ENG2000 Chapter 9
Thermal Properties of Materials
ENG2000: R.I. Hornsey
Thermal: 1
Thermal properties of materials
• The problem of heat generation in ICs is
becoming significant as there are more
transistors per chip and increasing numbers of
layers of metals and reduced dimensions
• The power (energy per unit time) delivered into a
conductor is
 P = IV = I2R = V2/R
• and the resistance is given by R = rL/A
• So longer tracks with a smaller area are more
susceptible to heating
• The conductor heats up until the temperature is
such that the heat lost per unit time balances the
power supplied
ENG2000: R.I. Hornsey
Thermal: 2
Convection, radiation, conduction
radiation
convection
conduction
ENG2000: R.I. Hornsey
Thermal: 3
• Conduction is the flow of heat through a solid,
analogous to electronic transport but with the
driving force being DT instead of DV
 the constant of proportionality is the thermal conductivity
rather than the electrical conductivity
• Radiation is the loss of energy by the emission of
electromagnetic radiation in the infra-red
wavelengths
• Convection is the transfer of heat away from a hot
object because the gas next to the object heats
up and becomes less dense
 hence it rises, setting up currents of gas flow
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Thermal: 4
• Clearly, the more isolated the conductor is, the
hotter it gets
• So consider a conductor in a chip:
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Thermal: 5
• But how hot does the conductor get?
• How fast is heat transported away from the region
of heating?
• How do we optimise the design to overcome
these issues?
• Read on!
• We will first cover a few basics and then talk
about heat capacity, thermal conductivity and
their origins
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Thermal: 6
Thermodynamics
• Pretty much everything to do with heat and heat
flow is covered by thermodynamics
• When two bodies of different temperatures are
brought in contact heat, Q, flows from the hotter
to the cooler
• Alternatively, a temperature increase can be
achieved by doing work, W, on the system
 e.g. electrical heating, friction, …
• In either case, there is a change of energy of the
“system”
 DE = W + Q
 where Q is the heat received from the environment
• The first law of thermodynamics
ENG2000: R.I. Hornsey
Thermal: 7
• Energy, heat and work all have the same units
 Joules, J
 Joule performed experiments to demonstrate the
equivalence of heat and mechanical work (1850)
• For our discussion of the intrinsic thermal
properties of materials, we will often take W = 0
• Note that heat transfer is like diffusion of a gas
 there is nothing in principle to stop heat or gas molecules
from piling up in one place but it is statistically extremely
unlikely
 look up Maxwell’s Demon!
ENG2000: R.I. Hornsey
Thermal: 8
Heat capacity
• The heat capacity of a material is used to indicate
that it takes different amount of heat to raise the
temperature of different materials by a given
amount
 e.g.4.18J to heat 1g of water by 1°C
 same energy heats 1g of copper by 11°C
• The exact value of the heat capacity depends on
the conditions under which you measure it
 C’p for constant pressure
 C’v for constant volume
• At room temperature, the difference for solids is
about 5%
ENG2000: R.I. Hornsey
Thermal: 9
• C’V is directly related to the energy of the system
E 
Cv   
T V
• It is easier to measure C’p though, and the two C’s
are related by
Cv  C p 
aTV
K
 where V = volume, a = coefficient of thermal expansion and
K = compressibility
• More frequently, we use the heat capacity per unit
mass for generality
ENG2000: R.I. Hornsey
Thermal: 10
• Where
c p,v 
Cp,v
m
 c is a material characteristic but is temperature-dependent
• Now we can write
DE  Q  mcv DT
 we have assumed that W = 0
 and the DE can be supplied by e.g. electrical power
ENG2000: R.I. Hornsey
Thermal: 11
Values
cp (J/gK)
Cp (J/mol.K)
Al
0.899
24.3
Fe
0.460
25.7
Ni
0.456
26.8
Cu
0.385
24.4
Pb
0.130
26.9
Ag
0.236
25.5
C
0.904
10.9
Water
4.184
75.3
ENG2000: R.I. Hornsey
Thermal: 12
Molar heat capacity
• We can also express heat capacity per mole of
molecules
C 
Cv  cv M  v
n
N
n
N0
 where M is the molar mass, N = # particles, N0 = Avogadro’s
number
• From the previous table, most of the metals have
a Cp ≈ Cv of about 25 J/(Mol.K)
 this is known as the Dulong-Petit “law” (1819)
• The variation of Cv with T is shown in the next
slide
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Thermal: 13
Cv
25 J/mol.K
Pb
Cu
C
T (K)
300K
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Thermal: 14
• The temperature at which cv reaches 96% of its
final value is known as the Debye temperature
 Pb: 95K
 Cu: 340K
 C: 1850K
• At room temperatures, classical theory can
explain heat capacities, but at low temperatures
quantum theories are needed
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Thermal: 15
Thermal conductivity
• If we have a bar of material with a difference in
temperature between the ends, heat will flow from
the hotter to the cooler
dT
dx
 where JQ is in units of J/(m2s), and K, the thermal
conductivity, has units of W/(mK)
JQ  K
• This is Fourier’s Law (1822). The negative sign
indicates the flow of heat from hot to cold
• This is exactly analogous to charge flow
JE  
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dV
dx
Thermal: 16
• Thermal conductivity is also slightly temperaturedependent and usually decreases for increasing T
• Typical values for K at room temperature are





