Transcript Thermal Properties of Materials

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
[email protected]
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)
2
Fourier’s Law for Heat Conduction
Q (heat flow)
Hot
Th
Cold
Tc
L
Th  Tc
dT
Q  kA
 kA
L
dx
Thermal conductivity
3
Heat Diffusion Equation
1st law (energy conservation)
Heat conduction = Rate of change of energy storage
 T
T
k 2 C
t
 x
2
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
4
materials processing…
1 km
Length Scale
Aircraft
Automobile
1m
Human
Computer
Butterfly
1 mm
Fourier’s law
Microprocessor Module
MEMS
Blood Cells
Wavelength of Visible Light
MOSFET, NEMS
Nanotubes, Nanowires
Width of DNA
1 mm
100 nm l
1 nm
5
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)
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
 mc
L
1


2
npD L n
: collision cross-sectional area
7
Mean Free Path for Gas Molecules
Number Density of
Molecules from Ideal
Gas Law:
n = P/kBT
Mean Free Path:
 mc
kB: Boltzmann constant
1.38 x 10-23 J/K
1 k BT


n P
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:
23
 mc
1.38 10  300
7


3

10
m or 0.3mm
5
19
10 1.3 10
At 1 Torr pressure, mc  200 mm; at 1 mTorr, mc  20 cm
8
Effective Mean Free Path
Wall
b: boundary separation
Wall
Effective Mean Free Path:
1
1
1


  mc  b
9
Kinetic Theory of Energy Transport
u: energy
u(z+z)
z + z
q
qz

z
Net Energy Flux / # of Molecules
1
q' z  v z u  z   z   u  z   z 
2
through Taylor expansion of u
u(z-z)
z - z


du
du
2
q ' z  v z  z
  cos q v
dz
dz
Integration over all the solid angles  total energy flux
1 du dT
1
dT
dT
q z   v
  Cv
 k
3 dT dz
3
dz
dz
Thermal conductivity:
1
k  Cv
3
Specific heat
Velocity
10
Mean free path
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
Free Electrons in Metals at 0 K
Fermi Energy – highest occupied energy state:

 2 k F2  2
EF 

3p 2 e
2m
2m
1

2
Fermi Velocity: v F 
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
F [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
Number and Energy Densities
Number density:
Energy density:
N 
e    f  E De  E dE;
V 0
Ee 
e 
  Ef  E De  E dE
V 0
Density of States -- Number of electron states available between
energy E and E+dE
De  E  
m
 p
2 2
2mE
2
in 3D
14
Electronic Specific Heat and Thermal Conductivity

d e
df
C


E
DE dE
Specific Heat
e

dT
dT
0
p 2  k BT 
Ce 
e k B
2  EF 
1
1
2
Thermal Conductivity ke  Ce vF  e  Ce vF
te
3
3
e
Bulk Solids
Increasing
Defect Concentration
Defect
Scattering
Mean free time:
te = le / vF
Electron Scattering Mechanisms
• Defect Scattering
• Phonon Scattering
• Boundary Scattering (Film Thickness,
Grain Boundary)
Phonon
Scattering
Temperature, T
in 3D
Grain
15
Grain Boundary
Thermal Conductivity of Cu and Al
Matthiessen Rule:
1
1
1


1
1
ke  Ce vF  e  Ce vF2 t e
3
3
te
t boundary
1
t phonon
1
1
1
1



 e  defect  boundary  phonon
10 3
Thermal Conductivity, k [W/cm-K]
t defect

Copper
10 2
1
1
Electrons dominate k in
metals
Aluminum
10 1
Phonon Scattering
Defect Scattering
10
0
10 0
10 1
10 2
T emperature, T [K}
10 3
16
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
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 
Solution
xn  xo exp  it exp inKa 
Eb
1-D Array of Spring Mass System
Spring constant, g
Mass, m
Equilibrium
Position
a
Deformed
Position
18
x n-1
xn
x n+1
Dispersion Relation
 2 m  g 2  exp  iKa   exp iKa   2 g 1  cos Ka 
1
2g
1  cos Ka 2

m
Group Velocity:
Frequency, 
d
vg 
dK
Speed of Sound:
d
vs  lim
K 0 dK
0
Wave vector, K
p/a
19
Two Atoms Per Unit Cell
Lattice Constant, a
2
m1
m2
d 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
0
TA
Wave vector, K
p/a
20
Phonon Dispersion in GaAs
LO
Frequency (1012 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
Energy Quantization and Phonons
Energy
Total Energy of a Quantum
Oscillator in a Parabolic Potential
Distance
1

u   n  
2

n = 0, 1, 2, 3, 4…; /2: zero point energy
h
Phonon: A quantum of vibrational energy,
, which travels through the lattice
Phonons follow Bose-Einstein statistics.
Equilibrium distribution:
1
n 
  
