Phonon Scattering Processes Affecting
Thermal Conductance at Solid-solid
Interfaces in Nanomaterial Systems
Patrick Hopkins
University of Virginia
Department of Mechanical and Aerospace Engineering
March 10, 2008
Microscale Heat Transfer Lab – University of Virginia
Rocket nozzle
107 W/m2
Nuclear reactor
106 W/m2
Hot plate
105 W/m2
Equivalent power density [W/m2]
Moore’s Law
45 nm
100 nm
500 nm
Transistor size
Microscale Heat Transfer Lab – University of Virginia
Field effect transistors
Heat
generated
Rejected heat
Thermal management is highly dependent on the
boundary between materials
Microscale Heat Transfer Lab – University of Virginia
Thermoelectrics
S T
ZT 
k
2
ZT = figure of merit
S = Seebeck coefficient
σ = electrical conductivity
k = thermal conductivity
T = temperature
Microscale Heat Transfer Lab – University of Virginia
Thermal boundary resistance
q  hBDDT
hBD  1
RBD
Temperature
•Thermal boundary resistance creates a temperature drop, DT,
across an interface between two different materials when a heat
flux is applied.
•First observed by Kapitza for a solid and liquid helium interface in
1941.
T
x
A typical resistance of
10-9-10-7 m2K/W
is equivalent to
~ 0.15-15 mm Si
~ 1-100 nm SiO2
Mismatch in materials causes a resistance to heat flow across an interface.
Microscale Heat Transfer Lab – University of Virginia
Two types of interface resistance
Thermal Boundary Resistance
• Due to difference in the acoustic
properties: Phonon reflection at the
interface
• Electron-phonon interaction
• Present even in the case of perfect
contact with no roughness
• Microscopic quantity
Thermal Contact Resistance
• Important for bulk surfaces
• Macroscopic quantity
• Due to imperfect contact or voids
in microstructure
Voids,
imperfect contact
Tcold
Thot
Tcold
Thot
.
A
Q
Thot
.
B
Q
hBD= thermal boundary conductance
1/hBD = thermal boundary resistance
.
Q
A
B
Thot
DT
DT
Tcold
Tcold
Distance
.
Q
Distance
Microscale Heat Transfer Lab – University of Virginia
Major research objectives
• the role of interface disorder on interfacial
heat transfer
• the effects of different phonon scattering
mechanisms on interfacial heat transfer
Microscale Heat Transfer Lab – University of Virginia
Outline of presentation
• Theory of phonon interfacial transport
• Measurement of interfacial transport with the
transient thermoreflectance (TTR) technique
• Influence of atomic mixing on interfacial phonon
transport
• Influence of high temperatures on interfacial
phonon transport
• Summary
Microscale Heat Transfer Lab – University of Virginia
Thermal conduction in bulk materials
Thermal conduction
T
z
T
Microscopic picture
L
T
q z  k
q
z
Z
k = thermal conductivity [Wm-1K-1]
q = thermal flux [Wm-2]
l = mean free path [m]
phonon-phonon scattering
length in homogeneous material
What happens if l is on the order of L?
Microscale Heat Transfer Lab – University of Virginia
Thermal conduction in nanomaterials
T
Microscopic picture
of nanocomposite
T
Z
Ln < ln
keffective of nanocomposite does not
depend on phonon scattering in the
individual materials but on phonon
scattering at the interfaces
q=hBDDT
Z
hBD = thermal boundary
conductance [Wm-2K-1]
Change in material properties gives rise to hBD
Microscale Heat Transfer Lab – University of Virginia
Theory of hBD
Phonon flux transmitted across interface
1
2
I
Ij 
q
1
D j ( ) f ( , z, t )v j
4
q1  hBDDT
q1  

c, j
 mI j dd
j   4 0
 / 2 1, j
c
1
q1   
2 j 0
 D1, j   f  , z, t v1, j1, j  , j,   cos sin  dd  hBD DT
0
Phonon
distribution
Spectral phonon
density of states
[s m-3]
Phonon
energy
[J]
Projects phonon
Phonon
Phonon
transport
speed interfacial perpendicular to
[m s-1] transmission
interface
Microscale Heat Transfer Lab – University of Virginia
Diffuse scattering
q1 
cutoff

