Scaling properties of polycrystalline graphene: a review

Four-terminal measurements across individual GBs show an enhanced electrical resistance compared to the surrounding grains, ${R}_{\mathrm{total}}={R}_{\mathrm{grains}}+{R}_{\mathrm{GB}}$ [18–21, 23–27]. The origin of this enhanced resistance has been probed with scanning tunneling spectroscopy (STS), and these measurements have revealed that GBs tend to be n-doped compared to the surrounding grains, such that they act as an electrical potential barrier to charge transport [40, 41]. Magnetotransport measurements [18, 19, 42], numerical simulations [43], and STS measurements [40, 41] have shown that GBs also induce weak localization, indicating that they are a source of intervalley scattering. Additionally, temperature-dependent measurements have shown that ${R}_{\mathrm{GB}}$ is independent of temperature, providing further support for the hypothesis that scattering at the GBs is dominated by structural defects and impurities [18–20].

The resistance of an individual graphene GB can be written as ${R}_{\mathrm{GB}}={\rho }_{\mathrm{GB}}/W$, where ${\rho }_{\mathrm{GB}}$ is the GB resistivity and W is the device width (or equivalently, the GB length). The GB resistivity thus provides an intrinsic measure of the charge transport properties across the GB, independent of the device or measurement technique. Beyond four-terminal electrical measurements, the resistivity of individual GBs has been measured using a.c. electron force microscopy (AC-EFM) [13], four-point scanning tunneling potentiometry (STP) [21], and Joule expansion microscopy [23]. The average ${\rho }_{\mathrm{GB}}$ of a series of polycrystalline samples can also be extracted by measuring the sheet resistance of each sample as a function of the average grain size [44–49] and then fitting to a simple ohmic scaling law, ${R}_{{\rm{S}}}={R}_{{\rm{S}}}^{0}+{\rho }_{\mathrm{GB}}/{l}_{{\rm{G}}}$, where ${R}_{{\rm{S}}}$ is the sheet resistance of the polycrystalline sample, ${R}_{{\rm{S}}}^{0}$ is the sheet resistance within the grains, and ${l}_{{\rm{G}}}$ is the average grain size [26].

A summary of the values of ${\rho }_{\mathrm{GB}}$ that have been reported in the literature is shown in figure 2. Markedly, ${\rho }_{\mathrm{GB}}$ spans more than three orders of magnitude, from less than 0.1 up to 100 kΩ μm. To account for this spread of values, there are several factors that should be considered. Factors that impact the intrinsic value of the GB resistivity include the structural quality of the GB, the position of the Fermi level, and the measurement technique. Other factors that can alter the GB resistivity are related to the cleanliness of the device, such as if the graphene was protected by a gate dielectric or if an exposed graphene film was measured in air or in vacuum.

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Tsen et al showed that poorly-connected GBs can significantly enhance ${\rho }_{\mathrm{GB}}$, in their case by one order of magnitude [20], and this behavior has also been seen in numerical simulations [50]. Kochat et al also showed a strong correlation between the GB quality and its resistivity, where GBs with a wide region of disorder were more resistive than narrower GBs [27]. Meanwhile, the numerical simulations in figure 2 were made with perfectly connected polycrystalline samples, and thus yielded relatively low values of ${\rho }_{\mathrm{GB}}$ [26].

When determining the GB resistivity, control of the Fermi level is also important, as four-terminal measurements have shown that the value of ${\rho }_{\mathrm{GB}}$ can vary by one order of magnitude as a function of gate voltage, with the maximum occurring at the charge neutrality point (CNP) [20, 27]. In figure 2, open circles represent measurements with the Fermi level at the CNP, closed circles are for measurements where the Fermi level is far from the CNP, and ⋆'s are for measurements where the Fermi level is unknown. In general, the largest measured values of ${\rho }_{\mathrm{GB}}$, on the far right of figure 2, were all made at the CNP, while the lowest measured values of ${\rho }_{\mathrm{GB}}$ were made far from the CNP. Scaling the lowest measured values of ${\rho }_{\mathrm{GB}}$ by a factor of 10 brings them into the range of the four-terminal measurements made at the CNP.

