patrick hopkins university of virginia department of mechanical and aerospace engineering

Microscale Heat Transfer Lab – University of Virginia

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


Moore’s Law

Hot plate

105 W/m2

Transistor size

Eq

uiv

alen

t p

ow

er d

ensi

ty [

W/m

2]

Nuclear reactor

106 W/m2

500 nm100 nm

45 nm

Rocket nozzle

107 W/m2


Heat generated

Rejected heat

Field effect transistors

Thermal management is highly dependent on the boundary between materials


Thermoelectrics

k

TSZT

2

ZT = figure of meritS = Seebeck coefficientσ = electrical conductivityk = thermal conductivityT = temperature


Thermal boundary resistance•Thermal boundary resistance creates a temperature drop, T, 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.

Thq BD

x

Tem

pera

ture

T

A typical resistance of

10-9-10-7 m2K/W

is equivalent to

~ 0.15-15 m Si

~ 1-100 nm SiO2

Mismatch in materials causes a resistance to heat flow across an interface.

BDBD Rh 1


Two types of interface resistance

Thermal Boundary ResistanceThermal 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

hBD= thermal boundary conductance

1/hBD = thermal boundary resistance TThothot

TTcoldcold

T

Distance

Thermal Contact ResistanceThermal Contact Resistance• Important for bulk surfaces• Macroscopic quantity• Due to imperfect contact or

voids in microstructure

Distance

TThothot

TTcoldcold

T

A B

TTcoldcold

.

Q.

Q

TThothot

.

QA

TThothot.

Q B

Voids, imperfect contact

TTcoldcold


Major research objectives

• the role of interface disorder on interfacial heat transfer

• the effects of different phonon scattering mechanisms on interfacial heat transfer


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


Thermal conduction in bulk materialsThermal conduction

Z

k = thermal conductivity [Wm-1K-1] = thermal flux [Wm-2]

T

qz

Tkqz

z

T

q

= mean free path [m]

phonon-phonon scattering length in homogeneous material

Microscopic picture

What happens if is on the order of L?

L


Thermal conduction in nanomaterials

n

Microscopic picture of nanocomposite

Ln

keffective of nanocomposite does not depend on phonon scattering in the individual materials but on phonon scattering at the interfaces

T

Z

Z

T

hBD = thermal boundary conductance [Wm-2K-1]

Change in material properties gives rise to hBD

q=hBDT


Theory of hBDPhonon flux transmitted across interface

ThddjvtzfDq BDjj

jj

cj

sincos,,,,2

1,1

2/

0 0,1,11

,1

Spectral phonon density of states[s m-3]

Phonon distribution

Phonon energy

[J]

Phonon speed[m s-1]

Phonon interfacial transmission

Projects phonon transport perpendicular to interface

jjj vtzfDI

),,()(4

1

Thq BD1

1 2I

q

j

j

jc

ddIq

4 01

,


Diffuse scattering

Diffuse Mismatch Model (DMM) E. T. Swartz and R. O. Pohl, 1989, "Thermal boundary resistance,” Reviews of Modern Physics, 61, 605-668.

j

jjBD

cutoffj

dvTfDT

h,1

0

1,1,11, ,4

1

diffuse scattering – phonon “looses memory” when scattered

• 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?

ThddjvtzfDq BDjj

jj

cutoffj

sincos,,,,2

1,1

2/

0 0,1,11

,1


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

PRL

DMM

j

jjBD

cutoffj

dvTfDT

h,1

01,1,11, ,

4

1

j

jjBD

cutoffj

dvTfDT

h,1

0,2,21, ,

4

1

Frequency [Hz]

Deb

ye d

ensi

ty o

f Sta

tes

[m-3

]

cutoff1

cutoff2

B

cD k


DMM and PRL calculations


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


Transient ThermoReflectance (TTR)Mira 900

p ~ 190 fs @ 76 MHz

= 720-880 nm16 nJ/pulse

PolarizerDetector

Lock-in AmplifierAutomated Data

Acquisition System

Verdi V5= 532 nm

5 W

RegA 9000

p ~ 190 fssingle shot - 250

kHz4 J/pulse

Verdi V10= 532 nm

10 W

Probe Beam

Sample

Dovetail Prism

Delay ~ 1500 ps

Lenses

/2 plate Beam Splitter

Acousto-Optic Modulator

VariableND Filter

Pump Beam


Transient ThermoReflectance (TTR)

