pubs.acs.org/NanoLett
Thickness-Dependent Thermal Conductivity of Encased Graphene and Ultrathin Graphite Wanyoung Jang,†,§ Zhen Chen,†,§ Wenzhong Bao,‡ Chun Ning Lau,‡ and Chris Dames*,† †
Department of Mechanical Engineering and ‡ Department of Physics and Astronomy, University of California, Riverside, California 92521 ABSTRACT The thermal conductivity of graphene and ultrathin graphite (thickness from 1 to ∼20 layers) encased within silicon dioxide was measured using a heat spreader method. The thermal conductivity increases with the number of graphene layers, approaching the in-plane thermal conductivity of bulk graphite for the thickest samples, while showing suppression below 160 W/m-K at room temperature for single-layer graphene. These results show the strong effect of the encasing oxide in disrupting the thermal conductivity of adjacent graphene layers, an effect that penetrates a characteristic distance of approximately 2.5 nm (∼7 layers) into the core layers at room temperature. KEYWORDS Graphene, graphite, thermal conductivity, heat spreader method, encased, experiment
A
mong graphene’s many remarkable properties,1 its expected very high thermal conductivity k has been suggested as a key advantage for applications in microelectronics and thermal management. Using a Raman method for both heating and temperature sensing,2 the thermal conductivity of suspended single-layer graphene (SLG) has recently been measured in the range ∼600 to ∼5000 W/m-K near and above room temperature,2-4 far higher than the k of copper (∼400 W/m-K). Although the variations in the reported optical absorbance2-4 remain to be clarified, these results suggest that suspended SLG has a k comparable to, if not exceeding, its very high-k carbonaceous cousins, including graphite (1950 W/m-K in-plane5), diamond (2310 W/m-K5), and suspended carbon nanotubes (up to ∼3400 W/m-K for micrometer-length tubes, according to modeling6 and experiments7-11). However, in most practical devices, graphene layers will be encased within dielectrics such as silicon dioxide, and it is essential to understand how this boundary interaction impacts thermal transport. Although traditional analysis of the in-plane k of thin films ignores the distinction between suspended and supported boundaries,12,13 recent measurements of SLG with one face supported on a substrate show that k is greatly reduced to ∼50 - 1020 W/m-K on copper (Raman method4), or ∼600 W/m-K on SiO2 (suspended platform method14). In contrast to suspended SLG, these values fall far below the in-plane k of bulk graphite,5 demonstrating the great importance of free versus supported boundaries. Here, we present the first measurements of graphene and ultrathin graphite (1 e NLayers e ∼20) in an encased config-
uration. The results show that the encasing SiO2 further reduces k well below that of supported graphene. Furthermore, these experiments capture the thickness-dependent transition from SLG to bulk graphitic behavior and imply that the surface-induced disruptions of k penetrate into the neighboring graphene layers by a characteristic distance of approximately 2.5 nm (7 layers) at room temperature. Figure 1a shows a schematic of the heat spreader method developed in this work to measure the in-plane k of encased graphene. A graphene flake is encased between SiO2 layers of thickness 320 nm (below) and 30 nm (above). Underneath this stack is a silicon wafer of high k acting as a heat sink. At the top of the stack, a metallic line heater dissipates Joule heat at a rate QH, which flows vertically through the stack into the Si heat sink, while simultaneously spreading laterally through the high-k graphene layer. The resulting temperature profile is measured by three resistance temperature sensors (T1, T2, T3) and fit using a numerical model described below to extract the in-plane k of the encased graphene. To fabricate these samples, graphene flakes are first mechanically exfoliated onto a thermally oxidized doped silicon wafer.15 A combination of optical and atomic force microscopy (AFM) is used to identify and determine the thickness of candidate as-deposited flakes.15-17 Then an oxygen plasma18 is used to trim irregular flakes into uniform rectangles, chosen from regions of constant thickness. The trimmed flakes are annealed to remove polymer residues (400 °C, 1 h, Ar 1.7 L/min, H2 1.9 L/min)19 and then the entire sample is blanketed by 30 nm of e-beam evaporated SiO2. Finally, electrodes (50 nm Au/5 nm Cr) are deposited by a standard e-beam lithography and lift-off process. We used two different patterns for the T sensors for the various samples in this work, 140 nm wide sensors spaced 350 nm center-to-center and 240 nm wide spaced 740 nm center-
* To whom correspondence should be addressed. E-mail:
[email protected]. § These authors contributed equally to this work. Received for review: 05/6/2010 Published on Web: 09/13/2010
© 2010 American Chemical Society
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FIGURE 1. (a) Schematic of the heat spreader method. Heat flows (red arrows) through the encased graphene and into the Si heat sink. (b) Top-view SEM image of one of the devices used in this work, including heater and three T sensors (white) and trimmed graphene flake (dark rectangle). An additional triangular flake can be seen in the upper-left corner. (c) Temperature profiles normalized to the heater power QH for validation experiments using oxide (blue) and a Pt film (red). k is extracted by fitting the experimental data (crosses) with the FEM model (circles), resulting in kox ) 1.43 W/m-K and kPt ) 25.4 W/m-K. Dashed lines are to guide the eye. Inset: Detail of a typical 3D FEM simulation.
