Dielectric properties

CHAPTER 15

Dielectric properties 15.1 Introduction When a direct current (dc) voltage V is applied to a large parallel plate capacitor having a narrow vacuum gap d between the plates of area A each, the plates are charged with +q and
q charges per unit area immediately. The electrical flux density or the electric displacement vector D is defined by the integral form of the Maxwell third field equation (Gauss’s law) stating that the surface integral of the normal component of D over an arbitrary enclosed surface is equal to the net charge enclosed by the surface: ð D  ndS ¼ Aq, (15.1) where n is a unit vector normal to a surface element dS. In a parallel plate capacitor, the flux lines originate from the charge on the positively charged plate and end at the negative charge on the opposite plate. Thus, the D lines are parallel to n with the result that Eq. (15.1) reduces to D0 ¼ qn,

(15.2)

where the subscript “0” has been used to denote vacuum. The electrical field strength vector E between the plates is defined by D0 ¼ ε0 E,

(15.3)

where ε0 is the absolute permittivity (dielectric constant) of free space, which in SI units has the value 8.854  10–12 C2 N
1 m
2 (¼ farad m
1). E is related to the potential through ðd (15.4) V ¼ E  jdy ¼ Ed, 0

where j is a unit vector in the direction of the plate separation (assumed to be along the y-axis). The capacitance C0 of the parallel plates is given by C0 ¼ Aq=V ¼ Aε0 =d, by substituting for q and V :

Fundamentals of Inorganic Glasses https://doi.org/10.1016/B978-0-12-816225-5.00015-8

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(15.5)

425

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Let us now replace the vacuum between the plates by a dielectric which is not a good conductor of electricity. The action of a dielectric between the plates is to reduce the applied voltage, or if the voltage were to be maintained constant then the charge on the + q plate should increase to +(q + P) and on the
q plate to –(q + P). The increase in charge P is called the polarization and can be considered to result from the appearance of
P “bound” charges on the dielectric surface adjacent to the + q plate. In vectoral representation. P ¼nP where n is a unit normal. The flux lines originating from the +P “bound” charges on the +(q + P) plate end on the
P “bound” charges on the adjacent dielectric surface (see Fig. 15.1). The capacitance C of the new system is now increased to A(q + P)/d and the new electric flux density is (q + P). The ratio C/C0 (which is greater than 1) is called the static relative permittivity εs (or the relative dielectric constant). Since the net electric displacement D may be written as D ¼ D0 + P,

(15.6)

we have, using the basic definitions: εs ¼

C q+P D D ¼ : ¼ ¼ C0 q D0 ε0 E

(15.7)

From Eqs. (15.6) and (15.7) we get several other useful relations: P P ¼ χ, εs
1 ¼ ¼ q ε0 E

(15.8)

where χ is called the dielectric susceptibility, and D ¼ ε0 E + P ¼ εs ε0 E ¼ εE,

(15.9)

where ε is called the electric permittivity or the dielectric constant of the medium (similar to ε0 of vacuum). Had we carried the vectoral representation, we would have arrived at D ¼ D0 + P ¼ ε0 E + P ¼ εs ε0 E ¼ εE:

(15.10)

In general, the electric displacement vector D need not be parallel to the electric field vector E, and hence, ε is a second rank tensor. However, in isotropic bodies such as glass, the vectors D and E are parallel, hence, ε may be treated as a scalar quantity. Also, from Eq. (15.10)

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427

Fig. 15.1 (A) A parallel plate capacitor connected to a potential difference V. With vacuum between the plates, the electric displacement is D0 (magnitude ¼ charge q). The flux lines start from the + charges on the positive plate and end on the
charges on the negative plate. (B) On introducing a dielectric between the plates, polarization charges  P appear on the dielectric facing the plates. This causes the charges on the plates to become (q + P) and
(q + P). Inset shows exaggeration of the plate-nearest dielectric face region. (Adapted from B.K.P. Scaife, Principles of Dielectrics, Oxford Science Publishers, Clarendon Press, Oxford, 1989, Fig. 1.2, p. 3. Reproduced with permission of Oxford University Press.)

