ENG2000 Chapter 9 Thermal Properties of Materials

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ENG2000 Chapter 9Thermal Properties of Materials
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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

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Convection, radiation, conduction
radiation
convection
conduction
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• 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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• Clearly, the more isolated the conductor is, the
  hotter it gets
• So consider a conductor in a chip

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• 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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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

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• 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 Maxwells Demon!

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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 1C
• same energy heats 1g of copper by 11C
• The exact value of the heat capacity depends on
  the conditions under which you measure it
• Cp for constant pressure
• Cv for constant volume
• At room temperature, the difference for solids is
  about 5

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• CV is directly related to the energy of the
  system
• It is easier to measure Cp though, and the two
  Cs are related by
• where V volume, a coefficient of thermal
  expansion and K compressibility
• More frequently, we use the heat capacity per
  unit mass for generality

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• Where
• c is a material characteristic but is
  temperature-dependent
• Now we can write
• we have assumed that W 0
• and the DE can be supplied by e.g. electrical
  power

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Values
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Molar heat capacity

• We can also express heat capacity per mole of
  molecules
• where M is the molar mass, N particles, N0
  Avogadros 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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• 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 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
• where JQ is in units of J/(m2s), and K, the
  thermal conductivity, has units of W/(mK)
• This is Fouriers Law (1822). The negative sign
  indicates the flow of heat from hot to cold
• This is exactly analogous to charge flow

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• Thermal conductivity is also slightly
  temperature-dependent 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)

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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 Boltzmanns 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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Kinetic energy of gases

• We consider a small volume of gas
• 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

x
unit area
dx
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• 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

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• The momentum transferred per unit time per unit
  area is thus
• We know that force is given by F  d(mv)/dt and
  that force per unit area is pressure, so
• But we also know that PV nRT nkN0T NkT, so

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• If we now put in Ekin mv2/2, we find
• 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

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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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• 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

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• From before, we know that Cv  (dEtotal /dT)v
  and Cv CvN0/N, so we find
• 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

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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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• There are four kinds of waves
• 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

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• 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

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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

Emax
OK
Emax - kT
no empty state
filled levels
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• We wont 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

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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?

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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 ...

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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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• So materials are divided into phonon conductors
  and electron conductors of heat
• 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 wont do that but we will quote the result

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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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• Consider the following bar with a temperature
  gradient dT/dx
• We are interested in a small volume of material,
  with a length 2?
• where l is the mean free path between collisions
  with the lattice

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• 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

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• 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
• Therefore, the thermal energy transferred per
  unit time per unit area is

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• But we also know that, by definition, JQ -
  K(dT/dx)
• So we can write
• This says that the thermal conductivity is larger
  if
• there are more electrons
• they move faster (more v)
• they move easier (fewer collisions, larger ?)

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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
• this works quite well for metals but not for
  phonon materials

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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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potential energy
atomic separation
average inter-atomic radius increases
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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Values
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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