The Nature of Dielectric MaterialsMost solid materials are classified as insulators because they offer very large resistance to the flow of electric current. Metals are classified as conductors because their outer electrons are not tightly bound, but in most materials even the outermost electrons are so tightly bound that there is essentially zero electron flow through them with ordinary voltages. Some materials are particularly good insulators and can be characterized by their high resistivities:

Resistivity (Ω-m)Glass 1012

Mica 9 x 1013

Quartz (fused) 5 x 1016

Resistivity (Ω-m)Copper 1.7 x 10-8

The familiar parallel plate capacitor equation with free space as an insulator under vacuum is given by: L

AVQC o

oε

== 0

εo absolute permittivity or the permittivity of a vacuum, the value of 8.85×10-12 F/mA is the areaL is the separation between the platesC is the capacitance (charge storage ability per unit voltage)

Plates connected to a constant voltage supply V.

Qo is the charge on the plates.

Co is the capacitance of the parallel plate capacitor in free space (vacuum).

The electric field E is defined as the gradient of the potential then:

LVE =

ELV

AQD ooO εε === 0

DO - is dielectric displacement = dipole moment per unit volume = charge per unit area, C/m2

If there is a material medium between the plates, then the capacitance (C) increases by a factor εr, where εr is called the dielectric constant or relative permittivity

LA

VQC o

oε

== 0

Under vacuum With a material medium

LA

VQC ε

==

ooor Q

QCC

===εεε

The increase in stored capacity is due to the polarization of the dielectric material by the applied field

Co+Qo –Qo

E

V

(a)

+Q –QC

E

V

(c)

Dielectric

i (t)

V

(b)

Fig. 7.1: (a) Parallel plate capacitor with free space between plates.(b) As a slab of insulating material is inserted between the plates,there is an external current flow indicating that more charge is storedon the plates. (c) The capacitance has been increased due to theinsertion of a medium between the plates.

From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)http://Materials.Usask.Ca

VQC o

o =VQC =

A dielectric is any polarizable material. All materials are polarizable, so all materials are dielectrics, even air.

The increase in capacitance is due to the polarization of the medium in which positives and negative charges are displaced with respect to their equilibrium position.

Polarization can be viewed as: Charge per unit area or Dipole per unit volume

EEDPDD

ED

Or

O

OO

εεε

ε

==+=

= Where P is the polarization (total dipole moment/volume)

( ) eOrO EEP χεεε =−= 1

Energy Stored in a CapacitorThe quantity of energy stored in a capacitor is given by the equation

2221 22 VC

CQVQEnergy ×

==×=C = 220 μFVmax = 400 V

LAC ε

= LEV =

( )( ) joulesVFVCEnergy MaxMax 6.17

240010220

2

262

=×

=×

=−

ALE

Energy2

2×=

ε

2__

2EDensityEnergyElectric Or ××

=εε

χe is the electrical susceptibility

The larger the dielectric constant the higher the energy that can be stored.

Dielectric strengthDesired Properties of a dielectric medium:

ability to increase capacitanceinsulating behavior or low conductivity so that the charges are not simply

conducted from one plate of the capacitor to the other through the dielectric.Many dielectrics can sustain very high internal electric fields before electrical breakdown. The maximum electric field that can be sustained is called the dielectric strength of the material.The voltage across the dielectric can not be increased without limit. Very high electric fields (>108 V/m) can excite electrons to the conduction band and accelerate them to such high energies that they can, in turn, free other electrons, in an avalanche process (or electrical discharge). The field necessary to start the avalanche process is called dielectric strength or breakdown strength. This is the maximum electric field to which a dielectric material can be subjected without breaking down or discharging

εr

Example: We want to make a simple parallel plate capacitor that can store 4×10-5 C at a potential of 1000 V. The separation between the plates is to be 0.2 mm. Calculate the area of the plate required if the dielectric is (a) vacuum, (b) polyethylene, (c) water, and (d) BaTiO3. The relative permittivity for polyethylene, water and BaTiO3 are 2.26, 78.3, and 3,000 respectively.

