Effect of uniaxial strain on the optical Drude scattering in graphene

We used commercially available CVD-grown monolayer graphene (Graphenea) transferred onto a 250 μm thick PET slab. This substrate is sufficiently transparent and flexible to allow precise and reproducible control of strain while doing optical and scanning-probe measurements. Large-area square samples ($11\times 11$ mm²) were used to ensure the uniformity of the strain in the middle of the sample and to avoid any diffraction effects at the lowest optical frequencies investigated.

Tensile strain was applied to the graphene by bending the substrate, with graphene on the convex side, using a home-made micrometer-based mechanical device, similar to the the method used in [9, 12], as shown in figures 1 and 2. The nominal applied strain (assuming that graphene follows the substrate without slipping) is calculated as s  =  d/2r, where d is the substrate thickness substrate and r is the bending radius. Assuming a uniform bending, the micrometer displacement x and the strain are related by the formula $x=L\left[1-(d/sL)\sin (sL/d)\right]\approx {{L}^{3}}{{s}^{2}}/\left(6{{d}^{2}}\right)$, where L is the substrate width, as shown in figure 1(b). The same bending method was used in all measurements presented in this paper.

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The atomic force microscopy (AFM) and Kelvin probe force microscopy (KPFM) measurements were performed with Ti/Ir coated tips (resonance frequency  ∼70 KHz) under controlled humidity ($30\pm 3$%) at room temperature. These measurements proceed in two-pass scan mode: in the first pass, a normal topography height image is obtained in tapping mode; in the second pass, the cantilever is lifted 30 nm above the surface and follows the previously obtained topography to record the contact potential difference (CPD) between the graphene and the tip.

The Raman measurements were performed at the excitation wavelength of 514.5 nm (2.41 eV) with a $50\times$ objective. The spot size was 1 μm and the laser power less than 2 mW to avoid laser-induced heating. Because of the dominance of PET-induced Raman peaks over the phonon G peak of graphene (∼1580 cm⁻¹), we have focused our analysis on the shift of the double-resonance 2D peak (∼2700 cm⁻¹), which is moreover very sensitive to strain [8–10].

Infrared transmission spectra were measured using a standard FT–IR spectrometer (Bruker Vertex 70 v) over the 40–700 cm⁻¹ (1.2–21 THz) frequency range, using a globar light source and a liquid-He cooled Si bolometric detector. A spot size of 2 mm was used to keep light well in the middle of the sample. Absolute transmission at every strain value was normalized against the empty sample holder at the same micrometer position x in order to keep the measurement conditions exactly the same. Spectra for both graphene/PET samples, ${{T}_{\text{g}}}\left(\omega \right)$ and bare reference substrates ${{T}_{\text{s}}}\left(\omega \right)$ were obtained separately for incident light polarized parallel (${{E}_{\parallel}}$) and perpendicular (${{E}_{\bot}}$) to the strain direction as sketched in figure 2.

All measurements were performed at room temperature. They were repeated during several strain cycles and showed excellent reproducibility.

