Low-loss dielectric ceramic materials and their properties

Abstract

In addition to the constant demand of low-loss dielectric materials for wireless telecommunication, the recent progress in the Internet of Things (IoT), the Tactile Internet (fifth generation wireless systems), the Industrial Internet, satellite broadcasting and intelligent transport systems (ITS) has put more pressure on their development with modern component fabrication techniques. Oxide ceramics are critical for these applications, and a full understanding of their crystal chemistry is fundamental for future development. Properties of microwave ceramics depend on several parameters including their composition, the purity of starting materials, processing conditions and their ultimate densification/porosity. In this review the data for all reported low-loss microwave dielectric ceramic materials are collected and tabulated. The table of these materials gives the relative permittivity, quality factor, temperature variation of the resonant frequency, crystal structure, sintering temperature, measurement frequency and references. In addition, the methods commonly employed for measuring the microwave dielectric properties, important from the applications point of view, factors affecting the dielectric loss, methods to tailor the dielectric properties and materials for future applications, are briefly described. The data will be very useful for scientists, industrialists, engineers and students working on current and emerging applications of wireless communications.

Introduction

Microwave dielectric materials play a key role in global society, with a wide range of applications from terrestrial and satellite communications, including Internet of Things (IoT), software radio, GPS and DBS TV, to environmental monitoring via satellite, etc. Today low-loss dielectric materials are all-pervasive. The mobile phone is one of the most widely spread technologies on the planet. In many countries, the number of mobile subscriptions exceeds the population. The IoT is posed to make an explosive growth in the near future. In this paradigm, many every-day objects will be networked via radio-frequency identification (RFID), printed electronics and sensor network technologies. According to GSMA intelligence, the revenue from interconnected devices for mobile network operators alone in the segments of automotive, health, utilities and consumer electronics will be $1.3 trillion by 2020. In order to meet the specifications of future systems, new designs and improved or new microwave dielectric components are required. The recent progress in the IoT, microwave telecommunications, satellite broadcasting and intelligent transport systems (ITS) has resulted in an increasing demand for low-loss dielectric materials. Indeed, low-loss dielectric oxide ceramics have revolutionised the microwave wireless communication industry by reducing the size and cost of filter, oscillator and antenna components in applications ranging from cellular phones to IoT. Wireless communication technology demands materials with highly specialised properties. The importance of miniaturisation cannot be overemphasised in any handheld communication application, as can be seen in the dramatic decrease in the size and weight of devices in recent years. This constant need for miniaturisation provides a continuing driving force for the discovery and development of ever smaller/lighter dielectrics which can outperform existing materials. Recently the demand for materials with low sintering temperature has increased not only to lower the energy cost of devices but also to integrate with polymers and silver-based electrodes. Several polymer-based (polymer–ceramic) composites have also recently been developed for wireless communication technology. In the present paper, we restrict our discussions to ceramic materials. For polymer-based composite dielectric materials, the reader is referred to the recent review by Sebastian and Jantunen.¹ The number of papers published on low-loss microwave materials and related devices has increased considerably over the years as shown in Fig. 1.

A dielectric resonator (DR) is an electromagnetic component that exhibits resonance for a narrow range of frequencies. The resonance is similar to that of a circular, hollow metallic waveguide except that the boundary is defined by a large change in permittivity rather than conduction. Dielectric resonators generally consist of a ceramic puck and require high values of relative permittivity (ε_r) and quality factor (Q) and near-zero temperature coefficients of resonant frequency (τ_f). The quality factor, which is a function of resonant frequency, is sometimes expressed as Qf, the product of Q and the resonant frequency (in GHz). While Qf is not technically a dimensionless figure of merit, the units (GHz) are almost invariably dropped. The resonant frequency is determined by the overall physical dimensions of the puck and the permittivity of the material and its immediate surroundings. Optimising these three properties simultaneously is difficult.

Oxide ceramics are critical elements in these microwave devices, and a full understanding of their crystal chemistry is fundamental to future development. Properties of microwave ceramics depend on several parameters including the processing conditions and the purity of starting materials. Design of the heating/cooling schedule requires knowledge of the formation mechanisms of various phases in multicomponent systems, and the starting powders must sinter to high density to obtain optimum electrical properties.

Low-permittivity ceramics are used for millimetre-wave communication and also as substrates for microwave integrated circuits. Medium-ε_r ceramics with ε_r in the range of 25–50 are used for satellite communications and in mobile phone base stations. High-ε_r materials are used in mobile phone handsets where miniaturisation is very important. For millimetre-wave and substrate applications, temperature-stable, low-permittivity and high-Q are required for high-speed signal transmission with minimum attenuation. The signal transmission speed increases as the relative permittivity decreases. High-Q dielectrics minimise circuit insertion losses and can be used to create highly selective filters. In addition, a high-Q suppresses the electrical noise in oscillator devices. Although several manufacturers may produce similar components for the same application, there are subtle differences in circuit design, construction and packaging. Since frequency drift of a device is a consequence of the overall thermal expansion drift of its unique combination of components, each design requires a slightly different τ_f for temperature compensation. Typically, ceramics with a specific τ_f in the range of − 15 to +15 ppm/°C are selected. In ceramic production, τ_f and ε_r specifications must be produced to within demanding tolerances typically ± 1%.²

Electronic circuits for the automotive industry, home electronics and telecommunications have to handle a steadily increasing amount of functionality within as tiny a space as possible. In the development of complex miniaturized circuits, flexible glass–ceramic composites, the so called low-temperature cofired ceramics (LTCCs), play a decisive role as a base material. LTCCs have become crucial in the development of various modules and substrates. This technology enables fabrication of three-dimensional ceramic modules with embedded silver or copper electrodes, and LTCCs with relative permittivity from ∼4 up to >100 have been developed showing low dielectric loss. These advantages make LTCC technology very attractive for a variety of micro- and millimetre-wave applications.³ The important characteristics required for LTCCs are (a) densification temperature < 950°C (b) ε_r in the range 5–70 (c) Q f >1000 (d) τ_f close to zero (e) high thermal conductivity (f) preferably low thermal expansion and (g) chemical compatibility with the electrode material. Low sintering temperatures are required to avoid melting metallic conductors like silver or gold in the fabrication of dielectric devices.³ Most conventional electroceramics do not meet the basic requirements with regard to sinterability for LTCC technology since they have relatively high sintering temperatures. The different methods used to reduce the sintering temperature of dielectrics include: (1) addition of low melting-temperature glass phases, (2) addition of low melting-point compounds such as Bi₂O₃, B₂O₃, V₂O₅ or CuO and (3) the use of chemical processing in order to achieve smaller particle sizes. The first method, while commonly found effective in decreasing the sintering temperature, usually results in a degradation of microwave dielectric properties. The selection of glass materials is very important for sintering glass–ceramic composites, since the liquidation of glass takes a dominant role in the viscous flow mechanism during sintering; hence, this method remains the focus of intense research. The dielectric table (supplementary file) lists the key property data of microwave dielectric materials available from published and, to a far lesser extent, reputable unpublished sources. These data are the relative permittivity (ε_r), the product of the Q factor and the frequency (Q f), the frequency of measurement (f), the temperature coefficient of the resonant frequency (τ_f), sintering temperature and crystal structure or structural family.

Measurement of microwave dielectric properties

The three important characteristics of an ideal low-loss dielectric material are application optimised value of relative permittivity (ε_r), low dielectric loss (loss tangent, tanδ) and low temperature coefficient of resonant frequency (τ_f). These three properties and different measurement methodologies to measure them are briefly discussed in the following sections.