The measurement technique also appears to have some impact on the estimated value of the GB resistivity. In particular, the values of ${\rho }_{\mathrm{GB}}$ extracted from the scaling law tend to be lower than those from four-terminal measurements, even at the CNP. When considering electrical transport through a large polycrystalline sample, charge will flow through a series of parallel conducting paths, and the sheet resistance will be determined by the lowest resistance path. For this reason, the value of ${\rho }_{\mathrm{GB}}$ extracted from the scaling law represents the lower end of the range of GB resistivities present in the sample, and not the average value. When considering all three factors together, it appears that the resistivity of well-connected GBs, measured at the CNP, typically falls in a range around 1 kΩ μm.

While GBs by themselves may not always give rise to significant scattering, the adsorption and reaction of adsorbates or impurities can adversely impact the electrical transport. While this may be detrimental for high-performance electronic devices, it could be advantageous for gas sensing applications [51, 52]. Four-terminal measurements of highly-resistive GBs showed that oxygenation can vary ${\rho }_{\mathrm{GB}}$ by a factor of two [26], while adsorption of dimethyl methylphosphonate gas molecules increased the resistivity of a low-resistance GB by one order of magnitude [25]. Numerical calculations also revealed a strong variation of ${\rho }_{\mathrm{GB}}$ with chemical functionalization [26], indicated by the spread of values on the far left of figure 2. In contrast to these results, recent work by Gao et al demonstrated a room temperature mobility of nearly 5000 cm² V⁻¹ s⁻¹ for polycrystalline samples with an average grain size on the order of 100 nm [53]. These promising results highlight the need to design experiments that can separate the intrinsic effect of the GBs from extrinsic factors like surface cleanliness. The dry transfer method recently reported by Banszerus et al, which yielded single-grain CVD graphene with mobilities up to 350 000 cm² V⁻¹ s⁻¹, is one such approach [54].

Beyond impeding charge transport, individual graphene GBs can impact the electrical properties of polycrystalline graphene in other ways. For example, Joule expansion microscopy measurements have revealed strong Joule heating localized at graphene GBs, which can be many times larger than Joule heating in the grains. This has important implications for the reliability of graphene devices, as localized device failure could occur without a significant increase in the average device temperature [23]. Other measurements have shown that electrical noise is greatly enhanced by graphene GBs, which is detrimental for low-noise devices but may be useful for sensor applications [27].

While knowledge of the resistivity of individual GBs is valuable from both a fundamental point of view and for particular applications, it is also important to have a clear picture of the global charge transport properties of polycrystalline graphene. This is especially important for large-area devices that utilize this material, such as flexible transparent electrodes. In polycrystalline graphene, charge transport is ultimately limited by two sources—scattering at the GBs, and scattering within the graphene grains. The competition between these two sources of resistance is simply captured by the ohmic scaling law mentioned above, ${R}_{{\rm{S}}}={R}_{{\rm{S}}}^{0}+{\rho }_{\mathrm{GB}}/{l}_{{\rm{G}}}$. For samples with large grains, the sheet resistance will be dominated by that of the grains, ${R}_{{\rm{S}}}^{0}$, and ${R}_{{\rm{S}}}$ will be independent of the grain size. For small-grained samples, ${R}_{{\rm{S}}}$ will be dominated by ${\rho }_{\mathrm{GB}}$ and will scale inversely with the grain size. The crossover from grain- to GB-dominated behavior occurs at a grain size of ${l}_{{\rm{G}}}\approx {\rho }_{\mathrm{GB}}/{R}_{{\rm{S}}}^{0}$. This general behavior is shown by the solid gray line in figure 3, where we plot the sheet resistance as a function of the grain size assuming ${R}_{{\rm{S}}}^{0}$ = 300 ${\rm{\Omega }}/\square$ and ${\rho }_{\mathrm{GB}}=0.3$ kΩ μm.

