Graphene Thermal Physics
Eric Pop
Dept. of Electrical & Computer Engineering
Micro & Nanotechnology Lab, Beckman Institute
http://poplab.ece.illinois.edu
E. Pop / DRC 2009
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Table of Contents
• Heat capacity of graphene
+ Background
• Thermal conductivity of graphene
• Graphene devices
• “Other” physics
(thermoelectrics, asymmetry, quantum thermal conductance…)
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Structure of Graphite
c-axis
(abab) or
Bernal stacking
~ 0.335 nm
ab plane
(2 atoms / unit cell)
• 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)
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Heat Capacity: Energy Stored ΔU=CΔT
• Heat capacity of a solid: C = Celectrons + Clattice
CL 
du
df ( )
 
g   d 
dT
dT
ω(k)
phonon dispersion
density of states
phonon distribution
(e.g. Bose-Einstein)
• High temperature: classically recall C = 3NAkB
Dulong-Petit Law (1819)  most solids  25 J/mol/K at high T
• Low temperature: experimentally C  0
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Phonons: Atomic Lattice Vibrations
Graphene Phonons [100]
2 atoms
per unit cell
Lattice Constant, a
200 meV
60
xn+1
yn
transverse
small k
transverse
max k
u(r, t )  A exp[i(k  r  it )]
vS 
50
Freq
(Hz) ω (cm-1)
Frequency
xn
Energy (meV)
optical
yn-1
160 meV
40
100 meV
30
Si optical
20
60 meV
d
dk
300
10 K =
26 meV
k
Wave vector qa/2p
• 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
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Fermi-Dirac vs. Bose-Einstein Statistics
1
T=0 K
fBE
fFD
T=300 K
0.6
0.4
T=0 K
1.5
0.8
T=1000 K
0.2
0
-0.2
∞
2
1.2
1
T=300 K
T=1000 K
0.5
0
-0.4
-0.2
0
0.2
0.4
0.6
0
E-EF (eV)
f FD ( E ) 
1
 E  EF 
exp 
 1
 k BT 
Fermions = half-integer spin
(e.g. electrons, protons)
f BE ( E ) 
0.2
E (eV)
0.4
0.6
1
 E 
exp 
 1
 k BT 
Bosons = integer spin
(e.g. phonons, photons)
• In the limit of high energy, both reduce to the classical
Maxwell-Boltzmann statistics: f MB ( E )  exp   E / kBT 
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Heat Capacity: Dielectrics vs. Metals
Dielectrics
Metals
C = bT
• 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)
* assuming linear ω=vk phonon dispersion
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Heat Capacity: Einstein Model (1907)
• Key assumption: all oscillators have same frequency
• High-T (correct, recover Dulong-Petit Law):
 
E
Frequency ω
CE (T )  3NkB
• Low-T (incorrect, drops too fast)
CE (T )  3NkB
 
E
k BT
2
e
0
k
2p/a
 E / k BT
But… Einstein model
OK for optical phonon
heat capacity
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Annalen der Physik 39(4)
p. 789 (1912)
Peter Debye (1884-1966)
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Heat Capacity: Debye Model (1912)
• Key assumption: oscillators have linear ω-k with cutoff ωD
D 
1/3
vs
6p 2 N 

kB
• … roughly corresponds to max acoustic
phonon frequency cutoff
T 
12p
CD (T ) 
Nk B  
5
 D 
• High-T: (> 0.8 θD)
CD (T )  3NkB
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  vs k
kB D  D
• Low-T (< θD/10):
4
Frequency, 
• Often in terms of Debye temperature
3
0
Wave vector, k
In D-dimensions
when T << θD:
2p/a
CD  T D
10
Debye Model at Low- and High-T
10 7
3 3-K)
(J/m
Heat
Specific
)
-K
C (J/m
Specific Heat,
• In practice: θD ~ fitting
parameter to heat capacity data
• θD is related to “stiffness” of solid
and melting temperature
C  3kB  4.7 10 6
10 6
J
3NkBT
m3 K
Diamond
Each atom
has
Diamond
a thermal
energy
of 3KBT
10 5
10 4
 TT3
CC 
3
10 3
Classical
Regime
10 2
 D  1860 K
10 1 1
10
2
10
10
Temperature, T (K)
3
10
4
Temperature (K)
Pb, θD = 102 K
Ag, θD = 226 K
Al, θD = 428 K
Diamond
θD = 1860 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)
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Phonon Dispersion of Diamond & Graphite
LO
TO
ZO
LA
TA
300 K =
26 meV
ZA
• 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
Tohei, Phys. Rev. B (2006)
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Heat Capacity of Diamond & Graphite
Dulong-Petit 3NAkB high-temperature limit
θD
300 K
Silicon
SiO2
0.80
0.71
• 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
Pierson (1993)
Tohei, Phys. Rev. B (2006)
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Phonon Dispersion in Graphene
LO
meV
TO
ZO
LA
vLA ~ 23 km/s
(silicon ~ 9 km/s)
Si optical
phonons
TA
ZA
vTA ~ 16 km/s
(silicon ~ 5 km/s)
ω ~ k2
vZA ~ 0
out-of-plane bending or flexing mode
important for heat capacity at very low T
may be affected (suppressed) by substrate
Yanagisawa et al., Surf. Interf. Analysis 37, 133 (2005)
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Table of Contents
• Heat capacity of graphene
+ Background
• Thermal conductivity of graphene
• Graphene devices
• “Other” physics
(thermoelectrics, asymmetry, quantum thermal conductance…)
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Thermal Conductivity of Solids
Unlike electrical conductivity, thermal spans “only” 4 orders of magnitude
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Thermal Conductivity: Kinetic Theory
u(z+z)

