**12. Analysis of the physical properties of MW dielectrics**

The main attention in the analysis of the physical properties of MW dielectrics is given to the temperature coefficient of permittivity (TCε) and to dielectric loss (tg δ). The expression for TCε can be derived directly from the Clausius-Mossotti equation [137]:

$$T\mathcal{C}\varepsilon = \frac{(\varepsilon - 1)(\varepsilon + 2)}{\varepsilon} \left[ \frac{\nu}{\alpha} \left( \frac{\partial \alpha}{\partial \nu} \right)\_T a\_l + \frac{1}{3\alpha} \left( \frac{\partial \alpha}{\partial T} \right)\_\nu - a\_l \right] \tag{4}$$

where al is linear thermal expansion coefficient.

136 Dielectric Material

7000).

electromagnetic scattering.

The arrows mark superlattice reflections.

**11. Multiphase MW dielectrics** 

an attractive host for engineering advanced temperature-stable microwave dielectric

**Figure 20.** [110] electron diffraction patterns of the Ba3Co1+yNb2O9+y samples with y = (a) –0.07 and (b) 0.

High Q value is usually observed in single-phase systems. In the case of complex cation sublattices, it is necessary that the ions should be ordered by a definite type [105, 106]. In multiphase systems, which are chemically inhomogeneous, considerable dielectric loss (relatively low Q) is generally observed. When investigating barium polytitanates, however, we showed that multiphase systems having a high Q and thermostability of electrophysical properties can be formed. When zinc oxide is added to barium polytitanates, an extra BaZn2Ti4O11 phase is formed [132] which does not interact chemically with the main phase. A multiphase system is formed, in which the main and extra phases have the dependence ε(T) of different sign, which ensures the realization of the volume temperature compensation effect and hence a high thermostability of electrophysical properties (TCε = ±2 × 10-6K-1) in the MW range. The multiphase dielectrics obtained have a high Q (Q10GHz ~ 6500-

One more example of multiphase MW dielectrics is TiO2 materials, viz the compounds MgTiO3 and Mg2TiO4, which have a high Q (Q10GHz ~ 5000-10000) and permittivity (14 and 16 respectively) [133]. A demerit of these materials is the temperature instability of electrophysical parameters (TCε = (40-50) × 10-6K-1). To increase the Q value, cobalt ions were partially substituted for magnesium ions, and to increase the temperature stability of electrophysical properties, small amounts of the paraelectric phase CaTiO3, which has a high negative value of TCε, were added. Investigations showed that in this case multiphase

This made it possible to obtain MW dielectrics with a permittivity of 18-20, high Q values (Q × ƒ ≥ 5000-10000) and thermostable electrical properties. It may be supposed that high Q values are due to the fact that the size of chemical inhomogeneity is much smaller than the electromagnetic wavelength in dielectric and does not cause, therefore, noticeable

systems of chemically noninteracting phases are formed (Fig 21) [134, , 136].

materials with Qf on the order of 80000–90000 GHz and τf = –2 to +3 ppm/K.

It should be noted that polarization a in Eq (4) is equal to the sum of polarizations of all atoms of cell, whose volume is ν, only if all atoms of the structure have a cubic environment. This is the case, e.g. for alkali halide crystals. In more complex structures, effective polarization aeff is used [138]. For example, for perovskite structure, effective polarization aeff is obtained by introducing an ionic component of polarization, Δ ai , in addition to the electronic and ionic polarization of all atoms in the unit cell. In this case, the last term al in expression (4) must be written as al Δ ai [137]. An analysis of the TCε value of different materials as a function of chemical composition showed [139] that the large positive value of TCε in alkali halide crystals probably arises from high ionic polarizability ai and great thermal expansion al. There are compounds, e.g. LaAlO3, SrZrO3, the behavior of whose ε(T) differs from that of paraelectrics. This is attributed to the presence of nonferroelectric phase transitions [137], which are coupled with the rotation of oxygen octahedra. Similar nonferroelectric phase transitions were found, e.g. in the system (BaxSr1-x)(Zn1/3 – Nb2/3)O3 [140]. It was shown that structural transitions are accompanied by the reversal of the TCε sign, though the ε quantity does not undergo noticeable changes, which are typical of

#### 138 Dielectric Material

spontaneously polarized state. Similar dependences were also found in the system (BaxSr1 x)(Mg1/3 –Nb2/3)O3 [141]. When x is changed, a correlation between the value of tolerance factor (t) and the inclination of oxygen octahedra is observed, which results in the reversal of the TCε sign.

