Introduction

Since the first isolation of graphene from graphite1, atomically thin membranes of various materials have been studied revealing novel electronic1,2, optical2,3, chemical4,5,6 and mechanical7 properties distinct from those of their bulk counterparts. Many of the findings led to proposals of unique applications, for graphene in particular, such as nanoelectronics8, transparent conductive electrodes9, high-performance composites10, and so on. Each of the applications requires controllable modification of the material properties of graphene, for instance, electronic band gap8, sheet resistance9, or dispersibility11. Mechanical strain () has also been predicted useful in implementing energy gaps in graphene under triangular stress12 and confinement effects13 like one-dimensional channels without physical cutting inducing unwanted charge localization on edges14. More recently, it was shown that non-uniform strain generates pseudo-magnetic field of 300 T pointing to a new application15.

On the other hand, strain can be induced uncontrollably during various processes involved in preparation of graphene sheets and devices. In particular, thermal treatments tend to generate in-plane strain due to difference in the thermal expansion coefficients (TEC) of graphene and underlying substrates. For example, graphene epitaxially grown on SiC16 above 1,100 °C exhibits substrate-induced compressive strain (~−1%) at room temperature17. On annealing at 300 °C, graphene on SiO2 substrates undergoes drastic structural deformation forming sub-nm high ripples with a lateral quasi-period of several nm (ref. 4), implying the presence of corrugation-induced strain18. A similar thermal rippling was observed on a larger length scale in graphene suspended over a trench19. Even pristine graphene on SiO2 substrates, prepared by mechanical exfoliation of graphite, is deformed4 on the nm-length scale owing to the ultrastrong adhesion20 with the undulating substrates. Although many scanning tunnelling microscopy (STM) studies have revealed corrugation18,21,22,23 and related strain18,23, however, quantitative characterization of the native strain and its behaviour on thermal stress has been rare4,24.

Raman spectroscopy has been a useful tool in characterizing strain in crystalline and semi-crystalline materials25 as changes in lattice constants lead to variations in phonon frequencies. The strain-sensitivity of the Raman frequencies of G (ωG) and 2D (ω2D) modes have been determined for graphene under uniaxial or biaxial stress by several groups with the resulting Gruneisen parameters in agreement with the theoretical predictions26,27,28,29,30,31. Both ωG and ω2D are also strongly dependent on the extra charges induced by either electrical32,33 or chemical methods4,6,34 owing to the static effects on the bond lengths and non-adiabatic electron–phonon coupling35. The bimodal sensitivity of ωG and ω2D complicates independent determination of either of strain or charge density (n), which typically requires prior knowledge of the other. While the wide distribution in ωG of mechanically exfoliated graphene samples were attributed to charge impurities36; for instance, it is not known how much native strain contributes to the variation of ωG.

Here we demonstrate that the concurrent native strain and charge doping in graphene can be determined separately from each other by Raman spectroscopy. Extensive two-dimensional Raman analysis shows that most of pristine graphene sheets exhibit strain in the range of −0.2 to 0.4%, which varies gradually on the length scale of several microns. The native strain is relieved and becomes compressive when annealed at 100 °C or above, showing rich thermal transformation behaviours.

Results

Strain- and charge-sensitivity of ωG and ω2D

First, we show that the pristine samples exhibit spatial inhomogeneity in ωG ranging from 1,573 to 1,586 cm−1. Figure 1a,b shows optical micrographs of two graphene samples deposited on SiO2/Si substrates. As shown in Fig. 1c, the Raman spectra of these samples contain the two prominent peaks typifying single-layer graphene, G and 2D peaks originating from the doubly degenerate zone-centre phonon E2g mode and overtone of the transverse optical phonon near K points in the Brillouin zone, respectively37. The absence of the disorder-related D peak near 1,350 cm−1 and symmetric Lorentzian line shape of the 2D peak with a width of ~25 cm−1 are the most salient features of defect-free single-layer graphene37. To investigate the spatial variations of the spectral features, hundreds of spectra were obtained per sample by raster-scanning the laser spot within the dashed boxes in Fig. 1a,b. The ωG-Raman maps shown in Fig. 1d,e exhibit significant random variations. Whereas the edge regions in Fig. 1d have ωG close to that of graphite (1,581.5±0.3 cm−1; see Methods) with ~4 cm−1 downshift in the central region, Fig. 1e exhibits an even wider distribution over an area of 35×15 μm2. Besides the pixel-to-pixel variations, there are long-range undulations in ωG occurring on a length scale of several μm. We attribute most of this frequency modulation to strain as will be shown below. In addition, we found that the density of native charges in strain-dominant pristine graphene is very low (≤4×1011 cm−2) when judged from various Raman spectroscopic features. (see Methods.)

