The latest process and challenges of microwave dielectric ceramics based on pseudo phase diagrams

The explosive process of 5G communication evokes the urgent demand of miniaturized and integrated dielectric ceramics filter. It is a pressing need to advance the development of dielectric ceramics utilization of emerging technology to design new materials and understand the polarization mechanism. This review provides the summary of the study of microwave dielectric ceramics (MWDCs) sintered higher than 1000 from 2010 up to now, °C with the purpose of taking a broad and historical view of these ceramics and illustrating research directions. To date, researchers endeavor to explain the structure-property relationship of ceramics with multitude of approaches and design a new formula or strategy to obtain excellent microwave dielectric properties. There are variety of factors that impact the permittivity, dielectric loss, and temperature stability of dielectric materials, covering intrinsic and extrinsic factors. Many of these factors are often intertwined, which can complicate new dielectric material discovery and the mechanism investigation. Because of the various ceramics systems, pseudo phase diagram was used to classify the dielectric materials based on the composition. In this review, the ceramics were firstly divided into ternary systems, and then brief description of the experimental probes and complementary theoretical methods that have been used to discern the intrinsic polarization mechanisms and the origin of intrinsic loss was mentioned. Finally, some perspectives on the future outlook for high-temperature MWDCs were offered based on the synthesis method, characterization techniques, and significant theory developments.


Introduction 
Over the past half century, semiconductor integration the huge amount and a wide variety of components with different functions are passive devices. The core materials of these components are various types of functional ceramic materials. Microwave dielectric ceramics (MWDCs) are the pivotal component of a passive device, which are mainly used as filters, resonators, RF antennae, frequency discriminators in electronic countermeasures, navigation, radar, home satellite live television receivers, and hand-held mobile phones. The applications of MWDCs in different frequency are directly plotted in Fig. 1. However, the development of microwave ceramics had gone through a sluggish procession because of the lack of suitable materials for dielectric resonator. The discovery of rutile (also known as titanium dioxide ceramics) in the 1970s makes it possible to synthesis dielectric resonator [1]. Various literature has been reported to explore the potential candidates of MWDCs after that, from single oxide, binary oxide, to ternary oxide. According to the data in the Web of Science, over 1000 papers were published about MWDCs around the world after 2000. Figure 2 presents the trend of published papers where more than 30% of investigations belong to China.
To evaluate the dielectric properties of ceramics, the relative permittivity (ε r ), dielectric loss (loss tangent or quality factor (Q×f value)), and temperature coefficient of resonant frequency (τ f ) are the three pivotal characteristics. As early as in 2006, the direction of development of microwave dielectric materials has been highlighted by Ohsato et al. [2], including high Q and low ε r ceramics for millimeter-wave application, high Q and high ε r ceramics for base station, and high ε r ceramics for miniaturization of mobile phone. Up to now, researchers have explored hundreds of ceramics to enrich the database of MWDCs, but only a dozen of those ceramics with unique properties have been commercially used to fabricate relevant devices because most of the ceramics lack stability or generate large loss in the electronic components. Booming development of millimeter technology and 5G communication have rendered a new round of requirement of MWDCs of low permittivity with a stable dielectric loss in the scope  of frequency up to 100 GHz. Especially, the emergency of COVID-19 makes video conferencing and telecommuting as a daily part in our lives. Consequently, the unprecedented growth of global data volume and huge demand for high data rates urge researchers to search more alternative materials for commercial electronic market. It is also a very significant issue for the industry to yield ceramics with ultra-low permittivity which are suitable for 5G and 6G communication system. However, it is still a "try and error" state in our experiments for discovering materials or optimizing the properties of the reported ceramics. The main difficulty in the development of MWDCs is to understand the fundamental relationship of compositionstructure-property and draw general trends throughout the field, after normalizing and comparing the various results. Despite long-term sustained attempts, there is no systematic or comprehensive theory which can provide common guidance in the experiments and drive currently reported ceramics toward commercialization applications.
With the exploration of MWDCs clusters and the development of modern experiment techniques, investigations about MWDCs have been largely scoped by the designs and search for new systems and reoptimizing their properties. It is paramount that an MWDCs candidate has an appropriate dielectric constant, low dielectric loss, and near-zero temperature coefficient of resonant frequency for applications. Generally, to tune the microwave dielectric properties, there are two parts that should be taken into consideration (extrinsic and intrinsic parts). Extrinsic part is usually regarded as the influence originated from the synthesis method and raw materials. MWDCs usually prepare by solid state reaction method, and the sintering conditions directly influence the microstructure and compactness of ceramics, which subsequently affect the microwave dielectric properties. Meanwhile, the selectivity of size distribution, purity, non-stoichiometric ratio, species of different compounds, and pretreatment of raw materials based on their physical and chemical properties are crucial for reaching optimal microwave dielectric properties. For example, the procedures to reduce pores are designed for ceramics containing the volatile element, evolving non-stoichiometric ratio in the chemical formula, and providing the compensation atmosphere of volatile element. The relevant attempts are mostly discussed for the rock salt structure ceramics such as Li 2 Mg 3 TiO 6 . Besides, various synthesis methods, namely sol-gel method, sink plasma sintering method, and high energy ball-milling method are gradually used for preparing the MWDCs, and numbers of studies analyze the discrepancy of microwave dielectric properties obtained with different methods. The intrinsic part stems from anharmonic lattice vibration, which primarily generates large dielectric loss. As yet, there is no technology or theory that could feasibly adjust the anharmonic lattice vibration to reduce dielectric loss. In the experiment, after carefully controlling the sintering conditions and selecting raw materials, the most pragmatic approach to optimize the properties is cation substitution with the consideration of the radii and the electronegativity of cations, contributing to reducing the dielectric loss or modifying the temperature coefficient of resonant frequency. Near-zero temperature coefficient of resonant frequency is also obtained by designing co-exited phase system with introduction of two ceramics with opposite τ f values, but the composite ceramics may lead to a poor Q×f value. More recently, the strategy of tri-layer structures of Zn 1.01 Nb 2 O 6 /TiO 2 /Zn 1.01 Nb 2 O 6 [3], MgTiO 3 /TiO 2 /MgTiO 3 [4], and Zn 3 Nb 2 O 8 /TiO 2 / Zn 3 Nb 2 O 8 [5] were verified as a method to obtain the temperature-stable ceramics with low dielectric loss.
Currently, the database of MWDCs is enriched by insightful information about the structure and properties, and the growing number of literature converts from description of phenomena to explanation of the theoretical mechanism of the dielectric materials. Thorough and comprehensive investigation of ceramics is gradually presented to estimate the extrinsic and intrinsic influence on the microwave dielectric properties. For instance, the common discussion of polarization mechanism is usually based on the ionic polarization, where the Clausius-Mossotti (C-M) equation is applied to evaluate the discrepancy of theoretical ε r and measured ε r . The popularization of Rietveld-refinement in the literature supports the analysis of lattice parameters, packing fraction, and chemical bond characteristic obtained by the complex chemical bond theory (P-V-L) theory. Especially, disassembling the crystal into the sum of sample binary compound based on the crystal parameters and coordinate numbers of each ions [6], the investigations about application of P-V-L theory into multi-type structure emerge in abundance. The origin of dielectric loss is quantified by lattice vibrational spectroscopy, and the contribution of each chemical bond to the microwave dielectric properties is verified by P-V-L theory. For some unique ceramics, researchers bend themselves to exploring the underlying mechanism for the observed phenomenon. The influence of long-range movement of charged defects in the grain and grain boundary was estimated by the impedance analysis, terahertz (THz) time-domain spectroscopy analysis, and the electron paramagnetic resonance spectra, which can explain the defect generation mechanism in doped Li 2 ZnTi 3 O 8 ceramics. The analysis of disordered-ordered crystal structure evolution and super-lattice in rock salt ceramics and complex perovskite ceramics gives evidence to explain the ultra-low dielectric loss. Both the development of experimental and theoretical method allows us to summarize the relevant experimental probes of different systems and propose the challenges and prospects of MWDCs.
While many great review and perspective articles have been published about MWDCs, they have finished the review by classified MWDCs based on the criteria of sintering temperature, dielectric constant, and crystal structure [1,[7][8][9]. Furthermore, the early works before 2010 are mainly concentrated on the description of phase composition, micrographic images, and variation of microwave dielectric properties. The topic about the MWDCs sintered lower than 950 is ℃ especially focused owing to the advantages of low-temperature co-fired ceramics (LTCC) technology where this approach guarantees the integration of electronic components. Considering either the timespan or topic covered, the mentioned ceramics, in this review, are all sintered higher than 1000 . The ℃ LTCC system including ceramics with a few sintering aids, glass-ceramics system, or glass-free system is not referred. To follow the development of new analysis methods, MWDCs, beginning with the first reported properties and upgrading the relevant references after 2010, were included. Additionally, because of so various structures and properties of MWDCs, pseudo phase diagram was used to classify the ceramics according to the composition, which will serve as the basis and link for each pseudo phase diagram of diversity composition. The organization of this review consists of a brief section detailing the phase evolution or structure transformation of oxide ceramics in the designed pseudo phase diagram, and then the chronological experimental probes for a unique system are summarized.