When MW dielectric loss is studied, three sources of loss are considered: 1) loss in perfect crystal, which is coupled with anharmonicity, which is due to interaction between crystal phonons, resulting in optical-phonon attenuation; such loss is usually called intrinsic loss; 2) loss in real homogeneous material, which is caused by deviation from lattice or defect periodicity (point defects, dopant atoms, vacancies or twin defects, which give rise to quasibound states); such defects give rise to phonon scattering; 3) loss in real inhomogeneous crystalline materials, which is caused by the existence of dislocations, grain boundaries, including minor phases; this loss is usually called extrinsic loss.

ε Dispersion for infrared polarization is usually described by the Drude-Lorentz equation:

$$\mathcal{E}\left(\phi\right) = \mathcal{E}\_{\phi} + \frac{\mathcal{E}\_0 - \mathcal{E}\_{\phi}}{1 - \left(\frac{\phi\nu}{\phi\_{\Gamma}}\right)^2 + i\Gamma \frac{\phi\nu}{\phi\_{\Gamma}}} \tag{5}$$

Microwave Dielectrics Based on Complex Oxide Systems 139




It is not less important to elucidate the effect of cation ordering on MW loss. It was shown earlier [58] that cation ordering in Ba(Zn1/3Ta2/3)O3 and Ba(Mg1/3Ta2/3)O3 allows MW loss to be reduced. When investigating (Zr1-xSnx)-TiO4 materials, however, it was found that the substitution of Sn4+ ions for Zr4+ ions results in the suppression of cation ordering [148], and that the Q value increases in this case [140]. It is likely that in the system (Zr1-xSnx)TiO4, the increase in Q on the substitution of Sn4+ ions with smaller radius for Zr4+ ions with larger

MW dielectrics are used in modern communication systems, for the manufacture of dielectric resonators (DR) of various types, substrates for MW hybrid integrated circuits. On the basis of DRs, radio-frequency filters are developed; they are also used in the manufacture of solid state oscillators. In the frequency range 150 MHz – 3 GHz, coaxial resonators are often used, whose surface is metallized (Fig 22). The height of quarter-wave

> <sup>0</sup> 1 4 *l*

where λ0 is free-space electromagnetic wavelength, ε is permittivity in the operating

At higher frequencies, open resonators are generally used (Fig 23), whose diameter is

<sup>0</sup> *D* 

The characteristic modes for coaxial and open dielectric resonators are usually the modes

The Q value of coaxial resonator is determined both by dielectric loss and by loss in the resonator metal coating, which may be high; therefore, the Q value of coaxial resonators is, as a rule, under 1000, which is their demerit. At the same time, Q of open resonators is determined only by dielectric loss (intrinsic and extrinsic). Therefore, Q of open resonators is

(7)

(8)


intrinsic loss.

vibration mode.

frequency range.

TE01s and H01s.

determined from the formula:

accordance with the microscopic theory [147].

radius is due to a decrease in intrinsic loss.

**13. Applications of MW dielectrics** 

coaxial resonator is determined from the formula:

where ε∞ is permittivity at optical frequencies, ωT is transverse optical mode frequency, (ε0 ε∞) is dielectric oscillator strength, Г is relative attenuation.

Using Eq (5), we can estimate dielectric loss (tg δ) pertaining to phonon attenuation in the MW range (intrinsic loss) (ω » ωT):

$$
tg \delta \approx \Gamma \frac{\alpha \nu}{\alpha\_{\Gamma}^{4}} \frac{\varepsilon\_{0} - \varepsilon\_{\alpha}}{\varepsilon\_{0}} \tag{6}
$$

Even in perfect crystal, loss may arise from the anharmonicity of vibration. In this case, three-phonon and four-phonon interactions may predominate, in which Г~T and Г~T2 respectively [142]. Equation (6) shows that if intrinsic loss predominates, the product of Q (Q = 1/tg δ) and frequency f(ω = 2πf) is a constant, which is employed for the analysis of MW dielectrics.

To determine intrinsic loss, IR spectroscopy is usually used [143, 144] since the value of intrinsic loss at IR frequencies is much larger than that of extrinsic loss. Having determined intrinsic loss at IR frequencies, one can approximate the quantity Q × ƒ = const to MW frequencies and calculate thereby extrinsic loss, which is due to ceramic imperfection; this loss can be reduced by improving the technology.

