Thermal Conductivity of Graphene and Graphite: Collective Excitations

Letter pubs.acs.org/NanoLett

Thermal Conductivity of Graphene and Graphite: Collective Excitations and Mean Free Paths Giorgia Fugallo,*,†,‡ Andrea Cepellotti,‡,§ Lorenzo Paulatto,† Michele Lazzeri,† Nicola Marzari,‡,§ and Francesco Mauri† †

IMPMC, UMR CNRS 7590, Sorbonne Universités − UPMC Univ. Paris 06, MNHN, IRD, 4 Place Jussieu, F-75005 Paris, France Theory and Simulations of Materials (THEOS), École Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland § National Center for Computational Design and Discovery of Novel Materials (MARVEL), École Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland ‡

ABSTRACT: We characterize the thermal conductivity of graphite, monolayer graphene, graphane, fluorographane, and bilayer graphene, solving exactly the Boltzmann transport equation for phonons, with phonon−phonon collision rates obtained from density functional perturbation theory. For graphite, the results are found to be in excellent agreement with experiments; notably, the thermal conductivity is 1 order of magnitude larger than what found by solving the Boltzmann equation in the single mode approximation, commonly used to describe heat transport. For graphene, we point out that a meaningful value of intrinsic thermal conductivity at room temperature can be obtained only for sample sizes of the order of 1 mm, something not considered previously. This unusual requirement is because collective phonon excitations, and not single phonons, are the main heat carriers in these materials; these excitations are characterized by mean free paths of the order of hundreds of micrometers. As a result, even Fourier’s law becomes questionable in typical sample sizes, because its statistical nature makes it applicable only in the thermodynamic limit to systems larger than a few mean free paths. Finally, we discuss the effects of isotopic disorder, strain, and chemical functionalization on thermal performance. Only chemical functionalization is found to play an important role, decreasing the conductivity by a factor of 2 in hydrogenated graphene, and by 1 order of magnitude in fluorogenated graphene. KEYWORDS: Thermal transport, graphene, graphite, chemical functionalization, strain, first-principles calculations graphite where a value of 2000 W m−1 K−112 has been obtained with high precision on perfect crystals of large sizes.12 In this paper, we try to elucidate these discrepancies, using state-of-the-art first-principles and transport calculations, that we verify and validate thanks to an excellent agreement with experiments on the thermal conductivity of graphite. In addition, we study how the thermal conductivity of graphene can be affected by the presence of extrinsic sources of scattering, by isotopic disorder, applied strain and chemical functionalization. Our analysis suggests that heat is carried by collective excitations of phonons characterized by a mean free path much longer than the phonon mean free path, which causes graphene to be in a non-Fourier regime for the sample sizes typically used in experiments.

G

raphene is undoubtedly one of the most fascinating and extensively studied materials of this decade. Among its remarkable properties, an extremely high thermal conductivity (k) has received great attention both in experiments and calculations.1−3 However, the exact increment in thermal conductivity achievable in going from three-dimensional (3D) graphite to 2D graphene is still a matter of debate. Even if the recent availability of high quality single-layer graphene samples has allowed for a large number of experimental studies,2−7 definitive results remain elusive. The difficulty is reflected in a variety of estimates for the thermal conductivity k, spanning a rather large range of values between 1500 and 5000 W m−1 K−1 at room temperature.2−6 This spread has been sometimes justified with the presence of mechanical strains on the samples studied.1 On the theoretical side, the picture is equally open8 with estimates varying in an even larger range of k between 1000 and 10 000 W m−1 K−1.8−11 The situation is different for © XXXX American Chemical Society

Received: June 2, 2014 Revised: October 23, 2014

A

dx.doi.org/10.1021/nl502059f | Nano Lett. XXXX, XXX, XXX−XXX

Nano Letters

Letter

Macroscopically, the thermal conductivity of a material is defined by the Fourier’s law as the ratio between the heat flux Q and the applied gradient of temperature ▽T, such that Q = −k▽T. A rigorous microscopic description of k requires a solution of the Boltzmann transport equation (BTE),13 giving for every phonon ν the deviation of the phonon population Fν with respect to the equilibrium distribution nν̅ (the Bose− Einstein), such that the total out-of-equilibrium distribution is nν = nν̅ + nν̅ (n̅ν + 1)Fν▽T (we assume here the linearized form). The thermal conductivity can then be expressed in terms of Fν, the phonon energies ℏων and the projection along ▽T of the phonon group velocities cν k=

