Letter pubs.acs.org/NanoLett

Bimodal Phonon Scattering in Graphene Grain Boundaries Poya Yasaei,† Arman Fathizadeh,‡ Reza Hantehzadeh,† Arnab K. Majee,§ Ahmed El-Ghandour,∥ David Estrada,⊥ Craig Foster,∥ Zlatan Aksamija,*,§ Fatemeh Khalili-Araghi,*,‡ and Amin Salehi-Khojin*,† †

Department of Mechanical and Industrial Engineering and ‡Department of Physics, University of Illinois at Chicago, Chicago, Illinois 60607, United States § Electrical and Computer Engineering Department, University of Massachusetts Amherst, Amherst, Massachusetts 01003, United States ∥ Department of Civil and Material Engineering, University of Illinois at Chicago, Chicago, Illinois 60607, United States ⊥ Department of Materials Science and Engineering, Boise State University, Boise, Idaho 83725, United States S Supporting Information *

ABSTRACT: Graphene has served as the model 2D system for over a decade, and the effects of grain boundaries (GBs) on its electrical and mechanical properties are very well investigated. However, no direct measurement of the correlation between thermal transport and graphene GBs has been reported. Here, we report a simultaneous comparison of thermal transport in supported single crystalline graphene to thermal transport across an individual graphene GB. Our experiments show that thermal conductance (per unit area) through an isolated GB can be up to an order of magnitude lower than the theoretically anticipated values. Our measurements are supported by Boltzmann transport modeling which uncovers a new bimodal phonon scattering phenomenon initiated by the GB structure. In this novel scattering mechanism, boundary roughness scattering dominates the phonon transport in low-mismatch GBs, while for higher mismatch angles there is an additional resistance caused by the formation of a disordered region at the GB. Nonequilibrium molecular dynamics simulations verify that the amount of disorder in the GB region is the determining factor in impeding thermal transport across GBs. KEYWORDS: Graphene, grain boundaries, thermal transport, electrical thermometry, Boltzmann transport modeling, molecular dynamics

T

hermal management1−11 of power dissipation in nanoelectronic devices5,8 is among the recently emerged potential commercial applications of graphene. In order to realize this potential, extensive research efforts have focused on the high-volume manufacturing of graphene. One of the most practical approaches for large-scale production of graphene is through chemical vapor deposition (CVD) and roll-to-roll processing techniques.12,13 In this approach, the outcome is primarily a polycrystalline film, where single crystalline domains are joined by defective grain boundaries (GBs).14 GBs have been shown to largely alter the physical properties of CVDgrown graphene films.9,14−22 So far, a wide range of research studies have been dedicated on how the GBs affect the electronic properties of graphene.14−20,22−25 In this regard, strong carrier scattering and suppressed electrical conductivity are reported at GBs as a result of modified density of states and formation of potential barriers in the vicinity of the GBs.17,18 © XXXX American Chemical Society

However, efforts on understanding the role of GBs in the thermal transport across the graphene lattice have been limited to theoretical works9,26−29 and a few indirect experimental measurements.30,31 This is mainly because the direct experimental thermal transport study of GBs requires a measurement platform capable of separating the contribution of the GB from the graphene lattice itself. In this paper, we report the temperature-dependent thermal transport of supported CVD-grown graphene for a single crystalline flake as well as two merged flakes having an individual GB. We employ a symmetric electrical thermometry platform fabricated on an ultrathin freely suspended silicon Received: March 20, 2015 Revised: May 25, 2015

A

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Figure 1. (a) Scanning electron microscopy (SEM) image of two merged hexagonally shaped single crystalline graphene grains forming an individual grain boundary (GB). The scale bar is 5 μm. (b) Raman spectra obtained from two points on graphene single crystalline grain and the vicinity of the GB (shown with red and blue circles in panel a, respectively). The spectra are normalized with the G-peak intensity at 1580 cm−1. (c) Low magnification optical microscopy image of a fabricated thermometry platform on a 100 nm thick silicon nitride (SiN) membrane. The scale bar is 100 μm. The inset shows the fabricated chip-on-a-chip carrier with wire bonded electrical connections. (d) High-magnification SEM image of a fabricated thermometry platform, comprised of a heater electrode in the center and two sensor electrodes on the sides having perfect symmetry. The scale bar is 5 μm. The inset shows the SEM image of the merged grains before oxygen plasma patterning and device fabrication. (e) Schematic side view of the thermometry platform. (f) A simple thermal resistance circuit that shows the paths of heat dissipation. The red star shows the heater position, and the green and blue triangles show the position of the grain sensor and GB region sensor, respectively. (g) Vector plot of the thermal flux in a half-model finite element (FE) simulation of the thermometry device, showing asymmetric heat dissipation through grain and GB regions. (h) Temperature distribution in the FE model. (i) Tilted view of the FE model, showing an isotherm contour to demonstrate the asymmetry of the temperature distribution.

