Transcript Thermal Properties of Materials Li Shi, PhD Assistant Professor Department of Mechanical Engineering & Center for Nano and Molecular Science and Technology, Texas Materials Institute The.

Thermal 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
[email protected]
Outline
• Thermal Issues in Nanoscale Devices
• Microscale Thermal Transport Theory – Particle Transport
-- Kinetic Theory of Gases
-- Electrons in Metals
-- Phonons in Insulators
• Nano-engineer Thermal Properties of Materials
-- Thin Films
-- Nanowires and Nanotubes
-- Bulk materials with embedded nanostructures
2
GMR
Cu Interconnects
3
Thermal Issues in Si Nanotransistors
Ju and Goodson, APL 74, 3005
IBM SOI Chip
Lines: BTE results
Hot spots!
4
Thermoelectric Cooling
Venkatasubramanian et al. Nature 413, 597
2.5-25nm
Bi2Te3/Sb2Te3 Superlattices
Harman et al., Science 297, 2229
•Coefficient of Performance (COP)
COP
2
Quantum dot superlattices
CFC unit
1
Bi2Te3
0
0
1
2
3
4
5
ZT
•Seebeck
ZT: Figure of Merit
coefficient
ZT 
S 2

Electrical
conductivity
T
Thermal conductivity
Nano-engineer thermal properties of materials!
5
Courtesy of A. Majumdar
Microscopic Origins of Heat Conduction
--The Particle Nature
Materials
Dominant energy carriers
Gases:
Molecules
Hot
Metals:
Electrons
Hot
-
Insulators:
Phonons (crystal vibration) Hot
p
Cold
N2
Cold
Cold
6
Mean Free Path for Intermolecular
Collision for Gases
D
D
Total Length Traveled = L
Total Collision Volume
Swept = pD2L
Number Density of Molecules = n
Total number of molecules encountered in
swept collision volume = npD2L
Average Distance between
Collisions, mc = L/(#of collisions)
Mean Free Path
L
1
 mc 

2
npD L n
: collision cross-sectional area
7
Effective Mean Free Path
 mc ~ 100 nm at atomospheric pressure
Wall
b: boundary separation
Wall
When b ~ mc, the effective mean free path:
1
1
1


  mc  b
8
Kinetic Theory of Particle Transport
Cold
u: energy
u(z+z)
qz
u(z-z)


z + z
Net Energy Flux / # of Molecules
z
1
q' z  v z u  z   z   u  z   z 
2
z - z
through Taylor expansion of u


du
du
2
q ' z  v z  z
  cos  v
dz
dz
Hot
Integration over all the solid angles  total energy flux
1 du dT
1
dT
dT
q z   v
  Cv
 
3 dT dz
3
dz
dz
Thermal conductivity:
1
  Cv
3
Specific heat capacity
Velocity
9
Mean free path
Drawbacks 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: u=u(T)
Breaks down when L  ; t  t
• Cannot handle non-equilibrium problems
Short pulse laser interactions
High electric field transport in devices
Boltzmann Transport Equation
• Cannot handle wave effects
Interference, diffraction, tunneling
10
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?
11
Wave-Particle Duality of Electrons
The double slit experiment
Probability of finding an electron at position x
  2 x 
Schrodinger eqn.
for free electrons
 2 d 2

 E x 
2
2m dx
: electron wave function
Electrons
m: electron mass
: Planck’s constant
E: electron energy
Traveling wave solution to
ikx



x

e
k
Schrodinger’s eqn:
k: wave vector = 2p/l
A metal ring with perimeter L
x
 k x   eikx
(x+L) = (x)
k = 2np/L; n = ±1, ± 2, ± 3, ± 4, …..
1-D k-space
 2k 2
E
2m
-6p/L -4p/L -2p/L
0
 Quantization of energy
2p/L
4p/L
12
Fermi Parameters
Fermi Energy – the highest occupied
energy state at 0 K:

 2 k F2  2
EF 

3p 2e
2m
2m
1

2
Fermi Velocity: vF 
3p e 3
m

Fermi Temp:
TF 
Metal
3
2

F: Work Function
EF
Vacuum
Level
Energy
EF
kB
Band Edge
Element Electron
Density, e
[1028 m-3]
Cu
8.47
Au
5.90
Fe
17.0
Al
18.1
Fermi
Energy
EF [eV]
7.00
5.53
11.1
11.7
Fermi
Temperature
TF [104 K]
8.16
6.42
13.0
13.6
Fermi
Wavelength
lF [Å]
4.65
5.22
2.67
3.59
Fermi
Velocity
vF [106 m/s]
1.57
1.40
1.98
2.03
Work
Function
F [eV]
4.44
4.3
4.31
4.25
Effect of Temperature
Occupation Probability, f
Fermi-Dirac equilibrium distribution
for the probability of electron
occupation of energy
level E at temperature T
1
f E  
 E  EF 
1  exp 

 k BT 
kBT
1
T=0K
Vacuum
Level
Increasing T
0
Electron Energy, E
EF
Work Function, F
Electronic Specific Heat and Thermal Conductivity
Ee 
  Ef E DE dE
Energy density e 
V 0
Bulk metals:
Density of states

d e
df
Ce 
 E
DE dE
dT
dT
0
Specific Heat
Thermal Conductivity
e
2  EF 
1
 e  Cev F  e
3
Bulk Solids
Increasing
Defect Concentration
Defect
Scattering
p 2  k BT 
Ce 
e k B
Electron Scattering Mechanisms
• Defect Scattering
• Phonon Scattering
• Boundary Scattering (Film Thickness,
Grain Boundary)
Phonon
Scattering
Temperature, T
Grain
15
Grain Boundary
Thermal Conductivity of Cu and Al
1
 e  Cev F  e
3
Thermal Conductivity, k [W/cm-K]
10
Matthiessen Rule:
1
1
1
1



 e  defect  boundary  phonon
3
Copper
10 2
1
1
10
1
10
0
 of a metal is
dominated by the
electronic contribution
Aluminum
Phonon Scattering
Defect Scattering
10 0
10 1
10 2
Temperature, T [K}
10 3
16
Conditions
• 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?
17
Crystal Vibration
Interatomic Bonding
Equation of motion with
nearest neighbor interaction
Energy
Parabolic Potential of
Harmonic Oscillator
ro
m
Distance
d 2 xn
dt
2
 g xn1  xn1  2 xn 
Traveling wave solution
xn  xo exp it expinKa
1-D Array of Spring Mass System
Eb
Spring constant, g
M ass, m
Equilibrium
Position
a
Deformed
Position
18
x n-1
xn
x n+1
Dispersion Relation
Spring constant, g
Mass, m
Equilibrium
Position
a
Deformed
Position
x n-1

1
2g
1  cos Ka  2
m
A
al
sti
u
o
ng
o
L
de
o
M
ti
s
ou
l
Ac
e
s
er
v
Edge of First
ns
a
r
T
Brillouin Zone
0
Wave vector, K
d
dK
d
Speed of Sound: vs  lim
K 0 dK
c
)
TA
(
c
vg 
e
LA
(
c
in
d
itu
x n+1
Group Velocity:
od
M
)
Frequency, 
xn
p/a
K
2p
l
19
Two Atoms Per Unit Cell
Lattice Constant, a
m1
m2
d 2 xn
dt
2
d 2 yn
dt
2
xn
 g  yn  yn1  2 xn 
 g  xn1  xn  2 yn 
xn+1
yn
LO
Frequency, 
yn-1
Optical
Vibrational
Modes
TO
LA
TA
TO
LO
0
Wave vector, K
Oscillating out of phase against each other. Vg ~0 little contribution to 
p/a
20
Phonon Dispersion in GaAs
LO
12
Frequency (10 Hz)
8
LO
TO
TO
6
LA
LA
4
2
0
L
TA
0.4
0.2
0
(111) Direction 
TA
0.2
Ka/p
0.4
0.6
(100) Direction
0.8
1.0
X
21
Allowed Wave Vectors
A linear chain of N=10 atoms with two ends jointed
u: atom displacement
Solution: us ~uK(0)exp(-it)sin(Kx), x =sa
B.C.:
us=0 = us=N=10
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/a=Np/L
22
The Wave-Particle Duality of Crystal Vibration
Energy
Parabolic Potential of
Harmonic Oscillator
ro
Distance
Total Energy of a Harmonic
Oscillator in a Parabolic Potential
1