SUBSTRATE

FILM

HEATING “PUMP”

PROBE

Thermal Diffusion

Ballistic Electron Transport

Electrons Transfer

Energy to the Lattice

Thermal Diffusion by Hot

Electrons

Thermal Equilibrium

Thermal Diffusion within Thin Film

Thermal Conductance Across the

Film/Substrate Interface

Electron-PhononCoupling (~2 ps)

Thermal Diffusion (~100 ps)

Thermal BoundaryConductance (~2 ns)

Thermal Diffusion within Substrate

Substrate Thermal Diffusion (~100 ps – 100 ns)

Focus of current analysis

Free Electrons Absorb Laser Radiation


TTR data

Thermal Conductance across the


Thermal Boundary Conductance (~1-10 ns)


50 nm Cr/Si

)](),0([)(

ttdC

h

dt

tdfs

f

bdf

2

2 ),(),(

x

tx

t

tx ss

s

)],0()([),0(

tthx

tk sfBD

ss

0

0,, )0( TT

TT

f

sfsf

0

),(

x

ts


Stevens, Smith, Norris, JHT, 2005Lyeo, Cahill, PRB, 2006Stoner, Maris, PRB, 1993New data

DMM compared to experimental data

Goal: investigate the over- and under-predictive trends of the DMM based on the single phonon elastic scattering assumption






• Summary


DMM assumptions

DMM Assumption Realistic Interface

Slight changes in deposition conditions can give rise to different elemental compositions around solid interfaces


AES depth profiles

0

0.2

0.4

0.6

0.8

1

30 40 50 60

Ele

me

nta

l fr

ac

tio

n

0

0.2

0.4

0.6

0.8

1

30 40 50 60

Ele

me

nta

l fr

ac

tio

n

Cr-1: no backsputter

Cr-2: backsputter

Cr/Si mixing layer9.5 nm

Cr/Si mixing layer14.8 nm

Depth under Surface [nm]

Ele

men

tal F

ract

ion Si change

9.7 %/nm

Si change16.4 %/nm


Results from AES dataSample

IDCr Film

Thickness [nm]

Mixing Layer[nm]

Slope of Si in Beginning of Mixing

Layer [%/nm]

Cr-1 38 ± 2.1 9.5 ± 0.6 9.7 ± 0.7

Cr-2 37 ± 0.4 14.8 ± 1.0 16.4 ± 0.7

Cr-3 35 ± 0.5 11.5 ± 0.7 16.6 ± 1.0

Cr-4 35 ± 2.8 10.8 ± 0.8 7.4 ± 1.0

Cr-5 39 ± 0.5 5.8 ± 0.5 24.1 ± 1.0

Cr-6 45 ± 0.5 7.0 ± 0.4 28.1 ± 1.2


TTR testing

P. E. Hopkins and P. M. Norris, Applied Physics Letters 89, 131909 (2006).


hBD results

DMM predicts a constant hBD = 855 MWm-2K-1

Hopkins, Norris, Stevens, Beechem, and Graham, to appear in the Journal of Heat Transfer, 2008

Decreasing hBD with increasing mixing layer thickness


Virtual crystal DMM

T. E. Beechem, S. Graham, P. E. Hopkins, and P. M. Norris, Applied Physics Letters 90, 054104 (2007)

int

int

D

multiple scattering events from interatomic mixing

ppVC

jppVCjpp

BDBD RRR

D

hR 2

,1

,int

int1

The disordered region is replaced by a homogenized virtual crystal of thickness Dint having effective properties based on the disordered medium with MFP= int.


Virtual crystal DMM

int

int

D


ppVC

jppVCjpp

BDBD RRR

D

hR 2

,1

,int

int1

Majumdar and Reddy, APL, 2006

eppep hGk

R11

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.

G = electron-phonon coupling factor


Virtual crystal DMM

1

2,

1,int

int 111

j

VCjpp

j

VCjppp

BDhh

D

Gkh

int

int

D


ppVC

jppVCjpp

BDBD RRR

D

hR 2

,1

,int

int1

Majumdar and Reddy, APL, 2006

eppep hGk

R11


VCDMMDMM 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.

Hopkins, Norris, Stevens, Beechem, and Graham, to appear in the Journal of Heat Transfer, 2008


Summary

• 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 over-

prediction 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

Investigate the role of interface disorder on interfacial heat transfer



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.