process,15,32,33 the crystalline misorientation angle between the merged grains can be estimated by comparing the edge orientations of the flakes15 (Figure 1a). Formation of a GB in the merging region of the flakes can be verified by Raman spectroscopy.15,21 We have acquired multiple Raman point spectra from different locations of the samples and show two representative spectra in Figure 1b. As indicated by circles in Figure 1a, the blue spectrum is obtained in the merging region of the flakes, while the red spectrum is obtained at central region of a hexagon. Clearly, the intensity of the defect-induced D-peak of the Raman signal (at 1350 cm−1) is higher in the blue spectrum, confirming the existence of a defective GB structure.34 Moreover, the I2D/IG intensity ratio of >2 corresponds to monolayer graphene with a well-known thickness of 0.335 nm.15,21,34 This is also consistent with our previous transmission electron microscopy (TEM) analysis.21 To determine the role of GBs on thermal transport in graphene structures, we designed a symmetric electrical thermometry platform that allows for simultaneous examination of the thermal conductance through a single crystalline graphene flake (graphene region) and a bicrystalline region, having an individual GB perpendicular to the direction of heat flow (GB region). To enhance the measurement sensitivity for in-plane thermal conduction, the platform is fabricated on a 100 nm thick suspended SiN membrane (500 × 500 μm2), which

nitride (SiN) membrane to extract the role of individual GBs having different crystalline mismatch angles between the grains. Our results demonstrate an order of magnitude variation in the heat conductance per unit area through the GBs, depending on their crystallographic mismatch angle. For instance, a fewnanometer-wide GB with a 21° mismatch angle adds a thermal resistance to the graphene lattice that is equivalent to a 3.25 μm length extension of the single crystalline region. In contrast, the equivalent length for a GB with a 3° mismatch angle is only 380 nm. Our phonon Boltzmann transport equation (pBTE) modeling together with molecular dynamics (MD) simulations uncover that the significant drop in the thermal conductance at high mismatch angle GBs can only be a result of multiple phonon boundary-scattering events, caused by formation of a disordered (amorphous) structure in the merging region of two adjacent flakes.17 The graphene used in this study is synthesized through atmospheric pressure CVD method, similar to previous reports.15,21 By precise control over the growth time, hexagonally shaped graphene domains grown from random nucleation points are merged together forming the GBs with high quality of stitching in the coalescing region of the grains (see section S1 in Supporting Information). The GBs are distinguished by the geometry of the as grown hexagonal flakes. Due to a predominant zigzag edge preference in the growth B

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Figure 2. (a) Temperature-dependent extracted thermal conductivity of the single crystalline graphene and the GB region with different mismatch angles (3, 8, and 21 degrees). The error bars represent the overall uncertainty of the measurements, as discussed in section S6 of the Supporting Information. (b) Thermal resistance of the GB regions and the single crystalline graphene grains. (c) The thermal conductance per unit area (G/A) of the different GBs. (d) The additional thermal resistance caused by an individual GB is shown as an equivalent extra length of the single grain with similar width.