Permittivity

When microwaves enter a dielectric medium, they are slowed down by a factor equal to ε_r ^− 1/2; therefore

At resonant frequency, λ = f ₀ and λ_d ∼ D (diameter of resonator); therefore Equation (2) is only valid in the case of resonators in free space. It fails for resonators in more realistic situations (e.g., on microstrips, in cavities, between shorting plates, etc.). In order to calculate permittivity in these geometries, several techniques have been developed and variously discussed. Perturbation techniques rely on the shift of f ₀ (and Q) of a resonant cavity caused by the presence of a dielectric disc or sphere. Optical methods at microwave frequencies are suited to measurements at which λ < 1 cm and require a large amount of material. Transmission-line methods have the practical difficulty of requiring a very small waveguide for λ < 4 mm. All of these methods have an accuracy of approximately ± 1%. The exact resonance method proposed by Karpova⁴ and further developed by Hakki and Coleman,⁵ Courtney⁶ and others yields errors of only ± 0.1% but is limited to the accuracy of the measurements of resonant frequency and sample dimensions. The reader is referred to the recent book² for details of these techniques. In this paper, we restrict the discussion to the measurement of the relative permittivity and loss tangents of low-loss dielectric materials.

Hakki–Coleman method

Karpova⁴ used a re-entrant cavity for the measurement of dielectric properties, but the physical size of the resonant structure required could be problematic for the low-millimetre range. In order to avoid the problem of physical size while maintaining high accuracy, Hakki and Coleman⁵ instead proposed an open-boundary resonant structure in which a dielectric rod was positioned between much larger conducting plates (Fig. 2).

The characteristic equation which describes this condition for an isotropic resonator in a TE_0mp mode: where J ₀(α) and J ₁(α) are Bessel functions of the first kind of orders zero and one, respectively. K ₀(β) and K ₁(β) are modified Bessel functions of the second kind of orders zero and one, respectively. The parameters α and β are functions of geometry, resonant wavelength and permittivity: where c is the speed of light, a is resonator radius and v _p is the phase velocity in the resonator such that: where p is the number of longitudinal variations of field along the axis and λ₀ = c/f ₀. Clearly v _p can be calculated from thickness and resonant frequency alone; and β can then be calculated from v _p, frequency and radius.

The characteristic equation (3) is transcendental and requires a graphical solution. Hakki and Coleman⁵ used analogue mode charts to relate various {α_m} to each corresponding value of β, resulting in somewhat limited accuracy (Fig. 3).

Although this technique is sometimes called the Courtney method,⁶ ‘Courtney, actually, only perfected and scrutinised a parallel-plate arrangement introduced [10 years] earlier’ by Hakki and Coleman.⁵ Courtney also adapted the technique to the use of coaxial probes (an innovation introduced 4 years earlier by Cohn and Kelly⁷), allowing a greater range of sample dimensions.

An improvement in accuracy over a purely graphical approach can be achieved by numerically solving for each Bessel/modified Bessel function rather than trying to read values off the mode charts of Hakki and Coleman⁵ or even relying on curve fits. With modern computers, ordinary Bessel functions and modified Bessel functions can be numerically calculated, and these numerical methods make it possible to solve equation (3) for β ≤ 10. The algorithm employed in the HakCol program⁸ starts by calculating β from the resonator radius and resonant frequency. Next an approximate corresponding value for α is calculated using a curve fit to the m = 1 (TE_01p) mode chart of Hakki and Coleman⁵ (Fig. 3). The polynomial which describes the curve in Fig. 3 is: The mode TE_nml, the integer n denotes the azimuthal variation, m radial variation and l the axial variation. The α so calculated is used as a first approximation in order to calculate ε_r. Next, equation (3) is evaluated and if the two sides are unequal then ε_r is adjusted accordingly, α re-calculated, and the process iterates until equation (3) is satisfied. The entire algorithm and the HakCol program is detailed in ref.⁹

The TE₀₁₁ mode is used for the measurements since this mode propagates inside the sample but is evanescent outside; therefore, a large amount of electrical energy can be stored in high-Q DRs.¹⁰ In the end-shorted condition, the E field becomes zero close to the metal wall and electric energy vanishes in the air gap.⁷ The TE and TM modes do not contain electric and magnetic fields in the axial (z) direction. For the TE₀₁₁ mode only the azimuthal component of the electric field exists and the error because of the air gap is practically eliminated.¹¹ For cylindrical resonators, TE and TM modes exist only if the azimuthal mode index m = 0 otherwise all other modes are hybrid, i.e., they have all six electromagnetic components. Hybrid modes are usually divided into two mode families: HE and TM. They are only occasionally used in measurements of dielectrics (e.g., for uniaxially anisotropic crystals). This method is proposed as one of the international standard IEC techniques¹² for measurements of the complex permittivity of low-loss solids. Hennings and Schnabel¹³ studied the reproducibility of the ε_r measured by this end-shorted method using 10 different samples prepared in a batch. Their results showed a maximum variation of 0.6% in ε_r. In this method, the ε_r is measured only at one resonant frequency corresponding to the TE₀₁₁ mode. If one can identify other resonant modes, then it is possible to measure ε_r at other resonant frequencies. By using the resonant modes TE₀₁₁, TE₀₂₁, TE₀₃₁ and TE₀₄₁, the ε_r of a sample can be measured over a range of frequencies. It may be noted that as the ε_r increases, the resonant frequency decreases and as the dimensions of the sample decrease the resonant frequency increases.

Shielded resonator in dielectric-rod waveguide method

For a high-Q material in a cavity, as proposed by Itoh and Rudokas¹⁴ and modified by Kajfezz and Guillon¹⁵ (Fig. 4), most of the electrical field is contained within the resonator itself (region 6), and very little exists in regions 1 and 2, and even less in regions 3 and 5. To a fair first approximation, then, the fields in regions 3 and 5 can be ignored. For the TE_01δ modes in this geometry, the requirement for continuity of fields leads to two simultaneous eigenvalue equations:

The symbol k represents the radial propagation constants in the different regions of the model, which are functions of both frequency and dielectric constant; and ρ is the radial distance from the geometric centre. The arguments of the various Bessel functions are the eigenvalues of the system, where k _ρ1 a is called the eigenvalue of the TE_0n mode, and k _ρ2 is given by: where k ₀ is called variously the propagation constant, wavenumber or phase constant of free space, and has units of m^− 1: In equation (9), β is the propagation constant of the resonator. If p is the number of axial variations of the field along the resonator's height, then p = l+δ, where l is an integer and δ is a non-integer number smaller than unity which depends in a complicated way on propagation constants and geometry. Whereas for the TE₀₁₁ mode discussed above, l = 1 and δ = 0, for the TE_01δ mode, l = 0 and δ ≠ .0. The symbols (φ₁ and φ₂ are called the phase angles and are complex hyperbolic functions of the cavity geometry and the propagation constant of the resonator. The entire algorithm and the ErCalc program is detailed in reference.⁹

Correction for porosity

The porosity in the sintered ceramic disc influences the measured ε_r and thus the measured ε_r should be corrected to isolate the actual dielectric permittivity. This correction can be performed in a variety of ways.

