Evaluating arbitrary strain configurations and doping in graphene with Raman spectroscopy

Before introducing our method for strain-doping evaluation, we discuss the approach suggested by Lee et al [20] which has been used in many studies [5, 7, 21–27]. A sketch of the correlation analysis is shown in figure 1(a). The frequencies expected for purely strained and purely doped graphene are plotted as lines resembling a coordinate system in the $\omega_{\rm 2D}-\omega_{\rm G}$ plot. Extracting strain and doping for a potential frequency pair $\lbrace \omega_{\rm G}, \omega_{\rm 2D} \rbrace_{\rm exp}$ is done by projecting the point onto the strain and doping axes. One thereby obtains the frequency shifts induced by doping and strain from which one can calculate the magnitude of strain and doping using reference values. The slope $\Delta \omega_{\rm 2D}/\Delta \omega_{\rm G}$ for pure doping is $<$1 because the G frequency depends much stronger on doping than the 2D frequency [17, 28–30]. This is mostly attributed to a nonadiabatic Kohn anomaly at the Γ point in the phonon dispersion relation [31, 32]. Strain leads to a slope of  ∼2 since the 2D mode is a phonon overtone that shifts with approximately twice the rate of a single phonon excitation (G line).

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The value of the strain-related slope depends sensitively on the strain configuration[18, 33–38]. This leads to large uncertainties when separating the frequency shifts from strain and doping, as is schematically illustrated for an $\omega_{\rm 2D}$, $\omega_{\rm G}$ pair in figure 1(a). Bronsgeest et al noticed the ambiguity when analyzing the Raman spectra of graphene on cobalt [26]. They demonstrated that their strain was determined as 0.4% assuming a biaxial strain configuration, but more than twice as large (1%) for uniaxial strain. The uncertainty in the magnitude of the strain carries over to the charge density of the doping. For example, we obtain a frequency pair of $\omega_{\rm G} = 1579\ {\rm cm}^{-1}$, $\omega_{\rm 2D} = 2619\ {\rm cm}^{-1}$ from figure 2(b), $\updownarrow\updownarrow$. Assuming biaxial strain as the underlying configuration we find a charge density of $p = 3\cdot 10^{13}\ {\rm cm}^{-2}$, see below for details. Assuming uniaxial strain, on the other hand, $\Delta \omega_{\rm 2D}/\Delta \omega_{\rm G}$ varies between 0.8 and 5.3 depending on which peak components were measured and in which crystallographic direction the strain was applied [33]. The corresponding p-type doping ranges from $5\cdot 10^{12}\ {\rm cm}^{-2}$ to $> 10^{14}\ {\rm cm}^{-2}$ [17]. As we will show below, the graphene giving rise to the spectrum in figure 2(b), $\updownarrow\updownarrow$ is uniaxially strained and p doped with a charge density of $5\cdot 10^{12}\ {\rm cm}^{-2}$. The correct value is obtained by a strain slope of 5.3.

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To overcome these ambiguities, we suggest decomposing the effect of strain on the Raman spectrum into its fundamental components, i.e. the frequency shift and the frequency splitting, prior to the correlation analysis. Let us consider a general, unknown strain tensor

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \varepsilon({\bf r}) = \left( \begin{array}{@{}cc@{}} \varepsilon_{xx}( {\bf r}) & \varepsilon_{xy}( {\bf r}) \nonumber \\ \varepsilon_{yx}( {\bf r}) & \varepsilon_{yy}( {\bf r}) \end{array} \right), \nonumber \end{align} \tag{ 1 }$$

with in-plane components $\varepsilon_{ij}$. By solving the secular equation, we calculate the phonon shift rates as [18, 19, 39, 40]

