Direct observation of resistive heating at graphene wrinkles and grain boundaries

Graphene, a monolayer of hexagonally arranged carbon atoms, has been the subject of intense research due to its thinness (∼3.4 Å), unique linear band structure,¹ and quasi-ballistic electrical and thermal transport up to micron length scales at room temperature.^2,3 Graphene applications typically rely on material growth by chemical vapor deposition (CVD) on metal substrates.⁴ This process can produce graphene up to meter dimensions,⁵ but typically of a polycrystalline nature, with the sheet being made up of a patchwork of grains connected by grain boundaries (GBs).⁶ In addition, various transfer processes from the metallic growth substrate onto other substrates (e.g., SiO₂, BN, and plastics) can lead to wrinkling of the monolayer material.⁷ Not surprisingly, GBs and wrinkles are expected to degrade the thermal,⁸ electrical,^9,10 and mechanical¹¹ properties of graphene. Recent work has measured the electrical resistance of graphene GBs,^9–14 which is important as they limit the overall electrical performance of graphene devices grown by CVD.⁶ However, the associated temperature rise resulting from nanometer-scale resistive heating of GBs is currently unknown. Understanding this aspect is important both from a graphene device perspective (e.g. reliability) and also as a unique platform directly connecting the technology of nanoscale thermometry tools with the science of atomic-scale heat generation at defects within realistic devices.

In this study, we measured the nanometer-scale temperature rise in CVD grown hexagonal graphene grains using scanning Joule expansion microscopy (SJEM),^15–18 a thermometry technique based on atomic force microscopy (AFM). We specifically study the resistive heating at graphene wrinkles and GBs, giving insight into the coupled electrical and thermal properties of such nanoscale defects. We observe a small temperature rise at wrinkles and a larger temperature rise at GBs (150%–300% greater than the surrounding graphene) due to the finite GB resistivity and to non-uniform current flow across GBs, visualized here with nanometer-scale resolution.

Figure 1(a) shows the optical image of a typical GB device used in this study, labeled as Device 1. Sample fabrication is summarized below, while details can be found in the supplementary material¹⁹ and in recent reports.^12,20–23 CVD graphene was grown by atmospheric pressure CVD (APCVD) on electropolished Cu foil.^12,20 Graphene was transferred²¹ from the Cu foil to SiO₂ (90 nm) on Si (highly p-doped) substrates. Suitable grains and GBs were located by optical microscopy, and electrical contacts were accomplished by electron beam lithography and deposition of 1/70 nm Cr/Pd contacts.²³ Fabrication was completed by spin coating the samples with 55–70 nm of poly(methyl methacrylate) (PMMA), which amplifies the thermo-mechanical expansions of the graphene and GB device for the SJEM technique.^16–18 

Figure 1(b) shows the surface thermo-mechanical expansion Δh measured by SJEM overlaid onto the device topography during operation. A sinusoidal waveform with amplitude V_DS was applied to the device at frequency ω = 61–230 kHz. The AFM cantilever was in contact with the surface and a lock-in amplifier at 2ω, with a bandwidth of 4–125 Hz, recorded the peak-to-peak surface expansion Δh. (The supplementary material further discusses the SJEM technique.¹⁹) The spatial and temperature resolution of our SJEM measurements are ∼50 nm and ∼0.2 K, respectively, based on our previous reports.^16–18 The peak-to-peak graphene temperature rise ΔT is proportional to the measured Δh, and the two are related by finite element analysis (FEA) modeling.^16,17 The hexagonal graphene shape and GB are evident from the measured Δh in Figure 1. We also observe a decrease (increase) in Δh as the graphene device laterally expands (contracts) due to its hexagonal shape, creating a non-uniform current density throughout the device. The measured Δh increases 100%–200% at the GB near the device center (x ≈ 0 μm) compared to the graphene sheet due to (1) localized Joule heating from the presence of the GB with finite resistivity ρ_GB and (2) the laterally constricting device shape. The supplementary material discusses simulations which show <25% increase in Δh at the GB compared to device center due to the constricting graphene shape;¹⁹ thus, we attribute the majority of the measured Δh increase to the GB resistivity ρ_GB. Figure 1(b) also reveals a local increase (∼25%) in Δh at a wrinkle, discussed below.

