LETTERS
PUBLISHED ONLINE: XX MONTH XXXX | DOI: 10.1038/NMAT2753
Dimensional crossover of thermal transport in
few-layer graphene
Suchismita Ghosh1 , Wenzhong Bao2 , Denis L. Nika1,3 , Samia Subrina1 , Evghenii P. Pokatilov1,3 ,
Chun Ning Lau2 and Alexander A. Balandin1 *
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Graphene1 , in addition to its unique electronic2,3 and optical
properties4 , revealed unusually high thermal conductivity5,6 .
The fact that the thermal conductivity of large enough
graphene sheets should be higher than that of basal planes
of bulk graphite was predicted theoretically by Klemens7 .
However, the exact mechanisms behind the drastic alteration
of a material’s intrinsic ability to conduct heat as its
dimensionality changes from two to three dimensions remain
elusive. The recent availability of high-quality few-layer
graphene (FLG) materials allowed us to study dimensional
crossover experimentally. Here we show that the roomtemperature thermal conductivity changes from ∼2,800 to
∼1,300 W mK−1 as the number of atomic planes in FLG
increases from 2 to 4. We explained the observed evolution
from two dimensions to bulk by the cross-plane coupling of the
low-energy phonons and corresponding changes in the phonon
Umklapp scattering. The obtained results shed light on heat
conduction in low-dimensional materials and may open up FLG
applications in thermal management of nanoelectronics.
One of the unresolved fundamental science problems8 , with
enormous practical implications9 , is heat conduction in lowdimensional materials. The question of what happens with thermal
conductivity when one goes to strictly two-dimensional (2D) and
1D materials has attracted considerable attention8,10–13 . Thermal
transport in solids is described by thermal conductivity K through
the empirical Fourier law, which states that JQ = −K ∇T , where the
heat flux JQ is the amount of heat transported through the unit
surface per unit time and T is temperature. This definition has
been used for bulk materials as well as nanostructures. Many recent
theoretical studies8,10–13 suggest an emerging consensus that the
intrinsic thermal conductivity of 2D and 1D anharmonic crystals
is anomalous and reveals divergence with the size of the system
defined either by the number of atoms N or linear dimension L.
In 2D this universal divergence leads to K ∼ ln(N ) (ref. 8). The
anomalous nature of heat conduction in low-dimensional systems is
sometimes termed as breakdown of Fourier’s law11 . The anomalous
K dependence on L in strictly 2D (1D) materials should not be
confused with the length dependence of K in the ballistic transport
regime frequently observed at low temperatures when L is smaller
than the phonon mean free path (MFP). Ballistic transport has been
studied extensively in carbon nanotubes14–16 . Here we focus on the
diffusive transport and thermal conductivity limited by the intrinsic
phonon interactions via Umklapp processes17 .
Experimental investigation of heat conduction in strictly 2D
materials has essentially been absent owing to the lack of proper
material systems. Thermal transport in conventional thin films
still retains ‘bulk’ features because the cross-sections of these
structures are measured in many atomic layers. Heat conduction in such nanostructures is dominated by extrinsic effects,
for example, phonon–boundary or phonon–defect scattering18 .
The situation has changed with the emergence of mechanically
exfoliated graphene—an ultimate 2D system. It has been established experimentally that the thermal conductivity of largearea suspended single-layer graphene (SLG) is in the range K ≈
3,000–5,000 W mK−1 near room temperature5,6 , which is clearly
above the bulk graphite limit K ≈ 2,000 W mK−1 (ref. 7). The upper
bound K for graphene was obtained for the largest SLG flakes
examined (∼20 µm × 5 µm). The extraordinarily high value of K
for SLG is related to the logarithmic divergence of the 2D intrinsic
thermal conductivity discussed above. The MFP of the low-energy
phonons in graphene and, correspondingly, their contribution to
thermal conductivity are limited by the size of a graphene flake
rather than by Umklapp scattering7,19 .
