid solution under the change in ionic occupancies in the absence of other donors. As shown by the XRD patterns in Fig. 44, orthorhombic and rutile solid solutions are formed in the ranges of x = 0–0.2 and x = 0.65–1. An increase in x value obviously leads to an expansion in cell volume and an enlargement in bond length. In section 4.3, it is noticed that occupancies in Li, Mg and Ti cations have different impacts on bond parameters. Thus, based on consideration of P–V–L bond theory, the binary bonding formula A_mB_n should vary with ionic occupancy, which can be found in Table 9S in ESI,† and the m value makes a large difference even in the same structure. Variations in average bond covalency are shown in Fig. 45. An increase in the x value in solid solution regions leads to a monotonous decrease in Af_c value for all types of bonds. However, a small variation in Af_c value is not enough to support the decrease in dielectric polarizability. Therefore, as indicated by the drop in total bond susceptibility value from 4.04 to 2.98 in the rutile structure and from 3.34 to 2.84 in the ixiolite structure, it is confirmed that the drop in average bond covalency is responsible for the decline in the ε_r value. Additionally, in the rutile solid solution, the variations in d_apical/d_equatorial reflect the cation rattling and ε_r value. Similarly, the ratio varying from 1.0089 to 1.0083 still supports the decrease in dielectric polarization. Intrinsically, the lattice energy shows a linearly increasing trend in both rutile and ixiolite solid solutions, showing an improvement in structural stability. It can clearly be seen that with the variation in Zn/Ti/Nb cations, the Q × f value is reasonably improved (from 10 889 to 13 017 GHz in rutile and from 42 181 to 47 829 GHz in ixiolite).

Fig. 44 X-ray diffraction patterns for all the compositions of xZTN-(1−x)ZNT ceramics for x = 0–1 (reprinted with permission from ref. 224. Copyright 2018 American Chemical Society).

Fig. 45 Average bond covalency as a function of the x value in the R-ZNT phase and O-ZTN phase (reprinted with permission from ref. 224. Copyright 2018 American Chemical Society).

It can be concluded that in orthorhombic Zn_0.5Ti_0.5NbO₄ and tetragonal Zn_0.15Nb_0.3Ti_0.55O₂ solid solutions, when the system is doped with larger sized cations, variations in dielectric polarization are affected by bond covalency.

Another case in point is the Zr_1−xZn_0.5xTi_1−0.5xNb_xO₄ ceramic, where both ZrTiO₄ and Zn_0.5Ti_0.5NbO₄ belong to the same orthorhombic structure. With the increase in x value, both volume cell and bond length are increased. The covalency of all types of bonds declines in the range x = 0.1–0.5, corresponding to a drop in bond susceptibility. As a result, the ε_r value decreases as well. Similarly, the structural stability of Zr_1−xZn_0.5xTi_1−0.5xNb_xO₄ is improved from 24 150 to 25200 kJ mol⁻¹, accompanied by an increase in Q × f value from 24 000 to 30558 GHz.²³⁰

In the rutile Zn_0.15Nb_0.3Ti_0.55O₂ system, the larger sized Zr⁴⁺ cation was used for Ti site substitution²³¹ in the Zn_0.15Nb_0.3(Ti_1−xZr_x)_0.55O₂ system, where the rutile phase transforms into the orthorhombic in the range x = 0.3–0.4. Still, the increase in cell volume and bond length eventually leads to a drop in covalency for all types of bonds, and the theoretical and experimental ε_r values decrease.

From the above cases, one can find that when the cation sites in Zn_0.5Ti_0.5NbO₄ and Zn_0.15Nb_0.3Ti_0.55O₂ systems are replaced with larger sized ions, there is a proportional relationship between average bond covalency and ε_r value.

However, different situations occur if the cation site is doped with smaller sized ions. For instance, in our previous study, Ta₂O₅ was chosen as a substitution source to replace Nb sites in a rutile Zn_0.15Nb_0.3−xTa_xTi_0.55O₂ (0 ≤ x ≤ 0.3) system,²³² where an infinite solid solution can be formed because of the close size of the radii between Nb⁵⁺ and Ta⁵⁺ cations. With the substitution of Ta⁵⁺ cations, cell volume and bond length decrease, leading to decreasing tendencies for all types of bond ionicity. Due to the asynchronous change in bond length, the ratio of d_apical/d_equatorial shows a decreasing trend from 1.0086 to 1.0075, suggesting a decreasing trend for the ε_r value. Thus, the bond ionicity is reasonably correlated with the ε_r value, as indicated in Fig. 46. The lattice energy consistently increases along with the x value in x = 0–0.2, corresponding to a great improvement in Q × f value from 11 012 to 22 050 GHz.

Fig. 46 ε _r and average Nb-site bond ionicity of Zn_0.15Nb_0.3−xTa_xTi_0.55O₂ (0 ≤ x ≤ 0.3) ceramics (reprinted with permission from ref. 232. Copyright 2019 Elsevier).

Furthermore, Luo used complex ions (Al_0.5Nb_0.5)⁴⁺ with small size to partially replace Ti sites in a ZnTi_1−x(Al_0.5Nb_0.5)_xNb₂O₈ system and found that increasing structural stability has a positive relationship with bond covalency, where a high bond covalency indicates a low dielectric loss. By using THz time domain spectra to study the intrinsic dielectric properties, it is found that the ε_r value and dielectric loss in the THz region both present a declining trend; the variation of ε_r value is shown in Fig. 47.²³³

Fig. 47 Dielectric constant of ZnTi_1−x(Al_0.5Nb_0.5)_xNb₂O₈ ceramics in the microwave and terahertz range (reprinted with permission from ref. 233. Copyright 2019 John Wiley and Sons).