• Summary


Single phonon elastic scatteringhBD from DMM limited by f1

1exp

1

Tk

f

B

B

cD k

*Kittel, 1996, Fig. 5-1

Linear in classical regime (T>D)

f=T/Df

j

jjBD

cutoffj

dvTfDT

h,1

0

1,1,11, ,4

1


Single phonon elastic scatteringElastic scattering – hBD is a function of f/T in lower D material

f

j

jjBD

cutoffj

dvTfDT

h,1

01,1,11, ,

4

1

B

cD k

DMM Predictions

T/D

DPb=105 K

DAl=428 K


Molecular dynamics simulations

Stevens, Zhigilei, and Norris, IJHMT, 2007

0

0.4

0.8

1.2

1.6

2

0 0.1 0.2 0.3 0.4 0.5

Temperature [T *]

h* B

D/ h

* BD

( T*

=0.

25)

R=0.2

R=0.5

Linear(R=0.2)Linear(R=0.5)

Debye Temperature Ratios

R=0.5 trendline

R=0.2 trendline

Chen, Li, Yang, Wu, Lukes and Majumdar, Physica B, 2004

Kr/Ar Superlattice NanowireLennard-Jones Potential with Different Atomic Sizes

Computational results indicate a linear increase in conductance (decrease in resistance) with temperature.


Mismatched samples at low temperatures

Lyeo and Cahill, PRB, 2006Stoner and Maris, PRB, 1993


hBD results at temperatures above D of the softer material

P. E. Hopkins, R. J. Stevens, and P. M. Norris, To appear in the Journal of Heat Transfer, HT (2008).

Pt/Al2O3 Pt/AlN


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

j

jjBD

cj

dT

TfDvh

,1

0,11,1

),()(

4

1

DMM

JOINT FREQUENCY DMM

j

jjBD

cj

dT

TfDvh

mod,

0mod,1mod,

),()(

4

1

P. E. Hopkins and P. M. Norris, Nanoscale and Microscale Thermophysical Engineering 11, 247 (2007)

f/

T

Film (Pb)

Subs

trat

e (d

iam

ond)


DMM vs. JFDMM

DMMJFDMM


Summary

• 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

Investigate the effects of different phonon scattering mechanisms on interfacial heat transfer



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 DMM in mismatched material systems with distinctly different Debye temperatures.


Outline of presentation

• Theory of phonon interfacial transport




• Summary


Conclusions

• 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 role of interface disorder on interfacial heat transfer

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


ImpactHow does the knowledge of phonon scattering affect

nanoapplications?

k

TSZT

2


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


Transient ThermoReflectance (TTR)

SUBSTRATE

FILM

HEATING “PUMP”

PROBE

Thermal Diffusion

Electrons Transfer



Electrons

Thermal Equilibrium

Thermal Diffusion within Thin Film

Thermal Conductance across the


Electron-PhononCoupling (~2 ps)

Thermal Diffusion (~100 ps)

Thermal BoundaryConductance (~2 ns)

Thermal Diffusion within Substrate

Substrate Thermal Diffusion (~100 ps – 100 ns)

Focus of current analysis

Focus of previous analysis

Ballistic Electron Transport



Electron-phonon (e-p) nonequlibrium

),(),( tzSTTGz

TTTk

zt

TT pe

epee

eee

pep

p TTGt

TC

Electron-phonon

coupling factor

Energy stored in e- system

Energy conducted through e- system Energy deposited

into e- system

Energy transferred from e- system to l

system

Energy stored in l system

Energy gained by l system from e-

system

z

time

PARABOLIC TWO-STEP

MODEL (PTS)

*Anisimov, 19740 50 100 150

300

320

340

360

380

400

420

440

lattice temperature

Te

mp

era

ture

Time [fs]

electron temperature


Relate temperature to reflectance pe

pp TTG

t

TC

)(tSTTGt

TT pe

eee

R/R = aTe + bTl – only valid for Te < 150 K

• Test at fluences up to 15 J m-2

• Te 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


Measure G in Au with TTR

Different e-p equilibration curves for different fluencesBut G should be a material property????

