Reexamination of basal plane thermal conductivity of suspended graphene samples measured by electrothermal micro-bridge methods

Reexamination of basal plane thermal conductivity of suspended graphene samples measured by electro-thermal micro-bridge methods Insun Jo,1,a Michael T. Pettes,2,3 Lucas Lindsay,4 Eric Ou,2 Annie Weathers,2 Arden L. Moore,5 Zhen Yao,1 and Li Shi2,b 1Department of Physics, The University of Texas at Austin, Austin, TX 78712, USA 2Department of Mechanical Engineering, The University of Texas at Austin, Austin, TX 78712, USA 3Department of Mechanical Engineering and the Institute of Materials Science, University of Connecticut, Storrs, CT 06269, USA 4Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA 5Department of Mechanical Engineering and the Institute for Micromanufacturing, Louisiana Tech University, Ruston, LA 71272, USA


I. INTRODUCTION
Among many fascinating properties discovered in graphene, 1,2 its high basal-plane thermal conductivity (κ) has attracted broad interests because of the underlying two dimensional (2D) phonon transport physics and the potential for thermal management applications.4][5] In addition, the absence of interlayer interactions leads to exceptionally weak Umklapp scattering for low-frequency phonons in suspended single-layer graphene.Consequently, the scattering mean free paths of these low frequency phonons are only limited by the micrometer-scale lateral size of the suspended graphene considered in theoretical calculations and experiments.Moreover, the density of states of low-frequency phonon modes is relatively large in this 2D system compared to three-dimensional crystals.Therefore, the presence of low-frequency ballistic phonons can result in increasing lattice κ with increasing lateral size in single layer graphene, as suggested by Klemens. 3 In addition, recent theoretical calculations have found large κ contributions from the out-of-plane transverse acoustic phonon modes, namely the flexural or ZA modes. 5hese intriguing and unique thermal properties of graphene have motivated the development of different Raman-spectroscopy-based opto-thermal techniques [6][7][8][9][10][11] and micro-fabricated electrothermal devices [12][13][14][15][16][17][18][19] for measuring the basal-plane κ of graphene.Some Raman-based measurements of suspended single-layer graphene have obtained higher basal-plane κ than graphite values, 6 and a trend of increasing κ with decreasing layer thickness. 9While such Raman-based κ measurement methods require relatively simple sample preparation procedures, the measurement results vary considerably, [6][7][8][9][10][11] and have large uncertainties caused by a large variation in the reported optical absorption values of graphene, limited temperature sensitivity, and complications due to strains and local non-equilibrium of graphene phonons inside the tightly focused laser spot.[13][14][15][16]18,19 As decreasing κ with increasing temperature is the hallmark of dominant Umklapp phonon scattering in a crystal, the much higher peak temperature is a clear indication that other extrinsic processes dominate the thermal resistance in supported and suspended samples measured by the electro-thermal devices.As such, the size dependence and relative contributions from different phonon polarizations to κ in these samples are expected to differ from the intrinsic behaviors predicted for clean suspended graphene.
