The chemical formula for three areas divided by two lines: α = 5γ/4 and α = γ/2, here, α and γ are in BaαTaγOα+5γ/2 and vacancies on the A-, B- and O-sites .
So far, many microwave dielectric materials have been investigated for a range of telecommunication applications. In dielectrics, the three main dielectric properties are quality factor (Q), dielectric constant and temperature coefficient of resonant frequency. Among these, the most essential dielectric property is Q. More specifically, Q is the inverse of the dielectric loss (tanδ); thus Q = 1/tanδ. There are two kinds of losses: those depending on crystal structure and losses due to external factors. The former is intrinsic losses such as ordering, symmetry, and phonon vibration. The latter is extrinsic losses due to factors such as grain size, defects, inclusions and distortion. In this chapter, the authors present the origin of dielectric losses based on the crystal structure. An ideal and well-proportional crystal structure constitutes a low loss material. Most dielectric materials are paraelectrics with inversion symmetry i and high symmetry. In general, it is believed that ordering gives rise to a high Q, on which many researchers are casting doubt. In the case of complex perovskites, the symmetry changes from cubic to trigonal. Ordering and symmetry should be compared with the structure. In this chapter, three essential conditions for the origin of high Q such as high symmetry, compositional ordering and compositional density are presented.
- microwave dielectrics
- pseudo tungsten-bronze
- complex perovskite
Microwave and millimetre-wave dielectric materials [1, 2, 3, 4, 5, 6] have been investigated for a wide range of telecommunication applications, such as mobile and smartphones, wireless local area network (LAN) modules and intelligent transport system (ITS). Millimetre-wave dielectric materials with high quality factor Q and low dielectric constant εr are required for the next 5G telecommunication applications used for noncondensed high data transfer on LAN/ personal area networks (PAN) and the higher frequency radar on autonomous cars.
In microwave dielectrics, there are three fundamental dielectric properties: quality factor (Q), dielectric constant (εr) and temperature coefficient of resonant frequency (TCf/τf) [1, 2, 6]. Microwave dielectrics have been used as the critical constituents of wireless communications [7, 8, 9, 10], such as resonators, filters and temperature-stable capacitors with a near zero ppm/°C TCεr (temperature coefficient of the dielectric constant). Among the dielectric properties, the most essential property is Q, the inversion of the dielectric loss (tanδ); thus Q = 1/tanδ. The dielectric losses of microwave dielectrics should be small. So, most of the microwave dielectrics are paraelectrics with inversion symmetry i, while most of the electronic materials are ferroelectrics with spontaneous polarity showing substantial dielectric losses [11, 12, 13]. The microwave dielectrics attract attention as a high potential material, which have an over-well-proportional rigid crystal structure with symmetry. That is, the structure should be without electric defects, nondistortion and without strain.
Under the influence of an electric field, four types of polarisation mechanisms can occur in dielectric ceramics, that is, interfacial, dipolar, ionic and electronic. In general, the microwave dielectric properties such as εr and Q are mostly influenced by ionic or electronic polarisation. The dielectric polarisation generates the dielectric losses in the presence of an electromagnetic wave. When the frequency is increased to millimetre-wave values, the dielectric losses may be increased or decreased depending on the polarisation mechanism. There are two kinds of losses: those depending on crystal structure and losses due to external factors. It was believed that the intrinsic losses are due to the ordering/disordering, symmetry and phonon vibration, while extrinsic losses are due to factors such as grain size, defects, inclusions, density and distortion from stress.
In this chapter, the origins of high Q are discussed based on the intrinsic factors related to the crystal structure, such as symmetry, compositional ordering and compositional density. Although it has previously been believed that ordering based on the order-disorder phase transition brings high Q , the authors propose that it is primarily a high symmetry that leads to high Q . The following focused studies relate to specific examples; indialite with high symmetry showing higher Q than cordierite with an ordered structure [16, 17, 18]; pseudo tungsten-bronze solid solutions without phase transition showing high Q based on the compositional ordering [19, 20, 21]; complex perovskite compounds with order-disorder transitions depending on density and grain size [22, 23] and complex perovskites with composition deviated from the stoichiometric depending on the compositional density showing a high Q [24, 25, 26, 27, 28, 29].
2. Focused studies
2.1. Indialite/cordierite glass ceramics
2.1.1. Indialite Q-factor improved by Ni-substitution
Cordierite (Mg2Al4Si5O18) has two polymorphs: cordierite and indialite, as shown in Figure 1(a) and (c), respectively [30, 31]. Cordierite is of low symmetry form: orthorhombic crystal system Cccm (No. 66), which has Si4Al2O18 six-membered tetrahedron rings with ordered SiO4 and AlO4 tetrahedra as shown in Figure 1(b). On the other hand, indialite is of high symmetry form: hexagonal crystal system P6/mcc (No. 192), which has disordered Si4Al2O18 equilateral hexagonal rings as shown in Figure 1(c).