u   n  
2


Eb
Phonon: A particle carrying a quantum of vibrational energy,
, which travels through the lattice
Phonons follow Bose-Einstein statistics.
Equilibrium distribution:
1
n 
  
  1
exp
 k BT 
n
T

23
Energy of Lattice Vibration
El  
p


1

  n  K , p  2  K , p
K
Dispersion Relation:
Energy Density:
p: polarization(LA,TA, LO, TO)
K: wave vector
K  g  
El
1

l 
    n   D d
V
2

p
D(): Density of States, number of allowed wave vectors between  and +d
Lattice Specific Heat:
d l
Cl 

dT
p

d n
dT
D d
24
  vs K
Debye Approximation:
g   dg

Debye Density
D





of States:
2p 2 d 2p 2vs3
2
2
Frequency, 
Debye Model
  vs K
LA or TA branch
Specific Heat in 3D:
0
 T
Cl  9k B 
D
3  D T
 

 


0

e x x 4 dx 
2
x
e 1 


In 3D, when T << D,
l  T , Cl  T
4
3
Wave vector, K
Debye Temperature [K]
D 
C(dimnd)
Si
Ge
B
Al

vs 6p 
kB
1860
625
360
1250
394
2

1
3
Ga
NaF
NaCl
NaBr
NaI
240
492
321
224
164
p/a
Phonon Specific Heat
10
7
C  3 kB  4.7 10 6
3
Specific Heat,
C (J/m
-K)3-K)
(J/m
Heat
Specific
10 6
10
J
3 k B
m3 K
Diamond
Each atom
has
Diamond
a thermal energy
of 3KBT
5
10 4
C  T 33
CT
10 3
10
Classical
Regime
2
 D  1860 K
10 1 1
10
10 2
10 3
Temperature, T (K)
10 4
Temperature (K)
In general, when T << D,
l  T d 1, Cl  T d
d =1, 2, 3: dimension of the sample
Phonon Thermal Conductivity
Kinetic Theory
1
kl  Cl vs  l
3
Phonon Scattering Mechanisms
• Boundary Scattering
• Defect & impurity Scattering
• Phonon-Phonon Scattering
1
1
1
1



 l  defect  boundary  phonon
Decreasing Boundary
Separation
kl
l
Increasing Defect
Concentration
Increasing
Defect
Concentration
kl  T d
Phonon
Defect
Boundary
Scattering
0.01
0.1
Temperature, T/D
Boundary Defect
1.0
0.01
Phonon
Scattering
0.1
Temperature, T/D
1.0
Phonon-Phonon 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.
n 
1
  
  1
exp
 k BT 
Decreasing Boundary
Separation
l
 phonon ~ exp(/bT)
Increasing
Defect
Concentration
phonon ~ exp(/bT)
Phonon
Boundary Defect Scattering
0.01
1.0
0.1
Temperature, T/D
28
Thermal Conductivity of Bulk Crystals
3

29
Effect of Impurity on Thermal Conductivity
10 3
Thermal Cond uctivity, k [W/cm-K]
Diam ond
10 2
10
1
10
0
Increasing
Defect Density
10 -1
10 -2 0
10
Defect
Scattering
Boundary
Scattering
10 1
10 2
10 3
Temperature, T [K]
30
Why the effect of impurity is negligible at low T?
Phonon-Impurity Scattering
• Impurity change of local spring stiffness (acoustic impedance)
• Scattering mean free path for phonon-impurity scattering:
i  r
Decreasing Boundary
Separation
where r is the impurity concentration,
and the scattering cross section
l
  R/l4 for l >> R
  R2 for l << R
l: phonon wavelength
Increasing
Defect
Concentration
R: radius of lattice imperfection
Phonon
Defect
Boundary
Scattering
0.01
1.0
0.1
Temperature, T/D
Impurity and alloy atoms scatter only short- l phonons that are absent at31low T!
k [W/m-K]
Bulk Materials: Alloy Limit of Thermal Conductivity
A
Alloy Limit
B
32
Phonon Scattering with Imbedded Nanostructures
Phonon Scattering
Spectral
distribution
of phonon
energy (eb)
/ group
velocity (v)
v
eb

