Thermal Conductivity of Graphene
Alexander A. Balandin
Nano-Device Laboratory
Department of Electrical Engineering and
Materials Science and Engineering Program
University of California – Riverside
Outline
Š Motivations for Thermal Conductivity Study
Š Temperature Coefficients of Graphene
Š Measured Thermal Conductivity of Graphene
Š Theory of Thermal Conduction in Graphene
Š Conclusions
2
Balandin Group, UCR
Raman Spectroscopy of Graphene
Graphene is a single atomic layer of carbon atoms by definition!
Optical visualization on Si/SiO2 substrates
D band: A1g (~1350 cm-1); G peak: E2g; 2D band
AFM inspection
Alternatives:
Æ low-temperature transport study
Æcross-sectional TEM
Æ few other exotic methods…
A.C. Ferrari et al., Phys. Rev. Lett. 97, 187401 (2006).
A. Gupta et al., Nano Lett., 6, 2667 (2006).
I. Calizo, A.A. Balandin et al., Nano Lett., 7, 2645 (2007)
D. Graf et al., Nano Lett., 7, 238 (2007).
Balandin Group, UCR
3
Raman Nanometrology of Graphene:
Counting the Number of Atomic Planes
24000
Deconvolution of 2D (G’) Band
Graphene @ 300K
λ exc = 488 nm
20000
Intensity (arb. units)
5 layers
16000
4 layers
12000
3 layers
8000
4000
2 layers
1 layer
0
2300
2400
2500
2600
2700
2800
2900
3000
-1
Raman Shift (cm )
I. Calizo, et al., Appl. Phys. Lett., 91, 201904 (2007).
I. Calizo, et al., Appl. Phys. Lett., 91, 071913 (2007).
Balandin Group, UCR
2D-band features of graphene are reproducible
and can be used to count the number of
graphene layers.
4
Graphene Temperature Coefficients:
Raman Spectrometer as “Thermometer”
Note: the sign is negative
Bi-Layer Graphene
Intensity (arb. units)
1584
G Peak Position
Linear Fit of Data
-1
G Peak Position (cm )
1582
12500 Bi-layer
10000
Graphene
λexc = 488 nm
7500
N. Mounet et al,
Phys. Rev. B 71,
205214 (2005).
5000
2500
1530
1580
1578
G Peak
1580 cm-1
1575
1620
Raman Shift (cm-1)
N. Bonini, et al.,
Phys. Rev. Lett., 99,
176802 (2007).
-1
slope = -0.015 cm /°C
1576
-150
-75
0
75
Temperature (°C)
Temperature is controlled externally; very low excitation
power on the sample surface is used (< 0.5 – 1 mW).
Phonon frequency downshift with T is unusual for optical mode
when the bond-bond distances shorten with T since normally
lattice contraction leads to the upward shift of the frequencies.
Balandin Group, UCR
5
Thermal Conductivity Measurements with
Micro-Raman Spectrometer
Idea of the Experiment:
Æ Induce locale hot spot in the middle of the
suspended graphene flake and monitor
temperature rise in the middle with increasing
excitation laser intensity.
G mode
D mode
Importance of the Suspended Portion of Graphene
Æ Formation of the specific in-plane heat front
propagating toward the heat sinks
Æ Reduction in the graphene – substrate coupling
Æ Determining the fraction of the power dissipated in
graphene through callibration procedure
Graphene Specifics:
Æ Atomic thickness: good for this method; bad for formal definition of thermal conductivity
Æ Heat transport: diffusive or partially diffusive thermal transport
Æ In-plane phonon modes: less effect from the substrate and possibility of graphite calibration
Balandin Group, UCR
Experimental Approach and Suspended
Graphene Layers
Graphene flakes suspended across trenches in
Si/SiO2 wafers
Trench
substrate
FLG
SLG
Trench
FLG
Longer flakes are preferable to be in the diffusive regime
Cooperation: C.N. Lau (UC – Riverside)
Balandin Group, UCR
7
Few-Layer Suspended Graphene
Measurements
Improvements in the
accuracy of the
measurements:
Æ Massive metal heat
sinks
ÆData extraction for
actual shape of the
Æ Cross-checking the
data for light
absorption in
graphene layers with
known results
Æ Monitoring SiO2
temperature: W2 and
W3 Si-O-Si stretching
bond position (800 –
1100 cm-1)
Collaboration with C.N. Lau (UC – Riverside)
Balandin Group, UCR
8
Extraction of the Thermal Conductivity Data:
Raman Spectrometer as a Thermometer
Excitation laser acts as a heater: ΔPG
Raman spectrometer acts as a
thermometer: ΔTG=Δω/χG
Thermal conductivity: K=(L/2aGW)(ΔPG/ΔTG)
4
-1
G PEAK POSITION SHIFT (cm )
SUSPENDED GRAPHENE
Thermal conductivity of rectangular flake (L is
the half-length):
K = ( L / 2 a G W ) χ G ( Δ ω / Δ PG ) − 1 .