ℏ2 ΩkBT 2

∑ nν̅ (nν̅ + 1)cνωνFν ν

(1)

where Ω is the volume, kB is the Boltzmann constant, and T is the temperature. In the great majority of cases, due to the difficulties associated with an exact solution of the BTE an approximation is used where the individual phonons are considered to be the main heat carriers. This is the single mode relaxation time approximation (SMA) of the BTE,14−16 obtained by approximating Fν with cνωντν, where τν is the lifetime or relaxation time of the phonon mode ν. The SMA thus estimates Fν as a quantity that depends only on the properties of each individual phonon mode ν; the exact Fν is instead qualitatively different because it takes into account also the presence of collective excitations.17 Only the exact solution of the BTE, obtained by inverting the full scattering matrix, is able to characterize the collective nature of the excitations that are present in the nonequilibrium states such as the one induced by a gradient of temperature. These effects are the outcome of the complex interplay of different scattering events in the whole Brillouin zone that affect both the depopulation of initial states and the repopulation of final states. Instead, the SMA is able to describe correctly the depopulation of a phonon mode after a scattering process but repopulates all final states isothermally, losing memory of the initial phonon distribution. On the other hand, thanks to recent progress18−20 we can now solve the BTE exactly21 following the method explained in ref 18 where harmonic and anharmonic properties of phonons are calculated fully from first-principles22 within density functional perturbation theory.23−26 Specifically, we used a recent generalization for metallic systems and arbitrary wavevectors, implemented starting from the QuantumESPRESSO distribution10,27 The anharmonic first-principles results are necessary to compute without parameters the scattering rates due to threephonon interactions. These processes are dominant at high temperatures and describe the 1/T part of the thermal conductivity. In addition, we also consider scattering due to the presence of isotopic disorder,28 treated within perturbation theory at the harmonic level and in its symmetric form as in ref 18, which has the effect of a global downscaling of thermal conductivity and extrinsic sources of scattering, such as interface scattering, mostly relevant to describe the T3 behavior at very low temperatures. We first consider the results of graphite (Figure 1) that we can even compare with high-precision measurements on perfect crystals of large sizes.12 The comparison shows a remarkable agreement between measurements and the results of our simulations (also tested on crystalline diamond in ref 18). Most

Figure 1. (Top panel) In-plane lattice thermal conductivity in bulk graphite, single- and bilayer graphene of crystalline-domain sizes (see text) L = 1 mm and (bottom panel) out-of-plane thermal conductivity of graphite with L = 0.3 μm. The data (EXP) of graphite are taken from ref 12, which were obtained by a compilation and analysis of several research papers with a widely accepted extrapolation above 300 K, a posteriori shown to be accurate by the present calculations. This extrapolation regime is highlighted by the use of filled diamonds. Solid lines are used for the exact solutions while dashed lines for the singlemode approximation (SMA) solutions. (Inset) Zoomed SMA results for the range T = 200−600 K, in which in-plane thermal conductivity is qualitatively wrong; graphite conductivity is found to be higher than single- and bilayer graphene.

importantly, if the SMA is used we find a severe underestimation by an order of magnitude for the in-plane conductivity, while keeping a good agreement with experiments for the out-of-plane conductivity. A full agreement with experiments is achieved only when the exact solution of the BTE (exact in Figure 1) is considered. The inadequacy of the SMA was found also in a recent first-principles work on graphene29 and in single and multilayer graphene described with an empirical potential.11,30−32 These results highlight how the description of these systems in terms of single-phonon properties is not sufficient and it is necessary to describe the collective excitations that arise in the exact BTE. The phonon representation used in the SMA is related to the eigenvalues of the dynamical matrix, while a suitable representation of the heat carriers is instead given by the eigenvalues of the scattering matrix17 (matrix A in ref 18), which we here refer to as the collective excitations. A much higher thermal conductivity with respect to the graphite case is found for single-layer graphene (top panel of Figure 1), as the experimental evidence suggests, and the bilayer is intermediate between the two extreme cases. We stress that this difference in thermal conductivity upon B

dx.doi.org/10.1021/nl502059f | Nano Lett. XXXX, XXX, XXX−XXX

Nano Letters

Letter

reduction of the thickness cannot be reproduced by the SMA, which actually would predict similar conductivities. Before commenting on the simulation results and the experiments on graphene, it is crucial to characterize the average mean free path λ of these collective excitations and determine under which conditions the thermodynamic limit is achieved and Fourier’s law is satisfied in experiment, that is, when the distance D between the source and the sink is much larger than the heat carriers mean free paths (D ≫ λ). If this is not the case, the BTE can still be used if D is much larger than the characteristic length L of an extrinsic source of scattering (D ≫ L). For example, in a ribbon geometry the characteristic length L of extrinsic scattering is the width (Figure 2, panel a).

