transport (i.e. why does the electrical conductivity vary over. >20 orders of ...... G.
Chan, Nanoscale Energy Transport and Conversion, eqn. 1.35 on p. 27,.
NCN Summer School: July 2011
Near-equilibrium Transport: Fundamentals and Applications Lecture 9: Phonon Transport Mark Lundstrom and Changwook Jeong Electrical and Computer Engineering and Network for Computational Nanotechnology Birck Nanotechnology Center Purdue University, West Lafayette, Indiana USA
1
copyright 2011 This material is copyrighted by Mark Lundstrom under the following Creative Commons license:
Conditions for using these materials is described at http://creativecommons.org/licenses/by-nc-sa/2.5/
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Lundstrom and Jeong 2011
heat flux and thermal conductivity 1) Electrons can carry heat, and we have seen how to evaluate the electronic thermal conductivity. = J xq πσE x − κ 0 dTL dx = J xq π J x − κ e dTL dx
E − EF ) ( κ0 = ∫ σ ′ ( E ) dE 2 2
q TL κ= κ 0 − π Sσ e
2) In metals, electrons carry most of the heat. 3) But in semiconductors and insulators, most of the heat is carried by lattice vibrations (phonons). 3
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lecture 9
J xq π J x − κ e dTL dx = J xQ = −κ L dTL dx
This lecture is a brief introduction to phonon transport. We also discuss the differences between electron and phonon transport (i.e. why does the electrical conductivity vary over >20 orders of magnitude while the thermal conductivity only varies only over ~3 orders of magnitude?) 4
outline
9.1 Introduction 9.2 Electrons and Phonons 9.3 General model for heat conduction 9.4 Thermal conductivity 9.5 Debye model 9.6 Scattering 9.7 Discussion 9.8 Summary
Lundstrom and Jeong 2011
5
electron dispersion E
Electrons in a solid behave as both particles (quasi-particles) and as waves.
E = 2 k 2 2m*
Electron waves are by a described “dispersion:” E k = ω k
( )
( )
BW
Because the crystal is periodic, the dispersion is periodic in k (Brillouin zone). Particles described by a “wavepacket.” The “group velocity” of a wavepacket is determined by the dispersion: υg k = ∇k E k
( )
( )
k −π a
k0
π a
6
phonon dispersion ω
Lattice vibrations behave both as particles (quasi-particles) and as waves.
Einstein
Lattice vibrations are described by a “dispersion:” ω (q) = E (q)
Debye
Because the crystal is periodic, the dispersion is periodic in k (Brillouin zone).
BW
Particles described by a “wavepacket.” The “group velocity” of a wavepacket is determined by the dispersion: υg ( q ) = ∇q ω ( q )
q −π a
q0
π a
7
mass and spring U=
ks
1 2 ks (x − x0 ) 2
F=−
dU = − ks (x − x0 ) dx
d2x M 2 = − ks (x − x0 ) dt
M
x (t ) − x0 = Aeiωt 1 2
En = n + hω
ω = ks M Lundstrom and Jeong 2011
8
general features of phonon dispersion LO and TO degenerate at q = 0 for non polar semiconductors
ω
low group velocity
ω0
LO (1) TO (2) LA (1)
ω = υ Sq υ S = cl ρ
TA (2)
q π a
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LO and LA degenerate at zone boundary for non polar semiconductors
real dispersion 3
LO (1)
0 -1 -2 0
LA (1)
0.04
ph
1
(eV)
cb
E
el
E (eV)
2
TO (2)
0.06
0.02
hh
TA (2)
lh
0.2 0.4 0.6 0.8 kkxx (x (× ππ/aa0))
1
0 0
0.2 0.4 0.6 0.8 qqxx(×(xπ π/a a ) 0)
1
note the different energy scales! electrons in Si (along [100]) 10
phonons in Si (along [100])
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wavelengths: electrons vs. phonons
= λBel
h 3m*k BTL
≈ 60A
= 300 K ) ( m* m= 0 , TL
4π υ S = λ ≈ 5A 3k BTL ph B
(υS ≈ 5000 m s )
(see appendix for derivation) 11
Lundstrom and Jeong 2011
outline
9.1 Introduction 9.2 Electrons and Phonons 9.3 General model for heat conduction 9.4 Thermal conductivity 9.5 Debye model 9.6 Scattering 9.7 Discussion 9.8 Summary
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general model for electronic conduction From Lecture 2: 2q = I Tel ( E ) M el ( E )( f1 − f 2 ) dE ∫ h
I I
0 f1 ( E ) =
13
channel
1 e( E − EF 1 ) kBTL + 1
f2 ( E ) =
Lundstrom and Jeong 2011
V 1 e(
E − EF 1 + qV ) k BTL
+1
for phonon conduction Thermal reservoir in equilibrium at temperature, TL1.