qz
compare to… J diff  qDn
z + z
λ
Integrate energy flux over all angles:
1
du dT
dT
J q   v
 k
3 dT dz
dz
z
u(z-z)
dn
dz
z - z
heat capacity
heat gen. per unit
volume (W/m3)
Thermal conductivity:
Heat diffusion eq:
1
k  Cv
3
phonon velocity
dω/dk
in 3-D
·(kT )  q '''  0
mean free
path (MFP)
1  2
1 
v 


3
v
v
LA  
  TA
Poisson eq:
compare to… ·( V )    0
1
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Thermal Conductivity and Phonon MFP
• 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
C
λ
k
low T
 Td
λ  L (size)
 Td
high T
3NkB
 1/T
 1/T
kkl
Increasing Defect
Concentration
Boundary Defect
1
1
k  Cv  Cv 2
3
3
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k ~ 1/T
kkl ~TTd d
0.01
Phonon
Scattering
0.1
Temperature, T/D
1.0
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Thermal Conductivity of Graphite*
* typical commercial pyrolitic graphite
Thermal Conductivity (Wm-1K-1)
400
comparable
to copper
Copper
300
comparable to
plastics, SiO2
Pyrolytic Graphite, Any
ab- Direction
200
100
500
1000
1500
2000
2500
Silicon
SiO2
2.0
1.0
Pyrolytic Graphite, Any
c- Direction
500
1000
1500
2000
2500
140
1.4
High (100-1000x) thermal
anisotropy in all graphite
Temperature (oC)
Pierson (1993)
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Graphene Thermal Conductivity (Theoretical)
k (Wm-1K-1)
CNT and graphene
“ballistic upper limit”
T (K)
• 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)
Berber et al, Phys. Rev. Lett. (2000); Mingo et al, Phys. Rev. Lett. (2005)
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The Imprint of Dimensionality
bulk Si at room T (~140 Wm-1K-1)
Si nanowires
CNT and graphene
“ballistic upper limit”
Li et al, Appl. Phys. Lett. (2003)
Li et al, Appl. Phys. Lett. (2003)
• 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
Thermometer
E. Pop / DRC 2009
Heater
• Ex: Si nanowires ~ from 1-D to 3-D features!
21
Graphene Thermal Conductivity (Experimental)
• Graphene flakes
suspended over
SiO2 trenches (~3
um wide)
• Thermometry using
Raman G-band
(1583 cm-1) shift
• Room temperature
results in-line with
existing CNT data
~3080-5150
Balandin et al.
Balandin et al, Nano Lett. (2008); Ghosh et al, Appl. Phys. Lett. (2008)
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Wiedemann-Franz Law
• 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
1p2 2 T  2
kB n
 vF
3 2
EF 
q 2
  q n 
n
m
 e p 2 kB2
taking the ratio: L 

0
 T 3q 2
Experimentally
e  
Remarkable: independent
of n, m, and even !
• Graphene electronic contribution
appears <10% throughout T range
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
8
L0  2.45  10 WΩ/K 2
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Table of Contents
• Heat capacity of graphene
+ Background
• Thermal conductivity of graphene
• Graphene devices
• “Other” physics
(thermoelectrics, asymmetry, quantum thermal conductance…)
E. Pop / DRC 2009
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Background on Thermal Resistance
P = I2 × R
∆T = P × Rth
Rth 
L
kA
∆ V = I × Rel
R = f(∆T)
Fourier’s Law (1822)
Rel 
L
A
Ohm’s Law (1827)
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Thermal-Electrical Cheat Sheet
J q  kT
·(kT )  q '''  0
Rth 
L
kA
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J diff  qDn
·( V )    0
Rel 
L
A
26
Thermal Resistance of Devices
Single-wall
nanotube SWNT
100000
High thermal resistances:
• SWNT due to small thermal
conductance (very small d ~ 2 nm)
RTH (K/mW)
10000
1000
100
• Others due to low thermal
conductivity, decreasing dimensions,
increased role of interfaces
GST
Phase-change
Memory (PCM)
Silicon-onInsulator FET
10
Cu
SiO2
Power input also matters:
Cu Via
• SWNT ~ 0.01-0.1 mW
1
• Others ~ 0.1-1 mW
Bulk FET
Si
0.1
0.01
0.1
L (m)
1
10
∆T = P × Rth
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).
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Modeling Device Thermal Response
• Steady-state models
100000
– Finite-Element models
10000
RTH (K/mW)
– Lumped: Mautry (1990), Goodson-Su
(1994-5), Pop (2004), Darwish (2005)
1000
100
SOI FET
10
1
• Transient models
0.1
0.01
Bulk FET
0.1
L (m)
1
10
D
L
tSi
W
tBOX
Bulk Si FET
RTH 
1
1