Figure 1: Raman maps of single-layer graphene revealing large frequency variations.
figure 1

(a) Optical micrograph of sample K1. Scale bar, 10 μm. (b) Optical micrograph of sample K2. Scale bar, 5 μm. (c) Representative Raman spectra of K2 obtained before (black line) and after (red line) thermal annealing at 400 °C for 2 h. The absence of the disorder-related D band (marked by the vertical arrow) indicates high crystalline order of the sample. (d) Raman map for the G-mode frequency (ωG) obtained from pristine K1. (e) Raman map for the G-mode frequency (ωG) obtained from pristine K2. (f) Raman map for the G-mode frequency (ωG) obtained from annealed K1. (g) Raman map for the G-mode frequency (ωG) obtained from annealed K2. Each of the Raman mapping was carried out in the areas specified by the blue dashed boxes spanning 20×20 and 35×15 μm2 for K1 and K2, respectively. Each Raman spectrum of the map data was obtained for 15 s.

Full size image

To investigate the effects of thermal perturbation on the native strain, the samples in Fig. 1 were annealed at 400 °C in vacuum for 2 h. The Raman spectrum obtained following the annealing reveals ~25 cm−1 increases in ωG and ω2D (Fig. 1c), which occurred throughout the whole graphene sheets as shown by the ωG–Raman maps in Fig. 1f,g. The annealing-induced stiffening of both Raman modes, first reported by Li et al.6, was attributed to hole doping caused by O2 in the presence of water4, although exact doping mechanisms still remain unclear4,38,39,40,41,42. The intensity decrease and line broadening in 2D mode of the annealed graphene (Fig. 1c) are also mainly attributed to the hole doping43,44. While Fig. 1f confirms the upshift in ωG across the entire area, it also reveals that the spatial variation of ωG has been removed on the annealing. On the contrary, the ωG-undulation in Fig. 1e remains almost unaffected by the thermal treatment despite the annealing-induced upshift (Fig. 1g).

The puzzling spectral variations above are now presented in a different perspective in Fig. 2 to show how the pixel-to-pixel variations in ωG correlate with those in ω2D. Interestingly, hundreds of data points from a given pristine sample, each corresponding to a spectral average over ~1 μm2, form a linear line. Remarkably, the data sets from eight pristine samples including three in Fig. 2 turned out to fall on a single line with a slope (Δω2D/ΔωG) of 2.2±0.2 (black dashed line). To determine intrinsic frequencies of the two modes (ωG0,ω2D0) not affected by strain or excess charges, we investigated a freestanding graphene (F1; Supplementary Fig. S1) suspended across a circular well (7 μm in diameter and 5 μm in depth). The green circle in Fig. 2 indicates the values averaged over a freestanding area of 16 μm2, (1,581.6±0.2, 2,676.9±0.7). Freestanding graphene is known to be virtually charge-neutral with a residual charge density less than 2×1011 cm−2 (ref. 34). Despite the possibility of pre-tension in the suspended graphene45, we conclude that F1 is essentially strain-free because ωG0 agrees well with the aforementioned value of graphite and that of electrically neutralized bilayer graphene46 within 0.5 cm−1 corresponding to a biaxial strain less than ~0.01% (ref. 30). Setting (ωG0,ω2D0) as the origin (O) of the ωG–ω2D space, the points near O in Fig. 2 originate from graphene areas that are nearly charge- and strain-free like the freestanding graphene and the rest are mechanically strained or charge-doped.