1
 /2 ,j
1
 
2 j 0
 D1, j   f , z, t v1, j1, j  , j,  cos sin  dd  hBD DT
0
Diffuse Mismatch Model (DMM)
E. T. Swartz and R. O. Pohl, 1989, "Thermal boundary resistance,” Reviews
of Modern Physics, 61, 605-668.
diffuse scattering – phonon “looses memory” when scattered
hBD,1 
1


4 j T
1cutoff
,j
 D   f , T v
1, j
1d
1, j
0
• Scattering completely diffuse
• Elastically isotropic materials
• Single phonon elastic scattering
T > 50 K and realistic interfaces
Averaged properties in different
crystallographic directions
Is this assumption valid?
Microscale Heat Transfer Lab – University of Virginia
-3
Debye density of States [m ]
Maximum hBD with elastic scattering
Phonon radiation limit (PRL)
Same assumptions as DMM
DOS side 1 (softer) in DMM
DOS side 2 (harder) in PRL
1

DMM hBD,1  
4 j T
PRL hBD,1 
1


4 j T

cutoff
1
D 
 c
kB
2cutoff
Frequency [Hz]
1cutoff
,j
 D1, j   f , T v1, j1d
0
1cutoff
,j
 D2, j   f , T v2, j d
0
Microscale Heat Transfer Lab – University of Virginia
DMM and PRL calculations
Microscale Heat Transfer Lab – University of Virginia
Outline of presentation
• Theory of phonon interfacial transport
• Measurement of hBD with the TTR technique
• Influence of atomic mixing on hBD
• Influence of high temperatures (T > D) on hBD
• Summary
Microscale Heat Transfer Lab – University of Virginia
Transient ThermoReflectance (TTR)
Verdi V10
l = 532 nm
10 W
RegA 9000
Mira 900
tp ~ 190 fs
tp ~ 190 fs @ 76 MHz
l = 720-880 nm
16 nJ/pulse
Verdi V5
l = 532 nm
5W
single shot - 250 kHz
4 mJ/pulse
Probe Beam
Delay ~ 1500 ps
l/2 plate
Beam Splitter
Sample
Dovetail Prism
Lenses
Detector
Polarizer
Pump Beam
Variable
ND Filter
Acousto-Optic
Modulator
Lock-in Amplifier
Automated Data
Acquisition System
Microscale Heat Transfer Lab – University of Virginia
Transient ThermoReflectance (TTR)
Free Electrons Absorb Laser Radiation
Ballistic Electron Transport
PROBE
HEATING
“PUMP”
Electrons Transfer
Energy to the Lattice
Thermal Diffusion
by Hot Electrons
Electron-Phonon
Coupling (~2 ps)
Thermal Equilibrium
Thermal Diffusion within Thin Film
FILM
Thermal Diffusion
Thermal Conductance Across the
Film/Substrate Interface
Thermal Diffusion (~100 ps)
Thermal Boundary
Conductance (~2 ns)
SUBSTRATE
Thermal Diffusion within
Substrate
Substrate Thermal
Diffusion (~100 ps – 100 ns)
Focus of current analysis
Microscale Heat Transfer Lab – University of Virginia
TTR data
Free Electrons Absorb Laser Radiation
Thermal Conductance across the
Film/Substrate Interface
Thermal Boundary
Conductance (~1-10 ns)
d f (t )
dt
50 nm Cr/Si
hbd

