Probing mechanical properties of graphene with Raman spectroscopy

The Raman spectrum of carbon-based materials is characterized by a set of common features in the region between 800 and 2000 cm⁻¹, in particular the so-called D and G bands, which lie at around 1330–1360 and 1580 cm⁻¹, respectively, for visible excitation [29‐31], as shown in Fig. 2a. Under these excitation conditions, the Raman spectra of carbon films are dominated by the sp² sites, because visible excitation always resonates with the π states. Due to the comparatively small cross section for the amorphous sp³ versus sp² C–C vibrations, a significant fraction of sp³ bonds is required in a sample for the sp³ peak at 1332 cm⁻¹ to be visible, as is the case in diamond (Fig. 2) [29].

The phonon dispersion curves of graphene (Fig. 3) are the key to understand its Raman spectrum. They consist of three acoustic phonon modes (A) and three optical (O) phonon modes since the graphene unit cell contains two carbon atoms (Fig. 1a). Among these modes, one acoustic branch and one optical phonon branch correspond to out-of-plane phonon modes (o), while for the other acoustic and optical phonon branches, the vibrations, and thus the phonon modes, are in-plane (i). Each in-plane mode has two branches, one longitudinal (L) and one transverse (T). Following the high symmetry ΓM and ΓK directions, the six phonon dispersion curves are assigned to LO, iTO, oTO, LA, iTA, and oTA phonon modes [30, 31]. In graphite the LO and iTO modes are degenerate at the center of the Brilloin zone, the Γ point. According to group theory, these modes are the only Raman active modes, corresponding to the two dimensional E_2g phonon. The G peak (located around 1580 cm⁻¹) corresponds to such doubly degenerate E_2g mode at the Brillouin zone center [30, 31]. In the “molecular” picture of carbon materials, the G peak is due to the bond stretching of all pairs of sp² atoms (Fig. 2b).

The D peak (∼1340 cm⁻¹) corresponds to modes associated with transverse optical (iTO) phonons around the edge of the Brillouin zone (K or Dirac point) [29]. In the molecular picture, it is associated with the breathing mode of the sp² aromatic rings (Fig. 2b) [36, 37]. The D peak is energy dispersive, so that its position is dependent on the excitation energy (Fig. 4) [38]. The D peak is usually very intense in amorphous carbon samples, while it is absent in perfect graphitic samples. Its overtone (2D, ∼2660 − 2710 cm⁻¹), however, is always visible even when the D peak is absent. Such peculiar behavior is due to the double resonance (DR) activation mechanism [39] of the D peak, which requires the presence of defects for its initiation [37, 40]. In a DR process, Raman scattering is a four-step process: (i) a laser-induced generation of an electron–hole pair; (ii) electron–phonon scattering with an exchanged momentum q ∼ K; (iii) electron scattering from a defect, whose recoil absorbs the momentum of the electron–hole pair; (iv) electron–hole recombination [29]. The requirements of conservation of energy and momentum can only be satisfied if a defect is present. In a perfect sample, momentum conservation would be violated by the DR mechanism, and thus the D peak is absent. Momentum conservation however is always satisfied in case of the 2D peak, without the need for defect activation, since the process involves two phonons with opposite momentum vectors [29]. A similar process is possible with scattering within the same K point. This intravalley process activates phonons with small momentum q, resulting in the so-called D′ peak, located around ∼1620 cm⁻¹ in defective graphite [41].

Scattering from holes can also occur in the Raman process. In graphene, under these circumstances, the electron is not scattered back by a phonon of momentum −q, but instead a hole is scattered forward by a phonon with momentum +q. In this case, during the electron–hole generation, both electron and hole scattering processes are resonant. The electron–hole resonant recombination at the opposite side with respect to the K point is also resonant, resulting in the triple resonance (TR) scattering process. It has been suggested that the higher intensity of the 2D peak relative to the G band in a graphene monolayer is due to the TR activation mechanism [31].