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \Delta \omega_{\rm pn}^\pm = &- \omega_{\rm pn}^0 \gamma_{\rm pn}\left( \varepsilon_{xx} + \varepsilon_{yy} \right) \nonumber \\& \pm \frac{1}{2} \omega_{\rm pn}^0 \beta_{\rm pn} \sqrt{\left( \varepsilon_{xx} - \varepsilon_{yy} \right)^2 + 4 \varepsilon_{xy}^2}, \label{eq:PeakShiftRates} \nonumber \end{align} \tag{ 2 }$$

with the phonon frequency in the absence of strain $\omega_{\rm pn}^0$, the Grueneisen parameter $\gamma_{\rm pn}$ and the shear-strain phonon deformation potential $\beta_{\rm pn}$. Any strain configuration can be decomposed into a hydrostatic component $\varepsilon_{\rm h} = \varepsilon_{xx} + \varepsilon_{yy}$ and a shear component $\varepsilon_{\rm s} = \sqrt{\left( \varepsilon_{xx} - \varepsilon_{yy} \right){\hspace{0pt}}^2 + 4 \varepsilon_{xy}^2}$ (assuming $\varepsilon_{xy} = \varepsilon_{yx}$) [40]. Hydrostatic strain corresponds to an isotropic increase or decrease in the size of the graphene lattice; it leads to a frequency shift $\Delta \omega_{\rm pn}^{\rm h} = -\omega_{\rm pn}^0 \gamma_{\rm pn} \varepsilon_{\rm h}$ determined by the Grueneisen parameter $\gamma_{\rm pn}$ [37]. Note that from its definition $\varepsilon_{\rm h} = \varepsilon_{xx} + \varepsilon_{yy}$ the hydrostatic strain is twice as large as the corresponding biaxial strain $\varepsilon_{\rm b} = \varepsilon_{xx} = \varepsilon_{yy}$. Shear strain corresponds to an anisotropic distortion of the graphene lattice leaving the area of the unit cell unchanged. It leads to a peak splitting $\Delta \omega_{\rm pn}^{\rm s} = \omega_{\rm pn}^0 \beta_{\rm pn} \varepsilon_{\rm s}$ that depends on the shear deformation potential $\beta_{\rm pn}$, while the mean position of the two peak components remains constant [18, 41]. Equation (2) is strictly valid for first-order Raman processes such as the G mode. For higher-order processes like the 2D mode, the electronic structure and its dependence on strain have to be considered as well. The general concept of peak splitting by shear strain and peak shift by hydrostatic strain, however, also applies to the 2D mode, as was shown experimentally and theoretically [33, 37, 42–44]. In the following, the peak components are labeled G⁻ and G$^+$ for the G mode and 2D⁻ and 2D$^+$ for the 2D mode.

We propose an evaluation of arbitrary strain configurations and simultaneous doping in graphene that is based on the phonon frequency shift induced by the hydrostatic strain component:

• 1.  
  To obtain the frequency shift by hydrostatic strain, we eliminate the peak splitting from shear strain $\Delta \omega_{\rm s}$ by considering the mean frequencies

  $$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{eq:MeanFrequencies} \overline{\omega_{\rm G}} = \frac{\omega_{{\rm G}^-}+\omega_{{\rm G}^+}}{2}, \quad \overline{\omega_{\rm 2D}} = \frac{\omega_{{\rm 2D}^-}+\omega_{{\rm 2D}^+}}{2}. \nonumber \end{align} \tag{ 3 }$$

  This leads to a data point in the correlation plot that is only affected by hydrostatic strain and doping (see figure 1(b)).
• 2.  
  The peak shifts from hydrostatic strain $\Delta \omega_{\rm pn}^{\rm h}$ and doping $\Delta \omega_{\rm pn}^{\rm d}$ are separated with a correlation analysis of $\overline{\omega_{\rm G}}$ and $\overline{\omega_{\rm 2D}}$, following [20]. The slope expected for hydrostatic strain is used as a reference (illustrated in figure 1(b)).
• 3.  
  Hydrostatic strain $\varepsilon_{\rm h}$ is calculated from the G mode shift $\Delta \omega_{\rm G}^{\rm h}$ and doping from $\Delta \omega_{\rm G}^{\rm d}$.
• 4.  
  Shear strain $\varepsilon_{\rm s}$ is calculated from the G splitting $\Delta \omega_{\rm G}^{\rm s}$.

The approach requires no knowledge of the strain configuration as an input parameter. The type of doping needs to be known for choosing the correct reference values. For many graphene-material combinations it is available in the literature; alternatively it may be obtained from reference measurements with other techniques.