One of the important questions concerning the determination of dielectric loss in MW ceramics is the possibility of determining intrinsic loss solely from IR spectroscopic data. To this end, the authors of [145, 146] investigated several Ba(B'1/2 B''1/2)O3 compounds in a wide frequency (102 – 1014 Hz) and temperature (20-600 K) range. On the basis of the data obtained, they came to the following conclusions:


It is not less important to elucidate the effect of cation ordering on MW loss. It was shown earlier [58] that cation ordering in Ba(Zn1/3Ta2/3)O3 and Ba(Mg1/3Ta2/3)O3 allows MW loss to be reduced. When investigating (Zr1-xSnx)-TiO4 materials, however, it was found that the substitution of Sn4+ ions for Zr4+ ions results in the suppression of cation ordering [148], and that the Q value increases in this case [140]. It is likely that in the system (Zr1-xSnx)TiO4, the increase in Q on the substitution of Sn4+ ions with smaller radius for Zr4+ ions with larger radius is due to a decrease in intrinsic loss.

### **13. Applications of MW dielectrics**

138 Dielectric Material

the TCε sign.

spontaneously polarized state. Similar dependences were also found in the system (BaxSr1 x)(Mg1/3 –Nb2/3)O3 [141]. When x is changed, a correlation between the value of tolerance factor (t) and the inclination of oxygen octahedra is observed, which results in the reversal of

When MW dielectric loss is studied, three sources of loss are considered: 1) loss in perfect crystal, which is coupled with anharmonicity, which is due to interaction between crystal phonons, resulting in optical-phonon attenuation; such loss is usually called intrinsic loss; 2) loss in real homogeneous material, which is caused by deviation from lattice or defect periodicity (point defects, dopant atoms, vacancies or twin defects, which give rise to quasibound states); such defects give rise to phonon scattering; 3) loss in real inhomogeneous crystalline materials, which is caused by the existence of dislocations, grain boundaries,

ε Dispersion for infrared polarization is usually described by the Drude-Lorentz equation:

2

 

 

where ε∞ is permittivity at optical frequencies, ωT is transverse optical mode frequency, (ε0 -

Using Eq (5), we can estimate dielectric loss (tg δ) pertaining to phonon attenuation in the

 

Even in perfect crystal, loss may arise from the anharmonicity of vibration. In this case, three-phonon and four-phonon interactions may predominate, in which Г~T and Г~T2 respectively [142]. Equation (6) shows that if intrinsic loss predominates, the product of Q (Q = 1/tg δ) and frequency f(ω = 2πf) is a constant, which is employed for the analysis of MW

To determine intrinsic loss, IR spectroscopy is usually used [143, 144] since the value of intrinsic loss at IR frequencies is much larger than that of extrinsic loss. Having determined intrinsic loss at IR frequencies, one can approximate the quantity Q × ƒ = const to MW frequencies and calculate thereby extrinsic loss, which is due to ceramic imperfection; this

One of the important questions concerning the determination of dielectric loss in MW ceramics is the possibility of determining intrinsic loss solely from IR spectroscopic data. To this end, the authors of [145, 146] investigated several Ba(B'1/2 B''1/2)O3 compounds in a wide frequency (102 – 1014 Hz) and temperature (20-600 K) range. On the basis of the data

0 4 *T* 0

*T T i*

(5)

(6)

<sup>0</sup>

 

 

*tg*

1

including minor phases; this loss is usually called extrinsic loss.

ε∞) is dielectric oscillator strength, Г is relative attenuation.

loss can be reduced by improving the technology.

obtained, they came to the following conclusions:

MW range (intrinsic loss) (ω » ωT):

dielectrics.

MW dielectrics are used in modern communication systems, for the manufacture of dielectric resonators (DR) of various types, substrates for MW hybrid integrated circuits. On the basis of DRs, radio-frequency filters are developed; they are also used in the manufacture of solid state oscillators. In the frequency range 150 MHz – 3 GHz, coaxial resonators are often used, whose surface is metallized (Fig 22). The height of quarter-wave coaxial resonator is determined from the formula:

$$l = \frac{\mathcal{A}\_0}{4} \frac{1}{\sqrt{\varepsilon}}\tag{7}$$

where λ0 is free-space electromagnetic wavelength, ε is permittivity in the operating frequency range.

At higher frequencies, open resonators are generally used (Fig 23), whose diameter is determined from the formula:

$$D = \frac{\lambda\_0}{\sqrt{\varepsilon}}\tag{8}$$

The characteristic modes for coaxial and open dielectric resonators are usually the modes TE01s and H01s.

