E. Pop / DRC 2009 1

Eric Pop

Dept. of Electrical & Computer Engineering

Micro & Nanotechnology Lab, Beckman Institute

http://poplab.ece.illinois.edu

Graphene Thermal Physics

E. Pop / DRC 2009 2

• Heat capacity of graphene

• Thermal conductivity of graphene

• Graphene devices

• “Other” physics(thermoelectrics, asymmetry, quantum thermal conductance…)

Table of Contents

+ Background

E. Pop / DRC 2009 3

• Graphite is highly asymmetric

• Strong in-plane covalent sp2 sigma bonds (524 kJ/mol)

• Weak out-of-plane van der Waals pi bonds (7 kJ/mol)

Structure of Graphite

~ 0.335 nm

(abab) orBernal stacking

c-a

xis

ab plane

(2 atoms / unit cell)

E. Pop / DRC 2009 4

• Heat capacity of a solid: C = Celectrons + Clattice

• High temperature: classically recall C = 3NAkB

Dulong-Petit Law (1819) most solids 25 J/mol/K at high T

• Low temperature: experimentally C 0

Heat Capacity: Energy Stored ΔU=CΔT

( )

L

du dfC g d

dT dT

ω(k)phonon dispersion

phonon distribution(e.g. Bose-Einstein)

density of states

E. Pop / DRC 2009 5

• Transverse (u ^ k) vs. longitudinal modes (u || k)

• Acoustic (transport sound & heat) vs. optical (scatter light & electrons)

• “Hot phonons” = highly occupied modes above room temperature

Phonons: Atomic Lattice Vibrations

)](exp[),( tiit rkAru

transversesmall k

transversemax k F

req

(H

z)

En

erg

y (m

eV

)

Wave vector qa/2p

10

20

30

40

50

60

k

Graphene Phonons [100]

200 meV

160 meV

100 meV

300 K =

26 meV

Fre

qu

en

cy ω

(cm

-1)

Si optical

60 meV

optical

Lattice Constant, a

xn ynyn-1 xn+1

2 atomsper unit cell

S

dv

dk

E. Pop / DRC 2009 6

0 0.2 0.4 0.6

0

0.5

1

1.5

2

E (eV)

f BE

• In the limit of high energy, both reduce to the classical

Maxwell-Boltzmann statistics:

Fermi-Dirac vs. Bose-Einstein Statistics

1( )

exp 1

FD

F

B

f EE E

k T

1( )

exp 1

BE

B

f EE

k T

Fermions = half-integer spin(e.g. electrons, protons)

Bosons = integer spin(e.g. phonons, photons)

-0.4 -0.2 0 0.2 0.4 0.6-0.2

0

0.2

0.4

0.6

0.8

1

1.2

E-EF (eV)

f FD

T=0 K

T=300 K

T=1000 K

T=0 K

T=300 K

T=1000 K

∞

( ) exp /MB Bf E E k T

E. Pop / DRC 2009 7

• Very high T: C = 3nkB (constant) both dielectrics & metals

• Intermediate T: C ~ aTD both dielectrics & metals in D dimensions*

• Very low T: C ~ bT metals (electron contribution only)

Heat Capacity: Dielectrics vs. Metals

C = bT

Dielectrics Metals

* assuming linear ω=vk phonon dispersion

E. Pop / DRC 2009 8

• Key assumption: all oscillators have same frequency

• High-T (correct, recover Dulong-Petit Law):

• Low-T (incorrect, drops too fast)

Heat Capacity: Einstein Model (1907)

( ) 3E BC T Nk

2

/( ) 3 E E B

B

k T

E B k TC T Nk e

But… Einstein modelOK for optical phonon

heat capacity

Fre

quency ω

0 2p/a

E

k

E. Pop / DRC 2009 9

Annalen der Physik 39(4)p. 789 (1912)

Peter Debye (1884-1966)