Hopkins and Norris, App. Surf. Sci., 2007

1),( pe

ep

eeRTpe TT

B

AGTTG

20 nm Au/glass 20 nm Au/glass


Single phonon elastic scattering

12,2

2,1

2,2

03,2

2

2

,2

03,1

2

2

,1

03,2

2

2

,2

1,2,1

,2

22

2)(

j jjj

jj

j jj

j jj

j jj

vv

v

dv

vdv

v

dv

v

cutoffj

cutoffj

cutoffj

Simplifies transmission coefficient

21 qq

j

jj

cutoffji

dvTfDq,

0

1,1,11 ,4

1

3,2/1

2

2

,2/1 2)(

jj v

D

121 1 R

Frequency [Hz]

Deb

ye d

ensi

ty o

f Sta

tes

[m-3

]

cutoff1

cutoff2


hBD results for Al/Al2O3

Stoner and Maris, PRB, 1993P. E. Hopkins, et al., International Journal of Thermophysics 28, 947 (2007)


Thermal model

0

0.2

0.4

0.6

0.8

1

0 200 400 600 800 1000 1200 1400

Time [ps]

R

[a.u

.]

h BD = 1.0x108 W m-2 K

h BD = 2.0x108 W m-2 K

h BD = 3.0x108 W m-2 K

Change in temperature across a 50 nm Cr film on Si substrate interface


Resolving TBC with TTR

2df

BD

fi h

dC

1f

BD

i

f

k

dh

Resolving TBC with TTR Al/Al2O3 interfaces kf = 237 Wm-1K-1

hBD = 2.0 x 108 Wm-2K-1

0

0.5

1

1.5

0 25 50 75 100

Film thickness [nm]

Tim

e co

nsta

nt [n

s]

0

5

10

15

20

0 250 500 750 1000

if


Thermal ModelLumped capacitance

BD

f

f

BD

h

kd

k

dhBi

1.01.0

T

x

Bi<<1

Bi = 1

Bi>>1

film substrate Al/Al2O3 interfaces kf = 237 Wm-1K-1

hBD = 2.0 x 108 Wm-2K-1

d =75 nm< 120 nm

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 25 50 75 100

Film thickness [nm]

Tim

e c

on

sta

nt

[ns]


Sample fabricationSample

IDBacksputter

EtchHeat Treat Prior

to DepositionDeposition Notes(on Si substrates)

Cr-1 none none 50 nm Cr @ 300 K

Cr-2 5 min none 50 nm Cr @ 300 K

Cr-3 5 min 20 min @ 873 K 50 nm Cr @ 300 K

Cr-4 5 min 50 min @ 873 K 50 nm Cr @ 300 K

Cr-5 5 min 20 min @ 873 K 50 nm Cr @ 573 K

Cr-6 5 min none 10 nm of Cr at 300 K;heating to 770 K;40 nm of Cr at 300 K


Interface characterizationAuger electron spectroscopy (AES)

Relaxation andAuger emissionIonizationElectron bombardment

Higher levels

Core level

Vacuum Energy

e- [3 keV]Monitor energy


AES depth profiling

Ar+ gun

e- gundetector

Si

O2

Cr

C

dN

/dE

Energy [eV]


AES depth profile

0

0.2

0.4

0.6

0.8

1

0 10 20 30 40 50 60

Depth into film [nm]

Ele

me

nta

l fra

cti

on

Cr/Si mixing regionCr

Si

C

O2


Energetic Electrons

k

E

Interband

Intraband

Valence Band

Conduction BandIntraband excitation – E<Eg

Interband excitation – E>Eg

Doped Semiconductors (T = Troom)

Ef

Eg


Energetic Electrons

Electron transport – Noble metals

Electron transport – Transition metals

Metals (T = Troom)

k

E

Ef

Ef

d

s

ITT

Noble Metal

d

Transition Metals

ITT


Band StructuresGaAs

* Swaminathan and Macrander, 1991

Gold

*Christensen, 1976

Nickel

*Weiling and Callaway, 1982

c

s

d

v ds


*Sun, et al., 1994*Fatti et al., 2000

Reflectance at ITTGoldSilver


G Measurement in Gold

*Smith and Norris, 2001*Hohlfeld et al., 2000

G @ energies lower than ITT G @ energies around ITT

Ef

d

s


Transient Thermoreflectance Data


Thermal Model

),(),(' txSTTGx

TTTk

xt

TTC le

elee

eee

lel

l TTGt

TC

nmG

kthermal 17

k = 91 Wm-1K-1

G = 3.6 x 1017 Wm-3K-1 -1 0 1 2 3 4 5

Time [ps]