Solution: 900, 400, 11.5 and 0.3 cm2

Dielectric Behavior

Mechanisms:dipole formation/orientationelectronic (induced) polarization: Applied electric field displaces negative electron “clouds” with respect to positive nucleus. Ionic materials (induced) polarization: Applied electric field displaces cations and anions in opposite directionsmolecular (orientation) polarization: Some materials possess permanent electric dipoles (e.g. H2O). In absence of electric field, dipoles are randomly oriented. Applying electric field aligns these dipoles, causing net (large) dipole moment.

Ptotal = Pe + Pi + Po

Types or Mechanism of PolarizationElectronic Polarization: It may be induced (to some degree) in all atoms.

No field Electric Field

+ +-

--

-----

-

-

-

--

-

--

- ---

Displacement of the center of the negative electron cloud off the nucleus (only present when there is an electric field)

Fig. 7.3: The origin of electronic polarization.From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)http://Materials.Usask.Ca

Electron cloud

Atomicnucleus

(a) A neutral atom in E = 0.

pinduced

E

Center of negativecharge

xC O

(b) Induced dipole moment in a field

The magnitude of the electric dipole moment is P = q x d, where d is the distance between dipoles

Ionic Polarization - Only occurs in ionic materials

- - -

- -

- - -

- -

+ +

+ +

+ +

+ +

+

+

No electric field

- - -

- -

- - -

- -

+ +

+ +

+ +

+ +

+

+

Electric Field

An applied field displaces cations in one direction and anions in another:

-

+

+-

++

E

Molecular polarization, occurs in all insulating molecules;

oils, polymers, H2O…

p+ p–

x

p'+ p'–

E

Cl– Na+

(a)

(b)

Fig. 7.8: (a) A NaCl chain in the NaCl crystal without an appliedfield. Average or net dipole moment per ion is zero. (b) In thepresence of an applied field the ions become slightly displacedwhich leads to a net average dipole moment per ion.From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)http://Materials.Usask.Ca

Orientation PolarizationOnly in materials which possess permanent dipole moments

No field Electric Field

H HO

δ+

δ-

example

Cl° H+

po

(a)

(b)

pav = 0

θ

–Q

F = Q E

F

po = aQ

τ

E

pav ≠ 0 E

(c)

(d)

+Q

Fig. 7.9: (a) A HCl molecule possesses a permanent dipole moment, po(b) In the absence of a field, thermal agitation of the molecules resultsin zero net average dipole moment per molecule. (c) A dipole such asHCl placed in a field experiences a torque which tries to rotate it toalign po with the field E. (d) In the presence of an applied field thedipoles try to rotate to align with the field against thermal agitation.There is now a net average dipole moment per molecule along thefield.From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)http://Materials.Usask.Ca

CapacitorsParallel plate capacitor. Apply a voltage; charge Q accumulates on the platesPlace a material between the plates - Q increases e.g. H2O

-q

+qdP

Polarization Vector, PElectric Dipole Moment (- to +) P = qdP=Zqd

In the presence of an electric field, a force will tend to orient

the electric dipole with the applied field

The process of dipole alignment is called Polarization

Surface charge density or Dielectric Displacement:D (C/m2) ∝ Ε (electric field V/m)D = εΕ D = dielectric displacement

Total Polarization: P = Pe + Pi + Po(electronic + ionic + orientation)

Example (Electronic Polarization)

Suppose that the average displacement of electrons relative to the nucleus in a copper atom is 1x10-8 Angstroms when an electric field is imposed on a copper plate. Calculate the electronic polarization. Data: Copper (Z=29 and lattice parameter = 3.6151 Angstroms)Solution

330310 10462

1061513294 melectronsatomelectronscellatomsZ /.