to-center. A top view scanning electron microscope (SEM) image of one of the completed samples is shown in Figure 1b. We use a three-dimensional finite element method (3D FEM) combined with a least-squares method20,21 to fit the experimental T profile and extract k of the encased film. A related analytical method has been reported previously for micrometer-thick silicon films22 but in the present graphene study we find that full 3D FEM is essential. A detail of a typical FEM simulation is shown in the inset of Figure 1c. These simulations include known 3D geometries (graphene flake, heater, T sensors, oxide layers, and at least (50 µm)3 of the Si substrate), thermal conductivities (top and bottom oxide layers, Si substrate, and metal electrodes), and the thermal contact resistance between graphene and SiO2.23 For more details please see the Supporting Information. Each FEM-simulated sensor temperature takes into account the local T averaging caused by the finite sensor length and width. The results are only sensitive to the in-plane (rather than cross-plane) k of the encased flake. Also, because the FEM method is based on the continuum diffusion equation, which cannot handle ballistic effects, it is most appropriate for flakes with phonon mean free paths smaller than the center-to-center spacing of our T sensors. As described in the Supporting Information, using kinetic theory we estimated that the phonon mean free paths in the graphene samples measured in this work are typically in the range of 20-100 nm, considerably smaller than the electrode spacings and thus justifying the use of the diffusion equation. To validate this new heat spreader method, we used the two experiments shown in Figure 1c. First, a control experiment was performed without the graphene layer. As shown by the lower curve, this measured baseline T profile is described very well by an FEM fit that treats k of SiO2 as the only free parameter. (In all other experiments, the oxide thermal conductivities are also considered to be known and fixed). The fit value for kSiO2 at 310 K is 1.43 W/m-K, agreeing to better than 1% with our separate measurement of this SiO2 using a standard 3ω method.24 © 2010 American Chemical Society
In the second validation experiment (upper curve of Figure 1c), we prepared a sample with a 38 nm thick film of evaporated Pt in the stack rather than graphene. The Pt film is an effective heat spreader, leading to significantly higher Ts than the oxide control experiment for the same heating power. Using k of Pt as the only free parameter, the best-fit FEM temperature profile (T1, T2, T3) is in very good agreement with the experimental measurements and was obtained using a best-fit value of kPt ) 25.4 W/m-K. To confirm this value of kPt, we used the Wiedemann-Franz law to estimate kPt ) 26.6 W/m-K from the resistivity of a fourprobe Pt line prepared during the same evaporation run, thus validating this heat spreader and 3D FEM method to within 5%. Figure 2 shows k of encased graphene and ultrathin graphite as a function of thickness at three different temperatures. The 95% confidence intervals (CI) are evaluated using a Monte Carlo method described in ref 25 and our estimates of the random and systematic uncertainties (see also the Supporting Information). This analysis reveals that our experiments are most sensitive for thick flakes of high k. Conversely, we find that k of encased SLG is so low that this heat spreader method can meaningfully give only the upper bound of k. For example, as shown in Figure 2a, at 310 K our measurements show with 97.5% confidence that k of encased SLG is below 160 W/m-K. This is well below the room-temperature values of 580 W/m-K reported for SiO2-supported SLG14 and ∼1000 to ∼5000 W/m-K reported for suspended SLG.2-4 The dominant feature of Figure 2 is the trend that k of encased graphene increases with the number of layers, approaching bulk graphite for the thickest sample at room temperature (Figure 2a). We will return to the interpretation of this result shortly in the context of Figure 4. Here we briefly note that this trend is opposite of that reported recently for suspended graphene,26 which for SLG shows k well above that of bulk graphite and with k decreasing with the number of layers to approach the bulk graphite value. 3910