E. Pop / DRC 2009 10

• Key assumption: oscillators have linear ω-k with cutoff ωD

• Often in terms of Debye temperature

• … roughly corresponds to max acoustic

phonon frequency cutoff

• Low-T (< θD/10):

• High-T: (> 0.8 θD)

Heat Capacity: Debye Model (1912)

Fre

quency,

0

sv k

Wave vector, k 2p/a

1/3

26sD

B

vN

k p

B D Dk

3412

( )5

D B

D

TC T Nk

p

( ) 3D BC T NkD

DC TIn D-dimensions

when T << θD:

E. Pop / DRC 2009 11

• In practice: θD ~ fitting parameter to heat capacity data

• θD is related to “stiffness” of solid and melting temperature

Debye Model at Low- and High-T

104

103

102

101

101

102

103

104

105

106

107

Temperature, T (K)

Sp

ecif

ic H

ea

t, C

(J

/m

-K)

3

C T3

C 3kB 4.7 106 J

m3 K

D 1860 K

Diamond

ClassicalRegime

Each atom has a thermal energy of 3KBT

Specific

Heat

(J/m

3-K

)

Temperature (K)

C T3

3NkBT

Diamond

DiamondθD = 1860 K

Ag, θD = 226 K

Al, θD = 428 K

Pb, θD = 102 K

Graphite 2480

180

in-plane (sp2)

out-of-plane (vdW)

to resolve low-temperature heat capacity “quandary”

since graphene data was neither 2-D (T2) nor 3-D (T3)

E. Pop / DRC 2009 12

• Diamond “like” silicon:

– Longitudinal & transverse (x2) acoustic (LA, TA)

– Longitudinal & transverse (x2) optical (LO, TO)

• Graphite is unusual:

– Layer-shearing, -breathing, and -bending modes (ZA, ZO)

– Higher optical freq. than diamond, strong sp2 bond stretching modes

– Graphite has more low-frequency modes

Phonon Dispersion of Diamond & Graphite

LO

TO

LA

TA

ZO

ZA

Tohei, Phys. Rev. B (2006)

300 K =

26 meV

E. Pop / DRC 2009 13

• Graphite has higher phonon DOS at low frequency higher

heat capacity than diamond at room T

• Both increase up to Debye temperature range, then reach “classical” 3NAkB limit

Heat Capacity of Diamond & Graphite

Silicon 0.80

SiO2 0.71

Pierson (1993)

300 K

Dulong-Petit 3NAkB high-temperature limit

θD

Tohei, Phys. Rev. B (2006)

E. Pop / DRC 2009 14

Phonon Dispersion in Graphene

Yanagisawa et al., Surf. Interf. Analysis 37, 133 (2005)

Si opticalphonons

meVLO

TO

LA

TA

vTA ~ 16 km/s(silicon ~ 5 km/s)

vLA ~ 23 km/s(silicon ~ 9 km/s)

ZO

ZA

ω ~ k2

vZA ~ 0out-of-plane bending or flexing modeimportant for heat capacity at very low Tmay be affected (suppressed) by substrate

E. Pop / DRC 2009 15

• Heat capacity of graphene

• Thermal conductivity of graphene

• Graphene devices

• “Other” physics(thermoelectrics, asymmetry, quantum thermal conductance…)

Table of Contents

+ Background

E. Pop / DRC 2009 16

Thermal Conductivity of Solids

Unlike electrical conductivity, thermal spans “only” 4 orders of magnitude

E. Pop / DRC 2009 17

Thermal Conductivity: Kinetic Theory

z

z - z

z + z

u(z-z)

u(z+z)

λqz

1

3q

du dT dTJ v k

dT dz dz

Integrate energy flux over all angles:

1

3k CvThermal conductivity:

phonon velocitydω/dk

mean freepath (MFP)

heat capacity

in 3-D

1

1 2 1

3 TA LA

vv v

'·( ' 0')Tk q

Heat diffusion eq:

· )( 0V

Poisson eq:

heat gen. per unitvolume (W/m3)

diff n

dnJ qD

dz

compare to…

compare to…

E. Pop / DRC 2009 18

• Heat capacity depends on ω(k) and T

• Phonon group velocity depends only ω(k)