Figure 2. Schematic representation of the different uses of the scattering rate term |cν|/L in the presence of a gradient of temperature between the source and the sink at distance D. (a) Ribbon of width L with D ≫ L. (b) Polycrystalline sample of domain size L with D ≫ L. (c) Ribbon or circular sample of diameter L with D ≃ L and comparable with the average heat carriers mean free path λ.

In a polycrystalline sample (Figure 2, panel b), the characteristic length L of extrinsic scattering is the size of crystalline domain size. In both cases, the heat carriers are not allowed to travel for a distance larger than the extrinsic scattering length L because they will undergo a scattering event at the grain boundary or at the surface of the ribbon, where we assume they are redistributed isothermally. A description of these systems is achieved by introducing an additional scattering rate of the form |cν|/L.33 Historically, this term was introduced to describe the electrical transport in connection with a Boltzmann equation for electrons along a thin wire, where L represented the diameter and the longitudinal direction D is considered infinite,34 supposing perfect absorption at the surface. In literature, this term has been mistakenly used2,3,11,29,31,32,35,36 to describe finite-sized sample with length L along the transport direction overlooking the effect of the width or in samples of circular geometry with D ≃ L ≃ λ (panel c), which instead would require the solution of the BTE in real space16 with the proper conditions of reflection/absorption at the boundaries of the sample. The typical procedure adopted in literature to analyze the mean free paths is the study of their distribution for every phonon mode.11,29,31,32 However, since the thermal conductivity is a collective property, the change of extrinsic sources of scattering does not merely scale down the largest mean free paths but it affects all the distribution at once. Therefore, we study the change in thermal conductivity as a function of the extrinsic scattering length L at room temperature. As reported in Figure 3, one moves from a region

Figure 3. Lattice thermal conductivity of naturally occurring (magenta line) and isotopically pure (blue line) graphene, bilayer graphene, inplane graphite, and out-of-plane graphite, at 300 K as a function of the crystalline-domain sizes L, in the SMA (left panel) and in the exact solution (right panel). Experimental points from the left panel of Figure 3 in ref 35. Dotted lines report the values of the thermodynamic limit.

in which the conductivity rises for increasing values of L (i.e., ballistic-like regime) to a plateau area (diffusive regime) above a length Ldiff, where k has reached the thermodynamic limit of the intrinsic thermal conductivity. The length Ldiff at which the diffusive regime is reached represents the longest mean free path of the heat carriers. The study of the heat carrier mean free paths for graphene, bilayer graphene, in-plane, and out-of-plane graphite is reported in Figure 3. Because the SMA uses individual phonon excitations to describe k, the Ldiff observed within the SMA is determined by the phonon mean free paths (in particular, the longest mean free path of the heat carrying phonons). For graphene, Ldiff falls then in a range of values between 1 and 10 μm, which is in agreement with what is commonly accepted in the literature.8,10 Crucially, the exact solution of the BTE reaches a diffusive limit for lengths that are 2 orders of C

dx.doi.org/10.1021/nl502059f | Nano Lett. XXXX, XXX, XXX−XXX

Nano Letters

Letter

not been found in this work, and the k of an infinite sample at finite temperature always converges to a finite value. We can thus provide the highest theoretical limit of the lattice thermal conductivity of graphene, that would be reached in a perfect infinite crystal where no grains or surfaces limit the mean free path. At room temperature, the highest predicted k has the value of 3600 W m−1 K−1 for naturally occurring graphene and 4300 W m−1 K−1 for the isotopically pure case. For comparison, these values are of 2200 and 2500 Wm−1 K−1 for a bilayer and 2000 and 2200 Wm−1 K−1 for graphite. In the rest of the manuscript, we will try to understand how the thermal conductivity of graphene can be tuned. The introduction of isotope disorder is a method that reduces thermal conductivity without altering the electronic transport. In Figure 5, we show the effect of different isotopic