Supporting Information). The thermal resistances associated with the chip carrier and its contacts with the coldfinger in one side and with the sample in the other side causes a drift in the actual temperature of the chip below 100 K, which is carefully corrected by using a reference cryogenic diode sensor (see section S2 in Supporting Information). We performed heat flow measurements at a temperature range of 44 to 305 K in an ultrahigh vacuum (∼10−7 mbar) on three devices, having GBs with mismatch angles of 3, 8, and 21°. Initially, the electrical resistances of all electrodes are calibrated in a four-probe measurement configuration over the entire temperature range (see section S3 in Supporting Information). This calibration enables us to accurately measure the temperature in the subsequent experiments by monitoring the changes in the resistance of the electrodes. Next, the ambient temperature is held constant and a range of electrical powers are dissipated into the heater. The temperature rise of the sensors are used to measure the temperature gradients as functions of the input thermal power (see section S3 in Supporting Information). The simplified equivalent thermal circuit of the device is shown in Figure 1f, demonstrating two parasitic heat dissipation paths other than the graphene films. After carrying out the thermometry measurements, the graphene flakes are etched in low-power oxygen plasma, and control experiments are repeated in identical conditions in order to determine the contribution of the underlying SiN membrane and the metal electrodes. To accurately extract the thermal properties of the graphene single crystals and the GB regions, we performed 3D finite element (FE) simulations using the actual geometrical dimensions of the fabricated device (see section S4 in Supporting Information). Initially, the data collected from the

practically confines the heat into a two-dimensional plane. Figure 1c shows a low-magnification optical image of the fabricated device under illumination from the top (reflection) and the bottom (transmission) of the membrane in order to produce maximum contrast. The targeted area for measurement is in the center of the window (within a 100 × 100 μm2 square) to maintain the symmetry of the heat dissipation through the electrodes and the underlying substrate. Figure 1d magnifies the central part of the device by scanning electron microscopy (SEM), showing three thermometry metal electrodes and the graphene flakes, purposely colored for clarity. The central (red) electrode serves as the heater, while the right and left (yellow) electrodes serve as thermometers, operating in very low power with negligible Joule heating. The inset of Figure 1d shows the as-grown flakes before oxygen plasma etching and electrode deposition. As indicated by dashed lines in Figure 1d, the electrodes are patterned in such a way that the role of an individual GB can be examined through a comparison between the left and right thermometers. The width of the graphene flakes are 11.5 μm and the distance between the electrodes are 5 μm. Figure 1e schematically shows the cross section (side view) of the platform. The electrodes consist of 5/5/40 nm of SiO2/Cr/Au which are 11.5 μm long and 500 nm wide (see Methods section). The SiO2 layer between the electrodes and graphene is essential for electrical insulation of the electrodes which are supposed to simultaneously operate without any interference. To carry out temperature-dependent thermal transport experiments, the devices are mounted on chip carriers using copper tape and thermally conductive grease (inset of Figure 1c). The devices are subsequently loaded in a cryostat system with a precise temperature controller (see section S2 in C

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80% of the Ballistic limit (∼4.2 GW·m−2·K−1), depending on the phonon transparency of the GB structure.9 While the G/A for the GB with 3° mismatch angle lies within the predicted range (∼53%), the results of the GBs with 8° and 21° mismatch are strikingly lower, in range of 24% and 6% of the ballistic limit, respectively. Figure 2d shows the equivalent length of single crystalline graphene per unit width which causes the same thermal resistance as of the GBs. This representation reveals that a fewnanometer-wide GB with 21° mismatch angle imposes a huge thermal resistance, equivalent to 3.25 μm of single crystal CVD graphene, while this equivalent length is only 380 nm for a GB with 3° mismatch angle. In order to explain the order of magnitude difference between our experiments and previous predictions,9 we turned to extensive theoretical modeling to elucidate the different phonon modes and their scattering mechanisms involved in thermal transport across the graphene GBs. To calculate the inplane thermal conductance across graphene GBs, we use the solution to phonon Boltzmann transport equation (BTE) in the presence of GBs40