Recent studies have led to improved understanding of the non-intrinsic processes that dominate phonon transport in supported graphene samples measured by the electro-thermal method.Specifically, it has been shown that the interaction between graphene and a disordered supporting material suppresses the basal-plane κ of supported graphene more than the effects of the inter-layer interaction in graphite. 12,13,17,18In fact, Klemens had pointed out that the graphene κ can be suppressed by a support layer due to phonons leaking across the interface. 3Consequently, the basal-plane κ of supported multi-layer graphene was found to increase with increasing layer thickness.However, the graphite value was not recovered even in 34-layer supported graphene due to long intrinsic mean free paths along both the in-plane and cross-plane directions.The long mean free paths result in ballistic phonon transport in supported few-layer graphene and phonon scattering from the support interface. 1816]19 Different explanations have been put forth.For example, it is suggested based on a molecular dynamics (MD) simulation of a 50 nm long graphene nano-ribbon that the thermal boundary resistance between the suspended and supported graphene segments is responsible for the lower measured κ. 20 However, the work failed to recognize that similar thermal boundary resistance exists in both the thermal-bridge measurement and Raman measurements.Further, the temperature drop at the contacts and the relative error caused by the thermal boundary resistance are expected to decrease as the length of the suspended graphene segment increases by two orders of magnitude from 50 nm considered in the MD simulation to more than 5 µm in some electrothermal measurements.The length dependence in the contact temperature drop has been shown in a recent theoretical study. 21Another explanation of the low κ measured by the electro-thermal method is based on the observation of polymer residue from the transfer process on the graphene samples fabricated for such measurements.It has been suggested that the polymer residue scatters graphene phonons in a similar way as an amorphous support layer. 15,16he κ of several suspended graphene samples measured by the electro-thermal methods has revealed a T 1.5 dependence at low-temperatures. 15,22In comparison, a dominant flexural phonon contribution to κ can lead to the T 1.5 dependence at low temperatures when a constant phonon mean free path is set by lateral boundary scattering of phonons.However, calculations show that the observed low-temperature κ values are much lower than predicted by this mechanism. 15Hence, non-intrinsic scattering processes other than lateral boundary scattering are likely dominant in this temperature range.In addition, the measured and calculated κ of single-layer graphene supported on a disordered oxide also exhibits approximately T 1.5 dependence at temperatures below about 150 K, although the calculated flexural phonon contribution was strongly suppressed by substrate scattering and lower than the contributions from the in-plane acoustic modes. 12Based on these results, it has been suggested that the T 1.5 dependence observed in the suspended graphene samples should not be simply attributed to a dominant contribution from flexural modes.Clarifying the actual mechanism responsible for the suppressed κ and temperature dependence of suspended graphene samples measured by the electro-thermal method is not only necessary for developing a better understanding of phonon transport in graphene, but also relevant to applications of graphene as an electronic material integrated with other systems and in composite materials for thermal management.
In this article, we examine the theory of the length dependence of the total thermal resistance rather than just the apparent κ of a system in the diffusive to ballistic phonon transport regimes.It is noted that the measured total thermal resistance (R s ) of single-layer graphene samples synthesized by chemical vapor deposition (CVD) and measured in a recent work 19 increases linearly with the suspended length (L) and does not contain clear nonlinear features expected for non-diffusive transport, despite the increase in the reported apparent κ with length. 19Besides reanalyzing the results reported for CVD single-layer graphene samples, we examine the influence of extrinsic and intrinsic contact thermal resistance in two exfoliated bi-layer graphene samples measured in this work and in an earlier work.The obtained apparent κ values of CVD and exfoliated graphene samples approach those for highly oriented pyrolytic graphite (HOPG) and the natural graphite source used for the exfoliation, respectively, above room temperature.However, our theoretical analysis of the experimental data indicates that surface polymer residue instead of lateral boundaries suppresses the κ of these suspended graphene samples to be considerably lower than graphite values at low temperatures.These findings suggest that there is still a need to experimentally observe large intrinsic κ and low peak temperature values in a clean graphene sample before the experimental results can be used for verifying various theoretical predictions of 2D phonon physics in clean suspended graphene, such as the peculiar length dependence and flexural phonon contributions.

II. THEORETICAL MODEL OF LENGTH-DEPENDENT THERMAL RESISTANCE
The measured thermal resistance (R s ) of a graphene sample by an electro-thermal method is the combination of the intrinsic thermal resistance (R i ) of the suspended graphene and the extrinsic thermal contact resistance (R c ) at its two ends, R s = R i + R c .In the diffusive phonon transport limit, R i approaches the diffusive thermal resistance (R d ) given by the Fourier law where L and A are the length and cross sectional area, and κ is the diffusive thermal conductivity Here, the summation is over different phonon modes denoted by λ, C λ is the volumetric specific heat contribution, and v x, λ and l x,λ are group velocity component and the mean free path component along the temperature gradient, respectively, for the phonon mode λ.