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In figure 3 we also plot a selection of values of sheet resistance versus grain size that have been reported in the experimental literature. Most measured values follow a general inverse scaling between ${R}_{{\rm{S}}}$ and ${l}_{{\rm{G}}}$, and the slope of this scaling can provide information about the relative values of ${\rho }_{\mathrm{GB}}$ and ${R}_{{\rm{S}}}^{0}$. For example, the shallow scaling reported by Yagi et al [46] suggests a low value of ${\rho }_{\mathrm{GB}}$ = 0.7 kΩ μm, but a relatively large grain sheet resistance of ${R}_{{\rm{S}}}^{0}$ = 6 ${\rm{k}}{\rm{\Omega }}/\square$. However, the measurements of Yang et al [49], while similar in magnitude, show a much steeper slope, giving a large value of ${\rho }_{\mathrm{GB}}$ = 18 kΩ μm but a smaller ${R}_{{\rm{S}}}^{0}$ = 600 ${\rm{\Omega }}/\square$. The measurements of Duong et al [45], Lee et al [47], and Venugopal [48] all reveal low values of ${R}_{{\rm{S}}}^{0}$ = 100–300 ${\rm{\Omega }}/\square$, while those of Lee et al have the smallest GB resistivity, 0.3 kΩ μm [47]. It should be noted that for the values reported by Vlassiouk et al, Raman spectroscopy was used to determine the size of defect-free regions, but this technique does not distinguish between disorder in the grains and disorder arising from the GBs [44]. As mentioned before, the spread in the numerical calculations of the sheet resistance (open squares) is due to varying degrees of chemical functionalization applied to the GBs [26].

It is clear that minimizing the impact of GBs is important for electronic applications of polycrystalline graphene, and today it is possible to grow single graphene grains with diameters on the order of centimeters [55–57]. However, there is a significant tradeoff between grain size, the required temperature, and the growth time in the CVD process, and when it comes to industrial applications these growth conditions should be minimized [58]. By understanding the scaling of the sheet resistance, one can determine the grain size needed for a particular application. For example, with a growth process that yields a grain sheet resistance of ${R}_{{\rm{S}}}^{0}$ = 100 ${\rm{\Omega }}/\square$ and a GB resistivity of ${\rho }_{\mathrm{GB}}$ = 1 kΩ μm, the grains only need to be larger than ${\rho }_{\mathrm{GB}}/{R}_{{\rm{S}}}^{0}$ = 10 μm for the contribution of the GBs to no longer matter. Thus, depending on the requirements, centimeter-sized grains may be unnecessary. This was recently demonstrated by Samsung and its collaborators, whose rapid thermal CVD process yielded polycrystalline graphene samples with a sheet resistance that was constant over grain sizes in the range of 1–10 μm [58].

In addition to direct charge transport, polycrystalline graphene has also been studied for its application to quantum Hall metrology. In single-crystal graphene, clear signatures of the quantum Hall effect have been measured at room temperature [59], owing to the relatively large splitting of the two lowest Landau levels. This makes graphene quite promising for the establishment of new resistance standards, as measurements can be made at higher temperatures and lower magnetic fields than those in traditional two-dimensional electron gases [60, 61]. However, in polycrystalline graphene the quantum Hall measurements are less ideal, with a degradation of the precision of the quantum Hall plateaus and the development of a finite longitudinal conductivity [42, 62, 63]. Various numerical simulations have shown that GBs can short circuit the quantum Hall measurement [43, 63–65], and the network of GBs provides a path for percolating transport through the bulk of the material [43]. Very small grains can also impede the onset of the quantum Hall regime until the magnetic field becomes large enough such that ${l}_{{\rm{B}}}\lt {l}_{{\rm{G}}}$, where ${l}_{{\rm{B}}}=\sqrt{{\hslash }/{eB}}$ is the magnetic length [43, 53]. In general, these results indicate that GBs should be avoided in order to achieve high-precision quantum Hall measurements.