P ¼ ðεs
1Þε0 E ¼ ðε
ε0 ÞE ¼ αE,

(15.11)

which defines the polarizability α of the medium. Again, the polarization vector is assumed to be parallel to the field vector E for glass such that the polarizability α is a scalar quantity. Henceforth, the use of bold letters in this chapter shall be dropped, in effect, treating all quantities as scalars. It may be recognized that dielectric polarization involves the appearance of induced charges as dipoles (equal charges of opposite sign separated by distance d). These dipoles create a depolarization field whose direction is opposite to the field that generates the polarization. With time some of the

428

Fundamentals of Inorganic Glasses

induced charges tend to migrate toward each other and hence the depolarization field will decay. A capacitor with a real dielectric medium filling the space may thus be equated to various combinations of resistance and capacitance. (The presence of vacuum between the capacitor plates implies the existence of a pure capacitor or a perfect dielectric.) For a series combination of capacitance Cs and resistance Rs, it may be shown [1] that for the charging of a capacitor with an applied step voltage, V0, the charge Q has a time dependence: h  t i Q ¼ Cs V0 1
exp
, (15.12) τ where τ ¼ RsCs is the time constant. The buildup of the full charge is, thus, not instantaneous. For discharging of the capacitor, the charge Q decays according to  t Q ¼ Cs V0 exp
: (15.13) τ The preceding discussion also implies that the sudden application of a constant field produces an instant polarization P∞ which approaches a static value Ps over a period of time. The two polarization values may be defined in terms of Eq. (15.11) as P∞ ¼ ðε∞
1Þε0 E and Ps ¼ ðεs
1Þε0 E:

(15.14)

Here, ε∞ may be called the instantaneous dielectric constant. The subscript ∞ refers to infinite frequency and not infinite time. If an alternating potential represented by V ∗ ¼ V0 exp ðiωt Þ, (15.15) pffiffiffiffiffiffiffi where i 
1 and ω is the angular frequency, is applied to the capacitor plates, then an alternating field develops between the plates with the field strength E* given by V ∗ V0 (15.16) ¼ exp ðiωtÞ ¼ E0 exp ðiωt Þ: d d The complex displacement flux D* for a real dielectric is, in general, not in phase with E*, and is given by E∗ ¼

D∗ ¼ D0 exp ½iðωt
δÞ,

(15.17)

where δ is the phase difference by which the electric displacement lags the electric field. For the alternating field, one can rewrite Eq. (15.10) in terms of D* and E* as D∗ ¼ ε∗ ε0 E ∗ ,

(15.18)

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429

where ε* is the complex (relative) dielectric constant of the form ε∗ ¼ ε0
iε00 :

(15.19)

0

The real part, ε , is the (static) dielectric constant, and the imaginary part, ε00 , is the loss factor. ε* approaches εs at low frequencies, and ε∞ at high frequencies. These terms may now be renamed as the zero-frequency and the highfrequency dielectric constants, respectively. The complex capacitance C* may be defined from C ∗ ¼ ε∗ C0 :

(15.20)

It follows that ε∗ ε0 ¼

D0 D0 exp ð
iδÞ ¼ ð cos δ
i sinδÞ, E0 E0

(15.21)

and hence, the ratio ε00 ¼ tan δ, (15.22) ε0 where δ is the phase angle defined by Eq. (15.17). The term tan δ is called the loss angle or the loss tangent. The frequency dependence of ε0 and ε00 is generally described in terms of the Debye relations. They are as follows: εs
ε∞ ε0 ¼ ε ∞ + (15.23) 1 + ω2 τ2 ωτ ε00 ¼ ðεs
ε∞ Þ (15.24) 1 + ω2 τ 2 where τ is called the relaxation time for the process. The term (εs
ε∞) ¼ Δε is sometimes called the dielectric relaxation strength. The variation of ε0 and ε00 as a function of frequency is shown in Fig. 15.2. These curves are referred to as dielectric dispersion and absorption (or loss), respectively, and are symmetric about ωτ ¼ 1. Note that the loss curve displays a single relaxation maximum at an angular frequency ω ¼ 1/τ. Most real dielectrics (dashed curves in Fig. 15.2) show broader distributions than those predicted by Eqs. (15.23) and (15.24), presumably because of a spectrum of relaxation times. Recall that the current I is the time derivative of the charge on the plate, _ In an alternating field, this leads to that is, I ¼ AD.   V ∗A 0 00 ∗ ∗ ∗ I ¼ Aiωε ε0 E ¼ ðiωε0 ε + ωε0 ε Þ : (15.25) d

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Fundamentals of Inorganic Glasses

Fig. 15.2 The dependence of the dielectric constant and dielectric loss on frequency. Solid curves are Eqs. (15.23) and (15.24); the dashed curves represent the behavior of real dielectrics.