n1 ( ω ) =
1 eω kBTL1 − 1
channel characterized by a dispersion
TL1
channel
TL 2
Thermal reservoir in equilibrium at temperature, TL2.
n2 ( ω ) =
x 2q = I Tel ( E ) M el ( E )( f1 −= f 2 ) dE ⇒ Q ? ∫ h 14
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1 eω kBTL 2 − 1
heat flux 2q = I Tel ( E ) M el ( E )( f1 − f 2 ) dE ∫ h Q
Assume ideal contacts, so that the transmission describes the transmission of the channel.
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1 ( ω ) Tph ( ω )M ph ( ω )( n1 − n2 ) d ( ω ) ∫ h
TL1
channel
TL 2 x
Lundstrom and Jeong 2011
near-equilibrium heat flux Q
1 ( ω ) Tph ( ω )M ph ( ω )( n1 − n2 ) d ( ω ) ∫ h
n2 ≈ n1 + ∂n0 ∂TL
∂n1 ∆TL ∂TL
( n1 − n2 ) ≈ −
∂n ∂n1 ∆TL ≈ − 0 ∆TL ∂TL ∂TL
ω k T ∂ 1 ω e B L = ∂TL eω kBTL − 1 k BTL2 ( eω kBTL − 1)2
∂n0 ∂ 1 1 eω kBTL = ω kBTL = − ∂ ( ω ) ∂ ( ω ) e − 1 k BTL ( eω kBTL − 1)2 ∂n0 ω ∂n0 = − ∂TL TL ∂ ( ω ) 16
( n1 − n2 ) ≈ −
∂n0 ω − ∆TL TL ∂ ( ω )
Lundstrom and Jeong 2011
Q= − K L ∆TL
lattice thermal conductance
Q= − K L ∆TL K L
∂n0 k B2TL ω Tph ( ω ) M ph ( ω ) d ( ω ) − ∫ h k BTL ∂ ( ω )
Recall the electrical conductance:
G
2q 2 ∂f 0 T E M E ( ) ( ) el el − dE ∫ h ∂E
“window function”:
Wel ( E ) =
( − ∂f0
∂E )
+∞
∫ ( − ∂f
−∞
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Lundstrom and Jeong 2011
0
∂E ) dE =1
lattice window function
Q= − K L ∆TL
+∞
ω ∫0 kBTL
KL
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2
KL
2 ∂n0 kT ω Tph ( ω ) M ph ( ω ) d ( ω ) − ∫ h k BTL ∂ ( ω ) 2 B L
∂n0 − d ( ω ) = ∂ ω ( )
π 2 k B2TL 3h
+∞
∫ 0
2 ω k T ( ω k BTL ) e B L 2 ω k BTL e − 1 ( )
ω d = k T B L
+∞
∫ 0
x π2 x 2e dx = x 2 3 ( e − 1)
3 ω 2 ∂n0 ∫ Tph ( ω )M ph ( ω ) π 2 kBTL − ∂ ( ω ) d ( ω ) W ph ( ω )
heat conduction 1) Fourier’s Law of heat conduction:
2) Thermal conductance: K L =
π 2 k B2TL
3) Quantum of heat conduction:
3h
Q= − K L ∆TL
∫ T ( ω )M ( ω )W ( ω ) d ( ω ) ph
ph
ph
π 2 k B2TL 3h
3 ω 2 ∂n0 W ph ( ω ) 2 4) Window function for phonons:= − π ∂ ω k T ( ) B L 19
Lundstrom and Jeong 2011
electrical conduction I = G ∆V
1) Electrical current:
2) Electrical conductance:
G=
2q2
h
∫ Tel (E )Mel (E )Wel dE
2 2 q 3) Quantum of electrical conduction:
h
4) Window function for electrons: 20
Wel (E ) = (− ∂f0 ∂E )
Lundstrom and Jeong 2011
window functions: electrons vs. phonons Electrons 60
( eV-1)
50K
W
20 0
300K
40
-0.1
20 0 0
0 0.1 EEel−-E EFF ((eV) eV)