  1
exp 
 k BT 
In 3D, allowable wave vector K:
2p 4p 6p
, , ,....
L L L
22
Lattice Energy
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
Density of States: Number of vibrational states between  and +d
g 2   dg
D  
2p 2 d
Lattice Specific Heat:
d l
Cl 

dT
p
in 3D

d n
dT
D d
23
  vs K
Debye Approximation:
g   dg

Debye Density
D





of States:
2p 2 d 2p 2vs3
2
2
Frequency, 
Debye Model
  vs K
Specific Heat in 3D:
0
 T
Cl  9k B 
qD
3 q D T
 

 


0

e x x 4 dx 
2
x
e 1 


In 3D, when T << qD,
l  T , Cl  T
4
3
Wave vector, K
Debye Temperature [K]
qD 
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kBT
m3 K
Diamond
Each atom
has
Diamond
a thermal energy
of 3KBT
5
10 4
10
3
C  T 33
CT
10 2
10 1 1
10
Classical
Regime
q D  1860 K
2
10
10
Temperature, T (K)
3
10
4
Temperature (K)
In general, when T << qD,
l  T d 1, Cl  T d
d =1, 2, 3: dimension of the sample
Phonon Thermal Conductivity
Kinetic Theory
1
1
kl  Cl vs  l  Cl vs2t l
3
3
Decreasing Boundary
Separation
Phonon Scattering Mechanisms
• Boundary Scattering
• Defect & Dislocation Scattering
• Phonon-Phonon Scattering
l
kl
Increasing Defect
Concentration
Increasing
Defect
Concentration
kl  T
Phonon
Defect
Boundary
Scattering
0.01
0.1
Temperature, T/qD
d
1.0
Boundary Defect
0.01
Phonon
Scattering
0.1
Temperature, T/qD
1.0
10
3
Thermal Conductivity of Insulators
• Phonons dominate k in insulators
Thermal Conductivity, k [W/cm-K]
Diamond
10 2
10 1
Increasing
Defect Density
10 0
10 -1
10 -2 0
10
Defect
Scattering
Boundary
Scattering
10
1
10
Temperature, T [K]
2
10
3
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() or t
• Cannot handle non-equilibrium problems
Short pulse laser interactions
High electric field transport in devices
• Cannot handle wave effects
Interference, diffraction, tunneling
28
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:
f
 f 
 v  r f  F   p f   
t
 t  scat
fo  f
 f 

Relaxation Time Approximation 

 t  scat t r, p 
Relaxation time
f  fo
e
t
t
t
Energy Flux
q
Energy flux in terms of particle flux
carrying energy:
v
dk
q
k
qr, t    vr, t  f r, k , t  k dk

Vector
Integrate over all the solid angle:
qr, t  
p
 
2p
Scalar
2






v
r
,
t
f
r
,
k
,
t

k
k
cos q sin qdqddk

k q 0  0
Integrate over energy instead of momentum:
p 2p
1
qr, t  
vr,   f r,  , t D  cos q sin qdqdd



4p  q 0  0
Density of States:
# of phonon modes per frequency range
Continuum Case
t  t
L  
f
 0; f  f o
t
Quasi-equilibrium
BTE Solution:
f  f o  tv  f o  f o  tv
Energy Flux:
df o
cosq
dx
Direction x is chosen
to in the direction of q
df o
1
2




v
r
,

t
r
,

D d

3
dx
df o
df o dT

dx
dT dx
qr, t   
df o
1 dT 2
dT
Fourier Law of
qr, t   
 v r,  t r,    D d  k
Heat Conduction:
3 dx
dT
dx
df o
1 2
k   v r,  t r,  
 D d t() can be treated using Callaway method
(Phys. Rev. 113, 1046)
3
dT
If v and t are independent of particle k  1 v 2t
energy, , then  Kinetic theory:
3
df o
1
2



D

d


Cv
t
 dT
3
At Small Length/Time Scale (L~l or t~t)
Define phonon intensity:
I r,  ,q ,  , s, t   v ,q ,  , s  f r,  ,q ,  , t  D , s 
From BTE:
0
f
f
f
 v  f  F 
 ( ) Scat
t
k
t
 vD 
Equation of Phonon Radiative Transfer (EPRT) (Majumdar, JHT 115, 7):
I  r,  ,q ,  , s, t 
 I  
 v  I  r,  ,q ,  , s, t   

t
 t  scat
Heat flux:
p 2p
1
qr, t  
   I cos qdqdd
4p  q 0  0
Acoustically Thin Limit (L<<l) and for T << qD
Acoustically Thick Limit (L>>l)
q   T
q  k l T
4
1
4
 T2