We may infer from the reports that in Zn_0.5Ti_0.5NbO₄ and Zn_0.15Nb_0.3Ti_0.55O₂ systems, variations in bond covalency should play a dominant role in determining the ε_r value when the cation site is replaced with large sized ions; otherwise, it is the bond ionicity value that is proportional to dielectric polarization. Still, this conjecture should be tested with large amounts of experimental results in the future.

4.12 Tetragonal trirutile A_0.75Ti_0.75Ta_1.5O₆ structure (A = Co, Ni, Zn, Mg)

It is reported that in the substitution of the ZnTi(Nb_1−xTa_x)₂O₈ system, the τ_f value is largely compensated for due to the existence of trirutile phases. Recently, pure tetragonal trirutile ceramics with chemical formula A_0.75Ti_0.75Ta_1.5O₆ have been investigated from the point of view of crystal structure and chemical bond features.

In general, the trirutile structure is regarded as an enlargement of the rutile structure along the c-axis length. Typical types of trirutile structures are shown in Fig. 48.²³⁴ All the cations are surrounded by six oxygens. However, the bond types and ionic occupancies are presented with large differences even though they share the same structural type, leading to different bonding formula expressions (A = Ni is similar to A=Co):²³⁴

Fig. 48 The schematic diagram of the crystal structure of (a) Ni_0.75Ti_0.75Ta_1.5O₆, (b) Co_0.75Ti_0.75Ta_1.5O₆, (c) Zn_0.75Ti_0.75Ta_1.5O₆, (d) Mg_0.75Ti_0.75Ta_1.5O₆ ceramics (reprinted with permission from ref. 234. Copyright 2020 Elsevier).

Therefore, it is found that Ta–O bonds have larger bond ionicity values than Ti–O and A–O bonds. The largest lattice energy also shows the same tendency. Combined with the bond susceptibility value, this indicates that Ta–O bonds provide the majority contributions of about 60% and 70% to dielectric polarization and structural stability, illustrating the significance of Ta–O bonds in a trirutile structure.²³⁵ In the synthesis of A_0.75Ti_0.75Ta_1.5O₆ (A = Co, Ni, Zn, Mg) ceramics, Wang also found a proportional relationship between the bond ionicity value of Ta–O bonds and the ε_r value, where with the increase in ionic radius, the increase in f_iTa–O value is mainly responsible for the ε_r value. The lattice energy, showing the maximum value in the A = Ni system (60941 kJ mol⁻¹), also corresponds to the largest Q × f value (25 051 GHz).

5. Summary and future work

The primary goal of applying P–V–L bond theory to microwave dielectric ceramics is to provide guidance for the exploration of practical novel systems and ion doping modifications on the basis of comprehensive knowledge of crystal structure and chemical bond properties. With the help of the binary bonding formula theory of complex polycrystals, it is very advantageous to provide basic chemical bond characteristics, such as bond ionicity, bond covalency and bond susceptibility. By applying these fundamental traits, the structural evolution and developments of microwave dielectric properties can be explained reasonably well. For instance, a preliminary understanding of the polarization ability and structural stability of crystals can be gained on the condition of knowing the crystal structure and structural parameters alone. Intrinsically, the crystal structure would be more lattice stable with a higher lattice energy. Far-infrared reflectance spectroscopy and THz time-domain spectroscopy may possibly be effective methods for examining the evolution of intrinsic dielectric properties. On the other hand, some conjectures based on experimental results are proposed, which require a large number of experiments to verify and supplement them in the future. For instance, it is found that Mg sites being doped using larger radius ions results in a similar variation tendency between bond ionicity and dielectric constant, while smaller radius ion doping at Ti sites is beneficial for improving the structural stability of and reducing the internal dielectric loss in cubic rock salt Li₂Mg₃TiO₆ systems. For most niobate systems, substitution of Nb sites by Ta ions obviously improves the structural stability of the crystal and further positively affects the Q × f value. Therefore, with the conclusions put forward, P–V–L bond theory can provide ideas for subsequent ion doping in other systems.

However, P–V–L bond theory is predictable theoretically, while the actual situation is still subject to influences from external factors, such as the experimental/fabrication process. Therefore, the use of bond theory is meaningful on the premise of improving the sinterability of materials and reducing the loss from external processes (such as the use of spark plasma sintering). Furthermore, we know that chemical bonds in polyhedra will be in stretched or compressed states, leading to discrepancies in ionic polarizabilities deviating from theoretical ionic polarization, as a result, affecting the dielectric constant of dielectric materials. In a binary system, the specific polarization ability of each kind of ion in a specific crystal can be determined according to bond theory given the crystal structure and dielectric constant. With the help of the binary bonding formula and calculation of bond susceptibility, the dielectric constant of a bond can be used to theoretically calculate the ionic polarizability and volume polarizability of the ions in the complex crystal. Combined with normalized chemical bond analysis, the variations in specific polarizability in the crystal can be quantitatively characterized, which is of great significance to discussing the change in dielectric polarization in microwave dielectric ceramics.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

Reproduced from ref. 199 with permission from the PCCP Owner Societies; Reproduced from ref. 217 with permission from the Royal Society of Chemistry. The authors acknowledge financial supports from the National Natural Science Foundation of China (Grant. 51872037). The authors would also like to thank Prof. Shu Zhang for her helpful discussions on the P–V–L bond theory.

Notes and references

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Footnote

† Electronic supplementary information (ESI) available. See DOI: 10.1039/d0qi00907e

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