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1 Thermal Conductivity of Isotopically Modified Graphene Shanshan Chen 1,, Qingzhi Wu, Columbia Mishra, Junyong Kang 1, Hengji Zhang 3 Kyeongjae Cho 3, Weiwei Cai 1, *, Alexander A. Balandin 4* and Rodney S. Ruoff * 1 Department of Physics, Fujian Key Laboratory of Semiconductor Materials and Application, Xiamen University, Xiamen , China Department of Mechanical Engineering and the Materials Science and Engineering Program, University of Texas at Austin, Austin, TX 7871 USA 3 Department of Physics, University of Texas at Dallas, Richardson, TX 75080, USA 4 Department of Electrical Engineering and Materials Science and Engineering Program, University of California at Riverside, Riverside, CA 951 USA I. Measurements of Thermal Conductivity A non-contact Raman optothermal method [1-7] was used to measure the thermal conductivity. During the measurement, a 53 nm wavelength laser beam is focused on the center of the suspended graphene using a 100 objective lens with a numerical aperture of 0.9. The temperature rise in the heated graphene T m was determined from the shift in the Raman D peak. To calibrate the Raman peak shift versus the temperature rise, the D peak position of each region was measured while the sample was on a hot plate with its temperature measured by a thermocouple. On the basis of about 10 measurements on each of the four regions, the D peak shifts with increasing temperature as 7.3, 7.05, 6.98 and cm -1 /K corresponding to 0.01%, 1.1%, 50% and 99.% 13 C-labeled graphene, respectively [4]. The power Q absorbed by the suspended graphene is the difference between the power measured at a power meter through an empty hole, and then through a suspended graphene NATURE MATERIALS 1

2 (i.e., a graphene membrane). The obtained optical absorption was.9 ± 0.% at 53 nm wavelength. The thermal conductivity K of graphene could be written as [3] R ln( ) r0 K = α (I.1) πtr g where R is the radius of the holes; r 0 is the laser beam size of 0.17 μm; t is the thickness of graphene; α is 0.98 for the 100 objective lens [3]; and R g is the graphene thermal resistance as Tm T 0 ; T0 is the room temperature; Q air is the heat loss in air as Q Q Q air = R air 0 π 0 m 0 r0 π g( T T ) rdr + r g( T T ). (I.) Here, g is the heat transfer coefficient of W/m K [4]. Table S1: Thermal conductivity K of graphene as a function of Raman-measured temperature 0.01% 13 C 1.1% 13 C 50% 13 C 99.% 13 C T m (K) K T m (K) K T m (K) K T m (K) K The units of K in this table are W/m-K. The error of the as-calculated K from Eq. (I.1) was calculated through the root sum square error propagation approach [8] as shown in Eq. (I.3). The error sources considered are the Raman peak position temperature calibration, temperature resolution of the Raman measurement method, and the uncertainty of the measured laser absorption. K = K ( Q ) Q + ( ΔT ) ΔT (I.3) NATURE MATERIALS

3 On the other hand, the error of the statistic of thermal conductivity in Figure 3 at ~380 K was calculated through the student s test [9]. Table S: Thermal conductivity K of suspended graphene membranes at 380 K 0.01% 13 C 1.1% 13 C 50% 13 C 99.% 13 C The units of K in this table are W/m-K. NATURE MATERIALS 3

4 II. Transfer of the Isotopically Modified Graphene The inner surface of the graphene-on-cu sample was spin-coated (4000 rpm 40 sec) with a thin layer of poly-methyl methacrylate (PMMA) (MW 350,000; 46 mg/ml in chlorobenzene). After dipping the Cu foil in an aqueous (NH 4 ) S O 8 solution (0.5 M) for 30 min, the graphene on the outside of the enclosure foil was removed by washing it off using deionized water. Then, after the Cu substrate was completely dissolved in the aqueous (NH 4 ) S O 8 solution, the PMMA-graphene was transferred onto the Au-coated surface of a 00-nm thick, mm, low-stress silicon nitride (SiN x ) membrane supported on a circular 3 3 mm silicon frame [3]. The SiN x membrane contains a array of.8 μm diameter holes at a pitch of.5 μm between holes. The whole PMMA-graphene-SiN x sample was dried in the air for 30 min and then under vacuum (10 - Torr) for 30 min in order to remove water between graphene and the substrate. Subsequently, PMMA was removed using acetone and graphene suspended on the holes in the SiN x substrate was obtained. III. Molecular Dynamics Simulations Phonon transport in graphene is affected by isotope impurities, scattering on rough edges, and scattering by other phonons. In order to describe thermal conductivity of graphene we set up a computational model, which accounts for all of these scattering mechanisms. Our computational model is composed of two parts. The first part is from MD simulation while the second part is from a Klemens-type analytical model [10]. Below we provide an explanation for such an approach. Usually, MD simulation can be used for thermal transport modeling of non-metallic crystal materials when an accurate empirical potential is available. For carbon materials, Tersoff and REBO potentials have been widely used for thermal transport modeling of diamond [11], carbon nanotubes [1-13] and graphene [14-16]. Although MD simulations can describe phonon-phonon and phonon-defect interactions with reasonable accuracy, they are not able to capture the boundary scattering effects. The latter is because MD simulation can be computationally expensive for simulating cells consisting of 4 NATURE MATERIALS