).()/)(/(

×=×

= −

27

10819330

/1094.3)/10)(10)(/106.1)(/1046.2(

mCPAmAelectronCmelectronsP

dqZPo

−

−−−

×=

××=

××=

Where Z is the number of electrons (electronic polarization) per unit volume

Example (Ionic Polarization)

Calculate the increase in separation of Cs+1 and Cl-1 in a CsCl crystal when an ionic polarization of 4x10-8C.m-2 is achieved by the application of an electric field. Data: lattice parameter a=0.402nm, ionic radii 0.165nmfor Cs+1 and 0.181nm for Cl-1.

Solution

Use the equation

Where Z is the number of charges per unit volume i.e. (dipoles per cell) x (charges per cell) per unit volume

dqZP ××=

( )( )( )

328

339

10541104020

11

−

−

×=

×=

meschZcellperm

dipoleperechcellperdipoleZ

.arg.__.

__arg___

( ) ( )nmmd

echCmeschmC

qZPd

817

19328

28

1062110621106110541

104

−−

−−

−−

×=×=

××××

=×

=

..arg/..arg.

.

Frequency Dependence of the Dielectric Constant

Alternating Current. (Applied voltage or electric field changes direction with time)

Dipoles try to reorient with field. (This requires time)

Relaxation Frequency = 1/time to reorient

+ + + + + +

+ + + + + +

- - - - - - - -

- - - - - - - -

Electric Field Electric Field

+ + + + +

- - - - -+ + + + +

- - - - -

Sometime dipoles can’t keep up with changing electric field:

As frequency increases, dielectric constant decreases as orientation and ionic components go to zero.

1 102 104 106 108 1010 1012 1014 101610–2

ƒ

Orientational,Dipolar

Interfacial andspace charge

IonicElectronic

εr'

εr''

Radio Ultraviolet lightInfrared

εr' = 1

From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)http://Materials.Usask.Ca

Fig. 7.14: The frequency dependence of the real and imaginary partsof the dielectric constant in the presence of interfacial, orientational,ionic and electronic polarization mechanisms.

Define the permittivity or dielectric constant of a material by:

H2O is a polar liquid; εr ~ 80Typical ionic solids; εr ~ 10Air; εr ~ 1BaTiO3 :-

vacr Q

Q=ε

Below 120°C, BaTiO3 is ferroelectric with aligned dipoles.Residual dipole disorder gives εr~200-1000At ~127°C, tetragonal → cubic phase transition.Dipoles randomise and εr increases to ~5,000-10,000

(a)

Al foils

Al case

Al2O3Anode Cathode

(b)

Electrolyte

Al Al

Fig. 7.31: Al electrolytic capacitor.From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)http://Materials.Usask.Ca

The anode of an Al electrolytic capacitor is an aluminum foil of extreme purity. The effective surface area of this foil is greatly enlarged (by a factor of up to 200) by electrochemical etching in order to achieve the maximum possible capacitance values. The type of etch pattern and the degree of etchingis matched to the respective requirements by applying specific etching processes. Etched foils enable very compact Al electrolytic capacitor dimensions to be achieved and are the form used almost exclusively nowadays. The electrical characteristics of Al electrolytic capacitors

The dielectric layer of an Al electrolytic capacitor is created by anodic oxidation (forming) to generate an aluminum oxide layer on the foil. The layer thickness increases in proportion to the forming voltage at a rate of approximately 1,2 nm/V. Even for capacitors for very high voltages, layer thicknesses of less than 1 μm are attained, thus enabling very small electrode spacings. This is one reason for the high volumetric efficiency achieved(e.g. in comparison to the minimum thickness of a paper dielectric, 6 to 8 μm).

Example

A 2mm thick porcelain dielectric is used in a 60 Hz circuit. Calculate the voltage required to produce a polarization of 5x10-7 C.m-2. Use Table.

Solution

( )

voltsVm

VEP rO

6.22

105102

1085.8)16(1 73

12

=

×=×

×××−=−= −−

−εε

CRYSTAL SYMMETRY:Crystal structures can be divided into 32 classes, or point groups, according to the number of rotational axes and reflection planes they exhibit that leave the crystal structure unchanged.