Raman spectroscopy, as a noninvasive probing technique, has been extensively employed to characterize graphene layer thickness [32, 42], domain grain size [29, 36, 43], doping levels [29, 44‐47], the structure of graphene layer edges [48‐51], anharmonic processes, and thermal conductivity [52, 53]. This has been possible through a combined investigation of the Raman peaks D, G, and 2D in graphite and graphene films of various thicknesses and morphologies. An indicative comparison of the Raman spectra of graphene and bulk graphite is made in Fig. 4a [32]. The most striking difference between the individual graphene layers and graphite resides in the change in shape and intensity of the 2D peak. While the 2D peak in graphite consists of two peaks 2D₁ and 2D₂ (with intensities of 1:4 and 1:2 compared to the G peak, respectively), the 2D peak in one single graphene layer has only one component with roughly four times the intensity of the G peak (Fig. 4a). For multilayer graphene (Fig. 4b), the evolution in the shape of the 2D peak has been used to determine the layer thickness [32, 42, 49]. The splitting of electronic bands in bilayer graphene is responsible for the splitting of the 2D peak into four components [37] (Fig. 4c). The two lower components further decrease while the higher wavenumber components increase as the film thickness approaches five layers. Above this threshold, however, the determination of the layer thickness with Raman becomes rather difficult, as the shape of the 2D peak is increasingly similar to that of bulk graphite.

Early investigations of disorder in graphitic carbon [36] show that the ratio of the D and G band intensities (I _D /I _G ) is inversely proportional to the in-plane crystallite size L _a , measured independently with X-ray diffraction. Such relation, known as the Tuinstra–Koenig (TK) relation, has been refined in recent years to provide an empirical method to determine the size of graphene domains from the Raman spectrum under a given excitation energy [30, 43]. There are known limitations in this approach, as the distribution of domains with different sizes is such that the smaller domains are weighted more, leading to an underestimation of the average size distribution. In addition, the use of peak intensity ratio instead of peak area ratio underestimates the average domain size, since the full-width-half-maximum (FWHM) of the D peak increases significantly in comparison to that of the G peak [29]. Furthermore, the ratio I _D /I _G is known to depend on the electron concentration (and thus on the film doping) [46], limiting the application of the TK relation when the doping concentration is unknown. Regardless of the limitations, the use of the TK relation allows an estimation of the degree of disorder in the graphene film.

The rate of change with strain of a given phonon frequency in a crystal is determined by its Grüneisen parameter [67, 68]. In metrology applications, accurate values of Grüneisen parameters are crucial for quantifying the amount of strain in the system, reflected in the change in phonon frequency from its value in the absence of strain. In presence of uniaxial strain, the Grüneisen parameter for a particular band m associated with in-plane Raman active phonon band (where m is either the D or G band in graphene), is defined as [67]:where ε _h  = ε _ll  + ε _tt is the hydrostatic component of the applied strain with l and t referring to the directions parallel and perpendicular to the applied strain, respectively, and \(\omega^0_m\) and \(\omega^h_m\) correspond to the phonon frequencies of peak m at zero strain and in presence of an applied strain, respectively. For a given shear component of strain, ε _s  = ε _ll  − ε _tt , the shear deformation potential β _m is defined as:

For the G band corresponding to the E_2g phonon, the shifts in the two components G⁺ and G⁻ relative to the position at zero strain, ω _G ⁰ , are given by:where \(\Updelta\omega^h_G\) and \(\Updelta\omega^s_G\) are the shifts associated with the hydrostatic and shear components of the strain, respectively. Under condition of uniaxial strain, ε _ll  = ε and ε _tt  = −νε, where ν is the Poisson ratio [67]. In case of graphene, if the layer adheres well to the substrate used for strain analysis, such as for example polyethyleneterephtalate (PET) [25], the Poisson ratio of the substrate must be used, instead of in-plane Poisson ratio for bulk graphite. Under uniaxial strain, Eq. 3 can be solved, yielding both the Grüneisen parameter and the shear deformation potential for the G band, as functions of the shifts in the positions of the two components G⁺ and G⁻:

Under the conditions of biaxial strain, ε _ll  = ε _tt  = ε, there is no shear deformation potential and no splitting of the G peak. In this case, Equation 3 can be solved to provide the Grüneisen parameter [25, 26]:

It is, however, possible that local anisotropies in the applied biaxial strain, possibly induced by the substrate over small domain size (such as in epitaxial graphene grown on SiC), may cause an increase in the FWHM as a result of a local splitting of the G band. It is also worth noting that under biaxial strain conditions, the shift in the peak position is independent of the presence of any substrate, because of the absence of a sheer deformation term and thus the absence of the Poisson term ν in Eq. 6 [25].

The Grüneisen parameter can be similarly derived for the D and D′ bands in graphene. Of the two, only the first is single-degenerate, and corresponds to A_1g phonons at the K point (Fig. 3). The D peak is thus not expected to split under uniaxial strain, and only the hydrostatic component of the stress is present. The Grüneisen parameter for the D peak (which is equivalent to that of the overtone 2D) can be written as:or(Note that the shear deformation potential β for the D and D′ bands cannot be extracted, because of the lack of shear component of the applied uniaxial strain). The D′ band is associated with an E symmetry mode, which is double-degenerate; as such, a splitting is expected under uniaxial strain. Experimentally the only study to report on the effects of strain on the D′ peak did not observe any splitting, due to the weak intensity of this peak and the small range of applied strain [25]. For small strains, the Grüneisen parameter for the D′ follows Eq. 8. In the case of biaxial strain, Eq. 7 is the same as Eq. 4, which can be generalized as:where m corresponds to the D, G, or 2D bands. It is worth mentioning that in all cases, the detection of strain effects is the most sensitive if the 2D band is considered. With a spectrometer resolution of ∼2 cm⁻¹, the sensitivity for uni- and biaxial strain is 0.03 and ∼0.01, respectively.

Mohiuddin et al. provided a complete characterization of the Grüneisen parameters for the G and 2D bands of exfoliated graphene [25]. In order to measure the Grüneisen parameters and the shear deformation potential of a single-layer exfoliated graphene, Eqs. (4), (5), and (7) were used to fit measured shifts in the positions of the G and 2D Raman bands as a function of applied uniaxial stress. The resulting Grüneisen parameters and shear deformation potential for the main Raman peaks for a single layer graphene are summarized in Table 1, along with previous theoretical and experimental studies. The values of γ _G and β _G from [25] are in good agreement with those calculated with density-functional theory (∼1.8 [68]) and first principle calculations (1.87 [25]). Recently Metzger et al. measured directly the Grüneisen parameters for biaxial strain, by placing a single-graphene layer onto a substrated prepatterned with shallow depressions [60]. The adhesion of graphene to the substrate across the prepatterned depression, despite the induced biaxial strain, allowed a controlled and precise determination of the biaxial strain. By using Eq. 9, the Grüneisen parameters for the G (2.4) and D (3.8) peak were extracted and found to be higher than those measured from uniaxial strain [25] or calculated [25, 68]. It has been speculated that the larger values of both the Grüneisen parameters and the peak shifts when compared to previous measurements were due to a better adhesion of graphene to the substrate in the latter studies. This leads to a measurement of the strain actually transferred to the graphene layer from the substrate (i.e., with no slippage). However, the large difference between the measured values of the Grüneisen parameters for the G band (1.8 [25] vs. 2.4 [60]) but not for that of the D peak (3.55 [25] vs. 3.8 [60]) remains unexplained.