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omitted from Figure 3. Figure 3 shows that for a given thickness, the suppression of k as compared to bulk graphite is even stronger at lower T. Over the range from 60 K < T < 150 K, seven of the samples follow a power law temperature trend between T1.5 and T2. The specific heat of bulk graphite27 also follows an approximately T1.5 power law in this temperature range, suggesting that these samples are in a boundary-scattering regime where k simply tracks the specific heat capacity. At higher T, the data in Figure 3 transition to a much weaker power law. We expect that all of these samples have a peak in k(T) near or just above room temperature, indicating the onset of significant Umklapp phonon scattering, although because of the limited T range measured in Figure 3 the peak is only clearly evident for the 19-layer flake. Recent measurements of supported and suspended SLG are also consistent with the existence of a peak in k(T) around room temperature.4,14 We now turn to the interpretation of the strong thickness dependence of Figure 2. In contrast to the free boundaries of suspended graphene, supported exfoliated graphene is constrained by its contacts to the corrugated topography of the supporting surface. Using AFM, we found that the thermal SiO2 has an rms roughness of ∼0.25 nm with an in-plane correlation length of ∼20-30 nm, consistent with previous reports.19,28 Furthermore, the presence of the evaporated oxide on top of the graphene is also known to cause defects in the adjacent graphene.29,30 The resulting interactions between the outermost graphene layers and their neighboring oxide layers are expected to reduce k of graphene through two mechanisms,14,31 phonon leakage into the low-sound-velocity oxide and additional phonon scattering by the inhomogeneous graphene-oxide interface. (However, note that like refs 2-4, 14, and 26 our experiments cannot distinguish between these two mechanisms of k suppression, nor can we distinguish between the relative contributions of in-plane (LA, TA) vs out-of-plane (ZA) modes.14,32) To confirm that the upper oxide layer causes additional suppression in graphene’s k beyond that induced by the bottom oxide alone, we prepared two samples designed to distinguish between the effective thermal conductivity with and without the upper oxide, thus giving k for both supported and encased configurations (the latter of which is also included in Figures 2 and 3). For details please see the Supporting Information. As shown in Figure 4a, at 310 K the k of three- and four-layer flakes was reduced by 64 and 38%, respectively, by deposition of the upper oxide. Because each supported and encased measurement was made on the same sample, most of the important sources of uncertainty are common to both configurations; thus the differences in k between supported and encased configurations are determined with much better accuracy than the accuracy in k itself. Thus, Figure 4a clearly confirms that the additional interactions with the second oxide layer indeed further
FIGURE 2. k vs thickness at (a) 310, (b) 164, and (c) 92 K. Because multiple samples were measured with NLayers ) 3, their thickness coordinates have been shifted slightly for clarity. The two-layer flake was only measured at 310 K. Error bars indicate 95% confidence intervals, and k of the encased SLG flake is so low that only the upper bound is significant. For comparison, literature values are also shown for bulk graphite (dashed lines5), suspended SLG (open square,2 open diamond3), and SiO2-supported SLG (open circles14).
FIGURE 3. Temperature dependence of k for encased graphene and ultrathin graphite. TPRC: ref 5.
Figure 3 shows k as a function of temperature for all of the encased samples measured in this work. Because the error bars are already given in Figure 2, for clarity they are © 2010 American Chemical Society
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FIGURE 4. (a) k of three- and four-layer flakes before (supported; open symbols) and after (encased; filled symbols) top oxide deposition, showing reductions by 64 and 38%, respectively, at room temperature. (b) Fit of eq 2 (solid line) to experimental data (points) at 310 K. TPRC: ref 5. (c) Best-fit values of k0 and δ as functions of temperature. kBulk(T) is graphite.5 (d) Dimensionless comparison of eq 2 with the thickness-dependent measurements from four different Ts, using the dimensionless conductivity (k - k0)/(kBulk - k0).
