ENG2000 Chapter 9 Thermal Properties of Materials

ENG2000: R.I. Hornsey Thermal: 1

ENG2000 Chapter 9Thermal Properties of Materials


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 = L/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


Convection, radiation, conduction

convection

conduction

radiation


• Conduction is the flow of heat through a solid, analogous to electronic transport but with the driving force being T instead of V 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


• Clearly, the more isolated the conductor is, the hotter it gets

• So consider a conductor in a chip:


• 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


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” E = W + Q where Q is the heat received from the environment

• The first law of thermodynamics


• 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!


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%


• C’V is directly related to the energy of the system

• It is easier to measure C’p though, and the two C’s are related by

where V = volume, = coefficient of thermal expansion and K = compressibility

• More frequently, we use the heat capacity per unit mass for generality

€

′ C v =∂E∂T

⎛ ⎝ ⎜

⎞ ⎠ ⎟ V

€

′ C v = ′ C p−αTVK


• Where

c is a material characteristic but is temperature-dependent

• Now we can write

we have assumed that W = 0 and the E can be supplied by e.g. electrical power

€

cp,v =′ C p,v

m

€

ΔE =Q =mcvΔT


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


Molar heat capacity• We can also express heat capacity per mole of

molecules

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

€

Cv =cvM =′ C v

n

n=NN0


25 J/mol.K

Cv

T (K)300K

PbCu

C


• 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


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 , the thermal conductivity, has units of W/(mK)

• 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

€

JQ =−ΚdTdx

€

J E =−σdVdx


• Thermal conductivity is also slightly temperature-dependent and usually decreases for increasing T

• Typical values for 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)


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)


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

dx

unit area

+x


• 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

p2 = -mv

p = p2 - p1 = -2mv


• 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

€

P* =2mvz=16

nvv2mv=13

NV

mv2

€

P =FA

=maA

=d mv( ) dt

A=P* =

13

NV

mv2

€

PV =13

Nmv2 =kNT


• 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

€

kNT=13

N212

mv2 =23

NEkin


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


• 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


• From before, we know that C’v = (dEtotal /dT)v and Cv = C’vN0/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


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


• 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

longitudinal transverse

optical

acoustic


• 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


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

Emax - kTOK

no empty statefilledlevels


• 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:

25 J/mol.K

Cv

T (K)300K

classical theoryexperimental

Einstein

electron component


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?


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 ...


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


• 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 won’t do that but we will quote the result

10-2 10-1 100 101 102 103

phonon conductors electron conductors

sulp

hu

r

wo

od

rub

be

rn

ylo

n

wa

ter

Na

Cl

SiO

2

Fe

Ge

Si

Al

Cu

Ag

[W/m.K]


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 [note: in the usual jargon, “hot” electrons are accelerated to

a high drift velocity by a high electric field]


• Consider the following bar with a temperature gradient dT/dx:

• We are interested in a small volume of material, with a length 2 where is the mean free path between collisions with the

lattice

x1 x0 x2

x

T

T1T0T2

E1 E2unit area


• 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

€

energy=32

kT1=32

k T0+λ −dTdx

⎛ ⎝ ⎜

⎞ ⎠ ⎟

⎡

⎣ ⎢

⎤

⎦ ⎥

number=16

nv

E1=nv6

32

k T0−λdTdx

⎛ ⎝ ⎜

⎞ ⎠ ⎟


• 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

€

E2 =nv6

32

k T0+λdTdx

⎛ ⎝ ⎜

⎞ ⎠ ⎟

€

JQ =E1−E2 =−nv2

kλdTdx


• But we also know that, by definition, JQ = - (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 )

€

Κ =nvkλ

2


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

• 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”

€

ΚσT

=π2k2

2q2=L, the Lorentz number

=2.443×10−8JΩ/ K2s( )


Thermal expansion• The majority of materials expand when their

temperature is increased• This is simply expressed as

L/L0 = T

where L0 is the original length and 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


• 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

increasing thermal energy

average inter-atomic radius increases

atomic separation

potential energy


Values

Material [°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


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

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ENG2000 Chapter 9 Thermal Properties of Materials

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