Moulson A.J., Herbert J.M. Electroceramics: Materials, Properties, Applications

Подождите немного. Документ загружается.

planes, with the whole comprising a so-called Magne

li phase (named after A.

Magne

li). In the case of TiO

, non-stoichiometric compositions can be shown to

belong to the stoichiometric series Ti

2n71

in which n ¼ 15, 16, 18, 19, 20, 22,

29, 31 etc. For example, TiO

1.95

is actually Ti

The change in conductivity with oxygen pressure can be exploited for gas

sensing as discussed in Section 4.6.2. Most compositions containing TiO

show

similar behaviour when ﬁred in reducing atmospheres, so that sintering in air or

oxygen is essential if they are to be used as low-loss dielectrics.

292 DIELECTRICS AND INSULATORS

Fig. 5.28 Dielectric properties of titania ceramic as a function of frequency and temperature.

TEAMFLY

Team-Fly

5.6.2 Degradation in titanium-containing oxides

The degradation of capacitor dielectrics has been discussed in general terms in

Section 5.2.2 and 5.4.1. Here the topic is ampliﬁed with regard to titanium-

containing oxides. Deterioration can occur under two diﬀerent sets of conditions

and most probably with diﬀering mechanisms.

MEDIUM-PERMITTIVITY CERAMICS 293

Fig. 5.29 Conductivity of titania ceramic in oxygen (101 kPa; 1 atm) as a function of

temperature.

Fig. 5.30 The conductivity measured in the c direction of a rutile single crystal as a function

of oxygen pressure and temperature (after R.N. Blumenthal et al.: see [7] p. 145).

In one circumstance, for capacitors with thin dielectrics (approximately 25 mm

thick) subjected to less than 5 V d.c. at room temperature, the resistance drops

rapidly during prolonged life tests. It can be restored to its initial level by the

brief application of a higher (106) voltage, or sometimes just by minor

mechanical disturbance. The reason for this behaviour is not known for certain

but it seems likely that silver has migrated in the form of a ﬁlament which results

in a low-resistance path bridging the electrodes. An increased current is able to

destroy the ﬁlament by Joule heating and fusion. The eﬀect of mechanical shock

is less easy to explain. Multilayer capacitors containing structural defects such as

cracks or laminations appear to be more likely to be subject to this behaviour.

The presence of moisture may also be a factor.

Another circumstance is when deterioration becomes apparent under ﬁelds in

excess of 0.5 MV m

at temperatures above 85 8C, and occurs more rapidly the

higher the ﬁeld or the temperature. The fall in resistance has been observed in

single crystals of rutile and barium titanate and so must be assumed to be a bulk

rather than a grain boundary eﬀect, although there is evidence that grain

boundaries play a part in degradation processes in ceramics.

The mechanisms responsible for degradation are complex with V

playing an

important role. Because they act as an eﬀective positive charge they migrate

under the inﬂuence of a d.c. ﬁeld towards the cathode with a corresponding

depletion in the anode region. Accompanying these changes are corresponding

changes in electron and hole concentrations in the bulk of the ceramic and it is

the steady build-up of electron currents which is responsible for the degradation

process. The degradation mechanism is discussed in detail in a series of papers by

R. Waser, T. Baiatu and K.-H. Ha

rdtl [8].

Degradation can be slowed down by suitable substituents. The presence of

donor ions at levels exceeding 2 mol.%, e.g. substituting Nb

for Ti

,La

for

or F

for O

, prolongs useful life. Donor ions reduce the concentration of

oxygen vacancies, which are relatively mobile, and increase the concentration of

cation vacancies. The latter have low mobilities at room temperature and, when

combined with holes in the valence band, behave as acceptors (see Section 2.6.2,

and Section 5.7.3).

Manganese at the 1% level in air-ﬁred dielectrics acts as a palliative. It is present

as both Mn

and Mn

. The former ion must be expected to act as an eﬀective

electron trap since it is readily converted into Mn

. However, the presence of

will result in a corresponding concentration of oxygen vacancies.

Humidity is known to accelerate degradation and it is argued that the

mechanism involves protons incorporated into the structure, as discussed in the

context of hydrogen sensors (see Section 4.6.1). Because an hydroxyl group can

occupy a vacant oxygen ion site without distorting the lattice, ‘water’ may be

incorporated into the structure according to the reaction:

O þ V

þ O

2

! 2OH



ð5:21Þ

294 DIELECTRICS AND INSULATORS

Proton conduction can then occur via a ‘hopping’ mechanism between hydroxyl

ions.