5 thousands of atoms, which is still far less than the actual size of the sample in the experiments. Since the longest phonon wavelength cannot exceed the size of the simulation cell, a sizable contribution from the long-wavelength phonon to the thermal conductivity is lost in MD simulations. In order to address this issue, we used a Klemens-type analytical model to estimate the long-wavelength phonon contribution to the thermal conductivity. In this sense, our model uses MD to describe accurately the short-wavelength-phonon contribution and a Klemens-type analytical correction for the long-wavelength-phonon contribution. In the MD simulation, we are using the Green-Kubo method to compute the thermal conductivity (TC) through the integration of the heat current autocorrelation function (HCAF), which can be calculated from the equilibrium MD simulation. The definition of heat current and its integration can be found in the following expression, 1 J = [ Eivi + rij ( Fij vi )] i j i (III.1) κ xy 1 = ΩK T B τ m < J xj y > dτ 0 (III.) The heat current is a function of atom energy E i, velocity, interatomic distance and forces, all of which can be determined in MD simulation. The thermal conductivity tensor is a function of the integration of HCAF. Ω is the system volume defined as the area of grapheme multiplied with the thickness of 3.4 Ǻ, K B is the Boltzmann constant, and T is the system temperature. The simulation cell with periodic boundary condition is first thermalized at 380 K for 00 ps. Then, the heat current is recorded every fs in ten NVE ensemble simulations with uncorrelated initial conditions. Each NVE ensemble runs up to 8 ns to ensure better averaging of HCAF. Usually, HCAF would decay to zero when τ m reaches 0.5 ns. To improve the accuracy we utilized the recently re-optimized REBO carbon potential [16]. Other important issues for MD simulations such as periodic boundary size and the HCAF plot have been addressed elsewhere [16]. In the present simulations for graphene with isotope defects we used the shown simulation cell, which ensured the convergence of the NATURE MATERIALS 5

6 thermal conductivity for graphene. The C 13 isotope atoms in graphene are randomly mixed with C 1 for the different C 13 concentrations. Figure S1: Periodic boundary cell for equilibrium MD simulations. The red-dot enclosed area represents the unit cell. The thermal conductivity of graphene has been converged when using N x =6 and N y =10. The detailed study of the simulation cell size effect can be found in Ref. [16]. Table S3: TC of graphene at 380K calculated from MD simulations C 13 % Concentration TC in armchair direction W/mK TC in zigzag direction W/mK Averaged TC from MD W/mK ±75 195± ± ±61 135± ± ±45 417±31 390± ± ±33 103±98 In Table S3, the thermal conductivity of graphene with different C 13 concentrations is calculated for the temperature of 380 K. As we can see from the DOS calculation shown in Figure S, the smallest frequency mode that can be excited in the current simulation cell (i.e. N x =6, N y =10) is about 0.7 THz. This suggests that all the long-wavelength phonon modes with frequency smaller than 0.7 THz cannot contribute to the thermal conductivity of graphene in the MD simulation. However these modes are important and need to be 6 6 NATURE MATERIALS

7 accounted for in the thermal conductivity as a long-wavelength correction. Below we outline the use of the Klemens' model as the long-wavelength correction. Figure S: Phonon DOS of pure C 1 graphene (simulation cell size N x =6, N y =10). The red vertical line corresponds to 0.7 THz, which is the minimum frequency mode in the MD simulation. In the Klemens model, a linear phonon dispersion relation is assumed so that a simple formula for TC of graphene can be derived as [10] λ f 1 m ( T ) = ( πγ ) ρ ln f mt f B v 4 (III.3) where γ is the Grüneisen parameter, ρ is the crystal density (.6 g/cm 3 ), v is the average group velocity, T is the temperature, f m is the upper-bond cut-off frequency (the analogue of the Debye frequency), which equals to 45.9 THz in the original model [10] but changed in our approach, and f B is the lowest cutoff frequency which is a function of graphene boundary size L and temperature T. The lowest cutoff f B is expressed as [10] f B = LT (III.4) Based on the Klemens model [10], the thermal conductivity can be calculated using the NATURE MATERIALS 7