control experiments were used to find the temperaturedependent thermal conductivity of the SiN membrane. In more details, thermal conductivity values of the metal electrodes (using Wiedemann−Franz law)10,35 and the SiO2 layer36,37 are introduced into the steady-state thermal analysis of FE model as known parameters, and the thermal conductivity of SiN membrane was found to match the experimental data (see section S5 in Supporting Information). Next, the average thermal conductivities of the grain and the GB regions are fitted into the model in order to obtain the best match between the FE simulations and the experimental temperature gradients on the thermometers. In our simulations, a thermal interface resistance of 4 × 10−8 m2·K·W−1 is assumed between SiO2-graphene and electrode-SiO2, which is in the range of typical values according to previous experimental reports.4,38,39 It is worth noting that owing to the large contact areas in the designed platform, the extracted quantities are just slightly sensitive to the interface resistances. Modulating contact resistances in the range of 5 × 10−9 to 7.5 × 10−8 m2·K·W−1 (±87%) causes less than 3% change in the extracted thermal conductivities at 305 K (see section S6 in Supporting Information). This range is adequately wide to account for the temperature dependence of the contact resistances. The results of the FE simulations are plotted in Figure 1g−i. Figure 1g shows the vector plot of the thermal flux in a half-size FE model, clearly showing that more heat is being dissipated through the graphene region (right) compared with the GB region (left). The steady-state temperature distribution in the FE model is shown in Figure 1i−h. An isotherm contour (red dashed line) is plotted in the tilted view of the model (Figure 1i) to demonstrate the asymmetry of the temperature distribution caused by existence of a single GB. Figure 2a shows the extracted average thermal conductivity values of the graphene and GB regions for the three tested devices having misorientation angles of 3, 8, and 21°. The thermal conductivity of single crystalline graphene at room temperature is extracted to be 836 ± 126 W·m−1·K−1. The error bars represent the overall uncertainties of the measurements and are fully discussed in section S6 in Supporting Information. This figure also reveals that a single GB can significantly decrease the thermal conductivity of the graphene structures, and its effect becomes more evident as the misorientation angle of the merged grains increases. To quantify the excessive thermal resistance caused by the GBs, we calculated the thermal resistance (R) of the measured films as a function of temperature in Figure 2b using a simple equation R = L/(KWt) where L, W, and t are the length (5 μm), width (11.5 μm), and thickness (0.335 nm) of the graphene channel, and K is the average thermal conductivity values from Figure 2a. The resistance of the GB regions (RGB‑Region) can be thought as a superimposition of two components: (i) the isolated GB (RGB) and (ii) its adjacent graphene lattice (RG). The contribution of the graphene lattice is directly found in the grain region of the device; hence, the contribution of the GB is obtained by subtracting the overall resistance of GB region from that of graphene region (RGB = RGB‑Region − RG). As a geometry-independent quantity, Figure 2c shows the thermal conductance per unit area (G/A) for the isolated GBs having different mismatch angles (G = 1/RGB). At room temperature, the measured G/A are 2.25, 1.02, and 0.26 GW·m−2·K−1, for the GBs with 3°, 8°, and 21° mismatch angles, respectively. A previous theoretical calculation predicted that thermal conductance through the GBs at room temperature varies 50% to

κ (T ) =

ℏ Sδ

∑ v b2(k ⃗)τtot(k ⃗)ω b(k ⃗) k ⃗,b

∂N0(ω , T ) ∂T

(1)

where S is the surface area, δ = 0.335 nm is the graphene thickness, ωb is the vibrational frequency, and vb is the group velocity of phonon branch b, computed from the full phonon dispersion relationship. In general, thermal transport in supported graphene is characterized by the complex interplay between boundary roughness scattering and various internal scattering mechanisms (substrate, phonon−phonon, impurity, isotope scattering)41 (see section S7 in Supporting Information). It has been found that, in wide supported graphene nanoribbons (GNRs), substrate scattering plays a dominant role over edge roughness scattering, while in narrow GNRs, edge roughness scattering plays a significant role. In the case of graphene GBs, it is crucial to determine the contribution of the phonon scattering at the GB compared to other scattering mechanisms.40 It is also necessary to determine how different characteristics of the GBs affect the thermal transport. Direct imaging of graphene GBs has shown the atomic structure of the GBs can be comprised of a highly disordered and corrugated region,14,17−19 which is in contrast to well-ordered and symmetric GB structures more commonly used in simulations. For low angle of grain mismatch, this disordered region is narrow and can be treated as a thin GB (single ring wide) having a root-mean-square (rms) value of edge roughness Δ, whose value depends on the mismatch angle. When phonons encounter such GBs, they may scatter with the roughness, and the probability of this interaction depends on the direction of phonon propagation and its wavelength. Small wavelength phonons interact with the GB more strongly and are scattered randomly, while large wavelengths can pass through the boundary unaffected. This interaction is captured by the momentum-dependent specularity parameter, representing the fraction of incident phonons with wavevector q⃗ that will pass through the boundary unaffected, and is given by p(q)⃗ = exp(−4Δ2q2 cos2θ). As Δ increases, the specularity p(q⃗) rapidly tends to zero, indicating that all incident phonons will scatter diffusely once at the GB. However, GB scattering is only one of several scattering mechanisms in supported graphene. Most notably, substrate D