The Maxwell Garnett¹⁶ approximation treats one of the components as a host in which inclusions of the other component are embedded. Lichtenecker's¹⁷ logarithmic mixing rule assumes a randomly connected second phase and, although it is much used, is actually one of the least accurate mixture rules available. By contrast, the Bötcher mixture rule¹⁸ assumes a dispersion of spherical porosity (or another second phase) in a mixture of both solid and air (or another second phase), like that of Bruggeman,¹⁹ thereby allowing for the interaction between the two phases and increasing the accuracy even for high values of porosity: where ε_rm, ε_r1, and ε_r2 are the permittivities of the mixture, phase 1 and phase 2, respectively, and δ₁ is the volume fraction of phase 1. This rule is also based in part on the work of Wiener²⁰ and Stratton²¹ and supported by Reynolds.²²

The various equations typically only diverge significantly for very high or very low densities of second phase and re-converge for densities of 0 and 100%. Maxwell's equation, in particular, slightly inflates the value at intermediate densities, presumably because it does not allow for the interaction between the two phases.

Measurement of loss tangent/quality factor

The measured Q value is commonly the loaded quality factor (Q_L) taking into account the external circuit (the network analyser with coupling probes). However, if the measurement is arranged under very weak coupling the Q_L is the same as unloaded one Qu and is obtained from the following equation. where f is the measured resonance frequency and Δf is 3 dB band width of the peak. There are various methods which enable measurement of the quality factors of low-loss dielectrics.^23–31 One should however keep in mind that not all of them take into account practical effects introduced by a real measurement system, such as noise, cross-talk, coupling losses, transmission-line delay and impedance mismatch. Inadequate accounting of these effects may lead to significant uncertainty in the measured Q-factor.

For example the quality factor can be measured by Hakki and Coleman's end-shorted method,^{2,5–7,25–28} but the quality factor measured by this method will be somewhat low since loss occurs because of the conducting plates and radiation effects. Fortunately, corrections for conductor losses can be applied knowing the surface resistance of the conducting plates.

TE_01δ mode DR method

To avoid the problems of the conduction and radiation losses, the Q of a DR sample can be measured by using the cavity method in which the DR is placed on a low-loss (e.g., single crystal quartz or Teflon) spacer inside the cavity. This method is proposed by Krupka et al. ^23,29 using a transmission-mode cavity. It enables measurement of the quality factor (Q), permittivity (ε_r) and temperature coefficient of resonant frequency (τ_f) of the DRs, which is placed inside a cylindrical metallic cavity usually made of copper. The inner surfaces are polished and gold or silver coated. A loop coupling is used to feed microwave to the DR Since the cavity has an infinite number of modes, the diameter and height ratio of the sample is commonly kept on the level 2–2.5 to get maximum mode separation. Since the electric field is symmetric in this measurement method, the sources of loss owing to the cavity are reduced. In this method the TE₀₁₁ mode is designated as TE_01δ, since the field confinement is not complete in the z direction. As shown in Fig. 5, the spacer isolates the sample from the effects of losses because of the finite resistivity of the metallic cavity.

After identifying the mode, the resonant frequency and 3dB bandwidth are determined. The network analyser is then calibrated and S₁₁ and S₂₂ are measured at the resonant frequency (Fig. 6). From these values, the coupling coefficients βc₁ and βc₂ for the coupling ports are determined using the relations βc₁ = (1 − S₁₁)/(S₁₁ + S₂₂) and βc₂ = (1 − S₂₂)/(S₁₁ + S₂₂), where S₁₁ and S₂₂ are reflection coefficients of ports 1 and 2.³⁰ Figure 7 shows the typical resonance spectra in reflection and transmission configuration of a Ba(Mg_1/3Ta_2/3)O₃ ceramic sample having ε_r = 38. The TE_01δ mode frequency is noted and the unloaded Q factor is measured.

From the measured Q_L, Q_U can be calculated as In some cases the desired mode (TE_01δ) may be close to other modes but this method allows slight change of the cavity volume by rotating the top screw, which enable separation of the modes. This action enables the identification of the desired resonant mode and in addition allows the cavity to measure samples of different dimensions. Figure 8 shows a typical test fixture manufactured by QWED, Warsaw, Poland. The evaluation of the permittivity and the dielectric loss tangent of the sample under test require rigorous electromagnetic analysis. QWED uses the Rayleigh–Ritz method in their software.

The TE_01δ mode DR method is one of the most accurate techniques for measuring especially loss tangent of isotropic low-loss materials.^29,31 The inverse of measured unloaded Q-factor is approximately equal to the dielectric loss tangent if all parasitic losses can be neglected (true in the cases when the permittivity of the sample is large) and if the electric energy filling factor can be assumed to be equal to unity. One must keep in mind that these assumptions are not valid when the sample has very low dielectric loss or permittivity value. In the first case the conductor losses must be taken into account. What comes to the low-permittivity materials, the electric energy filling factor in the sample is substantially smaller than 1. However, the advantages of the cavity method using the TE_01δ mode are easy mode identification, small parasitic losses and lack of mode degeneracy.²³ On the other hand, the evaluation of tanδ requires advanced numerical computations, which can only be done employing dedicated computer programs because of the absence of exact solutions of Maxwell's equation. The uncertainty in dielectric loss tangent using TE_01δ mode cavity method with optimized enclosure is of the order of 0.03 tanδ. The frequency band this method is feasible depends on the size and permittivity of the samples, and the cavity geometry. Higher frequency measurements are performed by using smaller cavities and samples, or by using several higher order quasi-TE_0nm modes.³²

Valant et al. ³³ reported the effect of the test cavity dimensions on the microwave dielectric properties of the ceramic resonator. The electromagnetic field could penetrate into the conducting walls of the test cavity (skin effect) lowering the Q factor. With large size of the test cavity this source of error can be avoided. Thus in order to derive the unloaded Q value, the test cavity should be large enough. A good practice is to select the test cavity size in such a way that the TE_01δ mode of the DR is the lowest resonance and hence it can be easily identified. This is especially true in the case DRs with permittivity >20 when increase of the size of the test cavity is needed to move the resonant modes of the cavity to lower frequencies. Figure 9 shows how the measured quality factor decreases with the cavity diameter/disc diameter ratio. Thus it is advisable to use 3–5 times larger cavity compared to the size of the test sample. In addition the surface resistance of cavity walls can be calculated from the quality factor of the TE₀₁₁ resonance of the empty cavity.¹⁵

Strip line excited by cavity method

Magnetic coupling of the DR to a 50 Ω microstrip line is used in the microstrip line excited cavity method, as shown in Fig. 10 along with the equivalent circuit.³⁴ In this method the Q factor is estimated through the so called coupling factor, β_c, which is the ratio of the resonator-coupled resistance R at the resonant frequency to the resistance external to the resonator. Here S₁₁₀ and S₂₁₀ are the real quantities of the reflection and transmission coefficients, respectively, at the resonant frequency.

When the coupling factor β_c is equal to one, the power dissipated in the external circuit is the same as the power dissipated in the resonator (P _d), which is equally divided into the power reflected to the generator (P _r = S ₁₁₀ ²) and the power transmitted to the load (P _t = S ₂₁₀ ²). In the shielded resonator configuration like in shielded cavities, the power dissipated in the resonator is given by

The coupling factor β_c is a function of the distance between the DR and the microstrip line under fixed shielding conditions. According to Khanna and Garault³⁴ the unloaded voltage transmission coefficient S_21u is S_21u corresponds to the voltage transmission coefficient of the unloaded resonator. The frequencies f₁ and f₂ corresponding to S_21u given by equation (17) are measured (Fig. 11) and their difference Δf = (f ₂ − f ₁) is calculated. When the resonance frequency, f, corresponds to the peak of the S ₂₁ curve and Δf is known, the unloaded quality factor, Q _u, is calculated using equation (13).