It is also established that ‘water’ is able to enter a vitreous phase and, acting as

a glass ‘network modiﬁer’, enhance ionic mobilities. There is no doubt that the

nature of the grain boundary phase, coupled with the presence of water vapour,

inﬂuences the diﬀusion of silver from electrodes and terminations in multilayer

capacitors leading to eventual inter-electrode shorting and component failure [9].

5.6.3 High-power capacitors

The capacitors in the output stage of high-frequency generators provide an

application for Class I dielectrics. These units isolate the d.c. voltage component

from the external load and the whole output from the generator passes through

them. Their values range up to 5000 pF and their frequency of operation ranges

up to 50 MHz. They may have to withstand up to 25 kV and pass 150 A. Their

shape and fabrication have been outlined in Section 5.4.3. The properties of the

principal dielectrics used are given in Table 5.5. The low loss tangents are

achieved by selecting raw materials which are low in impurities, especially

transition elements, and ensuring that none are introduced during processing.

It is instructive to consider important features of the design of high-power

capacitors in some detail, particularly with regard to power dissipation. The

average rate



PP at which heat is developed in a dielectric due to the dissipation of

electrical energy is



PP ¼ IU tan d ð5:22Þ

in which I and U are the root mean square (r.m.s.) current and voltage. Therefore

a 500 kW unit with tan d ¼ 2  10

4

has to dissipate 100 W as heat. Clearly a

particular application requires a particular capacitance value C and, since

I ¼ 2pfUC and C ¼ eA=h, Eq. (5.22) becomes



PP ¼ 2p fU

tan d ð5:23Þ

MEDIUM-PERMITTIVITY CERAMICS 295

Table 5.5 Properties of dielectrics for ‘transmitter’ capacitors

Principal

constituent

Relative

permittivity tan d=10

4

TCC/MK

Steatite 6 3–5 +100 to +150

MgTiO

12–15 1–3 +60 to +100

BaTi

36–40 1–3 730 to +36

TiO

80–90 2–4 7800 to 7700

If it is assumed that the rate at which heat can be removed from the unit is

proportional to its exposed surface area and to its temperature excess DT over

the surroundings, we obtain

DT / fU

tan d ð5:24Þ

Therefore, for a given application, the temperature rise increases with

permittivity and frequency, and decreases with the thickness of the dielectric.

An allowed DT deﬁnes the maximum power rating



for a given capacitor and

this, in turn, deﬁnes upper and lower bounds on the frequency range over which

the unit can be operated at maximum power level.

At low frequencies, say below f

, the power-handling capability is limited by

the high reactance 1/oC and therefore by the voltage rating U

of the capacitor.

Above a frequency f

the reactance is low, so that the power capability is limited

by the current rating I

. The frequencies are given by



2pU

and f

2pP

ð5:25Þ

It is also necessary to consider the power



dissipated by the electrodes, which

is given by



¼ R

ð5:26Þ

where R

is the electrode resistance. Since



¼ IU ¼ I

=2p fC, Eq. (5.26) can be

written



¼ R

2p fC



ð5:27Þ

is complicated by the ‘skin eﬀect’, by which high-frequency currents are

concentrated near the surface of a conductor. The eﬀect arises for the following

reason. When a direct current ﬂows down a wire the current density is distributed

evenly over the cross-section of the wire. The wire can be considered as being

made up of elementary current-carrying ﬁlaments, each of which has its

associated magnetic induction. A little consideration (or consultation of a

textbook on electricity) will show that a central ﬁlament is linked with more ﬂux

than a ﬁlament running along the outer surface of the wire. Therefore, when the

current alternates, the back e.m.f. is greater along the centre of the cross-section

than at the surface. Consequently the current density along the length increases

radially from the centre outwards towards the surface. In fact the current density

falls oﬀ exponentially with depth below the surface, and at a depth d

its value is

1/e of that at the surface. Clearly the eﬀect becomes more pronounced with

increasing frequency.

296 DIELECTRICS AND INSULATORS

The skin depth d

is inversely proportional to the square root of the frequency,

and at 1 MHz is 0.064 mm for silver, 0.066 mm for copper and 0.19 mm for a

typical solder. A surface resistivity r

can be deﬁned by

¼ r=d

ð5:28Þ

where r is the volume resistivity. Alternatively the variation in d

with frequency

can be introduced explicitly by deﬁning a surface resistivity r

, which is a

material property independent of frequency, such that

¼ r

1=2

ð5:29Þ

For silver, copper and solder r

is 2.5610

O Hz

71/2

, 2.6610

O Hz

71/2

and

7.7610

O Hz

71/2

respectively.

