Thermal Properties of Materials

1. Thermal Properties of Materials Li Shi Department of Mechanical Engineering & Center for Nano and Molecular Science and Technology, Texas Materials Institute The University of Texas at Austin Austin, TX 78712 www.me.utexas.edu/~lishi lishi@mail.utexas.edu

2. Outline • Macroscopic Thermal Transport Theory– Diffusion • -- Fourier’s Law • -- Diffusion Equation • Microscale Thermal Transport Theory – Particle Transport • -- Kinetic Theory of Gases • -- Electrons in Metals • -- Phonons in Insulators • -- Boltzmann Transport Theory • Thermal Properties of Nanostructures • -- Thin Films and Superlattices • -- Nanowires and Nanotubes • -- Nano Electromechanical System (NEMS)

3. Fourier’s Law for Heat Conduction Thermal conductivity Q (heat flow) Hot Th Cold Tc L

4. Heat Diffusion Equation 1st law (energy conservation) Heat conduction = Rate of change of energy storage Specific heat • Conditions: • t >> t  scattering mean free time of energy carriers • L >> l scattering mean free path of energy carriers • Breaks down for applications involving thermal transport in small length/ time scales, e.g. nanoelectronics, nanostructures, NEMS, ultrafast laser materials processing…

5. Length Scale 1 km Aircraft Automobile 1 m Human Computer Butterfly 1 mm Fourier’s law Microprocessor Module MEMS Blood Cells 1 mm Wavelength of Visible Light Particle transport l MOSFET, NEMS 100 nm Nanotubes, Nanowires 1 nm Width of DNA

6. Outline • Macroscopic Thermal Transport Theory– Diffusion • -- Fourier’s Law • -- Diffusion Equation • Microscale Thermal Transport Theory– Particle Transport • -- Kinetic Theory of Gases • -- Electrons in Metals • -- Phonons in Insulators • -- Boltzmann Transport Theory • Thermal Properties of Nanostructures • -- Thin Films and Superlattices • -- Nanowires and Nanotubes • -- Nano Electromechanical System (NEMS)

7. Mean Free Path for Intermolecular Collision for Gases D D Total Length Traveled = L Average Distance between Collisions, mc = L/(#of collisions) Total Collision Volume Swept = pD2L Mean Free Path Number Density of Molecules = n Total number of molecules encountered in swept collision volume = npD2L s: collision cross-sectional area

8. Mean Free Path for Gas Molecules kB: Boltzmann constant 1.38 x 10-23 J/K Number Density of Molecules from Ideal Gas Law: n = P/kBT Mean Free Path: Typical Numbers: Diameter of Molecules, D  2 Å = 2 x10-10 m Collision Cross-section: s  1.3 x 10-19 m Mean Free Path at Atmospheric Pressure: At 1 Torr pressure, mc 200 mm; at 1 mTorr, mc 20 cm

9. Effective Mean Free Path Wall b: boundary separation Wall Effective Mean Free Path:

10. Kinetic Theory of Energy Transport u: energy Net Energy Flux / # of Molecules u(z+z) z + z  q qz z through Taylor expansion of u u(z-z) z - z Integration over all the solid angles  total energy flux Thermal conductivity: Specific heat Velocity Mean free path

11. Questions • Kinetic theory is valid for particles: can electrons and • crystal vibrations be considered particles? • If so, what are C, v,  for electrons and crystal vibrations?

12. Free Electrons in Metals at 0 K Metal Fermi Energy – highest occupied energy state: Vacuum Level F: Work Function Fermi Velocity: EF Energy Fermi Temp: Band Edge

13. f k T B 1 T = 0 K Vacuum Occupation Probability, Level Increasing T 0 E F F Work Function, Electron Energy, E Effect of Temperature Fermi-Dirac equilibrium distribution for the probability of electron occupation of energy level E at temperature T

14. Number and Energy Densities Number density: Energy density: Density of States -- Number of electron states available between energy E and E+dE in 3D

15. e Bulk Solids Increasing Defect Concentration Defect Scattering PhononScattering Temperature, T Electronic Specific Heat and Thermal Conductivity Specific Heat in 3D Mean free time: te = le / vF Thermal Conductivity • Electron Scattering Mechanisms • Defect Scattering • Phonon Scattering • Boundary Scattering (Film Thickness, • Grain Boundary)

