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Maintaining the THERMO in Thermoelectrics
How the fundamentals of heat transport are still being worked out, and why it matters
Keith A. Nelson Research GroupMIT Department of Chemistry
Outline Motivations
Thermoelectric power generation How it works Requirements for materials
Thermal conductivity What mediates it How to model it
Measurements & results How to measure thermal diffusion Measuring thermal non-diffusion! Fundamental insights into thermal transport Practical exploitation of what we understand Work in the trenches supports the big picture
Thermoelectric generator
Thermal gradient ∆T Current I
Simple, rugged device
Thermoelectrics use waste heat or solar heat
Great opportunities for energy recovery
< 1/3 gasoline energy pushes the car forward!
Great opportunities for energy recovery
~ 1/8 gasoline energy pushes the car forward in city driving!
~ 10-20% waste heat recovery would have a huge impact
Run backward ⇒ Thermoelectric cooler
Thermoelectric generator
Thermoelectric (Peltier) cooler
Requirements for efficient thermoelectrics
1. High Seebeck Coefficient S = -∆V/∆T
• • • ••• •
•
• • •
• • •
• • • • • •
•
• • •
• • •
• • • •
• • • • •
•
• •
Built-In Potential
Temp. Gradient
V2 Tcold
V1 Thot
Seebeck effect 1821
2. High Electrical Conductivity σ
I = ∆V/R
Requirements for efficient thermoelectrics
Thermal gradient ∆T Current I X X
3. Low Thermal Conductivity k
Requirements for efficient thermoelectrics
What determines k?
1. High Seebeck Coefficient S = -∆V/∆T
2. High Electrical Conductivity σ
3. Low Thermal Conductivity k
2S TZTk
σ=
Figure of merit (unitless)
What moves thermal energy around?
Theories of thermal energy & transportEarly ideas
Phlogiston theory (J. Becher, 1667)Caloric theory (A. Lavoisier, 1770s)
Theories of thermal energy & transport
Heat as substance
Kinetic theory (mid-19th century)Heat is the energy of molecular motion
Explained specific heat, thermal conductivity of gasesDiffusive transport when mean free path is short
Molecular diffusion
J. Fourier 1768-1830
Fourier’s law (1822)
2T Tt
α∂= ∇
∂
Thermal diffusion
Solid-state thermal energy & transport
Thermal conductivity in non-metallic solids (Debye, Peierls, 1920s)
Heat is carried by lattice vibrations (acoustic phonons)
Sound speed v – how fast they go Mean free path Λ – how far they go
Specific heat c – how much energy they carry
Thermal conductivity is the sum of contributions of all phonons
( ) ( ) ( )max
013pk v c d
ωω ω ω ω= Λ∫
Solid-state thermal energy & transportPeierls theory (1929) still forms the basis for our understanding
k still ordinarily describes diffusive thermal transportFourier law is valid on length scales > mean free path Λ
Average MFP ~ 10-40 nm at 300 K
( ) ( ) ( )max
013
k v c dω
ω ω ω ω= Λ∫
kFourier law is valid on length scales > mean free path
Average is over all acoustic modesMany acoustic frequencies ω MHz-GHz-THz
Many acoustic wavelengths λ mm-µm-nmMost of the modes are high frequency ~ ω2
Solid-state thermal energy & transport Peierls theory (1929) still forms the basis for our understanding
But for ~ 80 years no one could calculate k accurately with it!