Cordierite shows a lower εr of 6.19 which depends on the silicates and a near-zero TCf of −24 ppm/°C  as compared to other silicates as shown in Figure 2(a). Based on these properties, Terada et al. carried out initiative research on these microwave dielectrics . They reported an excellent Qf by substituting Ni for Mg as shown in Figure 2(b). The Qf was improved from 40 × 103 GHz to 100 × 103 GHz by Ni substitution of x = 0.1 in (Mg1−xNix)2Al4Si5O18. The Ni substitution did not change the εr value considerably, but the TCf was degraded from −24 to −30 ppm/°C . For x > 0.1, the properties were affected by the formation of the secondary phase of NiAl2O4.
Terada et al. also analysed the crystal structure by the Rietveld method  to clarify the origin of the improved Qf value. The X-ray powder diffraction (XRPD) pattern was obtained by a multi-detector system (MDS)  in the synchrotron radiation “Photon Factory” of the National Laboratory for High Energy Physics in Tsukuba, Japan. Figure 3(a)–(d) shows the crystal structures of Ni-substituted cordierite (Mg1−xNix)2Al4Si5O18 with x = 0, 0.05, 0.1 and 0.15. The crystal structure showed a tendency to deform to indialite with high symmetry on the hexagonal ring composed of corner-sharing of (Si, Al)O4 tetrahedra in the a-b plane. Ni-substituted cordierite (Mg1−xNix)2Al4Si5O18 with composition x = 0.1 (Figure 3(c)) was obviously closer to equilateral hexagonal rings compared to (Mg0.95Ni0.05)2Al4Si5O18 (Figure 3(b)) and Mg2Al4Si5O18 (Figure 3(a)).
The transformation from cordierite to indialite, represented by the ratio of disordering between the SiO4 and AlO4 tetrahedra, is based on the volumes and covalencies of the SiO4 and AlO4 tetrahedra . The volume was calculated using atomic coordinates obtained by Rietveld crystal structural analysis as shown above. The covalency (fc) of the cation-oxygen bond was estimated from the following equation .
The empirical constants a and M depending on the inner-shell electron number 10 are 0.54 v.u. and 1.64, respectively , where s is the bond length obtaining from the following equation:
where, R is defined as the bond length, and R1 and N are the measured parameter reliant on the cation site and each cation-anion pair, respectively.
Figure 2(c) and (d) depicts the calculated volume and covalency of SiO4 and AlO4 octahedra, respectively. These figures show the phase changing from cordierite to indialite as substitution of Ni in the Mg site. In the cordierite Mg2Al4Si5O18 (Figure 1(a)), Si/Al ions in the tetrahedra are ordered. Therefore, the volume and covalency of tetrahedra are different values, but the values are becoming similar to the substitution of Ni in the Mg site. This is due to the disordering of Si/Al ion phase transition in the cordierite (Figure 1(a)) to indialite (Figure 1(c)). In the indialite, the disordered Si4Al2O18 equilateral hexagonal rings with 6-ford axis are the main framework as analysed by the Rietveld method as shown in Figure 3(d). The improvement of Qf as shown in Figure 2(b) should be based on the disordering due to high symmetry instead of an ordering of SiO4 and AlO4 tetrahedra by order-disorder transition. It is one example of high symmetry bringing a higher Q than ordering by the order-disorder transition .
2.1.2. Indialite glass ceramics with high Q
As described in the previous section, the Qf value of indialite derived by substituting Ni for Mg was improved to three times that of cordierite. Based on the new knowledge, Ohsato et al. proposed the synthesis of indialite with superior microwave dielectric properties . The indialite, being a high-temperature form, could not be synthesised by the solid-state reaction because the order-disorder phase transition is hindered by the incongruent melting to form mullite and liquid. On the other hand, indialite is an intermediate phase during the crystallisation process from glass with a cordierite composition to cordierite, as shown in Figure 4. Therefore, fabrication of indialite glass ceramics has been attempted [17, 39]. Although the indialite is a metastable phase transforming to cordierite at higher temperatures, it is a relatively stable phase which occurs in nature formed by the crystallisation of natural glass. As this occurrence is in India, the mineral was named indialite. Another phase of μ-cordierite precipitating in the early stage of the crystallisation of cordierite glass is β-quartz solid solutions. The naming of μ-cordierite is not correct because of the different crystal structure, so the name that should be used is β-quartz solid solutions .