In figures 3(a) and (b) we show a typical $4\times 4$ μm² topographical micrograph and contact potential difference map renormalized around the mean value of the same region of unstrained graphene. The topography, representative of measurements performed at multiple different locations on the sample, reveals a clean graphene surface almost free from chemical residues, and is characterized by wrinkles and folds of different heights and orientations. For the CPD, while absolute values could not be determined in the absence of a reference calibration, we note that only very small variations are observed across the graphene surface, with a root mean square (${{V}_{\text{rms}}}$) value of 3.4 mV, which is limited by the resolution of KPFM setup at ambient conditions, and a peak-to-peak variation of 50 mV between the highest (light beige) and lowest (dark brown) contrast areas. These high contrast areas appear similar to charge puddles observed with high resolution SET surface potential measurements [11], but do not appear to be strongly correlated with the presence of wrinkles or folds seen in the topography. When the same area is measured under higher nominal strain ($s=1.2 \%$, 1.6% and 2%) no significant evolution of either the CPD ${{V}_{\text{rms}}}$ values or the peak-to-peak variation is observed. However, as can be seen in figure 3(c) for the topographical micrograph at 2% nominal strain, a number of small folds and wrinkles in the graphene appear to have relaxed. To quantify this relaxation behavior, we performed a differential analysis of the topographical images using an in-house developed drift correction algorithm [17], which allows changes from one scan to another of the same area to be determined with sub-nm precision. Figure 3(d) shows the resulting differential image, with the dark contrast features corresponding to all the wrinkles and folds that relaxed after the application of 2% nominal strain. They show decreased height with respect to the unstrained graphene imaged in figure 3(a). We can see that relaxation occurs primarily for wrinkles aligned perpendicular to the axis of applied strain, indicated by the red arrow, as would be logically expected. The only clear increase in the height of a fold, occurring in the star-like feature at the upper right of the image, was observed for a wrinkle aligned parallel to the the axis of applied strain, and accompanied relaxation in the rest of the feature. These strong variations of the nanoscale topographic landscape through relaxation under the applied strain imply that the effective strain transferred to the sample might be significantly smaller than the nominal strain.

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The actual amount of strain applied to graphene was therefore calibrated with Raman measurements. At 2% nominal strain, we observed a 52 cm⁻¹ red shift caused by the elongation and weakening of C–C bonds, displacing the 2D peak from 2697 to 2645 cm⁻¹ as shown in figure 4(a). The solid black and green lines show the experimental data measured at 0% and 2% nominal strain, respectively, fitted with a Lorentzian function (red dashed line for 0% and blue dashed line for 2%). The observed red shift in the position of the 2D peak is smaller than the values of 64 cm⁻¹ per % uniaxial strain reported by Mohiuddin et al on a mechanically exfoliated single layer graphene flake free of wrinkles and grain boundaries associated with CVD-grown material [9]. In our case, we must therefore presume that the observed relaxation of folds/wrinkles leads to a sliding at the graphene-PET interface [7], and use the single-grain data as a reference to calibrate the effective versus nominal strain in our measurements. Figure 4(b) shows the 2D peak position (black circles) as a function of nominal strain s (bottom abscissa) and the corresponding effective strain (upper abscissa). The corresponding values of the red shift $\Delta {{\omega}_{2\text{D}}}$ were extracted from a fit of the data (red line). is then obtained using the relation $\Delta {{\omega}_{2\text{D}}}/\epsilon =64$ cm⁻¹ per % uniaxial strain, based on the exfoliated graphene results [9].

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As the PET substrate demonstrates strong photoelastic effects (as discussed in the supplementary information (stacks.iop.org/TDM/4/025081/mmedia)) it is necessary to fully characterize it optically as a function of the bending radius before the extraction of the intrinsic graphene properties. To this end we used a reference graphene-free substrate of the identical thickness, taken from the same batch. In figure 5 the absolute far-infrared transmission spectra ${{T}_{\text{s}}}\left(\omega \right)$ are presented for the polarization of light parallel (a) and perpendicular (b) to the strain axis. The spectra for a perfectly flat and a maximally bent substrate, corresponding to the effective graphene strain of 0.79%, are shown by the dashed blue and green lines, respectively. One can see that the substrate is sufficiently transparent everywhere except in the region between 300 cm⁻¹ and 540 cm⁻¹, where strong phonon absorption is present. The observed minima at 138 cm⁻¹ and 632 cm⁻¹ are due to weaker optical phonons in PET, which do not fully block the transmission. Periodic oscillations due to the Fabry–Perot interference in the substrate are seen, as expected, in the spectral regions of high transparency. For both polarizations, applying strain results in a frequency-dependent reduction of the transmission due to the above-mentioned photoelastic effects. Applying a Kramers–Kronig analysis [18] to these data allows us to extract the spectra of the complex refractive index $N=\sqrt{{{\epsilon}_{\text{PET}}}}$ for each polarization, at each value of the applied strain, as detailed in the supplementary information.

