Title: Thermal Properties of Materials
1Thermal Properties of Materials
Li Shi, PhD Assistant Professor 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_at_mail.utexas.edu
2Outline
 Thermal Issues in Nanoscale Devices
 Microscale Thermal Transport Theory Particle
Transport   Kinetic Theory of Gases
  Electrons in Metals
  Phonons in Insulators

 Nanoengineer Thermal Properties of Materials
  Thin Films
  Nanowires and Nanotubes
  Bulk materials with embedded nanostructures
3GMR
Cu Interconnects
4Thermal Issues in Si Nanotransistors
Ju and Goodson, APL 74, 3005
IBM SOI Chip
Lines BTE results
Hot spots!
5Thermoelectric Cooling
Venkatasubramanian et al. Nature 413, 597
2.525nm
Bi2Te3/Sb2Te3 Superlattices
Harman et al., Science 297, 2229
Quantum dot superlattices
 Coefficient of Performance (COP)
2
CFC unit
1
COP
Bi2Te3
0
0
1
2
3
4
5
ZT
Seebeck coefficient
Electrical conductivity
Thermal conductivity
Nanoengineer thermal properties of materials!
Courtesy of A. Majumdar
6Microscopic Origins of Heat Conduction The
Particle Nature
Materials Dominant energy carriers Gases
Molecules Metals
Electrons Insulators Phonons (crystal
vibration)
Cold
Hot
N2
Cold
Hot

Hot
Cold
p
7Mean 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 crosssectional area
8Effective Mean Free Path
Wall
?b boundary separation
Wall
When ?b ?mc, the effective mean free path
9Kinetic Theory of Particle Transport
Cold
u energy
u(z?z)
Net Energy Flux / of Molecules
z ?z
?
q
qz
z
through Taylor expansion of u
z  ?z
u(z?z)
Hot
Integration over all the solid angles ? total
energy flux
Thermal conductivity
Specific heat capacity Velocity Mean free
path
10Drawbacks of Kinetic Theory
 Assumes single particle velocity and single mean
free  path or mean free time.
 Breaks down when, vg(E) or ?(E)
 Assumes local thermodynamics equilibrium uu(T)
 Breaks down when L ? ? t ? t
 Cannot handle nonequilibrium problems
 Short pulse laser interactions
 High electric field transport in devices
 Cannot handle wave effects
 Interference, diffraction, tunneling
Boltzmann Transport Equation
11Questions
 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 WaveParticle Duality of Electrons
The double slit experiment
Schrodinger eqn. for free electrons
? electron wave function ? Plancks
constant m electron mass
E electron energy
Electrons
Traveling wave solution to Schrodingers eqn
k wave vector 2p/l
A metal ring with perimeter L
x
? Quantization of energy
13Fermi Parameters
Fermi Energy the highest occupied energy state
at 0 K
Metal
Vacuum Level
F Work Function
Fermi Velocity
EF
Energy
Fermi Temp
Band Edge
14Effect of Temperature
FermiDirac equilibrium distribution for the
probability of electron occupation of energy
level E at temperature T
15Electronic Specific Heat and Thermal Conductivity
Energy density
Bulk metals
Density of states
Specific Heat
Thermal Conductivity
 Electron Scattering Mechanisms
 Defect Scattering
 Phonon Scattering
 Boundary Scattering (Film Thickness,
 Grain Boundary)
16Thermal Conductivity of Cu and Al
Matthiessen Rule
k of a metal is dominated by the electronic
contribution
17Conditions
 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
Can we treat crystal vibration as particles just
like what we have done for electrons?
18Crystal Vibration
Interatomic Bonding
Equation of motion with nearest neighbor
interaction
Traveling wave solution
1D Array of Spring Mass System
19Dispersion Relation
Group Velocity
Speed of Sound
l
20Two Atoms Per Unit Cell
Optical Vibrational Modes
LO
TO
Frequency, w
TA
LA
TO
LO
0
p/a
Wave vector, K
Oscillating out of phase against each other. Vg
0 ?little contribution to k
21Phonon Dispersion in GaAs
22Allowed Wave Vectors
A linear chain of N10 atoms with two ends jointed
u atom displacement
Solution us uK(0)exp(iwt)sin(Kx), x sa B.C.
us0 usN10
x
a
K?2np/(Na), n 1, 2, ,N Na L
Only N wavevectors (K) are allowed (one per
mobile atom)
K 8p/L 6p/L 4p/L 2p/L 0
2p/L 4p/L 6p/L 8p/L p/aNp/L
23The WaveParticle Duality of Crystal Vibration
Total Energy of a Harmonic Oscillator in a
Parabolic Potential
?w
Phonon A particle carrying a quantum of
vibrational energy, ?w, which travels through
the lattice
Phonons follow BoseEinstein statistics.
Equilibrium distribution
T
w
24Energy of Lattice Vibration
p polarization(LA,TA, LO, TO) K wave vector
Dispersion Relation
Energy Density
D(w) Density of States, number of allowed wave
vectors between w and wdw
Lattice Specific Heat
25Debye Model
Debye Approximation
Debye Density of States
LA or TA branch
Specific Heat in 3D
Debye Temperature K
In 3D, when T
26Phonon Specific Heat
3?kB
Diamond
Each atom has a thermal energy of 3KBT
Specific Heat (J/m3K)
C ? T3
Classical Regime
Temperature (K)
In general, when T d 1, 2, 3 dimension of the sample
27Phonon Thermal Conductivity
Phonon Scattering Mechanisms
Kinetic Theory
 Boundary Scattering
 Defect impurity Scattering
 PhononPhonon Scattering
Decreasing Boundary Separation
?l
Increasing Defect Concentration
PhononScattering
Defect
Boundary
Temperature, T/qD
28PhononPhonon Scattering
 The presence of one phonon causes a periodic
elastic strain which modulates in space and time
the elastic constant (C) of the crystal. A second
phonon sees the modulation of C and is scattered
to produce a third phonon.
Decreasing Boundary Separation
?l
? ?phonon exp(?/bT)
Increasing Defect Concentration
?phonon exp(?/bT)
PhononScattering
Defect
Boundary
Temperature, T/qD
29Thermal Conductivity of Bulk Crystals
3
k
30Effect of Impurity on Thermal Conductivity
Why the effect of impurity is negligible at low T?
31PhononImpurity Scattering
 Impurity? change of local spring stiffness
(acoustic impedance)  Scattering mean free path for phononimpurity
scattering  ?i ? sr
 where r is the impurity concentration,
 and the scattering cross section
 s ? (R/l)4 for l R
 s ? R2 for l
 l phonon wavelength
 R radius of lattice imperfection
Impurity and alloy atoms scatter only short l
phonons that are absent at low T!
32Bulk Materials Alloy Limit of Thermal
Conductivity
33Phonon Scattering with Imbedded Nanostructures
Spectral distribution of phonon energy (eb) /
group velocity (v)
Longwavelength or lowfrequency phonons are
scattered by imbedded nanostructures!
34Nanodot Superlattice
Courtesy of A. Majumdar
Samples by J.M. Zide, D.C. Driscoll, M.P.Hanson,
J.D. Zimmerman, G. Zeng, J.E.Bowers, A.C. Gossard
(UCSB)
Images from Elisabeth Müller Paul Scherrer
Institut Wuerenlingen und Villigen, Switzerland
35Specular Phononboundary Scattering
Phonon Reflection/Transmission
TEM of a thin film superlattice
Acoustic Impedance Mismatch (AIM) (rv)1/(rv)2
36Phonon Bandgap Formation in Thin Film
Superlattices
Courtesy of A. Majumdar
37Diffuse Phononboundary Scattering
Specular
Diffuse
Diffuse Mismatch Model (DMM) Swartz and
Pohl (1989)
Acoustic Mismatch Model (AMM) Khalatnikov
(1952)
E. Swartz and R. O. Pohl, Thermal Boundary
Resistance, Reviews of Modern Physics 61, 605
(1989). D. Cahill et al., Nanoscale thermal
transport, J. Appl. Phys. 93, 793 (2003).
Courtesy of A. Majumdar
38SixGe1x/SiyGe1y Superlattice Films
Superlattice Period
AIM 1.15
Alloy limit
With a large AIM, k can be reduced below the
alloy limit.
Huxtable et al., Thermal conductivity of Si/SiGe
and SiGe/SiGe superlattices, Appl. Phys. Lett.
80, 1737 (2002).
39Thin 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
40Nanowire Materials
ZnO nanowires (Z.L. Wang, GaTech)
Sb2Te3 nanowires (potentially high ZT) (X. Li et
al., USTC)
Ge nanowires (B. Korgel, UT Austin)
SnO2 nanowires (Z.L. Wang, GaTech)
41Thermal Measurements of Nanowires
Suspended SiNx membrane
Long SiNx beams
Pt resistance thermometer
Kim, Shi, Majumdar, McEuen, Phys. Rev. Lett. 87,
215502 Shi, Li, Yu, Jang, Kim, Yao, Kim,
Majumdar, J. Heat Tran 125, 881
42Sample Preparation
 Dielectrophoretic trapping
Chip
SnO2 nanobelt
Nanotube bundle
Individual Nanotube
43Thermal Conductance Measurement

