Two-Dimensional Phonon Transport in Supported Graphene

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Science  09 Apr 2010:
Vol. 328, Issue 5975, pp. 213-216
DOI: 10.1126/science.1184014


The reported thermal conductivity (κ) of suspended graphene, 3000 to 5000 watts per meter per kelvin, exceeds that of diamond and graphite. Thus, graphene can be useful in solving heat dissipation problems such as those in nanoelectronics. However, contact with a substrate could affect the thermal transport properties of graphene. Here, we show experimentally that κ of monolayer graphene exfoliated on a silicon dioxide support is still as high as about 600 watts per meter per kelvin near room temperature, exceeding those of metals such as copper. It is lower than that of suspended graphene because of phonons leaking across the graphene-support interface and strong interface-scattering of flexural modes, which make a large contribution to κ in suspended graphene according to a theoretical calculation.

Since graphene was first exfoliated from graphite and studied on dielectric substrates in 2004 (1), the monatomic layer of carbon atoms has attracted great interest for electronic applications because of superior charge mobility (2) and mechanical strength (3), as well as its compatibility with existing planar silicon devices. Other carbon allotropes, including diamond (4), graphite (5), and carbon nanotubes (CNTs) (68), have the highest thermal conductivity (κ) values reported because the strong bonding of the light carbon atoms results in a large phonon contribution to κ despite a much smaller electronic component. For similar reasons, graphene is expected to possess a much higher κ than the silicon active layers and copper interconnects in current-generation electronic devices. This potential enhancement may provide a solution to the increasingly severe heat dissipation problem in nanoelectronics, which is caused by increased power density as well as reduced κ of electronic materials with decreasing feature sizes (9, 10).

Although electronic transport in graphene has been investigated extensively, there have been few studies on the thermal transport because of experimental challenges. Recently, the room-temperature κ of a suspended single-layer graphene (SLG) flake has been reported from a Raman measurement (11). The obtained values, 3000 to 5000 W m−1 K−1, exceed those of diamond (4) and graphite (5). However, because of the limited temperature (T) sensitivity of the Raman technique, this study has not yielded the κ − T relation that is important for understanding the intriguing two-dimensional (2D) behavior of phonons in SLG. Furthermore, SLG is usually supported on a dielectric substrate for device applications. The charge mobility in SLG supported on silicon dioxide (SiO2) is suppressed by about 10 times compared with clean suspended SLG because of scattering by substrate phonons and impurities (12). The effect of substrate interaction on thermal transport, however, has not been elucidated so far.

Here, we report thermal transport measurements on SLG supported on amorphous SiO2, which represents the most commonly used device configuration. Despite phonon-substrate scattering, we find that the room-temperature κ of the supported SLG approaches about 600 W m−1 K−1, considerably higher than the κ values of common thin-film electronic materials (9, 10). In addition, using full quantum mechanical calculations of the three-phonon scattering processes, we find a large κ contribution from the flexural (ZA) modes in suspended SLG. The measured κ − T relation can be explained by large suppression of the ZA contribution in the supported SLG.

We made thermal measurements on three SLG flakes exfoliated onto SiO2 using the device shown in Fig. 1, A to C (13). The patterned SLG of 1.5 to 3.2 μm in width (W) and 9.5 to 12.5 μm in length (L) covers the surface of a ~300-nm-thick SiO2 beam, the two ends of which are connected to four Au/Cr resistance thermometer (RT) lines on suspended SiO2 beams. The two straight inner RT lines cover the two ends of the SLG, which does not contact the two outer U-shaped RT lines.

Fig. 1

Graphene sample G2. (A to C) SEM images of the suspended device, the central beam, and the folded edge of the SLG ribbon near the right electrode. The inset in (C) is a thermal circuit of the measurement device, where Ro is the thermal resistance of the SiO2 joint between two adjacent RT lines. The scale bar is 10, 3, and 1 μm in (A), (B), and (C), respectively. (D) Raman spectra obtained from G2 immediately after the thermal measurement. The spectra (e.g., P1) obtained near the left end of the SLG indicates a monolayer sample, whereas those (such as P2 and P3) taken at the center and near the right electrode show a broadened 2D peak (centered at 2700 cm−1) and a decreased 2D peak to G peak (centered at 1580 cm−1) ratio because the folded-edge region of the SLG has a signal similar to that of bilayer graphene.

