Probing mechanical properties of graphene with Raman spectroscopy
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The use of Raman scattering techniques to study the mechanical properties of graphene films is reviewed here. The determination of Grüneisen parameters of suspended graphene sheets under uni- and bi-axial strain is discussed, and the values are compared to theoretical predictions. The effects of the graphene−substrate interaction on strain and to the temperature evolution of the graphene Raman spectra are discussed. Finally, the relation between mechanical and thermal properties is presented along with the characterization of thermal properties of graphene with Raman spectroscopy.
The growing interest in understanding the mechanical properties of graphene films is sparked by the ability to control such properties, and thus to modify the structure and electronic behavior for graphene-based applications. Raman spectroscopy is increasingly used to measure accurately and nondestructively graphene mechanical or thermal properties, such as strain or thermal conductivity. This review outlines the current state-of-the-art in the use of Raman spectroscopy to characterize the strain and temperature effects in exfoliated and epitaxial graphene. The relationship between strain and film morphology is also reviewed.
In “Graphene atomic structure” section, we review the basic atomic structure of graphene, with a brief overview of the methods used to isolate and prepare graphene films on various substrates. An overview of the mechanical properties of graphene films determined by nanoindentation methods is presented in “Graphene mechanical properties measured by nanoindentation” section along with the current limitations of such approach. The Raman spectrum of graphene in conjunction with its phonon spectrum is described in “Raman scattering in graphene and graphite” section. A detailed overview of the use of Raman spectroscopy for the determination of mechanical properties of graphene is presented in “Probing mechanical properties of graphene with Raman spectroscopy,” with particular emphasis on the characterization of strain and of the temperature effects in the graphene films.
Graphene atomic structure
The Brillouin zone for a single graphene layer is shown in Fig. 1b. It exhibits high symmetry points: the Γ point at the zone center, the M point in the middle of the hexagonal sides and the K and K′ points at the corners of the hexagons. K and K′ are inequivalent points, since they correspond to the two different and inequivalent sublattices in the graphene atomic structure.
Graphene samples can be prepared by mechanical exfoliation of highly oriented pyrolithic graphite (HOPG) [5, 6, 7], which leads to the production of micrometer scale single and multilayer graphene sheets with high degree of control over their thickness. Graphene can be also grown epitaxially on SiC surfaces by high temperature Si sublimation, in ultrahigh vacuum (UHV) [8, 9] and in controlled environment [10, 11, 12, 13]. Epitaxial graphene can also be grown on the surfaces of various metals such as Pt , Ni [15, 16], Ir [17, 18], Ru [19, 20, 21], and Cu . With this method, large domains can be obtained (domain size ∼10 μm) . Epitaxial graphene grown on metals can be transferred from the synthesis substrate to any chosen substrate . This procedure is suitable for investigation of large-scale graphene layers either suspended or transferred to various substrates. The graphene−substrate interaction strongly depends on the type of substrate due to the different degree of adhesion of graphene to the substrate (whether, for example, graphene is grown epitaxially on a substrate or mechanically transferred to it). Therefore, the choice of substrate and synthesis method have several implications in the mechanical properties of the epitaxial graphene film.
Graphene mechanical properties measured by nanoindentation
The mechanical behavior of graphene layers can be described macroscopically by continuum elasticity theory. In this spirit, nanoindentation techniques are well suited to measure the macroscopic mechanical properties of graphene, including Young’s modulus and bending stiffness. For example, by using nanoindentation methods on suspended multilayer graphene flakes, the bending stiffness has been measured and found to be in the range from 2 × 10−14 N/m to 2 × 10−11 N/m for 8 to 100 layers, respectively. Static nanoindentation experiments based on the deflection of AFM cantilevers pressed within 100 nm of the center of ∼1 μm long double-clamped graphene films, provided a measurement of the effective spring constant of multilayer graphene (1–5 N/m). The spring constant was found to scale with the dimensions of the suspended region and the layer thickness (from 5 to 30 layers), and of the extracted Young modulus of 0.5 TPa, independent of thickness . A significant limitation of the use of nanoindentation techniques is the requirement of a graphene layer to be suspended. The presence of a substrate, over which graphene may either be deposited (SiO2 , glass, and sapphire  or polymers [15, 25]) or directly grown epitaxially (e.g., SiC [2, 9] and metals [19, 20, 21]), makes it hard to separate by nanoindentation measurements the intrinsic mechanical properties of a graphene from that of the substrate.
In contrast to nanoindentation, Raman spectroscopy provides access to information relating to the underlying chemical bonds. Besides complementing the coarse-grained approach of macroscopic elasticity, the interrogation of bond vibrations by optical spectroscopy enables the retrieval of information about mechanical and structural properties of films that can have monolayer thickness and be strongly interacting with a substrate. Raman spectroscopy has thus been used to measure mechanical properties of graphene films, both freestanding and on a substrate [25, 26], at room and at elevated temperatures [27, 28].
Raman scattering in graphene and graphite
Raman spectroscopy of graphene
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 .
Graphene metrology with Raman scattering
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, 45, 46, 47], the structure of graphene layer edges [48, 49, 50, 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 . 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  (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  show that the ratio of the D and G band intensities (ID/IG) is inversely proportional to the in-plane crystallite size La, 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 . Furthermore, the ratio ID/IG is known to depend on the electron concentration (and thus on the film doping) , 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.