The expression within the square brackets is of the form (cos ϕ+ i sin ϕ), which implies that I leads V by a phase difference ϕ, causing a power loss. Since π  tan ϕ ¼ ε0 =ε00 ¼ cot δ ¼ tan
δ , 2 it follows that ϕ ¼ (π/2)
δ. In a loss-free dielectric, the current leads the voltage by π/2. In contrast, a “lossy” dielectric is a conductor where ϕ ¼ 0, that is, the current is in phase with the applied voltage. Also, I* ¼ iωC*V*. If we write Ohm’s law as V* ¼ Z*I*, where Z* is the complex impedance, then 1 : (15.26) iωC ∗ The reciprocal of complex impedance is the complex admittance Y*. As a material, dielectrics are basically insulators to semiconductors of electricity. When an alternating electric field is applied to a relatively more conducting dielectric slab, the equivalent circuit may be described by a resistance Rp (¼1/Gp; Gp being called the conductance) and a capacitance Cp in parallel (Fig. 15.3). The real parts of the complex electrical conductivity (defined below), σ 0 , and the dielectric constant, ε0 , are given by Z∗ ¼

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431

Fig. 15.3 Equivalent circuit of a real dielectric.

  d σ ðωÞ ¼ Gp ðωÞ , A

(15.27)

  Cp ðωÞ d , ε0 A

(15.28)

0

and ε0 ðωÞ ¼

where both the resistance and the capacitance are, in general, frequency dependent. For the parallel circuit, the admittances, Gp(ω) [from Eq. (15.27)] and iωCp(ω) [from Eq. (15.28)] of the two components are additive. The response of the circuit may be expressed in one of the following four mechanisms [2, 3]: (1) Complex admittance Y*(ω) or conductivity σ*(ω): Y ∗ ðωÞ ¼ Gp ðωÞ + iωCp ðωÞ ¼ Y 0 + iY 00   d σ ∗ ðωÞ ¼ Y ∗ ðωÞ ¼ σ 0 + iωε0 ε0 ¼ σ 0 + iσ 00 A

(15.29) (15.30)

(2) Complex capacitance C* or permittivity ε*: 1 ∗ i Y ðωÞ ¼ Cp ðωÞ
Gp ðωÞ ¼ C 0
iC 00 iω ω   C ∗ ðωÞ d iσ 0 ε∗ ðωÞ ¼ ¼ ε0
iε00 ¼ ε0
ωε0 ε0 A

C ∗ ðωÞ ¼

(3) Complex impedance Z* or resistivity ρ*:

(15.31) (15.32)

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Fundamentals of Inorganic Glasses

Z∗ ¼

Gp ðωÞ iωCp ðωÞ 1 ¼ 2
2 2 2 Y ∗ ðωÞ Gp ðωÞ + ω Cp ðωÞ Gp ðωÞ + ω2 Cp2 ðωÞ 0

¼ Z
iZ

(15.33)

00

ρ∗ ¼ Z ∗ 0

  A σ0 iωε0 ε0
¼ 02 2 2 d σ + ω2 ε20 ε0 σ 0 + ω2 ε20 ε0 2

(15.34)

00

¼ ρ
iρ

(4) Complex electric modulus M*:

  A 1 d M∗ ¼ ¼ ε∗ C ∗ ðωÞ ε0

  A ¼ ε0 iωZ ∗ ¼ ε0 iωρ∗ ¼ ε0 ωρ00 + iε0 ωρ0 ¼ M 0 + iM 00 d

(15.35)

To study dielectric relaxation in such conducting dielectrics, the loss factor, σ/ωε0, corresponding to the dc conductivity, σ, is subtracted from the observed total ε00 [see Eq. (15.32)]. The residual complex dielectric constant ε∗p is then written as   σ 0 00 ∗ ¼ ε0
iεp 00 : (15.36) εp ¼ ε
i ε
ωε0 For an “ideal solid electrolyte,” Gp and Cp are frequency independent. For such a case, ε0 ¼ εs, which is also frequency independent. Hence, from Eq. (15.32), ε∗ ¼ εs


iσ : ωε0

(15.37)

