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# Dimensional crossover of thermal transport in few-layer graphene

(Parte **1** de 2)

Dimensional crossover of thermal transport in few-layer graphene

Graphene1, in addition to its unique electronic2,3 and optical properties4, reveals unusually high thermal conductivity5,6. The fact that the thermal conductivity of large enough graphene sheets should be higher than that of basal planes of bulk graphite was predicted theoretically by Klemens7. However, the exact mechanisms behind the drastic alteration of a material’s intrinsic ability to conduct heat as its dimensionality changes from two to three dimensions remain elusive. The recent availability of high-quality few-layer graphene (FLG) materials allowed us to study dimensional crossover experimentally. Here we show that the roomtemperature thermal conductivity changes from ∼2,800 to ∼1,300Wm−1 K−1 as the number of atomic planes in FLG increases from 2 to 4. We explained the observed evolution from two dimensions to bulk by the cross-plane coupling of the low-energy phonons and changes in the phonon Umklapp scattering. The obtained results shed light on heat conduction inlow-dimensionalmaterialsandmayopenupFLGapplications inthermalmanagementofnanoelectronics.

One of the unresolved fundamental science problems8, with enormous practical implications9, is heat conduction in lowdimensional materials. The question of what happens with thermal conductivity when one goes to strictly two-dimensional (2D) and one-dimensional (1D) materials has attracted considerable attention8,10–13. Thermal transport in solids is described by thermal conductivity K through the empirical Fourier law, which states that JQ = −K∇T, where the heat flux JQ is the amount of heat transported through the unit surface per unit time and T is temperature.Thisdefinitionhasbeenusedforbulkmaterialsaswell as nanostructures. Many recent theoretical studies8,10–13 suggest an emerging consensus that the intrinsic thermal conductivity of 2D and 1D anharmonic crystals is anomalous and reveals divergence with the size of the system defined either by the number of atoms N or linear dimension L. In 2D this universal divergence leads to K ∼ ln(N) (ref. 8). The anomalous nature of heat conduction in low-dimensional systems is sometimes termed as breakdown of Fourier’s law11. The anomalous K dependence on L in strictly 2D (1D) materials should not be confused with the length dependence of K in the ballistic transport regime frequently observed at low temperatures when L is smaller than the phonon mean free path (MFP). Ballistic transport has been studied extensively in carbon nanotubes14–16. Here we focus on the diffusive transport and thermal conductivity limited by the intrinsic phonon interactions through Umklapp processes17.

Experimental investigation of heat conduction in strictly 2D materials has essentially been absent owing to the lack of proper

1Nano-Device Laboratory, Department of Electrical Engineering and Materials Science and Engineering Program, University of California–Riverside, Riverside, California 92521, USA, 2Department of Physics and Astronomy, University of California–Riverside, Riverside, California 92521, USA, 3Department of Theoretical Physics, State University of Moldova, Chisinau MD-2009, Republic of Moldova. *e-mail:balandin@e.ucr.edu.

materials systems. Thermal transport in conventional thin films still retains ‘bulk’ features because the cross-sections of these structures are measured in many atomic layers. Heat conduction in such nanostructures is dominated by extrinsic effects, for example, phonon-boundary or phonon-defect scattering18. The situation has changed with the emergence of mechanically exfoliated graphene— an ultimate 2D system. It has been established experimentally that the thermal conductivity of large-area suspended single-layer graphene (SLG) is in the range K ≈ 3,0–5,0Wm−1K−1 near room temperature5,6, which is clearly above the bulk graphite limit K ≈2,000Wm−1K−1 (ref.7).TheupperboundK forgraphenewas obtainedforthelargestSLGflakesexamined(∼20µm×5µm).The extraordinarily high value of K for SLG is related to the logarithmic divergence of the 2D intrinsic thermal conductivity discussed above7,8.TheMFPofthelow-energyphononsingrapheneand,correspondingly,theircontributiontothermalconductivityarelimited bythesizeofagrapheneflakeratherthanbyUmklappscattering7,19.

