interface effects on thermophysical properties in nanomaterial systems

Microscale Heat Transfer Lab – University of Virginia

Interface effects on thermophysical properties in nanomaterial systems

Patrick E. Hopkins

MAE Dept. Seminar

March 22, 2007


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


Thermal boundary conductance

k

TSZT

2

SuperlatticesField effect transistors

Heat generated

Rejected heat

Thermal management is highly dependent on the boundary of two materials


Today’s TalkPurpose: Determine the effects that the properties of the interface have on thermal boundary conductance, hBD

•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


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 Thq BD

T

Z

Z

T

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

Change in material properties gives rise to hBD


Particle theory of hBD

Phonon flux transmitted across interface

ThddjctznDq BDjj

jj

cutoffj

sincos,,,,2

1,1

2/

0 0

,1,1

,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


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

dcTfDT

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?

ThddjctznDq BDjj

jj

cutoffj

sincos,,,,2

1,1

2/

0 0

,1,11

,1


Single phonon elastic scattering events

12,2

2,1

2,2

03,2

2

2

,2

03,1

2

2

,1

03,2

2

2

,2

1,1,1

,1

22

2)(

j jjj

jj

j jj

j jj

j jj

cc

c

dc

cdc

c

dc

c

cutoffj

cutoffj

cutoffj

Simplifies transmission coefficient

21 qq

j

ijijii

cutoffji

dcTfDq,

0

,, ,4

1

3,2

2

2

, 2)(

jji c

D

121 1

Frequency [Hz]

Deb

ye d

ensi

ty o

f Sta

tes

[m-3

]

cutoff1

cutoff2


Single phonon elastic scattering eventshBD 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

dcTfDT

h,1

0

1,1,11, ,4

1


Single phonon elastic scatteringElastic Scattering – hBD is a function of df/dT

Df/

dT

j

jjBD

cutoffj

dcTfDT

h,1

0

1,1,11, ,4

1


Today’s TalkPurpose: Determine the effects that the properties of the interface has on thermal boundary conductance, hBD






Transient ThermoReflectance (TTR)Mira 900

p ~ 190 fs @ 76 MHz = 720-880 nm

16 nJ/pulse

Polarizer

Detector

Lock-in AmplifierAutomated Data

Acquisition System

Verdi V5= 532 nm 5 W

RegA 9000

p ~ 190 fssingle shot - 250 kHz

4 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

Free Electrons Absorb Laser Radiation

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 Boundary (~2 ns)Conductance

Thermal Diffusion within Substrate

Substrate Thermal Diffusion (~100 ps – 100 ns)


Thermal Model

)](),0([)(

ttdC

h

dt

tdfs

f

bdf

2

2 ),(),(

x

tx

t

tx ss

s

Initial conditions Boundary conditions

1)0( f

0)0,( xs

NondimensionalizedTemperature

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

tthx

tk sfBD

ss

0),(

x

ts

0

0,, )0( TT

TT

f

sfsf


DMM compared to experimental data

Ref 8. Stevens, Smith, and Norris, JHT, 2005Ref 63. Lyeo and Cahill, PRB, 2006Ref 65. Stoner and Maris, PRB, 1993

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

elastic scattering assumption


DMM Assumptions

DMM Assumption Realistic interface


Sample FabricationSample

IDBacksputter

EtchHeat Treat Prior

to DepositionDeposition Notes

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


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

Hopkins, and Norris, APL, 2006


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

0

0.2

0.4

0.6

0.8

1

0 200 400 600 800 1000 1200 1400

Time [ps]

No

rmal

ized

R

/R

Model fit pointt = 100 ps

Cr-2: h BD = 1.13x108 W m-2 K

Cr-1: h BD = 1.78x108 W m-2 K


hBD Results

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


Virtual Crystal DMM

VCVCBD RRD

R

2int1intint

int

1

21int

int 111

j

VCj

j

VCjp

BD hh

D

Gkh

BDhR

1int

Beechem, Graham, Hopkins, and Norris, APL, 2006

int

int

D

Multiple scattering events from interatomic mixing


VCDMM

Hopkins, and Norris, Beechem, and Graham, JHT, Submitted


Summary•DMM predicts hBD 850 MWm-2K-1 at room temperature

•Measured data varies from 1-2x108

•Multiple phonon elastic scattering could cause discrepancy

•DMM only takes into account single scattering event

•DMM assumes perfect interface

•Virtual Crystal DMM predicts same values and trendsfor Cr/Si at room temperature


Single phonon elastic scatteringElastic Scattering – hBD is a function of df/dT

j

jjBD

cutoffj

dcTfDT

h,1

0

1,1,11, ,4

1


Molecular Dynamics Simulations

Stevens, Zhigilei, and Norris, IJHMT, Accepted

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


Mismatched samples

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


TTR Testing


hBD results

Ref 65. Stoner and Maris, PRB, 1993

Hopkins, Salaway, Stevens, and Norris, IJT, 2007


hBD resultsHopkins, Stevens, and Norris, JHT, 2007


Analysis• Linear trend in MDS in classical regime• MDS calculates hBD with out assuming only elastic scattering

in interfacial phonon transport• Several samples show linear hBD trends around classical

regime

10 100 100010-3

100

103

106

h BD [W

m-2K

-1]

Temperature [K]

j

jjBD

cj

dT

TfDch

,1

0

,11,1

),()(

4

1

DMM

JOINT FREQUENCY DMM

j

jjBD

cj

dT

TfDch

mod,

0

mod,1mod,

),()(

4

1


JFDMM

j

jjBD

cj

dT

TnDch

mod,

0

mod,1mod,

),()(

4

1

3mod,

2

2

mod,2 j

jc

D

cjmod,

3/1

22112

mod,mod, 6 NNc jc

j

2211mod, ccc j

212

1

12

1

1

MMNN

MNN


DMM vs. JFDMM


Summary

• 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 phonon population in DMM improves prediction (JFDMM)


Conclusions & Acknowledgments

•Realistic interfaces – two phase regions, mixing, nonperfect junctions – multiple phonon scattering events that can decrease hBD

•Inelastic scattering can occur at elevated temperatures (T > D), increasing hBD

Purpose: Determine the effects that the properties of the interface have on thermal boundary conductance, hBD

•Thanks for the financial support from NSF GRFP, VSGC, U.Va. Faculty Senate and Double Hoo, and NSF grant CTS-0536744•Dr. Pam Norris, Dr. Samuel Graham, Thomas Beecham•Microscale Crew: Rich Salaway, Rob Stevens, Mike Klopf, Jenni Simmons, Thomas Randolph, Jes Sheehan


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]


hBD trends vs. sample mismatch

interface effects on thermophysical properties in nanomaterial systems

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