In the following, we present two examples for how to apply our methodology. These can be viewed as model cases for the strain analysis of graphene on arbitrary substrates. As a first example, we show that circular light polarization is ideal for measuring the mean frequencies $\overline{\omega_{\rm G}}$ and $\overline{\omega_{\rm 2D}}$. We use a Raman setup where linear and circular light polarization can be independently chosen for incoming and scattered light (figure 2(a), see methods for details). Tensile uniaxial strain is induced in exfoliated monolayer graphene flakes by the deflection of a poly(methyl methacrylate) (PMMA) beam (figure 2(a)) [18, 45–48]. We observe a shift of the phonon frequencies to lower wavenumber because of hydrostatic strain and a peak splitting because of shear strain (figure 2(b)). For linearly polarized incoming and scattered light, the $+/-$ components differ strongly in intensity for different polarization configurations (figure 2(b), $\updownarrow\updownarrow$, $\leftrightarrow\leftrightarrow$ and $\updownarrow\leftrightarrow$). From the intensity ratio of G⁻ and G$^+$ peak, we find that the uniaxial strain direction is primarily along the zigzag direction with a misorientation of 7.5% (see methods, equation (4)) [18, 19].

The G⁻ and $G^+$ peak are of equal intensity for circular corotating polarization as expected from the selection rules (figure 2(b), $\circlearrowright\circlearrowright$, see methods for discussion). A similar behavior was observed for the 2D mode. Both G⁻ and G$^+$ modes vanished for circular contrarotating polarization (figure 2(b), $\circlearrowright\circlearrowleft$). For this polarization, the 2D mode split into components of unequal intensity; the 2D⁻ component was consistently more intense than the 2D$^+$ component. A similar behavior was visible for strain along the armchair direction in the graphene lattice (supplementary figure S1 (stacks.iop.org/2DM/5/015016/mmedia)).

Recording Raman spectra with circular corotating light is ideal for measuring the mean frequencies $\overline{\omega_{\rm G}}$ and $\overline{\omega_{\rm 2D}}$ because the two peak components of the G and 2D mode have the same intensity. For linear light polarization, in contrast, one of the peaks may be absent prohibiting the evaluation of the mean frequencies. While equally intense G$^+$ and G⁻ peaks may also be realized with linearly polarized excitation and unpolarized detection [18], this is impossible for the 2D mode because of its peculiar polarization dependence. For example, only the 2D⁻ component was detected in [18] with unpolarized detection, which can be also seen when calculating the average intensity of the spectra $\updownarrow\updownarrow$ and $\updownarrow\leftrightarrow$ in figure 2(b).

We demonstrate the benefit of circular polarization for strain analysis with incrementally increasing uniaxial strain in figure 3(a). Both G and 2D modes were fit by two Lorentzian peaks of same spectral width. We thereby obtained the frequencies of all components, i.e. $\omega_{{\rm G}^-}$, $\omega_{{\rm G}^+}$, $\omega_{{\rm 2D}^-}$ and $\omega_{{\rm 2D}^+}$, which were used to calculate $\overline{\omega_{\rm G}}$ and $\overline{\omega_{\rm 2D}}$. For low strain levels, when no peak splitting was visible, we obtained the mean frequencies from a single Lorentzian fit. This is only possible because the strain-split G and 2D components have equal intensity, i.e. because we use the circular-corotating configuration in Raman scattering.

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In figure 3(b), we plot $\overline{\omega_{\rm 2D}}$ and $\overline{\omega_{\rm G}}$ for varying strain in the correlation plot of 2D and G frequency (green dots, sample U1). The data points follow a linear trend. This shows that we systematically varied the strain at constant doping. The same linear trend was observed for two other graphene flakes, where strain was induced along the armchair direction (samples U2 and U3 in figure 3(b); for spectra see supplementary figure S2). From a linear fit, we obtain the slope for hydrostatic or biaxial strain $\Delta \omega_{\rm 2D}^{\rm h} /\Delta \omega_{\rm G}^{\rm h} = 2.21\pm0.05$. The value is in excellent agreement with theoretical calculations of Mohr et al [38] and measured values on graphene blisters reported by Metten et al [35] and Lee et al [49]. It also agrees with the experimentally determined slope of $2.45 \pm 0.37$ measured by Zabel et al [37] due to its large margin of error.