[ s (0, t )   f (t )]
dC f
 s ( x, t )
 2 s ( x, t )
 s
t
x 2
 s (0, t )
 ks
 hBD [ f (t )   s (0, t )]
x
 s (, t )
0
x
 f ,s 
Microscale Heat Transfer Lab – University of Virginia
T f , s  T0
T f (0)  T0
DMM compared to experimental data
Stevens, Smith, Norris, JHT, 2005
Lyeo, Cahill, PRB, 2006
Stoner, Maris, PRB, 1993
New data
Goal: investigate the over- and under-predictive trends of the
DMM based on the single phonon elastic scattering assumption
Microscale Heat Transfer Lab – University of Virginia
Outline of presentation
• Theory of phonon interfacial transport
• Measurement of hBD with the TTR technique
• Influence of atomic mixing on hBD
• Influence of high temperatures (T > D) on hBD
• Summary
Microscale Heat Transfer Lab – University of Virginia
DMM assumptions
DMM Assumption
Realistic Interface
Slight changes in deposition conditions can give rise to
different elemental compositions around solid interfaces
Microscale Heat Transfer Lab – University of Virginia
AES depth profiles
1
Cr/Si mixing layer
9.5 nm
Elemental fraction
Elemental Fraction
Elemental fraction
0.8
Cr-1: no backsputter
0.6
Si change
9.7 %/nm
0.4
0.2
0
1
30
40
50
60
Cr/Si mixing layer
14.8 nm
0.8
0.6
Cr-2: backsputter
Si change
16.4 %/nm
0.4
0.2
0
30
40
50
60
Depth under Surface [nm]
Microscale Heat Transfer Lab – University of Virginia
Results from AES data
Sample
ID
Cr-1
Cr-2
Cr-3
Cr-4
Cr-5
Cr-6
Cr Film
Thickness
[nm]
38 ± 2.1
37 ± 0.4
35 ± 0.5
35 ± 2.8
39 ± 0.5
45 ± 0.5
Mixing Layer
[nm]
9.5 ± 0.6
14.8 ± 1.0
11.5 ± 0.7
10.8 ± 0.8
5.8 ± 0.5
7.0 ± 0.4
Slope of Si in
Beginning of Mixing
Layer [%/nm]
9.7 ± 0.7
16.4 ± 0.7
16.6 ± 1.0
7.4 ± 1.0
24.1 ± 1.0
28.1 ± 1.2
Microscale Heat Transfer Lab – University of Virginia
TTR testing
P. E. Hopkins and P. M. Norris, Applied Physics Letters 89, 131909 (2006).
Microscale Heat Transfer Lab – University of Virginia
hBD results
Decreasing hBD with
increasing mixing
layer thickness
DMM predicts a constant hBD = 855 MWm-2K-1
Microscale Heat Transfer Lab – University of Virginia
Virtual crystal DMM
The disordered region is replaced by a homogenized virtual crystal
of thickness Dint having effective properties based on the disordered
medium with MFP= lint.
RBD 
1
hBD

Dint
lint
Dint
T. E. Beechem, S. Graham, P. E.
Hopkins, and P. M. Norris, Applied
Physics Letters 90, 054104 (2007)
lint
R
1VC
pp, j

2
 RVC
 R pp
pp, j
multiple scattering
events from interatomic
mixing
Microscale Heat Transfer Lab – University of Virginia
Virtual crystal DMM
RBD 
1
hBD

Dint
lint
Dint
lint
1
1
Rep 

k pG hep
G = electron-phonon
coupling factor
Majumdar and Reddy, APL, 2006
R
1VC
pp, j

2
 RVC
 R pp
pp, j
multiple scattering
events from interatomic
mixing
In well-matched material
systems such as Cr on Si,
Rpp is very small and on the
same order as Rep, so this
additional resistance must be
considered and added in
parallel with Rpp.
Microscale Heat Transfer Lab – University of Virginia
Virtual crystal DMM
RBD 
1
hBD

Dint
lint
Dint
lint
Rep 
Majumdar and Reddy, APL, 2006
R
1VC
pp, j

2
 RVC
 R pp
pp, j
multiple scattering
events from interatomic
mixing
1
1

k pG hep

 1
D
hBD  
 int
 k p G lint




 
1
1



1VC
VC  2 
  hpp, j
 h pp, j  
 j
 
j
Microscale Heat Transfer Lab – University of Virginia
1
VCDMM
DMM predicts hBD that is
almost 8 times larger than
that measured on the
samples and no dependence
on mixing layer thickness
or composition.
The VCDMM calculations
are within 18% of the
measured values and show
the same trend with mixing
layer thickness as the
measurements.
Microscale Heat Transfer Lab – University of Virginia
Summary
Investigate the role of interface disorder on interfacial heat transfer
• Examined the effects of interfacial properties on hBD in the
acoustically matched Cr/Si system with TTR
• DMM predicts hBD 855 MWm-2K-1 at room temperature
• Measured data varies from 100-200 MWm-2K-1, depending
on deposition conditions
• Multiple phonon elastic scattering could cause this overprediction of the DMM
• DMM only takes into account single scattering events
• DMM assumes a perfect interface, but interface disorder
will increase the scattering thus decreasing the hBD
• VCDMM is introduced and predicts same values and
trends for Cr/Si at room temperature as experimental data
Microscale Heat Transfer Lab – University of Virginia
Summary
The presence of an interfacial
mixing region causing multiple
elastic scattering events which are
not accounted for and may be the
cause of the overestimation of the
DMM in well matched material
systems with Debye temperature
ratios close to one.
Stevens, Smith, Norris, JHT, 2005
Lyeo, Cahill, PRB, 2006
Stoner, Maris, PRB, 1993
New data
Goal: investigate the over- and under-predictive trends of the
DMM based on the single phonon elastic scattering assumption
Microscale Heat Transfer Lab – University of Virginia
Outline of presentation
• Theory of phonon interfacial transport
• Measurement of hBD with the TTR technique
• Influence of atomic mixing on hBD
• Influence of high temperatures (T > D) on hBD
• Summary
Microscale Heat Transfer Lab – University of Virginia
Single phonon elastic scattering
hBD from DMM limited by f1
 D   f , T v
1, j
1d
1, j
0
 c
D 
kB
1
f 
  
