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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
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.
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)
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
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)
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)
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
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
Extremely low thermal conductivity Extremely low thermal conductivity
WSe2
36A. 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)