Connect ΔPD Å Æ ΔPG through calibration
A.A. Balandin, S. Ghosh, W. Bao, I. Calizo, D. Teweldebrhan,
F. Miao and C.N. Lau, Nano Letters, 8: 902 (2008).
Balandin Group, UCR
EXPERIMENTAL POINTS
LINEAR FITTING
2
0
-2
-4
-6
-1
SLOPE: -1.292 cm /mW
0
1
2
3
POW ER CHANGE (mW )
4
9
Modeling Based Data Extraction for the
Arbitrary Shaped Graphene Flakes
Investigation of heat conduction in multi-layer graphene led to the use of the
irregular shaped graphene flakes due to the difficulty of the sample preparation.
The excitation source follows Gaussian
distribution:
⎛ x2 + y2 ⎞
⎟⎟
Q = P( x, y ) = P0 exp⎜⎜ −
2
2
σ
⎝
⎠
FWHM occurs at 0.5 μm
Finite-element solution of the heat
diffusion through the graphene flake
was obtained in order to take into
account the actual shape of the flake. 10
Balandin Group, UCR
Power Dissipated in Graphene:
Calibration Procedure
ΔI G = N σ G I o ,
Δ I G = ( N / A )(σ G / α G aG ) PG /(1 + RSi ).
ΔIHOPG
PD
REFERENCE HOPG
SUSPENDED GRAPHENE
5
10
-1
~ 1583 cm
INTENSITY (ARB. UNITS)
PG = α G a G (1 + R Si ) I o A ,
INTEGRATED INTENSITY (ARB. UNITS)
ΔIG
PG
6
10
4
10
G PEAK
3
LASER POWER: ~ 2 mW
10
1550
1600
1650
-1
RAMAN SHIFT (cm )
0
1
2
3
4
5
EXCITATION POWER ON SAMPLE (mW)
PD=IoA
Formula for data extraction:
∞
Δ I HOPG = N σ H I o ∑ exp( − 2α H a H n ),
n =1
Δ I HOPG = (1 / 2)( N / A)(σ H / α H a H ) PD (1 − R H ),
PG = (ς / 2)[σ H α G aG / σ Gα H aH ](1 + RSi )(1 − RH ) PD .
S. Ghosh, I. Calizo, D. Teweldebrhan, E.P. Pokatilov, D.L. Nika, A.A.
Balandin, W. Bao, F. Miao and C. N. Lau, Appl. Phys. Lett., 92, 151911 (2008)
Balandin Group, UCR
ς = Δ I G / Δ I HOPG
11
Thermal Conductivity of Single Layer Graphene:
Comparison with Carbon Nanotubes
Table I: Experimental RT Thermal Conductivity of Graphene and CNTs
Sample
K (W/mK)
Method
Comments
Reference
SLG
~ 3080 – 5300
optical
individual
Balandin et al., Nano Lett. (2008)
MW-CNT
~ 3000
electrical
individual
Kim et al., Phys. Rev. Lett. (2001)
SW-CNT
~ 3500
electrical
individual
Pop et al., Nano Lett. (2006)
SW-CNT
1750 – 5800
thermocouples
bundles
Hone et al., Phys. Rev. B (1999)
SW-CNT
3000 - 7000
thermocouples
individual
Yu et al., Nano Lett. (2005)
CNTs
~1500 - 2900
electrical
individual
Fujii et al., Phys. Rev. Lett. (2005)
Table II: Theoretical RT Thermal Conductivity of Graphene and CNTs
Sample
K (W/mK)
Method
Comments
Reference
CNTs
~6600
MD
predicted higher K for graphene
Berber et al. PRL (2000)
CNTs
~2980
MD
strong defect dependence
Che et al., Nanotech. (2000)
CNTs
1500-2250
MD
comparable with graphene
Osman et al., Nanotech. (2000)
graphite
~2000
C-K
basal plane (a-plane)
Klemens et al., Carbon (1994)
12
Balandin Group, UCR
Theoretical Interpretation of Experimental
Data: Simple Klemens Approximation
Æ Measured thermal conductivity of graphene: 3080 – 5300 W/mK
Æ Electron thermal conduction at RT: ~1% Wiedemann-Franz law
Æ Phonon MFP: ~ 775 nm at RT from K=(1/2)CVΛ
Umklapp-Limited Thermal
Conductivity in Graphene: Klemens
Approximation
Unlike in bulk graphite the phonon transport in
graphene is 2D all the way down to zero phonon
frequency. In bulk graphite it becomes 3D at ~4 THz.
N. Mounet et al, Phys. Rev. B 71, 205214 (2005).
Balandin Group, UCR
D.L. Nika, S. Ghosh, E.P. Pokatilov, A.A. Balandin, Thermal
conductivity of graphene flakes: Comparison with bulk
13
graphite, Appl. Phys. Lett., 94, 203103 (2009).