For the equivalent circuit, the current through the resistor IR∗ ¼ V ∗ =Rp , and that through the capacitor IC∗ ¼ Cp V ∗ . The total current I* is given by 

I ∗ ¼ IR∗ + IC∗ ¼

V∗ + Cp V ∗ : Rp 

(15.38)

For the applied alternating current (ac) voltage V ∗ ¼ V0 exp(iωt), Eq. (15.38) can be solved to give   V∗ 1 ¼ Rp Z∗ ¼ , (15.39) I∗ 1 + iωτ where τ ¼ RpCp, is the conductivity relaxation time characterizing the decay of the electric field due to the dc conduction process under the constraint of a constant displacement vector D. In other words,

Dielectric properties

 t E ¼ E0 exp
: τ

433

(15.40)

Since εs ¼ (Cp/ε0)(d/A), and Rp ¼ d/σA, we get τ ¼ Rp Cp ¼

εs ε0 : σ

(15.41)

Hence, from Eqs. (15.37) and (15.35), ε∗ ¼ ε s


iεs 1 + iωτ ¼ εs ωτ iωτ

(15.42)

and M∗ ¼

1 iωτ 1 ¼ Ms , where Ms ¼ : ∗ ε 1 + iωτ εs

(15.43)

Plots of the real and imaginary parts of the various quantities as a function of the angular frequency ω, and in the complex plane (by plotting the imaginary part against the real part; called the Cole-Cole plot) are shown in Fig. 15.4 for an ideal solid electrolyte where ε0 is frequency independent and ε00 varies inversely with ω (Fig. 15.4B). For convenience, Rp, Cp, and (d/A) have been set equal to 106 Ω, 10–12 F, and 1, respectively. Since d/ A ¼ 1, Y* ¼ σ*, and Z* ¼ ρ*, it may be noted that the plots for log(Y*) and log(ε*) are straight lines (e.g., Y 0 ¼ Gp, which is constant with ω, and Y00 ¼ ωCp) and ε0 and σ 0 do not show any dispersion. On the other hand, Z 0 , Z 00 , M 0 , and M 00 versus log ω display dispersion and absorption behavior of a single Debye relaxation process; the peak in Z00 and M00 being at ωm ¼ 1/τ; note that the abscissa in Fig. 15.2 is log ωτ whereas that in Fig. 15.4 left-hand side is log ω. Complex plane plots are semicircles with the peak being, again, at ωm ¼ 1/τ. The behavior of M00 or Z00 represents conduction loss. Since it is easier to spot trends when plots show peaks instead of just straight lines, many authors prefer not to use ε* formalism, but use M* and Z* (or ρ*). Further, M 0 ! 0 as ω ! 0 implies a lack of restoring force for the flow of electrical charge in a conducting dielectric in a constant electrical field. This is analogous to the shear modulus at low frequencies at constant shear stress. For this reason, M is called the electric modulus. All dielectrics suffer what is known as a breakdown when the applied voltage exceeds a critical value, called the dielectric strength. The breakdown results in a changeover from an insulator to a conductor behavior, usually in a matter of microseconds. Dielectric breakdown in glass generally results from two reasons: (i) intrinsic in an extreme case of polarization, where under

434 Fundamentals of Inorganic Glasses

Fig. 15.4 Plots of the various quantities associated with the measurement of dielectric properties. (The plots on the right-hand side are the Cole-Cole plots.) The dielectric is assumed to have an equivalent circuit as shown in Fig. 15.3. Assume Rp ¼ 106 Ω, and Cp ¼ 10
12 F. (A) Admittance Y. (B) Dielectric constant, ε. (C) Impedance, Z. (D) Electric modulus, M. (After Hodge et al. [2]. Reproduced with permission of Elsevier Sequoia S. A., Lausanne.)

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435

Fig. 15.5 Propagation delay time vs the dielectric constant. (After McDowell and Beall [4]. Reproduced with permission of the American Ceramic Society.)

the action of high electric fields the electrons are accelerated to the point of causing ionization of atoms, and (ii) thermal, where a material absorbs large amounts of energy by virtue of a high dielectric loss factor causing it to heat up, further lowering its electrical resistivity, increasing current flow, increasing heating, and so on. Dielectric constants are important for many applications. High magnitude capacitors obviously require high dielectric constants. On the other hand, in integrated circuit applications, the substrates are required to have low dielectric constant for higher interconnect signal speeds between chips. As shown in Fig. 15.5, the delay time per length of the interconnect can be decreased significantly by lowering the dielectric constant.