The experimental set-up for the Q measurement by the microstrip line excited by cavity method is shown in Fig. 12. In this a 50 Ω microstrip line of width 3 mm is etched on RT-Duroid 5880 (ε_r ∼2.2 and thickness 1.9 mm) to the bottom wall of a rectangular cavity made of copper. The cavity is then excited using 3.5 mm microstrip edge connectors. The DR is placed near the microstrip line and the TE_01δ mode is identified. After system calibration, the resonant frequency is measured, the transmission coefficient S ₂₁₀ corresponding to f is recorded, and S _21u is calculated using equation (17). Finally the unloaded quality factor is calculated through Δf corresponding S _21u and f.

Whispering gallery mode resonator method

TE_01δ, TM_01δ or HE_11δ modes of the DRs are normally measured by the end-shorted TE₀₁₁, TE_01δ (cavity) or stripline methods.^5,6,25,31 However, the measured Q of these modes depends not only on the material's tanδ but also on the radiation and conduction losses of the cavity as stated earlier. Thus simple measurement by the above methods for very low-loss dielectrics are not accurate enough. The Whispering Gallery modes (WGMs)^35–39 has been reported to be able to confine the entire field within the resonator which yields negligible radiation and conductor losses at microwave frequencies. The Q factor of WGM dielectric resonators is limited only by the intrinsic losses in the dielectric material leading to the situation where the measured WGM Q factor is approximately equal to 1/tanδ. In this method, most of the electromagnetic energy is confined to the dielectric near the perimeter of the air-dielectric.^37,38 One additional advantage of using the WGMs technique is that it allows measurements of two permittivity components of uniaxially anisotropic materials and is very useful, for example, in the case of several single-crystals. In this technique measurements of resonant frequencies and Q factors of two modes belonging to different mode families employing rigorous numerical analysis, e.g. mode-matching, are needed. The electrical energy filling factors for E (quasi-TM mode) and H (quasi-TE mode) modes are then calculated.^37,39 The dielectric tanδ can be solved³⁸ using the equation where R_s is the surface resistance of the cavity, ε_ll is the permittivity parallel to the anisotropic axis and ε_⊥is the one perpendicular to it. In general the conductor losses decrease as the surface resistance becomes smaller and as the geometric factor (G) increases.³⁹ However for the WGM the geometric factor, G, is sufficiently large and thus the effect of the cavity can be ignored. In addition the radiation losses are negligible for these modes.

The spurious modes in WGM method dominate since the propagation constant along the z axis is very small and unwanted modes leak out axially. The WGM dielectric resonators are classified as WGE_n,m,l or WGH_n,m,l.These correspond to the case where the electric field is essentially transverse, or axial. WGMs are periodic according to the azimuthal number, and with the diameter of the DR the number of modes in a bandwidth increases. This means that samples with small resonator diameter, the frequency interval between two successive modes will be large. Dielectric resonators acting in WGMs can be excited in different ways. In the low-frequency range, an electric or magnetic dipole is used. However, this type of excitation is stationary, and travelling WGMs cannot be excited. In the millimetre-wavelength frequency region dielectric image waveguides or microstrip transmission lines are used to excite travelling WGMs.

Split-postdielectric resonator method

The split-post-dielectric resonator (SPDR), shown in Fig. 13, is an accurate method for measuring the complex permittivity and loss tangent of substrates and thin films at a single frequency in the range of 1–20 GHz.^40–43 This arrangement allows the formation of an evanescent electromagnetic field, not only in the air gap, but also in the cavity region for radii greater than the radius of the dielectric resonators which simplifies the numerical analysis and reduces possible radiation effects. In the SPDR method, a flat sample of the test material is inserted through one of the open sides of the fixture and positioned between two low-loss dielectric rods or resonators kept in a metallic enclosure.

The sample is in contact with one of the resonators and separated from the other by a small air gap. The electric field in resonators is parallel to the surface. This means that the test sample should have strictly parallel faces, the thickness of the sample should be less than that of the fixture air gap, and the sample should have enough area to cover the inside of the fixture. In these conditions, the accuracy of the measurement is not affected by the air gap between the sample and the resonator. The required thickness of the sample also depends on the ε_r of the material and materials with high ε_r must be thinner. Several modes are commonly excited, but the TE_01δ mode is preferred since it is insensitive to the presence of air gaps perpendicular to z-axis of the fixture.

The complex permittivity is calculated based on electromagnetic modelling of the split-post-resonant structure using the Rayleigh–Ritz technique.⁴⁰ The real part of the complex permittivity can be iterated from the measured resonant frequencies and thickness of the test sample, h, using the following equation ⁴¹ where f₀ is the resonant frequency of the empty SPDR, f_s is the resonant frequency of the SPDR with the dielectric sample. K_ε is a function of ε’_r and h and is evaluated for a number of ε_r using the Rayleigh–Ritz technique.⁴⁰ The tanδ of the sample is calculated from the measured unloaded Q factors of the SPDR with and without the dielectric sample from where and denote losses of the dielectric and metallic parts of the resonator, respectively. P_e is the electric energy filling factor of the sample given by the following equation where P _ei and tanδ_i are the electric energy filling factor and the dielectric loss tangent for the ith dielectric region, respectively.

Uncertainty of the permittivity depends on the sample thickness h as follows: Δε_r/ε_r = ± (0.0015+Δh h^− 1) and the accuracy of the loss tangent is ∼ ± 0.03 tanδ. For the complex permittivity of a sample, the resonant frequencies and Q factors of the empty SPDR and the SPDR containing the test sample must be measured. The SPDR is operated in a particular mode with a particular resonant frequency which depends on resonator dimensions and to a limited extent the electrical properties of the test sample; thus each SPDR is designed for a particular nominal frequency and the actual measurement is taken close to this frequency. The size of the sample is determined by the nominal frequency. This means that, for example, a nominal frequency of 5–6 GHz requires a minimum sample size of 30 × 30 × 2.1 mm. QWED provides SPDRs (Fig. 14) with dedicated software for the evaluation of permittivity and loss tangents. Compared to the reflection-transmission methods, the SPDR provides superior accuracy and is convenient and fast for low-loss laminar dielectrics such as substrates or LTCCs, printed circuit boards and even for thin films.

Measurement of dielectric properties of powder samples

Low-loss dielectric ceramic powders are also more and more used to enhance dielectric properties of polymers. In the last 10 years interesting polymer ceramic composites have been introduced enabling free adjustment of permittivity of devices being important in several applications areas where design of telecommunication devices has limited space. Additionally one overwhelming example is advanced printed electronics, where inks are based on low temperature curing polymers with embedded dielectric ceramic particles. In these cases it is crucial to know the dielectric properties of powder particles, which can differ significantly if compared to the bulk properties. However, very few papers are published^44–50 on the measurement of powder samples. More recently Tuhkala et al. ^48–50 reported an indirectly coupled open-ended coaxial cavity resonator method operating in TEM mode at 4 GHz to estimate the relative permittivity and loss tangents of powder samples. In this method the open-ended coaxial cavity with optimised dimensions and conductivity is filled with dielectric materials in powder form and the effective dielectric properties are determined by the shift of resonant frequency and change in Q factor between an empty resonator and a filled resonator. A schematic set-up for the measurement is shown in Fig. 15.

For a completely filled cavity, the effective dielectric constant is given by where c is the speed of the light in vacuum, L is the resonator length and f_r is the resonant frequency. The dielectric constant of powder samples can be obtained from measured resonator response, volume fractions of each phase (powder, air and different phases) and calculating the effective dielectric constant using Bruggeman symmetric¹⁹ and the Looyenga⁵¹ mixing rules.

In the same manner the effective loss tangent of the powder sample (with different phases) is estimated from the difference in Q factor between an empty resonator and a filled resonator using the equation and utilising general mixing rules.