When the channel length L is reduced to be much smaller than the average mean free path, the transport is in the ballistic phonon transport regime, where local non-equilibrium exists between the carriers moving along two opposite directions between the two reservoirs. 23Such local non-equilibrium can result in a drop in the average carrier temperature at the interface between the conduction channel and each reservoir, even for a reflectionless interface. 21,23,24This temperature drop gives rise to ballistic thermal resistance, which can be obtained from Landauer's approach as 21,23,25 where α λ is the transmission coefficient across the two interfaces between the suspended graphene segment and the two supported graphene segments.This ballistic resistance is essentially a contact resistance in nature, 23 although it is the intrinsic resistance of a ballistic channel.Here, thermal resistance can be well defined in both the ballistic and diffusive regimes, although κ is inherently a diffusive property, ill-defined in the ballistic transport regime.
In the transition regime where the channel length is comparable to the mean free path, the transport consists of both ballistic and diffusive features.Accurate calculation of the intrinsic thermal resistance in this regime is the subject of a large body of current research.With the use of a modified transmission coefficient of (1/α λ + L/2l x, λ ) −1 , the intrinsic thermal resistance can be approximated as which yields R d and R b in the diffusive and ballistic limits, respectively.This result is similar to an expression derived from the McKelvey-Shockley flux method. 21In addition, the summation in the denominator of Equation ( 4) is equivalent to the apparent κ of the quasi-ballistic conductor.
Here, the finite length effect is essentially combined with phonon-phonon scattering with the use of the Matthiessen's rule, 26 which has also been used to account for the finite size effect in a recently reported full numerical solution of the phonon BTE for graphene. 5If l x, λ /α λ is constant, R i given by Equation ( 4) decreases with L linearly.Otherwise, the obtained R i can decrease with decreasing length faster than a linear behavior for L below l x, λ .As shown by the black circles in Fig. 1(a), such nonlinear behavior is evident in the thermal resistance of finite-length suspended graphene calculated by replacing the summation in Equation ( 4) with the apparent κ of finite length suspended graphene obtained from the numerical solution of the BTE. 5 A cross-over from linear behavior (diffusive regime) to non-linear behavior (quasi-ballistic regime) occurs below 2 µm.In comparison, a thermal transport measurement of SiGe nanowires has reported a different nonlinear behavior, 27 where the measured total thermal resistance R s saturates to a nearly constant value when the suspended length is shorter than about 8 µm, as shown in Fig. 1(a) by the red squares.Although the causes of these two different non-linear behaviors remain to be better understood, they show clear deviation from Fourier's law, which would yield linear R s versus L.
As the intrinsic phonon-phonon scattering mean free paths in suspended graphene are often longer than the 50 nm length of the suspended graphene ribbon considered in the aforementioned MD simulation, 20 the reported temperature drop and boundary resistance between the suspended segment and supported segment of the graphene ribbon are largely caused by quasi-ballistic transport phenomena, and increase with increasing coupling between the supported graphene segment and the support because of decreasing interface transmission between the suspended and supported graphene regions.In that work, the heat flux is reduced by a factor of five when the substrate coupling strength is increased from the lower limit to that for typical graphene-support interfaces.This result is equivalent to approximately 20% interface transmission between the suspended and supported regions of graphene.

III. REEXAMINATION OF CONTACT THERMAL RESISTANCE IN PRIOR MEASUREMENTS
Recently, Xu et al. 19 reported length-dependent κ data of suspended CVD single-layer graphene measured by micro-bridge devices.Here, the reported data are re-analyzed by plotting the original measured thermal resistance (R s ) as a function of the suspended length of the 1. 5   graphene samples.As shown in Fig. 1(b), even at the smallest suspended length of about 280 nm near room temperature, the thermal resistance data in Fig. 1(b) still follows a linear dependence on the suspended length, with the deviation from the linear behavior within the typical uncertainty of such measurements.We note that the deviation from a linear behavior appears to become most notable at the lowest temperatures (∼40 K) and at a length below 1 µm.Regardless, the rather linear behavior observed at room temperature is consistent with that predicted by Fourier's law for diffusive transport.In comparison, ballistic or other non-diffusive transport behaviors are expected to result in nonlinear dependence of the resistance with length, as illustrated in Fig. 1(a).Therefore, it appears that the measured room-temperature R s data do not demonstrate a failure of Fourier's law, despite the length dependence shown by the reported apparent κ, which was calculated by estimating and subtracting an extrinsic contact thermal resistance (R c ) value from the measured R s .