• Mean free path λ phonon scattering mechanisms:

– Boundary & edge scattering

– Defect & impurity scattering

– Phonon-phonon scattering ~ 1/T

Thermal Conductivity and Phonon MFP

21 1

3 3k Cv Cv Temperature, T/D

0.01 0.1 1.00.01 0.1 1.0

kl

dl Tk

Boundary

Phonon

ScatteringDefect

Increasing Defect

Concentration

C λ k

low T Td λ L (size) Td

high T 3NkB 1/T 1/T

k ~ 1/Tk ~ Td

k

E. Pop / DRC 2009 19

High (100-1000x) thermal

anisotropy in all graphite

Thermal Conductivity of Graphite** typical commercial pyrolitic graphite

100

200

300

400Copper

Pyrolytic Graphite, Anyab- Direction

Pyrolytic Graphite, Anyc- Direction

1000 1500500 2000 2500

1000 1500500 2000 2500

1.0

2.0

Temperature (oC)

Therm

al C

onductivity (

Wm

-1K

-1)

comparableto copper

comparable toplastics, SiO2

Silicon 140

SiO2 1.4

Pierson (1993)

E. Pop / DRC 2009 20

• Graphite ab- thermal conductivity generally lower because of interlayer interactions (could be same for graphene on substrate)

• Graphite ~T2.5 at very low T due to bending modes (ω ~ k2)

• Graphene ~T1.5 at very low T due to -------””-------””------- (ω ~ k2)

Graphene Thermal Conductivity (Theoretical)

Berber et al, Phys. Rev. Lett. (2000); Mingo et al, Phys. Rev. Lett. (2005)

T (K)

k (

Wm

-1K

-1)

CNT and graphene“ballistic upper limit”

E. Pop / DRC 2009 21

• Phonon dispersion phonon DOS heat capacity k ~ Cvλ

• Information about dimensionality of system can be obtained from low-temperature k-T (or C-T) measurements

• Ex: Si nanowires ~ from 1-D to 3-D features!

The Imprint of Dimensionality

Li et al, Appl. Phys. Lett. (2003)

Si nanowires

HeaterThermometer

CNT and graphene“ballistic upper limit”

Li et al, Appl. Phys. Lett. (2003)

bulk Si at room T (~140 Wm-1K-1)

E. Pop / DRC 2009 22

• Graphene flakes suspended over SiO2 trenches (~3 um wide)

• Thermometry using Raman G-band (1583 cm-1) shift

• Room temperatureresults in-line with existing CNT data

Balandin et al, Nano Lett. (2008); Ghosh et al, Appl. Phys. Lett. (2008)

Balandin et al.~3080-5150

Graphene Thermal Conductivity (Experimental)

E. Pop / DRC 2009 23

• Q: what is the electronic contribution to thermal transport?

• Wiedemann & Franz (1853) empirically saw ke/σ = const(T)

• Lorenz (1872) noted ke/σ proportional to T

• Graphene electronic contributionappears <10% throughout T range

Wiedemann-Franz Law

22 21

3 2e B F

F

Tk n v

E

p

2q

q n nm

taking the ratio:

Remarkable: independent

of n, m, and even !

2 2

0 23

e BkL

T q

p

8 2

0 2.45 10 WΩ/KL

L = /T 10-8 WΩ/K2

Metal 0 ° C 100 °C

Cu 2.23 2.33

Ag 2.31 2.37

Au 2.35 2.40

Zn 2.31 2.33

Cd 2.42 2.43

Mo 2.61 2.79

Pb 2.47 2.56

Experimentally

E. Pop / DRC 2009 24

• Heat capacity of graphene

• Thermal conductivity of graphene

• Graphene devices

• “Other” physics(thermoelectrics, asymmetry, quantum thermal conductance…)