In order to evaluate the order of magnitude of the power



dissipated in the

electrode, a value is required for R

in Eq. (5.27). Because R

¼ r

l=w, where l/w

is the length-to-width ratio, and l  w, then R

 r

. Therefore it follows that



 2p f

3=2



ð5:30Þ

For example, for a 500 kW 500 pF unit with r

¼ 4  10

7

O Hz

1=2



is 0.02 W

at 0.1 MHz and 7 kW at 500 MHz. Therefore it is evident that below 1 MHz the

major contribution to heat generation is dielectric loss, whilst at higher frequencies

a signiﬁcant proportion is due to electrode resistance, and that, because of the skin

eﬀect, this resistance cannot be reduced by making the electrodes or leads thicker

than a small fraction of a millimetre. However, the thicker the electrodes and leads

are the better is the heat transfer from the capacitor.

The TCC is not of the greatest importance for these units since the capacitance

values themselves may vary some 10% from nominal in most cases.

A higher-permittivity dielectric based on CaTiO

is sometimes used. It has a

relative permittivity of about 140 and a dissipation factor of 2610

. The TCC

is about double that of rutile-based dielectrics.

5.6.4 Low-TCC low-loss capacitors

A common function of circuits is the provision of an accurate resonance state.

For instance, for a resonance frequency to stay within a tolerance of 0.1% over a

temperature range of 100 K a temperature coeﬃcient of less than 10 MK

would be required. It might be achieved in the 10–100 kHz range by using a

manganese zinc ferrite pot-core inductor (see Section 9.5.1) with a small positive

temperature coeﬃcient of inductance combined with a ceramic capacitor having

an equal, but negative, temperature coeﬃcient. This is clear from the resonance

condition

¼ðLCÞ

1=2

MEDIUM-PERMITTIVITY CERAMICS 297

which when diﬀerentiated with respect to temperature yields

¼





ð5:31Þ

In most applications a resonance tolerance of 0.1% would only be useful if the

resonance were correspondingly sharp, e.g. with a Q in the neighbourhood of

1000 (tan d ¼ 10

3

). Thus low-TCC capacitors must also be low loss if they are

to be of practical value in such applications.

The parameters that contribute to the TCC can be identiﬁed by ﬁrst

considering a rectangular parallel-plate capacitor with sides of length x and y

and thickness z. Then, since the capacitance is given by

C ¼

e xy

diﬀerentiation with respect to temperature leads to



þ a

TCC ¼ TC

þ a

ð5:32Þ

in which TC

is the temperature coeﬃcient of permittivity and a

is the linear

expansion coeﬃcient. Eq. (5.32) is derived under the assumption that the

expansion coeﬃcients in the x, y and z directions are identical, i.e. the dielectric

has isotropic linear expansion characteristics.

The capacitance may change with temperature not only because the

dimensions of the capacitor change but also because the permittivity of the

dielectric changes. To gain some insight into the sources of the variation in

permittivity with temperature, the Clausius–Mosotti equation (Eq. (2.88)) can be

diﬀerentiated with respect to temperature to give

ðe

 1Þðe

þ 2Þ





ð5:33Þ

When e

52,

ðe

 1Þðe

þ 2Þ



and

298 DIELECTRICS AND INSULATORS

¼

¼3a

where V is the volume containing N polarizable units. Eq. (5.33) therefore

reduces to





 3a



ð5:34Þ

For a number of dielectrics with e

430, TC

is negative and within 15% of

a

as illustrated by the examples given in Table 5.6. Eq. (5.34) suggests that

the temperature variation of polarizability is small compared with the volume

expansion coeﬃcient in these cases. Lower-permittivity oxides have positive TC

and in their case the temperature coeﬃcient of polarizability can be assumed to

exceed the volume expansion coeﬃcient. However, the extent to which the

Clausius–Mosotti equation can be applied to ionic solids is open to debate.

In order to improve volumetric eﬃciency, dielectrics combining a small TC

with a high permittivity have been sought. Since, in many cases, it is not zero TC

but a controlled value that is required, combinations of two components with

diﬀerent TC

s in a series of ratios has provided ranges of useful dielectrics. A

typical combination has end-members with compositions corresponding to

BaTi

¼ 35, TC

¼þ35 MK

) and TiO

¼ 100,TC

¼750 MK

1

)

which covers the range of TC

and permittivity lying between these two extremes

as illustrated in Fig. 5.31.