By applying such parameters to Eq. 4, the gradients in Raman peak position per unit of applied strain are extracted. A summary of both theoretical and experimental studies is reported in Table 2. The use of the correct value for the Grüneisen parameter is extremely important, because it affects the estimated value of the Raman peak shift for a given strain. Often the Grüneisen parameter of CNT is used, leading to a questionable estimate for the gradient in the peak position. For example, γ_2D  = 1.24 [56] \(\partial\omega_{2D}^{\rm uniax}/\partial\varepsilon \sim -27.1\,\hbox {cm}^{-1}/\%\) [56] to be contrasted to ∼−83 cm⁻¹/% when γ_2D  = 3.55 is used per reference [25], obtained on a single-layer graphene. This result has been used to justify the measured value of the gradient in peak position for uniaxial strain. However, the absence of any splitting of the G peak and lack of any difference in Raman peak position between uni- and biaxial graphene [56], which are in contradiction with theory, suggests that the applied strain is either far from being uniaxial [56] or points to poor sample quality. As a further indication, the estimated gradient in the G peak position as function of applied strain (\(\partial\omega_G/\partial\varepsilon \sim-27.8\,\hbox {cm}^{-1}/\%\)) is consistent with the averaged value of the gradients of the shifts in the G⁺ and G⁻ peaks (\(\partial\omega_G^{\rm uniax}/\partial\varepsilon \sim -27\,\hbox {cm}^{-1}/\%\), [25]). This is also consistent with the average value obtained from measurements on carbon fibers (∼−25 cm⁻¹/%), where individual sub-bands cannot be distinguished due to the broad G band for amorphous carbon [73]. The similarity in such measurements between graphene and graphite indicates that the in-plane Young modulus for graphene and bulk graphite are similar [74].

While uni- and bi-axial strain can be artificially applied to suspended graphene layers, strain can arise in graphene heterostructures from the interaction between graphene layers and the underlying substrate. Initially, in the case of exfoliated graphene, no appreciable shifts were observed in the G band of a graphene layer transferred onto SiO₂/Si and GaAs substrates [24]. Small downshifts of about 5 cm⁻¹ were observed for graphene placed on sapphire and glass, with a splitting of the G band in the latter case [24]. The higher adhesion offered by sapphire substrates, which is sufficient to introduce a small amount of strain in the graphene layer during the mechanical placement, was attributed to the particular binding of carbon–sapphire [24]. The binding is also responsible for the growth of highly aligned CNT on sapphire substrates [75]. Recently, however, more comprehensive investigations of the evolution of the mechanical and morphological properties in graphene suspended over a microfabricated trench reported variations in the positions of the G and 2D bands, during and after thermal cycling [59]. Upon thermal cycling to 700 K, while in purely suspended regions no shifts were observed, large upshifts (∼23, ∼10, ∼5 cm⁻¹ for a single-, bi-, and tri-layer graphene, respectively, corresponding to a compressive strain of 0.39%, 0.18%, and 0.09%, respectively) were instead observed in the regions where graphene was in contact with the underlying substrate. While graphene was compressed in the region over the substrate, the compression was relieved and the formation of ripples was observed in the purely suspended region (Fig. 6) [59].

The role of the substrate on strain in graphene films has been also investigated extensively on graphene grown epitaxially on SiC surfaces (so called epigraphene) by high-temperature decomposition [26, 58, 61‐63]. Figure 7 shows representative spectra of a single crystal 6H–SiC(0001) surface and that with 1.5 layers of epigraphene. The Raman peak of zone-center optical (G) phonons in monolayer epigraphene is overwhelmed by the second order signal from the SiC substrate, a broad band occupying the same spectral region. This unfortunate coincidence limits the ability to measure precisely the position of the epigraphene G band itself. This limitation can be overcome by the use of a depolarized scattering configuration [26], as shown in Fig. 7.

Raman spectra of epigraphene on the Si-terminated 6H- and 4H-SiC (0001) substrates usually show a blueshift in the graphene epilayer peak positions, with respect to those on exfoliated graphene [26, 62]. The extent of shift is different for each Raman peak, as shown in Fig. 8. The blueshift recorded varies by up to 22 cm⁻¹ for the G peak and 64 cm⁻¹ for the 2D peak. Graphene on C-terminated SiC substrates have not been investigated in full details with Raman spectroscopy. However, it is speculated that the decoupling of the graphene layer grown on the C-termination may reduce the amount of strain in the film.