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Figure 3. (a) Thermal conductivity vs temperature calculated from the phonon Boltzmann transport model. The symbols in (a) and (c) represent experimental data from Figure 2a and c, while solid curves represent simulation results. Panel b shows the thermal conductivity at room temperature vs the grain boundary roughness Δ and width of the disordered boundary region WD. Solid curve shows total thermal conductivity plotted against both WD and Δ, while dashed line shows thermal conductivity vs Δ with WD = 0 and dotted line shows thermal conductivity vs WD keeping Δ = 0. Symbols represent experimental data at 300 K. The agreement with the experimental data is achieved by including a narrow strip of disordered graphene, whose conductivity is calculated from Cahill’s minimum thermal conductivity model, as explained in the text, and plotted in the inset of (b). (c) Thermal conductance of the grain boundary vs temperature. The agreement in (a) and (c) is achieved using the following values for grain boundary LER (Δ) and width of the disordered region at the GB (WD): (1) without grain boundary: Δ = 0 nm, WD = 0 nm; (2) 3° mismatch: Δ = 0.12 nm, WD = 0.12 nm; (3) 8° mismatch: Δ = 1.3 nm, WD = 1.3 nm; (4) 21° mismatch: Δ = 7.5 nm, WD = 7.5 nm. (d) Solid line shows thermal conductivity vs temperature for 3° grain mismatch angle, while dash, dash−dot, and dotted lines show its branch-wise components (ZA, TA, LA), respectively, with ZA carrying most heat at low temperatures, and in-plane modes (TA and LA) dominating at room temperature and above.

scattering adds considerably to scattering of phonons in supported graphene.10 The additional scattering from the thin GB can only reduce the total thermal conductivity from ∼850 W·m−1·K−1 to ∼750 W·m−1·K−1 due to the strong competing influence of the substrate. At higher angles of mismatch, atomic resolution images obtained from graphene GBs by scanning tunneling microscope (STM)17 and transmission electron microscopy (TEM)14 show that a disordered boundary region can form, which can be wide enough to add a considerable amount of additional resistance to the heat flow. The added thermal resistance due to the disordered graphene region of width WD in the GBs is calculated from Cahill’s minimum thermal conductivity model42 by setting the scattering rate to the maximum value reached when the phonon lifetime equals one-half of its period of vibration (τ−1 max = ω/π). This model provides what is believed to be a lower bound on the thermal conductivity of the disordered region; hence the WD calculated from the minimum thermal conductivity model can be thought of as a lower bound on the width of the disordered region at the GB. Figure 3a shows the variation of thermal conductivity with temperature for various grain mismatch angles, including the one with no GB (single crystal). The total thermal conductivity in Figure 3a has been calculated as K GB‑ R e g i o n = WG/(RGB‑RegionA), where RGB‑Region is the total resistance due to graphene with GB and amorphous patch and is given by RGB‑Region = [(WG − WD)/(AKG)] + [WD/(AKmin)]. Here, WG

denotes the width of the graphene between heater and sensor (5 μm), WD is the width of the disordered region at the GB, and KG denotes thermal conductivity of graphene without any GBs. Kmin is the thermal conductivity of the disordered region from Cahill’s minimum thermal conductivity model. Figure 3c shows the effective G/A through a GB as a function of temperature. The G is calculated as the reciprocal of the resistance arising due to GB alone (RGB) as (G = 1/RGB), where RGB = [WG/(AKGB‑Region)] − [WG/(AKG)] and KGB‑Region denotes total thermal conductivity in the GB region (as calculated above). It can be seen that the boundary conductance (G/A) results obtained from phonon BTE are in excellent agreement with the experimental data. The effect of the GB on thermal conductivity due to different rms roughness (Δ) and the effect of disordered region due to its varying width (WD) are complementary to each other because the effect of roughness saturates after Δ = 0.25 nm. Figure 3b shows that, for low values of Δ and WD, the total thermal conductivity (solid curve) follows the thermal conductivity curve due to GB scattering alone (dashed curve) and, for higher values of Δ and WD, follows the thermal conductivity trend due to the width of disordered strip alone (dotted curve). Hence, for small values of Δ and WD (up to 0.25 nm), thermal conductivity is largely dominated by the GB scattering alone with negligible effect from the narrow disordered strip, in agreement with the experimental data for low mismatch angle. For higher values of Δ and WD (beyond E

