MSE 131 Ceramics Lecture 3 2016 student.pdf

7/26/2019 MSE 131 Ceramics Lecture 3 2016 student.pdf

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MSE 131 CeramicsLecture 3

Dr. Benjamin O. Chan

Physics Department

Ateneo de Manila University2016


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Electrical Properties

Electrical Conductivity ( )

Insulators

Semi-conductorsConductors

Superconductors


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Conductivity

mho/m units

N= number of mobile charge carriers

q= charge per carrier

= mobility of charge carriers Nq= charge density

1

Nq


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Ceramics vs.Metals/Polymers


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Conducting Ceramics Ionic Crystals

Alkali halides

Strongly bound positively charged cations and

negatively charged anions immobilized Wide band gap = insulating

Ionic conduction possible!

Some ions can hop from lattice site to lattice site under

an applied electric field

where Nion= hopping ion density, ion= hopping ion mobility

ionionion eN


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Ionic Conduction

Ions must have sufficient energy to overcome

energy barrier

Electric field reduces energy

barrier

Neighboring site must be vacant

Schottky defect

Proceeds in a diffusion-like manner

assuming one charge unit per atom is transported

Tk

De

B

ion


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Ionic ConductivityArrhenius Equation

ion can now be written as

where

Taking the ln of each sideyields

Tk

QDD

B

oexp

TkQ

B

oion exp

Tk

DeN

B

oiono

2

Tk

Q

B

oion lnln

*slope = activation energy


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Vacancy Formation

Charge neutrality maintained

Schottky defect: cation-anion vacancy pair

Frenkel defect: vacancy-interstitial pair Charged impurity

monovalent host replaced by divalent impurity

Stoichiometric compounds

More vacancies formed atelevated temperature

Non-stoichiometric compounds

contain a large amount of

vacancies even at low

temperatures


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Covalently Bonded Ceramics

Intrinsic impurities

No such thing as pure semiconductor

Defective by nature

Doping = Intentionally Impure

Group III impurity

Missing electron = hole = p-type carrier

Group V impurity

Extra electron = n-type carrier


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Group V Impurity in Si


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Intentional Impurities

Usually in the ppm range

Group V

Donates an electron to the material Impurity energy level just below conduction band

Group III

Accepts an electron from the material

Impurity energy level just above valence band


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Band Representation

Fermi Energy

Conduction Band

Valence Band


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Donor and Acceptor Levels


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Majority and Minority Carriers

Majority carrier Usually from dopant

Extrinsic carriers

Minority carrier Usually from interband transitions

Intrinsic carriers

Valid for reasonably low T The picture can change at high T

Intrinsic carriers become the majority!

Ec

Ev


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Si Band Structure


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GaAs Band Structure


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Energy Gap at 0 K

!"#$#%& !'(#)*

+ (-./$0%-* 1234

5. 6267

8# 9273

5% (':/;* 9294


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The Energy Gap

Decreases with atomic number

Artificial diamond: C-based IC

Ge: good IR detector

Sn easily conducts at room temperature

Wavelength equivalent

)(

24.1)(

eVEm


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Temperature Dependence

where Eg(0) is the energy gap at 0 K

= 5 x 10-4 eV/K and

D = Debye temperature (Table 19.2)

Temperature needed to reach 96% of the final value forheat capacity CV T> D : classical region

T< D : QM consideration

D

ggT

TETE

2

)0()(


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Eg vs. T

Eg decreases with

increasing

temperature

For Si,

D=650K,

Eg reduction

= -2.4 x 10-4 eV/K


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Basic Test Circuit

In general,

E = J E = electric field across

sample (volt/m) J= current density

(amps/m2)

= resistivity (ohm-m)

Ohms law E J

is constant

Measurement V = Electric potential (volts)

I = Current (amps)


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Resistance R and Resistivity

V= EL, J= I/A

L = length of sample

A = cross-sectional area

A

LR

A

L

I

V

A

I

L

V

JE

(ohm)

Slope of V-I graph


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Problems with Resistivity Measurements

Geometrical shape required for specimen

Uniform cross-sectional area

Contact resistance Loading effects from meter

Voltmeter: should not draw current

High impedance required

Ammeter: should not impede current flow

Zero resistance desired

Stray capacitance and inductance


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Four Point Probe Technique(ASTM F84-84, F374-84)

Suitable for wafer/thin film geometry

Rectangular/circular/cylindrical/slab

Minimizes contact resistance and loadingeffects

Four probes separated equally by distance s

Outer probes: constant current supply

Inner probes: voltmeter

Sheet resistance Rs )(CFI

VRs


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Four Point Probe

Correction Factors Geometrical in

nature

Something to do withs and how big thesample dimensionsare with respect to it

Computing

resistivity

w=sample thickness

wRs


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Correction Factor Limit

Rs formula valid for w a or d

In the limit as ds, we can show that

CF = ( /ln 2) = 4.54


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Carrier Type Determination

Is the material n-type or p-type?