The large shift in epitaxial graphene layers on Si-terminated SiC was attributed to compressive strain in the graphene layer. This explanation may seem surprising, since no external strain was applied to the system. However, the only possible alternative explanation, charge transfer from the substrate, was ruled out, based on the fact that it could not account for the magnitude of the shifts in the G and 2D peaks. Indeed, while charging induces a shift in the G peak up to ∼20 cm⁻¹ for an electron concentration of 4 × 10¹³ cm⁻² [46], the shift in the G peak corresponding to charge measured in a monolayer graphene on 6H–SiC (1.4 × 10¹³ cm⁻² [76]) would only account for approximately 7 cm⁻¹. Similarly, shifts in the 2D band corresponding to the given amount of charge in monolayer graphene is negligible [46]. Hence, the observed shifts could only be explained in terms of strain in the system [25, 26, 58, 62]. By using the Grüneisen parameters evaluated under applied uni- and biaxial strain on suspended graphene layers (Table 1), the amount of intrinsic strain in epigraphene can be evaluated using Eq. 9. It is interesting to note that the shifts of the D and G peaks occur in the approximate ratio of 1:1.4 [26, 62], which is in good agreement with the ratio between the Grüneisen parameters for those peaks on exfoliated graphene in presence of biaxial stress (1.8:2.7, Table 1). Hence, for the maximum observed upshift of 22 and 64 cm⁻¹ for the G and 2D peaks, the corresponding strain in epigraphene is approximately 0.7–0.8% [26]. The shifts in the Raman spectra are found to decrease as the number of graphene layers increases. More specifically, the G and 2D peaks in the epitaxial graphene bilayer are found to be shifted by up to 7 and 22 cm⁻¹ (as opposed to 22 and 64 cm⁻¹ for the monolayer, respectively), to approach the unstrained values for films thicker than ∼6–9 layers [62].

The presence of strain in epigraphene was initially explained in terms of the difference between the lattice constant of the reconstructed 13 × 13 graphene layer supercell (α_graphene = 31.923 Å) and of the reconstructed SiC \(6\sqrt{3}\times6\sqrt{3}\) supercell (α_SiC = 31.935 Å) [77]. Such small difference cannot account for the significant amount of strain measured. Compressive strain at room temperature in the graphene layer was later attributed to the large difference in the coefficients of thermal expansion (Fig. 9) between graphene (α_gr, as measured and calculated in [68]) and SiC (α_SiC, as measured in [78]) during cooldown from the synthesis temperature [26, 62]. This difference Δα(T) is nearly constant between room temperature (RT) and the graphene synthesis temperature, T _s ≈ 1250 °C. If the epitaxial film is in mechanical equilibrium with the SiC surface, as a stress-free monolayer commensurate with the 6 \(\times\,\sqrt{3}\)-reconstructed SiC surface at T _S, a large compressive strain would develop in the film upon cooling, since SiC contracts on cooling, while graphene expands [26]:

Ferralis et al. found that the shift observed in the position of the Raman peaks strongly depends on the duration of the high temperature annealing [26, 61]. The evolution in the shift of the 2D peak as a function of the annealing time is shown in Fig. 10. It was observed that for short annealing times (up to 2 min) the G and 2D Raman peaks were almost unshifted from their unstrained values. Longer annealing times (up to 1 h) were found to produce the largest shifts (as high as 22 cm⁻¹ for the G band, corresponding to a strain of ∼0.8%, based on Eq. 9). It was argued that higher compressive stress at room temperature resulted from a lower stressed film at the synthesis temperature (T _S), while a nearly stress-free film at room temperature indicated that the film existed under high tensile stress at T _S. Within experimental accuracy, the strain measured at room temperature might well vanish for very short annealing times. In contrast, for long annealing times, the graphene layer reaches mechanical equilibrium with the substrate at the synthesis temperature T _S, and a compressive strain develops at room temperature film (up to ∼0.8%). This analysis suggests that mechanical equilibrium with the \(6\sqrt{3}\) SiC substrate at T _S is indeed achieved for annealing times longer than 10 min, while for shorter annealing times (∼5 min or less), graphene is under high tensile strain at T _S [26, 61].