The dimensional crossover of thermal transport, that is,
evolution of heat conduction as one goes from 2D graphene to
3D bulk, is of great interest for both fundamental science8 and
practical applications9 . We addressed this problem experimentally
by measuring the thermal conductivity of FLG as the number
of atomic planes changes from n = 2 to n ≈ 10. A high-quality
large-area FLG sheet, where the thermal transport is diffusive and
limited by intrinsic rather than extrinsic effects, is an ideal system
for such a study. A number of FLG samples were prepared by
standard mechanical exfoliation of bulk graphite1 and suspended
across trenches in Si/SiO2 wafers (Fig. 1a–c). The depth of the
trenches was ∼300 nm and the trench width varied in the range
1–5 µm. Metal heat sinks near the trench edges were fabricated by
shadow mask evaporation. We used our previous experience in
graphene device fabrication20 to obtain the highest quality set of
suspended samples. The width of the suspended flakes varied from
W ≈ 5 to 16 µm. The number of atomic layers in the graphene
flakes was determined with micro-Raman spectroscopy (InVia,
Renishaw) through deconvolution of the 2D band in graphene’s
spectra21 . The measurements of K were carried out using a steadystate optical technique developed by us on the basis of micro-Raman
spectroscopy5 . The shift of the temperature-sensitive Raman G peak
in graphene’s spectrum22 defined the local temperature rise in the
suspended portion of FLG in response to heating by an excitation
laser in the middle of the suspended portion of the flakes (Fig. 1d).
The thermal conductivity was extracted from the power dissipated
in FLG, the resulting temperature rise and the flake geometry
through the finite-element method solution of the heat diffusion
equation (Fig. 1e; see the Methods section).
1 Nano-Device
Laboratory, Department of Electrical Engineering and Materials Science and Engineering Program, University of California–Riverside,
Riverside, California 92521, USA, 2 Department of Physics and Astronomy, University of California–Riverside, Riverside, California 92521, USA,
3 Department of Theoretical Physics, State University of Moldova, Chisinau, Republic of Moldova. *e-mail: [email protected].
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LETTERS
NATURE MATERIALS DOI: 10.1038/NMAT2753
a
Laser light
b
Reflected light
Substrate
Metal heat sink
Metal heat sink
Heat sink
FLG
Heat sink
Graphene
Silicon oxide
Silicon oxide
Trench
Silicon substrate
c
d
1,581
G-peak shift for BLG
Linear fit to experimental data
G-Peak position (cm¬1)
1,580
e
1,579
1,578
1,577
Slope: |
1,576
Min: 300
G
1
Max: 350
/ PD| = 0.701 cm¬1 mW¬1
3
2
Laser power on the sample (mW)
Figure 1 | Samples and measurement procedure. a, Schematic of the thermal conductivity measurement showing suspended FLG flakes and excitation
laser light. b, Optical microscopy images of FLG attached to metal heat sinks. c, Coloured scanning electron microscopy image of the suspended graphene
flake to clarify a typical structure geometry. d, Experimental data for Raman G-peak position as a function of laser power, which determines the local
temperature rise in response to the dissipated power. e, Finite-element simulation of temperature distribution in the flake with the given geometry used to
extract the thermal conductivity.
a
b
Thermal conductivity (W mK¬1)
Integrated Raman intensity (a.u.)
6,000
105
2 layers
3 layers
4 layers
Multiple layers
Kish graphite
HOPG
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0
0.5
1.0
1.5
2.0
2.5
Laser power on the sample (mW)
3.0
5,000
Theory prediction
Theory prediction with increased roughness
Average value for SLG
4,000
3,000
High-quality bulk graphite
2,000
1,000
3.5
Room-temperature measurements
Maximum5
Regular bulk graphite
0
1
2
3
4
5
6
Number of atomic planes
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8
9
Figure 2 | Experimental data. a, Integrated Raman intensity of the G peak as a function of the laser power at the sample surface for FLG and reference bulk
graphite (Kish and highly ordered pyrolytic graphite (HOPG)). The data were used to determine the fraction of power absorbed by the flakes. b, Measured
thermal conductivity as a function of the number of atomic planes in FLG. The dashed straight lines indicate the range of bulk graphite thermal
conductivities. The black diamonds were obtained from the first-principles theory of thermal conduction in FLG based on the actual phonon dispersion and
accounting of all allowed three-phonon Umklapp scattering channels. The green triangles are Callaway–Klemens model calculations, which include
extrinsic effects characteristic for thicker films.