Twenty of the 32 crystal classes are piezoelectric. All 20 piezoelectric classes lack a center of symmetry.

Any material develops a dielectric polarization when an electric field is applied, but a substance which has such a natural charge separation even in the absence of a field is called a polar material.

Whether or not a material is polar is determined solely by its crystal structure. Only 10 of the 32 point groups are polar.

Under normal circumstances, even polar materials do not display a net dipole moment. As a consequence there are no electric dipole equivalents of bar magnets because the intrinsic dipole moment is neutralized by "free" electric charge that builds up on the surface by internal conduction or from the ambient atmosphere.

Polar crystals only reveal their nature when perturbed in some fashion that momentarily upsets the balance with the compensating surface charge.

The possibility of inorganic crystals being polar (pyroelectric or piezoelectric) is strictly a function of their point group symmetry.

Polar MaterialsSolid with a natural charge separation even in the absence of a fieldCrystals comprising cations and anions can be classified into four types, according to their polar behavior:

• Piezoelectric materials: There is coupling between electrical and mechanical energies. For example, an applied stress results in the generation of polarization.

• Pyroelectric materials: Pyroelectricity refers to the change in polarization by changes to the structure from thermal effects. A material with a temperature dependent polarization. This requires a unique polar axis.

• Ferroelectrics: A subgroup of pyroelectric materials in which the spontaneous polarization can be reoriented between “equilibrium” states by applying an electric field.

All ferroelectrics are both pyroelectric and piezoelectric.The possibility of inorganic crystals being polar (pyroelectric or piezoelectric) is strictly a function of their structure (point group symmetry)

•Ferroelectrics: Ferroelectric materials possess a natural electric polarization. A subgroup of pyroelectric materials in which the spontaneous polarization can be reoriented between “equilibrium” states by applying an electric field. All ferroelectrics are both pyroelectric and piezoelectric. Not all piezoelectric materials are pyroelectric. Ferroelectrics are materials which possess an electric polarization in the absence of an externally applied electric field such that the polarization can be reversed if the electric field is reversed. Normally materials are very nearly electrically neutral on the macroscopic level. However, the positive and negative charges which make up the material are not necessarily distributed in a symmetric manner. If the sum of charge times distance for all elements of the basic cell does not equal zero the cell will have an electric dipole moment which is a vector quantity. The dipole moment per unit volume is defined as the dielectric polarization.

Piezoelectric materials: Piezoelectricity refers to a materials property that the polarization (or electric field) of the material can be changed by mechanical perturbation of the structure. There is coupling between electrical and mechanical energies. For example, an applied stress results in the generation of polarization.

PIEZOELECTRIC EFFECT:The piezoelectric effect is a linear, reversible electromechanical interaction occurring in materials possessing the proper symmetry properties. The direct piezoelectric effect is the production of an electric polarization by a strain; the converse piezoelectric effect is the production of a stress by an electric field. Piezoelectric materials have wide applications as transducers -transferring mechanical motion into electricity or electricity into mechanical motion. One of the most wide spread examples is a quartz resonator. The quartz resonator converts the electrical potential energy of a battery into a steady beat that becomes the oscillator (counter) of a watch.

Pyroelectricity : It is a property of dielectric materials, which show a temperature-dependent, macroscopic (permanent or spontaneous) polarization P, i.e. they generate surface charges as a result of a temperature change ΔT(t). These charges can either be detected directly or as a pyroelectric current I(t).

PYROELECTRICITY: Spontaneous polarization is temperature dependent, so a good perturbation probe is a change in temperature which induces a flow of charge to and from the surfaces. This is the pyroelectric effect. All polar crystals are pyroelectric, so the 10 polar crystal classes are sometimes referred to as the pyroelectric classes.