magnitude larger, setting Ldiff in a range of values between 100 μm and 1 mm. On the basis of our results, we infer that the logarithmic divergence claimed for graphene in ref 35 is only originated by the experimental complexity of extending the study to larger samples. We remark that while other works11,29 have performed numerical simulations of the BTE on graphene, such studies were limited up to lengths 50 μm and were then not able to access or discuss the effects presented here. As shown in Figure 3, the diffusive limit is reached at lower values in going from 2D to 3D, indicating that the mean free paths are decreasing. In the out-of-plane direction, the difference between SMA and the exact solution is reduced so that the difference between the mean free paths of phonons is not too different from the heat carrier mean free paths. Eventually, the reduction of k with respect to the in-plane direction is also reflected in a corresponding reduction of the mean free paths. The extremely large values of the heat carriers mean free paths in graphene have some important consequences, given that Fourier’s law is valid only for systems large enough to accommodate at least a few mean free paths. Current measurements of k in graphene (see Figure 4) have been

Figure 5. Lattice thermal conductivity as a function of temperature for L = 100 μm (solid lines) and L = 5 μm (dashed lines) graphene with 13 C isotopic percentages of 0.1, 1.1, 50, and 99.2%. Solid and dashed lines represent the theoretical results to be compared with experimental points44 in the inset.

compositions for polycrystalline graphene at L = 100 μm and L = 5 μm. Isotopic scattering is stronger for the larger size, as it is clearly visible from the peak occurring at T ∼ 100 K. This is because as the more extrinsic scattering processes become negligible, the more sensitive the thermal conductivity becomes to the change of intrinsic parameters. At T ∼ 100 K, where only a few Umklapp processes are activated and size effects are marginal, the thermal conductivity becomes very sensitive to tiny variations in the isotopic content. An enhancement by a factor of 3.5 is observed for L = 100 μm, going from the naturally occurring case (12C = 98.9%, 13C = 1.1%) to the isotopically enriched one (12C = 99.99%, 13C = 0.01%). For L = 5 μm, this enhancement is reduced to a factor 1.4. For both sizes, the thermal conductivity becomes less sensitive to the presence of mass disorder as temperature is increased. Another effect that can modify the thermal conductivity is the strain. This topic has been object of debate, due to the presence of contradictory results, as some have predicted a divergence of k in the presence of strain that casts doubts on the definition of k itself,8,45 while others found a finite and sometimes even decreasing thermal conductivity.29,46−48 In Figure 6, we see how the exact solution of the BTE predicts a weak change of k with applied isotropic in-plane strain. Also, this effect does not show a strong size dependence, and for three particular cases studied in detail (L=5 μm, 100 μm, ∞) the ratio between the strained and unstrained sample is always

Figure 4. Lattice thermal conductivity as a function of temperature for different grain size dimensions of single-layer naturally occurring polycrystalline graphene (98.9 12C and 1.1 13C). Solid lines represent the theoretical results. Experimental points2,7,37 are not shown for comparison but to indicate the state of the art of experimental measurements.

performed in samples whose lateral size covered a range from 1 μm up to some tens of micrometers, which is smaller than the mean free path for the collective phonon excitations. Thus, the use of Fourier’s law for interpreting the measurements performed in these systems becomes questionable. Because the sample is small, the conditions in which thermal conductivity is defined from a macroscopic and statistical point of view, that is, as a diffusion coefficient, are no longer met. We can still use the ratio between the heat flux and the difference of temperature, namely the “thermal conductance”, defined also in absence of a diffusion equation. The correct procedure to evaluate the thermal conductivity in such cases requires the extension of BTE to a space-dependent case. In recent years, there has been an active debate about the possible divergence of thermal conductivity in two-dimensional materials (refs 30 and 38−43 to name a few) and recently in the experimental community of graphene.35 Although we report indeed very large mean free paths, such divergence has D

dx.doi.org/10.1021/nl502059f | Nano Lett. XXXX, XXX, XXX−XXX

Nano Letters

Letter

of graphene. The hydrogenated form of graphene is referred to as graphane while fluorographene refers to the fluorogenated equivalent form that we will call here after graFane. In Figure 7,