Seebeck Effect

Double thermocouple

Works for

Bulk material

Films on high resistivity material

Films on opposite type substrate


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Hot Probe (ASTM F42-77(87))

Voltmeter/Ammeter

+


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How it works

Hot probe has greater carrier concentration

Positive terminal of meter must be on the hot

probe If p-type, electrons flow from cold to hot junction:

meter records positive reading

If n-type, electrons from flow from hot to cold:

meter records negative reading If the tester leads are inverted, the deflections

will be inverted! (this can be confusing!?)


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Another Way to Determine Resistivity and

Carrier Type

Hall Effect Apparatus

Can measure concentrations down to 1012

electrons/cm3

Sensitivity is several orders of magnitude better

than any chemical analysis

Applies to bulk material

Pass a current in a material

immersed in a magnetic field


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Hall Effect

Consider a rectangular block of n-typesemiconductor immersed in a magnetic fieldwith a magnetic induction B in the z-direction

Let a current density j flow through thematerial in the +x direction

Flowing electrons

experience a Lorentzforce deflecting themfrom their path

BvqF


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Hall Effect

Lorentz force makes the electrons drift to one side of

the material

Accumulation of charge sets up an electric field

Hall field

Hall field builds up until it balances the Lorentz force

The current density j in the material is

eNvj x

zxy BvE

EeF


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N and the Hall Coefficient

We can calculate the number of conductionelectrons N (per unit volume)

Define the Hall constant/coefficient

coefficient is inversely proportional to N

sign indicates carrier type

NeRH 1

y

zx

eEBjN


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Limitations of the Hall Technique

Heating Effects

An electromagnet is usually used

Current required is in the AMPS rangetypically

Do the measurements quickly

Demagnetization is necessary whenrepeating the experiment


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Non-Linear Response

Diode

Rectifying effect

Conducting under forward bias

Non-conducting under reverse bias

Thermal effects

Resistance increases with temperature

Resistance decreases with temperature


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Diode I-V Characteristic


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Temperature Dependence of

ToT 1


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Superconductors

Resistivity drops to zero at low

temperatures


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Critical Temperatures


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Ceramic Superconductors

YBa2Cu3O7-x (YBCO)

Orthorhombic layered

perovskite containing

periodic O vacancies

Drawbacks

Brittleness

Cannot carry high currentdensities

Environmental instability


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Cooper Pair Formation

Electron drifts through crystal generating temporaryperturbations along its path

Displaced ion can be set oscillating by perturbation

generating a phonon Phonon interacts quickly with a second electron

lowering its energy due to lattice deformation

Second electron emits phonon which interacts withfirst electron

Phonon is passed back and forth, coupling theelectrons together, bringing them into a lower energystate


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Cooper Pairs and

Fermi SurfacesAll electrons on a Fermi surface having

opposite momentum and spin form Cooper

pairs Cloud of Cooper pairs is formed

Cooperative drifting through the crystal proceeds

This is an ordered state of the conduction

electrons!

Ordering eliminates scattering by lattice atoms,

resulting in zero resistance!


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Fermi Energy

EF lower in superconducting state

Superconducting state is separated from the

normal state by an energy gap Eg Energy gap stabilizes Cooper pair against

small changes of net momentum

they wont break apart!

Eg ~ 10-4eV

Observed via IR absorption measurements at

T


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DOS Superconductor


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Josephson Effect

Another way to measure Eg Two pieces of metal

One in superconducting state, the other in normal state

Energy bands in superconductor are raised by anappropriate voltage

If the applied voltage is big enough, filled states insuperconductor are opposite empty states in normalconductor Cooper pairs can tunnel through

Eg is determined from the threshold voltage fortunneling


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Josephson Junction


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Optical and Dielectric Properties

Optical properties = response of

materials to electromagnetic radiation

Dielectric properties = response toelectric fields of insulators

Both properties (including magnetic

properties) depend to large extent onthe polarization of the material


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Electromagnetic Spectrum


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MECHANISMS OF

POLARIZATION


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ELECTRONIC POLARIZATION

External electric field shifts the electron cloud

of an atom

Electric field arises from an applied potentialdifference (volts) across the material

Temporary or induced dipole results

Polarization vanishes when the electric field is

removed Can occur in all types of material regardless

of the type of bonding


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IONIC/ATOMIC POLARIZATION

Applied electric field causes chargeseparation Cations move toward negative electrode

Anions move toward positive electrode

Important for ionic materials

Example 11.2-2Calculate the increase in separation of Cs+

and Cl- when an ionic polarization of 4 10-8

C/m2 is achieved by the application of an

electric field.