Phase diagram
The phase diagram is a visual representation of the phase equilibrium, which defines the composition of multiphase system. It is an efficient and convenient technique to analyze the composition and their proportion, which plays a significant role in guiding the research and exploration of materials to reduce the manpower and material resources effectively. This section provides a broad context by summarizing the ceramics system based on pseudo phase diagram, and all the composition in the following pseudo phase diagram is in molar ratio. The endpoint of each pseudo phase diagram contains more than one component, and the labelled ceramics are the primary system reported by researchers. The summary of investigations in the same general formal is listed in detail.

1 Silicate and germanate
There is a low ε r (< 10) for silicates, owing to the low ionic polarizability of Si 4+ and half covalent bond in Si-O. In the binary silicate, the CaSiO 3 , Mg 2 SiO 4 , Zn 2 SiO 4 , and Re 2 SiO 5 are the main representatives, where CaSiO 3 usually appeared as the crystalized phase in CaO-B 2 O 3 -SiO 2 glass. Ternary silicate such as diopside-type CaMgSi 2 O 6 , melitile-type A 2 BC 2 O 7 and AB 2 C 2 O 7 (A = Sr, Ca, Ba; B = Mg, Zn, Co, Mn, Cu), and cuspidine-type Ca 3 SnSi 2 O 9 were highlighted by researchers, due to the diverse crystal structures in those systems. With the wake of exploration of new ceramics, the germanate gradually occurred as a candidate material with low dielectric loss despite of the expensive cost of GeO 2 as raw material. The pseudo phase diagram of the silicate and germanate is presented in Fig. 3, where the primary phases of binary and ternary silicate and germanate are listed in the phase diagram.

Binary silicate ceramics
Synthesis of dense SiO 2 ceramic is challengeable because of its complexity in polymorphs and phase transitions. Until 2012, microwave dielectric properties of SiO 2 ceramic were reported as ε r ≈ 3. 81 [48,49].
The unpresented ternary silicate and germanate phase in pseudo phase diagrams are summarized as well in this section. The rare earth-based ternary silicate oxides, such as apatite with general formula A 10 (MO 4 ) 6 O 2 (A = alkaline earth, rare earth, Pb; M = Si, Ge, P, V), have received much attention since the apatite structure allowed numbers of substitutions at all the three sites. The lattice parameters and the local charge compensation of apatite type compounds were determined in 1972 [50], and those ceramics were established as candidate of fluorescent lamp phosphors and laser technology. To improve the densification of lithium apatite LiRe 9 (SiO 4 ) 6 O 2 ceramics (Re = La, Pr, Nd, Sm, Eu, Gd, Er), relative density was higher than 90% for all samples after doping 1 wt% LiF [51]. The microwave dielectric properties of SrRE 4 Si 3 O 13 (RE = La, Pr, Nd, Sm, Eu, Gd, Tb, Dy, Er, Tm, Yb, and Y) were in the range of 9-16 for permittivity with the maximum of Q×f value ≈ 26,000 GHz [52], while the optimal microwave dielectric properties of CaRE 4 Si 3 O 13 (RE = La, Nd, Sm, and Er) were ε r ≈ 13.37, Q×f value ≈ 18,600 GHz, and τ f ≈ -17.8 ppm/ at Re = Er [53].
℃ To obtain new dielectric materials, some researchers pursued materials with the composition containing GeO 2 and Ga 2 O 3 and reported microwave dielectric properties of those materials firstly. With inverse spinel structure, LiGa 5 O 8 was verified as a cubic structure where Li + and Ga 3+ distributed in the octahedral B site with 1:3 ordering [54]. The large deviation between ε r and ε rth in Ba 2 MGa 11 O 20 (M = Bi, La) was ascribed to the "rattling" effect of cations and the existence of lone pair ions of Bi 3+ [55]. The different τ f values of AGe 4 O 9 (A = Ba, Sr) were ascribed to the distortion of [GeO 6 ] octahedron where τ f values were -44.2 ppm/ ℃ for the former and -11.7 ppm/ for the later [56]. ℃ Normal garnet A 3 Y 2 Ge 3 O 12 (A = Ca, Mg) ceramics possessed τ f ≈ 120.5 ppm/ for A = Ca and ℃ -40.6 ppm/ ℃ for A = Mg [57]. As doped ions, (Li 0.5 Ga 0.5 ) 3+ in Mg 2 Al 4 Si 5 O 18 would obtain the highest Q×f value of 50,560 GHz [58]. Ca 3 M 2 Si 3 O 12 (M = Yb, Y) ceramics were consistent with the general formula of garnet structure, and those ceramics crystalized as silico-carnotite structure with high-energy ball milling method [59]. The microwave dielectric properties were recorded as ε r ≈ 9.  [60][61][62]. Similarly, Sr 3 B 2 Ge 3 O 12 (B = Yb, Ho) were investigated by Li et al. [63] using vibration spectroscopy, and the τ f was tuned to near zero with CaTiO 3 ceramics. 0.8Y 3 MgAl 3 SiO 12 -0.2TiO 2 ceramic sintered at 1475 ℃ showed a τ f ≈ +5.2 ppm/ , where ℃ the co-existed phase contained Y 2 Ti 2 O 7 and TiO 2 along with Y 3 MgAl 3 SiO 12 phase [64]. Dense Mg 3 Ga 2 GeO 8 ceramics presented microwave dielectric properties of ε r ≈ 9.41, Q×f value ≈ 133,113 GHz, and τ f ≈ -63.54 ppm/ [65]. Single phase LiYSiO ℃ 4 ceramics could be obtained in 1100-1140 , and a near ℃ -zero τ f of (+4.52)-(+8.03) ppm/ was observed [66]. ℃ Furthermore, phase transition from A2/a to P2 1 /a was observed in new silicate in the formula of CaSn 1-x Ti x SiO 5 , where the variation of τ f values was ascribed to the Sn/TiO 6 octahedral distortion [67]. Secondary phase of SnO 2 and SrSiO 3 appeared at 0.2 ≤ x ≤ 0.45 in Ca 1-x Sr x SnSiO 5 ceramics, which could adjust the positive τ f of CaSnSiO 5 to -1.2 ppm/℃ [68]. CaSiO 3 and CaSnSiO 5 phases would improve the τ f to -7.2 ppm/ in Ca ℃ 2 (Hf 1-x Sn x )Si 4 O 12 when x = 0.4 [69].