A direct correlation between the strain distribution and graphene surface morphology was made using a combined Raman spectroscopy and electron channeling contrast imaging (ECCI) [61]. It was found that the roughness of the SiC substrate terraces from where epigraphene grows increased paralleling the increase in the Raman peak shifts under the same conditions, as shown in Fig. 10. This observation provides a possible mechanism for strain relaxation. For long enough annealing times, tensile strain developed at T _S is relieved by the roughening of the step edges to which graphene films are pinned. Such increase in roughness does not induce a significant change in surface coverage (±0.2 ML). For short annealing times, surface relaxation and roughening do not take place, leaving the SiC terraces morphologically unchanged. Similarly, large inhomogeneities in the distribution of strain within the same epigraphene layer were reported by combined Raman mapping and atomic force microscopy (AFM) [63]. Large shifts in the 2D Raman band (up to 74 cm⁻¹, corresponding to a strain of about 1.0%) were observed to correspond to regions with screw dislocations, step terraces, and macrodefects, while regions with less pronounced band shifts corresponded to large flat terraces (Fig. 11). The strain distribution map obtained with Raman spectroscopy appears to be correlated with the surface morphology of the graphene film, monitored by AFM, confirming that changes in the physical topography are related to changes in the strain of the graphene film [63].

Raman spectra have a significant temperature dependence, both in intensity and in position of the Raman peaks. For example, the ratio of the intensities of anti-Stokes and Stokes peaks is commonly used as a metrology tool to determine the actual temperature of the analyzed sample [79]. Since strain in the lattice also affects Raman peak positions, it is crucial to understand and discern the role played by the changes in lattice parameters (due to strain or thermal expansion) from purely isovolumetric thermal dependencies. Experimentally, separating the two contributions is complicated, especially if either mechanism is not easily controllable, or strictly depends on the position of the Raman peak for its determination. In complete absence of strain, shifts in Raman peaks observed in response to temperature changes reflect both elementary anharmonic processes (electron-phonon and phonon-phonon scattering) and changes in lattice parameters with temperature (thermal expansion). The temperature dependence of the G and 2D peak positions ω _m for single and bilayer suspended graphene is approximately described by [27, 28]: where ω _m ⁰ is the position of the peak m (either G or 2D) at T = 0 K, and χ _m is the first-order temperature coefficient of the same peak. By measuring the position of the G and 2D peaks as a function of sample temperature, the temperature coefficients are extracted for single and bilayer graphene (Fig. 12). The results are reported in Table 3, and compared with other carbon-based materials. It should be noted that the geometrical configuration employed in these experiments (a graphene sheet rigidly connected to the substrate) does not guarantee the conditions of a strain-free environment. Hence, the actual determination of the thermal evolution of the Raman spectrum through these experiments may include non-negligible contribution from strain.

The temperature dependence of the G peak for the single layer is found to be higher than for the bilayer. Both values are higher than that for HOPG, and are expected to approach the HOPG value for thicker graphene films. The temperature coefficient χ _m depends on the anharmonic potential constants, the phonon occupation number and the thermal expansion of the graphene two-dimensional lattice [84]. The contribution of anharmonic terms is most significant at high temperatures; hence, the overall thermal dependency is not expected to follow a linear trend [52]. The nonlinearity must be taken into account when using calibration of thermally induced shifts in the Raman spectra of graphene. Commonly used linear fits need to be accompanied by the temperature range used for the measurements, as reported in Table 3. In HOPG, χ _m is found to depend mostly on the anharmonic contribution, due to direct coupling of phonon modes. Since thermal expansion occurs primarily along the c-axis, its effect on the in-plane G and 2D Raman modes are not very pronounced [85].