  1
exp 
 k BT 
f=T/D
f
1

hBD,1  
4 j T
1cutoff
,j
Linear in
classical regime
(T>D)
*Kittel, 1996, Fig. 5-1
Microscale Heat Transfer Lab – University of Virginia
Single phonon elastic scattering
Elastic scattering – hBD is a function of f/T in lower D material
1

hBD,1  
4 j T
1cutoff
,j
 D1, j   f , T v1, j1d
0
 c
D 
kB
DAl=428 K
f
DMM Predictions
DPb=105 K
T/D
Microscale Heat Transfer Lab – University of Virginia
Molecular dynamics simulations
Lennard-Jones Potential with Different Atomic Sizes
Kr/Ar Superlattice Nanowire
2
Debye Temperature Ratios
R=0.2
h *BD /h *BD (T* =0.25)
1.6
R=0.2 trendline
R=0.5
Linear
(R=0.2)
Linear
(R=0.5)
1.2
0.8
0.4
R=0.5 trendline
0
0
0.1
0.2
0.3
0.4
0.5
Temperature [T *]
Stevens, Zhigilei, and Norris, IJHMT, 2007
Chen, Li, Yang, Wu, Lukes and Majumdar,
Physica B, 2004
Computational results indicate a linear increase in conductance
(decrease in resistance) with temperature.
Microscale Heat Transfer Lab – University of Virginia
Mismatched samples at low temperatures
Lyeo and Cahill, PRB, 2006
Stoner and Maris, PRB, 1993
Microscale Heat Transfer Lab – University of Virginia
hBD results at temperatures above D of
the softer material
Pt/Al2O3
Pt/AlN
P. E. Hopkins, R. J. Stevens, and P. M. Norris, To appear in the Journal of Heat Transfer, HT (2008).
Microscale Heat Transfer Lab – University of Virginia
Analysis
• Linear trend in MDS in classical regime (T>>D)
• MDS calculates hBD making no assumption of elastic scattering in
interfacial phonon transport
• Several samples show linear hBD trends around classical regime
DMM
f/T
hBD 
Film (Pb)
1c, j
1
f ( , T )
v