2 Niobate and tantalate based on ZnO-Nb 2 O 5 -TiO 2
There is a large body of niobate and tantalate dielectric ceramics, and the relevant researches highlight the phase evolution, structure transformation, and chemical www.springer.com/journal/40145 bond characteristics. In order to elucidate the influence of undercoordinated sites on the dielectric properties, analysis according to P-V-L theory and vibration spectra is verified as valid approach to understand the relationship of the state of chemical bond with polarization and stability of lattice. Indeed, it seems that researchers could identify the contribution of each chemical bond to dielectric properties by P-V-L theory and infrared reflectivity spectrum. However, reaching general conclusions about the effect of a unique chemical bond or Wycoff site on different properties may be difficult, since the P-V-L theory is just predictable theoretically. The actual dielectric properties of ceramics are still evaluated based on experiments, and thorough, quantitative, and multiperspective analysis is required. Figure 4 is the phase diagram of the mainly reported niobate and tantalate dielectric ceramics, where the rutile-type, ixiolitetype, and columbite-type structures were obtained after (Zn 1/3 Nb 2/3 ) 4+ was doped into TiO 2 . The detailed phase division of A 0.5 B 0.5 CO 4 and the relevant investigations of this binary system are summarized in the following.

Rutile-trirutile/ixiolite/wolframite-columbite type ceramics
Rutile, brookite, and anatase are the three types of TiO 2 in nature. TiO 2 itself possesses a high permittivity ≈ 100, a low dielectric loss tangent (tanσ) value (6×10 -5 at a frequency of 3 GHz), and a high τ f value of 450 ppm/ [70]. It is valid that TiO ℃ 2 phase is used to target the aim of near zero τ f value as a secondary phase in the system with a negative τ f value. Meanwhile, long-term focus has been paid on the structure transformation and property optimization of TiO 2 with substitution ions of different physicochemical properties. The cation substitution for Ti 4+ can reduce the dielectric loss or tune the τ f value, evolving monovalent, divalent, trivalent, tetravalent, or pentavalent cations, and their groups of two cations. Especially, the extensive elaboration of dependence of microwave dielectric properties on the crystal structure of (Zn 1/3 B 5 2 + /3 ) x Ti 1-x O 2 (B 5+ = Nb, Ta) ceramics was reported by Kim and Kang [71]. The phase relation of ternary system of ZnO-TiO 2 -Nb 2 O 5 was first discussed in 1992 [72]. It summarized that the solid solution of rutile phase appeared in the range of molar content of (Zn 1/3 Nb 2/3 ) 4+ lower than 58%, ixiolite ZnTiNb 2 O 8 exited in the range of 69%-85%, while columbite solid solution of ZnNb 2 O 6 formed when the content was higher than 93% [73], and the solid solution area of different types was marked with shadow in the pseudo phase diagram in Fig. 4 [78], where the expansion of bond length and cell volume renders the decline of covalency of all bonds. The decline of bond ionicity was obtained since the shrinking of cell volume and bond length in Zn 0.15 Nb 0.3-x Ta x TiZr 0.55 O 2 [79].
℃ In the family of AB 2 O 6 (A = Ca, Mg, Mn, Co, Ni, Zn; B = Ta, Nb), the relationship of permittivity with electronegativity was presented by Lee et al. [137]. Two structure classifications have been identified in this system, namely rutile-type (trirutile) and α-PbO 2type (tri-α-PbO 2 , columibite) [138,139]. Comprehensive studies of columbite niobates concluded that the ε r was in the range of 17-22, τ f value varied from -45 to -76,  [140,141]. The investigations about property optimization and preparation methods were concentrated on MgNb 2 O 6 , ZnNb 2 O 6 , and ZnTa 2 O 6 due to their potential of application. For sintering behavior, the sintering temperature can be reduced to 1150 of ℃ MgNb 2 O 6 by sol-gel method [142]. Doped ceramics of (Zn 1-x Ni x )Ta 2 O 6 [143], Zn(Ta 1-x Nb x ) 2 O 6 [144], Zn(Ta 1-x Sb x ) 2 O 6 [145], and composite ceramics composed of ZnO- and (1-x)ZnTa 2 O 6 -xNiNb 2 O 6 were designed successfully to reach near-zero τ f value [146][147][148][149]. Liu and Deng [150] proposed that the grain size of ZnNb 2 O 6 -(Zn 0.7 Mg 0.3 )TiO 3 ceramics became smaller with the ZnNb 2 O 6 content increasing. The secondary ZnV 2 O 6 was observed with higher than 1 wt% V 2 O 5 into ZnNb 2 O 6 [151]. The property comparison of MgTa 2 O 6 was obtained by sol-gel procession and solid reaction sintering by Wu et al. [152]. Liu et al. [153] verified that the unpaired d-electrons contribution to the room temperature loss should be taken into consideration of ZrTiO 4 -ZnNb 2 O 6 . It was interesting that the structure transformation was identified as tri-α-PbO 2 , α-PbO 2 , trirutile, and rutile in (1-x)ZnTa 2 O 6 -xTiO 2 along with the increase of x [154]. ZnNb 2 O 6 ceramics prepared by microwave sintering exhibited relative density of 94.3%, and the quality factor was dominated by the distribution of grain size [155]. Recently, the intrinsic dielectric properties were investigated using chemical bond theory and lattice vibrational spectra, which indicated that B 1u mode at 168.87 cm -1 was highly related to the dielectric properties [156], and the fitted results of infrared-related spectrum are presented in Fig. 7.

