Thermal Properties of MaterialsThermal Properties of Materials

Li Shi

Department of Mechanical Engineering &Center for Nano and Molecular Science and Technology,

Texas Materials InstituteThe University of Texas at Austin

Austin, TX 78712www.me.utexas.edu/~lishi

lishi@mail.utexas.edu

2

OutlineOutline

Macroscopic Thermal Transport Theory– Diffusion

-- Fourier’s Law

-- Diffusion Equation

• Microscale Thermal Transport Theory – Particle Transport

-- Kinetic Theory of Gases

-- Electrons in Metals

-- Phonons in Insulators

-- Boltzmann Transport Theory

• Thermal Properties of Nanostructures

-- Thin Films and Superlattices

-- Nanowires and Nanotubes

-- Nano Electromechanical System (NEMS)

3

Thermal conductivity

HotTh

ColdTc

L

Q (heat flow)

Fourier’s Law for Heat Conduction

dxdT

kAL

TTkAQ ch

4

Heat Diffusion Equation

tT

Cx

Tk

2

2

Specific heat

Heat conduction = Rate of change of energy storage

1st law (energy conservation)

•Conditions: t >> scattering mean free time of energy carriersL >> l scattering mean free path of energy carriers

•Breaks down for applications involving thermal transport in small length/ time scales, e.g. nanoelectronics, nanostructures, NEMS, ultrafast laser materials processing…

5

Length Scale

1 m

1 mm

1 m

1 nm

Human

Automobile

Butterfly

1 km

Aircraft

Computer

Wavelength of Visible Light

MEMS

Width of DNA

MOSFET, NEMS

Blood Cells

Microprocessor Module

Nanotubes, Nanowires

Particle transport100 nm

Fourier’s law

l

6

OutlineOutline

Macroscopic Thermal Transport Theory– Diffusion

-- Fourier’s Law

-- Diffusion Equation

Microscale Thermal Transport Theory– Particle Transport

-- Kinetic Theory of Gases

-- Electrons in Metals

-- Phonons in Insulators

-- Boltzmann Transport Theory

• Thermal Properties of Nanostructures-- Thin Films and Superlattices-- Nanowires and Nanotubes -- Nano Electromechanical System (NEMS)

7

D

D

Mean Free Path for Intermolecular Mean Free Path for Intermolecular Collision for GasesCollision for Gases

Total Length Traveled = L

Total Collision VolumeSwept = D2L

Number Density of Molecules = n

Total number of molecules encountered inswept collision volume = nD2L

Average Distance betweenCollisions, mc = L/(#of collisions)

Mean Free Path

nLDn

Lmc

12

: collision cross-sectional area

8

Mean Free Path for Gas MoleculesMean Free Path for Gas Molecules

Number Density ofMolecules from IdealGas Law: n = P/kBT

kB: Boltzmann constant 1.38 x 10-23 J/K

Mean Free Path:

PTk

nB

mc 1

Typical Numbers:

Diameter of Molecules, D 2 Å = 2 x10-10 mCollision Cross-section: 1.3 x 10-19 m

Mean Free Path at Atmospheric Pressure:

m0.3or m103103.110

3001038.1 7195

23

mc

At 1 Torr pressure, mc 200 m; at 1 mTorr, mc 20 cm

9

Wall

Wall

b: boundary separation

Effective Mean Free Path:

Effective Mean Free PathEffective Mean Free Path

bmc 111

10

Kinetic Theory of Energy TransportKinetic Theory of Energy Transport

z

z - z

z + z

u(z-z)

u(z+z)

zzzz zuzuvq 21

'qz

Net Energy Flux / # of Molecules

dzdu

vdzdu

vq zzz 2cos'

through Taylor expansion of u

u: energy

dzdT

kdzdT

CvdzdT

dTdu

vqz 31

31

Integration over all the solid angles total energy flux

Cvk31Thermal conductivity:

Specific heat Velocity Mean free path

11

• If so, what are C, v, for electrons and crystal vibrations?

• Kinetic theory is valid for particles: can electrons andcrystal vibrations be considered particles?

