Introduction to Dielectric Resonators 1.1 Introduction to dielectric resonators
1.5 Physics of DR materials
1.5.1 Polarization mechanisms in dielectrics
Polarization is an ordering in space of an electrically charged unit under the influence of an external electric field. This causes the formation of an electric moment in the entire volume of the dielectric and in each separate polarizing unit (atom, ion or molecule). Linear dielectrics show a direct proportionality between the induced electric dipole moment p acquired by the polarizable unit during the process of polarization and the intensity E of the field acting on it as given by p = αE. α is known as the polarizability. Polarizability reflects the properties of individual polarizable unit and not of a certain volume of matter and is the most important microscopic electrical parameter of a dielectric. During polarization, the charges that are displaceable will be brought in to motion. Piling up of mobile charge carriers at physical barriers such as grain boundary causes interfacial polarization or space charge polarization. At low frequencies (~1 kHz) this mechanism gives rise to a high dielectric constant. Dipolar polarization occurs due to the molecules containing permanent dipole moment or by the rotation of dipoles between two equilibrium positions, and this relaxes around 103 to 106 Hz. The ionic polarization occurs due to the displacement of the positive and negative ions against each other, and it relaxes in the frequency range of 1012 to 1013 Hz.
Electronic polarization occurs due to the displacement of electrons with respect to the atomic nucleus, and it relaxes at high frequencies ~ 1015 Hz. The mechanisms of these four polarizations are shown schematically in Figure 1.3. Each of these involves a short - range motion of charge and contributes to the total polarization of the material, and at microwave frequencies ionic and electronic polarizations contribute to the dielectric properties [19].
Figure 1.3: Different types of polarization mechanisms appear in a dielectric [19].
1.5.2 Dielectric losses
According to the material requirements for DRs, the dielectric losses and temperature dependence of dielectric properties assumes more significant.
Generally, three mechanisms can be distinguished for dielectric losses at microwave frequencies [7].
The dielectric losses related to phonon damping are usually described in the framework of classical dielectric function
( )
ω ε ω ω γ ωε
j
j jTO i
j + + −
= ∞
∑
2 2 (1.7)where, ε∞ is the permittivity at optical frequencies,
jTO
ω are the frequencies of the transverse optical (TO) lattice modes, and fj and γj are their oscillator strengths and damping constants, respectively. The dielectric loss at microwave frequencies (ω <<ωjTO, i.e.ω= 0) can be calculated from equation (1.7) as
( )
4tan
jTO j
j O
j ω
γ ω ε
δ =
∑
(1.8)For the practically important case of kβT being larger than the phonon energy, the anharmonic interactions between phonons caused by the third and fourth order terms of the lattice potential lead to γj ∝T and γ j∝T2, respectively [20]. The dominating dampening process among these is the decay of one TO phonon into two acoustical (thermal) phonons [21]. Equation 1.8 shows that, in cases of dominating intrinsic losses, the product of quality factor (Q = 1/tanδ) and frequency f (ω = 2πf ) is expected to be constant, a rule which holds, at least approximately for many ceramics. Recent days, it is widely accepted that intrinsic losses can be determined by far - infrared (FIR) spectroscopy, and therefore this method is now often used for the characterization of classical microwave ceramics and related compounds [22]. FIR spectroscopy greatly simplifies the search for new low loss ceramics because in the FIR frequency range, intrinsic losses out number extrinsic ones, and thus FIR data are much less sensitive to processing than the losses measured at microwave frequencies.
Understanding the intrinsic loss mechanisms are useful to examine the extent to which the losses of microwave ceramics can be reduced to the small intrinsic values by using advanced processing methods [23]. It is also interesting to note that the intrinsic losses are closely related to the value of permittivity a power law, tanδ ∝εa with a = 4 [7].
1.5.3 Temperature coefficients
To obtain temperature stable oscillators and filters, it is necessary that the resonant frequency (f0) of a resonator does not change much over a large temperature range (generally
in the range of -20 to + 80oC). The temperature coefficient of resonant frequency τf is given by [7]
T f f T
f
f f
∆
⋅∆
∂ ≈
⋅ ∂
= 0
0 0
1
τ 1 (1.9)
where T is the temperature. Because the resonant frequency of a resonator depends on its size and onε, the following equation holds true [7].
τf ≈ -(τε/2 + αL)
where τε is defined according to τf in equation (1.9) and αL is the linear thermal expansion coefficient, i.e. the temperature coefficient of one of the resonator dimensions (eg. length, l).
The requirement that τf = 0 translates then in to τε ≈ -2αL. In addition, it is often necessary that the τf in a material system be variable between positive and negative values by virtue of variation of the materials composition. This enables one to set the temperature drift of the composite element containing a resonator equal to zero [7].
From the equation 1.10, the discussion on temperature coefficients can be restricted to consider only the temperature coefficient of permittivity τε and τε is defined as [7]
τε =
( )( )
−
∂ + ∂
∂
∂ +
−
L L
T r
r r
T α
α α α
υ α α υ ε
ε ε
3 υ
1 2
1 (1.10)
The above equation directly follows from the Clausius-Mosotti equation, which takes the long-range coulomb interactions into account. From the equation 1.11 it is important to note that the polarizability α equals the sum of polarizabilities of all atoms in a unit cell with volume υ only if all the atoms in the structure have a cubic environment. This is the case for alkali halides but not for complicated structures. However, the additional dipole fields can be taken in to account by an effective polarizability α so that both the Clausius - Mosotti equation and equation 1.11 are valid. In the case of perovskites, for example, the effective polarizabilty α is obtained by adding an extra ionic polarizabilty ∆αi to the electronic and ionic polaraizabilities of all atoms in the unit cell. This supplemental
polarizability ∆αi caused by the structure contributes an additional term proportional to -αL αi
∆ in equation 1.11 [7].
In materials with more complicated structures, the additional ionic polarizabilty ∆αi is not only raises εr value but also shifts τε to negative values [7]. Both strong electronic polarizabilities and ∆αi also give rise to a softening of the lattice modes [7]. Thus the anharmonic contribution
υ
α
∂
∂
T to τε increases and makes τε more negative. In the case of microwave dielectrics, ionic polarizabilitiy dominates εr and thus τε is affected [7].