Figure 2: Correlation between the frequencies of the G and 2D Raman modes of graphene (ωG,ω2D).
figure 2

The data were obtained from Raman mapping of three graphene samples (K1, K2 and K3, respectively, in red, blue and black) before (+) and after (×) thermal annealing at 400 °C. Each Raman spectrum of the map data was obtained for 15 s. The green dot (denoted O) obtained from a freestanding graphene sample (F1) represents (ωG0,ω2D0) that is not affected by strain or charge doping. (see the text and also Supplementary Fig. S1) The red and blue solid lines represent (ωG,ω2D) of graphene doped with varying density of holes and electrons, respectively, induced by an electrical method (ref. 49). (It is to be noted that there is an equivalent work, ref. 43, which shows some discrepancy from ref. 49. For detailed discussion, see Supplementary Methods A.) The magenta dashed line is an average of experimental (ωG,ω2D) for strain-free graphene with varying density of holes (n) (refs 43,49). The black dashed line represents a prediction of (ωG,ω2D) for charge-neutral graphene under randomly oriented uniaxial stress. (see the text) Inset: decomposition of the effects of hole doping and strain using a vector model. Any given (ωG,ω2D), OP, can be decomposed into OH along the 'strain-free' unit vector, eH for hole doping, and OT along the 'charge-neutral' unit vector, eT for tensile strain (−eT for compressive strain), respectively. eH and eT divide the ωG-ω2D space into the four quadrants (Q1–Q4).

Full size image

Although ωG and ω2D are highly sensitive to both n and , we note that their fractional variation due to n, (Δω2D/ΔωG)n, is very different from that caused by , . First, biaxially strained graphene, either compressive or tensile, shows a fairly large ratio of : three groups reported experimental values of 2.45 (ref. 47), 2.63 (ref. 31), and 2.8 (ref. 30), whereas theory predicted slightly smaller values of 2.25 (ref. 48) and 2.48 (ref. 27). For graphene under uniaxial stress, however, depends on the direction of the strain with respect to the crystallographic axes of graphene. Because of the strain-induced symmetry breaking, the G (2D) mode of graphene under uniaxial stress splits into G− (2D−) and G+ (2D+)26,27. When graphene is strained along the zigzag (arm-chair) directions, =2.05 (1.89) and =2.00 (3.00)29. When the observed G (2D) peaks are resultant of the G− (2D−) and G+ (2D+) peaks that are not resolved because of insufficient splitting, for the zigzag (arm-chair) directions can be approximated as an average of and , 2.02 (2.44). However, as the native strain in mechanically exfoliated graphene can be aligned along any direction between the zigzag and arm-chair axes, is expected to lie in the range of 2.02–2.44, which is in an excellent agreement with 2.2±0.2 obtained from the 8 samples. Second, the effects of n are dependent on the sign of the charges because of their static effects on bond-length and are more pronounced for ωG than ω2D because of the non-adiabatic electron–phonon coupling32,33,35. Hole doping induced by electrical gating leads to a quasi-linearity ((Δω2D/ΔωG)nhole=0.75±0.04) between ωG and ω2D as shown by the red solid line in Fig. 2 (Supplementary Fig. S2 and Supplementary Methods A), whereas Δω2D/ΔωG for electron doping becomes more nonlinear for high charge density as depicted by the blue solid line49. However, we exclude the possibility of electron-doping in the current studies because many charge transport50 and Raman scattering36 studies have observed hole doping predominantly in pristine36 and annealed4 graphene. Thus, based on the negligible charge density in these samples (Methods), we conclude that the linear variations of ωG and ω2D are due to native strain in graphene. Although our results are in better agreement with the scenario of uniaxial strain, the presence of biaxial strain or mix of both could not be excluded viewing the disagreement in experimental and discrepancy between experiment and theory. The presence of native strain also leads to an interesting question on 'strain coherence length', how large the strained domains are or how far the direction of the strain is maintained, which is beyond the scope of the current studies. However, a recent STM study suggests that the length scale can be as small as a few nm for graphene on SiO2 substrates23, which is also consistent with the spatial distribution of in-plane atomic displacements resulting from thermal fluctuation51.

Vector decomposition of and n

Now, it is logical to extract the contribution by or n for a given point in the ωG–ω2D space, P(ωG,ω2D), using a simple vector model as depicted in the inset of Fig. 2: OP=a eT+b eH, where a and b are constants, eT and eH are unit vectors for tensile strain (=2.2±0.2) and hole doping effects ((Δω2D/ΔωG)nhole=0.70±0.05), respectively. The ωG–ω2D space is now divided into four quadrants (Q1–Q4) by eT and eH. As increasing n () of an intrinsic graphene, its values of (ωG,ω2D) will move from O along eH (eT). Whereas Q4 (Q1) is attributed to tensile (compressive) strain combined with hole doping, Q2 and Q3 are not allowed because both of electron and hole doping should lead to increase in ωG. Thus, the variations in ωG and ω2D of the pristine graphene in Fig. 2 are mostly due to strain with negligible charge doping (b~0) and the extent of tensile strain is typically a few times larger than that of compressive one for a given sample. Few pristine graphene sheets of smaller area, however, exhibited non-negligible doping concurrent with strain as will be shown below.

