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

• • • ••• •

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• • •

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• • •

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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