Graphene-based electronic devices have been the focus of

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Published: February 11, 2011

r 2011 American Chemical Society 1195 dx.doi.org/10.1021/nl104156y_|Nano Lett. 2011, 11, 1195–1200

LETTER

pubs.acs.org/NanoLett

Influence of Polymeric Residue on the Thermal Conductivity of

Suspended Bilayer Graphene

Michael Thompson Pettes,

†,||

Insun Jo,

‡,||

Zhen Yao,

‡,§

and Li Shi*

,†,§

†_{Department of Mechanical Engineering,}‡_{Department of Physics, and}§

Center for Nano and Molecular Science and Technology, Texas Materials Institute, The University of Texas at Austin, Austin, Texas 78712, United States

b

S Supporting Information

ABSTRACT:The thermal conductivity (κ) of two bilayer

graphene samples each suspended between two microre-sistance thermometers was measured to be 620( 80 and 560( 70 W m-1K-1at room temperature and exhibits aκ µ T1.5_{behavior at temperatures (T ) between 50 and 125 K.}

The lowerκ than that calculated for suspended graphene along with the temperature dependence is attributed to scattering of phonons in the bilayer graphene by a residual polymeric layer that was clearly observed by transmission electron microscopy.

KEYWORDS:Graphene, thermal conductivity, polymer, bilayer, transmission electron microscopy

G

raphene-based electronic devices have been the focus of intense investigation, driven by the hope that such devices can greatly outperform silicon devices due to the record-high electron mobility in graphene.1,2_{The high thermal conductivity,}_{κ, of graphene}

is another attractive attribute that can potentially find applications such as nanofillers in a polymeric matrix to enhance the effective thermal conductivity of the nanocomposites.3,4Micro-Raman spec-troscopy measurements have yielded thermal conductivity values for suspended single- to quadruple-layer graphene sheets above room temperature.5-8For single-layer suspended graphene, the reportedκ values are in the range of 1500-5800 W m-1 _K-1_{near room}

temperature and decrease with increasing temperature.5,6For sus-pended bilayer graphene, one Raman measurement obtained aκ value of ∼2800 W m-1K-1 near room temperature.8 _{In these}

measurements, a suspended graphene sheet was heated by a laser and the resulting temperature rise was obtained from either the red shift of the Raman G-band or 2D-band frequency,5,6 or from the Anti-Stokes/Stokes ratio.7_{Because of the limited temperature sensitivity of}

the Raman thermometry method, the temperature rises in the suspended graphene are often larger than 50 K. Consequently, accurate temperature dependence of the thermal conductivity has not yet been obtained, especially in the low-temperature range that is useful for understanding the fundamental physics behind the ob-served thermal conductivity.

Recently, the lattice thermal conductivity of freestanding graphene has been predicted to be dominated by contributions from the out-of-plane acoustic (ZA) phonons, that is, theﬂexural modes.9_{Using a}

suspended microresistance thermometer device, it has been reported that damping of the ZA phonons by a silicon dioxide support limits the thermal conductivity of supported single-layer graphene to∼600 W m-1K-1,10_{considerably lower than the basal-plane values of highly}

ordered pyrolytic graphite (HOPG).11,12It is of considerable interest

to employ similar sensitive microresistance thermometer devices to obtain the accurate thermal conductivity versus temperature (T ) relationship of suspended graphene. However, diﬃculty in the assembly of suspended graphene between two suspended micro-thermometers has made these much-needed measurements highly challenging. Most recently, some successes have been reported in the assembly of a 0.5μm long suspended monolayer graphene13_{and a 1}

μm long suspended ﬁve-layer graphene14_{between two suspended}

microthermometers. The measured thermal conductivity exhibits a T1.5_{dependence at temperatures lower than 150 K for these two}

suspended samples and was attributed to a dominant contribution from the ZA modes. However, the obtained room-temperature thermal conductivity is lower than 225 W m-1K-1in these samples, considerably lower than the theoretical prediction of the ZA contribution.