Figure 7. Thermal conductivity as a function of temperature for naturally occurring graphene (black), graphane (turquoise − solid line), and fluorographene (GraFane, violet − solid line) for L = 100 μm. Dashed lines show graphane with the mass of hydrogen atoms set to be equal to the fluorine mass and vice versa for graFane.

we see how hydrogenation reduces the thermal conductivity by a factor of 2 almost constant above room temperature. Intriguingly, graFane shows a suppression of an order of magnitude, which makes it worth being investigated for thermoelectric application, such as quantum wires or superlattices52,53 that recently received the attention of both experimental and numerical studies.49,54−56 We found that the difference between graFane and graphane is mainly controlled by the phonon dispersion. To show this we can change the masses of the functionalizing atom. The conductivities of the two materials roughly differ (at room temperature) by 850%, but if we set the fluorine mass to be the same of hydrogen, the difference is reduced to a mere 23%. Similarly, if we set the hydrogen mass to fluorine, the difference is of 19%. We attribute the residual difference of 20% to the differences in harmonic and anharmonic force constants. We observed therefore that the thermal conductivity seems to be mainly controlled by the phonon dispersion. In summary, we find that only the exact solution of BTE provides results for the thermal conductivity for graphite that are in agreement with experiments. Graphite bilayer and monolayer graphene mean free paths are studied obtaining that collective phonon excitations, and not single phonons, are the main heat carriers. The mean free paths of these excitations can reach hundreds of micrometers, which are larger than typical suspended graphene samples. This result makes then questionable the use of the Fourier’s law and highlights some of the discrepancies experienced in extracting reliable thermal conductivities. Finally, the effects of isotopic disorder, strain, and chemical functionalization are studied to examine the possibility of tuning graphene thermal response for nanodevices applications. While isotopes and strain lead to a modest reduction of thermal conductivity, chemical functionalization, especially for the fluorogenated case, leads to a sensible reduction, which makes it worth to be further investigated as a promising option for quantum wires and superlattices.

Figure 6. (Top panel) Thermal conductivity percentage variation (%) as a function of strain distribution and grain-size length for the exact solution of the BTE. The reference value is the unstrained value of thermal conductivity at various sizes, which are, for example, 356, 646, 831, and 1037 W/mK for sizes L of 0.1, 1, 10, and 100 μm, respectively. (Bottom panel) Ratio between strained (4%) and unstrained cases in the exact solution of the BTE and in the SMA for three different crystalline-domain sizes L = 5 μm, 100 μm, and ∞. (Inset) Acoustic phonon dispersion between Γ an K/2 for the two different strain limits.

close to 1 for all the temperatures considered here, changes being typically within 10%. The SMA instead provides a qualitatively different view, and predicts a very large increase in k with respect to strain, diverging for infinite sizes in the limit of zero temperature. In the SMA, the response to strain is strongly dependent on size and temperature, and a strain of 4% leads to a change in k typically larger than 200%. We notice also that the discrepancy between the SMA and the exact solutions increases with L and persists even in the limit of high temperatures, where the SMA would be typically considered to be reliable. Therefore, our simulation of an infinite sample contradicts the divergences claimed by ref 8 and the simulation at large strains is in contrast with the divergence found in ref 45 confirming instead a small effect. Another way of tuning the thermal conductivity consists in changing the hybridization of carbon atoms from sp2 to sp3 and consequently altering the electronic properties by opening a band gap. Graphene hydrogenation has been shown to be very effective on controlling thermal conductivity49−51 We study here for the first time (to our knowledge) with the ab initio BTE the effect of a complete hydrogenation and fluorogenation E