Next, we turn to our AFM and SJEM measurements of a single-grain graphene device with wrinkles, but without a GB. Figures 2(a) and 2(b) show the measured Δh and second resonance amplitude A₂ of such a device. Figure 2(b) shows A₂ measurements from dual alternating contact AFM measurements. (The supplementary material describes the dual alternating contact AFM technique¹⁹ which contrasts the graphene from the surrounding SiO₂.²⁴) The labeled wrinkle in Fig. 2(b) has a height of 3–12 nm which varies along its length. We observe no increase in the measured thermal expansion along the wrinkle, and, therefore, we do not expect a large increase in ΔT at this wrinkle. This observation agrees with recent theoretical work suggesting that tall (>5 nm) wrinkles have low electrical resistance.⁷ The measured thermo-mechanical expansion Δh at the contacts is due to current crowding at the graphene-metal interface.¹⁶ 

Figures 2(c) and 2(d) show the simulated Δh and ΔT for the single grain graphene device. A three-dimensional (3D) FEA model was used to interpret these SJEM measurements as described in the supplementary material.¹⁹ The Fourier transform of the heat diffusion and Poisson equations coupled with a thermo-mechanical model simulated the frequency response of Δh and ΔT. Fitting the measured and simulated Δh for two measurements each at V_DS = 0.54, 1.12, and 1.67 V yields the bulk graphene resistivity ρ = 0.11 ± 0.01 Ω μm (sheet resistance²⁵ R_S ≈ 330 Ω/sq) and graphene-metal contact resistivity¹⁶ ρ_C = 280 ± 90 Ω μm² (per unit area). These values are in-line with previous studies of monolayer graphene and graphene-metal contacts on SiO₂.^16,25 The model matches measurements with a coefficient of determination r²= 0.78 ± 0.06 for all V_DS and yields a total device resistance R = 388 Ω, close to the measured 371 Ω. More information for fitting measurements and simulations is in the supplementary material.¹⁹ 

We now return to a more in-depth investigation of AFM and SJEM measurements of Device 1 which had a single GB. Figures 3(a) and 3(b) show the measured Δh and A₂ of two coalesced graphene grains, the same device with one GB as in Fig. 1. Figure 3(a) shows a large, 100%–200% increase in Δh at the GB. Figure 3(b) also reveals multiple wrinkles, most being 1–3 nm tall and oriented parallel to the current flow direction (along the x-axis). However, the wrinkles show no measureable increase in Δh in Fig. 3(a). We only measure a 25% increase in Δh at one wrinkle (2–3 nm tall), which rests at a ∼56° angle to the current flow direction. However, the measured Δh is ∼4–8 times larger at the GB than the wrinkle, indicating the GB has a greater (detrimental) influence on device performance. Figures S4 and S5 of the supplementary material show similar behavior for the measured Δh at wrinkles and GBs for two other devices.¹⁹ 

Figures 3(c) and 3(d) show the simulated Δh and ΔT for Device 1. Fitting the measured and simulated Δh for two measurements each at V_DS = 0.56, 1.13, 2.34, and 2.95 V yields bulk graphene resistivity ρ = 8.3 ± 0.1 × 10⁻² Ω μm (sheet resistance R_S ≈ 250 Ω/sq), grain boundary resistance ρ_GB = 120 ± 60 Ω μm, and graphene-contact resistivity ρ_C = 30 ± 10 Ω μm². (The contact resistivity is underestimated, an artifact discussed in the supplementary material.¹⁹) The GB resistivity ρ_GB is commonly defined per unit width,^9–13 here the width of the GB being about 4.7 μm for Device 1. One can also define an effective GB length ℓ_eff = ρ_GB/R_S ≈ 490 nm for Device 1, corresponding to the length of graphene channel that would yield the same resistance as the GB.¹³ (The longer the effective GB length, the larger the resistive contribution of the GB relative to the total resistance of the device.) Wrinkles were shown to have a small effect on Δh and were not included in the simulation. The model matches measurements well (r²= 0.89 ± 0.03) and predicts the total device resistance R = 481 Ω, close to the measured value of 471 Ω. Figure 3(d) shows the simulated ΔT increases ∼150% at the GB center and ∼300% at the GB edge compared to the middle of the graphene grains. The ∼150% rise is due to ρ_GB, and the ∼300% rise is due to ρ_GB plus the additional effects of current crowding near the grain edges.

Table I summarizes the GB resistivity ρ_GB extracted from three devices in this study, compared to values reported in the literature.^9–13 The full range of ρ_GB is from ∼8 to 43 000 Ω μm for GBs from CVD-grown graphene on Cu and transferred to SiO₂ substrates. By comparison, reported ρ_GB for graphene directly grown on SiC range from 7 to 100 Ω μm.^12,26 Although we observe a notable 150%–300% temperature increase at the GB, we estimate relatively low ρ_GB for our devices compared to the range reported in the literature for graphene grown by CVD on Cu and transferred for measurements to SiO₂. Our devices were grown using similar methods to those of Clark et al.¹² and we report similar ρ_GB as their study. Interestingly, the results summarized in Table I show no evident trend between graphene grain type and the electrical properties of GBs.^9–13 

In order to facilitate a simpler yet physical understanding of power dissipation at GBs, we developed an analytical model to predict their temperature rise for the range of observed ρ_GB in all studies summarized by Table I.^9–13 Figure 4(a) shows the model geometry and associated electrostatic and thermal boundary conditions. The steady-state analytical model is different from our frequency-dependent measurements and FEA predictions. The solution to the electrostatics and heat diffusion problems with accompanying assumptions is provided in the supplementary material and was verified by an FEA model.¹⁹ For the analysis described below, we assume graphene properties similar to Device 1 with a channel length 2L = 10 μm. Figure 4(b) shows the predicted voltage V(x) and temperature T(x) profile of the device schematic from Fig. 4(a). The model predicts that the small voltage drop across the GB (V_GB) causes a large localized temperature rise (T_GB) due to the highly confined Joule heating at the grain boundary. Figure 4(c) shows the percent voltage drop V_% = V_GB/V₀ and percent power dissipated P_% = P_GB/P₀ at the GB for the geometry shown in Figure 4(a), where V₀ and P₀ are the total applied voltage and power dissipation of the entire device. Current continuity along the device yields V_% = P_%. We estimate V_% = P_% = 2.9% for Device 1 shown in Fig. 3.

