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. 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. 3 Thermal conductivity Hot T h Cold T c L Q (heat flow) Fourier’s Law for Heat Conduction

4. 4 Heat Diffusion Equation Specific heat Heat conduction = Rate of change of energy storage 1 st law (energy conservation)  Conditions: 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. 5 Length Scale 1 m 1 mm 1  m 1 nm Human Automobile Butterfly 1 km Aircraft Computer Wavelength of Visible Light MEMS Width of DNA MOSFET, NEMS Blood Cells Microprocessor Module Nanotubes, Nanowires Particle transport 100 nm Fourier’s law l

6. 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. 7 D D Mean Free Path for Intermolecular Collision for Gases Total Length Traveled = L Total Collision Volume Swept =  D 2 L Number Density of Molecules = n Total number of molecules encountered in swept collision volume = n  D 2 L Average Distance between Collisions, mc = L/(#of collisions) Mean Free Path  : collision cross-sectional area

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

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

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

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

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

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

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

15. 15 Electronic Specific Heat and Thermal Conductivity Specific Heat Thermal Conductivity Electron Scattering Mechanisms Defect Scattering Phonon Scattering Boundary Scattering (Film Thickness, Grain Boundary) e Temperature, T Defect Scattering Phonon Scattering Increasing Defect Concentration Bulk Solids Mean free time:  e = l e / v F in 3D

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

17. 17 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 Afterthought

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

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

20. 20 Lattice Constant, a xnxn ynyn y n-1 x n+1 Two Atoms Per Unit Cell Frequency,  Wave vector, K 0  /a LA TA LO TO Optical Vibrational Modes

21. 21 Phonon Dispersion in GaAs

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

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

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

25. Phonon Specific Heat Classical Regime In general, when T <<  D, d =1, 2, 3: dimension of the sample Each atom has a thermal energy of 3K B T Specific Heat (J/m 3 -K) Temperature (K) C  T 3 3kBT3kBT Diamond

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

27. Phonons dominate k in insulators Thermal Conductivity of Insulators

28. 28 Drawbacks of Kinetic Theory Assumes local thermodynamics equilibrium: u=u(T) Breaks down when L  ; t  Assumes single particle velocity and single mean free path or mean free time. Breaks down when, v g (  ) or  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 r and time t Relaxation Time Approximation t Equilibrium Distribution: f 0, i.e. Fermi-Dirac for electrons, Bose-Einstein for phonons Relaxation time Non-equilibrium, e.g. in a high electric field or temperature gradient:

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

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

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

33. 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. 34 Thin Film Thermal Conductivity Measurement I 0 sin(  t) L2b Thin Film Substrate Metal line 3  method (Cahill, Rev. Sci. Instrum. 61, 802) I ~ 1  T ~ I 2 ~ 2  R ~ T ~ 2  V~ IR ~3  V

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

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

37. 37 Coefficient of Performance where Thermoelectric Figure of Merit (ZT) Seebeck coefficient Electrical conductivity Thermal conductivity Temperature Bi 2 Te 3 Freon T H = 300 K T C = 250 K

38. 38 ZT Enhancement in Thin Film Superlattices EcEc EvEv x E Ge Quantum well (QW) Si Barrier Increased phonon-boundary scattering  decreased k + other size effects  High ZT = S 2  T/ k SiGe superlattice (Shakouri, UCSC)

39. 39 Thermal Conductivity of Si/Ge Superlattices Period Thickness (Å) k (W/m-K) Bulk Si 0.5 Ge 0.5 Alloy Circles: Measurement by D. Cahill’s group Lines: BTE / EPRT results by G. Chen

40. 40 Superlattice Micro-coolers Ref: Venkatasubramanian et al, Nature 413, P. 597 (2001)

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

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

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

44. 44 ZT Enhancement in Nanowires Ref: Phys. Rev. B. 62, 4610 by Dresselhaus’s group Top View Nanowire Al 2 O 3 template Nanowires based on Bi, BiSb,Bi 2 Te 3,SiGe Bi Nanowires k reduction and other size effects  High ZT = S 2  T/ k

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

46. 46 Thermal Transport in Carbon Nanotubes Few scattering: long mean free path l Strong SP 2 bonding: high sound velocity v  high thermal conductivity: k = Cvl/3 ~ 6000 W/m-K Below 30 K, thermal conductance  4G 0 = ( 4 x 10 -12 T) W/m-K, linear T dependence (G 0 :Quantum of thermal conductance) Hot Cold p Heat capacity

47. 47 Thermal Conductance of a Nanotube Mat Estimated thermal conductivity at 300K: ~ 250 << 6000 W/m-K  Junction resistance is dominant Ref: Hone et al. APL 77, 666 Linear behavior 25 K Intrinsic property remains unknown

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

49. 49 Nano Electromechanical System (NEMS) Thermal conductance quantization in nanoscale SiN x beams (Schwab et al., Nature 404, 974 ) Quantum of Thermal Conductance Phonon Counters?

50. 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)