The property of pyroelectricity is the measured change in net polarization (a vector) proportional to a change in temperature. The total pyroelectriccoefficient measured at constant stress is the sum of the pyroelectriccoefficients at constant strain (primary pyroelectric effect) and the piezoelectric contribution from thermal expansion (secondary pyroelectric effect). Pyroelectric materials can be used as infrared and millimeter wavelength detectors.

Piezoelectricity or pressure electricity: Unusual phenomena in which polarization is induced and an electric field is established across a sample when it is mechanically stressed. Similarly, the same crystal also exhibits mechanical strain when it experiences an electric field.

Piezoelectricity

P = 0

Force

P V

(b)(a)

The direction of mechanical deformation (extension or compression) depends on the direction of the applied field, or the polarity of the voltage.Only crystals with a special crystal structure can exhibit piezoelectricity that which has no center of symmetry.

V V

(c) (d)

Fig. 7.35: The piezoelectric effect. (a) A piezoelectric crystalwith no applied stress or field. (b) The crystal is strained by anapplied force which induces polarization in the crystal andgenerates surface charges. (c) An applied field causes the crystalto become strained. In this case the field compresses the crystal.(d) The strain changes direction when the field is reversed, andnow the crystal is extended. The dashed rectangle is the originalsample size in (a).From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)http://Materials.Usask.Ca

Consider the cubic unit cell. When unstressed, center of mass (c.m. of the –ve charges at the corner of unit cell coincides with +ve charge at center therefore, no net polarization occurs and P=0.Under stress, unit cell becomes strained. However, c.m. of the –ve charges still coincides with +ve charge and net polarization is still 0. P=0 for strained crystal. This is generally true for crystals with center of symmetry.

If we draw a vector from O (position of an arbitrary point charge) to any charge, then the reverse vector will point to the same type of charge: we call O, or any other point charge, a center of symmetry

P = 0O

From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)http://Materials.Usask.Ca

(a)

P = 0

Force

(b)

Fig. 7.36: A cubic unit cell has a center of symmetry. (a)In the absence of an applied force the centers of mass forpositive and negative ions coincide. (b) This situation doesnot change when the crystal is strained by an applied force.

If we draw a vector from O to any charge, then the reverse vector will point to an opposite charge. The unit cell is said to be non-centrosymmetric. When unstressed, c.m. of –vecharges coincides with c.m. of the +ve charges, both at O, therefore, no net polarization occurs and P=0.Under stress, the +ve charge at A and –ve charge at B both become displaced inwards to A’and B’ respectively. The two c.m.’s become shifted and there is now a net polarization P for the strained crystal.

P = 0 PO

y

x

(a) (b)

A

B

A'

B'

P = 0

P

(c)

A''

B''

Fig. 7.37: A hexagonal unit cell has no center of symmetry. (a) Inthe absence of an applied force the centers of mass for positiveand negative ions coincide. (b) Under an applied force along y thecenters of mass for positive and negative ions are shifted whichresults in a net dipole moment P along y. (c) When the force isalong a different direction, along x, there may not be a resulting netdipole moment in that direction though there may be a net P alonga different direction (y).From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)http://Materials.Usask.Ca

Generally, an applied stress in one direction can give rise to induced polarization in other crystal directions and reversing the stress reverses the polarization. Crystals with no center of symmetry exhibit piezoelectricity.

P = 0 PO

y

x

(a) (b)

A

B

A'

B'

P = 0

P

A''

B''

The direction of induced polarization depends on the direction of applied stress. In the above case, P appears in the same direction as applied stress along y. If the stress is applied along x, A and B are displaced outwards to A’’ and B’’respectively, resulting in shift of c.m.’s away from each other in y direction therefore P appears along y direction.

Piezoelectric and Ferroelectric MaterialsWhen mechanical pressure is applied to a piezoelectric material, the crystalline structure produces a voltage proportional to the pressure. Conversely, when a piezoelectric material is subjected to an electric field, the structure changes in shape, producing dimensional changes in the material. Examples of natural piezoelectric crystals:

Quartz (SiO2), Rochelle Salt Tourmaline.