Polycrystalline graphene has also been considered for thermoelectric applications. The figure of merit of a thermoelectric material is given by ${ZT}={S}^{2}\sigma T/\kappa$, where T is the temperature, S is the Seebeck coefficient, and σ (κ) is the electrical (thermal) conductivity. ZT is a measure of the efficiency of a thermoelectric engine, and its value can be tuned by separately tuning S, σ, and κ. Recent theoretical work has predicted that an individual GB can significantly enhance the ZT of graphene at the CNP, primarily through an enhancement of the Seebeck coefficient [66]. However, simulations of large-area polycrystalline graphene samples are less optimistic, as they show that σ and κ scale similarly with the grain size [50, 67–69], while the Seebeck coefficient is reduced compared to pristine graphene [70]. Together, these results suggest an overall reduction of ZT compared to single-grain graphene.

Finally, it is important to note that beyond the microstructure, the process of device fabrication can also have a strong impact on electrical transport. Exfoliated and CVD graphene are usually transferred onto a dielectric substrate using scotch tape or organic compounds such as PMMA, resulting in physisorbed or chemisorbed residues on the graphene surface [71]. If the graphene sheet contains point defects, dislocations, GBs, or wrinkles, the residues can attach more easily [72]. In fact, during the initial stages of graphene device fabrication, dielectric films were usually difficult to deposit on unfunctionalized or 'clean' graphene surfaces, and several techniques were introduced to overcome this problem [73–75]. The ability to deposit the dielectric uniformly on graphene protected the films from ambient adsorbates, which can have a significant effect on the transport properties [76], but these processes also introduced organics, ozone, metals, or metal oxides that tended to degrade the mobility in comparison to devices without a top gate [77, 78]. Therefore, when analyzing graphene mobility data it is critically important to know the quality of the dielectric substrate, the crystallinity of the graphene, and the gate dielectric. This was clearly demonstrated by the use of hexagonal boron nitride (h-BN) as a substrate material and gate dielectric [79], which can lead to very high mobilities in both single-crystal and bi-layer graphene [80, 81]. In cases where the surface of graphene is not exposed to any organic compounds during the transfer process, the mobility is also greatly improved [82] although in this case perhaps the benefit is also associated with low strain in the graphene films transferred using h-BN [83].

The elastic properties of graphene are commonly studied experimentally by means of nanoindentation experiments [85]. For numerical studies, MD is typically required as system sizes are in general too large for density functional theory (DFT) to be practical. In a nanoindentation measurement, the graphene devices have the form of a micron-sized suspended circular drum. By pressing down with an atomic force microscope (AFM) while recording applied force and tip deflection, the elastic properties are deduced. While experimental devices have micron linear dimension, numerical simulations are limited to much smaller systems (≲100 nm). In MD studies, the mechanical properties are probed by recording the stress as the system is uni- or bi-axially strained. This should be contrasted with the nanoindentation experiments where the strain is typically non-uniform even for a pristine sheet and where the details of the AFM tip shape may influence the results. For pristine graphene, both numerics (MD/DFT) and indentation experiments show good agreement, yielding a Young's modulus either quoted as a 2D entity ${E}_{2{\rm{D}}}\approx 340\,{\rm{N}}$ m⁻¹ or as the equivalent 3D modulus ${E}_{3{\rm{D}}}\approx 1\,\mathrm{TPa}$ [85, 86]. The two are related through the interlayer spacing of graphite (3.35 Å).

For polycrystalline sheets, however, there is a larger discrepancy between reported values. On the experimental side, one complication arises from the different methods of fabrication and transfer. Not only does this yield samples with different quality of GBs and different amounts of pre-stress, but also the amount of wrinkling differs between samples. Wrinkling or buckling can be either the result of the fabrication-transfer process or due to the boundaries themselves, which alleviate the extra strain incurred by the defects. This causes a softening of the Young's modulus for small strains. On the theoretical side, the choice of interaction potential and GB geometry can greatly influence the results.