ReTiCO 6 ceramics
The crystal structure of double tantalates of rare-earth elements with titanium tantalite compounds based on ReTiTaO 6 is sorted into two parts: orthorhombic aeschynite symmetry with rare earth atomic number in the range of 55-66 and orthorhombic euxenite symmetry with that of 67-71 [157,158]. Generally, high ε r and positive τ f were obtained for the former, while relatively low ε r and negative τ f were observed for the latter. The effect of microstructure on properties of RETiNbO 6 (RE = La, Sm, and Y) ceramics was presented by Lei et al. [159]. The dielectric constant of RETiNbO 6 system (RE = Ce, Pr, Nd, Sm, Eu, Gd, Tb, Dy, Y, and Yb) and RETiTaO 6 (RE = La, Ce, Pr, Nd, Sm, Eu, Gd, Tb, Dy, Ho, Y, Er, Yb, Al, and In) increases with the RE ionic radius [157,158]. It was reported that LaTiNbO 6 usually stabilized as a monoclinic structure, and Zhang and Zuo [160] proposed that ceramics with coexistence of O and M phases could be achieved after prolonging the annealing time. And then, they [161][162][163][164] conducted out the substitutions for La and Nb sites, in which the structure evolution, octahedral distortion, and vibrational spectrum were elaborated in detail. More recently, dielectric and optical properties of Ln 0.8 Lu 0.2 TiNbO 6 (Ln = Ce, Pr, Nd, and Sm) were presented by John and Solomon [165], where the optimal microwave dielectric properties were shown for Sm 0. 8 [167][168][169][170][171][172][173][174][175], and Nb site by Ta, Sb [176][177][178][179] were completed to adjust the microwave dielectric properties. In our recent reports, the groups of different isovalent cations of (A x B 1-x ) 5+ (A = Mg, Al, Si, Zr; B = W, Mo) have been listed as valid substitution for Nb site to reduce dielectric loss [180][181][182][183]. The analysis of combination of P-V-L theory and vibration spectrum suggested that doping into Nb site was beneficial to improving quality factor. Meanwhile, NdNbO 4 prepared in sol-gel method or composite ceramics composed of NdNbO 4 -CaTiO 3 [184], NdNbO 4 -CaF 2 [185], and NdNbO 4 -MgO [186] have also been reported to perfect the properties. Similarly, intrinsic dielectric properties of EuNbO 4 were studied by Liu et al. [187]. In the full range of La 2 O 3 -Nb 2 O 5 -V 2 O 5 system, four typical phase regions were verified, including monoclinic fergusonite, tetragonal sheelite, B-site ordered sheelite, and composite of monoclinic LaVO 4 and tetragonal sheelite phases [188].
Likewise, MgO was designed as an addition for LaNbO 4 ceramics and the excellent properties were listed as ε r ≈ 19.8, Q×f value ≈ 94,440 GHz, and τ f ≈ 6.1 ppm/ [189]. More recently, structure ℃ -property relationship of another A 3+ B 5+ O 4 binary oxide, zircontype AVO 4 (A = Eu, Y) ceramics, was discussed by packing fraction and bond valence [190]. Ferroelastic phase transition from monoclinic fergusonite to tetragonal scheelite was observed by in situ Raman spectroscopy and X-ray diffraction of La(Nb 0.9 V 0.1 )O 4 , and the schematic of ε r typical-ceramics versus temperature was shown by Zhou et al. [191]. NiO/CoO added into LaNbO 4 would distinctly optimize the quality factor since the larger and uniform grain was obtained [192]. Although the thermal properties [193][194][195][196] and the first-principles calculation of electronic structure and optic properties of RETaO 4 (RE = Y, La, Sm, Eu, Dy, Er) [197] have been investigated, the intrinsic dielectric loss has not been summarized in this system. Microwave dielectric properties of ErNbO 4 prepared by sol-gel method were reported by Devesa et al. [198], and the grain size varied from 31.27 to 86.65 µm and 40.96 to 78.23 µm by Rietveld refinement and Sherrer's formula, respectively. ZrTiO 4 followed the general formula of ABO 4 , and the intrinsic dielectric loss of Zr 0.8 Sn 0.2 TiO 4 was investigated by THz time domain spectroscopy [199].
The structure of corundum-like phase of Mg 4 Nb 2 O 9 was verified by Kumada et al. [200], where the cations were ordered by the stack of two layers of a mixture of Mg and Nb and one layer of Mg along the c-axis. Mg 4 (Nb 2-x Ta x )O 9 solid solution was synthesized in the sintering temperature range of 1350-1400 [201], ℃ which possesses a comparable quality factor (Q×f value ≈ 350,000 GHz for x = 2) to that of Al 2 O 3 . To deal with the limitation of high sintering temperature, both Mg 4 Nb 2 O 9 and Mg 4 Ta 2 O 9 were generated by sol-gel method and the variation of property with www.springer.com/journal/40145 sintering temperature was analyzed [202][203][204] [206,207], which presented that the appearance of Mg 4 Nb 2 O 9 pure phase was more easily with Mg(OH) 2 as raw materials. A dramatically improvement of quality factor was achieved by Ni and Ta co-doped into this system, and (Mg 0.95 Ni 0.05 ) 4 (Nb 1-x Ta x ) 2 O 9 shows satisfied properties of ε r ≈ 12.76, Q×f value ≈ 442,000 GHz, and τ f ≈ -54 ppm/ , when ℃ x = 1 and sintered at 1375 [208]. 5+ substitution at Nb 5+ site (B = Li, Mg, Al, Ti) in Mg 4 Nb 2 O 9 -based ceramics revealed that the τ f depended on the distortion of the oxygen octahedra, while (Ti 1/2 W 1/2 ) 5+ substitution had the highest quality factor of 233,000 GHz [209]. The investigation of y(Mg 0.95 Co 0.05 ) 4 Ta 2 O 9 -(1-y)CaTiO 3 ceramics provided a promising dielectric material for application with temperature stability, and the properties were shown as ε r ≈ 25.78, Q×f value ≈ 200,000 GHz, and τ f ≈ -4.69 ppm/ [210]. ℃ Zn 3 Nb 2 O 8 was another promising binary niobite compound, which could be successfully produced with 98% theoretical density sintered at 1100 [211]. A ℃ two-stage sintering method was proposed to optimize the microstructure of Zn 3 Nb 2 O 8 [212], where the sintering temperatures were 1150 and 1200 for the ℃ first time and the second sintering temperatures were 1050 and 1100 , respectively. Sintered based on this ℃ approach, ceramics presented denser grain packing and less abnormal grain growth. Adding secondary phase into ceramics to compensate for τ f value would introduce a large amount of second phase, which were ascribed to the large dielectric loss. Aiming to reduce the defects stemmed from the secondary phase, layercofired ceramic architectures were designed such as Zn 1.01 Nb 2 O 6 /TiO 2 /Zn 1.01 Nb 2 O 6 [3], MgTiO 3 /TiO 2 /MgTiO 3 [4], and Zn 3 Nb 2 O 8 /TiO 2 /Zn 3 Nb 2 O 8 [5]. High-Q value was remained and temperature-stable MWDCs were obtained for all the reported tri-layer co-fired ceramics.