The cordierite composition was melted at 1550°C and was cast into a cylindrical rod with the diameter φ = 10 mm and l = 30 mm in a graphite mould. In order to avoid fracture due to internal strain, the cast glass rod was annealed at 760°C below the glass transition point of 778°C . The 10 mm diameter glass rod was cut to form a resonator with a height of 6 mm. The glass pellets were crystallised at temperatures in the range 1200–1470°C/10 and 20 h. The crystallised pellets had two problems: deformation by the formation of glass phase and cracking by anisotropic crystal growth from the surface (Figure 5(a)) . Figure 5(b) and (c) shows photographs taken by a polarising microscope of a thin section of the crystallised samples. The needle-like crystals grown from the surface had an orientation with c-axis elongation. The microwave dielectric properties of the sample with cracking had a wide scattering range of the data [17, 39].
Figure 6(a) shows the volume of indialite/cordierite examined by the Rietveld method , which is estimated with two phases such as indialite and cordierite. Hereabout, the residual % is compared to that of cordierite. At 1200°C, the precipitated phase of indialite was about 96.7%. The volume of indialite reduced as the temperature and to 17.1% (82.9% for cordierite) at 1400°C. Figure 6(b) and (c) shows the microwave dielectric properties of indialite/cordierite glass ceramics and remarkably high Qf value of more than 200 × 103 GHz at 1300°C/20 h . This is much better than the highest Qf value of 100 × 103 GHz obtained by substitution with Ni using the conventional solid-state reaction as previously described (Figure 2(b)) and is feasible for millimetre-wave dielectrics. The Qf values decreased as crystallisation temperature. In comparison with the amount of indialite as shown in Figure 6(a)  and its Qf values as shown in Figure 6(b) and (c) , it is clear that the indialite glass ceramics present a higher Qf than that of cordierite. The εr was the lowest among the silicates, about 4.7 as shown in Figure 6(b) and (c), and the TCf was −27 ppm/°C as shown in Figure 6(b). Therefore, from these figures, indialite shows a higher Qf than cordierite. This TCf value of −27 ppm/°C is better than that of other silicates having a low TCf of approximately −60 ppm/°C .
2.1.3. Conclusions for indialite/cordierite glass ceramics
Indialite/cordierite glass ceramics are one of the examples of high symmetry bringing a higher Q than ordering by order-disorder transition. Indialite glass ceramics with disordered high symmetry have higher Qf properties than cordierite with ordered low symmetry.
Cordierite with substituted Ni for Mg synthesised by solid-state reaction exhibited an improved Qf from 40 × 103 to 100 × 103 GHz (Figure 2(b)). Rietveld crystal structure analysis showed that the cordierite was transformed to indialite .
A novel idea from glass ceramics suggested the fabrication of indialite as an intermediate phase. Glass ceramics crystallised at 1200°C were almost completely indialite at 96.7% with a high Qf of 150 × 103 GHz, and those crystallised at 1400°C were cordierite at 82.9% with a lower Qf of 80 × 103 GHz. (Figure 6) [17, 39].
2.2. Pseudo tungsten-bronze solid solutions: compositional ordering bringing high Q
The pseudo tungsten-bronze solid solutions Ba6−3xR8+2xTi18O54 (R = rare earth) located on the tie-line of BaTiO3-R2Ti3O9 are shown in Figure 7(a) and have been utilised in mobile phones because of their high dielectric constant of 80–90 [20, 21]. This solid solution was first reported by Varfolomeev et al. , based on Nd and Sm systems. The composition ranges 0.0 < < x < < 0.7 for R = Nd and 0.3 < < x < < 0.7 for Sm  were reported by Ohsato et al.  and Negas et al. . The composition range of the solid solutions becomes narrower with the decrease in the ionic radius of the R-ion, and Ga and Eu form only BaO·R2O3·4TiO2 composition .
Ohsato et al. and Negas et al. reported the microwave dielectric properties for the Sm, Nd, Pr and La systems as a function of composition x as shown in the Figure 8(a) [20, 43, 45] and Fukuda et al. reported the Pr system . On the solid solutions, the composition with x = 2/3 was found by Ohsato et al. , at which the Qf value becomes the highest due to the ordering in the rhombic and pentagonal sites. The dielectric constants εr and TCf (Figure 8(b) and (c)) are decreased as a function of the composition x and are affected by volume and tilting angle of the TiO6 octahedra and the polarizabilities of R and Ba ions . The Clausius-Mosotti equation determined the temperature coefficient of the dielectric constant TCεr as a function of the ratio of the mean radii (ra/rb) of A- and B-site ions by Valant et al. . Hither, ra/rb is connected to the tilting of the TiO6 octahedra. In this study, on the system without order-disorder phase transition that is without symmetry change, it is discussed that the ordering especially compositional ordering brings high Qf.