15.2 Measurement of dielectric properties Dielectric properties are measured as a function of the frequency of the alternating voltage using commercially available ac bridges similar to a Wheatstone bridge. An example is the bridge shown schematically in Fig. 15.6. The variable capacitors CB and CN are adjusted until the detector shows minimum current, first with the unknown specimen X attached in parallel to CB, and next without it. The real part C0 (ω) is given by the difference of CB values whereas tan δ is proportional to CB
CN. One should be careful of the electrode material used and the process by which the electrodes are applied to glass specimens. As shown in the inset in Fig. 15.1, applied electrode and the glass surface can act as dipoles.

436

Fundamentals of Inorganic Glasses

Fig. 15.6 Schematic diagram of a bridge circuit to measure dielectric properties. (After N.P. Bansal, R.H. Doremus, Handbook of Glass Properties, Academic Press, New York, 1986, Fig. 14.1, p. 451. Reproduced with permission of the publisher.)

The resulting electrode polarization presents itself as a low-frequency arc in the Cole-Cole plot (the beginning of the arc is shown in Fig. 14.5). Likewise, hydrated surface layers may additionally produce a mid-frequency range arc [4].

15.3 Data on dielectric properties and AC conduction in glass When an alternating potential is applied across a glass, it is expected that the low-frequency regime will yield the dc conductivity. If the glass were an “ideal solid electrolyte,” there would be no dielectric loss mechanism other than the dc conductivity [as indicated by Eq. (15.37)]. A typical plot of ε0 versus log ω for silicate glasses, however, shows dispersion and a plot of ε00 versus log ω shows a skewed Debye-type peak (Fig. 15.7), similar to Fig. 15.2. The dc conductivity, the relaxation strength Δε (¼ε00
ε0 ), and the frequency maximum ωm generally obey the Barton-Nakajima-Namikawa (“BNN”) relationship [6–8]: σ dc ¼ pεo Δεωm where p 1, see Fig. 15.8. The ωm varies with temperature as

(15.44)

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437

Fig. 15.7 Variation of dielectric constant ε0 and dielectric loss ε00 with frequency for a soda lime silicate glass. After Tomozawa [5].

8

–6 5

–7

7

4

6

6

log s (mho · cm–1)

–8

1

–9

13 3

–10

1

11

4 7,12

2 3

7

8

3

5

8

4

–11

2

5 2 4 1 10 3

13

9

–12

T – 1 ~ 13

12 14

–13

C–1~8

11

–14

H–3~5

9

ETL – 1 ~ 14

6

–15

5 10

–15

–14

–13

–12

–11

–10

–9

–8

–7

–6

log wme0De (E · cm · s ) Correlation between conductivity and dielectric relaxation. = Taylor (1957, 1959); = Charles (1962; 1963); Heroux (1958); = measured at Electrotechnical Laboratory, MITI, Japan. (After Nakajima, 1972.) –1

–1

Fig. 15.8 Confirmation of the BNN relationship. The symbols T, C, H, and ETL correspond to measurements at four different laboratories cited by Tomozawa [5].

438

Fundamentals of Inorganic Glasses

0.05

b = 0.48

222°C 131°C 154°C

109°C

0.04

177°C

200°C

0.03 M“ 0.02

0.01

0

1

2

3

4

5

6

log10 [w/2p ], Hz 00

Fig. 15.9 KWW fit of M versus log(ω/2π) for Li2O3B2O3 glass. After Ngai et al. Phys. Rev. B 30 (1984) 2133.

  1 ΔEm
1 ωm ¼ ¼ τ0 exp
RT τ

(15.45)

12 to 1013 Hz. It is observed that ΔEm is almost the same as, where τ
1 0 10 may be slightly less than, the activation energy ΔHD for dc conductivity 00 (Section 14.2.5). The shape of the M versus log ω curve, Fig. 15.9, is neither Debye type nor Lorentzian, but can be fitted to a distribution of conductivity relaxation times, which may be characterized by a Kohlrausch-WilliamsWatts (KWW) relationship (see Chapter 13) with the stretching exponent β 0.4–0.6. Many ionic conductive glasses have an internal friction peak at ωm. Thus, mechanical loss and dielectric loss may be interrelated. The magnitude of the total conductivity is given by Jonscher’s power law [9].