Table of Contents

+ Background

E. Pop / DRC 2009 25

Background on Thermal Resistance

Ohm’s Law (1827)Fourier’s Law (1822)

∆ V = I × Rel

∆T = P × Rth

P = I2 × R

R = f(∆T)

th

LR

kA el

LR

A

E. Pop / DRC 2009 26

Thermal-Electrical Cheat Sheet

qJ k T

'·( ' 0')Tk q · )( 0V

diffJ qD n

th

LR

kA el

LR

A

E. Pop / DRC 2009 27

Thermal Resistance of Devices

0.1

1

10

100

1000

10000

100000

0.01 0.1 1 10L (m)

RTH (

K/m

W)

Cu

GST

SiO2

Si

Silicon-on-Insulator FET

Bulk FET

Cu Via

Phase-change Memory (PCM)

Single-wall nanotube SWNT

Data: Mautry (1990), Bunyan (1992), Su (1994), Lee (1995), Jenkins (1995), Tenbroek (1996), Jin (2001), Reyboz (2004), Javey (2004), Seidel (2004), Pop (2004-6), Maune (2006).

High thermal resistances:

• SWNT due to small thermal

conductance (very small d ~ 2 nm)

• Others due to low thermal

conductivity, decreasing dimensions,

increased role of interfaces

Power input also matters:

• SWNT ~ 0.01-0.1 mW

• Others ~ 0.1-1 mW

∆T = P × Rth

E. Pop / DRC 2009 28

• Steady-state models

– Lumped: Mautry (1990), Goodson-Su (1994-5), Pop (2004), Darwish (2005)

– Finite-Element models

• Transient models

Modeling Device Thermal Response

L W

D

tBOX

tSi

Bulk Si FET SOI FET

0.1

1

10

100

1000

10000

100000

0.01 0.1 1 10L (m)

RTH (

K/m

W)

SOI FET

Bulk FET

1/ 2

1

2

BOXTH

BOX Si Si

tR

W k k t

1 1

2 2TH

Si Si

Rk D k LW

E. Pop / DRC 2009 29

Assumptions:tox = 300 nm

L = 25 umW = 6 um

kox = 1.3 Wm-1K-1

ksi = 80 Wm-1K-1

L WtOX

1538oxox

ox

t KRWk LW

1510

2Si

Si

KRWk LW

0.25Si Si

G ox Si

T R

T R R

-5 -4 -3 -2 -1 0

x 10-5

0

20

40

60

Y (m)

T (

K)

0.2 mW/μm2

0.1 mW/μm2

-3 -2 -1 0 1

x 10-6

0

20

40

60

Y (m)

T (

K)

-5 -4 -3 -2 -1 0

x 10-5

0

20

40

60

Y (m)

T (

K)

0.29Si

G

T

T

Graphene FET

SiO2

Si

Bae, Ong, Estrada, Pop (2009)

E. Pop / DRC 2009 30

• Spatially resolved (~0.5 um) Raman (2D ~ 2700 cm-1) thermal imaging

• Suggests power dissipation directly with SO phonons (60 meV) in SiO2

• Suggests thermal boundary RB (graphene-SiO2) ~ 4x10-8 m2K/W

Raman Thermal Imaging of Graphene FET

Freitag et al, Nano Lett. (2009)

2D peak thermal conductivity

E. Pop / DRC 2009 31

• Heat capacity of graphene

• Thermal conductivity of graphene

• Graphene devices

• “Other” physics(thermoelectrics, asymmetry, quantum thermal conductance…)

Table of Contents

+ Background

E. Pop / DRC 2009 32

• Thermal expansion coefficient (TEC) is

– Positive and large in c- direction

– Negative (at room temperature) in ab- plane

• Graphite thermal expansion anisotropy can result in large internal stresses

Thermal Expansion of Graphite

ab- direction(covalent sp2)

c- direction(van der Waals)

4

2

-2

Lin

ear

Therm

al Expansio

n (

x 1

0-6

1/K

)

Temperature (oC)

0

Pierson (1993)

