Classical Theory Expectations -...
Transcript of Classical Theory Expectations -...
© 2008 Eric Pop, UIUC ECE 598EP: Hot Chips
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
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Molar heat capacity @ high T 25 J/mol/K
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Heat Capacity: Real Metals
• 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?
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Cv = bT
3
VC bT aT
due to
electron gas
due to
atomic lattice
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Heat Capacity: Dielectrics vs. Metals
• 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)
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Cv = bT
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Phonons: Atomic Lattice Vibrations
• Phonons = quantized atomic lattice vibrations
• Transverse (u ^ k) vs. longitudinal modes (u || k), acoustic vs. optical
• “Hot phonons” = highly occupied modes above room temperature
CO2 moleculevibrations
)](exp[),( tiit rkAru
transversesmall k
transversemax k=2p/a
k
Graphene Phonons [100]
200 meV
160 meV
100 meV
26 meV =
300 K
Fre
qu
en
cy ω
(cm
-1)
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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
• The Fourier transform in k-space is also a lattice
• This reciprocal lattice has a lattice constant 2π/a
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k
Sa(k)
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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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Vibrations in a Discrete 1D Lattice
• Can write down wave equation
• Velocity of sound (vibration
propagation) is proportional to
stiffness and inversely to mass
(inertia)
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Fre
quency,
Wave vector, K0 p/a
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Two Atoms per Unit Cell
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Lattice Constant, a
xn ynyn-1 xn+1
2
1 12
2
2 12
2
2
nn n n
nn n n
d xm k y y x
dt
d ym k x x y
dt
Fre
quency,
Wave vector, K0 p/a
LATA
LO
TO
OpticalVibrational
Modes
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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
• All N oscillators same frequency
• Density of states in ω
(energy/freq) is a delta function
• Einstein specific heat
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Fre
quency,
0 2p/a
E
Wave vector, k
(3 )Eg N
( )
E
du dfC g d
dT dT
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Einstein Low-T and High-T Behavior
• High-T (correct, recover Dulong-Petit):
• Low-T (incorrect, drops too fast)
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2
2
1( ) 3 3
1 1
E
E
E
T
E B BT
T
C T Nk Nk
/2
2/
2/
( ) 3
3
E B
E
BE B
E E B
B
k T
E B k Tk T
k T
B k T
eC T Nk
e
Nk e
Einstein modelOK for optical phonon
heat capacity
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The Debye Model
• Linear (no) dispersion
with frequency cutoff
• Density of states in 3D:
(for one polarization, e.g. LA)
(also assumed isotropic solid, same vs in 3D)
• N acoustic phonon modes up to ωD
• Or, in terms of Debye temperature
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Fre
quency,
0
sv k
Wave vector, k 2p/a
2
2 32 s
gv
p
kD roughly corresponds to max lattice wave vector (2π/a)
ωD roughly corresponds tomax acoustic phonon frequency
1/3
26sD
B
vN
k p
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Annalen der Physik 39(4)p. 789 (1912)
Peter Debye (1884-1966)
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The Debye Integral
• Total energy
• 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
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Fre
quency,
0 Wave vector, k 2p/a
sv k
0
( ) ( ) ( )D
u T f g d
3 / 4
2
0
( ) 9( 1)
D T x
D B x
D
T x e dxC T Nk
e
people like to write:(note, includes 3x)
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Debye Model at Low- and High-T
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• At low-T (< θD/10):
• At high-T: (> 0.8 θD)
• “Universal” behavior for all solids
• In practice: θD ~ fitting parameter
to heat capacity data
• θD is related to “stiffness” of solid
as expected
3412
( )5
D B
D
TC T Nk
p
( ) 3D BC T Nk
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Experimental Specific Heat
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104
103
102
101
101
102
103
104
105
106
107
Temperature, T (K)
Sp
ecif
ic H
ea
t, C
(J
/m
-K)
3
C T3
C 3kB 4.7 106 J
m3 K
D 1860 K
Diamond
ClassicalRegime
Each atom has a thermal energy of 3KBT
Specific
Heat
(J/m
3-K
)
Temperature (K)
C T3
3NkBT
Diamond
In general, when T << θD
1,d d
L Lu T C T
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Phonon Dispersion in Graphene
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Maultzsch et al., Phys. Rev. Lett. 92, 075501 (2004) Optical
Phonons
Yanagisawa et al.,Surf. Interf. Analysis37, 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
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how do we explain this mess?
