2371-10

Advanced Workshop on Energy Transport in Low-Dimensional Systems: Achievements and Mysteries

Ali SHAKOURI

15 - 24 October 2012

Birck Nanotechnology Center, Purdue University West Lafayette

U.S.A.

Ultrafast Characterization of Heat Transport and Thermoelectric Energy Exchange at Interfaces (Ballistic and Diffusive Transport of Heat/Energy

Ali ShakouriAli ShakouriBirck Nanotechnology Center; Purdue University

shakouri@purdue.edu

Acknowledgement:

Advanced Workshop on Energy Transport

Acknowledgement:

in Low Dimensional Systems 17 October 2012

atur

e iz

ed)

Fourier

FourierTe

mpe

ra(n

orm

ali

Hyperbolic Heat (C tt )

Hyperbolic

e )

(Cattaneo)

Hyperbolic

Boltzmann

mpe

ratu

rerm

aliz

ed)

Boltzmann (Equation Phonon Radiative Transfer)Fourier

Tem

(nor

L=0.1 m Diamond

Fourier

Boltzmann

2A. Shakouri; 10/15/2012

A. A. Joshi and A. Majumdar; J. Appl. Phys. 74, p. 31 (1993)Distance (norm.)

Phonon Number: Nq(x,t)

mobility; D: diffusion coefficientmobility; D: diffusion coefficient

Luttinger J M 1964 Phys. Rev. 135 A1505

3A. Shakouri; 10/15/2012

Luttinger J M 1964 Phys. Rev. 136 A1481Shastry B S 2009 Rep, Prog, Phys 72, 016501

N Th G ’ f ti f th t t l d itN2: The Green’s function of the total energy density propagation K(t,x) in a solid material when there is delta-function excitation P(t)delta function excitation P(t)

P(t)N2( q) =

K( ,q)

K(t,x)N2( ,q)

P( )

N ( )+ q

N2( ,q) = – q

2+ DQq2

q total relaxation time of energy carriers (function of wavevector q)

D heat diffusion constant

B. S. Shastry, Rep, Prog, Phys72 016501 (2009)

4A. Shakouri; 10/15/2012

DQ heat diffusion constant 72, 016501, (2009)

Thermalcontribution to

Non-thermalcontribution to

energy transport (heat propagation)

energy transport

2 3 (1 )F

a fsv

Damped oscillation of period a: lattice constant.vF: Fermi velocity.

Oscillations in the total energy density at the top free surface of the metal:due to the Bragg reflection of ballistic electrons from the Brillouin Zone

boundaries.

5A. Shakouri; 10/15/2012

Y. Ezzahri and A. Shakouri; Phys. Rev. B, 79, 184303, (2009)

N Di i l=0.5

)

Non Dimensional Variables=1

FourierShastry

= )

=5=10

Shastry predicts temperature wavefronts similar to Cattaneo (ballistic heat

6A. Shakouri; 10/15/2012

Y. Ezzahri and A. Shakouri; Journal of Heat Transfer, 2011equation) but the momentum-dependence of relaxation time can affect the shape.

Total energy density propagation K(t,x) in a thermoelectric material P(t)

when there is delta-function excitation P(t)

K(t,x)

Heat Diffusion DQCharge Diffusion DC

coupling factor betweenh d d it

*

*

Z Tcharge and energy density

Z* high frequency limit of figure of merit

*1 Z T

7A. Shakouri; 10/15/2012

B. S. Shastry, Rep, Prog, Phys 72, 016501, (2009)

Observation of second soundObservation of second soundObservation of second soundObservation of second soundin bismuthin bismuth

V Narayanamurti and R C Dynes Phys Rev Lett 28 1461 (1972)V. Narayanamurti and R. C. Dynes, Phys. Rev. Lett, 28, 1461, (1972).

Experiment principleYounes Ezzahri

8A. Shakouri; 10/15/2012

Results Propagation along the C3 axis

Younes EzzahriYounes Ezzahri

9A. Shakouri; 10/15/2012

Observation of a ballistic transverse phonon propagation.Transition to second sound.