D
(

)
d
 1, j  1 1, j
4 j
T
0
JOINT FREQUENCY DMM
hBD 
1
 vmod, j
4 j
c
mod,
j

1Dmod, j ( )
0
f ( , T )
d
T
P. E. Hopkins and P. M. Norris, Nanoscale and Microscale
Thermophysical Engineering 11, 247 (2007)
Microscale Heat Transfer Lab – University of Virginia
DMM vs. JFDMM
DMM
JFDMM
Microscale Heat Transfer Lab – University of Virginia
Summary
Investigate the effects of different phonon scattering mechanisms
on interfacial heat transfer
• Measured hBD at different metal-dielectric interfaces with a
range of acoustic similarity
• Observed linear trend in hBD around D
• Evidence of inelastic scattering – not predicted with DMM
• JFDMM takes into account substrate phonons – and
provides better agreement with experimental data
Microscale Heat Transfer Lab – University of Virginia
Summary
The presence of inelastic scattering
events, which add an additional channel
of interfacial energy transport may be
the cause of the underestimation of the
Stevens,
Norris, JHT, 2005
DMM
in Smith,
mismatched
material systems
Lyeo, Cahill, PRB, 2006
with
distinctly different Debye
Stoner, Maris, PRB, 1993
New data
temperatures.
Goal: investigate the over- and under-predictive trends of the
DMM based on the single phonon elastic scattering assumption
Microscale Heat Transfer Lab – University of Virginia
Outline of presentation
• Theory of phonon interfacial transport
• Measurement of hBD with the TTR technique
• Influence of atomic mixing on hBD
• Influence of high temperatures (T > D) on hBD
• Summary
Microscale Heat Transfer Lab – University of Virginia
Conclusions
Investigate the role of interface disorder on interfacial
heat transfer
• Determined that interfacial mixing can play a role in phonon
transport by inducing multiple phonon scattering events
• Accurately described with VCDMM taking into account e-p
resistance
Investigate the effects of different phonon scattering
mechanisms on interfacial heat transfer
• Inelastic scattering contributes to hBD at temperatures close to
D of the softer material where substrate phonon population is
still quantum mechanically increasing
• Developed JFDMM to take into account some portion of
inelastic scattering
Microscale Heat Transfer Lab – University of Virginia
Impact
How does the knowledge of phonon scattering affect
nanoapplications?
S 2T
ZT 
k
Microscale Heat Transfer Lab – University of Virginia
Acknowledgments
• Pamela Norris, my doctoral advisor and head of the Microscale
Heat Transfer laboratory at UVA
• Funding from the National Science Foundation (NSF) Graduate
Research Fellowship Program (GRFP)
• Funding from the Virginia Space Grant Consortium (VSGC)
• Collaborators: Leslie Phinney (Sandia), Robert Stevens (RIT),
Samuel Graham (GaTech), Thomas Beechem (GaTech) Rob
Kelly (UVA), Avik Ghosh (UVA), Mikiyas Tsegaye (UVA),
David Cahill (UIUC), John Hostetler (Trumpf Photonics), Mike
Klopf (Jefferson Lab), Vickie Connors (NASA Langley)
• Microscale Heat Transfer Crew – Rich Salaway, Jennifer
Simmons, John Duda, Justin Smoyer
Microscale Heat Transfer Lab – University of Virginia
Transient ThermoReflectance (TTR)
Free Electrons Absorb Laser Radiation
Focus of current analysis
Ballistic Electron Transport
PROBE
HEATING
“PUMP”
Electrons Transfer
Energy to the Lattice
Thermal Diffusion
by Hot Electrons
Electron-Phonon
Coupling (~2 ps)
Thermal Equilibrium
Thermal Diffusion within Thin Film
FILM
Thermal Diffusion
Thermal Conductance across the
Film/Substrate Interface
Thermal Diffusion (~100 ps)
Thermal Boundary
Conductance (~2 ns)
SUBSTRATE
Thermal Diffusion within
Substrate
Substrate Thermal
Diffusion (~100 ps – 100 ns)
Focus of previous analysis
Microscale Heat Transfer Lab – University of Virginia
Electron-phonon (e-p) nonequlibrium
440
conducted Energy transferred
Energy stored in e- Energy
electron temperature
from e- system to l Energy deposited
through e- system
system
420
system
into e- system


Temperature
Te 400
Te 
 e Te
 380 ke (Te , Tp )
  G Te  T p  S ( z , t )
t 360
z
z 
Electron-phonon
PARABOLIC
Tp
340
coupling factor
TWO-STEP
C

G
T

T
p
e
p
320
t
MODEL (PTS)
Energy gained by l

300
*Anisimov, 1974

lattice temperature
Energy stored
in l 100
50
systemTime [fs]
0
150from esystem
system
time
z
Microscale Heat Transfer Lab – University of Virginia
Relate temperature to reflectance
 e Te


Te
 G Te  T p  S (t )
t
Cp
T p
t

 G Te  T p

DR/R = aDTe + bDTl – only valid for DTe < 150 K
• Test at fluences up to 15 J m-2
• DTe in Au of up to ~ 4000 K
• ITT 2.4 eV > 1.55 eV TTR energy
Christensen, PRB, 1976
Intraband reflectance model Valid
for all electron temperatures
Smith and Norris, APL, 2001
Microscale Heat Transfer Lab – University of Virginia
Measure G in Au with TTR
20 nm Au/glass
20 nm Au/glass
Different e-p equilibration curves for different fluences
But G should be a material property????
 Aee

G (Te , T p )  GRT 
Te  T p  1
 Bep



Hopkins and Norris, App. Surf. Sci., 2007
Microscale Heat Transfer Lab – University of Virginia
Single phonon elastic scattering
Simplifies transmission coefficient
 D   f , T v
1, j
1d
-3
1

4 j
Debye density of States [m ]
q1 
icutoff
,j
1, j
0
1  R2  1  1
q1  q2
D1/ 2, j ( ) 
2
2 2 v13/ 2, j
 2cutoff
,j
 