Theoretical studies of the simplest geometry, graphene sheets containing a single straight GB [87–89], found values around ${E}_{3{\rm{D}}}\sim 800$ GPa. This was confirmed by nanoindentation measurements using an AFM tip to press down on polycrystalline suspended CVD graphene drums which showed, within statistical error, a value of ${E}_{3{\rm{D}}}\sim 1$ TPa [84]. Although the spread in measured elastic constants was larger for polycrystalline graphene than for pristine graphene, the average value was still in agreement with that of pristine graphene. In the experiments by Lee et al [84], the diameter of the suspended graphene drums was similiar to the grain size.

Both the theory [87–89] and experiments [84] essentially studied local GB properties, i.e they were in the regime ${l}_{{\rm{G}}}/L\lesssim 1$. However, those findings are in contrast to theory taking into account more natural GB geometries. Such simulations typically result in lower values for ${E}_{3{\rm{D}}}$ [90–95]. In particular, Kotakoski and Meyer [90], who performed large-scale MD simulations on polycrystalline graphene, found significantly lower values of the Young's modulus ${E}_{3{\rm{D}}}\sim 600\,\mathrm{GPa}$. While smaller values were obtained experimentally using indentation measurements [13, 96] with values ${E}_{3{\rm{D}}}\sim 150$ GPa, these were attributed the to the out-of plane buckling associated with the GBs. Finally, not only GBs may affect the elastic properties. As highlighted by nanoindentation measurements [97, 98] on graphene with controlled creation of defects, the type of defects and their concentration can also impact the results.

As in 3D macroscopic materials, the characteristic grain size ${l}_{{\rm{G}}}$ plays an important role. From MD simulations, a clear grain-size dependence of the elastic modulus has been found, with ${E}_{3{\rm{D}}}$ increasing upon increasing the grain size [90–95] (see figure 4). In order to have a well-defined value of an elastic property, it is desirable that it be measured for a system size $L\gg {l}_{{\rm{G}}}$. For practical reasons, simulating structures with linear dimensions L much larger than the characteristic grain sizes has remained prohibitive except for very small grains (∼1 nm), and most studies have not studied the scaling with ${l}_{{\rm{G}}}/L$. Also, nanoindentation studies are performed on micrometer-sized systems comparable to the grain sizes ($L\sim {l}_{{\rm{G}}}\sim 1$ μm), and reporting the ensemble-averaged values over many devices. There is, however, currently no experimental work covering the case $L\gg {l}_{{\rm{G}}}\sim 1$ μm for the Young's modulus.

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Monocrystalline graphene is predicted to have an extraordinary fracture strength of 110–130 GPa [84, 88, 99, 100]. In presence of GBs it is natural to expect that this strength is diminished. However, using MD, it was shown [87] that in contrast to this expectation, higher-angle tilt GBs with a higher density of defects (heptagon–pentagon pairs) can be as strong as mono-crystalline graphene, while low-angle boundaries with lower defect concentrations showed a reduction in fracture strength. This behavior derives from the dipolar stress profile around individual pentagon–heptagon pairs surrounded by hexagons, with tensile stress on the heptagon side and compressive stress on the pentagon side (see figure 1(c)). With an increased density of aligned pentagon–heptagon pairs, stress cancellation from adjacent dipoles results in reduced built-in stress. These results were confirmed by studies [88, 89, 101] which further found that the detailed arrangement of defects in the GB played an important role. In particular, Wei et al [88] found a non-monotonic behavior of fracture strength with tilt angle for armchair GB configurations, which was also corroborated by a continuum theory in excellent agreement with simulations. Subsequent MD studies have revealed similar results [102–106]. While most studies used straight boundaries, Rasool et al [14] and Ophus et al [15] showed that more complex boundary structures also typically exhibited a high fracture strength ∼100 GPa. Using MD it has further been shown that the fracture strength of tilt GBs decreases significantly with increasing temperature [93, 102, 106, 107].