Narayanamurti, and Dynes PRLand Dynes, PRL 28, 1461 (1972)

10A. Shakouri; 10/15/2012

McNelly, et al, Phys. Rev. Lett. 24, 100 (1970).

11A. Shakouri; 10/15/2012

Jackson and Walker, Phys. Rev. B, 1428 (1971)

Different macroscopic equations Different macroscopic equations describing heat wave propagationdescribing heat wave propagation

Cattaneo-Vernotte type: Telegraph equation

T2

22

01

V

VTC div Qt

Q Q T

T T Tt t C

tt

T

Jeffreys type

22 2

11

2

0

1

V V

VTC div Qt

Q Q T T T TTt tt tt C

TC

1 2

V Vt tt tt CC

1: Effective thermal conductivity.2: Elastic thermal conductivity.

12A. Shakouri; 10/15/2012

2 y

Younes Ezzahri (UCSC, now at Univ. Poitier)

Macroscopic equations of Guyer and Krumhansl

T2 2

2 222

22

2

0

1 25

1 33 5

3

NR

V

N V

TC div Qt

vQ v T Q Q div Q

T T v TT vt C

tt t

53 Rt

Jeffrey’s type equation with and effective thermal diffusivity:

211

35 N

V

vC

Vi i f hViscosity of phonon gas

13A. Shakouri; 10/15/2012

Younes Ezzahri (UCSC, now at Univ. Poitier)

Cummulative ph vs. mfp at 300K100

80

100Si, 300K

MD cal. [1]

This work60

80

ph (%

)

40

um.

p

Phonons with mfp > 1 um contribute 50%

20cu 1 um contribute 50% of heat conduction.

10-310-210-1100 101 102 103 1040

( m)

14A. Shakouri; 10/15/2012

ph( m)

15A. Shakouri; 10/15/2012

50 10050-100 nm

0.2-5 m0.2 5 m

16A. Shakouri; 10/15/2012

David G. Cahill, Rev. Sci. Instrum. 75, 5119 (2004).

Time Domain ThermoReflectance Modulated & delayed femtosecond laser pulse used as a PumpModulated & delayed femtosecond laser pulse used as a Pump.A Probe beam measures reflectivity variation on the surfaceThe lock-in amplifier gives the In-phase (Vin) and Out-of-phase (Vout) signals.

17A. Shakouri; 10/15/2012

Gilles Pernot (UCSC, Bordeaux)

Thin Film Thermal Characterization

Thermal decay => KSi/SiGe(30%)=8.1W/mK KSi/SiG (60%)= 2 8W/mKKSi/SiGe(60%)= 2.8W/mK

Acoustic echoesAcoustic echoes

Y. Ezzahri et al. Appl. Phys. Letters 2005

Metal film is heated by laser pulse and it acts both as a heat source and a transducer (creates acoustic waves) It can

18A. Shakouri; 10/15/2012

source and a transducer (creates acoustic waves). It can characterize thermal interface resistances as well as interface quality (acoustic mismatch).

Phonon echoes in SiGe superlatticesPhonon echoes in SiGe superlattices

Superlattice

19A. Shakouri; 10/15/2012

Phonon minibands in SiGe superlatticesPhonon minibands in SiGe superlattices

Measured phonon

spectrum

527GHz527GHz

Calculation

20A. Shakouri; 10/15/2012

R/R(a.u.)

Pump probe delay

21A. Shakouri; 10/15/2012

Optical sampling by TDTR

Pump

Probe

T(t)

Response

22A. Shakouri; 10/15/2012

Gilles Pernot (UCSC/Bordeaux)

Optical sampling by TDTR30

ifier

(μV

)

Acoustic echo

20

k-in

am

pli

Acoustic echo

David G. Cahill

10

0ps 10ps 20ps

In-phase Vby th

e lo

ck Cahill, Rev. Sci.