Cu: 4 x 102 W/(mK)
Si: 1.5 x 102 W/(mK)
glass: 8 x 10-1 W/(mK)
water: 6 x 10-1 W/(mK)
wood: 8 x 10-2 W/(mK)
• An important fact to note for microelectronics is
that the above values are for bulk materials
 while those for thin films can be significantly different
 largely because of the relative importance of the boundary
 (the same is true of electrical conductivity, heat capacity etc)
ENG2000: R.I. Hornsey
Thermal: 17
Gases
• The behaviour of gases is of some relevance to
us because we can treat the electrons in a metal
as a gas
• We will discover later that, for metals, heat
conduction and electrical conduction are
intimately related
• An ideal gas follows PV = nRT
 where P = gas pressure, V = gas volume, n = N/N0
 also R = kN0, where k = Boltzmann’s constant
 R = 8.314 J/(mol.K)
• We will use this expression in the calculation of
the kinetic energy of gas molecules (or electrons
in a metal)
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Thermal: 18
Kinetic energy of gases
• We consider a small volume of gas:
+x
unit area
dx
• If the particles move randomly, then 1/3 of them
on average move in the x-direction
 or 1/6 on average move in the +x direction
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Thermal: 19
• The number of particles per unit time which hit
the end of the volume (per unit volume) will be on
average
 z = nvv/6
 where nv is the number of particles per unit volume = N/V
 and v is the velocity
• When a particle bounces off the wall, they
transfer a momentum of 2mv
p1 = mv
Dp = p2 - p1 = -2mv
p2 = -mv
ENG2000: R.I. Hornsey
Thermal: 20
• The momentum transferred per unit time per unit
area is thus
1
1N 2
*
P  2mvz  nv v2mv 
mv
6
3V
• We know that force is given by F = d(mv)/dt and
that force per unit area is pressure, so
F ma d mv dt
1N 2
*
P 

P 
mv
A A
A
3V
• But we also know that PV = nRT = nkN0T = NkT,
so
1
2
PV  Nmv  kNT
3
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Thermal: 21
• If we now put in Ekin = mv2/2, we find
1
1 2 2
kNT  N2 mv  NE kin
3
2
3
• Which, rearranged gives us the kinetic energy as
Ekin = (3/2)kT
• This is an average kinetic energy because we
assumed an average number of particles moving
in the +x direction at the start of the analysis
ENG2000: R.I. Hornsey
Thermal: 22
Classical theory of heat capacity
• We will now use results from the previous
sections to derive the Dulong - Petit law, Cv = 25
J/(mol.K)
• We are comfortable with the idea that an atom
vibrates about its ideal lattice position because of
its thermal energy
 with an amplitude of about 10% of the equilibrium atomic
spacing
• Such an atom can be thought of as being like a
sphere supported by springs
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Thermal: 23
• The atom acts like a simple harmonic oscillator
which “stores” an amount of thermal energy
E = kT
 this is really the definition of k
• In a 3-dimensional solid, the oscillator has
energy, E = 3kT
• This the energy per atom. The total internal
energy per mole is therefore E = 3N0kT
ENG2000: R.I. Hornsey
Thermal: 24
• From before, we know that C’v = (dEtotal /dT)v and
Cv = C’vN0/N, so we find
N 0  E.# moles N 0   N 
Cv 