In this Letter, we report a transfer method that can be used to assemble suspended graphene on suspended measurement devices, as well as combined thermal conductance, micro-Raman spectrosco-py and transmission electron microscospectrosco-py (TEM) measurements of two suspended bilayer graphene samples. The measured thermal conductivity of our samples is considerably lower than the Raman measurement results5,6,8and theoretical predictions9,15for clean,ﬂat, suspended graphene of similar dimensions. However, our values are considerably higher than those reported in refs 13 and 14 for both suspended and supported graphene samples and slightly higher than those measured previously for supported single-layer graphene in our earlier work.10A T1.5dependence was also observed in the thermal conductivity measured at temperatures lower than 125 K for the samples in the present work. Our thermal conductivity modeling

Received: November 29, 2010 Revised: January 21, 2011

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shows that the T1.5_{dependence cannot be attributed to a dominant}

ZA contribution, which is in contrast to the mechanism proposed in refs 13 and 14. Instead, we attribute our observations to the presence of a residual polymeric layer that can be clearly observed by TEM on the suspended graphene. The result suggests that the residual polymer layer scatters phonons in graphene in a similar fashion as the SiO2

support in the previously reported supported single-layer graphene.10 The thermal conductance measurements were performed using a suspended microresistance thermometer device that has been developed for thermal and thermoelectric measure-ments of individual nanotubes, nanowires, and nanoﬁlms.16-18

The device consists of two adjacent 500 nm thick low-stress silicon nitride (SiNx) membranes each suspended by six 420μm

long SiNxbeams. One 35 nm thick, 200 nm wide, and 350μm

long serpentine platinum (Pt) resistance thermometer and two 1 μm-wide Pt electrodes were patterned on each membrane. A through-substrate hole was etched beneath the suspended mem-branes to allow for TEM characterization of the nanostructure suspended between the two membranes.

Inspired by a method described by Meyer et al.19we developed the graphene transfer process shown schematically in Figure 1. Bilayer graphene sheets were located via optical microscopy20of graphitic crystals mechanically exfoliated21_{from natural graphite}_{ﬂakes (NGS}

Naturgraphit GmbH) onto a 90 nm thick poly(methyl methacrylate) (PMMA)film coated onto a silicon substrate. Rectangular crystals were chosen so that oxygen plasma patterning was not required. Suspended microresistance thermometer devices were then placed top-side down over the isolated graphene flake, allowing the as-exfoliated side of the grapheneflake to come in contact with the microdevice. A drop of isopropyl alcohol (IPA) was placed on the assembly to assist the alignment and adhesion between the micro-thermometer and the graphene flake. Electron beam lithography (EBL) at 20 kV accelerating voltage was then used to expose the PMMA under the central area of the microresistance thermometer device. The electron irradiation dose was kept low (∼110 μC cm-2₎

to prevent damage to the bilayer graphene.22The PMMA exposed to

the electron beam was selectively removed after the entire device assembly was submerged in (1:3) methyl isobutyl ketone/IPA, leaving the bilayer graphene attached to the suspended device. The device was then placed in acetone at∼60 °C for several seconds to dissolve the PMMA and dried under reduced surface tension. For several samples, the graphene was found to be successfully suspended between the two membranes of the microresistance thermometer device.

A total of two bilayer graphene samples are reported in this Letter, denoted as BLG1 and BLG2. Following thermal measurements, the crystal structure of the graphene sample was characterized with the use of micro-Raman spectroscopy at 532 nm laser excitation (WITec Alpha 300), which obtains Raman spectra that match well with those of bilayer graphene.23However, the use of electron irradiation and relatively short soaking time in acetone are likely responsible for the small D-band and reduced IG/I2Dratio (Supporting Information,

Figure S1) when compared with the Raman spectrum of the as-fabricated single-layer graphene sample supported on SiO2in ref 10,

which was cleaned in acetone overnight. For sample BLG1, the optical microscopy images that were obtained before and after Raman spectroscopy was performed suggest that folding of the edge of the graphene sample occurred during the Raman measurement (Supporting Information, Figure S2a,b). This sample was subse-quently characterized with the use of TEM (FEI Tecnai G2 F20) at 120 kV accelerating voltage. Although areas near the folded graphene edges were observed to be atomically clean after the sample was annealed in hydrogen during the thermal measurement (Figure 2a) following the procedure in a previous report,24phase contrast TEM resolved a thin layer of polymeric residue remaining on most of the surface area of the suspended bilayer graphene even after hydrogen annealing was conducted (Figure 2b). The stability of this layer under electron irradiation indicates that it is highly decomposed with most noncarbon constituent elements likely removed during the annealing process. Additionally, electron diﬀraction analysis was conducted using a 335 nm coherent electron beam and the diﬀraction pattern of BLG1 is shown in Figure 2c. The bilayer graphene is determined to be turbostratic with a rotation of 11.7° between the two graphitic layers.