The SLG samples were characterized with Raman spectroscopy and scanning electron microscopy (SEM) (13). Sample 1 (G1) consists of a 3.2-μm-wide ribbon and a 1.5-μm-wide ribbon in parallel. Although a portion of one edge of the 2.4-μm-wide sample 2 (G2) is folded near one electrode (Fig. 1C), the two edges of the 2.4-μm-wide sample 3 (G3) appear to be straight. In addition, the obtained Raman spectra show that G1, G2 (Fig. 1D), and G3 are monolayers with no indication of the D band associated with defects (14).

The two-probe electrical resistance (R) at room temperature was measured to be 160, 7, and 5400 kilohm for G1, G2, and G3, respectively. The much lower two-probe R of G2 is indicative of smaller contact resistance and a cleaner electrode-graphene interface than for the other two samples. For G2 (Fig. 2A), the two-probe, 2D electrical conductivity, σ ≡ L/WR, shows a T dependence similar to that reported for oxide-supported SLG (12).

Fig. 2

(A) Measured two-probe electrical conductivity (σ) and extracted electron mobility (μ) of G2 as a function of temperature. (B) Measured Seebeck coefficients as a function of temperature for G1 (triangles) and G2 (circles). The error bars reported in this work include both bias and random uncertainties, the latter of which are determined based on three to nine measurements.

We measured the Seebeck coefficient (S) of the SLG by electrically heating one outer U-shaped RT line (RT 1). Because of a linear temperature profile between the midpoint and the ends of the other three RT lines (RT 2, 3, and 4 in order of increasing distance from RT 1, respectively), the highest temperature rise at the midpoint (Tj,m) is twice the average rise (Tj¯) of each of the three lines (13), i.e., Tj,mTj,mT0=2Tj¯; j=2,3,4, where T0 is the substrate temperature. Using the thermal circuit in Fig. 1C and averaging the parabolic temperature profile of RT 1, we derive (13)T1,m=32T1¯12(T2¯+T3¯+T4¯)(1)where Tj¯is obtained from the measured four-probe electrical resistance of each RT line. The measurement yields S = V23/(∆T2,m – ∆T3,m), where V23 is the thermovoltage between the two inner electrodes.

The obtained room-temperature S values of G1 and G2 (Fig. 2B) are –79.7 ± 0.6 and –82.7 ± 0.2 μV K−1, respectively, which suggests that the Fermi energy (EF) is in the conduction band. We fit the experimental S−T curve using a theoretical model (15) for oxide-supported SLG where screened charged impurity scattering dominates electron transport. For G2, the best fit to the measured S is given by EF = 0.049 eV (13). The electron concentration is obtained as n = (EF/ħvF)2= 1.7 × 1011 cm−2, where ħ is the reduced Planck’s constant, and vF = 1 × 106 m/s is the Fermi velocity in SLG. Based on μ = σ/ne, with e being the elemental charge, a μ value of ~20,000 cm2/Vs (Fig. 2A) is obtained, comparable to the highest electron mobility values reported for oxide-supported SLG (12, 16), thus suggesting similar sample quality.

For the κ measurement, the thermal resistance of each RT line including the SiO2 beam is the same and was obtained as (13)Rb=2T1,m+T2,m+T3,m+T4,mQ(2)where Q is the electrical heating in RT 1. The thermal resistance of the central beam is found from the thermal circuit asRs=RbT2,mT3,mT3,m+T4,m(3)and was several orders of magnitude higher than the calculated interface thermal resistance between the SLG and the electrode or the SiO2 (13). The thermal conductance (G ≡ 1/Rs) of the central beam was measured before and after the SLG was etched away in oxygen plasma. The measured G after etching was considerably smaller than that before etching for T above 80 K, below which the difference decreases to near the measurement uncertainty (Fig. 3). The difference in G before and after the etching is attributed to the thermal conductance of the SLG, Gg. We follow the convention to obtain κ = GgL/Wt, where we use the interlayer spacing in graphite as the SLG thickness, t = 0.335 nm. The results are similar for the three samples (Fig. 4). For G2, the room-temperature κ of 579 ± 34 W m−1 K−1 is about a factor of 3.4 lower than the highest reported basal-plane value of pyrolytic graphite (PG) (5). The appearance of the peak κ position at a much higher T ≈ 300 K in the supported SLG than T ≈ 140 K in the PG suggests that phonon scattering is dominated by substrate interaction and umklapp scattering at below and above 300 K, respectively.