1 ( ) 
j
1cutoff
,j
 
j
0
0

d
2 3
2 v2, j
1cutoff
2cutoff
Frequency [Hz]
2
v2 , j
2
v1, j
d  
2 3
2 v1, j
j
 2cutoff
,j

0
v2 , j
2
d
2 3
2 v2, j

2
v
 2, j
j
2
1, j
v
 v
j
Microscale Heat Transfer Lab – University of Virginia
j
2
2, j
 1
hBD results for Al/Al2O3
Stoner and Maris, PRB, 1993
P. E. Hopkins, et al., International Journal of Thermophysics 28, 947 (2007)
Microscale Heat Transfer Lab – University of Virginia
Thermal model
1
Change in temperature
across a 50 nm Cr film
on Si substrate interface
D R [a.u.]
0.8
0.6
h BD = 1.0x108 W m-2 K
0.4
h BD = 2.0x108 W m-2 K
0.2
h BD = 3.0x108 W m-2 K
0
0
200
400
600
800
1000
1200
Time [ps]
Microscale Heat Transfer Lab – University of Virginia
1400
Resolving TBC with TTR
Al/Al2O3 interfaces
kf = 237 Wm-1K-1
hBD = 2.0 x 108 Wm-2K-1
Resolving TBC with TTR

ti 
Cf d
hBD
t f hBD d

<1
ti
kf
1.5
20
15
Time constant [ns]
tf 
d
2
10
1
5
0
0
250
500
ti
0.5
750
1000
tf
0
0
25
50
Film thickness [nm]
Microscale Heat Transfer Lab – University of Virginia
75
100
Thermal Model
Lumped capacitance
substrate
film
Bi<<1
Al/Al2O3 interfaces
kf = 237 Wm-1K-1
hBD = 2.0 x 108 Wm-2K-1
d =75 nm< 120 nm
Bi = 1
1.4
Time constant [ns]
T
0.1k f
hBD d
Bi 
< 0.1  d <
kf
hBD
Bi>>1
1.2
1
0.8
0.6
0.4
0.2
0
0
25
50
Film thickness [nm]
x
Microscale Heat Transfer Lab – University of Virginia
75
100
Sample fabrication
Sample
ID
Cr-1
Cr-2
Cr-3
Cr-4
Cr-5
Cr-6
Backsputter
Etch
none
5 min
5 min
5 min
5 min
5 min
Heat Treat Prior
to Deposition
none
none
20 min @ 873 K
50 min @ 873 K
20 min @ 873 K
none
Deposition Notes
(on Si substrates)
50 nm Cr @ 300 K
50 nm Cr @ 300 K
50 nm Cr @ 300 K
50 nm Cr @ 300 K
50 nm Cr @ 573 K
10 nm of Cr at 300 K;
heating to 770 K;
40 nm of Cr at 300 K
Microscale Heat Transfer Lab – University of Virginia
Interface characterization
Auger electron spectroscopy (AES)
Electron bombardment
e- [3 keV]
Ionization
Relaxation and
Auger emission
Monitor
energy
Vacuum
Energy
Higher
levels
Core
level
Microscale Heat Transfer Lab – University of Virginia
detector
AES depth profiling
e- gun
O2
Ar+ gun
C
dN/dE
Cr
Si
Energy [eV]
Microscale Heat Transfer Lab – University of Virginia
AES depth profile
1
Cr/Si mixing region
Cr
Elemental fraction
0.8
Si
0.6
O2
0.4
C
0.2
0
0
10
20
30
40
50
Depth into film [nm]
Microscale Heat Transfer Lab – University of Virginia
60
Energetic Electrons
Doped Semiconductors (T = Troom)
Intraband excitation – E<Eg
E
Conduction
Band
Intraband
Ef
Interband excitation – E>Eg
Interband
Eg
k
Valence Band
Microscale Heat Transfer Lab – University of Virginia
Energetic Electrons
Metals (T = Troom)
E
Noble Metal
Electron transport – Noble metals
s
Ef
ITT
d
Electron transport – Transition metals
s
Transition
Metal
Ef
ITT
d
k
Microscale Heat Transfer Lab – University of Virginia
* Swaminathan and Macrander, 1991
Band Structures
GaAs
Nickel
Gold
s
c
v
s
d
d
*Christensen, 1976
*Weiling and Callaway, 1982
Microscale Heat Transfer Lab – University of Virginia
Gold
Reflectance at ITT
*Sun, et al., 1994
*Fatti et al., 2000
Microscale Heat Transfer Lab – University of Virginia
Silver
G Measurement in Gold
G @ energies lower than ITT
G @ energies around ITT
s
*Smith and Norris, 2001
Ef
*Hohlfeld et al., 2000
d
Microscale Heat Transfer Lab – University of Virginia
Transient Thermoreflectance Data
Microscale Heat Transfer Lab – University of Virginia
Thermal Model
Te  
Te 
CT
  ke (Te , Tl )
  GTe  Tl   S ( x, t )
t x 
x 
'
e e
k
 thermal 
 17nm
G
k = 91 Wm-1K-1
G = 3.6 x 1017 Wm-3K-1
1.3 eV
Normalized Reflectance D R/R [arb. units]
Tl
Cl
 GTe  Tl 
t
20 nm
50 nm
-1
0
1
2
3
Time [ps]
Microscale Heat Transfer Lab – University of Virginia
4
5
G Measurements
R = aDTe + bDTl
30 nm Ni/Si
1.3 eV
G = 5.8 x 1017 Wm-3K-1
1.55 eV
G = 3.7 x 1017 Wm-3K-1
Microscale Heat Transfer Lab – University of Virginia
Analysis
•G measurement at 1.3 eV interband transition yields higher results
than measurements taken at other energies (~6.0 x 1017 Wm-3K-1)
•This study: ~3.7 x 1017 Wm-3K-1 at 1.55 eV
•Wellershoff et al., 1998: ~3.6 x 1017 Wm-3K-1 at 3.11 eV
•Previous Au measurements of G were same at ITT and other energies
1
Reflectance
•1.3 eV in Ni is not at ITT
but at a higher interband transition
lower d-band  Fermi level
Ni/Si
0.8
0.6
0.4
0.2
Ni/Glass
0
0
10
20
30
40
50
60
Thickness [nm]
Microscale Heat Transfer Lab – University of Virginia
70
80
90
100
Reflectance Model
  2