The dimensional crossover of thermal transport, that is, evolution of heat conduction as one goes from 2D graphene to 3D bulk, is of great interest for both fundamental science8 and practical applications9. We addressed this problem experimentally by measuring the thermal conductivity of FLG as the number of atomic planes changes from n = 2 to n ≈ 10. A high-quality large-area FLG sheet, where the thermal transport is diffusive and limited by intrinsic rather than extrinsic effects, is an ideal system for such a study. A number of FLG samples were prepared by standard mechanical exfoliation of bulk graphite1 and suspended across trenches in Si/SiO2 wafers (Fig. 1a–c). The depth of the trenches was ∼300nm and the trench width varied in the range

1–5µm. Metal heat sinks near the trench edges were fabricated by shadow mask evaporation. We used our previous experience in graphene device fabrication20 to obtain the highest quality set of suspended samples. The width of the suspended flakes varied from W ≈ 5 to 16µm. The number of atomic layers in the graphene flakes was determined with micro-Raman spectroscopy (InVia, Renishaw) through deconvolution of the 2D/G′ band in graphene’s spectrum21. The measurements of K were carried out using a steady-state optical technique developed by us on the basis of micro-Raman spectroscopy5. The shift of the temperaturesensitive Raman G peak in graphene’s spectrum22 defined the local temperature rise in the suspended portion of FLG in response to heatingbyanexcitationlaserinthemiddleofthesuspendedportion of the flakes (Fig. 1d). The thermal conductivity was extracted from the power dissipated in FLG, the resulting temperature rise and the flake geometry through the finite-element method solution of the heat diffusion equation (Fig. 1e; see the Methods section).

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LETTERS NATUREMATERIALSDOI:10.1038/NMAT2753

Laser light Reflected light

Metal heat sinkMetal heat sink

Silicon oxideSilicon oxide Silicon substrate

Substrate Heat sinkHeat sink

Trench Graphene

Min: 300Max: 350

G-peak shift for BLG

Linear fit to experimental data 1,580

Slope: | | / P| = 0.701 cm mW |

1,576 G-peak position (cm cd e

Figure 1 | Samples and measurement procedure. a, Schematic of the thermal conductivity measurement showing suspended FLG flakes and excitation laser light. b, Optical microscopy images of FLG attached to metal heat sinks. c, Coloured scanning electron microscopy image of the suspended graphene flake to clarify a typical structure geometry. d, Experimental data for Raman G-peak position as a function of laser power, which determines the local temperature rise in response to the dissipated power. e, Finite-element simulation of temperature distribution in the flake with the given geometry used to extract the thermal conductivity.

2 layers 3 layers 4 layers Multiple layers Kish graphite HOPG

10 Integrated Raman intensity (a.u.)Thermal conductivity (W m

Room-temperature measurements Maximum

Average value for SLG

High-quality bulk graphite Regular bulk graphite

Theory prediction Theory prediction with increased roughness

Figure 2 | Experimental data. a, Integrated Raman intensity of the G peak as a function of the laser power at the sample surface for FLG and reference bulk graphite (Kish and highly ordered pyrolytic graphite (HOPG)). The data were used to determine the fraction of power absorbed by the flakes. b, Measured thermal conductivity as a function of the number of atomic planes in FLG. The dashed straight lines indicate the range of bulk graphite thermal conductivities. The blue diamonds were obtained from the first-principles theory of thermal conduction in FLG based on the actual phonon dispersion and accounting for all allowed three-phonon Umklapp scattering channels. The green triangles are Callaway–Klemens model calculations, which include extrinsic effects characteristic for thicker films.

The power dissipated in FLG was determined through the calibration procedure based on comparison of the integrated

Raman intensity of FLG’s G peak IGFLG and that of reference bulk graphite IGBULK. Figure 2a shows measured data for FLG with n=2, 3, 4, ∼8 and reference graphite. Each addition of an atomic plane leads to an IGFLG increase and convergence with the graphite whereas the ratio ς = IGFLG/IGBULK stays roughly independent of excitation power, indicating proper calibration. In Fig. 2b we present measured K as a function of the number of atomic planes n in FLG. The maximum and average K values for SLG are also shown. As graphene’s K depends on the width of the flakes19,23 the data for FLG are normalized to the width W = 5µm to

556 NATURE MATERIALS | VOL 9 | JULY 2010 | w.nature.com/naturematerials

NATUREMATERIALSDOI:10.1038/NMAT2753 LETTERS

TA1 TA1

ZA1 ZA1

ZA2

Phonon energy (meV)Phonon energy (meV)

q/qmax

K(q) (W m¬1 acd

5,0 W m¬1 K¬1 400 Wm¬1 K¬1

10 W m¬1 K¬1

0.1 W m¬1 K¬1

K suppression BLG

Graphene by Umklapp scattering Graphene

Figure 3 | Theoretical interpretation. a, Phonon dispersion in BLG, which shows that phonon branches are double degenerate for q>0.2qmax (where qmax =14.749nm−1). b, Close-up of the phonon dispersion in graphene (solid curves) and BLG (solid and dashed curves) near the Brillouin zone centre.