Figure 4(c) shows the predicted temperature ratio between the GB and the rest of the graphene grain, T_% = T_GB/T(L/2), from the analytical model; here, the graphene temperature is taken at x = L/2, the halfway point between the GB and contacts. The analytical model overestimates T_% by 20%–50% compared to FEA simulations, as it does not account for the (two-dimensional) heat spreading through the substrate at the GB. The analytical model predicts T_% ≈ 300% for Device 1, close to the observed value of 150%–300%. In fact, the GB dominates the temperature rise (T_% > 200%) of Device 1 for any value ρ_GB > 60 Ω μm, yet the associated V_% = P_% = 1.6% because the GB is a highly localized heat source versus the 10 μm long device. These results suggest that, in relatively “large” (e.g., > 5 μm) interconnects with micron-size grains, GBs may not significantly affect electrical performance, but the GBs will dominate the temperature rise at such “hot spots” and therefore limit the interconnect reliability. A similarly small effect of GBs on large graphene interconnects has been predicted for their thermal conductivity,⁸ as long as grains are micron-size (or larger), i.e., greater than the graphene phonon mean free path which is of the order of 100s of nm.³ 

In order to understand the effect of GB temperature rise on “small” (e.g., <1 μm) graphene devices and interconnects we recall recent experiments which have shown that GBs perturb the electronic wave functions for <10 nm of the surrounding graphene.²⁷ The electron-phonon scattering mean free path is also of the order 20–80 nm in graphene on SiO₂ at room temperature.²⁵ These two length scales suggest that resistive heating only occurs within a few tens of nanometers from the GB itself. However, the length scale of heat flow (1/e temperature decay) away from the GB heat source is the lateral thermal healing length, L_H ≈ 0.1–0.2 μm for graphene on common SiO₂ (90–300 nm) substrates on Si.^19,28,29 Thus, the average temperature of a sub-micron graphene device with even a single GB will be significantly affected by the local power dissipation at the GB.

In both small and large graphene devices and interconnects with GBs, the temperature rise at such highly localized nanoscale defects could lead to premature device failure²⁹ before the average temperature of the graphene sheet has significantly increased. Both graphene oxidation or dielectric breakdown may be more likely to occur at GBs. These scenarios are similar to carbon nanotube (CNT) devices, where breakdown³⁰ and highly localized temperature rise at nanoscale defects have also been studied with SJEM.¹⁸ 

In conclusion, we directly observed nanometer-scale Joule heating of CVD-grown graphene using SJEM with ∼50 nm and ∼0.2 K spatial and temperature resolution. We noted a small increase in temperature at some wrinkles but a large 150%–300% increase in temperature at GBs. Comparing SJEM and electrical measurements with simulations we estimate ρ_GB = 8–150 Ω μm for our devices, among the lowest values reported for CVD graphene.^9–13 An analytical model is developed to predict power dissipation, voltage drop, and temperature rise at GBs for the range of ρ_GB reported in the literature. The model predicts that the GB may experience a large localized temperature rise which could lead to localized device or dielectric failure at GB locations, even before a significant increase of the average device temperature.

Finally, methods which measure nanometer-scale temperatures, such as SJEM and SThM (scanning thermal microscopy), have greater sensitivity to study graphene GBs than electrical techniques alone due to the large and localized temperature rise at GBs and similar atomic-scale defects. Knowledge of the nanoscale temperature rise and Joule heating at graphene GBs is important for understanding graphene devices and their reliability, as well as the physics of polycrystalline graphene sensors, which have increased sensitivity at their GBs and defects.^6,31

The authors gratefully acknowledge the help of J. C. Koepke. This work was supported in part by the National Science Foundation (NSF) Grant Nos. ECCS 1002026 and ECSS 1201982, the National Defense Science and Engineering Graduate Fellowship (D.E. and J.D.W.), and the Army Research Office (ARO) Grant No. PECASE W911NF-11-1-0066 (E.P.). The synthesis science performed at ORNL by G.E. was sponsored by the Materials Sciences and Engineering Division, Office of Basic Energy Sciences, U.S. Department of Energy (DOE). Research on graphene synthesis performed by I.V. was sponsored by the Laboratory Directed Research and Development Program of ORNL, managed by UT-Battelle, LLC, for the U.S. DOE.