Graphite

Graphite

E. Pop / DRC 2009 33

• Recent estimate of thermal contraction in single-layer suspended graphene*

• TEC ~ -7x10-6 K-1, 5-6x greater than in graphite ab- plane

Thermal Expansion of Graphene

4

2

-2

Lin

ear

Therm

al Expansio

n (

x 1

0-6

1/K

)

Temperature (oC)

0

*Bao et al, arxiv/cond-mat (2009)

Graphite

Graphite

Graphene

E. Pop / DRC 2009 34

Thermoelectric Properties of Graphene

Checkelski (2009)

• Measured graphene thermopower S ≤ 100 μV/K

– note: |Sgraphene| >> |SPd, SAu| at T > 300 K

• Peltier coefficient at metal junction: Π = (S1-S2)T

• Peltier heating/cooling Q = ΠI ≈ ΠgrapheneI

Zuev (2009)

Wei (2009)

E. Pop / DRC 2009 35

• Thermal conductivity variation with angle:

– ~1% variation in graphene (p/3 angle period)

– ~10% variation in dimerite

– ~25% variation in nanoribbon by Molecular Dynamics*

• Triangular graphene nanoribbon with 30o vertex up to 2x

thermal anisotropy (rectification)*

Thermal Anisotropy in Graphene

*Hu, arxiv/cond-mat (2009)**Jiang, Phys. Rev. B (2009)

ballistic simulation**

Graphene

Dimerite

E. Pop / DRC 2009 36

• Graphene nanoribbons: does the WFL hold in 1D? YES

• 1D ballistic electrons carry energy too, what is their equivalent thermal conductance?

Wiedemann-Franz Law in Nanoribbons

2 222 2

23 3

Bth e

B eG L T

k kT

hh

T

e

p p

(x2 if electron spin included)

Greiner, Phys. Rev. Lett. (1997)

0.28thG nW/K at 300 K

E. Pop / DRC 2009 37

• Same thermal conductance quantum, irrespective of the carrier statistics (Fermi-Dirac vs. Bose-Einstein)

Phonon Quantum Thermal Conductance

Phonon Gth measurement in

GaAs bridge at T < 1 K

Schwab, Nature (2000)

SWNT

suspended

over trench

Pt

Pt 1 μm

Single nanotube Gth=2.4 nW/K at T=300K

Pop, Nano Lett. (2006)

nW/K at 300 K2 2

0.283

tB

h

k TG

h

p

E. Pop / DRC 2009 38

• Thermal conductivity 3000-5000 Wm-1K-1 at 300 K (k↓↑T)

– Phonon-dominated, electron contribution < 10%

– Phonon mean free path ~0.7 μm

• Heat capacity (graphite) ~0.7 kJ/kg at room temperature (C↑↑T)

• Thermal boundary resistance with SiO2 ~4x10-8 m2K/W (SO phonon)

• Role of dimensionality:

– Neither “2-D” nor “3-D”

– Flexing mode (ω~k2) dominates heat capacity at T<50 K

Graphene Thermal Properties Summary

What we think we know…

Much we don’t know…

• Temperature dependence of k, C, TBR

• Role of substrate interaction on all of the above

• Role of edges, defects, doping, isotopes…

E. Pop / DRC 2009 39

• Post-doc:

– Dr. Myung-Ho Bae

• Grads:

– V. Dorgan, D. Estrada, A. Liao, Z.-Y. Ong

– B. Ramasubramanian, T. Tsafack, F. Xiong

• Undergrads:

– Aidee, Gautam, Ryan, Shreya, Sumit, Tim, Yang

• Collaborators:

– D. Jena (Notre Dame), M. Hersam (NWU), W. King (MechSE), J.-P. Leburton

(ECE), J. Lyding (ECE), N. Mason (Physics), U. Ravaioli (ECE), M. Shim (MatSE)

• Funding:

– NASA, NRI, UIUC (Beckman Award)

– ONR, NSF, DARPA Young Faculty Award

Acknowledgements