reduce the effective k of graphene, especially at higher temperatures. Previous measurements of supported graphene have focused on SLG.4,14 For multilayer flakes, the oxide-induced disruptions of the outermost graphene layers are expected to penetrate a finite distance into the core of the flake due to the weak van der Waals coupling between adjacent graphene layers. Thus, for thicker flakes, the innermost core layers will be less affected by the surface perturbations, and for sufficiently thick flakes k must recover to that of bulk graphite, consistent with the dominant trends of Figure 2. Therefore a key physical question is to identify the characteristic length δ by which these surface disruptions extend into the bulk of the flake. To address this question, we introduce the following phenomenological model. We assume that the in-plane k of an ultrathin graphite flake varies continuously across the flake thickness according to an unknown function kˆ(z,t), where kˆ is the local thermal conductivity, and z is measured from the midplane of the flake which has a total thickness t. This assumption of a local thermal conductivity function kˆ(z,t) is supported in part by the fact that the in-plane thermal conductivity of 3D graphite can be largely understood through analysis of a 2D phonon gas.33 Because the interlayer van der Waals coupling is much weaker than the inplane bonding, the most important effect of the adjacent © 2010 American Chemical Society
layers is in scattering the in-plane 2D phonons rather than major alterations of the phonon dispersion.33-35 Note that our experiments yield only the effective conductivity of the entire flake, equivalent to averaging, k(t) ) t/2 ˆ k(z,t)dz. Referring to Figure 2, at any given temper(1/t)∫-t/2 ature we expect k(t) to have the following three features: (i) For very thick flakes k should recover to the bulk graphite value, that is, k(tf∞) ) kBulk; (ii) for sufficiently thin flakes it appears that k tends to an approximately constant value k0, that is, k(tf0) ) k0; and (iii) there is some characteristic thickness at which the k(t) function transitions between the k0 regime and the kBulk regime. Consistent with these criteria, here we suggest a semiempirical form for the local thermal conductivity function
[ ] (δz ) t cosh( ) 2δ
1 - cosh
kˆ(z, t) ) k0 + (kBulk - k0)
(1)
where δ characterizes the distance that the oxide-induced disruptions of the outermost surface layers penetrate into the core of the flake. Note that the symmetry of eq 1 assumes that the upper and lower surfaces of the flake experience similar constraining effects by their respective 3912
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REFERENCES AND NOTES
adjacent oxides, which is not obvious considering that one interface arises from mechanical exfoliation and the other from evaporation, but nevertheless might be a good approximation if the dominant effect of the oxide is in quenching the out-of-plane ZA modes, as suggested in ref 14. For flakes much thicker than δ, eq 1 is simply a function that interpolates between the constrained kˆ ) k0 at the flake surfaces and kˆ ) kBulk deep within the core of the flake, over the exponential decay length δ. Averaging eq 1 over -t/2 e z e t/2 to compare with experiments gives
(1) (2) (3) (4) (5) (6) (7) (8)
k(t) ) k0 + (kBulk
t 2δ - k0) 1 tanh t 2δ
[
( )]
(2)
(9) (10) (11) (12) (13)
To apply this model to our data, k0 and δ are treated as T-dependent fitting parameters, while kBulk(T) is taken from ref 5. In Figure 4b, we use a χ2 minimization to fit this model to the thickness-dependent k from Figure 2a, using kBulk ) 1950 W/m-K5 and obtaining best-fit values of k0 )175 W/m-K and δ ) 2.5 nm (∼7 layers). The temperature dependence of these fit parameters is given in Figure 4c, which shows that the oxide-induced surface distortions penetrate much deeper into the core at low temperature, an observation that also explains why in Figure 4a the difference between supported and encased measurements becomes less prominent at low T. Finally, in Figure 4d we collapse the measurements from these four Ts and all thicknesses on to a single dimensionless plot of (k - k0)/(kBulk - k0) versus t/δ, which suggests that eq 2 is a reasonable description of this entire data set. In conclusion, we have developed and validated a heat spreader method to measure k of graphene and ultrathin graphite encased within SiO2. These results highlight the importance of layer thickness and interfacial coupling for in-plane heat transfer. In particular, to maximize k by ultrathin graphite encased in SiO2 at room temperature, layer thicknesses of at least 10 nm should be targeted to ensure k > 1000 W/m-K.
(14) (15) (16) (17) (18) (19) (20) (21) (22) (23) (24) (25) (26) (27) (28)
Acknowledgment. W.J. (experiments and microfabrication) and Z.C. (FEM, analysis, and some experiments) contributed equally to this work. This work was supported in part by the NSF (CBET/0756359 and 0854554). W.B. and C.N.L. acknowledge support by FENA and ONR N00014-091-0724.
(29) (30) (31) (32)
Supporting Information Available. Uncertainty analysis, fabrication of supported samples, and mean free path estimates. This material is available free of charge via the Internet at http://pubs.acs.org.
© 2010 American Chemical Society
(33) (34) (35)
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