Fig. 3

Measured thermal conductance of G2 before (solid downward triangles) and after (unfilled upward triangles) the SLG was etched, with the difference being the contribution from the SLG (circles).

Fig. 4

Measured thermal conductivity of G1, G2, and G3 together with the highest reported values of PG (5), the BTE calculation results of suspended SLG (black solid line) and supported SLG with KLATA = 0 and KZA = 0.73 N/m (blue solid line) or KLATA = KZA = 0.46 N/m (blue dashed line), the RTA calculation result (red dashed-dotted line) for supported SLG with KLATA = 0.8 N/m. Specular edges are assumed in the calculations. For supported SLG, s = d = 30 nm, and the calculation results are insensitive to edge specularity and can be reproduced with decreased d/s and increased KZA for the same KLATA/KZA.

Interestingly, Klemens (17) had envisioned the feasibility of our thermal measurements several years before SLG was first exfoliated onto SiO2 (1). He suggested that phonons leaking from supported SLG into the substrate would suppress the contribution of low-frequency phonons and reduce κ by 20 to 50%. In SLG, the longitudinal (LA) and in-plane transverse (TA) acoustic branches are linear, whereas the out-of-plane ZA branch shows a quadratic dependence of the frequency (ω) on the wave vector. The contribution to κ from the ZA branch would be negligible based on a relaxation time approximation (RTA) model (18) because of the small group velocity and large umklapp scattering rate (τu−1) calculated from an expression derived by Klemens and Pedraza (19).

In addition to a long wavelength approximation and the assumptions of a linear branch and high T, Klemens and Pedraza cautioned that the greatest uncertainty in their τu−1 expression stems from the inaccuracy of the assumed three-phonon scattering phase space that was not explicitly calculated (19). To address this problem, we carried out full quantum mechanical calculations of both normal and umklapp three-phonon scattering processes in SLG throughout the Brillouin zone (13). Through the calculations of the three-phonon matrix elements, we obtain a selection rule for three-phonon scattering, which requires that an even number of ZA phonons be involved in each process as a consequence of the reflection symmetry in flat 2D SLG (13). We note that this selection rule was not used in a recent calculation (20), whereas an analogous selection rule has been found for electron-ZA phonon scattering in SLG (21). This selection rule strongly restricts the phase space for umklapp scattering of ZA phonons in flat SLG. We also find that this selection rule applies in large-radius single-walled CNTs whose curvature is comparable to that of ripples that can form in SLG. We have incorporated this selection rule in an exact numerical solution of the linearized phonon Boltzmann transport equation (BTE) for SLG. Our BTE approach is similar to that used recently for nanotubes (22). We find that the ZA modes can contribute as much as 77% and 86% of the total calculated κ at 300 K and 100 K, respectively, for a 10-μm-long suspended SLG with specular edges and 1.1% C13 isotopic impurities (13). The calculated κ for the suspended SLG is about a factor of 5 and 1.5 higher than the measured κ of the supported SLG and the PG, respectively, at T ≈ 300 K (Fig. 4).

Scanning probe microscopy measurements have found that SLG exfoliated on SiO2 is partially conformal to the surface roughness (23) and partly suspended between hills on the surface (24). These experiments consistently obtained a substrate-induced correlation length in SLG of ~30 nm, which gives a measure of the average center-to-center separation (s) between adjacent hills in intimate contact with the SLG. Perturbation theory yields an approximate expression for the scattering rate due to phonon leakage back and forth across the contact patches as τsub,j1ρj(ω)Kj2/ω2, where ρj(ω) depends on the phonon density of states, and Kj is the average van der Waals (vdW) interatomic force constant between the SLG and the SiO2 support for polarization j = ZA, TA, or LA (13). A similar frequency dependence has been obtained for phonon transmission across a vdW interface between two half spaces (25). In our case, the expression accounts for parallel momentum not being conserved due to the amorphous structure of the SiO2 support. The obtained τsub,j1 increases with the diameter (d) of the contact patches and the d to s ratio (13).

