Download Thermal Conductivity

Classical Theory Expectations
• Equipartition: 1/2kBT per degree of freedom
• In 3-D electron gas this means 3/2kBT per electron
• In 3-D atomic lattice this means 3kBT per atom (why?)
• So one would expect: CV = du/dT = 3/2nekB + 3nakB
• Dulong & Petit (1819!) had found the molar heat capacity
of most solids approaches 3NAkB at high T
Molar heat capacity @ high T  25 J/mol/K
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Heat Capacity: Real Metals
CV  bT  aT 3
due to
electron gas
due to
atomic lattice
Cv = bT
•
•
•
•
So far we’ve learned about heat capacity of electron gas
But evidence of linear ~T dependence only at very low T
Otherwise CV = constant (very high T), or ~T3 (intermediate)
Why?
© 2008 Eric Pop, UIUC
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Heat Capacity: Dielectrics vs. Metals
Cv = bT
• Very high T:
CV = 3nkB (constant) both dielectrics & metals
• Intermediate T: CV ~ aT3 both dielectrics & metals
• Very low T:
CV ~ bT metals only (electron contribution)
© 2008 Eric Pop, UIUC
34
ECE 598EP: Hot Chips
Phonons: Atomic Lattice Vibrations
Graphene Phonons [100]
200 meV
CO2 molecule
vibrations
transverse
small k
transverse
max k=2p/a
Frequency ω (cm-1)
160 meV
100 meV
26 meV =
300 K
u(r, t )  A exp[i(k  r  it )]
k
•
Phonons = quantized atomic lattice vibrations
•
Transverse (u ^ k) vs. longitudinal modes (u || k), acoustic vs. optical
•
“Hot phonons” = highly occupied modes above room temperature
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A Few Lattice Types
• Point lattice (Bravais)
– 1D
– 2D
– 3D
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Primitive Cell and Lattice Vectors
• Lattice = regular array of points {Rl} in space repeatable
by translation through primitive lattice vectors
• The vectors ai are all primitive lattice vectors
• Primitive cell: Wigner-Seitz
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Silicon (Diamond) Lattice
• Tetrahedral bond arrangement
• 2-atom basis
• Each atom has 4 nearest neighbors and 12 next-nearest
neighbors
• What about in (Fourier-transformed) k-space?
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Position  Momentum (k-) Space
Sa(k)
k
• The Fourier transform in k-space is also a lattice
• This reciprocal lattice has a lattice constant 2π/a
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Atomic Potentials and Vibrations
• Within small perturbations from their equilibrium
positions, atomic potentials are nearly quadratic
• Can think of them (simplistically) as masses connected
by springs!
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• Can write down wave equation
• Velocity of sound (vibration
propagation) is proportional to
stiffness and inversely to mass
(inertia)
© 2008 Eric Pop, UIUC
Frequency, 
Vibrations in a Discrete 1D Lattice
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0
Wave vector, K
p/a
41
Two Atoms per Unit Cell
Lattice Constant, a
xn
xn+1
yn
d 2x
m1 2n  k  yn  yn 1  2 xn 
dt
d 2 yn
m2
 k  xn 1  xn  2 yn 
dt 2
LO
Frequency, 
yn-1
TO
LA
0
© 2008 Eric Pop, UIUC
Optical
Vibrational
Modes
TA
Wave vector, K
p/a
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Energy Stored in These Vibrations
• Heat capacity of an atomic lattice
• C = du/dT =
• Classically, recall C = 3Nk, but only at high temperature
• At low temperature, experimentally C  0
• Einstein model (1907)
– All oscillators at same, identical frequency (ω = ωE)
• Debye model (1912)
– Oscillators have linear frequency distribution (ω = vsk)
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The Einstein Model
• Density of states in ω
(energy/freq) is a delta function
Frequency, 
• All N oscillators same frequency
  E
g    3N (  E )
0
Wave vector, k
2p/a
• Einstein specific heat
CE 
du
df ( )
  
g   d  
dT
dT
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Einstein Low-T and High-T Behavior
• High-T (correct, recover Dulong-Petit):
CE (T )  3Nk B
1  
 
1   1
E
T
E
T
2
E
T
2
 3Nk B
Einstein model
OK for optical phonon
heat capacity
• Low-T (incorrect, drops too fast)
CE (T )  3Nk B
 