3 Rock-salt structure
Closely followed by the ever-growing explosion of global data volume and the rapid boost of millimeterwave technology, the requirement of materials with low permittivity (ε r ≤ 25) and high Q×f value is increasingly urgent. In the exploration of new composition ceramics, many rock-salt Li-containing compounds emerge as focal points. The general formula of rock-salt ceramics is A a B b O a+b (A + = Li, Na; B 4+ = Ti, Sn, Zr; B 5+ = Nb and Ta). Li 2 TiO 3 underwent an order-disorder phase transition at 1213 , in which ℃ the structure consisted of ordered (Li,Ti) layer, with the property of ε r ≈ 12.76, Q×f value ≈ 44,200 GHz, and τ f ≈ -54 ppm/ [213]. The sintering behavior of ℃ excess Li for non-stoichiometry Li 2+x TiO 3 ceramics was investigated by Bian and Dong [214] and Hao et al. [215] after the determination of pseudo-binary of Li 2 O-TiO 2 [216,217]. For co-doped substitution, Zn 1/3 Nb 2/3 , Mg 1/3 Nb 2/3 , and Co 1/3 Nb 2/3 addition into Li 2 TiO 3 could adjust the τ f from positive to negative [218][219][220]. Cu 1/3 Nb 2/3 doped ceramics with 3 wt% H 3 BO 3 were designed as a patch antenna and a dielectric resonator antenna [221]. The solid solution of Li 2 Table 1 [224,, and the phase diagram of rock-salt structure is plotted in Fig. 8, where the ordered-disordered range was summarized from Zhang et al. [246,247]. Simultaneously, Gu et al. [248] stated the two-phase and thermally stable ceramics of 0.8Li 3 NbO 4 -0.2Ca 0 [250], and a near zero τ f (-4.03 ppm/ ) was ℃ obtained for 0.1 mol Zn substitution for Mg [251].   [253].
Except for the sintering temperature, the heating rates and substation will directly influence the grain size, densification, and properties. Lu et al. [254] pointed out that the sintering rate increasing from 3 to 7 /min would deteriorate the quality f ℃ actor of Li 2 ZnTi 3 O 8 ceramics. If ball milling is applied for the raw materials at first for 4 h, then the sintering temperature of Li 2 ZnTi 3 O 8 ceramics could reduce from 1075 to 925 , and those ceramics were chemically ℃ compatible with Ag [255]. Sintering the ceramics in a box type electric furnace and in a microwave furnace would obtain Li 2 ZnTi 3 O 8 ceramics with the grain size of 38 and 7 µm, respectively [256]. Mg, Co, and Zn substitution for Zn in Li 2 ZnTi 3 O 8 increased the quality factor because of the more compact microstructure [257][258][259]. Whereas, the secondary phases were recorded after the introduction of Sr 2+ or (Sr x Ca 1-x ) into Li 2 ZnTi 3 O 8 [260][261][262]. Phase evolution of (1-x)Li 2 ZnTi 3 O 8 -xTiO 2 system indicated that pure Li 2 ZnTi 3 O 8 with cubic structure was observed when x ≤ 0.2 (the lattice parameter is similar to MgFe 2 O 4 with space group of Fm3m (227)), solid solution was exited in the range of 0.2 ≤ x ≤ 0.4 with cubic structure (the lattice parameters is similar to Zn 2 Ti 3 O 8 with space group of P4332 (212)), and rutile TiO 2 phase appeared when x ≥ 0.6 [263]. The τ f value moves from -15 to 102.4 in (1-x)Li 2 ZnTi 3 O 8 -xTiO 2 (0 ≤ x ≤ 0.4) [264]; meanwhile, near zero τ f value was also achieved by Bari et al. [265] in this system. 4 wt% TiO 2 was added into Li 2 ZnTi 3 O 8 with different particle sizes, where the nanoparticles and micron particles all generated a more uniform microstructure and relative density reached to 98.5% [266]. Similar to TiO 2 -doped Li 2 ZnTi 3 O 8 ceramics, phase composition and properties of Li 2 Mg(Ti 1-x Sn x ) 3 O 8 (x = 0.1-0.25) were concluded as with 0.10 ≤ x ≤ 0.15, the spinel and rutile were co-exited; with 0.20 ≤ x ≤ 0.25, the spinel, rutile, and ilmenite were obtained [267], and the optimal properties of Li 2 ZnTi 3 O 8 -0.2SnO 2 composite ceramics exhibited: ε r ≈ 20.9, Q×f value ≈ 89,500 GHz, and τ f ≈ -24 ppm/ [268]. The ℃ variation of dielectric properties with density of (1-x)Li 2 (Mg 0.95 Zn 0.05 ) 3 Ti 3 O 8 -xLi 2 TiO 3 (x = 0.727, 0.778, 0.821, and 0.889) was discussed systematically by Zhang et al. [269]. The concentration of oxygen vacancy, relative density, and decrease in damping behavior would influence the Q×f value of Li 2 ZnTi 3 O 8 -x wt% Nb 2 O 5 [270]. To trace the dielectric response of lattice vibration, the response process of dielectric loss in Li 2 ZnTi 3-x M x O 8 (M = Al 3+ , Nb 5+ , (Al 0.5 Nb 0.5 ) 4+ , www.springer.com/journal/40145 (Zn 1/3 Nb 2/3 ) 4+ , and (Li 1/4 Nb 3/4 ) 4+ ) was discussed systematically containing the conduction loss and lattice vibration loss [271]. The conduction loss which acts at frequency lower than terahertz is neglectful by researchers concentrating on MWDCs, while AC impedance analysis was used to identify the effect of dopants and the mechanism of conduction loss in this system. Combining the fitting THz time domain spectrum and far infrared reflectivity spectrum, the dielectric response was illustrated in depth based on lattice loss and conduction loss.
Ultra-low loss microwave dielectric materials of Li 2 Mg 3 TiO 6 -based ceramics are extensively studied via doping cations into Mg and Ti site. Bivalent cations [272] [275][276][277] proposed a reliable method which provided the Li-rich sintering atmosphere, and they obtained serial MWDCs based on Li-Mg-Sn/Ti oxides with excellent properties. The schematic representation of the devices provided with the Li-rich atmosphere is shown in Fig. 9, and this similar method was gradually popularized to other systems with volatilization element to obtain the ceramics with dense microstructure. The negative τ f values can be compensated by Ca 0.8 Sr 0.2 TiO 3 , and the sample with 0.91Li 2 Mg 3 TiO 6 -0.09Ca 0.8 Sr 0.2 TiO 3 showed a τ f value of -3.65 ppm/ [278]. ℃ The phase evolution of Li 2 O-3MgO-mTiO 2 (1 ≤ m ≤ 6) was summarized as the phase diagram shown in Fig. 10 [298] or non-stoichiometric Li 3 Mg 2+x SbO 6 [299] have been probed and analyzed to explain the variation of dielectric properties through current theory including P-V-L theory, packing fraction, and C-M equations. It was interesting that the "dark hole" phenomenon of Li 2 TiO 3 was cured by adding Li 3 Mg 2 NbO 6 and the τ f value of 0.96Li 2 TiO 3 -0.04 Li 3 Mg 2 NbO 6 was 2.6 ppm/ [300]. Since yet there ℃ was no literature about the structure transformation of Li 2 TiO 3 -Li 3 NbO 4 -MgO to renew the understanding of rock-salt ceramics, Zhang et al. [247,301,302] gradually updated the reports about Li 3 Mg 2 NbO 6 -based ceramics. The phase transitions among the orthorhombic, cubic, and monoclinic were verified by XRD (Fig. 12) and TEM analysis (Fig. 13). The systematical analysis of lattice evolution and ordering transformation indicated that the low dielectric loss of this system was mainly ascribed to the superlattice. The THz time-domain spectroscopy was firstly used in this system to evaluate the intrinsic dielectric loss associated with phonon oscillation. Meanwhile, the configurational entropy was calculated to explain the change of disordered and ordered crystal structures, where the disordering cubic phase generated much larger configurational entropy than the ordered orthorhombic and monoclinic phase (Fig. 14).