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The black and red solid curves in figure 5 represent the absolute transmission ${{T}_{\text{g}}}\left(\omega \right)$ of graphene on PET for zero and the highest effective strain (0.79%), respectively. The presence of the graphene monolayer noticeably reduces the transmission at low frequencies (below 200–300 cm⁻¹) with respect to the bare substrate, while it does not essentially affect the transmission at high frequencies. Qualitatively, this indicates that the transmission reduction is due to the absorption by Drude carriers in graphene, as observed previously [16, 19, 20].

In order to see the Drude absorption more clearly, in figure 6 we plot the extinction spectra $1-{{T}_{\text{g}}}\left(\omega \right)/{{T}_{\text{s}}}\left(\omega \right)$, where the effects of the substrate are largely suppressed as compared to the absolute transmission. Furthermore, in order to avoid any spurious effects originating from a possible light depolarization in the substrate, from this point on we discuss the polarization-averaged, instead of polarization-resolved, spectra. Figure 6(a) shows the extinction at zero strain (solid blue line), where a pronounced Drude peak is seen, with the maximum extinction reaching about 25% at the lowest frequencies. For an ultrathin film on a substrate the extinction is given by:

$$\begin{eqnarray}&&1-\frac{{{T}_{\text{g}}}}{{{T}_{\text{s}}}}=1-\frac{1}{|1+{{f}_{\text{s}}}{{Z}_{0}}\sigma /2{{|}^{2}}},\end{eqnarray} \tag{ 1 }$$

where ${{Z}_{0}}=4\pi /c\approx$ 377 $\Omega$ is the vacuum impedance, $\sigma ={{\sigma}_{1}}+\text{i}{{\sigma}_{2}}$ is the two-dimensional optical conductivity of graphene and

$$\begin{eqnarray}&&{{f}_{\text{s}}}=2\frac{(N+1)+(N-1)\exp (2\text{i}\omega Nd/c)}{{{(N+1)}^{2}}-{{(N-1)}^{2}}\exp (2\text{i}\omega Nd/c)}.\end{eqnarray} \tag{ 2 }$$

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The substrate factor ${{f}_{\text{s}}}$ reduces the extinction as compared to the case of free standing graphene, where it is equal to one, and adds some Fabry–Perot oscillations, clearly seen in the experiment. The Drude model for the optical conductivity reads as [19]:

$$\begin{eqnarray}&&\sigma \left(\omega \right)=\frac{D}{\pi}\frac{\tau}{1-\text{i}\omega \tau},\end{eqnarray} \tag{ 3 }$$

where D is the Drude weight and τ is the scattering time. The red curve in figure 6(a) presents the best fit using the equations (1), (2) and (3). One can see that overall the model works well: apart from perfectly reproducing the shape of the Drude peak, it also generates Fabry–Perot oscillations similar to the experimental curve. The amplitude of the oscillations is difficult to match using the model due to various dephasing effects in the substrate. Nevertheless, from this fit we can rather accurately extract the Drude parameters $D=1.39\pm 0.015$ k${{ \Omega }^{-1}}$ ps⁻¹, and $\tau =0.28\pm 0.005$ ps. We emphasize that the ability to measure D and τ independently is one of the advantages of the infrared spectroscopy as compared to the DC transport measurements. Within the theory of non-interacting Dirac fermions [21], the Drude weight is determined solely by the absolute value of the Fermi energy ${{E}_{\text{F}}}$ with respect to the Dirac point: $D=\left({{e}^{2}}/{{\hbar}^{2}}\right){{E}_{\text{F}}}$, where $\hbar$ is the reduced Planck constant and e is the elementary charge. Therefore, this measurement allows us to determine the absolute Fermi energy ${{E}_{\text{F}}}={{\hbar}^{2}}D/{{e}^{2}}=231\pm 2$ meV (the sign cannot be formally deduced in our case without extra experiments) and the scattering rate $\hbar /\tau =14.8\pm 0.2$ meV. By adopting a standard value for the Fermi velocity $v\approx {{10}^{6}}$ ms⁻¹, we can estimate the electronic mobility $\mu =e{{v}^{2}}\tau /{{E}_{\text{F}}}\approx 2000$ cm² V⁻¹ s⁻¹ and the carrier concentration $n=E_{\text{F}}^{2}/\left(\pi {{\hbar}^{2}}{{v}^{2}}\right)\approx 4\times {{10}^{12}}$ cm⁻². These quantities are consistent with typical values found in CVD graphene under ambient conditions [16, 19, 22].