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The power dissipated in FLG was determined through the
calibration procedure based on comparison of the integrated
G
Raman intensity of FLG’s G peak IFLG
and that of reference bulk
G
graphite IBULK
. Figure 2a shows measured data for FLG with n = 2,
2
3, 4, ∼8and reference graphite. An addition of each atomic
G
plane leads to an IFLG
increase and convergence with the graphite
G
G
while the ratio ς = IFLG
/IBULK
stays approximately independent
of excitation power, indicating proper calibration. In Fig. 2b we
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LETTERS
NATURE MATERIALS DOI: 10.1038/NMAT2753
a
Phonon energy (meV)
180
b 15
TO1 ,TO
2
LO1 , LO2
150
, LA 2
LA 1
ZO1 ,ZO2
120
90
TA 1
60
, TA 2
0
TA1
TA2
6
ZA1
3
ZA 1
0
LA2
9
ZA 2
30
LA1
ZA2
12
0
0.2
0.4
0.6
0.8
1.0
Normalized phonon wavevector q/qmax
c
0
0.05
0.10
Normalized phonon wavevector q/qmax
d
(I)
(II) (III)
q'
5,000 W mK¬1
q
400 W mK¬1
q'
q"
q"
BLG
Phonon flux suppression
10 W mK¬1 by Umklapp scattering
Graphene
K(q) (W mK¬1)
q"
q"
0.1 W mK¬1
K supression
q
M
BLG
q
0
1
q"
q'
Graphene
0.2
0.4
0.6
0.8
1.0
Normalized phonon wavevector q/qmax
q
M
1
Figure 3 | Theoretical interpretation. a, Phonon dispersion in BLG, which shows that phonon branches are double degenerate for q > 0.2qmax (where
qmax = 14.749 nm−1 ). b, Close-up of the phonon dispersion in graphene (solid curves) and BLG (added dashed curves) near the Brillouin zone centre. Note
that the longitudinal acoustic (LA2 ) and transverse acoustic (TA2 ) phonon branches in BLG have a very small slope, which translates to a low phonon
group velocity. c, Contributions to thermal conductivity of different phonons, indicating the range of phonon wavevectors where Umklapp scattering is the
main thermal-transport-limiting mechanism. d, Diagram of three-phonon Umklapp scattering in graphene and BLG, which shows that in BLG there are
more states available for scattering.
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present measured K as a function of the number of atomic planes
n in FLG. The maximum and average K values for SLG are also
shown. As graphene’s K depends on the width of the flakes19,23
the data for FLG are normalized to the width W = 5 µm to
allow for direct comparison. At fixed W the changes in the K
value with n are mostly due to modification of the three-phonon
Umklapp scattering. The thermal transport in our experiment is
in the diffusive regime because L is larger than the phonon MFP
in graphene, which was measured6 and calculated24 to be around
∼800 nm near room temperature. Thus, we explicitly observed
heat conduction crossover from 2D graphene to 3D graphite as n
changes from 2 to ∼8.
