SUPPLEMENTARY INFORMATION

Transcription

1 SUPPLEMENTARY INFORMATION I. Experimental Thermal Conductivity Data Extraction Mechanically exfoliated graphene flakes come in different shape and sizes. In order to measure thermal conductivity of the suspended graphene flake with different number of atomic planes n we had to extract the data from the flakes of various shapes. To this end we developed a numerical procedure, which solves the heat diffusion equation for a graphene flake of arbitrary geometry. This is preferred approach since attempts to cut suspended graphene flakes often result in excessive damage, which affects thermal conductivity. The heat diffusion model is applicable to our few-layer graphene (FLG) samples, which have lateral dimensions (tens of m) much larger than the phonon mean free path (MFP) in graphene, i.e. ~775 nm near room-temperature (RT). The exact shape of the flakes was taken from the optical and scanning electron microscopy (SEM) images. The heat diffusion equation was solved using the finite element method (FEM) for a given set of boundary conditions. We assumed that there was no heat exchange with environment from the flake edges and the temperature of the heat sinks maintained constant during the measurement. The results obtained with the metal heat sinks were consistent with the bulk graphite heat sinks, which attest to the heat sink quality and low thermal resistance between the flake and the sink. The flake-sink thermal contact resistance is much smaller for graphene than for carbon nanotubes owing to a much larger area of the contact between graphene and the sink (see Figure 1). The temperature of the SiO layer, which was monitored during the measurements using the micro-raman technique, stayed constant indicating the absence of any substantial thermal coupling from the flake to the Si wafer through the SiO layer. The heat propagated to the metal hit sinks as expected in this experimental setup. Figure 1: Optical microscopy image of the few-layer graphene flake suspended across a trench and a large area metal heat sink deposited on top of the flake end. nature materials 1

2 supplementary information The thermal conductivity was found through the following iteration procedure. For a certain power dissipated in graphene P G, which is obtained from the experiment, and initially assumed value of thermal conductivity K 0, one obtains the resulting modeled temperature rise T M through the solution of the heat diffusion equation (see Figure ). The obtained T M is compared with the experimental temperature rise T E. If T M is larger than T E, the value of the thermal conductivity K is increased in the next run. The procedure continues until the T M and T E become equal. This condition defines the true value of the thermal conductivity K of the graphene flake with a given number of atomic planes n and specific geometry (size). In the described procedure, the laser spot, which serves as a heat source, was modeled by a Gaussian beam with the intensity given as x y P ( x, y ) P 0 exp. (1) ere x and y are coordinates within the graphene plane, is the standard deviation and P 0 is the maximum power. This assumption is rather accurate and standard in laser optics [1-]. ( ( Figure : Simulated temperature distribution in two few-layer graphene flakes with different geometry and the number of atomic planes (n= for the right flake and n=3 for the left). The heat source power corresponds to the experimental detector power P D = mw. The hot spot is modeled by the Gaussian distribution. In an alternative, faster algorithm, for determining thermal conductivity we translated the iterations for K to the iterations for the slope =(/P D ) measured in the experiment. The measured power P D is related to the power dissipated in graphene P G through the calibration procedure. Figure 3 shows the simulated thermal conductivity as a function of the slope, and the nature MATERIALS

3 supplementary information point where the slope becomes equal to the measured one. This point defines the value of thermal conductivity for the flake with the given n and size. Thermal Conductivity K( W/mK) Figure 3: Illustration of the thermal conductivity data extraction using the solution of the heat diffusion equation by the finite-difference method. The thermal conductivity, which corresponds to the measured slop, is the true value for a flake with given number of atomic plane and specific shape. Slope (cm -1 /W) II. Thermal Conductivity Normalization The values of thermal conductivity obtained for FLG with small n are specific for a given flake size. In the flakes with narrow width (e.g. d~ 1 m; comparable to the phonon MFP) thermal conductivity can be reduced due to the phonon edge scattering [3]. To elucidate the effects behind thermal conductivity evolution with the changing number of planes n (changing thickness) we normalized the measured K to the same width d o =5 m. The simplest normalization to the same width can be done with the help of the expression for the thermal conductivity K 1 / 3 ) C, where CV is the specific heat, and are the average phonon ( V velocity and lifetime and Ziman s expression for the phonon life time limited by the boundary scattering ( 1/ ) ( / d )(( p 1) /( p 1)), which shows that K scales down with the decreasing lateral dimension d (here p is the specularity parameter defined by the surface roughness). We carried out a more accurate normalization within Klemens theory for graphene thermal conductivity, which explicitly includes the dependence on the flake size and temperature [4-5]. A brief summary of our derivation is given below. Let us assume that we need to recalculate thermal conductivity K 1, obtained for the graphene flake with the width d 1 at T = T 1 to the value K 1,norm which corresponds to flake width d 0 and temperature T 0. For this purpose, we can write thermal conductivity as [6] nature materials 3