For each temperature, a linear fit to the resistance versus length data for all samples extrapolates to a finite thermal resistance value (R 0 ) at vanishing suspended length, as shown in Fig. 1(b).The obtained R 0 ranges from 10 to 20 % of the R s of a 7-µm long single-layer graphene sample measured in that work. 19If diffusive transport persists down to vanishing length, R 0 would be the same as R c .However, a transition to quasi-ballistic transport is expected as the suspended length is reduced further below 500 nm, especially at low temperatures.As such, R 0 can consist of some contribution from R b , so that R c is expected to be smaller than R 0 .It is worth noting that R b calculated for reflectionless contacts is about 25 % and 98 % of R 0 at 300 K and 40 K, respectively, for the 7-µm-long single-layer CVD graphene sample shown in Figure 1(c).
For measurements with the micro-bridge device, R c consists of two extrinsic contact thermal resistance components.One of them is the interface thermal resistance (R c, f ) between the supported graphene segment and the support.The other is the diffusive thermal resistance (R c, m ) between the graphene-support interface and the thermometer.The interface heat transfer between the supported graphene segment and the support results in a hyperbolic temperature distribution in the supported graphene segment, similar to that found along a fin with the base connected to a heat source and the circumference surface exposed to a reservoir at a different temperature. 28Hence, a fin resistance model has been developed in prior works for the calculation of the sample-support interface thermal resistance as [29][30][31][32] where g i is the sample-support interface thermal conductance per unit area, w is the contact width, κ s is the κ of the supported segment of the sample, A is the cross sectional area of the sample, and L c,i is the contact length for each supported sample segment.With the use of the highest g i values reported for encased graphene 33 and the κ s values reported for supported single-layer graphene, 12 we calculate R c, f for the single-layer CVD sample of Xu et al. 19 The obtained R c, f is taken as the lower limit of the R c for the CVD sample, with the upper limit of R c taken as the R 0 value, as shown in Fig. 1(c).
The diffusive thermal resistance R c, m occurs in the SiN x membrane where the resistance thermometer is located.The temperature distribution in the membrane can be assumed to be uniform only when the internal thermal resistance in the membrane is much smaller than those of both the supporting beams and the sample suspended between two SiN x membranes.While this is the case for low-thermal conductance nanowire or nanotube samples, the thermal resistance for the suspended graphene sample is not large enough for ignoring the R c, m . 17,32As shown in a prior work, 32 the R c, m can be evaluated from a numerical heat transfer calculation for the case of infinite g i , infinite κ of the suspended sample segment, and finite κ of the supported sample segment.Using the same approach, we have calculated the R c, m for the device used in Pettes et al.'s measurement of bi-layer graphene with suspended length and width being 5 µm and 1.8 µm, respectively. 15The obtained R c, m ranges from 5 to 14 % of the measured R s . 154][35][36] The obtained R c = R c, m + R c, f is shown in Fig. 1(c).R c, m can be reduced by adding high κ materials to the membrane to improve the membrane temperature uniformity, as shown in a recent measurement of conducting polymer films. 37Alternatively, R c, m can be reduced by confining the lateral size of the temperature measurement region to be in close proximity of the graphene-support interface.Because the size of the serpentine Pt thermometer cannot be reduced much with existing lithography capabilities, resistance line thermometer devices have been employed in several recent works for graphene and h-BN measurements. 12,18,31In such line thermometer devices, the thermometer temperature at the contact point to the graphene or h-BN sample is obtained from the measured average temperature and the linear temperature profile in the thermometer line.Because of the high metal thermal conductance and small lateral size of the contact point, the R c, m in the metal thermometer side of the contact point can be reduced to be about 30−40 % smaller than the R c, m values for the serpentine resistance thermometer device, based on numerical heat conduction calculations conducted in this work.