Nanostructures
Atoms/Alloys
Frequency, 
max
33
Long-wavelength or low-frequency phonons are scattered by imbedded nanostructures!
Nanodot Superlattice
InGaAs
0.8ML
ErAs
5x1018 Si-doped InGaAs
0.6ML
Si-Doped ErAs/InGaAs SL (0.4ML)
0.4ML
Undoped ErAs/InGaAs SL (0.4ML)
0.2ML
Plan View
[11
0]
100 nm
Cross-section
10 nm
Images from Elisabeth Müller Paul Scherrer
Institut Wueren-lingen und Villigen, Switzerland
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
34
(UCSB)
Specular Phonon-boundary Scattering
Phonon Reflection/Transmission
TEM of a thin film superlattice
critical
angle
incident
l , t1 , or t2
l
t1
1
reflected
t2
interface
Acoustic Impedance
Mismatch (AIM)
= (rv)1/(rv)2
l
2
t1
t2
transmitted
35
nl  2d cos
lmin=50
100
50
frequency, 
Phonon Bandgap Formation in Thin Film Superlattices
(i)
wavevector, K
n=1, l=100
n=2, l=50
frequency, 
(i)
n=1, l=200
l
(ii) n=2, =100
n=3, l=66
n=4, l=50
(ii)
wavevector, K
(A)
(B)
36
Courtesy of A. Majumdar
Diffuse Phonon-boundary Scattering
Specular
critical
angle
incident
l , t1 , or t2
Diffuse
l
t1
1
reflected
t2
interface
l
2
t1
t2
transmitted
Acoustic Mismatch Model (AMM)
Khalatnikov (1952)
Diffuse Mismatch Model (DMM)
Swartz and Pohl (1989)
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).
37
Courtesy of A. Majumdar
SixGe1-x/SiyGe1-y Superlattice Films
Superlattice
Period
Thermal Conductivity (W/m-K)
AIM = 1.15
15
Si/Si0.7Ge0.3 SL's
300 Å
150 Å
10
75 Å
45 Å
Si0.9Ge0.1 (3.5 m)
5
Alloy limit
Si/Si0.4Ge0.6 SL (150 Å)
0
50
100
150 200 250 300
Temperature (K)
350
400
With a large AIM,  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).
38
Thin Film Thermal Conductivity Measurement
3 method
(Cahill, Rev. Sci. Instrum. 61, 802)
Metal line
L
Substrate
Thin Film
2b
I0 sin(t)
V
• I ~ 1
• T ~ I2 ~ 2
• R ~ T ~ 2
• V~ IR ~3
P  1  Ds 
1
ip 
Pd
T (2 ) 
ln 2     ln2    