2.2.2. Crystal structure of pseudo tungsten-bronze solid solutions
The crystal structure of the pseudo tungsten-bronze Ba6−3xR8+2xTi18O54 (R = rare earth) solid solutions [48, 49, 50, 51] includes the perovskite blocks of 2 × 2 unit cells with rhombic (A1) sites and pentagonal (A2) sites, as shown in Figure 9, which are named after the tetragonal tungsten-bronze structure with 1 × 1 perovskite blocks and pentagonal sites [20, 48, 50]. On this compound, two large sites including Ba- and R-ions are placed such as A1 and A2. The Ba-ions engaged on the pentagonal A2-sites and R-ions A1-sites on the perovskite blocks. Two more sites, B and C are positioned on the tungsten-bronze crystal structure. The B-site is the same as the TiO6 octahedral place in the perovskite, and the C-site is a triangular site which is usually empty. This crystal structure of this compound has a special relationship with the perovskite structure. If the two ions are the same size, the structure will change to perovskite with only an A1-site owing to the combination of the pentagonal A2-site and the trigonal C-sites [20, 52]. The crystal data are as follows: orthorhombic crystal system of space group Pbnm (No. 62), point group mmm and lattice parameters a = 12.13, b = 22.27, c = 7.64 Å, Z = 2 and Dx = 5.91 g/cm3.
The structure has a super lattice along the c-axis with a double lattice of perovskite as shown in Figure 7(b) of an oscillation photograph with super diffraction lines [53, 54]. The crystal data of the fundamental lattice are as follows: orthorhombic crystal system of space group Pbam (No. 55), point group mmm and lattice parameters a = 12.13, b = 22.27, c = 3.82 Å, Z = 1 and Dx = 5.91 g/cm3. The super lattice is depending on the tilting of TiO6 octahedra as shown in Figure 7(d). The tilting was endowed in the density map (Figure 7(c)) which is of the fundamental lattice superimposed on a superstructure framework. The top oxygen ions (O(1), O(4), O(6), O(8) and O(14)) of octahedra are separated into two along the c-axis. The left figure of Figure 7(d) shows the reason for splitting of the top oxygen . However, this super lattice is not depending on the order-disorder phase transition as complex perovskite as explained at 2.3 section. The tilting of octahedra might be depending on the size of A-ion in the perovskite block.
22.214.171.124. Compositional ordering
The chemical formula of the solid solutions is Ba6−3xR8+2xTi18O54, and the structural formula is [Ba4]A2[Ba2−3xR8+2x]A1Ti18O54. Here, the amount of Ba in the A1-sites becomes zero (2 − 3x = 0), that is, x = 2/3. This composition is special as the following sentence: the structural formula is [Ba4]A2[R8+4/3]A1Ti18O54 which is occupied separately by Ba in A2 and by R in A1 as shown in Figure 10(b). This special composition is called “compositional ordering” [20, 21, 42].
2.2.3. Microwave dielectric properties of pseudo tungsten-bronze solid solutions
Figure 8 shows the microwave dielectric properties of the solid solutions as a function of composition x of Ba6−3xR8+2xTi18O54 [20, 21, 42]. The quality factor (Qf) changes nonlinearly and has the highest value at particular point x = 2/3 with compositional ordering specified above . The highest Qf value might be depending on the internal strain. Figure 11(a) confers internal strain η obtained from the slope of equation βcosθ = r/t + 2ηsinθ. The internal strain η of the special point x = 2/3 is the lowest with the compositional ordering as a function of composition x as shown in Figure 11(b). The internal strain comes from the fluctuation of d-spacing of the lattice broadening the full width at half maximum (FWHM) [20, 21, 56].
The Qf value at the special point x = 2/3 shows the highest of 10.5 × 103 GHz in the Sm system, 10.0 × 103 GHz in the Nd system and 2.0 × 103 GHz in the La system as depicted in Figure 8(a) [20, 21, 56]. The Qf values reducing in the order of Sm, Nd, Pr and La are depending on the ionic radius relating size difference between Ba and R , and that of La is deviating from the Qf line through the Sm, Nd and Pr as shown in Figure 12. If the sizes are similar, the crystal structure should become perovskite structure. In the case of Sm, the difference is maximum which introduces the stability of the crystal structure. The size of La ion is similar to Ba, so the structure might be unstable to be low Qf.
2.2.4. Symmetry and ordering for Q
On the microwave dielectrics, high Q has been brought by a high potential material, which has an over-well-proportional rigid crystal structure with symmetry [11, 12, 13]. That is, the structure should be without electric defects, nondistortion and strain. Complex perovskites were described later, it is believed that ordering by long time sintering brings high Q, but we are pointing out symmetry is the predominant factor [14, 15]. In the case of indialite/cordierite, indialite with high symmetry shows higher Q than cordierite with ordering [17, 18, 39]. This case has an order-disorder phase transition. On the other hand, in the case of pseudo tungsten-bronze solid solutions which has no phase transition, one of ordering that is the compositional ordering brings high Q [20, 21]. In the case of no symmetry change, ordering is predominant.
2.2.5. Conclusions for pseudo tungsten-bronze
The pseudo tungsten-bronze solid solutions have been used for mobile phones for miniaturisation based on their high Qf and high εr.