4 Tungsten bronze structure and titanate based on BaO-TiO 2
Since 1970, the exploration of BaO-TiO 2 system has been continuous renewed. Among them, BaO-4TiO 2 and 2BaO-9TiO 2 are the most extensively investigated ceramics as the representative ceramics with medium dielectric constant. The pseudo phase diagram of tungsten bronze structure and binary system based on BaO-TiO 2 system is shown in Fig. 15. In contrast to other sections in this review, the investigations about the compounds within this phase diagrams are relatively less, because the study of ceramics in BaO-R 2 O 3 -TiO 2 (R = La-Gd) has been almost accomplished and widely used in the industry.

BaO-TiO 2 /Nb 2 O 5 /Ta 2 O 5 system and Re 2 TiO 5
The frequency dependence of Q×f value was observed for Ba 2 Ti 9 O 20 ceramics, which was ascribed to the extrinsic dielectric loss [304].  [307], which showed ε r ≈ 11-51, Q×f value ≈ 2400-88,000 GHz, and τ f ≈ (-73)-232 ppm/℃. Based on sol-gel method, Mg 5 Nb 4 O 15 nano-powders were obtained at 600 , ℃ and then the sintering temperature can be reduced to 1300 [204]. On the basis of P ℃ -V-L chemical bond theory, the relationship of chemical bond characteristic and microwave dielectric properties of Eu 2 TiO 5 was discussed deeply [308]. Meanwhile, the electron localization function (ELF) based on the first-principles calculation was evaluated to provide the information of bond covalency [309], which provided a strategy to estimate the chemical bond characterization.

Tungsten bronze structure
The different compositions of tungstenbronze-type with Ba 6-3x R 8+2x Ti 18 O 54 solid solution reported by Ohsato [310] in 2001, and the compounds were presented in Fig. 16. The relative permittivity of BaO-R 2 O 3 -TiO 2 (R = La-Gd) was higher than 80, and the solid solubility region became narrower as the ionic radius of rare earth increasing [311]. The doping effect and the determination of crystal structure of Ba 6-3x R 8+2x Ti 18 O 54 were summarized in the review of dielectric materials for wireless communication [1]. After 2010, there are only a few studies focused on this system. Three distinct phases were formed using variable size TiO 2 reagents into BaO-Nd 2 O 3 -TiO 2 [312]. Ba 6-3x R 8+2x Ti 18 O 54 (BRT, R = La, Pr, Nd, Sm) solid solution family was reported with high permittivity. When x = 2/3, Ba 4 Nd 9.33 Ti 18 O 54 was regarded as the most investigated ceramics to lower its τ f value and sintering temperature or improve its Q×f value. Yao et al. [313] and Chen et al. [314] proposed that with Al 2 O 3 added BaO-Nd 2 O 3 -TiO 3 ceramics, the Q×f value would increase obviously. The temperaturestable ceramics could be obtained by Pb and Sr substitution for Ba 3 [325], and Ba 4 (Pr 1-x Sm x ) 28/3 Al 4y/3 O 54 [326]. Among those reports, the analysis of Raman spectrum of Ba 3.75 Nd 9.5 Ti 18-z (Al 1/2 Nb 1/2 )O 54 enriched the theoretical study of tungstenbronze-type.

5 Perovskite related structure
The ideal perovskite (written as ABO 3 ) is cubically symmetric with a space group of Pm3m, and the represented material is SrTiO 3 . Due to the flexibility of ABO 3 perovskite, variants of perovskite have been investigated, and the classification of perovskiterelated structure with representative structure is summarized in Fig. 17. The perovskite-related structure contained cubic-type, orthorhombic-type, and hexagonaltype structures. For hexagonal-type structure, the twinned hexagonal structure means the closely packed AO 3 layers were stacked in the order of (ccch) 2 , while the shifted hexagonal structure corresponds to ccchhccc order. The typical representative of twinned structure is Ba 8 CoTa 6 O 24 and the shifted structure is Ba 8 CoNb 6 O 24 with eight-layer hexagonal perovskite structure [332]. The pseudo phase diagram of ABO 3 and complex ABO 3 type is provided in Fig. 18. From cubic and orthorhombic to hexagonal perovskite structure, researchers have proposed that tolerance factor, distortion of octahedron, and temperature of phase transition determined the variation of τ f value, and the ordered/ disordered cations were primarily related to quality factor.
With the same general formula of ABO 3 , the investigations of NdGaO 3 , NdNbO 3 , and AgTa/NbO 3 are listed adjacently to CaTiO 3 and SrTiO 3 . Phase composition was identified for NdGaO 3 -Bi 0.5 Na 0.5 TiO 3 system, and new temperature-stable ceramics with 0.4NdGaO 3 -0.6Bi 0.5 Na 0.5 TiO 3 was obtained [428]. Order-disorder transformation of A-site-deficient perovskites plays a significant role in conductivity of materials. The investigation of crystal structure and dielectric properties of the Nd (1-x)/3 M x NbO 3 (M = Li, Ag; 0 ≤ x ≤ 0.2) suggested that the dielectric loss majored by the lithium or silver ionic conduction at low frequencies [429]. Solid solution of AgNb/TaO 3based ceramics was then studied extensively [430,431]. Temperature-stable MWDCs with the formula of (La,Nd) 2/3 TiO 3 were studied by Saleem et al. [432].