( ) ( ) ( )max
013
k v c dω
ω ω ω ω= Λ∫
Because MFPs decrease at high ω, many low-frequency modes participate in thermal transport
Recent first-principles calculations by D. Broida, G. Chen, others Results of calculations still largely untested
Easy! Speed is nearly independent of ω Easy! Energy per mode is ħω
Nasty! Decreases sharply at high ω Complicated ω and T-dependences
Using & testing Peierls theory
Limited tests of theory were OK for familiar materials & long length scales What has changed now:
(i) Heat transport at very small distances is crucial (current “technology node” in microelectronics is 22 nm)
(ii) Need for new materials with tailored thermal transport properties
( ) ( ) ( )max
013
k v c dω
ω ω ω ω= Λ∫Need reliable calculations of Λ(ω)
Measurements of k on short length scales Measurements of acoustic MFPs
Measuring thermal conductivityTransient grating experiment sets length scale
θ
TG period2sin( / 2)2 TG wavevector
L
q L
λθ
π
= =
= =
L
Excitation
Diffusion length scale is specified optically
Spatially periodic pattern is imparted to sample
Measuring thermal conductivity
Thermal grating
Pattern acts like a diffraction grating…
Measuring thermal conductivityProbing
…that diffracts probe laser lightAs heat diffuses from grating peak to null,
the diffracted signal decays
Measuring thermal conductivityLength scale can be varied easily
( )2sin 2L λ
θ=
Move selected patterninto beam path to vary L
Measuring thermal conductivity
Interference fringe spacing L sets thermal transport distance D = L/2Heated
Unheated
1D thermal diffusion with spatially periodic heating
Initial condition: ( ) ( )2
2, ,T x t T x t
tx∂ ∂
α∂∂
∆ ∆= ( ) ( )0; 0 cosT x t T qx∆ = = ∆
( ) ( ) 20, cos with tT x t T qx e qγ γ α−∆ = ∆ =∆ = ∆∆ = ∆Simple solution:
As expected for diffusion
thermal diffusivity Vk Cα ρ= = 2 1 grating period, diffusion length
q L DL D
π= == =
Spatially periodic pattern never changes!It just decays exponentially as heat moves from grating peaks to nulls
Decay rate γ ∝ thermal diffusivity α
Decay rate γ = αq2 ⇒Decay time τ = 1/γ = D2/α
D ατ=
Thermal transport in a liquid
t te eγ τ− −=2 2
thermal diffusivityq q Lγ α π
α= ==
Decane nanofluid TG data
Schmidt et al, J. Appl. Phys.103, 083529 (2008).
Thermal diffusion
Decane nanofluid q-dependence
2
D t
t D
α
α
=
=
TG decay rate γ ∝ q2 ⇒ Diffusive transport
Thermal transport in a Si membrane
400µm
390 nm membrane
Thermal transport in a Si membrane
John Cuffe, Timothy Kehoe,Clivia Sotomayor Torres
Thermal transport in a Si membrane
γ ∝ q2 ⇒ Transport is diffusive on > 10 µm length scale
L = 11.5-25 µm
L=18 µm
L=11.5 µm
Thermal transport in a Si membraneThermal transport in a Si membraneThermal transport in a Si membraneThermal transport in a Si membrane
L = 2.4-25 µm
“Average” mean free path is useless!Acoustic modes with long MFP play outsize role in heat transport
γ ∝ q2 ⇒ Transport is NOT diffusive on < 10 µm length scalePhys. Rev. Lett. 110, 025901 (2013)
Thermal transport in bulk GaAs 295 K & 425 K
tTC et
γ−
Strong deviation from γ ∝ q2 !