σ ðtotalÞ ¼ σ dc + Aωs

(15.46)

where s 0.6 and A is a constant. This is shown in Fig. 15.10 over a range of temperatures and a very large range of frequencies. At low frequencies, the

Dielectric properties

439

Fig. 15.10 Conductivity spectra for Na2O3SiO2 glass over a range of temperatures and 15 decades of frequency variation. (Prior data replotted by H. Jain (personal communication).)

frequency-independent σ dc is the dominant contribution. The increase in conductivity with frequency at low temperatures is several orders of magnitude; much less so at higher temperatures. One can, in fact, scale the conductivity in a log-log plot to obtain a “master curve.” A large number of disordered solids (regardless of bond types, bond strengths, amorphous or glassy, organic or inorganic, single crystals, or polycrystals, films, or bulk solids) fall onto the same master curve. This behavior, common to dipoles and charge carriers at high frequencies, is referred to as “universal dynamic response” or “ac universality.” It may also be shown that the ratio of energy lost/energy stored is nearly independent of frequency. A significant amount of controversy exists regarding the dielectric behavior of disordered solids as a function of the temperature and the frequency. Fig. 15.11 shows the complexity of the behavior for a 7.4Li2O92.6GeO2 glass, taken from Ref. [10]. It has been suggested [11] that the ac conductivity should be written in an augmented Jonscher power law as     ΔHD ΔEac s1 + A1 ω exp
+ A 2 ωs2 T β + A 3 ωs3 σ ðT , ωÞ ¼ σ 0, dc exp
RT RT (15.47)

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Fundamentals of Inorganic Glasses

7.4Li2O-92.6GeO2

10 100

1000 10,000 50,000 –8

–8

–8.5

–8.5

–9 –9.5 –10 –10.5 –11 –11.5 –12 –12.5

–9 –9.5 log (s) –10 S/cm –10.5 –11 –11.5 –12 –12.5 –13 10 10

100

f (Hz)

1000

100 10,000

T (K)

400

50,000

Fig. 15.11 A three-dimensional plot of log σ, log f, and log T for 7.4Li2O92.6GeO2 glass. (Note frequency f ¼ ω/2π.) (After H. Jain, X. Lu, J. Am. Ceram. Soc. 80 (1997) 517–520.)

where A1, A2, and A3 are constants. The first term on the right-hand side is operative at high T and low ω and is the frequency-independent dc contribution of a single ion hop following an Arrhenius rate relationship; the second term is the universal power law with the exponent s1 0.6, applying also at high T but at intermediate ω (1 Hz  ω  several MHz after ω > ωm). It is also thermally activated and obeys Arrhenius relationship with ΔEac  ΔHD, hence, may be related to a single ion hop; the third term actually applies to two regions: (a) high T and high ω (MHz to GHz; useful for microwave applications) for which s2 1 and is weakly temperature dependent (β 0.1). Because of the linear dependence of conductivity on ω, this region is also referred to as the “nearly constant loss” (NCL) region (note ε00 ¼ σ/ε0ω) and (b) low T and low ω. The third term is ascribed by Jain [11] to cooperative vibrations of a group of atoms, much like the fauna of a jellyfish. The last term is T independent with s3 2 at higher frequencies (to far IR), and is ascribed to vibrational relaxation of the mean squared displacement of an ion about its own site, oscillating in a cage of surrounding atoms.

Dielectric properties

441

The reader is referred to a host of review articles for advanced understanding [5, 12–16]. The dielectric constant of glasses is generally in the range of 4–11 at 1 MHz and 20°C. Fused silica has the lowest dielectric constant, having ε0 ¼ 3.8. The dielectric constant decreases slightly by the addition of B2O3, is almost unaffected by the addition of TiO2, and increases considerably by the addition of alkalis, alkaline earths, and other network-modifying oxides. Typical values of ε0 for soda lime glasses are 7–10, whereas those for sodium borosilicates are around 4.5–8. Some internally nucleated glass ceramics in the B2O3-P2O5-SiO2 system have been shown [17] to have dielectric constant in the 3.8–4.5 range, suggesting potential application as substrates for microelectronic packages substituting for Al2O3 where ε0 is about 9. Dielectric constants may be increased sharply by the addition of heavy metal oxides such as BaO, CdO, PbO, and Bi2O3. The frequency dependence of the loss tangent tan δ often displays peaks whose positions depend on the composition. In alkali-containing glass, tan δ may be as little as 0.002 in the low-frequency region, compared to as much as 0.03 in the high-loss region. Tan δ may be reduced by adding low mobility network modifiers. Glasses of the type (BaO,CaO)-Al2O3-SiO2 and PbO-B2O3-SiO2 have values of tan δ as low as about 0.0012 at 25°C. Dielectric strengths are generally of the order of 107 V/cm. Because of ionic mobility, lower strengths are obtained at elevated temperatures.