16. Thermal Conductivity of Cu and Al Matthiessen Rule: Electrons dominate k in metals

17. Afterthought • Since electrons are traveling waves, can we apply kinetic • theory of particle transport? • Two conditions need to be satisfied: • Length scale is much larger than electron wavelength or • electron coherence length • Electron scattering randomizes the phase of wave function • such that it is a traveling packet of charge and energy

18. Crystal Vibration Interatomic Bonding Equation of motion with nearest neighbor interaction Solution 1-D Array of Spring Mass System

19. Longitudinal Acoustic (LA) Mode Frequency, w Transverse Acoustic (TA) Mode p/a 0 Wave vector, K Dispersion Relation Group Velocity: Speed of Sound:

20. Two Atoms Per Unit Cell Lattice Constant, a xn+1 yn-1 xn yn Optical Vibrational Modes LO TO Frequency, w TA LA p/a 0 Wave vector, K

21. Phonon Dispersion in GaAs

22. Energy Quantization and Phonons Total Energy of a Quantum Oscillator in a Parabolic Potential n = 0, 1, 2, 3, 4…; w/2: zero point energy Phonon: A quantum of vibrational energy, w, which travels through the lattice Phonons follow Bose-Einstein statistics. Equilibrium distribution: In 3D, allowable wave vector K:

23. Lattice Energy p: polarization(LA,TA, LO, TO) K: wave vector Dispersion Relation: Energy Density: Density of States: Number of vibrational states between w and w+dw in 3D Lattice Specific Heat:

24. Frequency, w p/a 0 Wave vector, K Debye Model Debye Approximation: Debye Density of States: Specific Heat in 3D: Debye Temperature [K] In 3D, when T << qD,

25. Phonon Specific Heat 3kBT Diamond Each atom has a thermal energy of 3KBT Specific Heat (J/m3-K) C  T3 Classical Regime Temperature (K) In general, when T << qD, d =1, 2, 3: dimension of the sample

26. 1.0 0.01 0.1 Phonon Thermal Conductivity Phonon Scattering Mechanisms Kinetic Theory • Boundary Scattering • Defect & Dislocation Scattering • Phonon-Phonon Scattering Decreasing Boundary Separation l Increasing Defect Concentration PhononScattering Defect Boundary Temperature, T/qD

27. Thermal Conductivity of Insulators • Phonons dominate k in insulators

28. Drawbacks of Kinetic Theory • Assumes local thermodynamics equilibrium: u=u(T) • Breaks down when L ; t  t • Assumes single particle velocity and single mean free • path or mean free time. • Breaks down when, vg(w) or t(w) • Cannot handle non-equilibrium problems • Short pulse laser interactions • High electric field transport in devices • Cannot handle wave effects • Interference, diffraction, tunneling

29. Boltzmann Transport Equation for Particle Transport Distribution Function of Particles: f= f(r,p,t) --probability of particle occupation of momentum p at location rand time t Equilibrium Distribution: f0, i.e. Fermi-Dirac for electrons, Bose-Einstein for phonons Non-equilibrium, e.g. in a high electric field or temperature gradient: Relaxation Time Approximation t Relaxation time

30. Energy Flux q v Energy flux in terms of particle flux carrying energy: dk q k f Vector Integrate over all the solid angle: Scalar Integrate over energy instead of momentum: Density of States: # of phonon modes per frequency range

31. Continuum Case BTE Solution: Quasi-equilibrium Direction x is chosen to in the direction of q Energy Flux: Fourier Law of Heat Conduction: t(e) can be treated using Callaway method (Phys. Rev. 113, 1046) If v and t are independent of particle energy, e, then  Kinetic theory:

32. At Small Length/Time Scale (L~l or t~t) Define phonon intensity: From BTE: 0 Equation of Phonon Radiative Transfer (EPRT) (Majumdar, JHT 115, 7): Heat flux: Acoustically Thin Limit (L<<l) and for T << qD Acoustically Thick Limit (L>>l)

33. Outline • Macroscopic Thermal Transport Theory – Diffusion • -- Fourier’s Law • -- Diffusion Equation • Microscale Thermal Transport Theory – Particle Transport • -- Kinetic Theory of Gases • -- Electrons in Metals • -- Phonons in Insulators • -- Boltzmann Transport Theory •  Thermal Properties of Nanostructures • -- Thin Films and Superlattices • -- Nanowires and Nanotubes • -- Nano Electromechanical System