It is striking that the measured K dependence on FLG thickness
h × n (h = 0.35 nm) is opposite from what is observed for
conventional thin films with thicknesses in the range of a few
nanometres to micrometres. In conventional films with thickness H
smaller than the phonon MFP, thermal transport is dominated by
phonon–rough boundary scattering. The thermal conductivity can
be estimated from K = (1/3)CV υ 2 τ , where CV is the specific heat
and υ and τ are the average phonon velocity and lifetime. When
the phonon lifetime is limited by the boundary scattering, τ = τB ,
one can write, following Ziman, (1/τB ) = (υ/H )((1 − p)/(1 + p)),
which shows that K scales down with decreasing thickness (here p
is a parameter defined by the surface roughness). In FLG with 2 or
3 atomic layer thickness there is essentially no scattering from the
top and back surfaces (only the edge scattering is present23 ). Indeed,
FLG is too thin for any cross-plane velocity component and for
random thickness fluctuations, that is, p ≈ 1 for FLG. To understand
the thermal crossover one needs to examine the changes in the
intrinsic scattering mechanisms limiting K : Umklapp scattering
resulting from crystal anharmonicity.
The crystal structure of bilayer graphene (BLG) consists of
two atomic planes of graphene bound by weak van der Waals
forces resulting in a phonon dispersion different from that
in SLG (Fig. 3a,b). By calculating the phonon dispersion in
FLG from first principles25 , we were able to study phonon
dynamics (Supplementary Movie) and the role of separate phonon
modes. Figure 3c shows the contributions �K (qi ) to the thermal
conductivity of phonons with wavevectors from the interval
(qi ,qi+1 ). These contributions, �counted over the
� whole Brillouin
q
zone, make up the total K = 0 max K (q) dq = m
1 �K (qi ), which
is, in our calculation, limited by the phonon Umklapp and edge
boundary scattering. In BLG the number of available phonon
branches doubles (Fig. 3a,b). However, the new conduction
channels do not transmit heat effectively because the group velocity
of the LA2 and TA2 branches is close to zero. From the other
side, an increase in the thickness h × n leads to a corresponding
decrease in the flux density: see region I of small q where thermal
transport is mostly limited by edge scattering (Fig. 3c). In region
III of large q, the number of phonon states available for threephonon Umklapp scattering in BLG increases by a factor of four
as compared with SLG (Fig. 3d). As a result, despite the increase
in the number of conduction channels in BLG, its K decreases
because the q phase space available for Umklapp scattering increases
even more (Supplementary Materials). One can say that in SLG,
phonon Umklapp scattering is quenched and the heat transport
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LETTERS
NATURE MATERIALS DOI: 10.1038/NMAT2753
25
is limited only by the edge (in-plane) boundary scattering. This
is in agreement with Klemens’s explanation of the higher thermal
conductivity of graphene compared with graphite7 . It is rooted in
the fundamental properties of 2D systems discussed above. Our
theory also explains molecular-dynamics simulations for ‘unrolled’
carbon nanotubes, which revealed a much higher K for graphene
than for graphite26 . The theoretical data points for n = 1–4 shown
in Fig. 2b are in excellent agreement with the experiment. They
were obtained by accounting for all allowed Umklapp processes
through extension of the diagram technique developed by us for
SLG (ref. 23) to n = 2 and 3. The lower measured K values for
n = 4 are explained by the stronger extrinsic effects resulting from
difficulties in maintaining the thickness uniformity in such samples.
The theoretical data point approaches the bulk limit for the ‘ideal’
graphite. It is possible that the crossover point may be related
to AB Bernal stacking and completeness of the graphite unit cell.
Practically, its position is affected by many factors including FLG’s
quality, the width of the flake and the strength of substrate coupling.
Thus, we have experimentally demonstrated 2D → 3Ddimensional crossover of heat conduction in FLG. We related the
increased thermal conductivity of graphene to the fundamental
properties of 2D systems and elucidated the physical mechanisms
behind the thermal conductivity reduction in few-atomic-thick
crystals. The obtained results are important for the proposed
graphene and FLG applications in nanoelectronics.
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Methods
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The trenches in the Si/SiO2 wafers were made using reactive ion etching (STS).