The compound has a unique point of x = 2/3 on the Ba6−3xR8+2xTi18O54 chemical formula which shows the highest Qf value.
The special point of x = 2/3 on the structural formula of [Ba4]A2[Ba2−3xR8+2x]A1Ti18O54 is the composition at which Ba-ions disappear on the A1-sites because 2 − 3x = 0. That is the point of compositional ordering.
The compositional ordering brings high Q by maintaining the stability of the crystal structure.
2.3. Complex perovskites
There are many kinds of complex perovskites such as 1:1, 1:2 and 1:3 type in B-site and 1:1 type in A-sites . In this chapter, 1:2 type complex perovskite compounds A2+(B2+1/3B5+2/3)O3 are presented such as Ba(Zn1/3Ta2/3)O3 (BZT), Ba(Mg1/3Ta2/3)O3 (BMT) and Ba(Zn1/3Nb2/3)O3 (BZN). These complex perovskite compounds have order-disorder phase transitions (Figure 13(a) and (b)) . The ordered phase that appears at low temperatures is a trigonal (rhombohedral) structure of space group Pm1 (No. 164), and the disordered phase appearing at high temperatures is a high symmetry cubic structure of Pmm (No. 221), as shown in Figure 13 . In the ordered form of BMT, Mg2+ and Ta5+-ions located among the adjacent packing layers of BaO3 are ordering as -Mg-Ta-Ta-Mg-Ta-Ta-Mg-, as shown in Figure 13. On the other hand, in the disordered form, Mg2+ and Ta5+-ions occupy B-sites statistically.
Kawashima et al.  presented that ordering brings a high Q based on BMT with long duration sintering, which showed high Qf and ordering. This has previously been believed to be the case because long duration sintering samples generally show high Qf and ordering. However, some examples have arisen that contradict this relation, such as BMT-Ba(Co1/3Ta2/3)O3 , BMT-BaZrO3 , BMT-BaSnO3  and BZT-(SrBa)(Ga1/2Ta1/2)O3 . Koga et al. [23, 24, 25, 26] presented the relationship between high Qf and the ordering ratio as determined by the Rietveld method, the high Qf samples with disordered structure synthesised by spark plasma sintering (SPS)  and the effects of annealing of disordered BZN with an order-disorder transition point of 1350°C . HRTEM and Rietveld studies confirmed the ordering and disordering of BZN samples . Partial ternary phase diagrams such as BaO-ZnO-Ta2O5, BaO-MgO-Ta2O5 and BaO-ZnO-Nb2O5 were studied on the composition with high Qf that deviated from the stoichiometric composition of BZT/BMT/BZN by Kugimiya et al. [22, 27], Koga et al. [24, 26] and Kolodiazhnyi . Kugimiya pointed out that the solid solutions with high density and high Qf located on the tie-line BMT-Ba5Ta4O15, which have completed the ideal chemical formula without oxygen defects. It is one of the conditions for high Q that the high compositional density brings high Qf.
2.3.2. Origin of high Q for microwave complex perovskite
126.96.36.199. Ordering ratio and Qf
The ordering of BZT was observed on the samples with high Qf sintered at 1350°C  over 80 h. Figure 14 presents the XRPD patterns (a) with super lattice lines (asterisked), and the high angle diffraction patterns (b) which depicts splitting of 420 cubic diffraction peak into two peaks, namely 226 and 422 in the trigonal system. These data are consistent with the report by Kawashima et al. .
Koga et al. investigated the amount of BZT ceramic as ordering ratio by the Rietveld method , which is shown in Figure 15(a). The ordering ratio saturates at about 80%, but the Qf values increase up to 100 × 103 GHz. This shows that the effect of ordering on the Qf is not so significant. However, the Qf values are affected by density and grain size as shown in Figure 15(b) and (c), respectively [15, 23].
188.8.131.52. BZN with a clear order-disorder transition
Many complex perovskites such as BMT and BZT have the order-disorder phase transition at high temperature, and the order-disorder transition is not so clear. On the other hand, BZN shows clearly the phase transition at lower temperature 1350°C . Figure 16(a) shows Qf as a function of sintering temperature. Under the transition temperature such as 1200 and 1300°C, the sintered samples show order with under 50 × 103 GHz of Qf. Moreover, at 1400°C, higher than the transition temperature, the Qf values increased to 90 × 103 GHz with disordering structure. This shows that the high symmetry form with disorder performs higher Qf than ordering form. Moreover, the sample annealed at 1200°C/100 h transformed to order form, but the Qf value did not improve and slightly decreased. Grain size and densities as shown in Figure 16(b) and (c) also increased as the sintering temperature from 1200 to 1400°C [15, 26]. As if the sample sintered at 1400°C annealed at 1200°C/100 h, the grain size and densities were not changed. Because of annealing, the slight decrease in Qf might be a result of the low symmetry that accompanies order. On the contrary, Wu et al.  presented annealing of BZN at 1300°C brings high Qf with ordering. The annealing temperature is high enough for sintering, so sintering was proceeded with ordering the same as Kawashima’s results .