While the assumption of constant (Δω2D/ΔωG)nhole is approximately valid on a wider frequency range, more accurate analysis of and n can be performed using a theoretical prediction35 (blue solid line in Fig. 3) and the experimental data49 for (Δω2D/ΔωG)nhole (red solid line in Fig. 3) as explained below. In addition, the variation of ωG caused by the change in n, thus the Fermi level (EF), becomes nonlinear with respect to ΔEF at low charge density (|n|~1×1012 cm−2) because of the anomalous softening35 of G phonon occurring when (Supplementary Methods B). As a refined approach over the original vector model, Fig. 3a shows a new trajectory of O(ωG0,ω2D0) for hole doping (blue solid line) theoretically predicted35 by considering the phonon anomaly for n ranging from 0 to 2.6×1012 cm−2. With increasing n, O(ωG0,ω2D0) first moves into the blue-shaded region, a forbidden area in Q2, and returns back into the doping-affected area (Q1) represented by the yellow shade at n~1.4×1012 cm−2. As further increasing n, the refined trajectory approaches eH very closely as can be seen in Fig. 3b. In addition, the area of the blue region due to the anomalous softening (ΔωGAnom in Fig. 3a) is very small compared with the yellow region on a larger frequency scale as shown in Fig. 3b. We note, however, that any given point in the blue area or on the eT line, A(ωG,ω2D), can be attributed to either A′ or A″ affected by strain. Thus, unambiguous determination of and n cannot be made for A in the blue shade and the associated errors turn out to be Δ ≤ 0.03% and Δn ≤ 1.5×1012 cm−2. In contrast, B(ωG,ω2D) in the yellow area can be unequivocally interpreted as B′ affected by compressive strain, thus enabling more accurate determination of and n. However, the experimental path (red solid line in Fig. 3a) does not enter the blue area because of the absence of the anomalous softening of G mode as explained in Supplementary Fig. S3. Over the wide range of n (0~1.6×1013 cm−2) shown in Fig. 3b, however, the experimental (red solid line) and theoretical (blue solid line) trajectories agree well with each other and can be well represented by the eH line. Thus, the refined approach in Fig. 3b enables one to disentangle the degree of strain from that of charge doping more accurately. For the pristine graphene shown in Fig. 3b, for example, it can be seen that ranges from −0.2 to 0.4% with n<1.0×1012 cm−2 for the majority of the data points. It should be noted, however, that few experimental data sets43,49 for (Δω2D/ΔωG)nhole available in the literature reveal non-negligible discrepancy (Supplementary Fig. S3) and thus refined experimental data will enhance the accuracy of the proposed analysis.

Figure 3: Refined analysis model considering the logarithmic phonon anomaly.
figure 3

(a) The effects of hole carriers with varying density n on (ωG,ω2D) of graphene. The red solid line presents the experimental trajectory of (ωG,ω2D) as a function of n obtained by Das et al. (ref. 49). The red vertical bars on the red solid line mark n, every 1.0×1012 cm−2. The blue solid line represents a theoretical prediction by Lazzeri et al. reported in ref. 35 that considered the effects of the logarithmic phonon anomaly. We used the relationship that ωG(n)=ωGo+ΔωG(n) and ω2D(n)=ω2Do+Δω2D(n), where (ωGo,ω2Do) corresponds to O of Fig. 2. The numerical values of Δω2D(n) were obtained assuming its linear relationship with ΔωG(n) as shown in Supplementary Fig. S3b. The hole density (n) for the blue solid line ranges from 0 (corresponding to O) to ~2.6×1012 cm−2 as marked by the yellow and magenta diamonds every 0.2×1012 and 1.0×1012 cm−2, respectively. (b) The theoretical and experimental trajectories of (ωG,ω2D) affected by n ranging from 0 to 1.6×1013 cm−2 with Raman map data of pristine (+) and annealed (×) K2. The red solid line is identical to the one in a . The black horizontal bars on the eT line designate uniaxial strain () ranging from −0.6 to 0.3%, each bar representing a step of 0.1%. The uniaxial strain-sensitivity of the G mode, ΔωG/Δ=−23.5 cm−1/%, was calculated from the work by Yoon et al. (ref. 29) considering the splitting of the G mode and random crystallographic orientation of strain. In case of biaxial strain, however, an averaged sensitivity factor of −69.1±3.4 cm−1/% can be used (refs 27,30,47).