The peak position for pristine graphene (i.e. no strain and no doping; yellow star in figure 3(b)) was extracted from electrostatic gating measurements by Das et al [17] for the case of zero doping as $\omega_{\rm G}^0 = 1583\ {\rm cm}^{-1}$ and $\omega_{\rm 2D}^0 = 2678\ {\rm cm}^{-1}$. $\omega_{\rm 2D}^0$ was recalculated to account for the different excitation wavelength in our experiment using a 2D mode dispersion of 100 ${\rm cm}^{-1}$ ${\rm eV}^{-1}$ [50]. We note that there is still a slight uncertainty on the spectral positions $\omega_{\rm G}^0$ and $\omega_{\rm 2D}^0$ because of a possible native strain in the measurements of [17]. The G mode frequency at zero doping typically scatters by  ±$1\ {\rm cm}^{-1}$ when comparing similar experiments [17, 28, 29], which is small compared to the frequency shifts by strain and doping observed in our experiments. ${\omega_{\rm G}^0, \omega_{\rm 2D}^0}$ lies outside the measured 2D versus G frequency line in figure 3(b). This is due to p-doping of the graphene flakes by the PMMA substrate [51–53]. $\Delta \omega_{\rm 2D}/\Delta \omega_{\rm G} = 0.55$ for p-type doping in the absence of strain is shown as a red line in figure 3(b) [30]. We used the expected G frequency shift under p-type doping to estimate the doping of the graphene flakes as  ∼$5\times10^{12}$ ${\rm cm}^{-2}$ ([17], see also [30]).

From the frequency shift and splitting of the G line we calculated the hydrostatic and shear strain giving rise to the spectra in figure 3(a). We used a Grueneisen parameter of $\gamma_{\rm G}=1.8$ [18, 34, 35, 37] and shear deformation potential of $\beta_{\rm G} = 0.99$ [18]. The shear strain increases linearly with hydrostatic strain, which is expected for uniaxial stress (figure 3(b) inset). Hydrostatic strain and shear strain under uniaxial tension are connected by the Poisson ratio ν as $\varepsilon_{\rm s} = [(1+\nu)/(1-\nu)] \varepsilon_{\rm h}$. The Poisson ratio is $\nu = 0.33$ for the case of ideal stress-strain transfer from the PMMA beam to graphene [18, 48]. The expected slope agrees well with the data points extracted for sample U1 and U2 (see dots and squares in the inset of figure 3(b)). For sample U3, the shear strain at a given hydrostatic strain is larger than expected. In this experiment the PMMA beam was deflected several times before the measurement was taken, which led to a non-ideal load resulting in a different strain configuration. Extracting hydrostatic and shear strain components is useful for determining the strain configuration and testing the stress-strain transfer from a substrate to graphene. For samples U1 and U2 we observed good agreement of the uniaxial strain calculated from the beam deflection and the uniaxial strain calculated from the measured hydrostatic strain (supplementary figure S3).

So far, we presented one application of our methodology for strain evaluation. Using circular light polarization turned out to be useful for unraveling the local strain and doping levels, especially for weak strain when no peak splitting is detectable. As a second example, we analyze complex and unknown strain configurations on the nanoscale. Exfoliated flakes of graphene were suspended on multiple lithographically-fabricated pairs of gold nanodiscs with diameters of  ∼100 nm, heights of 40–70 nm and gaps of 20–30 nm (figure 4(a)) [54, 55].

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After transfer on top of the nanodimers, the substrate adhesion pulled the graphene into the gap between the two gold nanoparticles. This induced strong local strain with completely unknown configurations (see AFM topography, figure 4(b)). The Raman spectrum from the graphene bridging the dimer gap is strongly enhanced by surface-enhanced Raman scattering [54–56]. The plasmon of the gold nanodimer induces strong electromagnetic near fields that increase the local Raman cross section by three to four orders of magnitude [54]. The near field enhancement provides nanoscale spatial resolution; it also fixes the polarization of the field to be parallel to the nanodimer axis.

