ENG2000 Chapter 9 Thermal Properties of Materials

1. ENG2000 Chapter 9Thermal Properties of Materials

2. 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

3. Convection, radiation, conduction radiation convection conduction

4. 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

5. Clearly, the more isolated the conductor is, the hotter it gets • So consider a conductor in a chip:

6. 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

7. 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

8. 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!

9. 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%

10. 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, a = coefficient of thermal expansion and K = compressibility • More frequently, we use the heat capacity per unit mass for generality

11. 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

12. Values

13. 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

14. Cv 25 J/mol.K Pb Cu C T (K) 300K

15. 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

16. 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 Fourier’s Law (1822). The negative sign indicates the flow of heat from hot to cold • This is exactly analogous to charge flow

17. 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)

18. 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)

19. 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

20. p1 = mv Dp = p2 - p1 = -2mv p2 = -mv • 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

21. 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

22. 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

23. 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

24. 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

25. 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

26. 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

27. longitudinal transverse optical acoustic • 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

28. 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

29. 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

30. Cv classical theory 25 J/mol.K experimental Einstein electron component T (K) 300K • 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:

31. 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?

32. 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 ...

33. 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

34. sulphur rubber wood water NaCl nylon SiO2 Ge Fe Cu Ag Si Al K [W/m.K] 10-2 10-1 100 101 102 103 phonon conductors electron conductors • 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

35. 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]

36. unit area E1 E2 T l l T1 T0 T2 x x1 x0 x2 • 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

37. 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

38. 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

39. 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 )

40. 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”

41. 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

42. potential energy atomic separation • 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 average inter-atomic radius increases increasing thermal energy

43. Values

44. 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