Full size image

Thermal modulation of and n

We demonstrate that thermal annealing in vacuum removes the native tensile strain and induces compressive strain. The most prominent change caused by the thermal annealing in Fig. 2 is that the concurrent stiffening of the G and 2D modes. The refined analysis model in Fig. 3b readily leads to quantification of strain and charge density: despite the wide distribution, most of the (ωG,ω2D) points of annealed K2 lie on a line parallel to eT with ~80% of the thermally induced changes in ωG found along the eH axis. This indicates that the O2-induced hole doping activated by annealing4 dominates the spectral changes and n remains relatively constant at (1.4±0.1)×1013 cm−2. The wide distributions in ωG and ω2D, instead, can be attributed to compressive strain (−0.3% ≤≤ 0), which contrasts with the fact that the native strain was mostly tensile in nature. This finding is in agreement with the annealing-induced slippage19,24 and buckling4 of graphene on SiO2 substrates caused by the difference in TECs of both materials52, because the former relieves the tensile stress24 and the latter accompanies compression19. We also note that thermal modulation of strain is sample-specific. For example, the spectral spreads of K1 and K3 in the ωG–ω2D space decreased greatly on annealing, while those of K2 underwent only minor changes as shown in Fig. 2.

Presenting the ωG−ω2D graph obtained from K4 in Fig. 4, we varied the annealing temperature stepwise to determine the critical temperatures where the native strain starts to relax and changes into compression. It can be seen that the pristine graphene has mostly tensile strain with negligible hole doping: each P(ωG,ω2D) of K4 lies on the eT axis in the range between (1,578, 2,670) and (1,582, 2,678). Following the first annealing at 100 °C, most of P(ωG,ω2D) moved into the range between (1,580, 2,676) and (1,585, 2,683), but still mostly along eT. This change demonstrates that heating at 100 °C can be sufficient to remove most of the native tensile strain and compress graphene, simultaneously causing slight hole doping at certain areas. Although subsequent annealing at 150 °C led to more obvious hole doping in addition to further compression, repeated annealing at 200, 250 and 300 °C resulted in less significant variations in P(ωG,ω2D); data for the treatments at 200 and 250 °C are not shown to avoid congestion. Further treatments at 400 and 500 °C, however, induced marked movement of P(ωG,ω2D) to (1,591.5±1.4, 2,693±2.3) and (1,598.4±1.4, 2,698±2.6), respectively. The changes are largely in parallel with eH, indicating emergence of strong hole doping.

Figure 4: Thermally induced variations in (ωG,ω2D) on successive annealing.
figure 4

The data were obtained from Raman mapping of a graphene sample (K4) before and after successive thermal annealing at various temperatures (Tanneal in °C), except those in dark yellow that were obtained from K3 following the one-time annealing at 400 °C (see Fig. 2). The horizontal and vertical error bars for Tanneal=400 & 500 °C represent s.d. in ωG and ω2D, respectively. Each Raman spectrum of the map data was obtained for 10 s.

Full size image

We also note hysteretic effects in annealed graphene samples. Compared with the one-time annealing at 400 °C (K1–K3 in Fig. 2), the sample (K4) that underwent cycles of prior annealing at lower temperatures exhibit much less changes but larger distributions in frequencies (Fig. 4) and linewidths (Supplementary Fig. S4). As the buckling of annealed graphene sheets responsible for the thermally induced hole doping4 is dictated by adhesion and slippage of graphene on silica19,24, it is likely that prior history of thermal treatments affects the buckling behaviours and thus ωG and ω2D. The spectral inhomogeneity increased by the repeated annealing cycles can be attributed to further structural deformation or in-situ reactions with residual gases in the vacuum system during annealing or post-annealing surface reactions6 occurring in the ambient conditions. As annealing is widely used in fabricating graphene transistors50 and preparing graphene samples for various fundamental research18,21, the exact chemical changes made by annealing deserve careful studies in the future. The Lorentzian linewidths of G (ΓG) and 2D (Γ2D) peaks, and the 2D/G peak area ratios (A2D/AG), determined following each annealing cycle, are also consistent with the scenario of thermally induced mechanical transformation concurrent with hole doping (Supplementary Fig. S4 and Supplementary Methods C).

