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Dielectric behavior

Topic 9

Reading assignment• Askeland and Phule, The Science

and Engineering of Materials, 4th

Ed., Sec. 18-8, 18-9 and 18-10.• Shackelford, Materials Science for

Engineering, Sec. 15.4.• Chung, Composite Materials, Ch. 7.

Insulators and dielectric properties

• Materials used to insulate an electric field from its surroundings are required in a large number of electrical and electronic applications.

• Electrical insulators obviously must have a very low conductivity, or high resistivity, to prevent the flow of current.

• Porcelain, alumina, cordierite, mica, and some glasses and plastics are used as insulators.

Dielectric strength• Maximum electric field that an insulator

can withstand before it loses its insulating behavior

• Lower for ceramics than polymers• Dielectric breakdown - avalanche

breakdown or carrier multiplication

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Polarization in dielectrics• Capacitor – An electronic device, constructed

from alternating layers of a dielectric and a conductor, that is capable of storing a charge. These can be single layer or multi-layer devices.

• Permittivity - The ability of a material to polarize and store a charge within it.

• Linear dielectrics - Materials in which the dielectric polarization is linearly related to the electric field; the dielectric constant is not dependent on the electric field.

• Dielectric strength - The maximum electric field that can be maintained between two conductor plates without causing a breakdown.

Polarization mechanisms in materials:(a) electronic, (b) atomic or ionic, (c) high-frequency dipolar or orientation (present in ferroelectrics), (d) low-frequency dipolar (present in linear dielectrics and glasses), (e) interfacial-space charge at electrodes, and (f ) interfacial-space charge at heterogeneities such as grain boundaries.

©2003 Brooks/Cole, a division of Thomson Learning, Inc. Thomson Learning™ is a trademark used herein under license.A charge can be stored at the conductor plates in a vacuum (a). However, when a dielectric is placed between the plates (b), the dielectric polarizes and additional charge is stored.

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, AQ = Do

, dV = ∑

Do = εo Σ

εo = 8.85 x 10-12 C/(V.m)

, dA

dA

VQC = Slope oo

oεε

=ΣΣ

==

4

, AQD =D om

κκ =

Dm = κ εo Σ = ε Σ

, Cd

Ad

AVQ = C o

oom κ

κεκεκ==

ΣΣ

=

P = Dm − Do= κ εoΣ − εoΣ= (κ − 1) εoΣ

E

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, 1 = Q

QQ−

− κκ

, P 1= oΣ

=−ε

κχ

(bound charge)d = (κ − 1) Qd

, PA1)Q(

Ad1)Qd( =

VolumemomentDipole

=−

=− κκ

mCQ = V κ

κQ = Dm A = ε Σ A

xA

xA

= C om

εκε=

x

xAA = V Σ=

Σεε

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Table 7.6 Values of the relative dielectric constant κ of various dielectric materials at 1 kHz (Data from Ceramic Source ’86, American Ceramic Society, Columbus, Ohio, 1985, and Design Handbook for DuPont Engineering Plastics).

3.6Polyester

3.7Nylon-66 reinforced with glass fibers4.1-5.3Cordierite

6.7BeO (99.5%)9.8Al2O3 (99.5%)