 3Nk B
 
© 2008 Eric Pop, UIUC
E
k BT
E
k BT
2
eE / kBT
e
2
E / k BT

2
e  E / kBT
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• Linear (no) dispersion
with frequency cutoff
• Density of states in 3D:
g   
Frequency, 
The Debye Model
  vs k
2
2p 2 vs3
0
(for one polarization, e.g. LA)
Wave vector, k
2p/a
(also assumed isotropic solid, same vs in 3D)
• N acoustic phonon modes up to ωD
• Or, in terms of Debye temperature
v
 D  s 6p 2 N
kB

© 2008 Eric Pop, UIUC

kD roughly corresponds to
max lattice wave vector (2π/a)
1/3
ωD roughly corresponds to
max acoustic phonon frequency
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Annalen der Physik 39(4)
p. 789 (1912)
Peter Debye (1884-1966)
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Frequency, 
The Debye Integral
• Total energy
u (T ) 
D
  f ( ) g ( )d
  vs k
0

• Multiply by 3 if assuming all
polarizations identical (one LA,
and 2 TA)
• Or treat each one separately
with its own (vs,ωD) and add
them all up
• C = du/dT
© 2008 Eric Pop, UIUC
0
Wave vector, k
2p/a
people like to write:
(note, includes 3x)
T 
CD (T )  9 Nk B  
 D 
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3  /T
D

0
x 4e x dx
(e x  1)2
48
Debye Model at Low- and High-T
• At low-T (< θD/10):
T 
12p 4
CD (T ) 
NkB  
5
 D 
3
• At high-T: (> 0.8 θD)
CD (T )  3NkB
• “Universal” behavior for all solids
• In practice: θD ~ fitting parameter
to heat capacity data
• θD is related to “stiffness” of solid
as expected
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Experimental Specific Heat
3 3
(J/m
Heat
Specific
-K ) -K)
C (J/m
Specific Heat,
10
7
C  3 kB  4.7 10 6
10
6
10
5
J
3NkBT
m3 K
Diamond
Each atom
has
Diamond
a thermal
energy
of 3KBT
10 4
 TT3
CC 
3
10
3
Classical
Regime
10 2
10
 D  1860 K
1
10 1
10 2
10 3
Temperature, T (K )
10 4
Temperature (K)
In general, when T << θD
© 2008 Eric Pop, UIUC
uL  T d 1 , CL  T d
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Phonon Dispersion in Graphene
Maultzsch et al.,
Phys. Rev. Lett.
92, 075501 (2004)
Optical
Phonons
Yanagisawa et al.,
Surf. Interf. Analysis
37, 133 (2005)
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Heat Capacity and Phonon Dispersion
•
Debye model is just a simple, elastic, isotropic approximation; be
careful when you apply it
•
To be “right” one has to integrate over phonon dispersion ω(k),
along all crystal directions
•
See, e.g. http://www.physics.cornell.edu/sss/debye/debye.html
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Thermal Conductivity of Solids
how do we explain this mess?
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Kinetic Theory of Energy Transport
u(z+z)

qz
λ
z + z
z
u(z-z)
z - z
Net Energy Flux / # of Molecules
1
q' z  v z u  z   z   u  z   z 
2
through Taylor expansion of u
q ' z  v z  z


du
du
  cos 2  v
dz
dz
Integration over all the solid angles  total energy flux
1
du dT
dT
qz   v
 k
3 dT dz
dz
Thermal conductivity:
© 2008 Eric Pop, UIUC
1
k  Cv
3
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Simple Kinetic Theory Assumptions
• Valid for particles (“beans” or “mosquitoes”)
– Cannot handle wave effects (interference, diffraction, tunneling)
• Based on BTE and RTA
• Assumes local thermodynamic equilibrium: u = u(T)
• Breaks down when L ~ _______ and t ~ _________
• Assumes single particle velocity and mean free path
– But we can write it a bit more carefully:
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Phonon MFP and Scattering Time
• Group velocity only depends on dispersion ω(k)
• Phonon scattering mechanisms
– Boundary scattering
– Defect and dislocation scattering
– Phonon-phonon scattering
1
1
k  Cv  Cv 2
3
3
Decreasing Boundary Separation
l
kl
Increasing Defect
Concentration
Increasing
Defect
Concentration
kl  T d
Phonon
Scattering
Phonon
Scattering
Defect
Boundary
Boundary Defect
0.01
1.0
0.1
0.01
Temperature, T/D
© 2008 Eric Pop, UIUC
0.1
Temperature, T/D
1.0
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Temperature Dependence of Phonon KTH
C
 T d low T
3Nk B
 ph ph
high T
C
low T