Spatial mapping of and n

In Fig. 5, we demonstrate that the native strain concurrent with spatially varying charge doping can be optically mapped out using the refined vector analysis. The Raman maps obtained from the graphene sample (K5 in Fig. 5a) reveal that ωG, ω2D and ΓG exhibit variations of several cm−1 across an area of 20×15 μm2, respectively, in Fig. 5b–d. Unlike the strain-dominated graphene (K1–K4), all P(ωG,ω2D) from K5 are scattered in Q4 instead of forming a line along eT (Fig. 5g), indicating the coexistence of tensile strain and hole doping. According to the vector analysis in Fig. 2, each point of P(ωG,ω2D) can then be decomposed into and T(ωG,ω2D)n=0, representing phonon frequencies which are not affected by or n, respectively. Shown in Fig. 5e,f are the resulting Raman maps of and ωG, n=0: resorting to Fig. 5g; the former reveals the spatial distribution of the hole density ranging up to 3.5×1012 cm−2, whereas the latter maps out the native strain (−0.03%<<0.17%). Fig. 5e,f also reveal that the long-range distribution of the strain does not necessarily coincide with the native charge distribution. It is also to be noted that and ΓG obey a reciprocal relation that conforms to the theoretical prediction for charge doping43, whereas ωG and ΓG exhibit a much broader distribution due to the coexisting strain (Supplementary Fig. S5). Similar improvement was obtained in the correlation between and A2D/AG (Supplementary Fig. S5).

Figure 5: Decomposition of the effects of strain concurrent with spatially varying charge doping in pristine graphene.
figure 5

(a) Optical micrograph of a graphene sample (K5) under tensile stress concurrent with charge doping. Scale bar, 10 μm. (b) Raman map of ωG. (c) Raman map of ω2D. (d) Raman map for ΓG. (e) Raman map of . (f) Raman map of ωG, n=0. (g) Correlation between ωG and ω2D that were given in b and c. The Raman map data in a–c were obtained from pristine K5 for 10 s/pixel from the area (20×15 μm2) indicated by the blue dashed box in a. The data in e and f were mathematically derived by the simple vector model described in Fig. 2 (see the text).

Full size image

Discussion

The current study shows that most of graphene on silica substrates are mechanically strained and tensile strain is more frequently found than compressive one in a given sample. In this regard, graphene can be envisaged as food wrap that tends to become strained forming ripples when clinging to flat surfaces. Although both tensile and compressive shear stresses can be applied to the membrane, facile buckling along the out-of-plane direction will make compressive strain less likely than tensile one. Interestingly, the extent of the native strain is the larger on average for the larger graphene flake, which may be due to the fact that larger contact area with substrates provides enhanced adhesion to resist slippage caused by in-plane stress. However, thermal perturbation as shown in the current study and varying interactions with other substrates should lead to diverse structural transformation of graphene and other newly discovered two-dimensional materials such as h-BN, MoS2, MoSe2, and so on. We also note that strain dominates the spectral variations over charge doping that has been considered mainly responsible for the spectral irregularities in graphene36. This suggests that the degree of mechanical deformation or charge doping depends on preparation methods. Insignificant chemical doping in our pristine samples could be due to different chemical and structural properties of the substrate surfaces38. In this regard, the presence of tensile strain in graphene may have an influence on the degree of charge doping. A recent STM study revealed that graphene is partly suspended between microscopic hills of the substrates when supported on SiO2 substrates22. Tensile stress is likely to induce localized suspension of graphene that otherwise would largely conform18 to the undulating substrates owing to the van der Waals interaction20. (Supplementary Methods D.) Such semi-freestanding graphene should be less sensitive to charge-doping that occurs via contact with the substrates34,38,40 or through the corrugation-mediated mechanisms4,6. We also note that strain may give rise to charge inhomogeneity through rehybridization of π–σ bonds and vice versa53,54. However, no clear correlation was found as shown in Figs 1 and 5 and Methods, presumably, because of the insufficient spatial resolution and limited sensitivity towards charge density and strain.

