Lecture 4.0Lecture 4.0

Properties of Metals

Importance to Silicon ChipsImportance to Silicon Chips

Metal Delamination– Thermal expansion failures

Chip Cooling- Device Density– Heat Capacity– Thermal Conductivity

Chip Speed – Resistance in RC interconnects

Electrical CurrentElectrical Current

Flow of Charged Particles due to applied voltage– Solids• Ions/holes are large and slow• electrons are small and fast

– Electrons are often responsible for conduction

Ohm's LawOhm's Law

Current density, J=I/A==/ =electric field[V/cm] =Conductivity, [=1/] =Resistivity =ne, =mobility, e=electron charge,

n=#/vol.

Resistance, R= L/AV=IR

Metal ConductionMetal Conduction

Drude’s theory– electron scattering

by lattice

Mobility, e/me = average time

between collisions of electron with ions

Bloch’s Quantum theory– no electron scattering

in perfect lattice only in a imperfect lattice

Scattering– lattice vibrations– impurities– dislocations

Remember Molecular OrbitalsRemember Molecular Orbitals

New Energy– Bonding– Anti Bonding

•• •

•1s 1s

Energy BandsEnergy Bands

Energy BandsEnergy Bands

Partially Filled

Distribution of Electrons in BandDistribution of Electrons in Band

Fermi-Dirac distributionProbability, – F(E)=1/(exp{[E-Ef]/kBT}+1)

– Ef is the Fermi Energy

Fermi EnergyFermi EnergyMetal Ef(eV)

Na 3.22

Cu 7.00

Ag 5.46

Au 5.49

Mg 7.05

Zn 9.38

Al 11.58

Sn 9.99

Work FunctionWork Function

Fermi-Dirac Probability Fermi-Dirac Probability DistributionDistribution

Density of States-Density of States-3D Schrodinger Eq.3D Schrodinger Eq.

3/222

2/12/3

22

0

3

2

2

2)(

)(

V

N

mE

EmV

Eg

dEEgN

ef

e

E f

ElectronElectronFilling inFilling inBand-Band-density of density of occupied occupied statesstates

Eletrical ConductivityEletrical Conductivity

=ne =mobility, e=electron charge, n=#/vol.

=(N/V) F(E)G(E) e2/me,

Thermal Properties - Chapter 7Thermal Properties - Chapter 7

Thermal ConductivityThermal ExpansionHeat CapacityThermoelectric effect– thermocouple

Thermal Properties - Chapter 7Thermal Properties - Chapter 7

Thermal Vibrations-phonons– Displacement, xmax=(3kBT/Yao)1/2

– Y ao

is the spring constant

Thermal Expansion (l/lo)(1/T), also volume->(V/Vo)(1/T)

Heat Capacity– Cp=1/2 kBT per degree of freedom– 6 degrees of freedom per ion, Cp=3R

• kinetic and potential

Variation of Conductivity with Temp. d /dT

Thermal Thermal ExpansionExpansion

Heat CapacityHeat Capacity-Effect of Phonons/electrons-Effect of Phonons/electrons

Einstein Model

Debye Model

Electrons– density of occupied

statesElectronsValenceofNumbertotalN

NE

TkC

TkkNC

Tk

Tk

TkkNC

f

Bp

BBAp

B

B

BBAp

2

3

max

4

2

2

2

9

5

12

)1)(exp(

)exp(

3

En=(n+1/2)h<E>= h/(exp(h/kBT)-1)

g()= 2V/(22v3)

T

dTk

g

T

UC Bp

]

1)/exp()(

3[max

0

Heat Capacity of ElectronsHeat Capacity of Electrons

ElectronsValenceofNumbertotalN

NE

TkC

f

Bp

2

2

9

Heat CapacityHeat Capacity

Thermal ConductionThermal Conduction

Transport of Phonons (vibrations)kthermal/(T)=constant

– thermal conductivity scales with electrical conductivity

kthermal=kelectrons + kphonons

ConductivitiesConductivities

Thermal Conductivity-PhononThermal Conductivity-Phonon

kphonons= Ne Cp ph Vph/3

– Ne number e-/volume,

– Cp=heat capacity of atoms =3kB

ph =mean free path,

– Vph=velocity

Thermal Conductivity - ElectronThermal Conductivity - Electron

ke= Ne Ce e Ve/3

– Ne number e-/volume,

– Ce=heat capacity of electrons

e =mean free path,

– Ve=velocity

ElectronsValenceofNumbertotalN

NE

TkC

f

Bp

2

2

9

Thermal ConductivityThermal Conductivity

Phonon InteractionsPhonon Interactions

With other phononsWith impurities– depends upon phonon wavelength

With imperfections in Crystal– depends upon phonon wavelength

Phonons travel at speed of sound

Phonon InteractionsPhonon Interactions