Figure 1. Schematic of the bilayer graphene (BLG) transfer procedure showing (a) exfoliation of BLG onto PMMA on Si, (b) attachment and alignment of microresistance thermometer device to BLG with the help of a drop of isopropyl alcohol, (c) electron beam exposure of PMMA near the two central membranes, and (d) BLG suspended between two microthermometers after wetting in methyl isobutyl ketone and acetone. (e) Optical micrograph of bilayer graphene sample BLG1 corresponding to (b) andﬁnal suspended BLG sample (f) BLG1 and (g) BLG2 corresponding to (d). Scale bars in (e-g) are 10 μm.

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Nano Letters LETTER

For sample BLG1, the TEM analysis veriﬁes that graphene folding occurred during the Raman measurement (Supporting Information, Figure S2c). Sample BLG2 was broken during the TEM sample preparation procedure after the thermal conductance and Raman measurements.

The thermal conductance of the graphene was measured using the same method described previously.16-18After thermal con-ductance measurements of the as-assembled sample, BLG1 was annealed for 3 h at 500°C under ﬂowing hydrogen (50 cm3min-1, 99.999%, 760 Torr), followed by vacuum annealing at a temperature of 215 °C and pressure of 10-6 Torr. Subsequently, thermal conductance measurements were repeated on the annealed sample. BLG2 underwent vacuum annealing at 215°C and 10-6Torr after initial thermal conductance measurements. The measured thermal conductance (Gs) for both samples is shown in Figure 3 along with

the much smaller background thermal conductance, which is due to parasitic heat transfer between the two membranes via radiation and residual gas molecules in the evacuated sample space as well as heating of the substrate. Because of the very low thermal conduc-tivity of PMMA25 and amorphous carbon thin-ﬁlms,26,27 the

thermal conductance of the polymeric residue is calculated to be 2 orders of magnitude lower than the measured thermal conduc-tance of the suspended bilayer graphene samples, and hence negligible.

The contact thermal resistances between the suspended bilayer graphene and the two suspended membranes are calculated using a ﬁn thermal resistance model developed in previous works18,28

Rc,j ffiffiffiffiffiffiffiffiffiffiffiffi kAsws R}_c s tanh ffiffiffiffiffiffiffiffiffiffiffiffi ws kAsR}c r Lc,j ! 2 4 3 5 -1 ð1Þ where Rc,j and Lc,j are the thermal contact resistance and the

contact length at each of the two graphene-membrane contacts, respectively, κ is the thermal conductivity of the supported graphene, Rc00 is the interfacial thermal resistance between the

graphene and the membrane per unit area, and As= 2δwsis the

cross-sectional area, where wsis the sample width andδ = 0.34

nm is the interplanar spacing of graphite.29 _{Here, the bilayer}

graphene thickness is taken as twice the interlayer spacing of graphite, or 0.67 nm. Assuming that the thermal conductivity of the supported bilayer graphene is similar to that reported for supported single-layer graphene,10 _{and the thermal interface}

resistance is in the range of (4.2-12) 10-9_m2

K W-1reported for the graphene-SiO2interface,30,31(2.3-5.3) 10-8m2K W-1

reported for the graphene-Au interfaces,6_{and 4}₁₀-8_m2_{K W}-1

reported for the Au/Ti-graphene-SiO2interfaces,32the calculated

contact thermal resistance is only 1.6- 5.9% of the total measured thermal resistance at room-temperature. At low temperature, assuming Rc00 is in the range of (1-3) 10-8m2K W-1at

42 K reported for the graphene-SiO2interface,30and 1 10-7

m2 _{K W}-1 _{at 50 K reported for the Au/Ti-graphene-SiO} 2

interfaces,32_{the contact resistance is only 1.2}_{- 5.2% of the total}

measured thermal resistance at 50 K. Hence we do not expect thermal contact resistance to limit thermal transport in the samples over the measured temperature range.