The spatial distribution of native strain in mechanically exfoliated graphene samples has rarely been quantified23. Moreover, the strain has not been systematically considered in interpreting Raman spectra owing to the competing effects of extra charges, despite the well-characterized strain-sensitivities of the G and 2D bands26,27,28,29,30,31. Our studies demonstrate that the G and 2D Raman modes of graphene can be highly reliable in determining mechanical strain and charge density even when both coexist. The bimodal sensitivity of both modes, however, requires careful interpretation as suggested in this paper. Although many STM studies revealed structural irregularities such as buckling and strain in graphene18,21,22,23, the method is not practically useful in achieving statistical information on a length scale larger than microns. Moreover, the current studies demonstrate that graphene undergoes sample-specific hysteretic structural deformation on thermal treatments and possibly other external perturbations. As typical microfabrication50 and STM measurements18,21 of graphene and its devices involve various sample treatments such as annealing, transfer to substrates, polymer coating, wetting-drying, and so on., their effects need be considered in interpreting results. Our studies also suggest that not only graphene but also other two-dimensional materials supported on solid substrates are generally susceptible to native strain and thermal deformation owing to zero-dimension along the z axis and different TECs of involved materials.

Despite providing a systematic analysis, however, the current study also shows some limitations. In principle, simultaneous vector decomposition into strain, p-type, and n-type doping cannot be made unambiguously requiring that contribution of at least one component should be known or assumed. The non-zero dispersion of ω2D limits the current approach to the Raman measurement obtained with 514 nm as an excitation source. Follow-up studies are being carried out with other excitation wavelengths. This work is also limited to single-layer graphene, and analysis of few-layer graphene will require separate set of data. Finally, the accuracy of this approach will be directly affected by the spectral accuracy of employed spectrometers that can be tested against O(ωG0 ω2D0).

In conclusion, we have demonstrated that the native strain can be unambiguously determined by Raman spectroscopic analysis notwithstanding the interference from the coexisting charge-doping effects. Most of the pristine graphene sheets deposited onto SiO2 substrates, by the mechanical exfoliation method, were shown to be under in-plane stress with the resulting strain in the range of −0.2~0.4%. The native tensile strain was relieved by thermal annealing at a temperature as low as 100 °C and converted to compressive strain by annealing at higher temperatures, which also induced strong hole doping clearly resolved in the analysis. The proposed analysis should be useful for fast and reliable characterization of strain and excess charges in graphene materials and devices.

Methods

Preparation and treatment of samples

High-quality graphene samples were prepared by the micromechanical exfoliation method50 using kish graphite (Covalent Materials) and adhesive tape (3 M). The Si substrates with 285-nm-thick SiO2 layers were cleaned with piranha solutions before the deposition of graphene. For freestanding graphene samples, substrates with micron-scale circular wells (diameter: 2–7 μm, depth: 5 μm) were employed45. For thermal annealing, samples in a tube furnace evacuated to a pressure of 3 mTorr were heated to a target temperature (Tanneal) within 30 min, maintained at Tanneal for 2 h, and then cooled down to 23 °C for ~3 h.

Raman spectroscopy

The number of layers and crystallinity of the prepared samples were characterized by Raman spectroscopy37. All the Raman spectra were obtained in a backscattering geometry using a X40 objective lens (numerical aperture=0.60) in the ambient conditions. An Ar ion laser operated at a wavelength of 514.5 nm was used as an excitation source. Whereas the spectral width of the instrument response function was determined to be 3.0 cm−1 from the Rayleigh scattering peak, the spectral precision and accuracy were better than 1.0 cm−1 from repeated measurements of Raman standards. (see below for detailed analysis.) For two-dimensional Raman maps, spectra were obtained every 1 μm using an x–y motorized stage. The average power of the excitation laser beam was 1.5 mW that was focussed onto a spot of ~0.5 μm in diameter. No irreversible photoinduced change was detected during the measurements.