Experimental determination of fracture strength has been carried out by means of nanoindentation measurements. Early results [13, 96] revealed a significantly decreased fracture strength ∼35 GPa, which was attributed to vacancies and shear stress along the boundaries. However, by using different post-processing methods, it was shown, in accordance with the theoretical predictions, that well-stitched boundaries could have nearly the same fracture strength ($\approx 100$ GPa) as pristine graphene [84]. Rasool et al [108] characterized the GB angles in their measurements, confirming the angle dependence of the fracture strength, and finding high values close to 100 GPa for large angle boundaries. To explain the observed spread in fracture strength in a nanoindentation measurement, Sha et al [109] made explicit MD simulations of nanoindentation measurements of polycrystalline graphene, revealing that the relative positioning of the tip and the GBs could yield vastly different fracture strength and different failure paths.

The dependence of fracture strength on tilt angle was predicted by simulations of bi-crystal systems with a single boundary, but more realistic fracture strength simulations of polycrystalline graphene have shown a significant reduction in fracture strength [90] and dependence on grain size [91–93, 95, 110]. In these simulations, failure is typically initiated at the boundary, or more commonly at meeting points between three different grains. The dependence of fracture strength on grain size for the studies in [91–93, 95] are shown in figure 5. Notably, three studies [92, 93, 95] report increasing strength with increasing grain size, while Song et al [91] find the opposite trend. The fracture strengths in the study by Kotakoski and Meyer [90] and the paper of Yang et al [94], however, showed no significant dependence on grain size.

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Nanoindentation measurements by Suk et al [111] give support to a decreased fracture strength of polycrystalline graphene, with the fracture strength being smaller for smaller grains. For the comparison between theory and experiment, it is again worth pointing out that most numerical studies have $L\gtrsim {l}_{{\rm{G}}}\sim 1\,\mathrm{nm}$, while experimental works are in the regime $L\sim {l}_{{\rm{G}}}\sim 1\,\mu {\rm{m}}$. In a recent study by Shekhawat and Ritchie [112], a weakest-link argument is used together with MD simulations of $\sim 2\times {10}^{4}$ GB configurations to obtain a functional form for the fracture strength. They find a scaling for the fracture strength σ, $\sigma -{\sigma }_{0}\propto \nu {({l}_{{\rm{G}}}/L)}^{2/m}$. Here ${\sigma }_{0}=19.5$ GPa, $\nu =53.2$ Gpa and $m\approx 10.1$, showing that the fracture strength of large sheets may be significantly lower than previously predicted.

As described in section 2, a single GB can be created in the honeycomb lattice by tilting two crystalline graphene sheets of the same length by a certain angle α (see figure 1(b)). Although such simple geometries correspond to an idealized configuration of a perfectly straight, periodic and isolated GB, they nevertheless represent a paradigmatic situation for investigating the role of GBs in thermal transport.

Both the strain-induced corrugation and the lattice misorientation between the two neighboring grains deeply affect the heat transport along the direction normal to the GB, resulting in an effective GB thermal resistance ${R}_{\mathrm{GB}}$. This can be described as a series of resistances [67] according to

$$\begin{eqnarray}&&{R}_{\mathrm{tot}}=\displaystyle \frac{{L}_{\mathrm{tot}}}{{\kappa }_{\mathrm{tot}}}=\displaystyle \frac{{L}_{\mathrm{left}}}{{\kappa }_{\mathrm{left}}}+{R}_{\mathrm{GB}}+\displaystyle \frac{{L}_{\mathrm{right}}}{{\kappa }_{\mathrm{right}}},\end{eqnarray} \tag{ 1 }$$

where ${R}_{\mathrm{tot}}$, ${L}_{\mathrm{tot}}$, and ${\kappa }_{\mathrm{tot}}$ are the thermal resistance, length, and thermal conductivity of the total sample, while ${L}_{\mathrm{left},\mathrm{right}}$ and ${\kappa }_{\mathrm{left},\mathrm{right}}$ represent the length and thermal conductivity of the left and right crystallites, respectively (see figure 1(b)). By selecting a suitable simulation cell with ${L}_{\mathrm{right}}={L}_{\mathrm{left}}={L}_{\mathrm{tot}}/2$ and ${\kappa }_{\mathrm{left}}={\kappa }_{\mathrm{right}}={\kappa }_{\mathrm{sg}}$ (here ${\kappa }_{\mathrm{sg}}$ represents the thermal conductivity of a single-grain sample of pristine graphene), equation (1) allows one to predict the GB thermal resistance as