The eﬀective thermal conductivity is obtained as κ = GsLs/As,

where Gs and Ls are the measured thermal conductance and Figure 2. (a) Transmission electron micrograph of bilayer graphene

sample BLG1 after hydrogen annealing at 770 K showing a clean triple-folded edge. (b) Lower resolution reveals a relatively uniform layer of organic residue approximately 10 nm from the quintuple-folded edge. (c) Diffraction pattern for BLG1 using a 335 nm coherent electron beam. The two hexagons are added to highlight the two sets of{1010} reflections rotated by 11.7°. (Inset) Intensity profiles along AA0_{and BB}0 reveal that each set of reflections result from an individual graphene layer. Scale bars in (a) and (b) are 2 and 10 nm, respectively.

Figure 3. Measured total thermal conductance (Gs) versus temperature

(T) for the two bilayer graphene samples in this work. The open and ﬁlled symbols are results measured before and after thermal annealing was performed and cannot be distinguished from each other for sample BLG 1. The thermal conductance of both samples exhibits T1.5_behavior

between 50 and 125 K. Also shown is the background thermal conductance measured using a blank device without a graphene sample between the two thermometers.

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suspended length of the BLG samples, respectively, and Asis the

cross-sectional area. Figure 4 shows that the as-obtained eﬀective thermal conductivity of BLG1 and BLG2 is comparable to both the measured κ and the κ calculated by a numerical solution of the Boltzmann transport equation (BTE) for SiO2-supported single-layer

graphene reported in ref 10. These values are considerably lower than the basal plane values reported for high-quality HOPG11,12_{and some}

theoretical9or Raman-measured5,6values of suspended single-layer graphene with the peak thermal conductivity shifted to a higher temperature in the bilayer graphene samples than for the calculation results and graphite. In addition, the measurement results exhibit an approximate Gsµ T1.5orκ µ T1.5behavior at temperatures between

50 and 125 K, which is shown in the Gsversus T plot (Figure 3).

In order to establish the underlying mechanisms responsible for the temperature-dependentκ observed in the two samples, weﬁrst consider the eﬀect of phonon scattering by the edge of the graphene ribbon. Neglecting the weak interlayer van der Waals interaction in the bilayer graphene sample, we calculate the thermal conductivity of the graphene according to

k ¼

∑

p¼ LA,TA,ZAkp ð2Þ The thermal conductivity contributions from the longitudinal acoustic (LA), transverse acoustic (TA), and ZA branch are obtained as kp ¼ 1 4πδkBT2p¼ LA

∑

,TA,ZA Z kmax k¼ 0 v2pðpωpÞ2τp epω=kBT ðepω=kBT- 1Þ2k dk ð3Þ whereδ is the interplanar spacing of graphite, kBandp are the

Boltzmann and reduced Planck constant, respectively,τpis the

relaxation time,ωpis the phonon frequency, k is the wavevector

and vp dωp/dk is the group velocity. If the edge scattering with

a constant mean free path lbis the dominant phonon scattering

process for the acoustic phonon modes at low temperatures, the

scattering rateτp-1≈ vp/lbis approximately constant for the LA

and TA branches since vpis nearly independent ofωp, and is

proportional toωp1/2for the quadratic ZA branch. Under these

approximations, the diﬀerent thermal conductivity contributions are calculated as kp ¼ lb 4πδkBT2p¼ LA

∑

,TA,ZA Z kmax k¼ 0 vpðpωpÞ2 epω=kBT ðepω=kBT_{- 1Þ}2 k dk ð4Þ The full phonon dispersion in graphene by Nika et al.15_{is used to}

evaluate eq 4. To match the measured thermal conductivity values at low temperatures, lbin eq 4 needs to be approximately 155 nm for

BLG1 and 210 nm for BLG2 (Supporting Information, Figure S4). These values are considerably smaller than the 1.8μm (BLG1) to 6.5 μm (BLG2) width or the 5 μm length of the suspended graphene ribbons measured in this work. Hence, even though edge scattering with a constant mean free path can in principle give rise to the T1.5behavior at low T because of the dominant contribution from the ZA modes (Supporting Information, Figure S4), it cannot explain the large suppression in the observedκ values and thus is not expected to be the dominant scattering process in our samples.