Based on the interlayer vdW energy Γ0 ≈ 0.1 J/m2 in graphite and the measured average SLG-SiO2 separation h0 ≈ 0.42 nm reported in (23), we calculate KZA = 27SaΓ0/h02 ≈ 0.4 N/m, where Sa is the area occupied by one carbon atom in graphene (13). The interface force constant KLATA for the in-plane LA and TA modes is generally smaller than KZA (26), especially for an amorphous substrate. At the upper limit of KLATA = KZA = 0.46 N/m and d = s = 30 nm, the BTE model obtains a κ value close to the measured value at 300 K, and yields a large ZA contribution (13). However, the calculated κ increases with T at a steeper rate than the measurement result (Fig. 4). In comparison, at the lower limit of KLATA = 0, KZA = 0.73 N/m, and s = d = 30 nm, the BTE solution obtains a κ − T curve in good agreement with the measurement (Fig. 4). In this case, the ZA contribution to κ becomes smaller than the corresponding TA and LA contributions (13), which are reduced only by the suppression of the ZA + ZA → LA or TA mode conversions because of leakage of ZA phonons.

In 2D and at the low T limit, if the scattering rate τj −1 is proportional to ωα, the κ contribution is proportional to T2-α for j = TA, LA, or ZA polarization. For KLATA = KZA = 0.46 N/m, τj −1 of all the three polarizations is dominated by τsub,j −1, for which the α exponent is negative and results in an increased κ with T. In comparison, for KLATA = 0 and KZA = 0.73 N/m, boundary scattering with α = 0 and isotope and umklapp scatterings with positive α exponents play an increased role for the dominant in-plane polarizations at low and intermediate T, respectively, so that the calculated κ − T slope in this T range is relatively small, in agreement with the measured data. On the other hand, we have also fit the measured κ at 300 K by including substrate scattering of KLATA = 0.8 N/m in the RTA model (18) that neglects the ZA contribution (13). The as-calculated κ increases with T at a much more rapid rate than both the BTE results and the experimental data (Fig. 4), because of the negative α exponent of the dominant substrate scattering rate for the LA and TA phonons and the high zone-boundary frequencies of these two branches.

The theoretical analysis suggests that the ZA contribution to κ is large in suspended SLG and that the measured κ − T relation can be explained by much stronger substrate scattering of ZA phonons than LA and TA phonons. Although the strong ZA scattering can be caused by the expected behavior of KZA > KLATA, another possibility is that umklapp scattering of ZA phonons can be enhanced by the substrate interaction, which breaks the reflection symmetry. Indeed, our calculations show that the ZA contribution is reduced in suspended bilayer graphene because of interlayer interactions. Although future theoretical work is needed to clarify this issue, our experimental results clearly show that graphene exfoliated on SiO2 still conducts heat rather efficiently despite phonon-substrate interaction. However, the substrate effect could be quite different for few-layer graphene or SLG grown by thermal decomposition (27) or chemical vapor deposition (28, 29) on other substrates because of different interface interactions. These intriguing questions are expected to stimulate further experimental and theoretical investigations of phonon transport in suspended, supported, and embedded graphene.

Supporting Online Material

SOM Text

Figs. S1 to S11


References and Notes

  1. Materials and methods are available as supporting materials on Science Online.
  2. This work is supported in part by National Science Foundation awards CBET-0553649 and 0933454 (J.H.S. and L.S.), CBET 0651381 (D.A.B. and L.L.), 0651310 (N.M.), and CMMI-0926851 (Z.H.A. and R.H.); Office of Naval Research award N00014-08-1-1168 (A.L.M. and L.S.); Department of Energy Office of Science award DE-FG02-07ER46377(M.T.P. and L.S.); and The University of Texas at Austin (R.S.R.).
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