The BZN samples A and B are also studied by XRPD and HRTEM, which sintered at 1400°C/100 h above the order-disorder phase transition point and subsequently annealed at 1200°C/100 h below the transition point, respectively [26, 64]. The two samples were identified by conventional XRPD as shown in Figure 17(a). As the super lattice lines are not clear, the high angle XRPD patterns around 2θ~115° were measured (Figure 17(b)). On the XRPD pattern, the sample A shows a single peak of the 420 diffraction, so it was confirmed as disorder phase. On the other hand, the sample B shows the peak splitting of 422 and 226 depending ordering. These results are comparable with Koga’s data . These two samples were analysed by the Rietveld method.
HRTEM figures as shown in Figure 18 for most area of sample A (Figure 18(a)) and B (Figure 18(c)) are disordered and ordered area along the c direction, respectively. A fast Fourier transform (FFT) image is inserted in Figure 18(a) of a disordered area without further reflections along the c direction and in Figure 18(c) of a ordered area with additional two reflection points for super lattice. In the both sample A and B, mixed area of disordered and ordered area existed in Figure 18(b), and in the sample B, ordered area showing twin-related anti-phase domain boundary also existed as shown in Figure 18(d). The FFT image of twin area shows superimposed of ordered diffractions with four additional points.
Figure 19 depicts the high-resolution XRPD pattern of sample A and B using synchrotron radiation . The super lattice diffraction 100 t peaks (reciprocal lattice plane 100 in the trigonal crystal system) are observed in both samples. The diffraction intensity of sample A is lower than that of sample B. These super lattice diffraction intensity peaks are comparable with the ordering ratios, that is the sample A and B have the value of 27.6 and 54.2%, respectively, obtained by the Rietveld method. Although the degree of ordering of sample B is large compared to that of sample A, it was assumed about 80% ordering for a whole sample, as in the case of BZT .
It is revealed that the degree of ordering increased from 27.6 to 54.2% due to the annealing. However, the Qf values, grain size and the density have no influence on the degree of ordering (Figure 16). While the disordered area of sample A (sintered above the transitional temperature) changes to the low-temperature phase with ordering by the annealing, the Qf values were expected to be increased. However, the Qf values changed only somewhat from 95.7 × 103 GHz to 95.0 × 103 GHz . The effect of ordering is not acceptable to change the Qf value considerably.
184.108.40.206. BZT with disordering leaded high Qf by SPS
The ordered and disordered BZT ceramics can be achieved by varying the sintering duration in the conventional solid-state reaction (SSR). A high density and high Q ceramics of ordered BZT were obtained by SSR with a long sintering time of over 80 h, while the disordered BZT was not possible to fabricate by using SSR. Koga et al.  reported the high density disordered BZT ceramics for a short sintering time of 5 mins by using spark plasma sintering (SPS). Figure 20(a) presents the Qf as a function of the densities of BZT fabricated using SSR and SPS [15, 63]. The fabricated SPS samples were shown to be disordered cubic type of perovskite as depicted in the XRPD pattern (Figure 20(b)) with a peak of 420 reflection in compared with the ordered trigonal type with peaks separations of 422 and 226 when sintered using SSR (1400°C 100 h). The ceramics were sintered at the temperature between 1150 and 1300°C under 30 Mpa pressure .
This may result in the disordered BZT with a high density of 7.62 g/cm3, which is approximately 20% higher than that of low-density samples of 5.0–6.0 g/cm3 synthesized by conventional SSR. The full width at half maximum (FWHM) of the 420 peak became narrower with an increase in the temperature from 1100 to 1300°C (Figure 20(b)) indicates that the degree of crystallisation of the disordered cubic phase is improved without the need for conversion to the ordered trigonal phase. Regardless of the method of synthesis, Qf is strongly dependent on density, and Qf values were improved with increasing density. The dense disordered BZT ceramics synthesized by SPS showed a significantly high Qf (= 53.4 × 103 GHz) comparable to that of the ordered BZT with the same density (= ca. 7.5 g/cm3) synthesized by SSR. The crystallisation with densification of BZT ceramics should play a more critical role in the improvement of the Q factor in the BZT system rather than the structural ordering.
2.3.3. Deviated compositions with high Qf from stoichiometric complex perovskite composition
In a BaO-Mg/ZnO-Ta2O5 partial ternary ceramic (BMT/BZT system), complex perovskite such as BMT and BZT are forming solid solutions, and the Qf values varied intrinsically based on the crystal structure in the solid solutions depending on the density and defects. In this section, the crystal structure and properties on the varied compositions from the stoichiometric complex perovskite composition are reviewed for high Qf research.