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Kinetic Theory of Energy Transport
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z
z - z
z + z
u(z-z)
u(z+z)
λqz zzzz zuzuvq 2
1'
Net Energy Flux / # of Molecules
2' cosz z z
du duq v v
dz dz
through Taylor expansion of u
1
3z
du dT dTq v k
dT dz dz
Integration over all the solid angles total energy flux
1
3k CvThermal conductivity:
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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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l
Temperature, T/D
Boundary
Phonon
Scattering
Defect
Decreasing Boundary Separation
Increasing
Defect
Concentration
0.01 0.1 1.0
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
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21 1
3 3k Cv Cv
Temperature, T/D
0.01 0.1 1.00.01 0.1 1.0
kl
dl Tk
Boundary
Phonon
ScatteringDefect
Increasing Defect
Concentration
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Temperature Dependence of Phonon KTH
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C λ
low T Td
nph 0, so
λ , but then
λ D (size)
Td
high T 3NkB 1/T 1/T
C /11kT
ph ph
ph
en
3
d
B
T lowT
Nk high T
ThighkT
Tlow
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Ex: Silicon Film Thermal Conductivity
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Bulk single-crystal silicon:
Touloukian et al. (1970)
d = 0.44 cmsingle
crystal
110 100
Temperature (K)
10
100
1000
Th
erm
al C
on
du
cti
vit
y (
W m
-1K
-1)
104
bulk
1
21
3
1),(
i
G
G nAd
ACvndk
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
undoped poly-
crystal
doped
films
undoped
doped
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
McConnell, Srinivasan, and Goodson, JMEMS 10, 360-369 (2001)
size
effect
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Ex: Silicon Nanowire Thermal Conductivity
• Recall, undoped bulk
crystalline silicon k ~ 150
W/m/K (previous slide)
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Li, Appl. Phys. Lett. 83, 2934 (2003)
Nanowire diameter
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Ex: Isotope Scattering
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~T3
~1/T
isotope~impurity
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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
– Boundary & grain boundary scattering
– Phonon-phonon scattering
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224
2
1
4
i
ph i s
nV M
v M
p
1 s
ph b
v
D
1exp
ph ph B
A T Bk T
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What About Electron Thermal Conductivity?
• Recall electron heat capacity
• Electron thermal conductivity
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0
e
du dfC E g E dE
dT dT
2
2
Be e B
F
k TC n k
E
p
at most Tin 3D
ek Mean scattering time:e = _______
e
Temperature, T
Defect
Scattering
Phonon
Scattering
Increasing
Defect Concentration
Bulk Solids
eElectron Scattering Mechanisms
Grain Grain Boundary
• Defect or impurity scattering
• Phonon scattering
• Boundary scattering (film
thickness, grain boundary)
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Ex: Thermal Conductivity of Cu and Al
• Electrons dominate k in metals
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103
102
101
100
100
101
102
103
Temperature, T [K
Th
erm
al
Co
nd
uc
tiv
ity
, k
[W
/cm
-K]
Copper
Aluminum
Defect Scattering Phonon Scattering
1
1
Matthiessen Rule:
1 1 1 1
1 1 1 1
e defect boundary phonon
e defect boundary phonon
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Wiedemann-Franz Law
• Wiedemann & Franz (1853) empirically saw ke/σ = const(T)
• Lorenz (1872) noted ke/σ proportional to T
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22 21
3 2e B F
F
Tk n v
E
p
FE where
2qq n n
m
recall electrical
conductivity
taking the ratio ek
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Lorenz Number
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This is remarkable!
It is independent of n,
m, and even !
2 2
23
e BkL
T q
p
L = /T 10-8 WΩ/K2
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
8 22.45 10 WΩ/KL
Agreement with experiment is
quite good, although L ~ 10x
lower when T ~ 10 K… why?!
Experimentally
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Amorphous Material Thermal Conductivity
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Amorphous (semi)metals: bothelectrons & phonons contribute
Amorphous dielectrics:K saturates at high T (why?)
a-Si
a-SiO2
GeTe
© 2008 Eric Pop, UIUC ECE 598EP: Hot Chips
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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