Figure 4(c) shows two representative Raman spectra with different levels of intrinsic strain. They belong to the same graphene flake but were recorded on different nanodimers. Both G and 2D modes are split into three components of different intensity. The two peaks with the lowest Raman shift arise from the strained graphene in the interparticle gap [54]. The third peak is only slightly shifted from the peak position expected for pristine graphene (grey dashed lines in figure 4(c)). It arises from the unstrained graphene around the nanodimer that contributes to all Raman spectra without plasmonic enhancement. In the following, we focus on the two peaks that are shifted to lower wavenumbers and thereby analyze the local strain in the dimer cavity. We identify these as the G⁻ and G$^+$ or 2D⁻ and 2D$^+$ components. The spectra show clear signatures of hydrostatic strain (peak shift) and shear strain (peak splitting).

We now turn to a statistical analysis of the strain and doping locally induced in different positions of a graphene flake by various nanodimers. The dimers were individually addressed with a confocal Raman microscope. We recorded the Raman spectra of 60 graphene-covered gold nanodimers, all of which showed surface-enhanced Raman scattering. We calculated the mean frequencies $\overline{\omega_{\rm G}}$ and $\overline{\omega_{\rm 2D}}$ from fits similar to figure 4(c) for each nanodimer. The contributions of doping and hydrostatic strain were separated with a correlation plot of the mean 2D and G frequencies shown in figure 4(d). The peak positions are strongly shifted compared to the reference measurements 1 μm away from each nanodimer (grey triangles) and of pristine graphene (yellow star). To determine the local hydrostatic strain, we needed to analyse the contribution of local charge doping first. We assumed an n-type doping in the graphene bridging the dimer cavity, as previously observed for graphene on gold nanoparticles [57–60]. The $\omega_{\rm 2D}/\omega_{\rm G}$ positions for n-type doping were obtained from a fit of the experimental data in [30] and are plotted as a green line in figure 4(d). The local n-type doping was calculated from the expected G frequency shifts [28]. It reached levels of up to $10^{13}$ ${\rm cm}^{-2}$, see figure 4(e). This is more than one order of magnitude higher than the maximum doping levels of $6\times 10^{11}$ ${\rm cm}^{-2}$ for a comparable plasmonic system reported in [60]. This is expected because we probe the local doping in graphene in the vicinity of the gold nanoparticles, whereas Fang et al [60] probed the charge doping of the entire graphene sheet by transport measurements. Our approach complements the macroscopic measurements with an opportunity to probe local doping levels in an area of  ∼$20\times20$ nm². Interestingly, the local charge doping increased statistically with the hydrostatic strain (figure 4(e), calculated using a Grueneisen parameter of $\gamma_{\rm G} = 1.8$ [18, 34, 35, 37]). We expect that with increasing local strain, graphene is pulled deeper into the dimer cavity. This leads to a stronger interaction between graphene and the gold nanostructure and explains the increase in doping with strain.

The nanodimers induced strong local strain in the graphene lattice with a hydrostatic component ranging from 1% to 2.5% (figure 4(e)). The shear component was calculated from the G splitting using a shear deformation potential of $\beta_{\rm G} = 0.99$ [18]. With a magnitude from 0.75% to 1.8% it was weaker than the hydrostatic component (figure 4(e)). This differs from uniaxial strain, where the shear component was twice as large as the hydrostatic component (figure 3(b) inset) and biaxial strain where the shear component vanishes. Shear strain and hydrostatic strain are largely uncorrelated for the nanodimers (figure 4(e)), from which we conclude that the nanodimers induce different strain configurations in the suspended graphene. Our methodology for strain evaluation is even applicable for this complicated case because it does not require any assumption of the strain configuration as an input parameter.

Finally, we demonstrate our analysis of local strain and doping for several different graphene flakes covering arrays of gold nanodimers. Representative Raman spectra of three graphene flakes (samples D1–D3) are shown in figure 5. While the intensity ratio of the 2D⁻ and 2D$^+$ components is similar in all spectra, the G⁻/G$^+$-intensity ratio varies from one flake to the other. This is explained by the different lattice orientation of the graphene flakes with respect to the direction of the local shear-strain component (see methods section) [18, 19]. Based on symmetry arguments as discussed in detail in [54] we assume that the local shear strain is directed along and perpendicular to the dimer axis. We determined the angle between the zigzag direction of graphene and the dimer axis for 86 nanodimers covered by five different graphene flakes (supplementary figure S4(a)). The estimated angles range from $(9\pm2){\hspace{0pt}}^\circ$ for sample D1 to $(19\pm1){\hspace{0pt}}^\circ$ for D3. We also repeated the correlation analysis of the 2D and G mean frequencies for all graphene flakes (supplementary figure S4(b)). The peak positions nicely follow the same trend as for sample D1 in figure 4(d). We found statistically the same strain and doping levels in all graphene flakes.