We note that recent thermal conductivity measurements on suspended single-layer andﬁve-layer graphene samples in refs 13 and 14 also showed theκ µ T1.5_{dependence, which the authors}

attributed to a primary contribution from the ZA modes to the thermal conductivity and a dominant edge scattering process. However, theκ values in those samples are much lower compared to our samples. According to our calculations based on eq 4, the boundary scattering mean free path in those samples needs to be as small as 55 and 30 nm to match the below room-temperature values reported in refs 13 and 14, respectively. These mean free path values are 9-33 times smaller than the smallest lateral dimension of those samples, and also much smaller than the grain sizes of exfoliated33

and CVD grown34,35graphene samples. Because of this discrepancy, we believe that the proposed mechanism for the observations in those works deserves further investigation.

We next consider phonon scattering by the residual polymer layer on the suspended graphene sample. For single-layer graphene supported on SiO2, the quantum mechanical calculation reported

in ref 10 shows strong suppression of the ZA modes in the supported graphene by the substrate. Because of a much weaker eﬀect of the substrate scattering on the in-plane LA and TA modes than on the ZA mode, the LA and TA contribution becomes dominant in the supported graphene. For the LA and TA modes, Umklapp scattering is still important for the supported graphene. In 2D and in the low T limit, if the scattering rateτp-1is proportional toωR, whereω is the

phonon frequency, the contribution toκ is proportional to T2-Rfor all polarizations. The Umklapp scattering with a positiveR on the LA and TA modes together with the substrate scattering on the ZA mode gives rise to the approximateκ µ T1.5behavior at low temperatures observed in the calculation result for the single-layer graphene supported on SiO2.10Since the magnitude and temperature

depen-dence of the measured thermal conductivity of the suspended bilayer graphene in this work are rather similar to those reported for single-layer graphene on a SiO2 support,10 we attribute the κ µ T1.5

behavior and the comparableκ values observed in the suspended bilayer graphene samples of this work mainly to scattering of phonons and especially the ZA modes in the graphene by the residual polymer layer that acts essentially as a support layer for the bilayer graphene. Recent works have shown that the polymer residue can be reduced by current heating or vacuum annealing.2,36,37Using a nanomechanical

Figure 4. As-measured eﬀective thermal conductivity (κ) versus tem-perature (T) for the two bilayer graphene samples in this work. For sample BLG2, the open andﬁlled symbols are results measured before and after thermal annealing was performed, respectively. Shown in comparison are the reported κ values of a graphene single-layer supported on SiO2along with the numerical solution to the Boltzmann

transport equation (BTE) for supported (solid blue line) and suspended (dashed black line) graphene from ref 10.

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resonator made from graphene it was experimentally probed that the as-prepared suspended single-layer graphene was up to 13 times more massive than a single sheet of carbon atoms, and that the mass density could be reduced close to the value for atomically pristine graphene after current annealing.36In fact, the high thermal con-ductivity values measured by the micro-Raman spectroscopy meth-ods on suspended graphene samples5-8could indicate that residual polymer or other adsorbates on these samples might have been reduced by the high temperature caused by laser heating. Likely because of a relatively thick starting residual polymer layer on the two samples reported here as a result of a relatively short exposure to acetone during theﬁnal step of the graphene assembly process, the polymer layer was not eﬀectively removed by the annealing process, leading to the unexpected yet important observations in this work. In addition, the graphene sample preparation process in refs 13 and 14 involved a transfer process by attaching graphene to PMMA support. Because no sample annealing process above 325°C was mentioned, it is conceivable that some polymer residue could be present on those samples similar to lithographically patterned graphene samples prior to thorough cleaning,36_{which would naturally explain the observed}

large thermal conductivity suppression.

Besides phonon scattering by the residual polymer layer, interlayer coupling in the bilayer samples can also break the reﬂection symmetry of single-layer graphene, reducing the ZA contribution and resulting in a suppressed thermal conductivity based on a theoretical calculation.9,10However, this eﬀect is not strong enough to suppress the thermal conductivity to a value much lower than that of the basal plane of graphite according to previous reports.8_{In addition, the reported thermal conductivity}

values of the encased few-layer graphene samples increases with the layer thickness because of weaker eﬀect of interlayer coupling than graphene-medium interaction on phonon transport.38 _In

our work, the low-temperature thermal conductivity values of the two bilayer samples are slightly higher than those reported in ref 10 for the three single-layer graphene samples supported on SiO2, and 3-14 times higher than the few-layer graphene

samples encased in SiO2reported in ref 38. This diﬀerence could

be attributed to a weaker eﬀect of the interlayer coupling than that of interaction with the polymer residue that is present only on the top surface of the bilayer graphene, or to weaker interaction between the polymer residue and the bilayer sample than that between the previously reported single-layer samples and the SiO2support.