QuestionsQuestions

Free Electrons in Metals at 0 KFree Electrons in Metals at 0 K

EF

Work Function

Energy

Fermi Energy – highest occupied energy state:

Fermi Velocity:

3

12

32

2222

3

322

eF

eF

F

mv

mmk

E

Element Electron Density, e [1028 m-3]

Fermi Energy EF [eV]

Fermi Temperature TF [104 K]

Fermi Wavelength F [Å]

Fermi Velocity vF [106 m/s]

Work Function [eV]

Cu 8.47 7.00 8.16 4.65 1.57 4.44 Au 5.90 5.53 6.42 5.22 1.40 4.3 Fe 17.0 11.1 13.0 2.67 1.98 4.31 Al 18.1 11.7 13.6 3.59 2.03 4.25

VacuumLevel

Band Edge

Fermi Temp:B

FF k

ET

Metal

Effect of TemperatureEffect of Temperature

TkEE

Ef

B

Fexp1

1Fermi-Dirac equilibrium distribution for the probability of electron occupation of energy level E at temperature T

0

1

EFElectron Energy, E

Occ

upat

ion

Pro

babi

lity,

f

Work Function,

Increasing T

T = 0 K

k TB

Vacuum Level

14

dEEDEEfV

E

dEEDEfVN

ee

e

ee

0

0;

Number and Energy DensitiesNumber and Energy Densities

Density of States -- Number of electron states available betweenenergy E and E+dE

222

2

mEm

EDe

Number density:

Energy density:

in 3D

15

Electronic Specific Heat and Thermal ConductivityElectronic Specific Heat and Thermal Conductivity

dEEDdT

dfE

dT

dC e

e

0

BeF

Be k

E

TkC

2

2

eFeeFee vCvCk 2

3

1

3

1

Specific Heat

Thermal Conductivity

Electron Scattering Mechanisms• Defect Scattering• Phonon Scattering• Boundary Scattering (Film Thickness,

Grain Boundary)

Grain Grain Boundary

e

Temperature, T

Defect Scattering

PhononScattering

IncreasingDefect Concentration

Bulk Solids

Mean free time:e = le / vF

in 3D

1610 310 210 110 010 0

10 1

10 2

10 3

Temperature, T [K}

Th

erm

al

Co

nd

uc

tiv

ity,

k

[W/c

m-K

]

Copper

Aluminum

Defect Scattering Phonon Scattering

1

1

eFeeFee vCvCk 2

3

1

3

1

Matthiessen Rule:

Thermal Conductivity of Cu and AlThermal Conductivity of Cu and Al

phononboundarydefecte

phononboundarydefecte

1111

1111

Electrons dominate k in metals

17

• Since electrons are traveling waves, can we apply kinetic theory of particle transport?

Two conditions need to be satisfied:• Length scale is much larger than electron wavelength or

electron coherence length• Electron scattering randomizes the phase of wave function

such that it is a traveling packet of charge and energy

Afterthought

18

Crystal VibrationCrystal Vibration

Energy

Distancero

Parabolic Potential of Harmonic Oscillator

Eb

Interatomic Bonding

a

Spring constant, g Mass, m

xn xn+1xn-1

Equilibrium Position

Deformed Position

1-D Array of Spring Mass System

nnnn xxxg

dt

xdm 2112

2

Equation of motion withnearest neighbor interaction

inKatixx on expexp

Solution

19

Dispersion RelationDispersion Relation

21

cos12

cos12expexp22

Kam

g

KagiKaiKagm

Fre

que

ncy,

Wave vector, K0 /a

Longitudinal A

cousti

c (LA) M

ode

Transverse

Acousti

c (TA) M

ode

Group Velocity:

dK

dvg

Speed of Sound:

dK

dv

Ks

0

lim

20

Lattice Constant, a

xn ynyn-1 xn+1

nnnn

nnnn

yxxgdt

ydm

xyygdt

xdm

2

2

12

2

2

12

2

1

Two Atoms Per Unit CellTwo Atoms Per Unit Cell

Fre

que

ncy,

Wave vector, K0 /a

LATA

LO

TO

OpticalVibrationalModes

21

0 0.2 0.4 0.6 0.8 1.00.20.40

2

4

6

8

(111) Direction (100) Direction XL Ka/

LA

TATA

LA

LO

TO

LO

TO

Freq

uenc

y (

10

Hz)