$$\begin{eqnarray}&&{R}_{\mathrm{GB}}=\displaystyle \frac{{L}_{\mathrm{tot}}}{2}\left(\displaystyle \frac{1}{{\kappa }_{\mathrm{tot}}}-\displaystyle \frac{1}{{\kappa }_{\mathrm{sg}}}\right).\end{eqnarray} \tag{ 2 }$$

All quantities appearing in equation (2) are straightforwardly calculated by an AEMD simulation [68, 130], which provides the value of room-temperature ${R}_{\mathrm{GB}}$ as a function of the tilt angle α. As shown in figure 6, the trend is not monotonic, which is likely due to a complex interplay among buckling effects, the occurrence of coordination defects, and bond-network reconstruction. This non-monotonic behavior has not been seen experimentally, but the values of ${R}_{\mathrm{GB}}$ are of the same order of the available experimental data reported in [52]. Furthermore, the same experiments confirm the general trend of increasing thermal boundary resistance with increasing tilt angle.This is consistent with previous findings reporting an increasing effective thermal conductivity with decreasing mismatch angle [141]. We remark that increased thermal boundary resistance in multi-grain graphene has been attributed to larger out-of-plane buckling [69, 142]. This, however, is not valid for the single isolated GB; the highest thermal boundary resistance occurs for the largest tilt angle where the lowest out-of-plane buckling is observed, as shown in figure 1(d).

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The thermal conductivity in model nanocrystalline graphene samples has been calculated in [68, 69] and the main results are summarized in figure 7, where the room-temperature thermal conductivity κ is reported as a function of the average radius of gyration ${r}_{{\rm{G}}}$ of the grains (measuring the average granulometry of the nanocrystalline sample) or, equivalently, as a function of the concentration ${c}_{\mathrm{GB}}$ of atoms in the GB (defined as belonging to non-hexagonal rings in the graphene lattice). As expected, samples with a higher density of GBs (or, equivalently, a higher concentration of GB atoms) correspond to smaller thermal conductivity values, consistent with the analysis presented in the previous section. In all cases, κ is very much reduced with respect to the value 262 W m⁻¹ K⁻¹, found for a single-grain graphene sample with same length.This is in good agreement with [140] where a reduction of thermal conductivity to about 20% of the value in pristine graphene is observed for an average grain size of ∼2.5 nm. This highlights the major role played by GBs in scattering phonons, whose mean free path (MFP) is comparable with the average GB spacing. Based on the accumulation function of the thermal conductivity, the average phonon MFP has been estimated to vary from 451 nm for pristine samples to ∼30 nm for nanocrystalline ones [68, 69].

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An inverse rational function can be used to describe the scaling of κ with ${r}_{\mathrm{GB}}$ and ${c}_{\mathrm{GB}}$, as shown in figure 7 (where ${\kappa }_{{\rm{c}}-\mathrm{Graph}}$ is the thermal conductivity of the crystalline grains). A similar argument has been used previously in [67], where the thermal conductivity of polycrystalline graphene has been calculated by EMD simulations. The good agreement between the calculated data and interpolation functions indicates that the effective thermal conductivity of nanostructured graphene can be estimated considering GBs and grains as a connection of resistances in series. Given this, the average GB resistance extracted from figure 7 is ∼0.1 ${{\rm{m}}}^{2}$ K GW⁻¹ This is in reasonably good agreement withthe predictions reported in [139] as well as with the results of the previous section if one considers that the value extracted from the grain size scaling represents the lower end of the GB resistances in the sample. Further details are found in [68]. In this scheme, phonon scattering by the GBs is mimicked as an effective interface thermal resistance, while the thermal conductivity of crystalline domains is described by the thermal conductivity of the crystal without GBs.