In principle, point defects caused by electron beam irradiation can also play a role in the observed thermal conductivity of the suspended bilayer sample. However, point defects are not eﬀective in scattering long wavelength phonons39

that dominate the thermal conductivity of pristine graphene at low tempera-tures. In comparison, phonon scattering caused by interaction between graphene and a support layer increases with decreasing frequency and increasing wavelength of graphene phonons.10 Because the thermal conductivity at the low-temperature limit of the two bilayer graphene samples of this work is more than 1 order of magnitude smaller than that calculated for pristine single-layer graphene10

and the basal-plane value measured in high-quality graphite,12 scattering by the polymer residue is expected to dominate over point defect scattering especially in the low temperature range of the measurements.

The results of this work suggest that the interaction between graphene and surrounding residual polymer can eﬃciently suppress phonon transport in graphene. Nevertheless, the room-temperature thermal conductivity value of close to

600 W m-1K-1is still considerably higher than those of common thinfilm materials. While this finding provides an important insight for the use of graphene asfiller materials in polymeric nanocompo-sites, it also points to the need of obtaining ultraclean suspended graphene samples for fundamental studies of the intrinsic phonon transport properties of two-dimensional graphene.

’ASSOCIATED CONTENT

b

S Supporting Information. Additional micro-Raman spec-troscopy, TEM, and thermal conductivity modeling. This material is available free of charge via the Internet at http://pubs.acs.org.

’AUTHOR INFORMATION

Corresponding Author *E-mail: lishi@mail.utexas.edu. Author Contributions

)

These authors contributed equally to this work.

’ACKNOWLEDGMENT

This work was supported by the U.S. Department of Energy award DE-FG02-07ER46377 and the National Science Foundation Grad-uate Research Fellowship Program (M.T.P.). Z.Y. and L.S. conceived the graphene transfer and measurement method. I.J. conducted graphene exfoliation and transfer. M.T.P. conducted microfabrication of the measurement device, thermal measurements, micro-Raman and TEM characterization, data analysis, and modeling. All authors contributed to the writing of this manuscript. The authors also acknowledge Professor Rodney Ruoﬀ and Dr. Weiwei Cai for the use of and assistance with their Raman microscope.

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1 Supporting Information for:

Influence of Polymeric Residue on the Thermal

Conductivity of Suspended Bi-Layer Graphene

Michael Thompson Pettes,

1†

Insun Jo,

2†

Zhen Yao,

2,3

and Li Shi

1,3*

1

_{Department of Mechanical Engineering,}

2

_{Department of Physics and}

3

_{Center for Nano and Molecular}

Science and Technology, Texas Materials Institute, University of Texas at Austin, Austin, Texas 78712,

USA

†

_{These authors contributed equally to this work.}

*

_{Corresponding author, lishi@mail.utexas.edu.}

I. Raman spectroscopy.

Figure S1. Raman spectrographs of BLG1 (a)

and BLG2 (b) showing the D peak to G peak intensity

ratios (I

D

/ I

G

) of 3.1 % and 8.5 %, respectively. The small D-band and lowered intensity of the 2D-band

with respect to the G-band can be attributed to polymer residue on the sample and also possibly the use

of electron irradiation in the sample fabrication process.

S1-S3

The full width at half maximum (FWHM)

of the 2D peak of BLG1 (c) and BLG2 (d) is approximately 67 cm

-1

for both samples.

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2 Figure S2. Optical microscopy images of bi-layer graphene sample BLG1 (a) before and (b) after

micro-Raman spectroscopy. The arrow indicates a noticeable morphological change after heating the

suspended section with the Raman laser. (c) TEM image of sample BLG1 taken after the micro-Raman

measurement reveals edge folding.