The Q value of coaxial resonator is determined both by dielectric loss and by loss in the resonator metal coating, which may be high; therefore, the Q value of coaxial resonators is, as a rule, under 1000, which is their demerit. At the same time, Q of open resonators is determined only by dielectric loss (intrinsic and extrinsic). Therefore, Q of open resonators is over 1000. Since open resonators have a high Q value, they can be used in the decimeter wave band (about 1 GHz), though in this case their size becomes large.

Microwave Dielectrics Based on Complex Oxide Systems 141

Dielectric resonators are used for the frequency stabilization of oscillators, which are used, in turn, in radars, various communication systems. At frequencies below 200 MHz, quartz resonators are often used; in the frequency range 1000-3000 MHz, coaxial resonators and

On the basis of dielectric resonators, miniature bandpass filters, frequency separators are

**Figure 24.** Monolithic ceramic blocks for the utilization in radiofilters operating in the decimetre wavelength band (a), and low-noise microwave oscillator for the frequency of around 9 GHz (b)..

different radiative characteristics using different resonator modes.

temperature compensation effect [55-57, 124, 132, 135].

Dielectric resonators are also used in the creation of antennas of the new generation. The advantages of such antennas are: small size, simplicity, relatively broad emission band, simple scheme of coupling with all commonly used transmission lines; possibility to obtain

As was mentioned above, the use of dielectric resonators operating on the TE01s and H01s modes is limited in the millimeter wave band since the size of resonators becomes too small. Therefore, it is expedient to use in the millimeter wave band dielectric resonators operating on whispering gallery modes [150]. Besides, it is relatively easy to suppress spurious modes in such resonators. It should be noted that the Q value in the resonators using whispering gallery modes is limited only by intrinsic loss in the material in contrast to coaxial and open

High-Q MW dielectrics with high thermostability of electrophysical properties can be developed on the basis of single-phase and multiphase systems. Single-phase MW dielectrics are produced on the basis of solid solutions [6, 15, 98] using heterovalent substitutions in one of the crystal sublattices and influencing thereby the phonon spectrum [10] and by making one of the sublattices "mobile" [151]. At the same time, high-Q thermostable MW dielectrics based on multiphase systems are developed using the volume

above 3000 MHz open resonators are used.

developed (Fig 24) [149].

resonators.

**14. Conclusion** 

**Figure 22.** Coaxial dielectric resonators

**Figure 23.** Open dielectric resonators

The size of open dielectric resonators operating on the characteristic modes TE01s and H01s becomes very small at frequencies above 30 GHz, which makes their use in this band impossible. Therefore, at frequencies above 30 GHz, it is expedient to use extraordinary vibration modes such as whispering gallery modes.

The modern communication systems in which dielectric resonators are used operate in a temperature range of – 40-80 0C. Therefore, high thermostability of dielectric resonator resonance frequency is required. It is necessary that the temperature coefficient of frequency should tend to zero; it is defined as:

$$T\mathbf{K}\boldsymbol{\varepsilon} = \frac{1}{f\_p} \frac{\Delta f\_p}{\Delta T} \tag{9}$$

where ƒr is the resonance frequency of dielectric resonator, Δ ƒr is change in the resonance frequency of dielectric resonator in the temperature range ΔT.

Dielectric resonators are used for the frequency stabilization of oscillators, which are used, in turn, in radars, various communication systems. At frequencies below 200 MHz, quartz resonators are often used; in the frequency range 1000-3000 MHz, coaxial resonators and above 3000 MHz open resonators are used.

On the basis of dielectric resonators, miniature bandpass filters, frequency separators are developed (Fig 24) [149].

**Figure 24.** Monolithic ceramic blocks for the utilization in radiofilters operating in the decimetre wavelength band (a), and low-noise microwave oscillator for the frequency of around 9 GHz (b)..

Dielectric resonators are also used in the creation of antennas of the new generation. The advantages of such antennas are: small size, simplicity, relatively broad emission band, simple scheme of coupling with all commonly used transmission lines; possibility to obtain different radiative characteristics using different resonator modes.

As was mentioned above, the use of dielectric resonators operating on the TE01s and H01s modes is limited in the millimeter wave band since the size of resonators becomes too small. Therefore, it is expedient to use in the millimeter wave band dielectric resonators operating on whispering gallery modes [150]. Besides, it is relatively easy to suppress spurious modes in such resonators. It should be noted that the Q value in the resonators using whispering gallery modes is limited only by intrinsic loss in the material in contrast to coaxial and open resonators.