dx.doi.org/10.1021/nl502059f | Nano Lett. XXXX, XXX, XXX−XXX

Nano Letters

■

Letter

(24) Debernardi, A.; Baroni, S.; Molinari, E. Phys. Rev. Lett. 1995, 75, 1819. (25) Lazzeri, M.; de Gironcoli, S. Phys. Rev. B 2002, 65, 245402. (26) Deinzer, G.; Birner, G.; Strauch, D. Phys. Rev. B 2003, 67, 144304. (27) Giannozzi, P.; Baroni, S.; Bonini, N.; Calandra, M.; Car, R.; Cavazzoni, C.; Ceresoli, D.; Chiarotti, G. L.; Cococcioni, M.; Dabo, I.; Dal Corso, A.; de Gironcoli, S.; Fabris, S.; Fratesi, G.; Gebauer, R.; Gerstmann, U.; Gougoussi, C.; Kokalj, A.; Lazzeri, M.; Martin-Samos, L.; Marzari, N.; Mauri, F.; Mazzarello, R.; Paolini, S.; Pasquarello, A.; Paulatto, L.; Sbraccia, C.; Scandolo, S.; Sclauzero, G.; Seitsonen, A. P.; Smogunov, A.; Umari, P.; Wentzcovitch, R. M. J. Phys.: Condens. Matter 2009, 21, 395502. (28) Garg, J.; Bonini, N.; Kozinsky, B.; Marzari, N. Phys. Rev. Lett. 2011, 106, 045901. (29) Lindsay, L.; Li, W.; Carrete, J.; Mingo, N.; Broido, D. A.; Reinecke, T. L. Phys. Rev. B 2014, 89, 155426. (30) Lindsay, L.; Broido, D. A.; Mingo, N. Phys. Rev. B 2010, 82, 115427. (31) Singh, D.; Murthy, J. Y.; Fisher, T. S. J. Appl. Phys. 2011, 110, 094312. (32) Singh, D.; Murthy, J. Y.; Fisher, T. S. J. Appl. Phys. 2011, 110, 044317. (33) Scattering term from eq 11 in ref 18 with F = 1. (34) Dingle, R. B. Proc. R. Soc. A 1950, 201, 545. (35) Xu, X.; Pereira, L. F. C.; Wang, Y.; Wu, J.; Zhang, K.; Zhao, X.; Bae, S.; Tinh Bui, C.; Xie, R.; Thong, J. T. L.; Hong, B. H.; Loh, K. P.; Donadio, D.; Li, B.; Ö zyilmaz, B. Nat. Commun. 2014, 5, 3689. (36) 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. Science 2010, 328, 213. (37) 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. ACS Nano 2011, 5, 321. (38) Nika, D.; Ghosh, S.; Pokatilov, E.; Balandin, A. Appl. Phys. Lett. 2009, 94, 203103. (39) Lepri, S.; Livi, R.; Politi, A. Phys. Rep. 2003, 377, 1. (40) Yang, L.; Grassberger, P.; Hu, B. Phys. Rev. E 2006, 74, 062101. (41) Dhar, A. Phys. Rev. Lett. 2001, 86, 5882. (42) Mu, X.; Wu, X.; Zhang, T.; Go, D. B.; Luo, T. Sci. Rep. 2014, 4, 3909. (43) Basile, G.; Bernardin, C.; Olla, S. Phys. Rev. Lett. 2006, 96, 204303. (44) Chen, S.; Wu, Q.; Mishra, C.; Kang, J.; Zhang, H.; Cho, K.; Balandin, A. A.; Ruoff, R. S. Nat. Mater. 2012, 11, 203. (45) Pereira, L. F. C.; Donadio, D. Phys. Rev. B 2013, 87, 125424. (46) Ma, F.; Zheng, H. B.; Sun, Y. J.; Yang, D.; Xu, K. W.; Chu, P. K. Appl. Phys. Lett. 2012, 101, 111904. (47) Alofi, A.; Srivastava, G. P. Phys. Rev. B 2013, 87, 115421. (48) Li, X.; Maute, K.; Dunn, M. L.; Yang, R. Phys. Rev. B 2010, 81, 245318. (49) Kim, J. Y.; Lee, J.-H.; Grossman, J. C. ACS Nano 2012, 6, 9050. (50) Pei, Q.-X.; Sha, Z.-D.; Zhang, Y.-W. Carbon 2011, 49, 4752. (51) Barbarino, G.; Melis, C.; Colombo, L. Carbon 2014, 80, 167. (52) Donadio, D.; Galli, G. Phys. Rev. Lett. 2009, 102, 195901. (53) Li, D.; Wu, Y.; Kim, P.; Shi, L.; Yang, P.; Majumdar, A. Appl. Phys. Lett. 2003, 83, 2934. (54) Boukhvalov, D. W.; Katsnelson, M. I. J. Phys.: Condens. Matter 2009, 21, 344205. (55) Singh, A. K.; Yakobson, B. I. Nano Lett. 2009, 9, 1540. (56) Sessi, P.; Guest, J. R.; Bode, M.; Guisinger, N. P. Nano Lett. 2009, 9, 4343 pMID: 19883050.