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Answer

Lattice parameter of CsCl

Since there is one pair of ions per unit cell,

Increase in separation distance

Increase is less than 50 ppb

nmnmRr

ao 400.03

)181.0165.0(2

3

)(2

328

339 /1056.1

)/()10400.0(

)/1)(/1(mcharges

cellm

cellchargecelldipoleZ

nmchargeCmcharges

mC

Zq

Pd

8

19328

28

1060.1)/106.1)(/1056.1(

/104


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MOLECULAR/DIPOLE POLARIZATION

Alignment of permanent dipoles parallel tothe applied electric field

Common in ionic ceramics, silicates and polarpolymers

Can be retained after removal of electric field Can freeze the aligned structure (e.g. apply field

above TG of polymer and let it cool down with theapplied field)

Electrets are polymers that retain this poledstructure for long times Air filter: charged dust particles removed from airstream

by passing them through a web of electret fibers


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INTERFACIAL POLARIZATION

Piling up of mobile charge carriers at a

physical barrier (grain boundaries,

phase boundaries, free surfaces, filminterfaces)


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NET POLARIZATION

The sum of electronic, ionic, molecular

and interfacial polarization

Materials are differentiated by their levelof response from each mechanism

Ionic/molecular: type of atoms, type of

bonding Interfacial: planar defect density


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Frequency of Applied EM Field

The larger the masses involved, the slowerthe response Electrons can react to visible light ~1015 Hz

Ions can react to IR: 1012-1013 Hz

Molecules can react to sub-IR: 1011-1012 Hz

Interfaces: 10-3-103 Hz Depends on grain size/dimension from one interface to

the next

Optical properties from Electronic and Ionicpolarization (fast processes)

Dielectric properties from Molecular andInterfacial polarization (slow processes)


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Example 11.2-4

Describe what effect, if any, each of the

following processes would have on the

net polarization of a solid: a) increasethe grain size of a ceramic polycrystal,

b) increase the concentration of network

modifiers in a silicate glass, and c)change the frequency of the excitation

field from UV to microwave


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Solution

a) Interfacial polarization shifts to lowerfrequencies

Larger grain size means more time to cross the

grain

b) Net polarization increases in the sub-IRrange

Primary bond density decreases, number of

secondary bonds (dipoles) increasesc) Dominant mechanism shifts from electronic

to molecular polarization UV ~1016 Hz, microwave ~1011 Hz


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Perovskite Revisited

BaTiO3 Above 120 C, cubic crystal

Below 120 C, central Ti4+ cant fit position and

displaced 0.006 nm to one side of the cell while O2- are

displaced 0.008 nm in the opposite direction

120 C = ferroelectric Curie temperature (below which

permanent polarization occurs)


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Piezoelectric Effect

Mechanical deformation (e.g., uniaxialcompression) may decrease the separationbetween anion and cation. The motion of the

ion produces a current and induces aninternal field.

Converse effect: application of an electricfield along the correct crystallographic

direction changes the anion-cation spacing(Electrostriction)An applied electric field changes the dimensions

of the material


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Piezoelectric Effect Let = field produced by the stress. Then,

and,

Where g and d are constants related to themodulus of elasticity

Examples: quartz, various titanates, lead

zirconate, cadmium sulfide and zinc oxide.

g

d

gd1E


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PIEZOELECTRIC CONSTANTS

*d is in 10-12 m/V


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Applications of Piezoelectrics

Usually used as a transducer Phonograph cartridges

Convert mechanical vibrations to electrical signal to

acoustic waves Telephones

Electrets are however taking over the telephone market

Electrets exhibit the same behavior as piezoelectrics:applied stress cause deformation and charge motion

Poly(vinylidene fluoride) is popular choice

Pressure Gages

High frequency sound generators


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Piezoelectric Materials Quartz

One of the first piezoelectrics in common use

Precise timing of electrical circuits and watchesusing vibrational frequency control

Barium Titanate Low temperature applications

Solid solutions of PbZrO3 and PbTiO3 (PZTs)

High temperature applications (up to 490 C) Ferroelectricity = a dipole causes neighboring

dipoles to align spontaneously (even in theabsence of an electric field)


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Example 11.2-5

A compressive force of 5N (~1 lb) is applied toa slice of quartz with dimensions 3 mm x 3mm x 0.25 mm. If the elasticity modulus forquartz is 70 GPa, calculate the voltagecreated by the stress.