Probing mechanical properties of graphene with Raman spectroscopy
Any changes in the atomic structure in a crystalline solid due to plastic deformation, strain, or thermal expansion are reflected in the phonon spectrum of the crystal. By probing the phonon spectrum with Raman spectroscopy, such changes can be detected, thus providing insight into the mechanical and thermal properties of materials such as graphene.
Strained semiconductors have received significant interest in the past because of the wide ranging implications of strain, such as the ability to engineer the electronic structure and to affect the carrier mobility in silicon-based materials for electronic device application . The application of an external stress on a crystal results in a lattice strain, i.e., in a change in interatomic distances and consequent redistribution of electronic charge. Isotropic compression (hydrostatic pressure) generally results in an increase in the frequency of the vibrational mode (phonon hardening), while isotropic tension results in the decrease in the vibrational frequency (phonon softening). Application of anisotropic stress has more complex effects, and can result in lifting of the degeneracy of phonon frequencies.
In graphene, changes in the Raman spectra have been observed as a consequence of the presence of stress, either induced artificially on suspended or exfoliated graphene [25, 55, 56, 57, 58, 59, 60] or provided by the interaction with the substrate for graphene grown epitaxially on SiC substrates [26, 58, 61, 62, 63]. Such changes consisted of a systematic upshift in the position of the main Raman D (when present), G, and 2D peaks, by up to 30, 31 and 64 cm−1, respectively , for an applied strain of up to 1.3%.
In spite of specific changes in the electronic and vibrational band structure, the strain-induced frequency shifts of the Raman active E2g and 2D modes are independent of the direction of strain, which has been observed experimentally  and confirmed by ab initio calculations . Thus, the amount of strain can be directly determined from a single Raman measurement .
The Grüneisen parameter for uni- and biaxial strain
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 .
Determination of the Grüneisen parameter in graphene
Grüneisen parameter and shear deformation potential for a single layer graphene
Gradients in Raman peaks position per units of applied strain (cm−1/%), for a single layer graphene
−46 … 54
Substrate-induced strain on graphene
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 , the shift in the G peak corresponding to charge measured in a monolayer graphene on 6H–SiC (1.4 × 1013 cm−2 ) 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 . 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% . 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 .
Characterization of thermal properties of graphene with Raman spectroscopy
Temperature coefficients for the G and 2D peaks in suspended graphene layers
Temperature range, K
Single layer suspended
Single layer suspended and supported on Au/SiO2
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 χm depends on the anharmonic potential constants, the phonon occupation number and the thermal expansion of the graphene two-dimensional lattice . The contribution of anharmonic terms is most significant at high temperatures; hence, the overall thermal dependency is not expected to follow a linear trend . 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, χm 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 .
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 χm 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 . 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.
The thermal conductivity of a single and multilayer layer graphene is measured optically via Raman spectroscopy
Single layer (CVD)
Conclusions and Outlook
Raman spectroscopy is currently used as a metrology tool to determine the extent, the quality and the uniformity of graphene films. This review has illustrated the applications of Raman spectroscopy to probing the mechanical properties of graphene films. The direct measurement of Raman peak shifts, for example, has enabled the determination of parameters such as the Grüneisen parameter and the shear deformation potential, and thus to a measurement of the strain in graphene films. While such shifts, in general, can be attributed to other causes (e.g., induced charge, doping), under precise experimental conditions (thermal equilibrium, constant pressure, and with fixed Fermi level) lattice strain can be directly measured from peak changes in the Raman spectra . Understanding the evolution of strain in graphene films is important, as it allows for a deeper understanding of how graphene interacts with the environment, and particularly with a substrate. The ability to monitor and control strain in graphene could be crucial during device fabrication, as it affects the electronic properties of the material itself . For example, it has been recently shown that modulation in electrical  and optical  conductance can be induced by strain. It has been suggested that by properly modulating strain locally in graphene may lead to a controlled tuning of the electronic band gap . Such studies are in their infancy, however. The vast majority of investigations have been performed either on exfoliated graphene, or on epitaxial graphene grown on SiC. More investigations are needed to understand the presence and the evolution of strain in graphene grown, for example on transition metals via chemical vapor deposition, or as an effect of the mechanical transfer in the case of exfoliated graphene. Since deposition or synthesis methods strongly affect the graphene interaction with the substrate, further studies are needed to highlight and establish a connection between the strength of this interaction and the thermal evolution of the Raman spectra of graphene. While attempts to correlate strain to other structural properties of graphene (such as surface morphology) have been proposed, more work is needed to be able to connect strain with the electrical, optical, and thermal properties of the material. As doping strongly affects strain in thin films , more investigations are required to determine how doping affects the strain in graphene films.
From a fundamental standpoint, Raman spectroscopy can provide accurate in situ measurements of thermal properties such as the thermal conductivity. Such approach allows for the characterization of the role of geometry, chemistry, and morphology, and of their effects on thermal properties. Such capabilities need to be extended to other graphene-related materials, such as graphene oxide [102, 103] and graphane . When applied to graphene in a controlled environment, these measurements, may prove suitable for sensing applications. Overall, the characterization of mechanical properties of graphene with Raman spectroscopy will promise to be valuable in the determination of the optimal growth conditions, and even more in the optimization of fabrication methods of graphene-based devices.
The author is grateful to Roya Maboudian and Carlo Carraro for useful suggestions and their critical reading of this review. This work was supported by the National Science Foundation under Grants CMMI-0825531 and EEC-0832819 through the Center of Integrated Nanomechanical Systems.
This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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