Piezoelectric materials are anisotropic, that is their mechanical, electrical and electromechanical properties depend strongly on the crystal orientation.

Man-made piezoelectric ceramic example: lead-zirconate-titanate (PZT), lead-titanate (PbTiO2), lead-zirconate (PbZrO3), and barium-titanate (BaTiO3).

Strictly speaking, these ceramics are not actually piezoelectric but rather exhibit a polarized electrostrictiveeffect. PZT exhibits a cubic structure above a critical (Curie) temperature. During cooling (below the Curie temperature) the cubic structure transform to a tetragonal or rhombohedralstructure. Due to the non-centrosymmetry of this structure exhibits a dipole moment. Regions of the crystal with the dipoles having the same direction are called “domains”.

In a PZT, the direction of polarization among neighboring domains is random, so the ceramic element has no net polarization. Net polarization is induced by exposing the PZT to a strong direct electric field, to align all the individual domains towards one specific direction, the poling direction. With this treatment the ceramic element increases in size in the direction of the electric field. When the electric field is removed, most of the dipoles are locked into a configuration near alignment. The element now shows a permanent or net polarization.

Analogous to ferromagnetic materials, piezoelectric materials exhibit a hysteresis loop. An electric field is applied to the piezoelectric until a maximum polarization is achieved, then the electric field is removed and the material exhibit a “remanentpolarization”. To eliminate the remanentpolarization, an reverse electric field is induced, in opposite direction.

A compressive stress in the same direction of poling, creates a voltage of the same polarity as the poling voltage. Voltage of the same polarity as the poling voltage is produced by a tensile stress perpendicular to the direction of poling.

A tensile stress along the direction of poling produces a voltage with polarity opposite to the poling voltage.

Conversely, if a voltage of the same polarity as the poling voltage is applied to the piezoelectric, it will increase in size, while if the polarity of the applied voltage is reverse it will shorten in size. Finally if a cyclic voltage is applied in the direction of poling, the piezoelectric will change in dimensions cyclically at the frequency of the applied voltage.

Piezoelectric effect basicsApply mechanical stress -> Electric charge producedApply electric field -> Mechanical deformation producedDipole: each molecule has a polarization, one end is more negatively charged and the other end is positively charged.Monocrystal: the polar axes of all of the dipoles lie in one direction. --SymmetricalPolycrystal: there are different regions within the material that have a different polar axis. -- Asymmetrical

How to produce piezoelectric effect

a) Material without stress / charge

b) Compress -> same polarityc) Stretched -> opposite polarityd) Opposite voltage -> expande) Same voltage -> compressf) AC signal -> vibrate

Applications of piezoelectric materials is based on conversion of mechanical strain into electricity (microphones, strain gauges, sonar detectors, audible alarms, ultrasonic imaging, speakers)Piezoelectric materials include barium titanate BaTiO3, lead titanate, lead zirconate PbZrO3, quartz, ammonium dihydrogenphosphate (NH4H2PO4).

Piezoelectric Igniters

Hydrophones: A "Hydrophone" is a device which will listen to, or pick up, the acoustic energy underwater. A hydrophone converts acoustic energy into electrical energy and is used in passive underwater systems to listen only.

Liquid Atomization Devices

Piezoelectric AudiotoneTransducers

Fig. 7.38: Piezoelectric transducers are widely used to generateultrasonic waves in solids and also to detect such mechanicalwaves. The transducer on the left is excited from an ac sourceand vibrates mechanically. These vibrations are coupled to thesolid and generate elastic waves. When the waves reach theother end they mechanically vibrate the transducer on the rightwhich converts the vibrations to an electrical signal.

Oscillator

Elasticwaves in thesolid Oscilloscope

A B

Mechanicalvibrations

Piezoelectrictransducer

From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)http://Materials.Usask.Ca