___κ__Material

Σ = ( ),sinicosˆeˆ ti tt ωωω += ΣΣ

Dm = ( ) ( )[ ],)tsin(itcosD̂eD̂ mti

m δωδωδω −+−=−

( ) ,eˆeD̂ titim

ωδω εΣ=−

( ),sinicosˆD̂

eˆD̂ mi-m δδε δ −==

ΣΣ

, ofpart Real ofpart Imaginary tan

κκδ −=

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, dtdvC

dt dQic ==

ν = V sin ωt

, T

2f2 ππω ==

, cosCVdtdvCic tωω==

cosC1/

V tωω

=

⎟⎠⎞

⎜⎝⎛ +

2sin πωt

2

sincos2

cossin πωπω tt +=

, cos tω=

, 2

sinC1/

Vic ⎟⎠⎞

⎜⎝⎛ +=

πωω

t

, sinRV

RiR tων

==

, 2

sinC1/

Vi c ⎟⎠⎞

⎜⎝⎛ +=

πωω

t

, sinRV

RiR tων

==

, CR1

CVV/Rtan

ωωδ ==

8

Energy stored = , di0

C∫τ

ν t

, dcossinCV0

2 ttt ωωωτ

∫=

, d2sin2

CV

0

2

tt∫=τ

ωω

[ ] , 2cos4

CV0

2τω

ωω t−=

( ) , 12cosCV41 2 −−= ωτ

Maximum energy stored = ½ CV2

This occurs when cos 2ωt = -1

Energy loss per cycle due to conduction through the resistor R

Energy loss = dsinsinR

V /2

0

2

ttt∫ωπ

ωω

( ) ( ) d2cos121

RV 2

0

2

tt ωωω

π

∫ −=

2sin21

21

RV

2

0

2 π

ωωω ⎥

⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −= tt

( ) 000221

RV2

⎥⎦⎤

⎢⎣⎡ +−−= π

ω

. R

V2

ωπ

=

9

The smaller is R, the greater is the energy loss.

2/CV2

R/Vstoredenergymaximum2

cycleperlostEnergy2

2

πωπ

π=

×

δω

tan CR1

==

10

Frequency dependence of polarization mechanisms. On top is the change in the dielectric constant with increasing frequency, and the bottom curve represents the dielectric loss.

Quartz – polarization only under stress

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©2003 Brooks/Cole, a division of Thomson Learning, Inc. Thomson Learning™ is a trademark used herein under license.

(a) The oxygen ions are at face centers, Ba+2 ions are at cube corners and Ti+4 is at cube center in cubic BaTi03.

(b) In tetragonal BaTi03 ,the Ti+4 is off-center andthe unit cell has a net polarization.

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Different polymorphs of BaTiO3 and accompanying changes in lattice constants and dielectric constants.

Table 7.3 Contribution to dipole moment of a BaTiO3 unit cell by each type of ion.

Total = 17 x 10-30

4.2 x 10-30-0.13(10-10)(-2)(1.6 x 10-19)O2- (top and bottom of cell)

6.4 x 10-30-0.10(10-10)2(-2)(1.6 x 10-19)2O2- (side of cell)6.4 x 10-30+0.10(10-10)(+4)(1.6 x 10-19)Ti4+

00(+2)(1.6 x 10-19)Ba2+

Dipole moment (C.m)

Displacement (m)

Charge (C)Ion

3302

30

m103.984.03C.m1017

−

−

×××

=P

= 0.27 C.m-2

13

-Ec E

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(b) single crystal.

c) Polycrystalline BaTiO3 showing the influence of the electric field on polarization.

The effect of temperature and grain size on the dielectric constant of barium titanate. Above the Curie temperature, the spontaneous polarization is lost due to a change in crystal structure and barium titanate is in the paraelectric state. The grain size dependence shows that similar to yield-strength dielectric constant is a microstructure sensitive property.

Effect of grain size Ferroelectric domains in polycrystalline BaTiO3.

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E

,tt

logru

uu

1

2

1

12 =−

Piezoelectric aging rate rDepoling

u : parameter such as capacitancet: number of days after polarization

Ferroelectric - A material that shows spontaneous and reversible dielectric polarization.

Piezoelectric – A material that develops voltage upon the application of a stress and develops strain when an electric field is applied.

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©2003 Brooks/Cole, a division of Thomson Learning, Inc. Thomson Learning™ is a trademark used herein under license.The (a) direct and (b) converse piezoelectric effect. In the direct piezoelectric effect, applied stress causes a voltage to appear. In the converse effect (b), an applied voltage leads to development of strain.