No

rma

lize

d R

efl

ec

tan

ce

R

/R [

arb

. un

its

]

1.3 eV

20 nm

50 nm


G Measurements

30 nm Ni/Si

1.3 eVG = 5.8 x 1017 Wm-3K-1

1.55 eVG = 3.7 x 1017 Wm-3K-1

R = aTe + bTl


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

•1.3 eV in Ni is not at ITTbut at a higher interband transitionlower d-band Fermi level

0

0.2

0.4

0.6

0.8

1

0 10 20 30 40 50 60 70 80 90 100

Thickness [nm]

Re

fle

cta

nc

e

Ni/Glass

Ni/Si


Reflectance Model

R = aTe + bTl

ll

ee

ll

ee

TT

TT

iTT

TT

i 221121

12

22

11

1

RR

RR

R

eT

Ra

&lT

Rb


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 Aut > ee

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

1.0f

BD

k

dhBi

hBDe-=1E8 Wm-2K-1

d=50 nmke=317

Bi = .02

very conservative


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)

Box is 4 long

Consider 1D conduction

Can resolve local temperature

Cannot resolve local temperature

Kinetic Theory

CMean free path

speed

#Collisions/time

*Since equilibrium is achieved through multiple collisions

lepeeeepee TBTA 2at Te=Tl=300, ~1E13, ~1E4 m/sC

~1 nm

~15 nm


Joint frequency DMM

j

jjBD

cj

dT

TnDvh

mod,

0mod,1mod,

),()(

4

1

3mod,

2

2

mod,2 j

jv

D

c

jmod,

3/12211

2mod,mod, 6 NNv j

cj

2211mod, vvv j

212

1

12

1

1MM

NN

MNN

j

jjBD

cj

dT

TfDvh

,1

0,11,1

),()(

4

1

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)

DMM JFDMM


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)

j

jjBD

cj

dT

TfDvh

,1

0,11,1

),()(

4

1

DMM

JFDMM

j

jjBD

cj

dT

TfDvh

mod,

0mod,1mod,

),()(

4

1

PRL

j

jjBD

cutoffj

dvTfDT

h,1

0,2,2 ,

4

1

Is there an upper limit to inelastic scattering?

jjjBD

cutoffj

dvTfDT

h,2

0,2,2 ,

4

1

Inelastic phonon radiation limit (IPRL)


IPRL

)()( ThhTh inelBD

elBDBD )()()( ThTBAhTh IPRL

BDPRLBDBD


Elastic and inelastic contributions

In classical limit

)()()()( ThAhThThTB inelBD

PRLBDBD

IPRLBD

PRLBD

elBD

h

hA

Pb/diamond Pb/diamondPb/diamond

)()()( ThTBAhTh IPRLBD

PRLBD

LimBD


Elastic and inelastic contributions

1)()(

PRLBD

BDelBD

inelBD

Ah

Th

h

Th

Relative contribution to hBD of inelastic scattering compared to elastic scattering increases with sample mismatch and with temperature

DPb/Ddiamond

~0.05

DPt/DAl2O3

~0.23

Hopkins, Norris, and Stevens, Submitted to the Journal of Heat Transfer


Future directionsThermal testing in novel nanostructures

k

TSZT

2

CNT and nanocomposites


Future directionsSteady 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).


Future directions

11 nnnnAnAn uuuuKuMF

NonEquilibrium Green’s Function (NEGF) modeling

0102

0 uuKuMF BB


Future directions

Hopkins et al., MHT08-52244

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

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.


Future directions

Electrons Transfer



Electrons


20 nm Au/glass ),(),( tzSTTG

z

TTTk

zt

TT pe

epee

eee

pep

p TTGt

TC

Insulated boundary conditions always assumed

More TTR applications – electron-phonon scattering

P. E. Hopkins and P. M. Norris, Applied Surface Science 253, 6289 (2007)


Future directions

Different e-p equilibration curves for different fluences

But G should be a material property

z

1),( pe

ep

eeRTpe TT

B

AGTTG

P. E. Hopkins, et al., Submitted to Phys Rev B.

More TTR applications – electron-phonon scattering


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

patrick hopkins university of virginia department of mechanical and aerospace engineering

Documents