Wel (E ) = (− ∂f0 ∂E ) 21
50K
ph
40
el
W ( eV-1)
60
Phonons
300K
0.1 E ω ((eV) eV) ph
0.2
3 ω 2 ∂n0 = W ph ( ω ) 2 − π ∂ ω k T ( ) B L
Lundstrom and Jeong 2011
outline
9.1 Introduction 9.2 Electrons and Phonons 9.3 General model for heat conduction 9.4 Thermal conductivity 9.5 Debye model 9.6 Scattering 9.7 Discussion 9.8 Summary
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diffusive heat transport (3D) Q = − KL ∆TL KL =
π 2 k B2TL 3h
(Watts)
∫ T ( ω )M ( ω )W ( ω ) d ( ω ) ph
ph
ph
λ ph ( ω ) λ ph ( ω ) Tph ( ω ) = → λ ph ( ω ) + L L M ph (hω ) ∝ A
(diffusive phonon transport)
(large, 3D sample)
L ∆TL Q = − KL A A L 23
(Watts/K)
Q dTL J = = −κ L A dx Q x
Lundstrom and Jeong 2011
L κ L = KL A
(W m-K )
diffusive heat transport (3D) J xQ = −κ L κ L =
dTL dx
π 2 k B2TL 3h
(Watts / m2)
∫ λ ( ω )
d (Fn q ) Jx = σ dx
ph
M ph ( ω ) A
W ph ( ω ) d ( ω )
(Amperes / m2)
M el (E ) σ= λel (E ) Wel (E )dE ∫ h A 2q2
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(1/Ohm-m)
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(Watts/m-K)
thermal conductivity again κ L =
κ L =
π 2 k B2TL 3h
∫ λ ( ω ) ph
π 2 k B2TL ∫ λ ph ( ω ) 3h
M ph ( ω )
∫
A M ph ( ω ) A
M ph ( ω ) A
W ph ( ω ) d ( ω ) W ph ( ω ) d ( ω )
κ L =
25
3h
M ph A
λ ph
M ph ( ω ) A
W ph ( ω ) d ( ω ) M ph A ≡ ∫
π 2 k B2TL
(Watts/m-K)
λ ph
≡
M ph ( ω ) A
∫ λ ph ( ω )
Lundstrom and Jeong 2011
∫
W ph ( ω ) d ( ω )
M ph ( ω ) A
M ph ( ω ) A
W ph ( ω ) d ( ω )
W ph ( ω ) d ( ω )
diffusive heat transport (3D) J xQ = −κ L
dTL dx
(Watts / m2)
π 2 kB2TL κ Ll = M ph A 3h d (Fn q ) J =σ dx σ=
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2q2
h
M el A
λ ph
(Watts/m-K)
(Amperes / m2)
λel
(1/Ohm-m)
Lundstrom and Jeong 2011
diffusive heat transport (3D) J xQ = −κ L κ L =
dTL dx
π 2 k B2TL 3h
(Watts / m2)
∫ λ ( ω ) ph
M ph ( ω ) A
W ph ( ω ) d ( ω )
(Watts/m-K)
κ l
To evaluate the lattice thermal conductivity, we must specify: 1) the mean-free-path for phonon scattering 2) the number of channels per unit area for phonon conduction. Before we do that……the lattice thermal conductivity is often related to the lattice specific heat. Let’s see how that works. 27
Lundstrom and Jeong 2011
specific heat The total energy (per unit volume) of the lattice vibrations is: ∞
EL = ∫ ( ω ) D ph ( ω ) n0 ( ω ) d ( ω ) 0
where Dph is the phonon density of states per unit volume.