III. Thermal conductivity calculation – Full phonon dispersion.

To calculate lattice thermal conductivity in the basal plane, we first neglect the weak inter-layer

interaction and treat the system as two isolated single-layer graphene sheets. The heat flux in the

transport direction, J

q,x

, can then be expressed as

∫ ∫

∑

₌ ₌ =

=

π θ ω ω

π

θ

ω

θ

δ

2 0 0 ZA TA, LA, , max ,

2 d

d

)

(

)

(

cos

1

p p p p p p p p x q

v

f

D

J

=

(S1)

where δ = 0.34 nm is the interplanar spacing of graphite,

S4

_v

p

≡ d

ω

p

/ dk is the branch-specific phonon

velocity,

ω

p

and k are the branch-specific phonon frequency and wavevector, respectively, ħ is the

reduced Planck constant, f

p

is the branch-specific non-equilibrium phonon distribution function, and D

p

is the branch-specific two-dimensional (2D) phonon density of states. The subscript p = LA, TA, ZA

denotes the longitudinal acoustic (LA), transverse acoustic (TA) and out-of plane (ZA) phonon modes

respectively.

Neglecting transient and external force terms, f

p

is obtained from the Boltzmann Transport Equation

(9)

3 x

T

f

v

f

p p p

∂

−

=

0 0

τ

cos

θ

(S2)

where τ

p

is the branch-specific relaxation time and f

0

is the equilibrium phonon distribution, i.e., the

Bose-Einstein distribution

1

0

=

e

kBT

₋

f

₌_ω

(S3)

where k

B

is the Boltzmann constant. The 2D density of states (D

p

) can be obtained from

π

ω

2 d

1 )

2 (

)

2 (

d

2 d

)

(

d

)

(

k

L

k

D

k

D

y x y x p p p

=

(S4)

where L

x

and L

y

are the physical dimensions, length and width respectively, of the suspended graphene

samples. Substituting Eq. S2 – S4 into Eq. S1, and carrying out the integration over 0 ≤

θ

≤ 2π, the term

resulting from the first term in f

p

becomes zero, which is to be expected since there should be no net

energy transport for the equilibrium system. Noting that J

q,x

= -

κ

∂T / ∂x yields

( )

₍

₎

∫

∑

= = =

−

=

max 0 2 2 2 , , 2 , ,

d

1

4

1

k k k T T k p p p ZA TA LA p B p ZA TA LA p

k

e

v

T

k

B B ω ω

τ

ω

δ

π

κ

= =

=

(S5)

In 2D and at the low T limit, if the scattering rate

−1

p

τ is proportional to ω

α

_{, where ω is the phonon}

frequency, the contribution to

κ

is proportional to T

2-α

for all polarizations. If phonon-edge scattering

with a constant mean free path l

b

is the dominant phonon scattering process at low temperatures,

b p

p

≈

v

l

−1

τ

is constant for the LA and TA branches since the group velocity is approximately

independent of ω, and is approximately proportional to ω

1/2

for the quadratic ZA branch with v ∝ ω

1/2

_in

the low temperature limit. Making this substitution, the thermal conductivity can be expressed as

( )

₍

₎

∫

∑

₌ = =

−

=

max 0 2 2 , , 2 , ,

d

1

4

k k k T T k p p ZA TA LA p B b p ZA TA LA p

k

e

v

T

k

l

B B ω ω

ω

δ

π

κ

= =

=

(S6)

To calculate the thermal conductivity according to Eq. S6, we use the exact phonon dispersion of

Nika et al.

S6

In this case, the ZA contribution to

κ

is proportional to T

1.5

and the LA and TA

contributions are proportional to T

2

in the low T limit. Hence, a

κ

∝ T

1.5

_{behavior is expected if the ZA}

(10)

4 scattering in the low-T limit.

S1, S7

However, if the width of BLG1 is taken as l

b

, the results from Eq. S6

are considerably higher than the measured values, as shown in Fig. S4. In order to match the

low-temperature thermal conductivity values obtained in this work, the phonon mean free path must be

reduced to approximately 155 nm and 210 nm for BLG1 and BLG2, respectively. Hence, we conclude

that scattering by the edges of the graphene ribbon is not expected to be the dominant scattering process

at low temperatures.

Figure S4.

As-measured effective thermal conductivity (

κ

) versus temperature (T) for the two bi-layer

graphene samples in this work. For sample BLG 2, the open and filled symbols are results measured

before and after thermal annealing was performed, respectively. The lines are

κ

values calculated from

Eq. S6 using the full phonon dispersion of Nika et al.

S6

for constant phonon mean free path, l

b

= 1.8 μm

(solid black line), 155 nm (solid orange line) and 210 nm (solid purple line). The

κ

contributions from

the LA (short dashed green line), TA (medium dashed red line), and ZA (long dashed blue line) acoustic

phonon modes are also shown for the l

b

= 1.8 μm case.

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