12

Phonon Dispersion in GaAsPhonon Dispersion in GaAs

22

Energy Quantization and PhononsEnergy Quantization and Phonons

h

Energy

Distance

Total Energy of a QuantumOscillator in a Parabolic Potential

2

1nu

n = 0, 1, 2, 3, 4…; /2: zero point energy

Phonon: A quantum of vibrational energy, , which travels through the lattice

Phonons follow Bose-Einstein statistics.

Equilibrium distribution:

1exp

1

Tk

n

B

In 3D, allowable wave vector K: ,....6

,4

,2

LLL

23

Lattice EnergyLattice Energy

pKpKp

l nE ,, 2

1

K

p: polarization(LA,TA, LO, TO)K: wave vector

Dispersion Relation: gK

Energy Density: dDnV

E

p

ll

21

d

dggD

2

2

2

Density of States: Number of vibrational states between and +d

Lattice Specific Heat: dDdT

nd

dT

dC

p

ll

in 3D

Debye ModelDebye Model

Fre

quen

cy,

Wave vector, K0 /a

KvsKvsDebye Approximation:

32

2

2

2

22 svddgg

D

Debye Density

of States:

B

sD k

v 31

26

C(dimnd) 1860 Ga 240Si 625 NaF 492Ge 360 NaCl 321B 1250 NaBr 224Al 394 NaI 164

Debye Temperature [K]

Specific Heat in 3D:

TD

x

x

DBl

e

dxxeTkC

02

43

19

In 3D, when T << D,

34 , TCT ll

Phonon Specific HeatPhonon Specific Heat

10 410 310 210 110 1

10 2

10 3

10 4

10 5

10 6

10 7

Temperature, T (K)

Sp

ec

ific

He

at,

C (

J/m

-K

)3

C T 3

C3kB 4.7106 Jm3 K

D 1860 K

Diamond

ClassicalRegime

In general, when T << D,

dl

dl TCT ,1

d =1, 2, 3: dimension of the sample

Each atom has a thermal energy of 3KBT

Sp

ecifi

c H

ea

t (J/

m3-K

)

Temperature (K)

C T3

3kBT

Diamond

Phonon Thermal ConductivityPhonon Thermal Conductivity

lsllsll vCvCk 23

1

3

1 Kinetic Theory

l

Temperature, T/D

BoundaryPhononScatteringDefect

Decreasing BoundarySeparation

IncreasingDefectConcentration

Phonon Scattering Mechanisms

• Boundary Scattering• Defect & Dislocation Scattering• Phonon-Phonon Scattering

0.01 0.1 1.0

Temperature, T/D

0.01 0.1 1.00.01 0.1 1.0

kl

dl Tk

Boundary

PhononScatteringDefect

Increasing DefectConcentration

10 310 210 110 010 -2

10 -1

10 0

10 1

10 2

10 3

Temperature, T [K]

Th

erm

al

Co

nd

uc

tiv

ity,

k [

W/c

m-K

]

Diamond

BoundaryScattering

DefectScattering

IncreasingDefect Density

• Phonons dominate k in insulators

Thermal Conductivity of InsulatorsThermal Conductivity of Insulators

28

Drawbacks of Kinetic TheoryDrawbacks of Kinetic Theory

• Assumes local thermodynamics equilibrium: u=u(T)Breaks down when L ; t

• Assumes single particle velocity and single mean free path or mean free time. Breaks down when, vg() or

• Cannot handle non-equilibrium problemsShort pulse laser interactionsHigh electric field transport in devices

• Cannot handle wave effectsInterference, diffraction, tunneling

Boltzmann Transport Equation for Particle TransportBoltzmann Transport Equation for Particle Transport