The specific heat is the change in energy per degree change in TL:
∂EL ∂ = CV = ∂TL ∂TL
∞
∫ ( ω ) D ( ω ) n ( ω ) d ( ω ) ph
0
0
∂ n0 ( ω ) CV ≈ ∫ ( ω ) D ph ( ω ) d ( ω ) ∂TL 0 ∞
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Lundstrom and Jeong 2011
specific heat (ii) ∂ n0 ( ω ) CV ≈ ∫ ( ω ) D ph ( ω ) d ( ω ) ∂TL 0 ∞
Recall:
ω ∂ n0 2 CV ≈ k BTL ∫ D ph ( ω ) d ( ω ) − k BTL ∂ ( ω ) 0 2
∞
CV ≈
29
π 2 k B2TL 3
∞
∫ D ( ω )W ( ω ) d ( ω ) ph
ph
0
Lundstrom and Jeong 2011
∂n0 ω ∂n0 = − ∂TL TL ∂ ( ω )
specific heat and thermal conductivity
CV =
κ L =
π 2 k B2TL 3
π 2 k B2TL 3h
∞
∫ D ( ω )W ( ω ) d ( ω ) ph
ph
0
∫ λ ( ω )
M ph ( ω )
ph
A
W ph ( ω ) d ( ω )
one can show….(see appendix) κ= L
30
1 Λ ph 3
υ ph CV
Lundstrom and Jeong 2011
specific heat and thermal conductivity κ L =
π 2 k B2TL 3h
∫ λ ( ω )
1 κ= Λ ph L 3
Λ ph
λ ph (= ω )
M ph ( ω )
ph
A
W ph ( ω ) d ( ω )
υ ph CV
Λ υ D W d ( ω ) ∫ ph ph ph ph ≡ ∞ υ D W d ( ω ) ∫0 ph ph ph
υ ph
∞ υ ω D W d ph ph ph ( ) 1 ∫0 = ∞ 3 D phW ph d ( ω ) ∫ 0
( 4 3) Λ ph ( ω )
Λ ph ( ω ) ≡ υ ( ω )τ ( ω ) 31 Lundstrom and Jeong 2011
why? κ L =
π 2 k B2TL 3h
1 κ Ll = Λ ph 3
∫ λ ( ω ) ph
M ph ( ω ) A
W ph ( ω ) d ( ω )
υ ph CV
Why did we do this? Because this expression can be simply derived from kinetic theory and is widely-used. But, we now have a precise definition of the mfp and average phonon velocity. 32 Lundstrom and Jeong 2011
references
κ= L
1 Λ ph 3
υ ph CV
G. Chan, Nanoscale Energy Transport and Conversion, eqn. 1.35 on p. 27, Oxford Univ. Press, Oxford, U.K., 2005. C. Kittel, Introduction to Solid State Physics, 4th Ed., eqn. 58 on p. 225, John Wiley and Sons, New York, 1971.
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outline
9.1 Introduction 9.2 Electrons and Phonons 9.3 General model for heat conduction 9.4 Thermal conductivity 9.5 Debye model 9.6 Scattering 9.7 Discussion 9.8 Summary
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effective mass model for electrons
E
E = 2 k 2 2m*
As long as the BW >> kBTL, the effective mass model generally works ok.
BW
k −π a
π a
This is the typical case for electronic dispersions. Only states near the bottom of the conduction band or top of the valence band matter, and these regions can be described by an eff mass model.
Lundstrom and Jeong 2011
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Debye model for acoustic phonons Linear dispersion model
ω
ω = υD q Debye
D ph ( ω ) =
BW
q −π a
3 ( ω ) Ω 2
( υ D ) 2 3( ω ) A M ph ( ω ) = 2 2π
2
2 π υ D
3
( no. J ) ( no. J )
π a If acoustic phonons near q =0 mostly contribute to heat transport, Debye model works work well. Lundstrom and Jeong 2011
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caution Most textbooks derive the phonon DOS in frequency space, not energy space as we have.
