Lecture 4.0 Properties of Metals. Importance to Silicon Chips Metal Delamination –Thermal...
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![Page 1: Lecture 4.0 Properties of Metals. Importance to Silicon Chips Metal Delamination –Thermal expansion failures Chip Cooling- Device Density –Heat Capacity.](https://reader036.fdocuments.us/reader036/viewer/2022062805/5697bfbb1a28abf838ca0aae/html5/thumbnails/1.jpg)
Lecture 4.0Lecture 4.0
Properties of Metals
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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
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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
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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
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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
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Remember Molecular OrbitalsRemember Molecular Orbitals
New Energy– Bonding– Anti Bonding
•• •
•1s 1s
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Energy BandsEnergy Bands
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Energy BandsEnergy Bands
Partially Filled
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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
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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
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Work FunctionWork Function
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Fermi-Dirac Probability Fermi-Dirac Probability DistributionDistribution
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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
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ElectronElectronFilling inFilling inBand-Band-density of density of occupied occupied statesstates
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Eletrical ConductivityEletrical Conductivity
=ne =mobility, e=electron charge, n=#/vol.
=(N/V) F(E)G(E) e2/me,
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Thermal Properties - Chapter 7Thermal Properties - Chapter 7
Thermal ConductivityThermal ExpansionHeat CapacityThermoelectric effect– thermocouple
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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
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Thermal Thermal ExpansionExpansion
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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
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Heat Capacity of ElectronsHeat Capacity of Electrons
ElectronsValenceofNumbertotalN
NE
TkC
f
Bp
2
2
9
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Heat CapacityHeat Capacity
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Thermal ConductionThermal Conduction
Transport of Phonons (vibrations)kthermal/(T)=constant
– thermal conductivity scales with electrical conductivity
kthermal=kelectrons + kphonons
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ConductivitiesConductivities
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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
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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
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Thermal ConductivityThermal Conductivity
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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
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Phonon InteractionsPhonon Interactions