The method expects careful estimation of the volume fraction of the powder, homogeneous distribution of the powder throughout the cavity, management of measurement environment (mainly humidity) and probe coupling, which should be loose enough not disturbing the measurement itself and producing symmetric resonance peak.

Tuhkala et al. ^52,53 estimated the microwave dielectric properties of several materials with reasonable accuracy by this technique. The method is shown to be useful for studying the properties of powders in several ways. They also reported estimation of humidity level of the powders,⁵³ effect of surfactant treatment,⁵⁴ evaluation of the amount of two powder phases⁵² using the above method.⁴⁸

Measurement of temperature coefficient of resonant frequency (τ_f)

The temperature coefficient of resonant frequency, τ_f, is the parameter which indicates how much the resonant frequency drifts with temperature. In many cases microwave devices in order to operate correctly require τ_f value close to zero. The τ_f relates to the linear expansion coefficient of the sample itself, α_L, and dependence of the material's dielectric permittivity with temperature.² Mathematically τ_f relates to the temperature coefficient of the permittivity, τ_ε, as follows It may be noted that equation (31) is valid for 100% electric energy storage in the sample and thermal expansion of the metal cavity enclosing the DR is negligible. In general the electronic ceramic materials has α_L close to +10 ppm/°C meaning a significant influence of τ_ε on τ_f.

The τ_f value is measured by following the drift in the resonant peak frequency (f _o) as a function of temperature. Choosing an arbitrary temperature to use as standard, say 25°C, the ratio f _o(T )/f _o(25°C) can be defined at each temperature. Then, τ_f is obtained from the slope of a graph of f _o(T )/f_o(25°C) versus the reduced temperature T′ = T − 25°C: The actual curve of the graph is never linear but very nearly parabolic and is easily fitted with a quadratic equation in terms of f _o (T)/f _o(25°) and T − 25°. In this equation, the first-order coefficient is τ_f, which is the slope of the curve at 25°C. The second-order coefficient is the so-called non-linearity or double-derivative factor (τ_f’). Its sign shows whether the parabola is concave up (+) or down ( − ), and its magnitude indicates the severity of curvature. Both numbers are usually scaled up by 10⁶ and reported in parts per million per degree (ppm/°C) or MK^− 1.

If τ_f is measured by the cavity method, then the thermal expansion of the cavity during heating (or contraction during cooling – measurements are best done upon cooling) limits the accuracy of the method, in which case very low-α_L materials (e.g., invar, α_L = 1.2 ppm/°C) can be used for the cavity. The WGMs and also for TE_01δ mode resonant structures are reliable methods for τ_f measurements since the thermal expansion of the cavity is negligibly small especially if the relative permittivity of the sample is large and the sample is situated away from cavity walls.

The temperature coefficient of dielectric permittivity τ_ε can be obtained by the parallel-plate capacitor method using an LCR meter at low frequency.

Factors affecting dielectric losses

The tanδ is known to be very sensitive to humidity^6,55 meaning that the microwave measurements should be done in a humidity-controlled room. Before the experiments the samples should be heated in an oven to remove adsorbed moisture. However, the content of the dielectric has the main effect on the loss values. When especially low-loss materials are desired, the starting materials should be selected the way they have the lowest possible concentration of dipoles and charge carriers with the lowest possible mobility.⁵⁶ Dielectric loss is also affected by disordered charge distributions in the crystal lattice^56,57 which occur if the charge distribution in a crystal deviates from perfect periodicity. In 1964 Schlömann⁵⁶ reported that the loss tangent increases in ionic non-conducting crystals when ions are disordered in such a way that they violate periodicity. The loss tangent thus depends strongly on the spatial correlation between charge deviations and is negligible if the disordered charge distribution in the crystal maintains charge neutrality within a short range of the order of the lattice constant.

The intrinsic quality factor (Q _U = 1/tanδ) of any given material is frequency dependent. For many materials tanδ almost linearly increases as the frequency increases and thus often the intrinsic quality factor is reported as (Q_U f = f /tanδ) (in GHz) as a first approximation. This is most valid for well-densified ceramics within a limited frequency range. In practice, higher Q _U f values for samples measured at higher frequencies (5–12 GHz) than at lower frequencies. More recently Li and Chen reported⁵⁸ that the product Q f is frequency dependent and increases with frequency. The frequency dependence of Qf value is attributed to the presence of defects-induced extrinsic dielectric loss. It may be noted that larger samples resonating at lower frequencies statistically contain more imperfections than smaller ceramic discs resonating at higher frequencies.

The presence of porosity decreases the Q factor further because of presence of moisture in the pores. A fundamental theory of intrinsic losses set the lower limit of losses found in pure defect-free single crystals.⁵⁹ In a dielectric several phonon processes contribute to intrinsic losses and their importance depends on the ac field frequency, temperature range and symmetry of the crystal under consideration. The loss mechanisms are different for a crystal with and without a centre of symmetry.

Gurevich & Tagantsev⁵⁹ obtained numerical estimates of tanδ of ideal crystals.

For an ideal crystal with a hexagonal symmetry when T≪T_D. and for rhombohedral or cubic symmetry where η is a dimensionless anharmonicity parameter ranging between 1 and 100, ω is the angular frequency, k = Boltzmann constant, T = absolute temperature, v _s = sound velocity, T _D = Debye temperature and ρ is the mass density.

Owing to the complicating factors introduced by a variety of extrinsic mechanisms, there is no predictive theory to account for the microwave loss in dielectric ceramics meaning that finding new dielectric resonator materials is largely done by trial and error and involves the preparation and testing a large number of samples. This is a laborious and time-consuming job. The Q factor is highly dependent on not only the extrinsic and intrinsic quality of the ceramic sample but also the method of measurement, the measurement environment and the frequency at which the sample is measured. A given material sample may exhibit greatly differing Q values when tested in different test fixtures and environments which may vary in size, shape, conductor quality, coupling, type of sample support, ambient temperature and relative humidity.

Low-loss dielectric ceramics

A list of low-loss ceramic dielectric materials with sintering temperature, crystal structure, relative permittivity, quality factor-frequency product, measurement frequency, temperature variation of resonant frequency and references are given in the supplementary file. In tabulating these data, we make no judgement on the measurement method and the reliability of the result. The ceramic properties such as porosity, grain size, raw materials used, measurement methods and equipment used for measurements affect the dielectric properties and readers should be aware that exact comparison of data on materials of identical composition and manufactured in different laboratories using different processing conditions would be expected to lead to small variations in properties. The dielectric data measured by impedance methods at low frequencies are not included in the Table since it is unreliable when the loss tangent is less than 10^− 3. The Table shows nearly 4000 low-loss dielectric ceramic compositions reported in the literature. About 35% of them belong to the interesting, widely applicable perovskite family. The analysis of crystal systems shows that many of them enable interesting low-loss dielectric properties. The most common one is orthorhombic (35%), followed by hexagonal (18%), monoclinic (12%), cubic (12%) and tetragonal (10%) crystal systems. About 60% of the reported low-loss dielectric ceramics are based on alkaline earth metals like Ba, Sr, Ca or Mg. Additionally, titanates (46%) and compositions containing rare earths (40%) or tantalates/niobates (39%) are widely reported. Silicates and tungstates are also well represented. Understanding the relation between bonding mechanisms and the microwave dielectric properties is essential. The silicates, to mention one example, have in general predominantly covalent bonding which geometrically restricts the movement of atoms and leads to low dielectric loss. On the other hand, the low dielectric polarisability of silicon and the strong covalent bonds in silicates yield low ε_r. Thus, in general, the silicates and tungstates have low ε_r, niobates and tantalates have medium ε_r, and titanates have relatively larger ε_r. Another example is formed by an octahedral arrangement of anions within a perovskite family of materials where octahedral tilting, brought on by a geometrical instability related to the relative sizes of A and B cations, is accompanied by symmetry lowering and affects the dielectric loss. As expected the reported quality factors of the microwave dielectric ceramics decrease significantly with increasing relative permittivity as shown in Fig. 16. The inset in Fig. 16 shows the variation of quality factor frequency product with relative permittivity in the logarithmic scale.