GaAs TG data & q-dependence GaAs effective thermal diffusivity
2D thermal diffusion
γ ∝ q2 ⇒ Transport is NOT diffusive on < 15 µm length scale
Shorter light λ ⇒ Shorter length scales EUV & X-ray TG measurements
Collaboration w/ U Colo M. Murnane & H. Kapteyn group
Fs HHG EUV pulses
Optically pump patterned surface Probe by EUV diffraction
All-EUV TG measurement planned
Soft x-ray TG measurements planned at Trieste
Periods as short as 35 nm 1D and 2D patterns
Nature Materials 9, 26 (2010)
What determines thermal conductivity?Nature of transport depends on length scale d relative to phonon MFP λ
d < λ: Ballistic d ~ λ: Quasi-ballistic d > λ: Diffusive
A.S. Henry & G. Chen, J. Comp. Theor. Nanoscience 5, 1 (2007)
Phonons of many frequencies play a role
Need measurement of thermal transport & phonons on all length scales
Analyzing Si membrane thermal conductivity Effective Grating Conductivity (Alex Maznev)
Membrane Reduction (Fuchs, Sondheimer)
Here - Combination
keff =13
A qΛ( )kΛdΛ0
∞∫ A =3
q2Λ2 1−arctan qΛ( )
qΛ
keff −membrane = ASkΛFpdΛ AS =3
q2 ΛFp( )2 1−arctan qΛFp( )
qΛFp
0
∞∫
Λm = ΛbulkFpd
Λbulk
Fp χ( ) =1−
32χ
1− p( ) 1t 3 −
1t 5
1− e−χt
1− pe−χt dt1
∞∫
E.H. Sondheimer, “The mean free path of electrons in metals”, Phil. Mag. 1, 1 (1952). K. Fuchs, Proc. Cambridge Philos. Soc. 34, 100 (1938).
Analyzing Si membrane thermal conductivity
keff −membrane = ASkΛFpdΛ0
∞∫
Non-diffusive transport + boundary scattering
Membrane thermal conductivity vs thicknessSuspended membranes 15-1500 nm
max
0
13film
dk C v F dω
ω ω = Λ Λ ∫( ) ( )3 51
3 1 11 12
tF e dtt t
χχχ
∞ − = − − − ∫
12 14 16 18 20 220.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
α (c
m2 s
-1)
L (µm)
Bulk
1518 nm984 nm759 nm400 nm194 nm143 nm99 nm47 nm30 nm17.5 nm15 nm
10-8 10-7 10-6 10-50
40
80
120
160
Ashegi (1998)Ju (1999)Liu (2006)Hao (2006)Aubain (2010)Aubain (2011)
This Work Theory
k (W
m-1 K
-1)
d (m)
Bulk
Reconstructing the MFP distribution
10-9 10-8 10-7 10-6 10-5 10-40.0
0.2
0.4
0.6
0.8
1.0
Reconstruction 1st Principles10
Holland MD
k acc
Λc (m)Assuming diffuse boundary scattering
MHz-GHz-THz photoacousticsλ= 1-200 µm
ν = 10-1000 MHz Multiple pulses ⇒
time (ns)0 200 400 600 800
inte
nsity
glycerol 330 K
glycerol 275 K
glycerol 195 K
Supercooled ωτ ≈ 1
Liquid
ωτ > 1
Solidωτ < 1
165 GHz36 nmqd < 1
MHz: Select wavevector GHz: Select frequency
Optical probe pulse
Signal
20 nm metal films
Sample
Optical pulse
sequenceAcoustic
waveSapphire substrate
Sample
Crossed beams ⇒ λ= 5-500 nmν = 10-1000 GHz
Commercial TG metrology for 300 mm wafers
50 cm
Excitationlaser
Probelaser Detector
Vis
ion
syst
em
Optics Head
Summary Non-diffusive thermal transport readily measured in common materials
Fundamental understanding of thermal transport Applications to thermoelectrics, nanoelectronics… It’s all about phonon mean free paths!
Nanometer length scales now accessible and getting easier Full range of thermal transport regimes accessible
Acoustic wave spectroscopy determines MFPs directly Complete comparison to theory then possible
Thermoelectric design now taking non-diffusive transport into account Need measurements on full range of length scales, down to nm
A lot of work in the trenches! A lot of applications as a result
Acknowledgements
DOE EFRCNSF
Alexei MaznevJeremy JohnsonJeff Eliason
Thomas PezerilChristoph Klieber
Austin MinnichMaria LuckyanovaKim Collins Gang ChenMayank BulsaraGene FitzgeraldJohn CuffeTimothy KehoeClivia Sotomaor Torres
Solid State Solar Thermal Energy Conversion
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