Note that the longitudinal acoustic (LA2) and transverse acoustic (TA2) phonon branches in BLG have a very small slope at low q, which translates to a low phonon group velocity. c, Contributions to thermal conductivity of different phonons, indicating the range of phonon wavevectors where Umklapp scattering is the main thermal-transport-limiting mechanism. The dashed line corresponds to the rough-edge scattering alone. d, Diagram of three-phonon Umklapp scattering in graphene and BLG, which shows that in BLG there are more states available for scattering owing to the increased number of phonon branches.

allow for direct comparison. At fixed W the changes in the K value with n mostly result from modification of the three-phonon Umklapp scattering. The thermal transport in our experiment is in the diffusive regime because L is larger than the phonon MFP in graphene, which was measured6 and calculated24 to be around ∼800nm near room temperature. Thus, we explicitly observed heat conduction crossover from 2D graphene to 3D graphite as n changes from 2 to ∼8.

It is illustrative that the measured K dependence on FLG thickness h×n (h = 0.35nm) is opposite from what is observed for conventional thin films with thicknesses in the range of a few nanometrestomicrometres.InconventionalfilmswiththicknessH smaller than the phonon MFP, thermal transport is dominated by phonon-rough-boundary scattering. The thermal conductivity can be estimated from K = (1/3)CVυ2τ, where CV is the specific heat and υ and τ are the average phonon velocity and lifetime. When the phonon lifetime is limited by the boundary scattering, τ = τB,

is a parameter defined by the surface roughness). In FLG with 2 or 3 atomic layer thickness there is essentially no scattering from the top and back surfaces (only the edge scattering is present23). Indeed, FLG is too thin for any cross-plane velocity component and for randomthicknessfluctuations,thatis,p≈1forFLG.Tounderstand the thermal crossover one needs to examine the changes in the intrinsic scattering mechanisms limiting K: Umklapp scattering resulting from crystal anharmonicity.

The crystal structure of bilayer graphene (BLG) consists of two atomic planes of graphene bound by weak van der Waals forces resulting in a phonon dispersion different from that in SLG (Fig. 3a,b). By calculating the phonon dispersion in FLG from first principles23,25, we were able to study phonon dynamics (Supplementary Movie) and the role of separate phonon modes. Figure 3c shows the contributions 1K(qi) to the thermal conductivity of phonons with wavevectors from the interval is, in our calculation, limited by the phonon Umklapp and edge boundary scattering (m is the number of intervals (qi,qi+1) and p = 0.92 for the edges). In BLG the number of available phonon branches doubles (Fig. 3a,b). However, the new conduction channels do not transmit heat effectively in the low-energy regime becausethegroupvelocitiesoftheLA2 andTA2 branchesarecloseto zero at small q. On the other hand, an increase in the thickness h×n leads to a corresponding decrease in the flux density: see region I of small q where thermal transport is mostly limited by edge scattering (Fig. 3c). In region I of large q, the number of phonon states available for three-phonon Umklapp scattering in BLG increases by a factor of four as compared with SLG (Fig. 3d). As a result, despite the increase in the number of conduction channels in BLG, its K decreases because the q phase space available for Umklapp scattering increases even more (Supplementary Materials). One can say that in SLG, phonon Umklapp scattering is quenched and the heat transport is limited mostly by the edge (in-plane) boundary

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LETTERS NATUREMATERIALSDOI:10.1038/NMAT2753 scattering. This is in agreement with Klemens’s explanation of the higher thermal conductivity of graphene compared with graphite7. It is rooted in the fundamental properties of 2D systems discussed above. Our theory also explains molecular-dynamics simulations for ‘unrolled’ carbon nanotubes, which revealed a much higher K for graphene than for graphite26. The theoretical data points for n = 1–4 shown in Fig. 2b are in excellent agreement with the experiment. They were obtained by accounting for all allowed Umklapp processes through extension of the diagram technique developedbyusforSLG(ref.23)ton=2and3.Thelowermeasured K values for n = 4 are explained by the stronger extrinsic effects resulting from difficulties in maintaining the thickness uniformity in such samples. The theoretical data point approaches the bulk limit for the ‘ideal’ graphite. It is possible that the crossover point may be related to AB Bernal stacking and completeness of the graphite unit cell. Practically, its position is affected by many factors including FLG’s quality, the width of the flake and the strength of substrate coupling.

(Parte **1** de 2)