Spectral accuracy of the Raman measurements

Although single-grating spectrometers of Czerny–Turner type, including the one (SP2300, Princeton Instruments) employed in the current study, provide high throughput and small footprints55, careful calibration is required to achieve instrument-limited spectral accuracy because of the wavelength dependence of its reciprocal linear dispersion56. More specifically, wavelength (λ) of each charge-coupled device detector pixel needs to be expanded in series of the pixel position (x): λ=λ0+a1(x−x0)+a2(x−x0)2+a3(x−x0)3+...+an(x−x0)n, where an, λ0 and x0 are constants, the centre wavelength and its position on the detectors, respectively. Supplementary Fig. S6a presents the wavelengths of 13 plasma lines from the Ar laser and 3 Hg atomic lines as a function of their pixel positions recorded in the charge-coupled device detector. Although the data seem to be well described by the linear line in blue, the first order calibration with an=0 (n > 1) leads to non-negligible error of −0.15~0.25 nm in wavelength as can be seen in Supplementary Fig. S6b. We note that this amount of deviation translates into Raman shift error of 5.0~−6.8 cm−1 that is even larger than the linewidth of the Rayleigh line (3.0 cm−1). It is also to be noted that a first order calibration using only 3 Hg lines (546.075, 576.961 & 579.067 nm) instead of the above 16 lines generates even larger error up to 0.68 nm or 18 cm−1 across the entire detector area. When the quadratic term was included in the calibration as shown in Supplementary Fig. S6b, however, the deviation remained within ±0.01 nm or ±0.3 cm−1, which is sufficient in accuracy for the employed spectrograph with a focal length of 300 mm and a grating with 1,200 grooves mm−1. This suggests that extra caution needs to be paid when comparing Raman G and 2D frequencies recorded by different spectrographs with modest spectral resolving power. We further tested the accuracy by measuring the G band frequency (ωG) of thick graphite flakes: ωG from 12 different spots out of four different samples was 1,581.5±0.3 cm−1, which turned out to be within ~0.5 cm−1 from the literature values of 1,581–1,582 cm−1 (refs 57,58). Thus, the accuracy of our measurements was conservatively claimed as 1.0 cm−1.

Negligible native charge density in pristine graphene

The scheme of vector decomposition proposed in this article assumes that the variation in (ωG,ω2D) of the employed pristine samples except K5 is mostly due to strain and that the contribution of charge doping is sufficiently small for graphene under tensile stress. This hypothesis is well supported by a few different spectral features of the pristine samples as shown below. Supplementary Figure S7a shows that (ωG,ω2D) of the three samples lies on the black-dashed line (eT) representing graphene affected by strain, but not charge. Moreover, the 2D/G peak area ratio (A2D/AG=5.8±0.3) in Supplementary Fig. S7b remains constant and very close to that (A2D/AG0=6.2±0.2) of the charge-neutral freestanding sample (F1) while ω2D varies by more than 10 cm−1. (see Supplementary Methods E for the optical artefact caused by the substrates that affects the apparent A2D/AG.) As A2D/AG decreases drastically as increasing the charge density (n) for either type of charges43, Supplementary Fig. S7b supports that the presented pristine graphene samples have negligible charge density regardless of the widely varying native strain.

The behaviour of ΓG that is not affected by the optical artefact will be useful in judging the native charge density. As shown in Supplementary Fig. S7c, ΓG of the three pristine samples lies in the range of 13.1±0.7 cm−1, which is only slightly smaller than that of the freestanding sample, ΓG0=13.9±0.2 cm−1. As ΓG is sensitive to low level of charge doping owing to the blockage of the non-adiabatic decay channel of the G phonon, the native charge density in the pristine samples are generally very small34. A quantitative estimation of n according to the model proposed by Berciaud et al.34 leads to a conclusion that ΓG=13.1±0.7 cm−1 translates into |n|<4×1011 cm−2. (equation (1) and Supplementary Fig. S4 of ref. 34 were employed.) This level of charge density is an order of magnitude lower than the variations reported for graphene supported on SiO2 substrates by others34,36.

The distribution of Γ2D as a function of ω2D shown in Supplementary Fig. S7d also supports the assumption that the pristine graphene samples of which (ωG,ω2D) lies on eT in Supplementary Fig. S7a are not affected by significant level of charge doping. Das et al.43 showed that Γ2D increases by ~30% in contrast to decreasing ΓG when |n| is raised to ~2×1013 cm−2 by an electrical method. Several groups reported that Γ2D of supported graphene samples with some level of p-type doping lies in the range of 28–30 cm−1 (refs 36,59), which is significantly larger than that of freestanding graphene (22.5–24 cm−1)34. We confirmed that Γ2D of the charge neutral freestanding graphene (F1) is 23.1±0.2 cm−1, and found that Γ2D of the supported samples in Supplementary Fig. S7d also remains at very small values, 23.5±1.2 cm−1, indicating low level of native charge density.

Additional information

How to cite this article: Lee, J. E. et al. Optical separation of mechanical strain from charge doping in graphene. Nat. Commun. 3:1024 doi: 10.1038/ncomms2022 (2012).