1
1
 2
D  D1  iD 2 
DTe 
DTl  i 
DTe 
DTl 
Te
Tl
Tl
 Te

D2
D1

DR 1  R
R
 
D1 
D 2 
R R  1
 2

R = aDTe + bDTl
R
a
Te
&
R
b
Tl
Microscale Heat Transfer Lab – University of Virginia
How do we define temperature?
•Temperature is an equilibrium property
•How can we define temperature when there is a nonequilibrium (TTM)?
Consider case of homogeneous heating in Au
t > tee
Temporally
•Both e- and phonon systems are in local thermal equilibrium
•When e- scatters and emits phonon, e- system redistributes into a
new temperature distribution (lower T).
What about e- substrate scattering work?
•Homogeneous heating so lumped capacitance
hBD d
Bi 
< 0.1
kf
hBDe-=1E8 Wm-2K-1
d=50 nm
ke=317
Bi = .02
very conservative
Microscale Heat Transfer Lab – University of Virginia
How do we define temperature?
•However, how can we determine temperature spatially if there is a
thermal gradient?
•Can temperature, which is an equilibrium concept, still be invoked in a
nonequilibrium process such as heat conduction? (Cahill, JAP, 2003)
Consider 1D conduction
Mean free path
#Collisions/time
Box is 4l long
Cannot resolve
local temperature
Can resolve local
temperature
*Since equilibrium is achieved
through multiple collisions
Kinetic Theory speed
C
l

   ee  ep  AeeTe2  BepTl
at Te=Tl=300, ~1E13, C ~1E4 m/s
l~1 nm
~15 nm
Microscale Heat Transfer Lab – University of Virginia
Joint frequency DMM
DMM
hBD 
JFDMM
1c, j
1
f ( , T )
d
 v1, j  1D1, j ( )
4 j

T
0
hBD 
1
 vmod, j
4 j
c
mod,
j
 1Dmod, j ( )
0
vmod, j  1v1  2v2

1/ 3
c
2
1N1   2 N 2 
mod,

v
6

j
mod, j
Dmod, j 
2
2 2vmod, j 3
Weighting factor  is simply a
percentage of the composition of
each material in the unit volume
(M=atomic mass)
(N=number of oscillators per unit volume)
c
  mod,
j
1 
N1
M1
N2
N1
M1  M 2
N2
Microscale Heat Transfer Lab – University of Virginia
n( , T )
d
T
Analysis
•
Inelastic scattering – DMM does not account for this
•
Data at solid-solid interfaces taken at temperatures around Debye
temperature show linear trend
•
DMM predicts flattening of predicted hBD around Debye temperature
•
Accounting for substrate phonons in DMM improves prediction (JFDMM)
PRL
DMM
hBD 
1c, j
1
f ( , T )
v