It is, however, important to note that the interaction with the substrate may strongly affect thermal expansion of graphene, resulting in a different value of χ _m for purely suspended versus strongly interacting graphene layers. This might be the cause of the different value measured by Cai et al. for a single-layer graphene grown by CVD and pressed against a Au/SiO₂ thin film on Si [80]. As a further indication of a strong interaction with the substrate, the same value of χ was found on regions of the same graphene layer either supported and suspended over circular microfabricated holes.

Raman measurements on suspended nanostructures can be used to determine their thermal conductivity. This method has been employed to measure the thermal conductivity of single layer graphene [53, 80, 86] (Fig. 13). In one experiment, a single layer of graphene is mechanically placed across microfabricated SiO₂ trenches, to remove any interaction of the graphene layer with the substrate. The laser source used for the Raman measurement is also used as the local heating probe. By monitoring the shift in the G peak as a function of the change in laser power P, the thermal conductivity K of a graphene layer can be obtained according to [53]: where χ _G is the temperature coefficient of the G peak, L is the distance from the middle of the suspended graphene layer to the heat sink, h and W are the thickness and width of the graphene layer, respectively. Equation 12 is valid under the assumption that the front wave is nonspherical, as is usually the case when the laser spot size (∼0.5–1.0 μm) is of the same order as the graphene strip lateral size [53]. Although the interaction with the substrate is minimized across the trenches, residual strain may still be present in the supporting regions. The amount of strain in the suspended region, however, was considered negligible, as the Raman peak position in this region, at room temperature, corresponds to that of unstrained suspended graphene (Fig. 6) [52, 59]. Furthermore, the coefficient χ _G in Eq. 12 is measured on an unsuspended graphene layer, while the experiment is performed on a suspended layer. χ _G for an unsuspended graphene monolayer is expected to be lower than that for the suspended layer because of the interaction between the graphene layer and the substrate. Therefore, the measured thermal conductivity is underestimated. As previously noted, Cai et al. performed a similar experiment where graphene synthesized via chemical vapor deposition was pressed against a Au/SiO₂ thin film on Si [80]. The significant difference in the value of χ _G measured in these experiments may be due to an enhanced interaction of graphene with the substrate, possibly due to the graphene synthesis and deposition method. Under these conditions, the coefficient of thermal expansion of graphene is strongly affected by that of the substrate, leading to a value of χ _G which is significantly different from that of a purely suspended graphene film. Further investigations are needed to quantify how the thermal evolution of the graphene Raman spectra is affected by the graphene−substrate interaction and in particular by the difference in the coefficients of thermal expansions of graphene and the substrate.

The measured thermal conductivity is compared to those of other carbon-based materials in Table 4. The extremely high value of phonon thermal conductivity in the strictly two-dimensional graphene layer is in sharp contrast with the reduced phonon thermal conductivity (as compared to bulk values) in quasi-one-dimensional systems such as nanowires [93], or quasi-two-dimensional semiconducting thin films [94]. The net reduction in the phonon thermal conductivity observed in these systems is explained in terms of rough boundary scattering or phonon spatial confinement effects. Given the high values measured for a single layer graphene, such effects appear not to be present. Furthermore, when comparing the thermal conductivity of a single layer graphene to other graphitic materials such as CNT, graphene exhibits a higher value, possibly due to a reduced number of structural defects, and a reduced intralayer scattering. In a comparison with bulk graphite, thermal conductivity approaches that of bulk as the number of atomic planes in graphene films increases from 2 to 4 [88]. It has been shown that Umklapp-limited thermal conductivity of graphene grows with the increasing linear dimensions of graphene flakes and can exceed that of the basal planes of bulk graphite when the flake size is on the order of a few micrometers [95, 96].