2kSi D 2kSi LW
E. Pop / DRC 2009
SOI FET
RTH 
1
2W
1/ 2
 t BOX 


 k BOX kSi tSi 
28
Graphene FET
SiO2
Assumptions:
tox = 300 nm
L = 25 um
W = 6 um
kox = 1.3 Wm-1K-1
ksi = 80 Wm-1K-1
Si
TSi
 0.29
TG
60
L
W
20
RSi 
1
 510 K
2kSi 0-5LW-4 -3 -2W
Y (m)
TSi
RSi

 0.25
TG Rox  RSi
0.2 mW/μm2
40
0.1 mW/μm2
20
-1
0 0
-5
-4
-5
x 10
-3
-2
Y (m)
-1
0
x 10
-5
60
T (K)
T (K)
tOX
T (K)
60
t
Rox  ox40  1538 K
W
kox LW
40
20
Bae, Ong, Estrada, Pop (2009)
0
-3
-2
-1
Y (m)
E. Pop / DRC 2009
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x 10
1
-6
29
Raman Thermal Imaging of Graphene FET
2D peak
thermal conductivity
• 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
Freitag et al, Nano Lett. (2009)
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Table of Contents
• Heat capacity of graphene
+ Background
• Thermal conductivity of graphene
• Graphene devices
• “Other” physics
(thermoelectrics, asymmetry, quantum thermal conductance…)
E. Pop / DRC 2009
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Thermal Expansion of Graphite
c- direction
(van der Waals)
• Thermal expansion coefficient (TEC) is
– Positive and large in c- direction
Linear Thermal Expansion (x 10-6 1/K)
ab- direction
(covalent sp2)
Graphite
4
– Negative (at room temperature) in ab- plane
2
• Graphite thermal expansion anisotropy
can result in large internal stresses
-2
Pierson (1993)
E. Pop / DRC 2009
Graphite
0
Temperature (oC)
32
Graphene
• Recent estimate of thermal contraction
in single-layer suspended graphene*
• TEC ~ -7x10-6 K-1, 5-6x greater than
in graphite ab- plane
Linear Thermal Expansion (x 10-6 1/K)
Thermal Expansion of Graphene
Graphite
4
Graphite
2
0
-2
Temperature (oC)
*Bao et al, arxiv/cond-mat (2009)
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Thermoelectric Properties of Graphene
Wei (2009)
Zuev (2009)
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
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Thermal Anisotropy in Graphene
Graphene
Dimerite
• Thermal conductivity variation with angle:
– ~1% variation in graphene (p/3 angle period)
– ~10% variation in dimerite
ballistic simulation**
– ~25% variation in nanoribbon by Molecular Dynamics*
• Triangular graphene nanoribbon with 30o vertex  up to 2x
thermal anisotropy (rectification)*
*Hu, arxiv/cond-mat (2009)
**Jiang, Phys. Rev. B (2009)
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Wiedemann-Franz Law in Nanoribbons
• Graphene nanoribbons: does the WFL hold in 1D?  YES
• 1D ballistic electrons carry energy too, what is their
equivalent thermal conductance?
 p 2 k 2  e2 
p 2 kB2T
Gth  L eT   2B   T 
3h
 3e   h 
(x2 if electron spin included)
Gth  0.28 nW/K at 300 K
Greiner, Phys. Rev. Lett. (1997)
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Phonon Quantum Thermal Conductance
• Same thermal conductance quantum, irrespective of the
carrier statistics (Fermi-Dirac vs. Bose-Einstein)
Phonon Gth measurement in
GaAs bridge at T < 1 K
Schwab, Nature (2000)
Gth 
p 2 kB2T
3h
 0.28 nW/K at 300 K
Pt
1 μm
Single nanotube Gth=2.4 nW/K at T=300K
SWNT
Pop, Nano Lett. (2006)
suspended
over trench
Pt
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Graphene Thermal Properties Summary
What we think we know…
• 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
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…
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Acknowledgements
•
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
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