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

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
Q (heat flow)
Hot Th
Cold Tc L
Thermal conductivity

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 l 1 km 1 m 1 mm 1 mm 100 nm 1 nm Aircraft Automobile
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 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 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 qz q  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
Fermi Energy – highest occupied energy state:
Vacuum
Level
F: Work Function
Fermi Velocity: EF
Energy
Fermi Temp:
Band Edge

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

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. Electronic Specific Heat and Thermal Conductivity
in 3D
Mean free time:
te = le / vF
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

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

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

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 in 3D 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. Debye Model Debye Approximation: Debye Density of States:
Frequency, w
Wave vector, K
p/a
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. Phonon Thermal Conductivity
Phonon Scattering Mechanisms
Kinetic Theory
Boundary Scattering
Defect & Dislocation Scattering
Phonon-Phonon Scattering
Decreasing Boundary
Separation l
Increasing
Defect
Concentration
Phonon Scattering
Defect
Boundary
0.01
0.1
1.0
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 r and 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 Energy flux in terms of particle flux carrying energy: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:
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. Thermoelectric Figure of Merit (ZT)
Coefficient of Performance
where
Seebeck coefficient
Electrical conductivity
Thermal conductivity
Temperature
Bi2Te3
Freon
TH = 300 K
TC = 250 K

38. ZT Enhancement in Thin Film Superlattices
SiGe superlattice
(Shakouri, UCSC)
Increased phonon-boundary scattering
decreased k
+ other size effects
 High ZT = S2sT/k
Si Barrier
Ge Quantum well (QW) 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-coolers Ref: Venkatasubramanian et al, Nature 413, P. 597 (2001)

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

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

44. ZT Enhancement in Nanowires
Top View
Nanowire
Al2O3 template
Nanowires based on
Bi, BiSb,Bi2Te3,SiGe
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 p
Cold
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. Thermal Conductance of a Nanotube Mat
Linear behavior
25 K
Ref: Hone et al.
APL 77, 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 ~ 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

49. Nano Electromechanical System (NEMS)
Thermal conductance quantization in nanoscale SiNx beams
(Schwab et al., Nature 404, 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)