A 2 B′B″O 6 formula
Due to the flexibility and adjustability of the crystal structure of perovskite, the investigation of complex perovskite with various cations occupying Ti site gradually emerged. The structural studies of A 2 B′B″O 6 (A = Ba, Sr, Ca; B′ = lanthanide, Mg, Cr, Bi; B″ = Nb, Ta, Sb, W) indicated that phase transitions were ascribed to the tilting of B′O 6 /B″O 6 . In the Ba 2-2x Sr 2x SmSbO 6 system, phase transitions of Fm3m, I 2 /m, and P2 1 /n were observed and the τ f value shifted from +25 to -50 ppm/ [473]. Effect of non ℃ stoichiometry Ba 1+x (MgW) 1/2 O 3 , Ba(Mg 1+y W) 1/2 O 3 , and Ba(MgW 1+z ) 1/2 O 3 and the sintering temperature on microwave dielectric properties was systematically investigated by Wu and Bian [474] and Chen et al. [475], respectively. A zero τ f value ceramic was obtained in Ba 2 Mg 1-x Ca x WO 6 system with x = 0.1 [474]. First-principles calculation of assignment for vibrational spectra of Ba(Mg 1/2 W 1/2 )O 3 MWDCs is shown in Fig. 21 [476], which proposed that F 1u (2) modes originated from Mg-O 6 vibrations had the largest contribution to the dielectric properties. The investigation of microwave dielectric properties of giant permittivity ceramics with a A 2 B′B″O 6 formula (Ba(Fe 1/2 Nb 1/2 )O 3 and Sr(Fe 1/2 Nb 1/2 )O 3 ) indicated that the permittivity was independent of frequency [477].

A(B′ 1/3 B″ 2/3 )O 3 formula
℃ The variation of τ f values for 1:1 and 1:2 complex perovskites was clarified to be mainly relevant with tolerance factors, which are summarized in Fig. 24 [524]. It has been verified that samples with non-stoichiometric Mg 2+ and Ta 5+ in Ba(Mg 1/3 Ta 2/3 )O 3 exhibited a wide temperature stability [525,534], and the correlations between Q×f versus ε r and τ f versus ε r of high-Q (≥ 100,000 GHz) MWDCs are presented in Fig. 25.

A n B n O 3n+2 formula
Perovskite-related oxides of series A n B n O 3n+2 = ABO x (x = 3+2/n) (A = Ca, Sr, or La and B = Ti or Nb) with n = 4, 4.33, 4.5, 5, 6, and 7 have been a focus owing to their electronic and dielectric properties. The crystal type and the physical properties rely on the value of n, which descripts the number of octahedral layers in the slabs [535]. Besides Ca 5 TaO 11 is an member of n = 3 in this series, and the textured La 3 Ti 2 TaO 11 was fabricated by spark plasma sintering, showing that grain-orientation control was an effective way to tailor the properties of this ceramic [541]. SrCa 4 Nb 4 TiO 17 and Ca 5 Nb 4 TiO 17 sintered at their optimal temperature presented an elongated and plate-like grain [542]. From 0 to 4, the τ f value shifted from -117 to 415 in NaCa 4-x Sr x Nb 5 O 17 [543], while the τ f value changed in the range of (-117)-473 ppm/ for ℃ Na 1-x K x Ca 4 Nb 5 O 17 [544].

Ca 4 La 2 Ti 5 O 17
The dielectric properties of Ca 4 La 2 Ti 5 O 17 were firstly reported by Rejini et al. [545], which were crystalized as perovskite structure and the XRD results were matched well based on the formula of Ca 0 [551,552] through spectroscopic methods. In the microwave frequency region, the Q×f value and τ f values of Ba 8 (Mg 1-x Zn x )Ta 6 O 24 ceramics decreased with the augment of x [553]. Similarly, a single phase with hexagonal 8H perovskite structure of Ba 8 Ti 3 Nb 4-x Sb x O 24 ceramics was prepared, and τ f value declined from 110 to 2 ppm/ [554]. ℃ BaWO 4 was used to adjust the large τ f value of 8H hexagonal perovskite Ba 4 LiNb 3 O 12 , and the properties of ε r ≈ 16.9, Q×f value ≈ 75,500 GHz, and τ f ≈ +8.7 ppm/ were obtained [555]. Phase transformation in ℃ the sequence of hexagonal, hexagonal along with cubic, and cubic was observed in Ba 4 LiNb 3-x Sb x O 12 and Ba 4 LiTa 3-x Sb x O 12 system. Especially, the optimal microwave dielectric properties were achieved for Ba 4 LiNb 2 SbO 12 with a zero τ f [556,557]. τ f value dropped from positive to negative in Ba 3 LiTa 3-x Sb x Ti 5 O 21 [558], and Ba 3 LiNb 3-x Sb x Ti 5 O 21 [559], while the τ f value just reduced from 205 to 70 ppm/ for Ba ℃ 3 LiNb 3-x Ta x Ti 5 O 21 [560]. A-site deficient perovskite structure was well matched for LiSmTa 4 O 12 ceramics with tetragonal perovskite structure (A-site deficient perovskite structure), and the optimal microwave dielectric properties were ε r ≈ 59.60, Q×f value ≈ 7760 GHz, and τ f ≈ +41.8 ppm/℃ [561].

Sr n+1 Ti n O 3n+1 (n = 1, 2, 3 4, ∞) formula
Researchers paid their attention to Ruddlesden-Popper (R-P) structure until the dielectric properties of CaReAlO 4 (Re = Nd, Sm, Y) were reported. The general formula of R-P compounds was written as (A,A′) n+1 B n O 3n+1 , where the structure was built by corner-sharing (BO 6 ) octahedral and interlayer of ((A,A′)O). MLnAlO 4 and SrLn 2 Al 2 O 7 (M = Ca, Sr; R = Y, Sm, Nd, La) belong to the R-P series with n = 1 and 2, respectively. The crystal structures of SrLaAlO 4 and SrLa 2 Al 2 O 7 are presented in Fig. 26. Single crystals of ABCO 4 layered compounds with K 2 NiF 4 structure were used as substrates for high-temperature superconductive thin films, while dielectric properties in this system were mainly investigated by Chen and his co-workers [562][563][564][565][566][567][568][569][570][571][572][573][574]. They contributed to analyze the relation between the intrinsic dielectric properties and crystal structure of MRAlO 4 (M = Ca, Sr; and R = Y, Sm, Nd, La). Combining the compression/dilation effects of different cation-oxygen bonds and the stability of crystal structure with vibrational spectrum, they emphasized that the drop of the quality factor was ascribed to the abnormal variations of axial bonds and the theoretical dielectric loss was obtained after fitted the infrared reflectivity spectra. With (Zn 0.5 Ti 0.5 ) 3+ substituted for Al 3+ of SrLaAlO 4 , the best combination of microwave dielectric properties was ε r ≈ 23.5, Q×f value ≈ 102,000 GHz, and τ f ≈ -3.4 ppm/℃ [572]. In the SrLaAlO 4 -Sr 2 TiO 4 system, some diffraction peaks shifted toward higher angles along with the variation of x, while some of them shifted toward lower angles, as shown in Fig. 27 [569]. This phenomenon was explained by the opposite change of a-axis and c-axis, where the octahedron elongated in the ab plane with the shrinkage in the c direction. The tolerance factor (t) of perovskite layer was used to evaluate the stability of those compounds, and the relation of t and r(M 2+ )/r(Ln 3+ ) was plotted in Fig. 28 [573]. Sr 0.6 Ca 0.4 LaAlO 4 with 10 wt% TiO 2 presented a near zero τ f ≈ -2.5 ppm/ [575].   On the other hand, the R-P structure such as Sr n+1 Ti n O 3n+1 (n = 1, 2) [576], SrLn 2 Al 2 O 7 (Ln = La, Nd, Sm) [577][578][579][580][581], was also established as K 2 NiF 4 structure. The interlayer polarization was verified to influence the microstructure and internal stress, and the complete structure information of SrLn 2 Al 2 O 7 ceramics was obtained by TEM. Solid solution of (Sr 1-x Ca x ) 2 TiO 4 [582], Sr 2 Ti 1-x Sn x O 4 [583], Sr 2 [Ti 1-x (Al 0.5 Nb 0.5 ) x ]O 4 [584], and (Sr 1-3x/2 La x ) 2 Ti 1-y Ce y O 4 [585] was prepared to reduce the large τ f value of Sr 2 TiO 4 . Moreover, Sr 2 CeO 4 was obtained by Dai and Zuo [586], and the substitution of Ti 4+ for Ce 4+ in Sr 2 CeO 4 generated a ceramic with excellent properties of ε r ≈ 20.7, Q×f value ≈ 115,550 GHz, and τ f ≈ -1.8 ppm/ . ℃