Lpk s  2  b 
2
4  2 Lbk f
39
Nanowire Materials
Sb2Te3 nanowires (potentially high ZT)
(X. Li et al., USTC)
Ge nanowires
(B. Korgel, UT Austin)
ZnO nanowires
(Z.L. Wang, GaTech)
SnO2 nanowires
(Z.L. Wang, GaTech)
40
Thermal Measurements of Nanowires
1.5
Th (K)
Suspended SiNx membrane
Long SiNx beams
T0 = 54.95 K
1.0
0.5
0.0
I
-6
Q
-4
-2
0
2
4
6
4
6
Current (A)
0.10
Pt resistance thermometer
 Ts (K)
0.08
T0 = 54.95 K
0.06
0.04
0.02
0.00
-6
Kim, Shi, Majumdar, McEuen, Phys. Rev. Lett. 87, 215502
Shi, Li, Yu, Jang, Kim, Yao, Kim, Majumdar, J. Heat Tran 125, 881
-4
-2
0
2
Current (A)41
Sample Preparation
• Dielectrophoretic trapping
• Wet deposition
Pipet
Nanostructure
suspension
Chip
Spin
• Direct CVD growth
SnO2 nanobelt
Nanotube bundle
42
Individual Nanotube
Thermal Conductance Measurement
Heating membrane
Sensing membrane
Th
Ts
Rs
Ts
Rh
Sample
Gs
Qh
t
Q
QL=IRL
Beam, Gb
Beam,Gb
Q’
Environment
T0
Q
Q
G
Th  Ts
VTE
V
v
I
i
Gb-1 Th
G-1
Ts
Gb-1
T0
T0
2QL
Q
Qh
Q  Gb (Ts  T0 )
Qh  QL  Gb[(Th  T0 )  (Ts43 T0 )]
Thermal conductivity (W/m-K)
SnO2 Nanobelts
15
64 nm
64 nm
10
53 nm
39 nm
Collaboration:
N. Mingo, NASA Ames
5
53 nm
53 nm,
ti-1 =10t-1i, bulk
0
0
100
200
300
Temperature (K)
Circles: Measurements
Lines: Simulation
•Diffuse phonon-boundary scattering is the primary effect determining the
suppressed thermal conductivities
44
Shi et al., Appl. Phys. Lett. 84, 2638 (2004)
Thermal Conductivity (W/m-K)
Si Nanowires
kbulk  135 W m  K at 300K
60
Solid line: Theoretical prediction
50
115 nm
Diameter

40
Si Nanowire
56 nm
30
Carbon Deposits
37 nm
20
10
22 nm
0
0
50
100
150
200
250
Temperature (K)
300
350
Li et al., Appl Phys Lett 83, 2934 (2003)
• Phonon-boundary scattering is the primary effect determining the suppressed thermal
conductivity except for the 22 nm sample, where boundary scattering alone can45not
account for the measurement results.
Si/SiGe Superlattice Nanowires
Si/Si0.95Ge0.05
Si
Li et al., Appl
Phys Lett 83,
3186 (2003)
Thermal Conductivity (W/m-K)
SiGe
16
(a)
14
Si/Si0.7Ge0.3 superlattice film [7]
12
Si0.9Ge0.1 alloy film [7]
10
8
Alloy limit
6
4
83 nm Si/SixGe1-x nanowire
2
58 nm Si/SixGe1-x nanowire
0
0
50
100
150
200
250
Temperature (K)
300
350
46
Carbon Nanotubes
Nanotube Electronics (Avouris et al., IBM)
• Atomically-smooth surface, absence of defects: Long mean free
path l & Strong SP2 bonding: high sound velocity v
 high thermal conductivity:  = Cvl/3 ~ 6000 W/m-K
47
Thermal Conductivity (W/m-K)
Carbon Nanotubes
105
104
1-3 nm CVD SWCN
103
14 nm MWCN bundle
~T2
102
101
100
10-1
~ T 2.5
~ T 1.6
10-2
100
10 nm SWCN CVD SWCN
bundle
148 nm SWCN
bundle
101
102
Temperature (K)
103
• An individual nanotube has a high  ~ 2000-11000 W/m-K at 300 K
•  of a CN bundle is reduced by thermal resistance at tube-tube junctions
48
Challenges
• Synthesis of high-thermal conductivity carbon nanotube films/composites for
thermal management
• Designing interfaces for low thermal conductance at high temperatures
• Fabrication of thermoelectric coolers using low-thermal conductivity, high-ZT
nanowire materials
• Large-scale manufacturing of bulk materials with imbedded nanostructures to
suppress the thermal conductivity
AgPb18SbTe20
ZT = 2 @ 800K
Hsu et al., Science 303, 818 (2004)
AgSb rich
49