The evaporated metal heat sinks ensured proper thermal contact with the flakes
and a constant temperature during the measurements (Fig. 1b,c). FLG samples
were heated with a 488 nm laser (argon ion) in the middle of the suspended part
(Fig. 1a). The size of the laser spot was determined to be 0.5–1 µm. The diameter
of the strongly heated region on the flake was larger owing to the indirect nature
of energy transfer from light to phonons. The laser light deposits its energy to
electrons, which propagate with Fermi velocity vF ≈ 106 m s−1 . The characteristic
time for energy transfer from electrons to phonons is of the order of τph ∼ 10−12 s
(refs 27, 28). The measurement time in our steady-state technique is a few
minutes, which is much larger than τph but at the same time, much smaller than
the time required to introduce any laser-induced damage to the sample. The
correction to the hot-spot size is estimated as le–ph = vF τph ∼ 1 µm. Strain, stress
and surface charges change the G-peak position in Raman spectra and may lead
to its splitting29,30 . Strain and doping may also affect the thermal conductivity.
To minimize these effects, we selected samples where a symmetric and narrow G
peak was in its ‘standard’ location (∼1,579 cm−1 ) characteristic for the undoped
unstrained graphene21,22,29,30 . No bias was applied to avoid charge accumulation30 .
It was not possible to mechanically exfoliate FLG flakes with different numbers
of atomic planes n and the same geometry. To avoid damage to the graphene,
the obtained flakes were not cut to the same shape. Instead, we solved the heat
diffusion equation numerically for each sample shape to extract its thermal
conductivity (Fig. 1e). This was accomplished through an iteration procedure
(Supplementary Information) for the Gaussian-distributed laser intensity and an
effective spot size corrected to account for le–ph . The errors associated with the
laser spot size and intensity variation were ∼8%, that is, smaller than the error
associated with the local temperature measurement by Raman spectrometers
(∼10–13%) determined by the spectral resolution of the instrument and random
experimental data spread (see Display 2). The effects of the thermal contact
resistance were eliminated by using large-sized metallic heat sinks and ensuring a
large FLG–sink contact area.
59
Received 22 October 2009; accepted 24 March 2010;
published online XX Month XXXX
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References
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Acknowledgements
A.A.B. acknowledges support from ONR through award N00014-10-1-0224,
ARL/AFOSR through award FA9550-08-1-0100 and SRC - DARPA through the FCRP
Center on Functional Engineered Nano Architectonics (FENA) and the Interconnect
Focus Center (IFC).
Author contributions
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29. Mohiuddin, T. M. G. et al. Uniaxial strain in graphene by Raman spectroscopy:
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A.A.B. conceived the experiment, led the data analysis, proposed theoretical interpretation
and wrote the manuscript; S.G. carried out Raman measurements; S.S. carried out
finite-element modelling for thermal data extraction; W.B. prepared most of the
samples; C.N.L. supervised the sample fabrication; D.L.N. and E.P.P. assisted with theory
development and carried out computer simulations of thermal conductivity.
Additional information
The authors declare no competing financial interests. Supplementary information
accompanies this paper on www.nature.com/naturematerials. Reprints and permissions
information is available online at http://npg.nature.com/reprintsandpermissions.
Correspondence and requests for materials should be addressed to A.A.B.
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Page 1
Query 1: Line no. 1
Please provide post code for third affiliation.
Query 2: Line no. 6
Please check that the intended meaning of the
sentence ‘However, the exact ... remain elusive.’ in
the first paragraph has been retained after editing.
Query 3: Line no. 20
Note that the style states ‘The style guide states
‘In general, avoid using authors’ names and phrases
such as ‘A. N. Author has shown that’ in the text. For
example, say ‘Water flows downhill2’ rather than,
‘Garwin has shown2 that water flows downhill.’
There are occasions when this rule can be broken,
for example when two authors’ results are being
compared or when the significance of an author’s
contribution needs emphasis.’. Please check the use
of ‘Klemens’ throughout.