6 Other system and machine learning in MWDCs
Although the pseudo phase diagrams contain various primary systems, some ceramics such as CeO [591]. Vibrational spectroscopy and microwave dielectric properties of Ca 3 Ln 2 W 2 O 12 (Ln = La, Sm) were analyzed by Liu and Song [592], and the ε r of those two phases were 18.7 and 19.5. Ln 2 MoO 6 (Ln = La, Y) ceramics possessed a relative permittivity of 14.1-17.1, and the quality factor was 67,090 GHz for La 2 MoO 6 and 27,760 GHz for Y 2 MoO 6 , respectively [593].
In the wake of the update of computer science, date-driven approaches including data mining and machine learning have been applied in many disciplines for obtaining the obscure quantitative relationships. For material science, machine learning was used to realize the property prediction, composition optimization, and experimental design [594][595][596][597][598][599][600]. Qin et al. [601] employed five commonly-used algorithms with 32 intrinsic chemical, structural, and thermodynamic features for modeling to predict low permittivity materials, where a database of 3300 materials has not been reported and the distribution of permittivity in virtual space of materials was shown in Fig. 29. Quantitative prediction of the Q×f value of gillespitetype ACuSi 4 O 10 (A = Ca, Sr, Ba) ceramics was obtained by machine learning, and the results of (Ca x Sr 1-x )CuSi 4 O 10 and (Ba y Sr 1-y )CuSi 4 O 10 ceramics matched well with the experimental Q×f value, as shown in Fig. 30 [602].

Conclusions and further outlook
MWDCs with a suitable permittivity, low dielectric loss, and temperature stability are a perpetual pursuit for researchers. Those ceramics offer technoeconomic advantages including integration, lightweight, and reliability. With the continuous exploration, significant progress is presently being made in designing new compounds, analyzing the polarization mechanism  along with the origin of dielectric loss, and predicting the microwave dielectric properties by theoretical model of machine learning. The relevant computational and experimental methods currently used to probe, predict, and understand intrinsic mechanisms are covered in this review. Because target ceramic system and their associated investigations are so diverse, we provide a brief classification on the composition of ceramics using pseudo phase diagram. The exploration of substitution of the given ceramics or new compounds is listed briefly following the pseudo phase diagram. Experimentally, it appears that substitution and composite ceramics are the most common used methods to optimize the microwave dielectric properties for a given system (reduce dielectric loss or adjust the τ f value to near zero). The previous doping researches are concentrated on single ion substitution, while more development of the co-doping (group of two aliovalent cations with a certain mole ratio) appears recently. For the probe of new dielectric materials, the new system usually belongs to germanate and gallate, besides the familiar system of silicate, titanate, niobate, and tantalate. Comparing with conventional solid state reaction method, fabrication techniques containing solution-processed sol-gel method, high energy ball milling method, spark plasma sintering, and microwave sintering have been demonstrated as the promising approaches to improve the properties or sintering behaviors so far. Providing the atmosphere with the volatile element in the sintering procession is a valid method to reduce the pores. Multi-layer ceramic architecture has been verified as a design for temperature-stable ceramics, and the wide application for more system or in the industry is waiting for the exploration.
The influence factor of microwave dielectric properties evolves extrinsic and intrinsic parts. The defects such as porosity, microstructure, and secondary phase are related to the relative density and grain size, which are extrinsic factors. Those results of a unique ceramics can be easily obtained by XRD and SEM, while the investigation of dielectric responded mechanism of intrinsic part is difficult due to the restrain of characterization techniques and the lack of general theory. Theoretically, from Clausius-Mossotti equation, packing fraction, cation valence, distortion of octahedron to the combination of P-V-L theory, lattice dynamics, and THz time-domain spectroscopy with the first-principles calculation, the intrinsic mechanism for MWDCs is gradually created. Recent efforts to employ P-V-L theory and infrared reflectivity spectra to understanding the intrinsic mechanism seem to be an easy and potential approach to draw conclusions for prediction the microwave dielectric properties. However, the development of "try and error" situation in experiments is a long-term procession. Toward this state end, greater fundamental understanding of dielectric response mechanism and increased practical performance metrics are required. The experimental trials and theoretical calculation serve as a database of MWDCs, and then, the machine learning is applied to predict new materials and their microwave dielectric properties. There has been an emerging trend about machine learning to provide new insight to draw a general conclusion to verify the effect of each factor on the variation of microwave dielectric properties. Challenges remain in the reconciliation of conclusion between existing theoretical approaches, the evaluation of P-V-L theory on microwave dielectric properties, and the advancement of first-principles calculation for describing the state of bond. Based on the theoretical analysis of MWDCs and the careful control of extrinsic influence, more comprehensive applicationspecific analyses to justify their adoption in electronic market may be able to complete.
While there is always a need for fundamental research, the acceleration of the commercial application of new materials and property optimized ceramics is another persistent target for researchers. This includes ending the limitation of currently available system and exploration materials with stable and excellent properties for electronic market. For example, alternative materials with satisfied microwave dielectric properties equal to perovskite ceramics are required in the industry. With the development of 5G and 6G, there is an urgent need for ceramics with ultra-low dielectric constant (< 5), low dielectric loss, and excellent temperature-stability in high frequencies. The compounds of borate, aluminate, silicate, and fluoride with low polarization should take into consideration as promising candidate. It may be a direction for discovering composite materials consisted of ceramics and organics. Meanwhile, reducing the sintering temperature of ceramics for meeting the need of LTCC is a highly challenging issue owing to its advantages in fabrication of electronic devices. On the other hand, the repeatability of microwave dielectric properties and the normalized evaluation method should be emphasized. The advancement of preparation method with simplified procedures should be taken into consideration as well. The investigation combining the discussion of the performance of a simulated and fabricated device with the analysis of fundamental mechanism of structure-property relationship should be more popularized to provide an entire and systematical exploration. As a summary, the microwave dielectric properties listed in the references are presented in Figs. 31(a) and 31(b). Lastly, we hope this brief progress report helps to understand the recent experimental methods and suggests an insight to take a new research direction for MWDCs.