4 supplementary information K M F( s,min ) 4Th, (1) sta, s,max s s where s,min exp( s,min / kt ) F( s,min ) ln{ exp( s,min / kt ) 1 } k T exp( / k T ) 1 s,min () and s,min s s s,max (3) s M kt d The difference between,min and TA,min is small (few mev), so we can assume that (, ) M,max,min TA,min min Td. kt d Then setting F( TA,min ) F(,min ) F ( min ) and we can rewrite Eq. (1) as follows K F( ( Td, )) A T min, (4) where M s A =const. (5) 4h sta, s,max s Using the numerical value from Ref. [6],max =38.5 Tz, d in m and T in o K we obtain =1.3 km/s, =1.8, M = kg, (, ) M,max min Td = / dt Tz, (6) kt d and 4 nature MATERIALS

5 supplementary information F( ( T, d), T ) ln{ exp( / k T ) 1 } min exp( / k T ) min min min kt exp( min / kt ) 1 exp( y( T, d)) ln{ expytd ( (, )) 1 } ytd (, ), exp ( y ( T, d )) 1 (7) where yt (, d ) = / 3 dt. (8) Using Eq. (5) one can write for thermal conductivities K 1 and K 1,norm the following K 1 =AF(T 1,d 1 )/T 1 and K 1,norm =AF(T 0,d 0 )/T 0. Then, the recalculated value is determined as K 1,norm = K T1 1 T 0 F( T0, d0) F( T, d ) 1 1 with the values of FT ( 0, d0) and FT ( 1, d1) given by Eqs. (7-8). III. Dissipated Power Calibration Only fraction P G of the laser light focused on graphene flake will actually be dissipated in the graphene. Most of the light will be reflected back after the light travels through the flake to the trench bottom and reflected back (see Figure 4). The power, which is measured by the detector placed at the position of the flake, is the total power P D, part of which goes into the graphene flake after two transmissions (incident pass and reflected pass) and the rest is lost in the silicon wafer P Si. It is known that the fraction of the power absorbed by graphene is.3% per layer for light wavelength > 500 nm [7]. Our measurements were performed at smaller wavelength (=488 nm) where the absorption can be enhanced [7-8]. For this reason, it was important to determine the absorbed power in the specific conditions of our experiment. The power P G was measured through the calibration procedure with the bulk graphite serving as a reference as illustrated in Figure 4. It is based on the comparison of the experimentally determined integrated Raman intensity for G peak from the few-layer graphene and bulk graphite. The derivation starts with the single-layer graphene (SLG). The integrated Raman intensity from SLG is given as [9-10] I N I, where N is the number of the scattering atoms in the surface area A and G is G G o the Raman scattering cross section. The integrated Raman intensity can be expressed through the nature materials 5

6 supplementary information absorbed power as IG ( N/ A)( G / GaG ) PG /(1 R Si ). Focusing the same laser beam on the reference bulk graphite we get P D =I o A. The integrated Raman scattering intensity from graphite is obtained by summation over all n graphene layers, which make up bulk graphite, i.e. I ULK N I o n1 exp a n, where is the absorption coefficient and a is the thickness of each monolayer. This leads to I 1/ )( N / A )( / a ) P (1 R ), where ULK ( D R is the reflection coefficient for graphite. Defining the ratio of the integrated intensities as I / I, we express the power absorbed in SLG through the power measured by the G OPG detector as PG ( / )[ GaG / G a](1 RSi)(1 R) P D. The term in the square brackets is about unity because the Raman cross-section and absorption coefficient per layer are the same for graphene and graphite, which consists of graphene atomic planes weakly bound by van der Waals forces. y measuring one completes the calibration for SLG. Extending the same procedure to FLG with n atomic planes, one can derive the following expression for the power absorbed P(n) through the power on the detector P D : P ( n ) / P D ( n )(1 R 1 (1 R ( n )) R exp a n 1 exp a n 1 exp a n 1 ) G Si, G where ( n ) I FLG / I G ULK is the measured integrated intensity ratio for FLG and bulk graphite. P G I G P D I ULK D (a) (b) (c) Figure 4: Illustration of the calibration procedure. (a) Integrated Raman intensity is related to the power absorbed in FLG through the scattering cross-section and absorption coefficient. (b) Detector is placed at the sample location to measure the power at the surface. (c) Integrated Raman intensity is measured in bulk graphite used as reference for the calibration. 6 nature MATERIALS