IV. CURRENT THERMAL CONDUCTIVITY MEASUREMENTS OF BI-LAYER GRAPHENE
In order to reduce R c, m , a line thermometer device is used in this work to make thermal measurements of a bi-layer graphene sample, which has been prepared using a poly(methyl methacrylate) (PMMA) assisted transfer technique as demonstrated in a previous measurement of h-BN. 31he graphene flakes are isolated using mechanical exfoliation from the same natural graphite source as that measured in a recent work. 18The bilayer sample reported here was defined with oxygen plasma etching to be 13 µm in total length and 5 µm in width.
9][40][41] Thermal annealing in a hydrogen and argon environment above 340 o C has been used to remove residues from graphene devices in some prior works. 42,43However, doping by remnant molecules after annealing has been shown to yield degraded electron transport characteristics. 38,43,44Other studies using transmission electron microscopy (TEM) have shown that PMMA residues can be burned off relatively effectively with oxygen in air at elevated temperatures of 250-350 o C. 38,45 Thus the transferred graphene samples were annealed in air at 300 o C for 3 hours in this work.
Figures 2(a)-2(c) show an optical image of the electro-thermal device and a scanning electron microscopy (SEM) image of the suspended bi-layer graphene sample, denoted as G2 hereafter.The electron microscopy characterization was performed after thermal measurements were concluded.The suspended length of the sample is 7.5 µm along the horizontal direction in the figures.The  patterned width (w) of the graphene ribbon is 5 µm, which is larger than the distance between the two 400-nm-wide SiN x bars under the graphene.Hence, over 84% of the graphene is suspended, while the remaining 16% could be in contact with the underlying SiN x bars.Micro-Raman spectroscopy with 488 nm laser excitation shows bi-layer signatures, including similar G band and 2D band intensities, I G ≈ I 2D , and a full width at half maximum (FWHM) of about 52 cm −1 for the 2D band. 46lthough the FWHM of the 2D band in suspended bilayer graphene was measured to be as small as 17 cm −1 , a FWHM of about 55 cm −1 was observed in a supported bilayer graphene measured using a laser wavelength of 488 nm, 46 and is similar to our result.The Raman spectra show a small defect-related D band appearing at 1354.7 cm −1 with an intensity ratio of I D /I G = 0.07.Although the Raman spectrum was obtained in the central area of the sample with a 1 µm laser spot, the structural defect observed near the right vertical metal line in the SEM image of Fig. 2(b) is likely responsible for the small D peak in this bilayer graphene sample.In addition, it has been known that anisotropic strain in graphene can result in the splitting of the G band. 47 Absence of this signature in our Raman data indicates that the strain in the suspended graphene is negligible.