Query 4: Line no. 43
Can ‘via’ be changed to ‘through’ here?
Query 5: Line no. 49
To ensure that the text is formatted
correctly, please confirm the function of ‘–’ in
‘phonon–boundary’ and ‘phonon–defect’ here. Is it
a simple hyphen, or an en-rule taking the place of a
conjunction or a preposition?
Query 6: Line no. 59
‘extraordinary’ changed to ‘extraordinarily’ here.
OK?
Query 7: Line no. 63
Should ‘a graphene flake’ be ‘the graphene flake’
here?
Query 8: Line no. 82
Please check that the intended meaning of the
sentence ‘The number of ... graphene’s spectra’ has
been retained, and see the query below.
Query 9: Line no. 83
Is ‘spectra’ correct here, or should it be
‘spectrum’?
Page 2
Query 10: Line no. 1
Unit changed to ‘cm−1 mW−1 ’ in figure 1d. OK?
Query 11: Line no. 5
In figure 2b’s caption, should ‘black diamonds’
be ‘blue diamonds’?
Query 12: Line no. 5
Please check ‘An addition of each’. Should it be,
for example, ‘Each addition of an’?
Query 13: Line no. 6
According to style, ‘while’ can be used only in
reference to time. Is it best changed to ‘and’ or
‘whereas’ here?
Page 3
Query 14: Line no. 1
Please check that the intended meaning of the
sentence ‘Figure 3c shows ... interval (qi ,qi+1 ).’ has
been retained after editing.
Query 15: Line no. 1
Please check that the intended meaning of figure
3c’s caption has been retained after editing.
Query 16: Line no. 1
Can ‘Normalized phonon wave vector’ be
removed from the x axes of figure 3a–c, to leave
‘q/qmax ’?
Query 17: Line no. 12
In the sentence ‘The black diamonds ... Umklapp
scattering channels.’ can ‘accounting of’ be changed
to ‘accounting for’?
Query 18: Line no. 16
Please check that the intended meaning of
the sentence ‘It is striking ... nanometres to
micrometres.’ has been retained after editing.
Query 19: Line no. 18
In line with the query above, please indicate
the function of ‘–’ here. (If it takes the place of a
conjunction/preposition, it will be changed to a
solidus (/), according to style, as the second element
seems to consist of two words (rough boundary).)
Query 20: Line no. 22
Is a reference required here for ‘Ziman’? (In
addition, see the note above about the use of author
names in the text.)
Query 21: Line no. 31
‘due to’ changed to ‘resulting from’ here. OK?
Query 22: Line no. 32
Text here changed from ‘BLG’ to ‘bilayer
graphene (BLG)’, to define ‘BLG’, and ‘bilayer
graphene’ changed to ‘BLG’ throughout for
consistency. OK?
Query 23: Line no. 47
Can ‘From the other side’ be changed to ‘On the
other hand’ here?
Page 4
Query 24: Line no. 2
Please check that the intended meaning of the
sentence ‘This is in ... compared with graphite’ has
been retained after editing.
Query 25: Line no. 13
‘due to’ changed to ‘resulting from’ here. OK?
Query 26: Line no. 23
Can ‘in few-atomic-thick crystals’ be changed to,
for example, ‘in crystals that are a few atomic layers
thick’ or ‘in few-atom-thick crystals’ here?
Query 27: Line no. 27
Please check that the intended meaning of the
sentence ‘The trenches in ... ion etching (STS).’ has
been retained after editing.
Query 28: Line no. 33
‘somewhat’ deleted here before ‘larger’, according
to style. If necessary, please provide an alternative
qualifier.
Query 29: Line no. 45
Please check that the intended meaning of the
sentence ‘It was not ... the same geometry.’ has been
retained after editing.
Query 30: Line no. 55
What is ‘Display 2’ referring to here?
Query 31: Line no. 69
Please provide page range for ref. 4.
Query 32: Line no. 80
Please provide volume number for ref. 9.