D ph (ω ) =
3ω
2
2π υ D 2
3
( no.
Hz )
D ph (ω ) =
D ph ( ω ) =
3 ( ω ) 2π
2
2
( υ D )
3
( no. J )
D ph ( ω )
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Debye model: cutoff frequency / wavevector For phonons, BW ~ kBTL (recall slide 10) No. of states in a band = N.
3 ( ω ) N = ∫0 2π 3 ( υ )3 d ( ω ) 3 Ω D
ωD
ω D
= ∫ Dph ( ω )d ( ω ) 0
1/3
6π N = ωD υ D Ω 2
1/3
ωD 6π N = q= D υD Ω 2
38
≡ k BTD
2
Debye model: cutoff frequency / wavevector
ω
1/3
6π 2 N ω D = υ D Ω
ωD ω = υ D q
q= D BW
q −π a
1/3
ωD 6π N = υD Ω 2
k BTD ≡ ωD
qD π a Debye model valid when TL kBTL which for phonons, BW ~ kBTL. Most of the modes are occupied for phonons but only a few for electrons. Lundstrom and Jeong 2011
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quantized heat flow Both the charge and heat currents are quantized. KL =
π 2 k B2TL
∫ T ( ω ) M ( ω )W ( ω ) d ( ω ) ph
ph
ph
3h π 2 k B2TL ≈ Tph ( 0 ) M ph ( 0 ) at low temp 3h
Nanostructure at low temperatures can have nearly ballistic phonon transport with a small number of modes occupied. See the paper by Schwab, et al. for experimental confirmation of quantized heat flow. K. Schwab, E. A. Henriksen, J. M. Worlock, and M. L. Roukes, “Measurement of the quantum of thermal conductance,” Nature, 404, 974-977, 2000. Lundstrom and Jeong 2011
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outline
9.1 Introduction 9.2 Electrons and Phonons 9.3 General model for heat conduction 9.4 Thermal conductivity 9.5 Debye model 9.6 Scattering 9.7 Discussion 9.8 Summary
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summary 1) Our model for electrical conduction can readily be extended to describe phonon transport. The mathematical formulations are very similar. 2) Just as for electrons, phonon transport is quantized. 3) The difference BW’s of the electron and phonon dispersions has important consequences. For electrons, a simple dispersion (effective mass) often gives good results, but for phonons, the simple dispersion (Debye model) is not very good. 4) There is no Fermi level for phonons, so the lattice thermal conductivity cannot be varied across many orders of magnitude like the electrical conductivity. Lundstrom and Jeong 2011
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for more about heat transport C. Kittel, Introduction to Solid State Physics, 4th Ed., eqn. 58 on p. 225, John Wiley and Sons, New York, 1971. Gang Chen, Nanoscale Energy Transport and Conversion, Oxford Univ. Press, New York, 2005. J. Callaway, “Model for lattice thermal conductivity at low temperatures,” Phys. Rev., 113, 1046-1051, 1959. M.G. Holland, “Analysis of lattice thermal conductivity,” Phys. Rev., 132, 2461-2471, 1963.
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quantized thermal transport
K. Schwab, E.A. Henriksen, J.M. Worlock, and M.L. Roukes, “Measurement of the quantum of thermal resistance,” Nature, 404, 974-977, 2000.
Lundstrom and Jeong 2011
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for more about this lecture C. Jeong, S. Datta, M. Lundstrom, “Full Dispersion vs. Debye Model Evaluation of Lattice Thermal Conductivity with a Landauer approach,” J. Appl. Phys. 109, 073718-8, 2011.
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questions
9.1 Introduction 9.2 Electrons and Phonons 9.3 General model for heat conduction 9.4 Thermal conductivity 9.5 Debye model 9.6 Scattering 9.7 Discussion 9.8 Summary
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