Thermal conductance measurements of the suspended graphene samples, as well as that of a blank device with no graphene were made in a temperature range between 40 and 450 K, as shown in Fig. 3.The thermal conductance of the blank device is primarily due to heat conduction through two narrow supporting bars made of SiN x , and the measured values are consistent for similar devices fabricated from the same wafer batch.The thermal conductance of the bi-layer graphene sample, G s , is obtained after subtracting the contribution of the SiN x bars from the measured conductance.The relative thermal conductance contribution from the SiN x bars is in the range of 25-58 % of the measured conductance values, with the upper limit found at low temperatures.Figure 1(c) compares the corresponding total thermal resistance R s = 1/G s of this sample, together with the R c calculated based on the same approach used above for the bi-layer sample measured by Pettes et al. 15

V. THERMAL CONDUCTIVITY
The corresponding R c for each sample in Fig. 1(c) is eliminated from the measured R s to obtain the intrinsic thermal resistance R i .In comparison, the thermal boundary resistance between the suspended and supported graphene regimes is due to the ballistic resistance R b , which is an inherent component of R i and cannot be isolated from R i .However, the R b component is expected to be negligible in the diffusive transport regime.The 7-µm long CVD graphene sample measured by FIG. 4. Basal-plane thermal conductivity of an exfoliated and suspended bi-layer (blue triangles) graphene sample measured in this work.Also shown are the reanalyzed basal-plane thermal conductivity data of a 7-µm-long, 1.5-µm wide, suspended CVD single-layer graphene sample (black diamonds) measured by Xu et al., 19 a 5-µm-long, 1.8-µm wide, exfoliated and suspended bi-layer graphene sample (brown down triangles) measured by Pettes et al., 15 highly oriented pyrolytic graphite (HOPG) in Touloukian et al., 48 the natural graphite source (red filled diamonds) and an exfoliated and supported bi-layer graphene sample (green circles) measured by Sadeghi et al., 18 an exfoliated and supported single-layer graphene sample (purple right triangles) measured by Seol et al., 12 and a CVD suspended single-layer graphene sample (red unfilled diamonds) measured by Chen et al. 11 The red solid, and black, brown, blue, and purple dashed lines are theoretical calculation results for 10-µm-long suspended clean graphene, 12 the CVD single-layer graphene sample of Xu et al., 19 the bi-layer sample measured by Pettes et al., 15 the bi-layer sample measured in this work, and the supported single layer sample measured by Seol et al., 12 respectively.The two straight solid lines show the slope of T 2 and T 1.5 dependence.
Xu et al. 19 is in the diffusive transport regime based on the linear behavior shown in Fig. 1(b).In addition, the suspended segments of the two exfoliated bi-layer samples are as long as 5 and 7.5 µm, respectively, with lower κ values than those of CVD graphene, as discussed below.Hence, phonon transport in these two samples is also likely to be in the diffusive regime.Here, the apparent κ of these samples is obtained as κ = L/R i A. The results are shown in Fig. 4 together with the κ data of a natural graphite sample 18 and HOPG. 48he obtained room temperature basal-plane apparent κ values of the exfoliated and suspended bi-layer graphene samples measured here and by Pettes et al. 15 are determined to be 880 ± 60 Wm −1 K −1 and 730 ± 60 W m −1 K −1 , respectively, and approach values above room temperature reported in an earlier work 18 for the natural graphite source that has been used for exfoliating the graphene samples, as indicated in Fig. 4. In comparison, the obtained apparent κ for the 7-µm-long suspended CVD single-layer graphene sample reaches as high as 1680 ± 180 Wm −1 K −1 at room temperature, close to the reported HOPG value 19 and the Raman measurement value of a CVD graphene sample reported by Chen et al., 11 as shown in Fig. 4.However, the low-temperature κ of these graphene samples is much lower than measurement values for the HOPG, natural graphite as well as theoretical calculations for suspended clean graphene with a lateral size of 10 µm, 5,12 with the κ peaks of these graphene samples occurring at much higher temperatures.This difference suggests that the low-temperature behavior is still dominated by an extrinsic scattering mechanism rather than the intrinsic lattice anharmonicity.
To understand whether such extrinsic scattering is caused by scattering at the lateral boundary, we adjust the suspended graphene length in the boundary scattering term of the full numerical solution of the phonon BTE, 5 which has accounted for both three-phonon scattering and phonon-isotope scattering.To fit the calculation with the measured κ at the low temperature limit, the suspended length in the calculation is reduced to 650, 500, and 350 nm for the single-layer CVD graphene, and the two bi-layer exfoliated graphene samples, respectively.Although these values are much smaller than the actual suspended lengths and widths of these samples, a low interface transmission coefficient between the supported and suspended graphene segments can reduce the effective boundary length to α λ L, as illustrated in Equation ( 4).In addition, the calculation result becomes higher than the measurement data at temperatures above 250 K for the bi-layer graphene sample of Pettes et al. 30 However, the discrepancy can still be attributed to either a frequency-dependent α, which is lower for the higher frequency phonons, 12 or additional point defect scattering, which is also more pronounced for the higher frequency phonons populated at higher temperatures.In particular, among the three samples examined here, only this bi-layer sample was exposed to electron beam irradiation during the sample preparation process.While Raman spectroscopy data on this sample did not show a noticeable D peak, it is still possible that the electron beam irradiation had caused point defects, which has not been considered in the theoretical calculation.