1

1
T
G
G
T

1
G
h
b
s
b
T
T
0
0
Q
2QL
Q
h
44SnO2 Nanobelts
64 nm
64 nm
53 nm
39 nm
Collaboration N. Mingo, NASA Ames
53 nm
53 nm, ti1 10t1i, bulk
Circles Measurements Lines Simulation
 Diffuse phononboundary scattering is the primary
effect determining the suppressed thermal
conductivities
Shi et al., Appl. Phys. Lett. 84, 2638 (2004)
45Si Nanowires
Solid line Theoretical prediction
Li et al., Appl Phys Lett 83, 2934 (2003)
 Phononboundary scattering is the primary effect
determining the suppressed thermal conductivity
except for the 22 nm sample, where boundary
scattering alone can not account for the
measurement results.
46Si/SiGe Superlattice Nanowires
Alloy limit
Li et al., Appl Phys Lett 83, 3186 (2003)
47Carbon Nanotubes
Nanotube Electronics (Avouris et al., IBM)
 Atomicallysmooth surface, absence of defects
Long mean free path l Strong SP2 bonding high
sound velocity v  ? high thermal conductivity k Cvl/3 6000
W/mK
48Carbon Nanotubes
CVD SWCN
 An individual nanotube has a high k 200011000
W/mK at 300 K  k of a CN bundle is reduced by thermal
resistance at tubetube junctions
49Challenges
 Synthesis of highthermal conductivity carbon
nanotube films/composites for thermal management  Designing interfaces for low thermal conductance
at high temperatures  Fabrication of thermoelectric coolers using
lowthermal conductivity, highZT nanowire
materials  Largescale manufacturing of bulk materials with
imbedded nanostructures to suppress the thermal
conductivity