E

N
T
N T  N 0 
E
Cv 
 3kN 0
T
• Since N0 = 6.02 x 1023 (g.mol)-1 and k = 1.38 x 10-23
J/K
 3kN0 = 24.9 J/(mol.K)
 which agrees very well with Dulong-Petit
• The only problem is that this predicts Cv to be
independent of T, which we saw is not the case
ENG2000: R.I. Hornsey
Thermal: 25
Phonons
• Einstein (as usual) came up with the solution to
this difficulty
• In 1903 he proposed that the energies of the
“atomic oscillators” was quantized
 such quantized lattice oscillations are called phonons
• When we think of atoms vibrating due to their
thermal energy, we assumed they moved
independently
• However, because of bonding, the motions are
connected
 leading to a wave behaviour
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Thermal: 26
• There are four kinds of waves:
longitudinal
transverse
optical
acoustic
• Using wave-particle duality, we can say that these
waves can also act like particles
 these are called phonons
 they can scatter etc. just like particles
ENG2000: R.I. Hornsey
Thermal: 27
• Einstein proposed that, unlike electrons, the
number of phonons increases with temperature
 or, conversely phonons are eliminated with decreasing
temperature
 the energy of each phonon is constant
• For electrons, it is their energy that increases
with temperature, not their number
• The average number of phonons at any
temperature was found to obey a distribution
 the Bose-Einstein distribution
• So the average energy stored in phonons was
calculated. It simplified to Dulong-Petit at higher
temperatures but agreed with the experiments at
lower temperatures
ENG2000: R.I. Hornsey
Thermal: 28
Electrons
• We know that increasing the temperature also
increases the K.E. of the electrons
• So how much of the heat capacity is contributed
by the electrons?
• In fact, it turns out that electrons play a small part
in the heat capacity
• Only those electrons within a kT of the highest
occupied energy levels can gain extra thermal
energy:
OK
Emax
Emax - kT
no empty state
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filled
levels
Thermal: 29
• We won’t do the calculation, but it turns out that
only a small fraction of the total number of
electrons can gain thermal energy
 about 1% of Cv is contributed by the electrons at room
temperature
• The contributions from each mechanism can be
summarized in the following graph:
Cv
classical theory
25 J/mol.K
experimental
Einstein
electron component
ENG2000: R.I. Hornsey
300K
T (K)
Thermal: 30
Heat conduction
• We know that heat flows from the hot to the cold,
but what does the transferring?
• In a solid, only two things can move
 electrons and phonons
• Depending on the material involved, one or other
species tends to dominate
• It was found that good electrical conductors
tended also to be good thermal conductors
• But what about insulators?
ENG2000: R.I. Hornsey
Thermal: 31
Electrons (again)
• The connection between electrical and thermal
conductivities for metals was expressed in the
Wiedeman - Franz Law in 1953
 suggesting that electrons carry thermal energy as well as
electrical charge
• Because of electrical neutrality, equal numbers of
electrons move from hot to cold as the reverse
 but their thermal energies are different
• However, it is observed that electrical
conductivity varies over ~25 orders of
magnitude, while thermal varies over just 4
orders ...
ENG2000: R.I. Hornsey
Thermal: 32
Phonons (again)
• In electrical insulators, there are few free
electrons, so the heat must be conducted in some
other way
 i.e. lattice vibrations = phonons
• As we stated earlier, there is a major difference
between heat conduction by electrons and by
phonons
 for phonons, the number changes with the temperature, but
the energy is quantised
 while for electrons, the number is fixed but the energy varies
• Both movements are due to diffusion
 of particle numbers for phonons, of particle energies for
electrons
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Thermal: 33
10-2
10-1
100
101
phonon conductors
102
Cu
Ag
Al
Si
Fe
Ge
SiO2
NaCl
water
rubber
nylon
wood
sulphur
• So materials are divided into phonon conductors
and electron conductors of heat:
103 K [W/m.K]
electron conductors
• We will now derive the thermal conductivity using
a classical argument
• To prove the Wiedeman-Franz Law, we need to
treat both the heat capacity and the conductivity
in quantum terms
 we won’t do that but we will quote the result
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Thermal: 34
Thermal conductivity – classical derivation
• To do this, we consider a bar of material with a
thermal gradient
• We calculate the flow of energy through a volume
due to the temperature gradient and use this to
calculate the number of “hot” electrons
• An equal number of “cold” electrons must flow
the opposite way, so we can solve for K
 [note: in the usual jargon, “hot” electrons are accelerated to
a high drift velocity by a high electric field]
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Thermal: 35
• Consider the following bar with a temperature
gradient dT/dx:
E1
E2
T