However, the calculated κ with these short length values become much lower than the corresponding measurement results of the single-layer graphene and the bi-layer sample measured in this work as temperatures increase to above 140 and 80 K, respectively, as shown in Fig. 4.This discrepancy suggests that the mean free path for the dominant extrinsic scattering mechanism is higher for the higher frequency phonons, which play an important role in the peak κ, than for the low-frequency phonons, which dominate the low-temperature κ.In comparison, α is expected to increase with decreasing frequency, 12 which is opposite to that of the dominant extrinsic scattering mechanism.Hence, lateral boundary scattering cannot explain the discrepancy.In addition, side edge scattering has not been included in the calculations, as such scattering alone cannot yield a finite thermal conductivity in the 2D system, 49 and very small κ difference was measured in Xu et al.'s suspended graphene samples with 1.5 and 2.5 µm widths. 19The specularity parameter for side edge scattering increases with decreasing phonon frequency, 26 so that the side edge scattering mean free path can also increase with decreasing frequency near the low-frequency limit.Since the mean free paths for phonon-phonon, phonon-point defect, and phonon-lateral boundary scattering processes all decrease with increasing frequency, such phonon scattering mechanisms cannot explain the discrepancy observed in these two samples.In comparison, it has been suggested that phonon scattering by an amorphous support increases with decreasing frequency 12 for the same reason giving a higher phonon transmission coefficient across a weak van der Waals interface for lower frequency phonons. 50Hence, the suppressed κ found in the two suspended graphene samples at low temperatures can be attributed mainly to scattering by polymer residue, which has been found on all samples in our TEM analysis and is expected to play a similar role as an amorphous support.
In addition, it has been shown that the approximate T 1.5 dependence observed in a supported single layer graphene (see Fig. 4) and some suspended graphene samples at temperatures up to ∼100 K can be caused by a combination of strong frequency-dependent scattering of the flexural modes by the support and phonon-phonon scattering, 12,15 which is not negligible at temperatures as high as ∼100 K.In this case, the flexural modes make a smaller contribution to the κ than the in-plane modes.Moreover, the T 1.5 dependence does not differ drastically from the T 2 dependence associated with the contribution from the in-plane modes subject to a constant mean free path.Hence, to conclude convincingly that the flexural modes dominate the low-temperature κ of these graphene samples, one would need to clearly show that the low-temperature κ is dominated by phonon scattering by the lateral boundary of the suspended graphene sample.
Finally, we discuss the origin of the higher κ shown in Fig. 4 for the CVD single-layer graphene sample than the two exfoliated bi-layer graphene samples.Elimination of inter-layer scattering in single-layer graphene is expected to yield a higher κ in suspended clean single layer graphene than in few-layer graphene. 5However, the measured low-temperature κ of the three suspended graphene samples is well below the values of either the HOPG or the natural graphite source, so that scattering by polymer residue should dominate over inter-layer scattering in the bi-layer graphene.Thus, the observed κ difference between the CVD single-layer graphene and exfoliated bi-layer graphene samples is mainly caused by the difference in the sample cleanness, as well as the limited crystal quality of the natural graphite source used to exfoliate the bi-layer samples.In particular, the measured κ of the natural graphite source used for this study is considerably lower than that reported for HOPG.This difference may be the main reason for the lower κ at and above room temperature found for the exfoliated bi-layer graphene samples than for the single-layer CVD sample.