high T
3NkB
© 2008 Eric Pop, UIUC
Td
1

 e / kT  1
n ph
  low T

high T
kT
λ

nph  0, so
λ  , but then
λ  D (size)
 Td
 1/T
 1/T
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Ex: Silicon Film Thermal Conductivity
Thermal Conductivity (W m-1K-1)
McConnell, Srinivasan, and Goodson, JMEMS 10, 360-369 (2001)
10
4
bulk
1000
100
single
crystal
films
size
effect
doped
10
1
Undoped single-crystal film:
Asheghi et al. (1998)
d = 3 m
Doped single-crystal film:
Asheghi et al. (1999)
d = 3 m
n = 1·1019 cm-3 boron
undoped
doped
Bulk single-crystal silicon:
Touloukian et al. (1970)
d = 0.44 cm
undoped
10
polycrystal
100
Temperature (K)

1 A
k (d G , n)  Cv 1  A2 ni 
3  dG

1
© 2008 Eric Pop, UIUC
Doped polysilicon film:
McConnell et al. (2001)
d = 1 m
dg = 350 nm
n = 1.6·1019 cm-3 boron
Undoped polysilicon film:
Srinivasan et al. (2001)
d = 1 m
dg = 200 nm
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Ex: Silicon Nanowire Thermal Conductivity
Nanowire diameter
• Recall, undoped bulk
crystalline silicon k ~ 150
W/m/K (previous slide)
Li, Appl. Phys. Lett. 83, 2934 (2003)
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Ex: Isotope Scattering
isotope
~impurity
~T3
~1/T
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Why the Variation in Kth?
• A: Phonon λ(ω) and dimensionality (D.O.S.)
• Do C and v change in nanostructures? (1D or 2D)
• Several mechanisms contribute to scattering
– Impurity mass-difference scattering
1
 ph i
2
nV
 M  4
i
 
2 
4p vs  M 
2

– Boundary & grain boundary scattering
1

 ph b
vs
D
– Phonon-phonon scattering
1
 ph ph
© 2008 Eric Pop, UIUC

 
 A T  exp   B

k BT 

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What About Electron Thermal Conductivity?
• Recall electron heat capacity

Ce 
du
df
 E
g  E  dE
dT 0 dT
p 2  k BT 
Ce 
at most T
in 3D

 ne k B
2  EF 
• Electron thermal conductivity
Mean scattering time:
e = _______
ke 
e
e
Electron Scattering Mechanisms
Bulk Solids
• Defect or impurity scattering
• Phonon scattering
• Boundary scattering (film
thickness, grain boundary)
Increasing
Defect Concentration
Defect
Scattering
Phonon
Scattering
Temperature, T
Grain
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Grain Boundary
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Ex: Thermal Conductivity of Cu and Al
• Electrons dominate k in metals
Thermal Conductivity, k [W/cm-K]
10
Matthiessen Rule:
1
1
1
1



3
e
1
Copper
10 2
e
1
1

1
defect
 boundary

1
boundary
 phonon

1
 phonon
Aluminum
10 1
Phonon Scattering
Defect Scattering
10
 defect
0
10
0
10
1
10
2
10
3
Temperature, T [K}
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Wiedemann-Franz Law
1p2 2 T
kB n
3 2
EF
e  
 2
 vF

recall electrical
conductivity
  q n 
taking the ratio
ke

where
EF 
q 2
n
m

• Wiedemann & Franz (1853) empirically saw ke/σ = const(T)
• Lorenz (1872) noted ke/σ proportional to T
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Lorenz Number
Experimentally
L = /T 10-8 WΩ/K2
L
e p k

 T 3q
2
2
B
2
L  2.45  108 WΩ/K 2
This is remarkable!
It is independent of n,
m, and even  !
Metal
0°C
100 °C
Cu
2.23
2.33
Ag
2.31
2.37
Au
2.35
2.40
Zn
2.31
2.33
Cd
2.42
2.43
Mo
2.61
2.79
Pb
2.47
2.56
Agreement with experiment is
quite good, although L ~ 10x
lower when T ~ 10 K… why?!
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Amorphous Material Thermal Conductivity
a-SiO2
GeTe
a-Si
Amorphous (semi)metals: both
electrons & phonons contribute
© 2008 Eric Pop, UIUC
Amorphous dielectrics:
K saturates at high T (why?)
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Summary
• Phonons dominate heat conduction in dielectrics
• Electrons dominate heat conduction in metals
(but not always! when not?!)
• Generally, C = Ce + Cp and k = ke + kp
• For C: remember T dependence in “d” dimensions
• For k: remember system size, carrier λ’s (Matthiessen)
• In metals, use WFL as rule of thumb
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