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Figure 4. (a) Size dependency of the Kaptiza conductance (G/A) for two types of grain boundaries. The red line shows the conductance of a perfect 5−7 ring GB with a roughness of Δ = 0.1 nm, and the blue line shows the conductance of a GB with a Δ = 0.15 nm between two graphene sheets with a mismatch angle of 21.5°. Snapshots of the two GBs after stabilization in the MD simulations are shown in b and c, respectively. (d) Temperature gradient along the graphene sheet containing perfect grain boundaries (red) and two disordered grain boundary regions with widths of 0.8 nm (blue) and 1.6 nm (green). Structure of these two grain boundaries are shown in e and f, respectively. (g) Schematic representation of the simulation procedure for calculating the effect of SiN substrate on thermal transport of graphene. The total heat that is transferred through graphene/substrate system can be divided into the heat passing through the substrate (Qsub) and the heat passing through the graphene (Q′G). A comparison of Q′G with the transferred heat from a system of suspended graphene with the same GBs and the same size QG provides an estimate of how much heat transfer is affected by the substrate.

corresponding temperature gradients for the disordered GB regions (Figure 4e−f) in comparison with the one for the perfect GB shown in Figure 4b. As clearly demonstrated, the temperature drops at the disordered GB regions are dramatically larger than the perfect linear GB. The thermal conductance calculations for the regions of 0.8 and 1.6 nm at the steady state lead to thermal conductance of 12.2 GW·m−2· K−1 and 8.8 GW·m−2·K−1, respectively, which are 55% and 37% of the Kapitza conductance of a perfect GB (red curve, Figure 4a). Next, we perform similar NEMD simulations for a graphene with two GBs on a 2-nm-thick amorphous SiN to include the effect of substrate on thermal transport in graphene. Figure 4f schematically shows the simulated system and method of extraction of the substrate effect. Considering computational limitations, we have used a periodic simulation box with a length of 42 nm. Applying a temperature gradient on the system by thermostatting the central part to 310 K and the left side to 290 K, the total rate of heat transfer through the combined substrate/graphene system is extracted. In the steady state, this value is Q = 6.1 × 10−8 W. Moreover, we have separately simulated the same substrate structure as well as the graphene with GBs independently and obtained their corresponding heat transfer rates to be Qsub = 4.4 × 10−8 W and QG = 8.9 × 10−8 W, respectively. Assuming that the thermal conductivity of substrate is not being affected by the graphene, one can obtain Q′G = 1.7 × 10−8 W to be the rate of heat transfer of graphene on the substrate.6 Comparing this value with the rate of heat transfer in the suspended graphene with the same size leads to QG/Q′G = 5.2, which shows that the thermal transport of the graphene is 5.2 times smaller on the amorphous SiN substrate compared with suspended graphene. Considering this scaling factor, the calculated value for the thermal conductance from MD simulations is within the range of experimental results measured for the mismatch angle of 21°. In summary, we use a symmetrical electrical thermometry technique to directly extract the temperature-dependent thermal resistance of graphene GBs with different crystalline mismatch angles. We observed that the thermal resistance at

0.25 nm), GB scattering becomes completely diffuse (p = 0), and the effect of boundary roughness saturates. There is a natural crossover occurring at 1 nm, beyond which the thermal conductivity is governed dominantly by the additional resistance of the disordered region. This is in good agreement with the experimental data for large angle mismatch. Using the explained methodology, the experimental data for thermal conductance of the GB is well-reproduced for the entire temperature range (Figure 3c). The individual contributions to thermal conductivity from each of the branches are shown in Figure 3d, with the out-of-plane acoustic (ZA) branch contributing at low temperatures, and in-plane (LA and TA) dominating above 150 K. We use MD simulations to verify our BTE results and to investigate the dependence of thermal conductance to the width of the disordered strip (WD), as well as the GB roughness (Δ). MD simulations have been widely used to study thermal transport in pristine graphene43,44 as well as polycrystalline graphene consisting of GBs.27,45 We use nonequilibrium molecular dynamics (NEMD), in which a temperature gradient is applied on a graphene sheet across the GB and the flowing heat is monitored, in order to obtain the GB Kapitza conductance (G/A) 29 (see section S8 in Supporting Information). We use suspended graphene sheets with different lengths 42−656 nm, having two GBs with 21.4° crystallographic mismatch angle. Figure 4a shows the length dependence of the thermal conductance across the GBs shown in Figure 4b,c. The two GBs in each structure have an rms roughness (Δ) of 0.1 and 0.15 nm. When the system size is comparable to the mean free path of phonons (∼700 nm46), we observe that the GB consisting of a linear arrangement of 5 and 7 carbon rings with a roughness of 0.1 nm (Figure 4b) has a larger Kapitza conductance in comparison to the one with a higher roughness (Δ = 0.15 nm; Figure 4c) and also reaches its plateau in a longer length scale (blue line in Figure 4a). To explore the dependence of thermal conductance on the width of the disordered strip (WD), we broaden the disordered region to 0.8 and 1.6 nm and performed similar NEMD simulations with the length of 656 nm. Figure 4d shows the F