unit area

T1
T0
T2
x
x1
x0
x2
• We are interested in a small volume of material,
with a length 2
 where  is the mean free path between collisions with the
lattice
Thermal: 36
ENG2000: R.I. Hornsey
• The idea is that an electron must have undergone
a collision in this space and hence will have the
energy/temperature of this location
• To calculate the energy flowing per unit time per
unit area from left to right (E1), we multiply the
number of electrons crossing by the energy of
one electron
 dT 
3
3 
energy  kT1  k T0   
 dx 
2
2 
1
number  nv
6
nv 3 
dT 
E1 
k T0   
6 2 
dx 
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Thermal: 37
• Now, we know that the same number of electrons
must flow in the opposite direction to maintain
charge neutrality
 but the energy per electron is lower
nv 3 
dT 
E2 
kT0   
6 2 
dx 
• Therefore, the thermal energy transferred per unit
time per unit area is
nv dT
JQ  E1  E2   k
2
dx
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Thermal: 38
• But we also know that, by definition,
JQ = - K(dT/dx)
• So we can write:
nvk
K
2
• This says that the thermal conductivity is larger
if:
 there are more electrons
 they move faster (more v)
 they move easier (fewer collisions, larger )
ENG2000: R.I. Hornsey
Thermal: 39
Wiedeman-Franz law
• Quantum mechanically, we could calculate the
energy of electrons at the uppermost filled energy
levels
• And we could multiply by the number of electrons
there to get another expression for K
• We could then compare this expression to that
for the electrical conductivity
• And we would find
K  2k 2

 L, the Lorentz number
2
T 2q
 
 2.443108 J/ K 2 s
 this works quite well for metals but not for “phonon materials”
ENG2000: R.I. Hornsey
Thermal: 40
Thermal expansion
• The majority of materials expand when their
temperature is increased
• This is simply expressed as
 DL/L0 = aDT
 where L0 is the original length and a is the coefficient of
(linear) thermal expansion
• Clearly, a material whose length is constrained
becomes strained when the temperature is
changed
• The expansion is a result of the bonding energy
diagram we saw a long time ago …
 page ATOM 20
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Thermal: 41
potential energy
average inter-atomic
radius increases
atomic
separation
increasing
thermal
energy
• Because the curve is not symmetric, the
increased energy of the atoms leads to a change
of average atomic spacing
• If the curve is more symmetric, the effect is
reduced
 and the coefficient of thermal expansion is lower
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Thermal: 42
Values
ENG2000: R.I. Hornsey
Material
a [°C-1 x10-6]
Al
23.6
Cu
17.0
Fe
11.8
Ag
19.7
W
4.5
stainless steel
16.0
glass
9.0
polyethylene
~150
nylon
144
Thermal: 43
Summary
• Not surprisingly, the thermal properties of
materials are intimately connected with atomic
bonding and electronic effects
• We found that energy is stored in ‘atomic
oscillators’
 classical treatments lead to an approximate value for the
heat capacity
 a full treatment involves phonons
• Phonons are quantised units of lattice vibration
 effectively heat particles
• Thermal conductivity takes place either by
electrons or phonons, depending on the material
• Thermal expansion is related to atomic bonding
ENG2000: R.I. Hornsey
Thermal: 44