34. Thin Film Thermal Conductivity Measurement 3w method (Cahill, Rev. Sci. Instrum. 61, 802) Metal line Thin Film L 2b V • I~ 1w • T ~ I2 ~ 2w • R ~ T ~ 2w • V~ IR ~3w I0 sin(wt) Substrate

35. Silicon on Insulator (SOI) Ju and Goodson, APL 74, 3005 IBM SOI Chip Lines: BTE results Hot spots!

36. Thermoelectric Cooling • No moving parts: quiet and reliable • No Freon: clean

37. Seebeck coefficient Electrical conductivity TH = 300 K TC = 250 K Freon Temperature Bi2Te3 Thermal conductivity Thermoelectric Figure of Merit (ZT) Coefficient of Performance where

38. ZT Enhancement in Thin Film Superlattices SiGe superlattice (Shakouri, UCSC) • Increased phonon-boundary scattering • decreased k • + other size effects  High ZT = S2sT/k Ge Quantum well (QW) Si Barrier Ec E Ev x

39. Thermal Conductivity of Si/Ge Superlattices k (W/m-K) Bulk Si0.5Ge0.5 Alloy Circles: Measurement by D. Cahill’s group Lines: BTE / EPRT results by G. Chen Period Thickness (Å)

40. Superlattice Micro-coolersRef: Venkatasubramanian et al, Nature413, P. 597 (2001)

41. Nanowires 22 nm diameter Si nanowire, P. Yang, Berkeley • Increased phonon-boundary scattering • Modified phonon dispersion •  Suppressed thermal conductivity • Ref: Chen and Shakouri, J. Heat Transfer 124, 242 Hot p Cold

42. Q I Themal conductance: G = Q / (Th-Ts) Thermal Measurements of Nanotubes and Nanowires Suspended SiNx membrane Long SiNx beams Pt resistance thermometer Kim et al,PRL 87, 215502 Shi et al, JHT, in press

43. Si Nanotransistor (Berkeley Device group) Si Nanowires Gate Drain Source Nanowire Channel D. Li et al., Berkeley Symbols: Measurements Lines: Modified Callaway Method Hot Spots in Si nanotransistors!

44. Nanowires based on Bi, BiSb,Bi2Te3,SiGe Al2O3 template Top View Nanowire ZT Enhancement in Nanowires • k reduction and other size effects  High ZT = S2sT/k Bi Nanowires Ref: Phys. Rev. B. 62, 4610 by Dresselhaus’s group

45. Nanotube Nanoelectronics TubeFET (McEuen et al., Berkeley) Nanotube Logic (Avouris et al., IBM)

46. Thermal Transport in Carbon Nanotubes Hot Cold p • Few scattering: long mean free path l • Strong SP2 bonding: high sound velocity v •  high thermal conductivity:k = Cvl/3~ 6000 W/m-K • Below 30 K, thermal conductance  4G0 = ( 4 x 10-12T) W/m-K, linear T dependence (G0 :Quantum of thermal conductance) Heat capacity

47. Linear behavior 25 K Thermal Conductance of a Nanotube Mat Ref: Hone et al. APL77, 666 • Estimated thermal conductivity at 300K: ~ 250 << 6000 W/m-K •  Junction resistance is dominant • Intrinsic property remains unknown

48. Thermal Conductivity of Carbon Nanotubes CVD SWCN CNT • An individual nanotube has a high k ~ 2000-11000 W/m-K at 300 K • k of a CN bundleis reduced by thermal resistance at tube-tube junctions • The diameter and chirality of a CN may be probed using Raman spectroscopy

49. Nano Electromechanical System (NEMS) Thermal conductance quantization in nanoscale SiNx beams (Schwab et al., Nature404, 974) Quantum of Thermal Conductance Phonon Counters?

50. Summary • Macroscopic Thermal Transport Theory – Diffusion • -- Fourier’s Law • -- Diffusion Equation • Microscale Thermal Transport Theory – Particle Transport • -- Kinetic Theory of Gases • -- Electrons in Metals • -- Phonons in Insulators • -- Boltzmann Transport Theory • Thermal Properties of Nanostructures • -- Thin Films and Superlattices • -- Nanowires and Nanotubes • -- Nano Electromechanical System (NEMS)