8 following equation, λ f 1 m ( T ) = CvΛdf f B (III.5) where C is the modes dependent heat capacity, v is the group velocity and Λ is the mean free path which can be expressed as, 1 1 = Λ Λ i 1 + Λ p 1 + L eff (III.6) Here, Λ i is the intrinsic mean free path defined in Ref. [10], L eff is the effective boundary size of the graphene sample, and Λ p is the mean free path limited by the point mass defect scattering. Within such a formula, the phonon-phonon interaction, isotope defect scattering, and boundary scattering effect are combined to obtain the mean free path from Matthiessen's rule. The analytical expression for Λ p is given as [10] 1 Λ p = Γ4π vf m f 3 (III.7) Γ = i mi m Ci m (III.8) where Г is a parameter to measure the mass variance, C i is the concentration and m i the atomic mass of isotope type i atoms, respectively, and m is the average mass. In order to estimate the long-wavelength correction to the thermal conductivity at 380 K, the f m value in E. (3) is set to 0.7 THz, which is the minimum frequency mode in the MD simulation. The integration results of the long-wavelength thermal conductivity correction for different isotope concentrations are shown in Table S4. We found that the long wavelength corrections for different concentrations are similar. This suggests that isotope defects are almost transparent for long wavelength phonon modes. This can be explained with the mean 8 NATURE MATERIALS

9 free path in Eq. (III.7) showing that the smaller the phonon frequency is (i.e., the phonon wavelength is longer), the less it would be scattered by isotope defects. This provides additional justification to our simulation approach, which combines MD computation with the long-wavelength Klemens correction to the thermal conductivity value. Table S4: MD data with the long wavelength TC correction C 13 % Long wavelength Averaged TC from MD Corrected MD TC correction W/m-K W/m-K W/m-K ±19 859± ± ± ± ± ± ±98 References [1] Balandin, A.A., Ghosh, S., Bao, W., Calizo, I., Teweldebrhan, D., Miao, F. & Lau, C.N. Superior thermal conductivity of single layer graphene. Nano Lett. 8, (008). [] Ghosh, S., Calizo, I., Teweldebrhan, D., Pokatilov, E.P., Nika, D.L., Balandin, A.A., Bao, W., Miao, F. & Lau, C.N. Extremely high thermal conductivity in graphene: Prospects for thermal management application in nanoelectronic circuits. Appl. Phys. Lett., 9, (008). [3] Cai, W., Moore, A.L., Zhu, Y., Li, X., Chen, S., Shi, L. & Ruoff, R.S. Thermal transport in suspended and supported monolayer graphene grown by chemical vapor deposition. Nano Lett., 10, (010). [4] Chen, S., Moore, A.L., Cai, W., Suk, J.W., An, J., Mishra, C., Amos, C., Magnuson, C.W., Kang, J., Shi, L. & Ruoff, R.S. Raman measurement of thermal transport in suspended monolayer graphene of variable sizes in vacuum and gaseous environments. ACS Nano, 5, (011). [5] Ghosh, S., Bao, W., Nika, D.L., Subrina, S., Pokatilov, E.P., Lau, C.N. & Balandin, A.A. Dimensional crossover of thermal transport in few-layer graphene. Nature Materials, 9, (010). NATURE MATERIALS 9

10 [6] Seol, J.H., Jo, I., Moore, A.L., Lindsay, L., Aitken, Z.H., Pettes, M.T., Li, X., Yao, Z., Huang, R., Broido, D., Mingo, N., Ruoff, R.S. & Shi, L. Two-dimensional phonon transport in supported graphene. Science, 38, (010). [7] Balandin, A.A. Thermal properties of graphene and nanostructured carbon materials, Nature Materials, 10, (011). [8] Figliola, R.S.; Beasley, D.E. Theory and Design for Mechanical Measurements; John Wiley & Sons: New York, (000). [9] [10] Klemens, P.G. Theory of the A-Plane thermal conductivity of graphite. Journal of wide bandgap materials, 7, (000). [11] Che, J., et al. Thermal conductivity of diamond and related materials from molecular dynamics simulations. J. Chem. Phys., 113, 6888 (000). [1] Che, J., et al. Thermal conductivity of carbon nanotubes. Nanotechnology, 11, (000). [13] Donadio, D., Galli, G. Thermal condcutivity of isolated and interacting carbon nanotubes: comparing results from molecular dynamics and the Boltzmann transport equation. Phys. Rev. Lett., 99, 5550 (007). [14] Evans, W.J., et al. Thermal conductivity of graphene ribbons from equilibrium molecular dynamics: Effect of ribbon width, edge roughness, and hydrogen termination. Appl. Phys. Lett. 96, 0311 (010). [15] Hu, J., et al. Thermal conductivity and thermal rectification in graphene nanoribbons: A molecular dynamics study. Nano Lett., 9, (009). [16] Zhang, H., et al. Thermal transport in graphene and effects of vacancy defects. Phys. Rev. B. 84, (011). [17] Lindsay, L., et al. Flexural phonons and thermal transport in graphene. Phys. Rev. B 8, (010). [18] Lindsay, L., et al. Optimized Tersoff and Brenner empirical potential parameters for lattice dynamics and phonon thermal transport in carbon nanotubes and graphene. Phys. Rev. B 81, (010). 10 NATURE MATERIALS