The 0.993MgO–0.007B₂O₃ material has the highest quality factor (Qf = 773 700 GHz, with ε_r = 9.3 and τ_f of − 55 ppm/°C). On the other hand, the composition 0.8SiO₂–0.2B₂O₃ has the lowest relative permittivity (ε_r = 3.6, Qf = 70 600 GHz and τ_f of − 11 ppm/°C). Its relatively low Qf value can be explained by the glassy nature of this material. AlPO₄ is difficult to densify and it has a very low permittivity of 3.0. These low-ε_r materials are important for increasing the signal speed in communication systems. At the other end of the scale Ba_0.6Sr_0.4TiO₃+0.5 wt-% MgCo₂(VO₄)₂ composite represents the highest relative permittivity (Qf = 300 GHz, with ε_r = 2763). Figure 17 shows the variation of τ_f with relative permittivity.

In general the materials with lower relative permittivity show negative τ_f and high permittivity materials have a positive τ_f. The Bi₆Ti₅TeO₂₂ has the highest temperature variation of resonant frequency (Qf = 220 GHz, with ε_r = 350 and τ_f of +2600 ppm/°C) and BaNb₂O₆ (hexagonal) has the highest negative τ_f (Qf = 4000 GHz, with ε_r = 42 and τ_f of − 800 ppm/°C); however, this is a question of optimisation since there are several ways to tune the τ_f value such as by forming composites with positive and negative τ_f materials. The table (supplementary file) shows the availability of materials with almost any desired relative permittivity especially in the range of 5–100; however, simultaneously satisfying a desired relative permittivity with excellent Qf and τ_f values is difficult.

Tailoring microwave dielectric properties

Microwave dielectric properties can be tailored by chemical methods like doping, slight deviations from stoichiometry, or the formation of composites of dielectrics with oppositely signed τ_f values.^60–66 Luiten et al.^64,67 used paramagnetic effects of impurity ions to compensate for the permittivity–temperature dependence (τ_ε), which is related to τ_f by equation (28); but this technique is not applicable at cryogenic temperatures or even room temperature because of the finite energy gap of paramagnetic resonance. Hartnett et al.^68,69 proposed a method of compensating for the frequency–temperature dependence (τ_f) of high-Q monolithic sapphire resonators near liquid-nitrogen temperatures by doping single-crystal sapphire with Ti³⁺ ions. Breeze et al.⁷⁰ reported a new method of achieving temperature compensation by coating a film of TiO₂ on the surface of an alumina disc. The composite resonators obtained by firing at 1400°C showed a temperature compensation depending on the volume fraction of TiO₂. Materials having negative τ_f are usually tailored by adding TiO₂, CaTiO₃ or SrTiO₃, all of which have high positive τ_f values.^66,71–79 Similarly, positive τ_f materials can be tailored by adding negative τ_f materials.^80,81 For example, the addition of about 17 mol% TiO₂ in ZnAl₂O₄ results in a nearly zero τ_f as shown in Fig. 18. The quality factor and the relative permittivity also vary with TiO₂ content.

Of course, this technique can only be used when the additive material does not react with the parent material. It is also possible to tailor τ_f by stacking positive and negative τ_f resonators. The resultant properties depend on the volume fraction or thickness of the two different resonator materials.^82–85 Fig. 19 shows a sketch of the stacking and the variation of τ_f as a function of the volume fraction of the negative τ_f ( − 66 ppm/°C) resonator Sr(Y_1/2Nb_1/2)O₃ in a composite with the positive τ_f (+78 ppm/°C) resonator Ba₅Nb₄O₁₅. It was reported⁸³ that the properties slightly change on reversing the bottom and top resonator samples. The samples can be joined using low-loss adhesives, but the use of adhesives lowers the quality factor.⁸⁴

It is also possible to tailor properties by solid-solution formation between positive and negative τ_f materials provided they have similar crystallographic structures.^86–91 If the end members have different crystal structures, then a phase transition at some intermediate composition may result in sudden change in the dielectric properties.⁸⁷ Fig. 20 shows the variation of the dielectric properties of (1 − x)CaTiO₃–xNdAlO₃ solid solution. A zero τ_f is observed for x = 0.3. The solid solution can be represented by Ca_1 − xNd_xTi_1 − xAl_xO₃.

A slight non-stoichiometry is also sometimes found to improve the densification and microwave dielectric properties.^92–96 The presence of vacancies can facilitate atomic diffusion and thereby increase densification. For example, slight Ba or Mg deficiencies in Ba(Mg_1/3Ta_2/3)O₃ is found to improve densification, order parameter and quality factor, as shown in Fig. 21.

The addition of suitable dopants can improve the microwave dielectric properties, and a study of the dielectric table reveals that the microwave dielectric properties can also be tailored to some extent by suitable chemical substitution.^97–102 Usually, the dopant partially substitutes at appropriate sites in the parent material. For example, it is reported that the quality factor reaches a maximum when the ionic radius of the dopant is close to the average ionic radius of the B-site ion in Ba(Mg_1/3Ta_2/3)O₃ (BMT) and Ba(Zn_1/3Nb_2/3)O₃ (BZN) ceramics.^103,104 In BMT the Qf reaches a maximum when the ionic radius of the dopant is between 0.6 and 0.7 Å, and the weighted average ionic radius of Mg and Ta is 0.653 Å. Figure 22a shows the variation of Qf in BMT ceramics as a function of the concentration of various dopants. A very small amount of dopant is found to improve the quality factor, with slight changes in relative permittivity and τ_f. Figure 22b shows the variation of Qf in BMT as a function of the ionic radius of the dopant.

Many of the materials are difficult to densify even sintering at high temperatures, and such materials are usually densified by adding a small amount of low-melting-temperature compounds or glasses. The high sintering temperatures can also be lowered and the densification improved by liquid-phase sintering via the addition of low-melting-temperature compounds such as V₂O₅, Bi₂O₃, CuO, LiF, MgF₂, CuO, B₂O₃, Nb₂O₅, Li₂CO₃, BaCuB₂O₅, MoO₃, Li₂WO₄, CuV₂O₆, PbO, etc. and glasses such as Li₂O–B₂O₃–SiO₂, Li₂O–MgO–ZnO–B₂O₃–SiO₂, MgO–B₂O₃–SiO₂, ZnO–B₂O₃, CaO–B₂O₃–SiO₂, B₂O₃–P₂O₅, MgO–CaO–Al₂O₃–SiO₂, ZnB₂O_4, Bi₂O₃–B₂O₃, Al₂O₃–B₂O₃–SiO₂, ZnO–B₂O₃–SiO₂, BaO–B₂O₃–SiO₂, Bi₂O₃–B₂O₃–ZnO–SiO₂, PbO–B₂O₃, PbO–B₂O₃–SiO₂, Li₂O–Zn–B₂O₃, Li₂O–B₂O₃–SiO₂, Li₂O–MgO–B₂O₃, Bi₂O₃–B₂O₃–ZnO–SiO₂, BaO–B₂O₃–SiO₂–CaO–Al₂O₃, BaO–B₂O₃–Li₂O–CuO–, PbO–Al₂O₃–SiO₂, La₂O₃–ZnO–B₂O₃, PbO–Bi₂O₃–B₂O₃–ZnO–TiO₂. The addition of a small amount of several glasses is found to be effective in lowering the sintering temperature and improving microwave dielectric properties of BMT ceramics. Figure 23 shows the effect of some selected glasses on the Qf and τ_f of BMT ceramics. Although the addition of larger amounts of glass considerably lowers the sintering temperature, it also degrades the microwave dielectric properties. In general glasses have negative τ_f values, and glass addition improves the τ_f of materials with positive values of τ_f.