7 supplementary information IV. Theoretical Approach for Thermal Conductivity in FLG Display 3c shows function K(q), which illustrates how different intervals of the phonon wave vector q contribute to thermal conductivity. Specifically, we defined three different q intervals. 1) Region I is an interval characterized by an absence of the phonon Umklapp scattering where the phonon mean free path (MFP) is determined by the scattering on the flake boundaries and defects. ) Region II is an interval of q values characterized by an onset of the phonon Umklapp scattering. 3) Region III is an interval of q values characterized by the strong phonon Umklapp scattering. The increase in the number of atomic plane by a factor of s leads to the corresponding increase in the number of heat conducting channels by a factor of s (i.e. the increase in the number of the phonon branches by s). Simultaneously, the thickness of the sample increases by a factor of s. The thickness enters a denominator of the expression for the thermal conductivity. In few-layer graphene, the increase in the number of atomic planes lead to degeneracy of and TA phonon modes (see Display 3a) for all phonon wave vectors q with the exception of a narrow interval q < 0.1 q max. The and TA branches make dominant contributions in heat transfer in graphene. This trend is illustrated in Display 3a, which shows the phonon dispersion for bi-layer graphene (n=). In the case of FLG with n=3 one gets 3 transverse phonon branches TA 1, TA, TA 3 originating from a single TA in graphene, and longitudinal branches 1,, 3 originating from a single in graphene. For FLG with n=4 one get 4 phonon branches of each polarization. It is important to note that only one and one TA branches will be true acoustic phonons with the zero energy at q=0 and high phonon group velocity (see TA 1 and 1 for bi-layer graphene in Display 3a). The rest of the branches will be quantized in the interval q < 0.1 q max (see TA and for bi-layer graphene in Display 3a). The quantized modes have low group velocity, which slowly increases with increasing q. The modes with small group velocity make little contribution to heat transfer. nature materials 7

8 supplementary information The region - I phonon contribution to the total thermal conductivity will be slowly decreasing with the increasing number of layers n as a result of low velocity of modes j TA j, for j. This can be clarified by comparing SLG and LG. Let us represent the region - I thermal conductivity as (K + K TA )/h, where h is the thickness of the atomic layer, K and K TA are contributions of branches TA in region I. In LG a corresponding quantity will be written as (K 1 + K TA1 + K + K TA )/(h), where K 1 = K, K TA1 = K TA while K is smaller than K and K TA is smaller than K TA because of the low phonon velocities near q = 0. For this reason the region I phonon thermal conductivity of LG will be lower than that of SLG. This can be seen in Display 3c where the K(q) curve for LG is slightly lower than that for SLG in region I. The same trend will continue in region I for FLG as the number of atomic plane increases to n=3 and n=4. The decrease in the contributions to the total thermal conductivity of the phonons from region III is even more pronounced. In LG the number of Umklapp processes increases by a factor of four (as seen in Display 3) as the number of channels increases by a factor or and the thickness increases by a factor of. As a result, the region III thermal conductivity in LG decreases by a factor of 4 compared to that in SLG. In FLG with n=3 the region III thermal conductivity decreases by a factor of 9 and in FLG with n=4 by a factor of 16 as compared to SLG. This effect constitutes the Umklapp quenching in FLG with decreasing number of atomic planes n due to the restriction in the scattering phase space. It is not possible to describe quantitatively the region II thermal conductivity without the exact calculations. Qualitatively, it is clear that in region II one also has a substantial decrease in the thermal conductivity, which nevertheless is not as strong as in region III because there are q intervals in region II where Umklapp scattering is absent and the MFP is only determined by scattering on boundaries and defects. The combined contribution of these three q-intervals (three regions) lead to the resulting theoretical trend as the number of atomic planes changes from n=1 to n=4 as shown by the theoretical (diamond) data point in Display b. We indicated two more theoretical data points (green triangles) obtained via simplified Calaway Klemens model, which include scattering on rough boundaries. The transport in FLG with n=8 becomes mode similar to that in graphite films and has to include the top and bottom surface roughness. 8 nature MATERIALS

9 supplementary information References [1].E.A. Saleh and M.C. Teich, Fundamentals of Photonics (New York, John Wiley & Sons, 1991), pp. 80. [] L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge, Cambridge University Press, 1995) pp. 67. [3] D.L. Nika, E.P. Pokatilov, A.S. Askerov and A.A. alandin, Phys. Rev., 79, (009). [4] P.G. Klemens, J. Wide andgap Mater., 7, 33 (000). [5] P.G. Klemens, Int. J. Thermophys.,, 65 (001). [6] D.L. Nika, S. Ghosh, E.P. Pokatilov and A.A. alandin, Appl. Phys. Lett., 94, (009). [7] R.R. Nair, P. lake, A.N. Grigorenko, K.S. Novoselov, T.J. ooth, T. Stauber, N.M.R. Peres and A.K. Geim, Science, 30, 1308 (008). [8] K.S. Kim, Y. Zhao,. Jang, S.Y. Lee, J.M. Kim, K.S. Kim, J.-. Ahn, P. Kim, J.-Y. Choi and.. ong, Nature, 457, 706 (009). [9] M. C. Tobin, Laser Raman Spectroscopy (Wiley-Interscience, Toronto, 1971). [10] M. M. Sushchinskii, Raman Spectra of Molecules and Crystals (Nauka, Moscow, 1969). nature materials 9