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Our methodology is generally also applicable to other 2D materials than graphene. It requires two Raman modes $\omega_{\rm pn1}$ and $\omega_{\rm pn2}$ with a different shift $\Delta \omega_{\rm pn1}^{\rm h}/\Delta \omega_{\rm pn2}^{\rm h}$ induced by hydrostatic strain than the shift $\Delta \omega_{\rm pn1}^{\rm d}/\Delta \omega_{\rm pn2}^{\rm d}$ induced by doping. $\Delta \omega_{\rm pn1}^{\rm d}/\Delta \omega_{\rm pn2}^{\rm d}$ is particularly large for graphene because of the nonadiabatic Kohn anomaly of the G mode [31, 32]. The separation of frequency shifts from strain and doping is, however, still feasible as was shown in recent works for MoS₂ [61, 62].

For the measurements under uniaxial strain, monolayer graphene flakes were obtained by micromechanical exfoliation and deposited on a flexible PMMA beam. Prior to graphene transfer, the PMMA substrate was spin coated with SU8 photoresist (SU8 2000.5, MicroChem). The samples were soft-cured at 80 °C for 30 min and exposed to UV radiation (366 nm, 30 s). Graphene was transferred onto the samples with the scotch-tape method. Finally, the samples were covered with PMMA (1% in anisole) to improve the strain transfer efficiency. Uniaxial stress was induced by bending the PMMA beam with a jig (see e.g. [45, 46]). We assume that the doping does not change when inducing strain in graphene by deflection of the PMMA substrate, as we observed no systematic linewidth change of the G peak components with increasing strain. Moreover it was recently shown by Kelvin probe force microscopy measurements that the PMMA substrate most likely acts as a charge reservoir maintaining the doping level constant [64]. To avoid heating and accompanying structural change of the graphene-PMMA interface, we used laser powers below 150 μW for Raman characterization (532 nm laser excitation).

The graphene-covered gold nanodimers were fabricated using the same procedure as described in [54]. In short, gold nanodisk dimers were produced with electron-beam lithography, followed by metallization (5 nm Cr or Ti and 40 to 80 nm Au) and lift-off in an ultrasonic bath. Micromechanically cleaved large flakes of single-layer graphene were then transferred on top of the plasmonic nanostructures using a dry transfer method. The spacing between the nanodimers beneath the graphene flakes was much larger than the laser-spot size. This enabled us to record Raman spectra of graphene interacting with an individual nanodimer.

Raman spectra were recorded with a Horiba XploRA single-grating confocal Raman spectrometer, equipped with a 1200 groves per mm grating, which leads to a spectral resolution of 2–3 ${\rm cm}^{-1}$. The spectrometer was calibrated with the Raman response of diamond. 532 nm laser excitation was used for measurements on graphene under uniaxial stress ($2\times120$ s acquisition time, 130 μW laser power, spot size of 340 nm). Reference spectra of the PMMA substrate were recorded with the same parameters next to the graphene flakes and subtracted from the Raman spectra of graphene. The graphene-covered gold nanodimers were characterized with 638 nm laser excitation ($2\times 120$ s acquisition time, 280 μW laser power, spot size of 605 nm); at this wavelength large plasmonic enhancement is expected [54]. The laser was focused on each plasmonic nanodimer using a piezo stage and steps of 100 nm in x-, y- and z direction until a maximum intensity of the 2D mode was achieved. Linear light polarization was chosen along the dimer axis. Reference spectra were recorded 1 μm away from each nanodimer.