Distribution Function of Particles: f = f(r,p,t)--probability of particle occupation of momentum p at location r and time t

scatp t

fff

t

f

Fv r

pr,ff

tf o

scat

Relaxation Time Approximation

t

off

t

e

Equilibrium Distribution: f0, i.e. Fermi-Dirac for electrons, Bose-Einstein for phonons

Relaxation time

Non-equilibrium, e.g. in a high electric field or temperature gradient:

Energy flux in terms of particle flux carrying energy:

k

dk

q

kkkrrvrq dtftt ,,,,

k

dkddkktkftvt

0

2

0

2 sincos,,,, rrrq

0

2

0

sincos,,,4

1, dddDtfvt rrrq

v

Energy Flux

Integrate over all the solid angle:

Integrate over energy instead of momentum:

Density of States: # of phonon modes per frequency range

Vector

Scalar

Continuum CaseContinuum Case

tL off

t

f

;0 BTE Solution: cos

dx

dfvffff o

ooo v

Direction x is chosento in the direction of qEnergy Flux:

dDdx

dfvtq o,,

3

1, 2 rrr

dx

dT

dT

df

dx

df oo

dxdT

kdDdT

dfv

dxdT

tq o ,,31

, 2 rrrFourier Law ofHeat Conduction:

dDdT

dfvk o,,

3

1 2 rr

If v and are independent of particleenergy, , then

22

3

1

3

1CvdD

dT

dfvk o

Quasi-equilibrium

Kinetic theory:

() can be treated using Callaway method(Phys. Rev. 113, 1046)

At Small Length/Time Scale (At Small Length/Time Scale (LL~~ll or or tt~~))Define phonon intensity:

sDtfsvtsI ,,,,,,,,,,,,, rr

Scattff

ftf

)(

kFv

0

Dv

scatt

ItsI

t

tsI

,,,,,

,,,,,rv

rEquation of Phonon Radiative Transfer (EPRT) (Majumdar, JHT 115, 7):

From BTE:

42

41 TTq Acoustically Thin Limit (L<<l) and for T << D

Acoustically Thick Limit (L>>l) Tkq l

0

2

0cos

41

, dddItq rHeat flux:

33

OutlineOutline

Macroscopic Thermal Transport Theory – Diffusion

-- Fourier’s Law

-- Diffusion Equation

Microscale Thermal Transport Theory – Particle Transport

-- Kinetic Theory of Gases

-- Electrons in Metals

-- Phonons in Insulators

-- Boltzmann Transport Theory

Thermal Properties of Nanostructures

-- Thin Films and Superlattices

-- Nanowires and Nanotubes

-- Nano Electromechanical System

34

Thin Film Thermal Conductivity Measurement

I0 sin(t)

L 2b

Thin Film

Substrate

Metal line

f

s

s LbkPdi

b

D

kLP

T24

2ln21

ln21

)2(2

3 method(Cahill, Rev. Sci. Instrum. 61, 802)

• I ~ 1• T ~ I2 ~ 2• R ~ T ~ 2• V~ IR ~3

V

35

Silicon on Insulator (SOI)

IBM SOI Chip

Ju and Goodson, APL 74, 3005

Lines: BTE results

Hot spots!

36

Thermoelectric Cooling

• No moving parts: quiet and reliable• No Freon: clean

37

11

/1COPmax

m

chm

ch

c

zT

TTzT

TT

T

Coefficient of Performance

where

Thermoelectric Figure of Merit (ZT)

TS

ZT2

Seebeck coefficientElectrical conductivity

Thermal conductivity

Temperature0

1

2

0 1 2 3 4 5

ZT

CO

Pm

ax

Bi2Te3

Freon

TH = 300 KTC = 250 K

38

ZT Enhancement in Thin Film Superlattices

Ec

Ev

x

E

Ge Quantum well (QW)Si Barrier

•Increased phonon-boundary scattering

decreased k

+ other size effects

High ZT = S2T/k

SiGe superlattice(Shakouri, UCSC)

39

Thermal Conductivity of Si/Ge Superlattices

Period Thickness (Å)

k (W/m-K)