220.127.116.11. The highest Qf composition with intrinsic compositional density by Kugimiya’s research
Kugimiya [22, 27] presented the highest Qf composition with intrinsic compositional density on the Ta and Ba rich side near the BMT-Ba5Ta4O15 tie-line in a BaO-MgO-TaO5/2 partial system (BMT system), as shown in Figure 21. He presented three areas divided by the following two lines as shown in Table 1 and Figure 21.
|α > 5γ/4||Ba1+α(Mg1/3Ta2/3+γVα −γ)O3+α+5γ/2 V2α −5γ/2||A: fill, B, O: vacancy|
|α = 5γ/4||Ba1+α(Mg1/3Ta2/3+4α/5Vα/5)O3+3α||A, O: fill, B: vacancy|
|5γ/4 > α > γ/2||Ba1+αV5γ/6 −2α/3(Mg1/3Ta2/3+γ Vα/3−γ/6) O3+α+5γ/2||A, B: vacancy, O: fill|
|α = γ/2||Ba1+αVα(Mg1/3Ta2/3+γ)O3+6α||A: vacancy, B, O: fill|
|α < γ/2||Ba1+αVγ−α(Mg1/3Ta2/3+γ)O3+α+5γ/2 Vγ/2−α||A, O: vacancy, B: fill|
Here, α and γ are as written in the formula αBaO·γTaO5/2. On the α = 5γ/4 line, Ba1 + α(Mg1/3Ta2/3 + 4α/5Vα/5)O3+3α solid solutions are formed as the ideal compositions without vacancies in the A- and O-sites. In the B-site, the vacancy is neutralized and without charge.
In Figure 21, the composition with intrinsic compositional high density shows the highest Q of 50.0 × 103 on the tie-line between BMT and Ba5Ta4O15 (𝛼 = 5𝛾/4). The contour lines in Figure 21 show Q values from 2.0 × 103 in the outer area to 25.0 × 103 in the centre. The contour is elongated parallel to the Q max line as drawn in Figure 21, and it changes steeply on the perpendicular to this line. So, the compositions without oxygen vacancy and with neutralised charge vacancies are ideal for microwave dielectrics, and the density is high due to the partial substitution of Ta in the site of Mg, which is denoted as intrinsic compositional density . Other regions have some defects degrading the Qf values, which were explained on the references [21, 22, 27, 28].
18.104.22.168. Phase conditions in the vicinity of BZT by Koga’s research
Koga et al. [24, 25] showed the highest Qf composition shifted from the stoichiometric BZT composition. The ordering ratio of the deviated composition was not higher than that of the stoichiometric composition, which was calculated by the Rietveld method. These results were presented by the study of the phase relations in the vicinity of BZT in the BaO-ZnO-Ta2O5 ternary system, as shown in Figure 22 [24, 25]. These samples were sintered at 1400°C/100 h as reported in Koga’s paper. These diffraction patterns fit the Rietveld method well [23, 24]. Ordering ratios obtained are shown in Figure 23(a). Three areas in the vicinity of BZT are presented as shown in Figure 22. 1st one (I) is ordering area with BZT single phase, the 2nd one (II) is ordering area with secondary phase and 3rd one (III) is disordering area with BZT single phase.
The first area (I) is characterised as a BZT single phase with an ordered structure and a high Qf. The varied compositions E and K have high Qf values about 50% higher than that of the stoichiometric BZT composition A. The ordering ratios at E and K are lower than that of stoichiometric BZT at A, but the density at E is the same as that of A . The second (II) is composed by an ordered BZT accompanied by a secondary phase BaTa2O6 with a specific amount of Zn determined by X-ray microanalyser (XMA). The ordering ratio in this area is high at about 70–80% (Figure 23(a)). Although the structure is ordered, the Qf values decrease in the order of A-N-O-P from stoichiometric BZT (Figure 23(b)). The ordered BZT with the secondary phase is located on the Ta2O5 rich side as a eutectic phase diagram system. The third (III) with a disordered single phase shows low Qf and low density (Figure 23(c)). The low density comes from the numerous pores.
22.214.171.124. Phase conditions in the vicinity of BZT by Kolodiazhnyi’s research
Kolodiazhnyi  also found the highest Qf of 330 × 103–340 × 103 GHz positions deviated from the stoichiometric BMT composition which is located in the BMT-Ba5Ta4O15-Ba3Ta2O8 compositional triangle (CT) as shown in Figure 24. The positions located in the single-phase BMT, which was indicated by green line. The position is close to the BMT-Ba5Ta4O15 tie-line. A to H eight CTs are formed by BMT and five stable compounds, such as Ba5Ta4O15, MgO, BaO, Ba9MgTa14O45 and Mg4Ta2O9, and three metastable compounds, Ba6Ta2O11, Ba4Ta2O9 and Ba3Ta2O8. In A, B and C-CTs, although the samples demonstrated high density and a high degree of order, they showed low Qf values, attributed to the possible presence of the Ba9MgTa14O45 second phase. Moreover, in D, E and F-CTs, as the samples were very low density, no electromagnetic resonance peaks were detected.