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model using Solid70 and Shell57 elements, with same geometry as in the fabricated devices. Owing to symmetrical design, the model represents half of the actual body to improve computational efficiency. The layers in the 3D model are illustrated in section S4 in the Supporting Information. Heat generation as given from the experiments is used as input, and the nodal temperatures were recorded as the primary output. The optimization parameter is thermal conductivity of SiN (in control experiment) and the average thermal conductivities of graphene region and grain region (in the main experiment). Boltzmann Transport Equation (BTE) Modeling. The model used for this study is based on solving the phonon Boltzmann transport equation (pBTE). In the steady-state regime, the pBTE is

highly misoriented GBs can be remarkably higher than previous theoretical predictions owing to the larger disorder found in the atomic structure of GBs in CVD grown graphene. BTE calculations indicate that a bimodal scattering mechanism governs the phonon transport through the GB: for small mismatch angles, the thermal resistance of the GB can be captured through phonon scattering from the GB roughness, while for higher mismatch angles, the GB roughness effect is saturated. The lower thermal conductivity at higher mismatch angles can be explained through the presence of a narrow strip of disordered graphene at the GB. For the highest mismatch angle of 21°, we calculate the width of the disordered region to be 7.5 nm based on the minimum thermal conductivity model. For intermediately misoriented GBs, thermal conductivity is affected by a complex interplay between the amount edge roughness and amount of disorder in the disordered patch. Further study is required to probe the structure and morphology of the GB region. Our results highlight the important structure−property processing correlations of 1D defects on thermal transport in 2D crystals, emphasizing the importance of engineering such correlations in emerging 2D materials and devices. Methods. Graphene Growth. Partial coverage graphene films with distinguishable GBs are synthesized by ambient pressure CVD process on copper foils (from Alfa Aesar, product no. 46365). At first, copper foils are initially cleaned by 10 min soaking in 10% hydrochloric acid, rinsed in acetone and IPA, dried under nitrogen flow, and loaded into the chamber. The CVD chamber is then evacuated (down to 1 mTorr) and then purged with forming gas (5% hydrogen diluted in Argon) up to the atmospheric pressure. The graphene growth takes place at 1050° by introducing 20 ppm diluted methane (in Ar) for ∼30−50 min. The samples are then rapidly cooled down in protection of forming gas. Raman Spectroscopy. The Raman spectra are obtained using a Horiba Jobin Yvon Xplora confocal Raman microscope with a 532 nm laser wavelength. Device Fabrication. The graphene films (on Cu) were transferred to standard SiN membrane windows (from Silson Ltd.) using PMMA-assisted technique,47 followed by an annealing process at 350 °C for 90 min in flow of forming gas (5% H2 in Ar). The GBs are located by the geometrical shape of the merged flakes in the partially covered graphene film relative to predefined alignment marks. Electron beam lithography (EBL) followed by O2 plasma cleaning is used to remove excess of graphene flakes and define the target flakes into a rectangle with desired geometry. In another e-beam step, electrode patterns are transferred onto the sample, and SiO2/ Cr/Au (5/5/40 nm) electrodes are deposited using sputtering and electron beam evaporation systems. The devices are then annealed again to remove polymer residues by the fabrication process and are mounted on chip carriers by wire bonding. The samples are finally loaded in the cryostat chamber to perform the calibration process and the heat flow measurements. After the experiments, the samples are again loaded in an O2 plasma cleaning chamber, and all the graphene flakes are etched away; then, the control experiments are repeated in identical conditions. Finite Element (FE) Simulations. The model consists of four layers of solid elements representing gold, chromium, SiO2, and SiN, from top to bottom, respectively. Graphene is included by using a shell element between the SiN and the other layers. The commercial FE software ANSYS is employed to make the