Compounds like CeO₂, MnCO₃, SnO₂, NiO, ZnO, WO₃, TiO₂, Yb₂O₃, ZrO₂ etc. have also been used to aid solid-state sintering and improving the dielectric properties.^{103,106–112} Partial substitution by elements with higher dielectric polarisability can increase the relative permittivity.¹¹³ For example, as shown in Fig. 24, substituting 44% of the Sr²⁺(α_i = 4.24 Å³)¹¹⁴ in Sr₉Ce₂Ti₁₂O₃₆ with Pb²⁺(α_i = 6.58 Å³)¹¹⁴ increases the relative permittivity from 183 to about 800.¹¹³

In several complex perovskites an order–disorder transition is found to affect the microwave dielectric properties. Improvement in ordering by annealing or doping is found to improve the quality factor considerably.^103–105 The purity and origin of the initial raw materials can also influence the phase formation, densification and microwave dielectric properties. The presence of porosity decreases the relative permittivity and a correction for porosity can be performed using mixture rules, as discussed in section Correction for Porosity. The presence of porosity considerably increases the loss tangent for otherwise dense ceramics, as shown in Fig. 25 for alumina.¹¹⁵

A dense ceramic usually optimises the microwave dielectric properties. Figure 26 shows a typical microstructure of thermally etched dense ceria ceramic sintered at 1675°C. Ceria has a relative permittivity of 24 and Qf of 65 000 GHz.¹¹⁶ All types of defects contribute to extrinsic dielectric losses. For an ideal material, loss is mainly a manifestation of the interaction of the phonons with microwaves, hence it is possible to improve the quality factor by suppressing the phonons by cooling the ceramics. Figure 27 shows the variation of the quality factor of ceria ceramic as a function of cooling. The quality factor reaches a maximum of about 10 000 at 6 GHz at 50 K.

Applications of low-loss dielectric ceramics

Materials for LTCC applications

High and low temperature co-fired ceramics, HTCC and LTCC respectively, have created a new generation of small and lightweight electronic multilayer components with application area such as capacitors and microwave products. The LTCC tapes are fabricated from suitable choice of low-temperature sinterable dielectric materials. Currently these LTCC substrates are being developed by industrial organisations like DuPont, Ferro and Motorola. The developmental activities (basic, applied and product development) of dielectric materials have shown substantial increase in the last decade resulting in a variety of dielectric materials for choosing the required compositions with respective properties.

Ceramic composition that has a sintering temperature from 700 to 950°C can be categorised to belong to LTCCs. The upper limit comes from the requirement that the tapes made of it should densify in co-firing with high conductive electrode material like Ag or Cu. At lower sintering temperatures than 700°C, other electrode material like Al, Pd or different mixtures should be selected and the resistance of the electrodes increases highly. One must note in order to enable multilayer co-fired structures, the tapes made of the LTCC have to be co-fired with Ag or Cu pastes without excess reactions.

Low-melting and low-loss glasses are usually added to low-loss dielectric ceramics in order to decrease the sintering temperature below the melting point of the silver electrode. The addition of glasses degrades the dielectric and mechanical properties. Another option is to add sintering aids which is the most common way with LTCCs having high relative permittivity. The Table includes several compositions such as vanadates, telleurates, tungstates, molybdnates and phosphates based on Li, Mg, suitable for glass free LTCC applications and several ceramic glass composites. The reader is referred to the review on LTCC for more details in reference.³ It may be noted that many of these reported LTCC materials are not prepared in tape form and the reactivity with electrode, thermal expansion, thermal conductivity, etc. are not reported in the literature. Although several glass free LTCC materials are available, tapes of very few glass free materials are reported in the literature.^117,118 The important characteristics required for LTCC applications are as follows:

(a) relative permittivity ε_r>4

(b) tanδ < 10^− 2 at 5 GHz

(d) no reactivity with the electrode materials

(e) coefficient of linear thermal expansion less than 20 ppm/°C or matching with that of silicon

(f) high thermal conductivity.

Materials for ULTCC applications

There is a clear need for electroceramic compositions feasible for co-firing with organic or semiconductive structures expecting sintering temperatures less than 650°C using aluminium or less than 400°C using nano silver ink electrodes. In semiconductors, metal electrode should be deposited on top of the dielectric layers with low temperature process. Additionally, multilayer packages similar nowadays to those made by LTCC technology but with much lower sintering temperature would enable co-firing of semiconductor devices into the package. In the recent decade several electroceramic compositions with sintering temperature below 700°C have been reported as shown in the Table. These materials fall in two categories. The first one (category II) covers compositions having sintering temperature over 400 up to 700°C. This category is justified since in these temperatures only Al, Pd or different metal mixture electrodes with relative low conductivity can be used. These ULTCC II category materials can be used on some metal, glass or ceramics substrates, but their feasibility for real multilayer applications is somewhat limited although there are many interesting compositions like Li₂Mo₄O₁₂ sintered at 630°C has the highest Qf of 108 000 GHz with ε_r = 8.8 and τ_f = − 89,¹¹⁹ Zn₂Te₃O₈+30 wt-% TiTe₃O₈ sintered at 610°C has the lowest τ_f of 3 ppm/°C with ε_r = 19.8 and Qf = 50 000 GHz.¹²⁰ The [(Li_0.5Bi_0.5)xBi_x][Mo_xV_1 − x]O₄ with x = 0.098 when sintered at 650°C has the highest ε_r of 81 with Qf of 8000 GHz, τ_f of 10 ppm/°C.¹²¹ The main application areas can be found at moderately low frequency areas. Important application fields could be multilayer capacitors and packages.

The category I, with sintering temperature at 400°C or below, should be feasible with commercially available highly conductive nano silver inks in co-firing. These compositions have commonly ultra-low sintering temperature inherently. Although only very few compositions so far belonging to this category are reported, they will in the future provide great opportunities with integrated applications with semiconductor devices or on organic substrates. The NaAgMoO₄ has the lowest sintering temperature of 400°C among the reported materials. It has a relative permittivity of 7.9 with Qf of 33 000 GHz and τ_f of − 120 ppm/°C.¹²² On the other hand, most of these ULTCC I materials are based on vanadates and molybdates which are soluble in water. This means the ultimate device needs suitable encapsulation. The research and development of ULTCC materials are still in their initial stage. There is an urgent need for developing materials with sintering temperature less than 400°C for future applications.

Materials for dielectric resonators

Ceramic dielectric resonators are widely used for commercial and military purposes from MHz frequencies up to 50 GHz. Their main advantages are compact size, temperature stability and high unloaded Q factor. Commonly used products are dielectric resonator oscillators (DROs), low-loss filters and combiners, and in unmetallized form are intended to operate in the TE_01δ mode. With different kind of cavity shielding or metallic coating, the performance of the DR is adjusted for the product demands. In the case of DROs a low phase noise is needed and thus materials with high-Q factor are used. In commercial DROs, the ceramic resonators have relative permittivity from 20 up to 50 with Qf values as high as 100 000 GHz. The dielectric properties are commonly measured in the frequency range from 2 GHz up to 10 GHz. Regardless of the application, thermal stability of the resonant frequency ( − 10 < Tf < 10 ppm/°C) is desirable.