Bulk

Si0.5Ge0.5 Alloy

Circles: Measurement by D. Cahill’s groupLines: BTE / EPRT results by G. Chen

40

Superlattice Micro-coolers Ref: Venkatasubramanian et al, Nature 413, P. 597 (2001)

41

Nanowires

• Increased phonon-boundary scattering

• Modified phonon dispersion

Suppressed thermal conductivity

Ref: Chen and Shakouri, J. Heat Transfer 124, 242

Hot Coldp

22 nm diameter Si nanowire,P. Yang, Berkeley

42

Pt resistance thermometer

Suspended SiNx membraneLong SiNx beams

Current (A)

-6 -4 -2 0 2 4 6

T

s (K

)0.00

0.02

0.04

0.06

0.08

0.10

T0 = 54.95 K

QI Current (A)

-6 -4 -2 0 2 4 6

Th

(K

)

0.0

0.5

1.0

1.5T0 = 54.95 K

Thermal Measurements of Nanotubes and Nanowires

Kim et al, PRL 87, 215502Shi et al, JHT, in press

Themal conductance: G = Q / (Th-Ts)

43

Si Nanowires

Temperature (K)

0 50 100 150 200 250 300 350

The

rmal

Con

duct

ivit

y (W

/m-K

)

0

10

20

30

40

50

60

22 nm

37 nm

56 nm

115 nm

Source DrainGate

Nanowire Channel

Si Nanotransistor (Berkeley Device

group)

D. Li et al., BerkeleySymbols: MeasurementsLines: Modified Callaway Method

Hot Spots in Si nanotransistors!

44

ZT Enhancement in Nanowires

Ref: Phys. Rev. B. 62, 4610 by Dresselhaus’s group

Top View

Nanowire

Al2O3 template

Nanowires based on

Bi, BiSb,Bi2Te3,SiGe

Bi Nanowires

• k reduction and other size effects

High ZT = S2T/k

45

Nanotube Nanoelectronics

TubeFET (McEuen et al., Berkeley)

Nanotube Logic (Avouris et al., IBM)

46

Thermal Transport in Carbon Nanotubes

• Few scattering: long mean free path l

Strong SP2 bonding: high sound velocity v

high thermal conductivity: k = Cvl/3 ~ 6000 W/m-K

• Below 30 K, thermal conductance 4G0 = ( 4 x 10-12T) W/m-K,

linear T dependence (G0 :Quantum of thermal conductance)

Hot Coldp

Heat capacity

47

Thermal Conductance of a Nanotube Mat

• Estimated thermal conductivity at 300K: ~ 250 << 6000 W/m-K Junction resistance is dominant

Ref: Hone et al. APL 77, 666

Linear behavior

25 K

• Intrinsic property remains unknown

Thermal Conductivity of Carbon Nanotubes

CVD SWCN

Temperature (K)100 101 102 103T

herm

al C

ondu

ctiv

ity

(W/m

-K)

10-2

10-1

100

101

102

103

104

105

148 nm SWCNbundle

10 nm SWCN bundle

1-3 nm CVD SWCN

14 nm MWCN bundle

~ T 1.6

~ T 2

~ T 2.5

• An individual nanotube has a high k ~ 2000-11000 W/m-K at 300 K

•k of a CN bundle is reduced by thermal resistance at tube-tube junctions

•The diameter and chirality of a CN may be probed using Raman spectroscopy

CNT

49

Nano Electromechanical System (NEMS)

Thermal conductance quantization in nanoscale SiNx beams

(Schwab et al., Nature 404, 974 )

Quantum of Thermal Conductance

Phonon Counters?

50

SummarySummary

Macroscopic Thermal Transport Theory – Diffusion

-- Fourier’s Law

-- Diffusion Equation

Microscale Thermal Transport Theory – Particle Transport

-- Kinetic Theory of Gases

-- Electrons in Metals

-- Phonons in Insulators

-- Boltzmann Transport Theory

Thermal Properties of Nanostructures

-- Thin Films and Superlattices

-- Nanowires and Nanotubes

-- Nano Electromechanical System (NEMS)