Lecture 6.0Lecture 6.0

Properties of Dielectrics

Dielectric use in Silicon ChipsDielectric use in Silicon Chips

Capacitors– On chip– On Circuit Board

Insulators– Transistor gate– Interconnects

Materials– Oxides

–SiO2

– Boro-Silicate Glass

– Nitrides–BN

– polymers

Importance of Dielectrics to Silicon ChipsImportance of Dielectrics to Silicon Chips

Size of devices– Electron Tunneling dimension

Chip Cooling- Device Density– Heat Capacity– Thermal Conductivity

Chip Speed – Capacitance in RC interconnects

Band theory of DielectricsBand theory of Dielectrics

Forbidden Zone–Energy Gap-LARGE

ValenceBand

ConductionBand

Difference between Difference between Semiconductors and Semiconductors and DielectricsDielectrics

Material Eg(eV)

Ge 0.67

Si 1.12

GaAs 1.43

SiO2 8

UO2 5.2

Ga2O3 4.6

Fe2O3 3.1

ZnO 3.2

NiO 4.2

Al2O3 8

kBT =0.0257 eV

at 298˚K

Fermi-Dirac Probability Fermi-Dirac Probability Distribution for electron energy, EDistribution for electron energy, E

Probability, F(E)=

(e{[E-Ef]/k

BT}+1)-1

–Ef is the

Fermi Energy

Number of Occupied StatesNumber of Occupied States

Fermi-Dirac

Density of States

T>1000K only

Probability of electrons in Probability of electrons in Conduction BandConduction Band

Lowest Energy in CBE-Ef Eg/2

Probability in CBF(E)= (exp{[E-Ef]/kBT} +1)-1 )

= (exp{Eg/2kBT} +1)-1

exp{-Eg/2kBT} for Eg>1 eV @ 298K

exp{-(4eV)/2kBT}= exp{-100} @ 298KkBT =0.0257 eV

at 298˚K

Intrinsic Conductivity of DielectricIntrinsic Conductivity of Dielectric

Charge Carriers – Electrons– Holes– Ions, M+i, O-2

= ne e e + nh e h # electrons = # holes

ne e (e+ h)– ne C exp{-Eg/2kBT}

Non-Stoichiometric DielectricsNon-Stoichiometric Dielectrics

Metal Excess M1+x O Metal with Multiple valence

Metal Deficiency M1-x O Metal with Multiple valence

Reaction Equilibrium Keq (PO2)±x/2

)(2

122

)(2

2..'

222

gOVTiOTi

gOx

TiOTiO

OTiOTi

x

ONigOx

NiO

OZngOx

ZnO

x

x

12

12

)(2

)(2

+4

+2

+3

+3

Density Changes with PoDensity Changes with Po22

SrTi1-xO3

Non-Stoichiometric DielectricsNon-Stoichiometric Dielectrics

ExcessM1+x O

DeficientM1-x O

Non-Stoichiometric DielectricsNon-Stoichiometric Dielectrics

Ki=[h+][e-]

K”F=[O”i][V”O]

Conductivity=f(Po2 )

Density =f(Po2 )

Dielectric Conduction due to Non-stoichiometryDielectric Conduction due to Non-stoichiometryN-type P-type

Dielectric Intrinsic Conduction due to Non-stoichiometryDielectric Intrinsic Conduction due to Non-stoichiometryN-type P-type

ExcessZn1+xO

DeficientCu2-xO

+ h

+ h

Extrinsic ConductivityExtrinsic Conductivity

Donor Doping Acceptor Dopingn-type p-type

Ed = -m*e e4/(8 (o)2 h2)Ef=Eg-Ed/2 Ef=Eg+Ea/2

Extrinsic Conductivity of Non-stoichiometry oxidesExtrinsic Conductivity of Non-stoichiometry oxidesAcceptor Dopingp-type

p= 2(2 m*h kBT/h2)3/2 exp(-Ef/kBT)

Law of Mass Action, Nipi=ndpd or =nndn

@ 10 atom % Li in NiO conductivity increases by 8 orders of magnitude@ 10 atom % Cr in NiO no change in conductivity

ONiNiLiOx

NiOxOLix

xxx )(4

)1(2

322122

CapacitanceCapacitance

C=oA/d

=C/Co

=1+e

e =electric susceptibility

PolarizationPolarization

P = e E

e = atomic polarizability

Induced polarizationP=(N/V)q

Polar regions align with E fieldPolar regions align with E field

P=(N/V) Eloc

i(Ni/V) i=3 o (-1)/(+2)

Local E FieldLocal E Field

Local Electric Field

Eloc=E’ + E

E’ = due to surrounding dipoles

Eloc=(1/3)(+2)E

Ionic PolarizationIonic Polarization

P=Pe+Pi

Pe = electronic

Pi= ionic

Pi=(N/V)eA

Thermal vibrations prevent Thermal vibrations prevent alignment with E fieldalignment with E field

Polar region follows E fieldPolar region follows E field

opt= (Vel/c)2

opt= n2

n=Refractive index

Dielectric ConstantDielectric Constant

Material (=0) opt=n2

Diamond 5.68 5.66

NaCl 5.90 2.34

LiCl 11.95 2.78

TiO2 94 6.8

Quartz(SiO2) 3.85 2.13

Resonant Absorption/dipole relaxationResonant Absorption/dipole relaxation

Dielectric Constantimaginary number

’ real part dielectric storage

” imaginary partdielectric loss

o natural frequency

Dipole RelaxationDipole Relaxation

Resonant frequency,o Relaxation time,

22"

22'

1

)(

1

opts

optsopt

22222

222"

22222

222'

)(

)(

o

o

io

opto

o

io

m

e

V

N

m

e

V

N

tiem

ex

dt

dx

dt

xd 202

2

Relaxation Time, Relaxation Time,

Dielectric Constant vs. Dielectric Constant vs. FrequencyFrequency

Avalanche BreakdownAvalanche Breakdown

Avalanche BreakdownAvalanche Breakdown

Like nuclear fission