D
(

)
d
 1, j  1 1, j
4 j
T
0
hBD 
1


4 j T
1cutoff
,j
 D2, j   f , T v2, j d
0
JFDMM
hBD 
1
 vmod, j
4 j
Inelastic phonon radiation limit (IPRL)
Is there
an upper limit to
2cutoff
,j
f ( , T )
1




D
(

)
d

inelastic
 1
mod, j
hBD  
D2, j   f  , T v2, j d
scattering?
T
c
mod,
j
0
4 j T
0
Microscale Heat Transfer Lab – University of Virginia
IPRL
el
inel
hBD (T )  hBD
 hBD
(T )
PRL
IPRL
hBD (T )  AhBD
 B(T )hBD
(T )
Microscale Heat Transfer Lab – University of Virginia
Elastic and inelastic contributions
Lim
PRL
IPRL
hBD
(T )  AhBD
 B(T )hBD
(T )
Pb/diamond
Pb/diamond
el
hBD
A  PRL
hBD
In classical
limit
Pb/diamond
IPRL
PRL
inel
B(T )hBD
(T )  hBD (T )  AhBD
 hBD
(T )
Microscale Heat Transfer Lab – University of Virginia
Elastic and inelastic contributions
DPb/Ddiamond~0.0
5
inel
hBD
(T ) hBD (T )

1
el
PRL
hBD
AhBD
DPt/DAl2O3~0.23
Hopkins, Norris, and Stevens, Submitted to the Journal of Heat Transfer
Relative contribution to hBD of inelastic scattering compared to elastic
scattering increases with sample mismatch and with temperature
Microscale Heat Transfer Lab – University of Virginia
Future directions
Thermal testing in novel nanostructures
S 2T
ZT 
k
CNT and nanocomposites
Microscale Heat Transfer Lab – University of Virginia
Future directions
Steady state and 3 electrical resistance techniques
Hopkins and Phinney, MNHT2008-52293
B. W. Olson, S. Graham, and K. Chen, Review
of Scientific Instruments 76, 053901 (2005).
Microscale Heat Transfer Lab – University of Virginia
Future directions
NonEquilibrium Green’s Function (NEGF) modeling
Fn  M A u n  K A u n 1  u n  u n  u n 1 
F0   M B 2u0  K B u1  u0 
Microscale Heat Transfer Lab – University of Virginia
Future directions
NEGF to calculate phonon conductivity in nanostructures
from first principles
Relies on basic quantum
mechanics
No assumptions based on
scattering or transport
Can be extended to any
nanostuctures
Hopkins et al., MHT08-52244
Si wire data from: D. Li, Y. Wu, P. Kim, L. Shi, P. Yang and
A. Majumdar, 2003, "Thermal conductivity of individual
silicon nanowires," Applied Physics Letters, 83, 2934-2936.
Microscale Heat Transfer Lab – University of Virginia
Future directions
More TTR applications – electron-phonon scattering
Free Electrons Absorb Laser Radiation
Thermal Diffusion
by Hot Electrons
Electrons Transfer
Energy to the Lattice
20 nm Au/glass
 e Te


Te  
T 
  ke (Te , Tp ) e   G Te  T p  S ( z , t )
t
z 
z 
Cp
Tp
t

 G Te  Tp

Insulated boundary conditions always assumed
P. E. Hopkins and P. M. Norris, Applied Surface Science 253, 6289 (2007)
Microscale Heat Transfer Lab – University of Virginia
Future directions
More TTR applications – electron-phonon scattering
z
Different e-p equilibration
curves for different fluences
But G should be a material property
 Aee

Te  Tp   1
G (Te , Tp )  GRT 
 Bep

P. E. Hopkins, et al., Submitted to Phys Rev B.
Microscale Heat Transfer Lab – University of Virginia
Future directions
• Extend nanoscale thermophysics to realistic low dimensional
nanostructures
• Electrically based resistance techniques to measure thermal
transport and thermophysical properties of nanomaterials
• NEGF formalism for accurate modeling of real nanosystems
• TTR technique to measure electron-phonon coupling and
interfacial thermal transport
Microscale Heat Transfer Lab – University of Virginia