N ( r ⃗ , k ⃗) − N0(ω , T ) v (⃗ k ⃗)∇r⃗ ⃗ N ( r ⃗ , k ⃗) = − τi(ω)

(2)

where N0(ω,T) is the equilibrium distribution function, given by the Bose−Einstein distribution, and τi(ω) is the intrinsic relaxation time, comprised of anharmonic interaction of threephonon processes (both normal and umklapp), substrate roughness, and 1.1% of 13C isotope concentration.48 Fourthnearest neighbor force constant model, as described by Saito,49 has been used to calculate phonon dispersion.50 Phonon dispersion of an isolated sheet has been considered for this study, which is a good approximation.6 The interaction with the silicon nitride substrate is modeled through perturbations to the scattering Hamiltonian using the model for graphene on SiO23. As silicon nitride is a smoother substrate than that of SiO2, thus, a lower value of force constant for out-of-plane modes (0.011 N/m) has been considered to capture the weaker van der Waals coupling force between silicon nitride substrate and graphene. In this model, the two-dimensional graphene sheet is considered to be in contact with the substrate in the form of small circular patches with radius of 16.3 nm. Molecular Dynamics Simulations. Molecular dynamics simulations have been carried out by the Large-scale Atomic/ Molecular Massively Parallel Simulator (LAMMPS).51 We used nonequilibrium molecular dynamics (NEMD) to extract the thermal conductivities.52 Tersoff potential53 with its optimized parameters for thermal properties54 have been used to describe interaction of the carbon atoms. For simulations involving graphene and the substrate, we have used the Tersoff potential to describe the interactions of the Si and N atoms in SiN.55 The interactions between carbon and substrate atoms are described by a 6−12 Lennard−Jones potential.6,56 The time step has been chosen to be 0.1 fs in all simulations to reduce probable numerical errors (see SI file for more details).

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ASSOCIATED CONTENT

S Supporting Information *

Details of graphene GB characterization, measurement setup, measurement process, control experiments, FE simulations, uncertainty analysis, BTE modeling, and MD simulations. The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.nanolett.5b01100.

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected]. *E-mail: [email protected]. *E-mail: [email protected]. G

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Nano Letters Author Contributions

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A.F., R.H., and A.K.M. contributed equally to this work. Author Contributions

A.S.K., R.H., P.Y., and F.K.A. conceived the idea. A.S.K. led the synthesis, fabrication, characterization, and experiments. F.K.A. led the MD simulations. Z.A. led the BTE modeling and calculations. C.F. led the FE simulations. D.E. contributed on the fabrication and experimental procedures. P.Y. and R.H. synthesized graphene flakes and developed the fabrication process and experimental setup. P.Y. and R.H. fabricated final devices and performed the experiments. A.E. developed the FE model and carried out the parameter optimization and data extraction process (with P.Y.). A.F. performed the MD simulations. A.K.M. carried out the BTE modeling and calculations. All authors contributed to the write-up of the manuscript. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS A.S.K. work was supported by University of Illinois at Chicago through the Start-up budget. F.K.A. was supported by the startup funds from University of Illinois at Chicago. Computer simulations were carried out at the HPC cluster at University of Illinois at Chicago, as well as the Blue Waters machine at NCSA (allocated through GLCPC). This research is part of the Blue Waters sustained-petascale computing project, which is supported by the National Science Foundation (awards OCI0725070 and ACI-1238993) and the state of Illinois. Blue Waters is a joint effort of the University of Illinois at Urbana− Champaign and its National Center for Supercomputing Applications. D.E. acknowledges support through startup funds from Boise State University as part of a gift from the Micron Foundation.

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DOI: 10.1021/acs.nanolett.5b01100 Nano Lett. XXXX, XXX, XXX−XXX

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DOI: 10.1021/acs.nanolett.5b01100 Nano Lett. XXXX, XXX, XXX−XXX