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
33
Thin Film Thermal Conductivity Measurement
3 method
(Cahill, Rev. Sci. Instrum. 61, 802)
Metal line
L
Substrate
T (2 ) 
Thin Film
2b
I0 sin(t)
V
• I ~ 1
• T ~ I2 ~ 2
• R ~ T ~ 2
• V~ IR ~3
P  1  Ds 
1
ip 
Pd
ln



ln

2




 2

Lpk s  2  b 
2
4  2 Lbk f
34
Silicon on Insulator (SOI)
Ju and Goodson, APL 74, 3005
IBM SOI Chip
Lines: BTE results
Hot spots!
35
Thermoelectric Cooling
• No moving parts: quiet and reliable
• No Freon: clean
36
Thermoelectric Figure of Merit (ZT)
Coefficient of Performance
COPmax
1  zTm  Th / Tc
Tc

Th  Tc
1  zTm  1
Seebeck coefficient
Electrical conductivity
ZT 
S 
2

T
Temperature
COPmax
where
2
1
TH = 300 K
TC = 250 K
Bi2Te3
0
0
Thermal conductivity
Freon
1
2
3
ZT
4
5
37
ZT Enhancement in Thin Film Superlattices
SiGe superlattice
(Shakouri, UCSC)
•Increased phonon-boundary scattering
decreased k
+ other size effects
 High ZT = S2T/k
Si Barrier
Ge Quantum well (QW)
Ec
E
Ev
x
38
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)
39
Superlattice Micro-coolers
Ref: Venkatasubramanian et al, Nature 413, P. 597 (2001)
40
Nanowires
22 nm diameter Si nanowire,
P. Yang, Berkeley
• Increased phonon-boundary scattering
Hot
p
Cold
• Modified phonon dispersion
 Suppressed thermal conductivity
Ref: Chen and Shakouri, J. Heat Transfer 124, 242
41
Thermal Measurements of Nanotubes and Nanowires
Themal conductance: G = Q / (Th-Ts)
1.5
Th (K)
Suspended SiNx membrane
Long SiNx
beams
T0 = 54.95 K
1.0
0.5
0.0
-6
Q
I
-4
-2
0
2
4
6
4
6
Current (mA)
0.10
Pt resistance
thermometer
 Ts (K)
0.08
T0 = 54.95 K
0.06
0.04
0.02
0.00
-6
Kim et al, PRL 87, 215502
Shi et al, JHT, in press
-4
-2
0
2
Current (mA)42
Si Nanotransistor
(Berkeley Device group)
Si Nanowires
Gate
Drain
Source
Thermal Conductivity (W/m-K)
Nanowire Channel
D. Li et al., Berkeley
Symbols: Measurements
Lines: Modified Callaway Method
115 nm
60
50
Hot Spots in Si
nanotransistors!
40
56 nm
30
37 nm
20
10
22 nm
0
0
50
100
150
200
250
Temperature (K)
300
350
43
ZT Enhancement in Nanowires
Nanowires
based on
Bi, BiSb,Bi2Te3,SiGe
Al2O3 template
• k reduction and other size effects
 High ZT = S2T/k
Top View
Bi Nanowires
Nanowire
Ref: Phys. Rev. B. 62, 4610
by Dresselhaus’s group
44
Nanotube Nanoelectronics
TubeFET (McEuen et al., Berkeley)
Nanotube Logic (Avouris et al., IBM)
45
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
Heat capacity
• Below 30 K, thermal conductance  4G0 = ( 4 x 10-12T) W/m-K,
linear T dependence (G0 :Quantum of thermal conductance)
46
Thermal Conductance of a Nanotube Mat
Linear
behavior
Ref: Hone et al.
APL 77, 666
25 K
• Estimated thermal conductivity at 300K: ~ 250 << 6000 W/m-K
 Junction resistance is dominant
• Intrinsic property remains unknown
47
Thermal Conductivity (W/m-K)
Thermal Conductivity of Carbon Nanotubes
105
104
1-3 nm CVD SWCN
103
102
14 nm MWCN bundle
~T2
101
100
~ T 2.5
~ T 1.6
10-1
10-2
100
10 nm SWCN
bundle
148 nm SWCN
bundle
101
102
Temperature (K)
CVD SWCN
CNT
103
• 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
Nano Electromechanical System (NEMS)
Thermal conductance quantization in nanoscale SiNx beams
(Schwab et al., Nature 404, 974 )
Quantum of Thermal Conductance
Phonon Counters?
49
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)
50