Solution

Determine g so Ecan be calculated

Pa

m

N

A

F 523

1056.5

)103(

5

Edand

Edg

1


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Solution

The voltage developed is quite large

Small loads can be easily detected

Phonograph cartridge

mVVmPa

Pa/105.3

)/103.2)(1070(

1056.5 6129

5

VmmVEl 875)1025.0)(/105.3( 36


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Capacitance*

Investigate the ability of a material to polarizeand hold charge

Capacitance, C

Q = charge stored

on either plateV = applied voltage

C units: Farad (C/V)

V

QC

*Review Chapter 24 of University Physics by

Young and Freedman, 11th ed.


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Capacitance

Material factor = permittivity,

Reflects the ability of the material to be polarized

by the electric field

Geometric factor = area/plate separation

Dielectric = vacuum

d

AC

d

ACo o o = 8.85 10

-12 F/m


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Polarization and Dielectric Constant

Polarization and permittivity

where = electric field strength

Dielectric constant = permittivity normalized tothat of vacuum

Usually a function of frequency (polarization is!)

oP

o


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Dielectric

Constants


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Example 11.3-1

Polyethylene has a polarization of 5.75

x 10-8 C/m2 in a field of 5000 V/m.

Calculate the dielectric constant of thispolymer.

Solution

3.2)/5000)(/1085.8(

/1075.51

1

12

28

o mVmF

mCP


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Example 11.3-2

A simple parallel plate capacitor that can

store a charge of 10-4 C at 3 kV is required

for a particular application. The thickness of

the dielectric is 0.020 cm. Calculate the areaof the plates if the dielectric is vacuum, PTFE

(Teflon), BaTiO3, mica.

Solution

FV

C

V

QC

84

1033.33000

10


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Solution

Vacuum, =1 and A = 7525 cm2

PTFE, =2 and A = 3763 cm2

BaTiO3, = 3000 and A = 2.5 cm2

Mica, = 7 and A = 1075 cm2

2

14

8

o

7525

)/1085.8(

)02.0)(1033.3( cm

cmF

cmFCdCdA


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Dielectric Strength and Breakdown

The maximum electric field to which adielectric material can be subjected to withoutbreaking down or discharging

Dielectric breakdown is the electrical

equivalent of mechanical failure Pits, holes, channels bridging conductors are

formed

Dependent on the thickness of the material

maxmax

d

V


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Dielectric Strength

Determined by the probability of finding

a critical flaw in the test region

Strength increases with decreasingthickness!

Maximum charge storage: high

dielectric constant and high dielectricstrength required

Mica (+ good thermal stability)


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Dielectric

Strength


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Example 11.3-3 Which material (vacuum, PTFE (Teflon),

BaTiO3, mica) may be used for a simple parallelplate capacitor that can store a charge of 10-4 Cat 3 kV? The thickness of the dielectric is 0.02cm.

SolutionThe applied field strength is

Compared to the dielectric strengths given in theprevious table, Teflon and mica are usable althoughthe capacitor will be significantly large

cmkV

cm

kV/150

02.0

3


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Example 11.3-4

Suggest an alternative design for the

capacitor needed to fulfill the requirements of

storing a charge of 10-4C at 3 kV, assuming

that the raw material for the dielectric is 0.020cm-thick sheets of BaTiO3.

Solution

Applied field strength must be reduced below

dielectric strength of 120 kV/cm corresponding toa minimum dielectric thickness of

cm

cmkV

kVd 025.0

/120

3


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Solution BaTiO3 sheets must be stacked together to achieve a

total dielectric thickness of 0.040 cm. The required

area for the capacitor is calculated

For BaTiO3, = 3000 and C = Q/V = 3.33x10-8 F so

that

oCdCdA

2

14

8

02.5)/1085.8)(3000(

)04.0)(1033.3(cm

cmF

cmFA


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Surface Breakdown

Breakdown due to high fields may occur atthe surface (and not exclusively to the bulk) ofthe material

Can be aggravated by surface moisture andcontamination

E.g., spark plugs should withstand hightemperature, high fields

Fabricated from ceramic steatite (SiO2 + MgO +Al2O3) coated with a glassy silicate (ceramicglaze)

Most ceramics are porous and hygroscopic

MSE 131 Ceramics Lecture 3 2016 student.pdf

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