Lecture 4.0
Properties of Metals
Importance to Silicon Chips

Metal Delamination
– Thermal expansion failures

Chip Cooling- Device Density
– Heat Capacity
– Thermal Conductivity

Chip Speed
– Resistance in RC interconnects
Electrical 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 Law

Current density, J=I/A==/
– =electric field[V/cm]
– =Conductivity, [=1/] =Resistivity
– =ne, =mobility, e=electron charge, n=#/vol.
Resistance, R=  L/A
 V=IR

Metal Conduction

Drude’s theory

– electron scattering
by lattice

– no electron scattering
in perfect lattice only in
a imperfect lattice
Mobility, e/me
–  = average time
between collisions
of electron with
ions
Bloch’s Quantum
theory

Scattering
– lattice vibrations
– impurities
– dislocations
Remember Molecular Orbitals

New Energy
– Bonding
– Anti Bonding

1s •
•
• •

1s
Energy Bands
Energy Bands
Partially Filled
Distribution of Electrons in Band
Fermi-Dirac distribution
 Probability,

– F(E)=1/(exp{[E-Ef]/kBT}+1)
– Ef is the Fermi Energy
Fermi Energy
Metal
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 Function
Fermi-Dirac Probability
Distribution
Density of States-3D Schrodinger Eq.
Ef
N   g ( E )dE
0
3/ 2
V  2me  1/ 2
g (E)  2  2  E
2   
  3 N 


Ef 
2me  V 
2
2
2/3
Electron
Filling in
Banddensity of
occupied
states
Eletrical Conductivity

=ne

=mobility, e=electron charge, n=#/vol.

=(N/V) F(E)G(E) e2/me,
Thermal Properties - Chapter 7
Thermal Conductivity
 Thermal Expansion
 Heat Capacity
 Thermoelectric effect

– thermocouple
Thermal 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
Expansion
Heat Capacity-Effect of Phonons/electrons

Einstein Model
En=(n+1/2)h
<E>= h/(exp(h/kBT)-1)

2V/(22v3)
 max

exp(

)
k BT


 (exp(  )  1) 2
k BT
Debye Model
g()=
Cp 
 
C p  3 N A k B 
 k BT
2
U

T
[3
0
g ( )
d ]
exp(  / k BT )  1
T
Electrons
– density of occupied
states
12
4  k BT
Cp 
N A k B 
5
  max



3
9 k B2T
Cp 
N
2 Ef
N total Number of Valence Electrons
Heat Capacity of Electrons
9 k B2T
Cp 
N
2 Ef
N total Number of Valence Electrons
Heat Capacity
Thermal Conduction
Transport of Phonons (vibrations)
 kthermal/(T)=constant

– thermal conductivity scales with electrical conductivity

kthermal=kelectrons + kphonons
Conductivities
Thermal 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 - Electron

ke= Ne Ce e Ve/3
– Ne number e-/volume,
– Ce=heat capacity of electrons
– e =mean free path,
– Ve=velocity
9 k B2T
Cp 
N
2 Ef
N total Number of Valence Electrons
Thermal Conductivity
Phonon Interactions
With other phonons
 With impurities

– depends upon phonon wavelength

With imperfections in Crystal
– depends upon phonon wavelength

Phonons travel at speed of sound
Phonon Interactions