Callaway – Klemens Type Approach to
the Phonon Thermal Conductivity
The phonon heat flux:
r
r r
r
r
r
r r
r r
W = ∑ v ( s, q )hωs (q ) N (q , ωs (q )) = ∑ v ( s, q )hωs (q )n(q , ωs )
r
s ,q
r
s ,q
r
r ∂N (ω ) r r
r
W = −∑ (∇T ) β ∑ τ vβ ( s, q ) 0 s v ( s, q )hωs (q )
r
∂T
β
s ,q
Definition of the thermal conductivity:
Scattering processes included
into consideration:
Wα = −κ αβ (∇T ) β
The expression for thermal conductivity
Is obtained via Boltzmann’s equation:
κ=
1
4π kT h
2
Key challenge: capture the
specifics of the 2D material
∑ ∫ {hω (q) v (q)) τ
2
s
s =1...6
0
Balandin Group, UCR
1
1
1
1
=
+
+
.
τ tot ( s, q) τ U ( s, q) τ B ( s, q) τ Pd ( s, q)
×
qmax
×
Relaxation-time approximation
s
tot
( s, q )
Exp ( hω s ( q ) / kT )
( Exp ( hωs ( q ) / kT ) − 1)
2
q}dq
14
Phonon Dispersion and Gruneisen
Parameter in Graphene
Valence Force Field (VFF) calculation of
the phonon dispersion in graphene
The mode-dependent Gruneisen
parameters are measure of
sensitivity of the phonon frequencies
to changes in the system volume.
γ λ ,k = −
∂ ln(ωλ ,k )
∂ ln V
N. Mounet et al, Phys. Rev. B 71, 205214 (2005).
γ(graphite) = 1.59 – 2.0
Calculations are after D. Nika, E.P. Pokatilov, A.A.
Balandin, Phys. Rev. B (2008).
γ(CNT) = 1.24 after Reich et al., Phys Rev. B (2000)
γ(graphene)=1.06 after Hanfland et al., Phys. Rev. B
(1989); Popov et al., Diamond (2003)
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15
Phonon Transport in Graphene:
Accurate Theory vs Experiment
ÆSensitivity of the thermal
conductivity to the Gruneisen
parameter γ value
Æ Excellent agreement with the
mode-dependent γ(s, q)
Æ No hidden fitting parameters
D.L. Nika, E.P. Pokatilov, A.S.
Askerov and A.A. Balandin,
"Phonon thermal conduction in
graphene: Role of Umklapp and
edge roughness scattering,"
Physical Review B 79, 155413
(2009) - Editors' Selection
Calculations are performed for width W=5 mm and p=0.9
Balandin Group, UCR
T.M.G. Mohiuddin, et al.,, arXiv: 0812.1538
measurement of γ value for G peak agrees
with calculations of N. Mounet et al, Phys.
Rev. B 71, 205214 (2005).
16
Thermal Conductivity of Graphene
over Wide Temperature Range
Thermal conductivity in the lowtemperature limit is proportional to T2.
At T=100 K, TA modes transfer about
~28.5% of heat while LA modes
carry about 71.0%.
At T=400 K, TA and LA modes carry
~49% and 50% of heat,
correspondingly. The rest of the
modes, including out-of-plane
phonons, carry ~1% of heat in
graphene.
D. Nika, E.P. Pokatilov, A.S. Askerov and
A.A. Balandin, Phonon thermal conduction in
graphene: The role of Umklapp and edge
roughness scattering, Phys. Rev. B (2008);
also as arXiv:0812.0518
17
Balandin Group, UCR
Diffusive vs Ballistic Phonon Transport
in Graphene
Comparison of our experimental and theoretical
results with recent works of other groups
J.-W. Jiang, J.-S. Wang and B. Li, Directional
dependent thermal conductance of graphene,
arXiv: 0902.1836 (2009).
∞
σ (T ) = (1 / 2π )∫ T (ω )hω
0
df (T , ω )
dω
dT
Æ T(ω) is the number of phonon branches at ω in
the ballistic regime; phonon dispersion is in VFFM
Recalculation to thermal conductivity with typical
MFP gives values slightly higher than our result.
ÆRecalculation to thermal conductivity with the
characteristic lengths from our experiment gives
substantially higher values (factor of 5-10) than
our result.
Æ Good agreement with S. Berber,Y.-K. Kwon,
and D. Tomanek, Phys. Rev. Lett., 84, 4613
(2000).
σ/S=K/L; S=hW
Balandin Group, UCR
Conclusion: the phonon transport is most likely partially
diffusive in graphene; the ballistic limit is in agreement
18
Conclusions
Æ The measured thermal conductivity of graphene is in the range 3000 –
5000 W/mK at room temperature
Æ Experimental values agree well with the calculations based on the
Klemens approximation
Æ Thermal conductivity depends on the size of the flake (flake width)
Æ The thermal conductivity decreases with the addition of layers
approaching the bulk graphite limit
Æ Changes in three-phonon Umklapp scattering are likely responsible for the
evolution of thermal conductivity
Æ Excellent thermal properties of graphene are beneficial for all proposed
electronic applications and can lead to a thermal management
applications
19
Balandin Group, UCR
Acknowledgements
Nano-Device Laboratory (NDL) Group in from of the laboratory,
UCR 2007
Balandin Group, UCR
20