Instruments 75 5119

0

In-phase Vin

als

give

n b

Out-of-phase Vout

75, 5119 (2004)

0 1 2 3

0

Sign

a pout

23A. Shakouri; 10/15/2012

0 1 2 3

Pump-Probe delay(ns)

24A. Shakouri; 10/15/2012

25A. Shakouri; 10/15/2012

10

6

789

InGaAsm

K)

4

5

6

0.3%ErAsity (W

/m

3

0.3%ErAs

ondu

ctiv

i

1.8%ErAs2

herm

al C

o

1000 nm 200 nm

1 101

Th

26A. Shakouri; 10/15/2012

Frequency (MHz)

Thermal penetration length

1000

m)

Th l i d h12kl f

600

800

on d

epth

(nm Thermal penetration depth 2 ml f

C

400

600

measurementsal p

enet

ratio

200

measurements

Ther

ma

SAMPLE THICKNESS (UDEL)

SAMPLE THICKNESS (UCSB)

1 100

Modulation Frequency (MHz)

( )

27A. Shakouri; 10/15/2012

Modulation Frequency (MHz)

Gilles Pernot (UCSC)

28A. Shakouri; 10/15/2012

29A. Shakouri; 10/15/2012

Thermal Boundary Conductance Thermal Boundary Conductance

A B.

Q.

Q Qh.

T TA

h= Thermal boundary conductance

1/h = Thermal b d i t

30A. Shakouri; 10/15/2012

boundary resistance

Transmission ProbabilitiesTransmission Probabilities

1

l t1 t2 lt1

t21

l t1 t2 lt1

t222

l

t1t2

2

l

t1t2

2

4 ZZ )]()([ N2

21

2121 )(

4ZZZZ

)]()([

)]()([)(

,,

,2,2

21

jijiji

jjj

cN

cN

31A. Shakouri; 10/15/2012

iii cZ , ji

Interface thermal conductanceInterface thermal conductance

32A. Shakouri; 10/15/2012

D( )Debye approximation

(100)

LA

TA

ddcNjQ jj )cos()sin()(),,()(21

)2(

,1,12121

. max

jSpeedPhonon

j

DensityPhonon

j

vityTransmissiEnergyPhonon2 0 0

..

33A. Shakouri; 10/15/2012

)()()(

21

221121

TTATQTQh

Thermal Resistance of Metal-Nonmetal InterfacesThermal Resistance of Metal-Nonmetal Interfaces

M t lMetal Non-Metal

Rep

Electron

Rpp

Ph Ph

DMM

ep

Phonon Phonon

pepppep Gkhhh

hhh ;

Ph th l

ppep

pepppep

RRR

hh Phonon thermal conductivity of metal

1

mKWk

KmWG p3

1716 2010;1010hep

1 Ev

vCk

p

pppp 31

34A. Shakouri; 10/15/2012

KmGWhep 24.13.0Tep p

**

Th 1

T

35A. Shakouri; 10/15/2012

Extremely low thermal conductivity Extremely low thermal conductivity

WSe2

36A. Shakouri; 10/15/2012

37A. Shakouri; 10/15/2012

SummarySummaryA. Shakouri 8/15/2012

• Ballistic-diffusive heat transportBallistic diffusive heat transport– Cattaneo’s and Shastry’s formalisms– Second sound

• Phonon mean free paths• Time-domain thermoreflectance

– Picosecond acoustics– Frequency-dependent thermal conductivity

• Interface thermal resistance (DMM, AMM, experimental results)

Cahill, Ford, Goodson, Mahan, Majumdar, Maris, Merlin, and Phillpot, "Nanoscale thermal transport," J. Appl. Phys. 93, 793 (2003)

D id G C hill R S i I t 75 5119 (2004)

38A. Shakouri; 10/15/2012

David G. Cahill, Rev. Sci. Instrum. 75, 5119 (2004)

Y. Ezzahri and A. Shakouri; Phys. Rev. B, 79, 184303, (2009)

39A. Shakouri; 10/15/2012