The Raman intensities of the G components can be calculated with the macroscopic theory of Raman scattering as $I_{{\rm G}^\pm} \propto \vert {\bf e}_{\rm i} R_{{\rm G}^\pm} {\bf e}_{\rm s}\vert^2$. ${\bf e}_{\rm i}$ is the polarization of the incoming light and ${\bf e}_{\rm s}$ of the Raman-scattered light. $R_{{\rm G}^-}$ is the Raman tensor for the G⁻ peak and $R_{{\rm G}^+}$ for the G$^+$ peak [19]. For uniaxial stress applied at an angle ϑ with respect to the crystallographic zigzag direction in graphene, the intensities are calculated with the Raman tensors given in [19] as

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle I_{{\rm G}^+} &\propto d^2 \cos{\left( \varphi + \psi + 3 \vartheta \right)}^2, \nonumber \\ I_{{\rm G}^-} &\propto d^2 \sin{\left( \varphi + \psi + 3 \vartheta \right)}^2, \label{eq:GmodeIntLinLightPol} \nonumber \end{align} \tag{ 4 }$$

where d is the Raman tensor component of the G mode for unstrained graphene. φ is the angle of the incoming light and ψ the angle of the Raman scattered light to the strain axis. From equation (4) it is apparent that $I_{{\rm G}^-}$ and $I_{{\rm G}^+}$ can be very different; for $\varphi + \psi + 3\vartheta = n \cdot \pi/2$ (n integer) one of the components vanishes. The situation is different for circular light polarization with polarization vectors [65]

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {\bf e}_\circlearrowright = \frac{1}{\sqrt{2}} \left( \begin{array}{@{}c@{}} 1 \nonumber \\ - {\rm i} \end{array} \right), \quad {\bf e}_\circlearrowleft = \frac{1}{\sqrt{2}} \left( \begin{array}{@{}c@{}} 1 \nonumber \\ {\rm i} \end{array} \right). \nonumber \end{align} \tag{ 5 }$$

For circular corotating polarization (i.e. $\circlearrowright\circlearrowright$ or $\circlearrowleft\circlearrowleft$ as combinations for ${\bf e}_{\rm i}$ and ${\bf e}_{\rm s}$) we obtain $I_{{\rm G}^-} \propto d^2$ and $I_{{\rm G}^+} \propto d^2$; the G mode splits into components of equal intensity. When using circular contrarotating polarization (i.e. $\circlearrowright\circlearrowleft$ or $\circlearrowleft\circlearrowright$), both G mode components vanish.

The polarization dependence of the 2D mode is more complex because it is a second-order Raman process. The intensities of the 2D⁻ and 2D$^+$ components are determined by double-resonance processes and the simple theoretical treatment that was used for the G mode cannot be applied [33, 42, 44]. If incoming- and outgoing light are linearly polarized along the strain axis ($\updownarrow\updownarrow$), $I_{{\rm 2D}^-}>I_{{\rm 2D}^+}$. The opposite case $I_{{\rm 2D}^-}< I_{{\rm 2D}^+}$ is observed for light polarizations perpendicular to the strain axis $(\leftrightarrow\leftrightarrow)$. In general there is a complicated functional dependence of $I_{{\rm 2D}^-}$ and $I_{{\rm 2D}^+}$ on the polarization directions of incoming and outgoing light [33]. A reliable fit of the 2D peak components is only possible for a large peak splitting induced by shear strain. For circular light polarization, the overall peak intensity is determined by the selection rules of the Raman process. The 2D mode has symmetry $A_{1g}\oplus E_{2g}$ [66]. Under circular corotating polarization processes with $A_{1g}$ symmetry vanish whereas for circular contrarotating polarization processes with $E_{2g}$ symmetry are zero [65]. This explains the non-vanishing 2D mode for both, circular corotating and contrarotating light polarization.

For Raman measurements with circular light polarization, we used an optical setup as schematically depicted in figure 2(a). The laser beam with initially linear light polarization was directed through a quarter-wave plate ($\lambda/4$) to induce circular light polarization. The light was focused onto the sample with an optical microscope which was also used to collect the backscattered light. After the beam splitter (BS) and notch filter (NF), a second quarter-wave plate switched back to linear light polarization. With an analyzer, we selected either the circular-corotating or the circular-contrarotating polarization in the Raman experiment. In front of the spectrometer entrance we placed a half-wave plate ($\lambda/2$) to ensure that the Raman-scattered light entered the spectrometer always with the same light polarization because the spectrometer response depends on polarization. To realize co- and contrarotating configurations, it is also possible to use a single $\lambda/4$ plate between beam splitter and microscope objective if the setup permits [65]. Combinations of linear light polarization were realized by using $\lambda/2$ plates instead of $\lambda/4$ plates.