Summary Electric permittivity or the dielectric constant ε (units: farad m
1) of a material is defined by D ¼ εE, where D and E are the electric displacement and electric field vectors, respectively. The ratio ε/ε0 ¼ εs, where ε0 is the absolute permittivity of free space, and εs is called the relative dielectric constant. In an alternating field, the complex (relative) dielectric constant ε* is written as ε∗ ¼ ε0
iε00 , where the real part, ε0 , is the (static) dielectric constant, and the imaginary part, ε00 , is the loss factor. ε0 εs at low frequencies, and is about 3–10 for most silicate glasses. The ratio ε00 /ε0 ¼ tan δ. Values of tan δ, the loss tangent, are generally around 0.002. The temperature (T) and frequency (ω) dependence of ac conduction in glass is usually given by an augmented Jonscher’s power law comprising at least four terms. The first of these is the thermally activated, frequencyindependent dc conductivity at low ω and high T. The second term, also

442

Fundamentals of Inorganic Glasses

thermally activated, is the universal power law applicable to nearly all disordered solids in an intermediate frequency range, up to several MHz, and is proportional to ω0.6. The third, weakly T-dependent term with linear ω-dependence applies either at low T-low ω or at high T-high ω region and is ascribed to the cooperative vibration of a small group of atoms in a manner similar to jellyfish. The fourth term arises from mean squared displacement vibrational relaxation, applies to very high frequencies, and is roughly proportional to ω2.

References [1] See, for example, V.V. Daniel, Dielectric Relaxation, vol. 2, Academic Press, New York, 1967. [2] I.M. Hodge, M.D. Ingram, A.R. West, J. Electroanal. Chem. 74 (1976) 125. [3] P.B. Macedo, C.T. Moynihan, R. Bose, Phys. Chem. Glasses 13 (1972) 171. [4] E.N. Boulos, A.V. Lesikar, C.T. Moynihan, J. Non-Cryst. Solids 45 (1981) 419. [5] M. Tomozawa, in: M. Tomozawa, R.H. Doremus (Eds.), Treatise on Mat. Sci. & Technol., vol. 12, Glass I, Academic Press, New York, 1977, pp. 283–345. [6] J.L. Barton, Verre et Refrac. 20 (1966) 328. [7] T. Nakajima, Conf. on Electrical Insulation and Dielectric Phenomena, National Academy of Sciences, Washington, DC, 1972, pp. 168–176. [8] H. Namikawa, J. Non-Cryst. Solids 18 (1975) 173. [9] A.K. Jonscher, Nature 267 (1977) 673. [10] H. Jain, X. Lu, J. Am. Ceram. Soc. 80 (1997) 517. [11] H. Jain, Metals, Materials and Processes, vol. 11 (3&4), Meshap Science Publishers, Mumbai, 1999, pp. 317–328. [12] A.K. Jonscher, Dielectric Relaxation in Solids, Chelsea Dielectrics Press, London, 1983. [13] A.K. Jonscher, Universal Relaxation Law: A Sequel to Dielectric Relaxation in Solids, Chelsea Dielectrics Press, London, 1996. [14] J.C. Dyre, T.B. Schrøder, Rev. Mod. Phys. 72 (2000) 873. [15] A.K. Jonscher, J. Phys. D. Appl. Phys. 32 (1999) R57. [16] J.R. Macdonald, Phys. Rev. B 71 (2005) 184307. [17] J.F. McDowell, G.H. Beall, in: K.M. Nair, R. Pohanka, R.C. Buchanan (Eds.), Ceram. Trans., vol. 15, Amer. Ceram. Soc., Columbus, OH, 1990, pp. 259–278