VI. SUMMARY
The above analysis of the results of suspended graphene samples measured by various electrothermal methods in this and two prior works clarify several questions on such measurements.First, the small resistance obtained at vanishing length for the CVD graphene samples show that extrinsic contact thermal resistance and the intrinsic thermal boundary resistance between the suspended and supported graphene contributes up to ∼ 20% of the total sample thermal resistance when the suspended length is more than 5 micrometers long.Although this contact thermal resistance is not negligible, it is not the cause of the lower κ values measured by the electro-thermal method compared to some Raman measurement results, which are for temperatures higher than room temperature and also subject to similar extrinsic and intrinsic contact thermal resistance.With the extrinsic contact thermal resistance properly accounted for, these existing thermal measurement results of suspended graphene samples do not contain clear signatures of the failure of Fourier's law at room temperature.On the contrary, the κ values obtained by the electro-thermal method at and above room temperature in the single-layer CVD graphene sample are already comparable to the high values of HOPG and some CVD graphene samples measured by a Raman technique, 8,11 whereas the high-temperature κ values obtained by electro-thermal measurements for exfoliated bi-layer graphene already approach that for the natural graphite source used for the exfoliation.Compared to the Raman measurement results, the additional information revealed in the electro-thermal measurements is the suppression of κ at low temperatures compared to the corresponding graphite values.Our analysis shows that this suppression is mainly caused by scattering from polymer residue on the graphene surface rather than lateral boundary scattering.In particular, the measured κ in the temperature range between 200 and 300 K is significantly higher than calculation results used to fit the low-temperature data with a short boundary-scattering mean free path compared to the lateral sample dimension of two samples.This feature suggests that the scattering mean free path by the polymer residue is more pronounced for low-frequency phonons than for high-frequency phonons, in agreement with a previous theoretical model. 12As quasi-ballistic transport is largely due to low-frequency phonons in clean suspended graphene, strong scattering of low-frequency phonons by polymer residue explains the lack of ballistic transport features in the linear thermal resistance -length behavior observed in Xu et al.'s CVD samples. 19These findings provide a better understanding of thermal transport measurement results and phonon transport physics in graphene.

FIG. 1 .
FIG. 1.(a) Thermal resistance per unit cross section (R x A) calculated for suspended graphene (black circles) by Lindsay et al. 5 and measured for SiGe nanowires (red squares) by Hsiao et al. 27 as a function of the suspended length.The black solid and red dashed lines represent linear fitting of the graphene and SiGe data in the diffusive transport regime, respectively.(b) Thermal resistance as a function of the suspended length of the 1.5-µm-wide single-layer graphene samples reported by Xu et al. 19 The dotted lines are linear fitting of the measurement data for a sample length longer than 2 µm.(c) Different thermal resistance components as a function of temperature, including the measured total thermal resistance (R s , squares the extrinsic contact thermal resistance (R c , filled triangles), and the ballistic thermal resistance (R b , unfilled triangles) with reflectionless contacts for the 7-µm-long, 1.5-µm-wide single-layer graphene sample (black symbols) measured by Xu et al., 19 the 5-µm-long, 1.8-µm-wide bi-layer sample (red symbols) measured by Pettes et al., 15 and the 7.5-µm-long, 5-µm-wide bi-layer sample G2 measured in this work (blue symbols).For the 7-µm-long sample measured by Xu et al., 19 the error bars are those in the estimated R c .

FIG. 2 .
FIG. 2. (a) Optical micrograph of a suspended device used for thermal conductivity measurement.(b) SEM images of suspended graphene sample G2 measured with micro-bridge devices.(c) Raman spectra of G2 obtained with a 488 nm excitation laser.This sample is determined to be bi-layer graphene from micro-Raman spectroscopy.

FIG. 3 .
FIG.3.Measured thermal conductance of suspended bi-layer graphene (blue diamonds) after subtracting the thermal conductance of the blank device (grey triangles).