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Author Contributions

G.F. and A.C. contributed equally to this work. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS We gratefully acknowledge support from the ANR project Accattone (G.F., M.L., and F.M.), the EU Graphene Flagship (F.M.), and the Swiss National Science Foundation and National Supercomputing Center CSCS under the project ID s337 (A.C., N.M.)

■

REFERENCES

(1) Balandin, A. A. Nat. Mater. 2011, 569. (2) Balandin, A. A.; Ghosh, S.; Bao, W.; Calizo, I.; Teweldebrhan, D.; Miao, F.; Lau, C. N. Nano Lett. 2008, 8, 902 pMID: 18284217. (3) Ghosh, S.; Calizo, I.; Teweldebrhan, D.; Pokatilov, E. P.; Nika, D. L.; Balandin, A. A.; Bao, W.; Miao, F.; Lau, C. N. Appl. Phys. Lett. 2008, 92, 151911. (4) Cai, W.; Moore, A. L.; Zhu, Y.; Li, X.; Chen, S.; Shi, L.; Ruoff, R. S. Nano Lett. 2010, 10, 1645. (5) Ghosh, S.; Bao, W.; Nika, D. L.; Subrina, S.; Pokatilov, E. P.; Lau, C. N.; Balandin, A. A. Nat. Mater. 2010, 9, 555. (6) Ghosh, S.; Nika, D. L.; Pokatilov, E. P.; Balandin, A. A. New J. Phys. 2009, 11, 095012. (7) Faugeras, C.; Faugeras, B.; Orlita, M.; Potemski, M.; Nair, R. R.; Geim, A. K. ACS Nano 2010, 4, 1889 pMID: 20218666. (8) Bonini, N.; Garg, J.; Marzari, N. Nano Lett. 2012, 12, 2673. (9) Evans, W. J.; Hu, L.; Keblinski, P. Appl. Phys. Lett. 2010, 96, 203112. (10) Paulatto, L.; Mauri, F.; Lazzeri, M. Phys. Rev. B 2013, 87, 214303. (11) Lindsay, L.; Broido, D. A.; Mingo, N. Phys. Rev. B 2011, 83, 235428. (12) Ho, C. Y.; Powell, R. W.; Liley, P. E. J. Phys. Chem. Ref. Data 1972, 1, 279. (13) Peierls, R. Ann. Phys. 1929, 3, 1066. (14) Callaway, J. Phys. Rev. 1959, 113, 1046. (15) Klemens, P. Thermal Conductivity and Lattice Vibrational Modes; Seitz, F., Turnbull, D., Eds.; Solid State Physics; Academic Press, New York, 1958. (16) Ziman, J. Electrons and Phonons: The Theory of Transport Phenomena in Solids, Oxford Classic Texts in the Physical Sciences; Oxford University Press, New York, 2001. (17) Guyer, R. A.; Krumhansl, J. A. Phys. Rev. 1966, 148, 766. (18) Fugallo, G.; Lazzeri, M.; Paulatto, L.; Mauri, F. Phys. Rev. B 2013, 88, 045430. (19) Omini, M.; Sparavigna, A. Phys. Rev. B 1996, 53, 9064. (20) Omini, M.; Sparavigna, A. Il Nuovo Cimento D 1997, 19, 1537. (21) For the resolution of the BTE, for in-plane calculations of singleand bilayer graphene a Gaussian smearing of σ = 10 cm−1 along a 128 × 128 × 1 q-point phonon grid on the BZ mesh has been found to lead to converged relaxation times, while for the out-of-plane graphite K a smearing of σ = 25 cm−1 and a grid of 64 × 64 × 3 is sufficient. Note that for comparing 2D layers with the 3D bulk material, it is necessary to use an effective thickness, which was taken to be the experimental interlayer distance of graphite, namely 3.32 Å. (22) Harmonic and anharmonic forces for graphene (at each strain value considered), graphite, and bilayer have been obtained with the parameter described in ref 10. (23) Baroni, S.; de Gironcoli, S.; Dal Corso, A.; Giannozzi, P. Rev. Mod. Phys. 2001, 73, 515. F

dx.doi.org/10.1021/nl502059f | Nano Lett. XXXX, XXX, XXX−XXX