Combinations of high positive TC

with high permittivity and low loss are

rare. The antiferroelectric compound PbZrO

has e

¼ 110 and TC

¼1400 but

tan d ¼ 28  10

4

. Sphene (CaSiTiO

), which is also known as titanite, has

¼ 45, TC

¼ 1200 and tan d ¼ 5  10

4

. A combination of sphene and rutile

gives a dielectric with zero TC

, e

¼ 60–70 and low loss. The crystal structure of

sphene consists of chains of corner-sharing TiO

octahedra interlinked by SiO

tetrahedra by corner sharing. The Ti

ions are displaced from the centres of the

octahedra by about 10 pm but in opposite directions in alternate groups;

MEDIUM-PERMITTIVITY CERAMICS 299

Table 5.6 Temperature coeﬃcient of permittivity of Class I dielectrics

/MK

Composition e

=MK

1

Reported  e

TiO

100 7.5 7750 7750

SrTiO

230 10.3 72400 72370

CaTiO

170 10.8 71600 71836

MgTiO

17 7 +100 7119

10 8.5 +120 785

MgO 10 13.5 +190 7135

therefore it is an antipolar structure. It is not antiferroelectric since there is no

transition to a paraelectric state in which the Ti

ions have zero displacements.

As pointed out earlier (Chapter 2, Section 2.7.4), Lichtenecker’s rule for

mixtures leads to the prediction that the TC

of a mixture will be equal to the

volume average of the TC

s of its constituents. This presupposes that there is no

reaction leading to the formation of new compounds. The permittivity of the

mixture can also be predicted approximately by Lichtenecker’s rule.

5.6.5 Microwave ceramics

‘Wireless technologies’ have already penetrated many aspects of modern life and

the trend is set to continue. This has in turn been a strong stimulant to the

development of specially tailored microwave dielectrics.

Figure 5.32 gives an indication of speciﬁc technologies and of the

corresponding frequency bands.

Dielectric resonators

The rapid growth of satellite and mobile radio communications has led to a

requirement for narrow band, frequency-stable ﬁlters and oscillators. Selectivity

and stability are necessary to ensure that signals are conﬁned to closely deﬁned

allotted frequency bands and to prevent the intrusion of unwanted signals which

would interfere with the satisfactory performance of the system. Antennae are

also critical components of any wireless communications system. The need for

compactness in satellite and hand-held mobile systems is self-evident.

300 DIELECTRICS AND INSULATORS

Fig. 5.31 Temperature coeﬃcient versus TiO

content for BaTi

–TiO

mixtures.

The solution to providing stable ﬁlters and oscillators in the past lay in bulky

coaxial and cavity resonators fabricated from the temperature-stable metal

alloy Invar. The dielectric resonator (DR) oﬀers a means of miniaturizing the

device.

In its simplest form a DR is a cylinder of ceramic of relative permittivity e

suﬃciently high for a standing electromagnetic wave to be sustained within its

volume because of reﬂection at the dielectric–air interface. The electric and

magnetic ﬁeld components of the fundamental mode of a standing electro-

magnetic ﬁeld are illustrated in Fig. 5.33.

The wavelength l

of the standing wave in the dielectric approximates to the

diameter, D, of the cylinder, i.e. l

 D. If the resonance frequency is f

then in

free space f

¼ c=l

where c and l

are respectively the free space velocity and

wavelength. In a non-magnetic dielectric medium the velocity

¼ c=e

1=2

(see

Eq. (2.120)) so that

¼ c=ðl

1=2

Þc=ðDe

1=2

Þð5:35Þ

MEDIUM-PERMITTIVITY CERAMICS 301

Fig. 5.32 Frequency bands covered by the various wireless technologies (*includes

‘Bluetooth’; Harald Blatand (Bluetooth) was a famous King of Denmark c.960 who is

known for encouraging communication between people).

Fig. 5.33 Fields in a microwave resonance dielectric in the simplest standing wave mode: (a)

magnetic ﬁeld; (b) electric ﬁeld; (c) variation in E

’

and E

with r at z ¼ 0, with reference to

cylindrical coordinates (the z axis is perpendicular to the plane of the disc and the origin is at

the disc centre).