126.96.36.199. Koga’s and Kolodiazhnyi’s data comprehended in Kugimiya’s data
Koga’s data  and Kolodiazhnyi’s  data are comparable with Kugimiya’s BMT data . The area (I) and the H-CT with the highest Qf as shown in Figures 22 and 24, respectively, are located on the opposite side of Kugimiya’s data against the BMT-Ba5Ta4O15 tie-line (Figure 21). These compositions will be comparable with that of the ideal crystal structure Ba1+α(Mg1/3Ta2/3+4α/5Vα/5)O3+3α, as stated before in section (188.8.131.52) . The formula is rewritten as Ba(Mg1/3−α/3Ta2/3+2α/15Vα/5)O3 solid solutions on the tie-line BMT-Ba5Ta4O15. The crystal structure in the composition region is ideal, without defects, and with an intrinsic high compositional density as described above. Surendran et al.  also reported a composition with high Qf deviated from stoichiometric BMT reviewed in detail in Intech Open Access Book .
2.3.4. Conclusions: important points concerning complex perovskites
Ordering brings high Qf in the complex perovskite because of the long duration sintering. This situation has been bereaved for a long time. However, many examples contradicting this relation were presented.
Koga et al. presented that Qf values of BZT did not depend on the ordering, preferably depending on the density and grain size.
BZN with an order-disorder transition point at 1350°C (sample A) showed high Qf in the high-temperature disordered form. Moreover, annealing of the disordered sample B brings the ordered form, but the Qf does not improve. The both samples are analysed by the Rietveld method and HRTEM. The HRTEM presented the order form, disorder form and anti-phase domain by the FFT.
Disordered samples with high density could not be synthesised by the solid-state reaction, but could be by SPS. The samples with disordered structure showed high Q. The ordering phenomenon is the only barometer of sintering in the solid-state reaction.
Compositions deviated from stoichiometric complex perovskites such as BZT and BMT showed higher Qf and lower ordering than the stoichiometric composition. Based on these points, the ordering is not the reason for high Qf, and it is the only barometer of sintering.
Intrinsic compositional density brings high Qf. On the BMT-Ba5Ta4O15 tie-line, solid solutions are formed by the substitution Ta for Mg, which include high Qf compositions. The chemical composition with the highest Qf is Ba1+α(Mg1/3Ta2/3+4α/5Vα/5)O3+3α, which is an ideal solid solutions without oxygen defects and neutralised vacancies (Table 1).
Compositions deviated from stoichiometric BMT/BZT towards BaO and the Ta2O5 rich areas showing high Qf, as presented by Koga et al. , Kolodiazhny  and Surendran et al. , are comparable with intrinsic compositional density with high Qf as presented by Kugimiya .
The microwave dielectrics are the perfect, ideal and well-proportional crystal structures for low dielectric losses. Most of them belong to paraelectrics with inversion symmetry i and showing high symmetry and nondefects. In this chapter, the effects of ordering and symmetry were presented as follows: there are two types of ordering conditions. One is a case of nonphase transition such as pseudo tungsten-bronze solid solutions. These compounds show compositional ordering at a unique point of x = 2/3 on the Ba6−3xR8+2xTi18O54 system, which shows the highest Qf without degradation of crystal symmetry. The other is a case of order-disorder phase transition such as indialite/cordierite. Indialite with a disordered structure and a high symmetry of 6/mmm point group has a higher Qf than cordierite with an ordered structure and low symmetry of mmm point group. It is clarified that the effect of high symmetry is predominant for high Qf. In the case of complex perovskite, a long sintering time of more than 80 h brings a high Qf accompanying ordering. It was clarified that the ordering is not necessary for high Qf and only a barometer of sintering in the solid-state reaction. Moreover, compositions deviated from stoichiometric complex perovskite showed higher Qf than the stoichiometric composition which has substituted Ta-ions for Mg/Zn-ions. The substitution brings a high density that is the compositional density. It was clarified that high compositional density brings high Qf.
The authors are grateful to Professors and graduate students of NIT, Meijo University and Hoseo University, and Doctors and researchers in the many companies, which collaborated with NIT. Visiting Professor Hitoshi Ohsato is grateful to the following projects: (1) support industries of Japan by Ministry of Economy, Trade and Industry (METI), Japan, (2) JSPS KAKENHI Grant Number 22560673, 25420721, JP16K06735 and (3) Nokia Foundation 2016 for Nokia Visiting Professors Project 201700003. Professor Heli Jantunen and Dr. Jobin Varghese are grateful to European Research Council Project No. 24001893 for financial assistance. The authors would also like to thank Honorary Research Professor Arthur E Hill of Salford University for valuable discussion and improving English during the preparation of this manuscript.