Direct piezoelectric effect

Reverse (converse) piezoelectric effect

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E

E

18

P = dσ∂P = d ∂σ

σκε

∂∂

Σ= od

d: Piezoelectric coupling coefficient (piezoelectric charge coefficient)

Direct piezoelectric effect

Table 7.1 The piezoelectric constant d (longitudinal) for selected materials

80 x 10-12PbNb2O6

250 x 10-12PbZrTiO6

100 x 10-12BaTiO3

2.3 x 10-12Quartz

Piezoelectric constant d (C/N = m/V)Material

( )1o −∂

=Σ∂κε

P

( )1o −∂

=Σ∂κεσd

∂V = l∂Σ,

( )1o −∂

=∂κεσdV l

P = Dm − Do

= κ εoΣ − εoΣ

= (κ − 1) εoΣ

( )1o −∂

=∂κεσdV l

( ) 1 oεκ −

=dg

∂V = lg∂σg: piezoelectric voltage coefficient

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Reverse piezoelectric effect

S = dΣ∂S = d∂Σ

PS

=Σσ

PS∂∂

=∂Σ∂σ

Reverse piezoelectric effect

( ) 1 o Σ∂−∂

=∂Σ∂

εκσS

( ) 1 oΣ

=Σ

εκσ -S

( ) 1 oεκσ −

=Σ d

( ) 1 oεκσ −

=∂Σ∂ d

S = dΣ

( ) 1 oΣ

=Σ

εκσ -S

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Σ = gσ∂Σ = g∂σ

( ) 1 oεκσ −

=Σ d

( ) 1 oεκ −

=dg

σ = ES

Σ = gES

Hooke’s law

Σ = gσ

gE

S Σ=

1gE

d =

1gd

E =

S = dΣ

energy electricalinput

energymechanicaloutput=2k

energy mechanicalinput energyelectricaloutput

=2k

Electromechanical coupling factor(electromechanical coupling coefficient) k

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Substitution of A and B sites in BaTiO3PZT: PbZrO3-PbTiO3 solid solution or lead zirconotitanate

Table 7.4 Properties of commercial PZT ceramics

0.710.675k15

-0.334-0.388k31

0.700.752k33

Piezoelectric coupling factors496741d15

-123-274d31

289593d33

Piezoelectric coefficients (10-12 m/V)328193Curie temperature (Tc, °C)

0.0040.02Dielectric loss (tan δ at 1 kHz)13003400Permittivity (κ at 1 kHz)

PZT4(hard)

PZT-5H(soft)Property

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Table 7.2 Measured longitudinal piezoelectric coupling coefficient d, measured relative dielectric constant κ, calculated piezoelectric voltage coefficient g and calculated voltage change resulting from a stress change of 1 kPa for a specimen thickness of 1 cm in the direction of polarization.

15151024136PZT

8.58.5493.62 ± 0.40Cement paste with carbon fibers

8.78.72700208 ± 16Cement paste with steel fibers and PVA

2.22.2350.659 ± 0.031Cement paste (plain)

Voltage change (mV)†g (10-4 m2/C)†κ†d (10-13 m/V)*Material

Piezopolymer

Bimorph (bi-strip)

Cantilever beam configuration for actuation

Composites with piezoelectric/ferroelectric material sandwiched by metal faceplates fo enhancing the piezoelectric coupling coefficient

Moonie

Cymbal

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Pyroelectric - The ability of a material to spontaneously polarize and produce a voltage due to changes in temperature.

,dTd

dTdPp o

κε Σ==

p = pyroelectirc coefficientP = polarization

Table 7.5 Pyroelectric coefficient (10-6 C/m2.K)

0.002Cement paste

27PVDF

380PZT

20BaTiO3

o 1)-(Px = V

εκ

σεκσεκσ dd

dd

dd P

)1(xx

1)-(P = V

oo −+

Voltage sensitivity

Compliance Piezoelectric coupling coefficient d

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Piezoelectric composite • When any material undergoes polarization (due to an applied electric field), its ions and electronic clouds are displaced, causing the development of a mechanical strain in the material. polarization.

• This phenomenon is known as the electrostriction.

Examples of ceramic capacitors. (a)Single-layer ceramic capacitor

(disk capacitors). (b) Multilayer ceramic capacitor

(stacked ceramic layers).