Materials for dielectric ceramic antennas

The ceramic dielectric resonators are often enclosed inside metal cavities to confine radiation and to maintain a high-quality factor which is important for filter and oscillator applications. When the metallic shield is removed and with suitable feeding to excite appropriate mode, the resonators could become efficient radiators. McAllister and Long¹²³ proposed the use of resonators for antenna applications. For details of dielectric resonator antennas (DRA), the reader is referred to the recent reviews.^124–126 There is no inherent conductor loss in dielectric resonators which leads to high radiation efficiency. Simple coupling schemes can be used for DRA to most of the transmission lines used in microwave and millimetre-wave frequencies. Experimental and theoretical studies are extensively done on DRA with different shapes or geometries such as cylindrical, rectangular, circular, ring, conical, hemispherical and square-shaped structures. The dimension of the resonator is related to free space wavelength and ε_r by equation (2) and by choosing a high ε_r, the size of the DRA can be significantly reduced at the expense of bandwidth. The operating band width can be varied for a wide frequency range by suitably selecting the resonator parameters and a band width of 117% have been reported.¹²⁷ The lowest frequency of DRA reported is 55 MHz¹²⁸ and the highest 94 GHz.¹²⁹ DRA's with dimensions ranging from a few millimetre with ε_r in the range of 6–100 have been reported.^124–126 Very thin ( < 4 mm) structures are needed especially when integrated to portable terminals. Ceramic dielectric materials are widely used for GPS patch antennas leading to high performance and miniaturisation. Ceramic antennas have also been proposed for multi-purpose targets like machine-to-machine communication.¹³⁰ They are supposed to operate in the Zigbee, ISM and cellular bands including LTE in the frequency band between 700 and 2500 MHz. Ceramic chip antenna is calculated to provide 80% reduction in PCB space for 2.45 GHz applications.

Materials for millimetre-wave applications

The millimetre-wave radio spectrum is expected to be used in the future (e.g. 5G networks) since higher carrier frequencies are possible as compared to the current systems, such as 4G and Wi-Fi. For millimetre-wave applications the relative permittivity should be approximately in the range of 6–20 having very high-quality factors greater than 75 000 GHz with temperature stable dielectric properties. High permittivity materials in general have lower quality factors and also have the problem of fabricating extremely small sized resonators. ZnAl₂O₄–TiO₂-, Mg₂SiO₄-, Mg₄Ta₂O₉- and Al₂O₃-based materials are some of the examples for possible millimetre-wave communication in radars, space, 5G and military applications.

Materials for future applications and conclusions

The DR table indicates a large number of materials with very useful microwave dielectric properties. However, important emerging technologies will require seamless co-firing with plastics or paper substrates (printed electronics) or semiconductor devices. The DR table exhibits about 120 materials with Ultra Low Sintering Temperature (ULTCC – sintering temperature less than 700°C) and with materials like NaAgMoO₄ sintering temperature even lower than 400°C. However, further research is required to enable low-loss dielectric microwave ceramics to integrate with plastics and feasible with nano silver and other electrode materials. Recently room temperature curable silica ink has been screen printed on flexible mylar substrate for printed applications.¹³¹ The screen printed silica has a relative permittivity of 2.4 and tanδ of 0.003 at 15 GHz. Another field of application, the health care systems or monitoring, requires suitable antenna materials for bio-implantable communication devices.¹³² Apatite-type materials with reasonably good microwave dielectric properties are reported¹³³ (see the Table). However, their biocompatibility needs to be investigated for practical applications.

One interesting field for microwave ceramics is low-loss polymer–ceramic dielectric composites reported for antenna and printed circuit board applications.¹ However, they are rigid and not bendable or stretchable especially if the loading level of ceramic is high. Flexible, bendable and stretchable dielectrics which can cover even curved surfaces are important for applications in electronic control systems, consumer electronics, heart pacemakers, body worn antenna, etc. The requirements for a material to be used as a flexible dielectric waveguide are mechanical flexibility, high relative permittivity, low dielectric loss, high thermal conductivity, low coefficient of thermal expansion (CTE), etc. Recently low-loss ceramic-filled butyl rubber and silicon rubber-based composites have been reported,^134–136 but further work is needed for device optimisation.

Several applications areas like the semiconductor industry are in constant need for low-loss materials with ultra-low relative permittivity (low k materials) to reduce RC signal delay. Lowering the relative permittivity decreases power consumption and reduces cross-talk between nearby interconnects. Silica, which has the lowest permittivity of about 4.0, is commonly used as the low k material. Further decrease of the permittivity can be achieved by introducing porosity. However, the presence of porosity degrades the mechanical properties. Fluorination of silica (SiOF) lowered the permittivity to about 3.6.¹³⁸ SiCOH with k of about 2.4 have also been reported.¹³⁷ Recently several organic or hybrid dielectric materials have been developed¹³⁷ with even lower permittivities, but they are not suitable for very large-scale integration (VLSI) chips because of their poor chemical, mechanical and thermal properties. There remains a need to develop materials with lower relative permittivities and good mechanical, chemical and thermal properties in order to increase the signal speed.

As a conclusion, the study of the DR table reveals that many tantalates, niobates, titanates, silicates, tungstates, molybdanates, vanadates or tellurates based on alkali earth metal and rare earths show low dielectric loss. It seems that most of the low-loss dielectric microwave ceramic materials have an octahedral or tetrahedral arrangement of atoms. However, further investigation, including especially spectroscopic and XRD studies, is needed to understand the relationship between chemical bonding, lattice vibrations, atomic coordination, secondary phases, impurities and microwave dielectric properties. Such studies would be useful for finding new low-loss dielectric ceramic compositions for present and future applications. Attempts should be also done to lower the cost of production of microwave materials with emphasis on use of environment friendly materials with the possibility of recycling.¹³⁸ In the near future, the new emerging communication applications like 5G network machine-to-machine connection and IoT will need novel dielectric ceramics with feasible component fabrication technologies. This means that the low-loss microwave ceramics will continue to be an active area of research in years to come. The future will show their importance for improved performance with cost-efficient and miniaturised devices. The operational details of 5G networks and the IoT are still not available and hence the material requirements are yet to be determined.

Footnotes

Acknowledgement

The authors are grateful to European Research Council (ERC project) and the US National Science Foundation (DMR 1052788) for financial support.

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Low-loss dielectric ceramic materials and their properties

Abstract

Introduction

Measurement of microwave dielectric properties

Permittivity

Hakki–Coleman method

Shielded resonator in dielectric-rod waveguide method

Correction for porosity

Measurement of loss tangent/quality factor

TE01δ mode DR method

Strip line excited by cavity method

Whispering gallery mode resonator method

Split-postdielectric resonator method

Measurement of dielectric properties of powder samples

Measurement of temperature coefficient of resonant frequency (τf)

Factors affecting dielectric losses

Low-loss dielectric ceramics

Tailoring microwave dielectric properties

Applications of low-loss dielectric ceramics

Materials for LTCC applications

Materials for ULTCC applications

Materials for dielectric resonators

Materials for dielectric ceramic antennas

Materials for millimetre-wave applications

Materials for future applications and conclusions

Footnotes

Acknowledgement

References

TE_01δ mode DR method

Measurement of temperature coefficient of resonant frequency (τ_f)