The effect of strain on the extinction spectra is shown in figure 6(b), where the blue and green curves correspond to $\epsilon =0 \%$ and 0.79%, respectively. One can see that the Drude peak somewhat flattens under strain, showing a smaller (higher) value below (above)  ∼120 cm⁻¹. This indicates that the scattering rate, which determines the width of the Drude peak, increases with strain, although a detailed analysis is required to quantify this effect, as carried out below. One can also notice that the amplitude of the Fabry–Perot oscillations at high frequencies decreases with strain, which is likely due to the substrate bending, not included in the flat-multilayer model, which may suppress the phase coherence between internally reflected rays.

Before proceeding with a detailed analysis of the effect of strain, it is convenient to convert the extinction spectra into the optical conductivity of graphene using a model-independent Kramers–Kronig constrained analysis [18]. Figure 7(a) shows the extracted spectrum (blue line) as well as the best Drude fit (red line) of the real part of the optical conductivity, ${{\sigma}_{1}}\left(\omega \right)$ at zero strain, normalized to the universal value ${{\sigma}_{0}}={{e}^{2}}/4\hbar$ [23, 24]. The model matches the experiment well, apart from some remnant Fabry–Perot oscillations in the experimental curve, which appear due to the already-mentioned difficulty in precisely reproducing the oscillations in the extinction spectra. As one can see from figure 7(b), the Drude model works equally well at the maximum strain. The zero-frequency limit of the optical conductivity ${{\sigma}_{\text{dc}}}=D\tau /\pi$ (obtained from the fit) shows a reduction from 20 ${{\sigma}_{0}}$ to 18 ${{\sigma}_{0}}$, i.e. by about twice the total optical conductivity of graphene in the infrared and visible ranges [23, 24]. Thus the measured effect of strain on the optical conductivity in the far infrared regime is about two orders of magnitude larger than in the visible range [12]. In order to emphasise the effect of strain, in figure 7(c) we plot the experimental frequency-dependent differential conductivity $\Delta {{\sigma}_{1}}={{\sigma}_{1}}\left(0 \% \right)-{{\sigma}_{1}}\left(0.79 \% \right)$ (dark yellow curve) and the corresponding difference between the Drude fits (red curve), where the remnant Fabry–Perot oscllations are Fourier-filtered. In spite of the unavoidable noise in the differential spectra, the match between experiment and the Drude model is obvious and a sign inversion is clearly observed at about 120 cm⁻¹. This fully justifies our preliminary conclusion that the Drude peak broadens under strain.

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Using the same procedure, we extracted the Drude parameters independently at each value of strain. Furthermore, this was done separately for measurements carried out while increasing and decreasing the applied strain. Solid symbols in figures 8(a) and (b) show the Drude weight ${{\hbar}^{2}}D/{{e}^{2}}$ and the scattering rate $\hbar /\tau$, respectively, as a function of the effective strain for both phases of the cycle (red and blue). The dashed lines of the corresponding color are the linear fits. The central observation of this paper is that the Drude weight remains constant within the experimental error bars, while the scattering rate increases significantly with , by more than 10% at only 0.79% of strain. It is important to note that the behavior of both the Drude weight and the scattering rate is reproducible and reversible.

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Apart from monitoring the Drude parameters, it is interesting to follow the strain dependence of the optical conductivity itself. Figure 8(c) shows the measured value of the optical conductivity, ${{\sigma}_{1}}$, normalized to zero strain and spectrally averaged between 40 and 100 cm⁻¹ (1.2–3 THz), i.e. the part of the experimental range where it decreases most significantly with . One can see that the optical conductivity decreases reversibly by about 5% for the maximum applied strain, which is driven in the present case, as we have shown, by the strain dependent Drude scattering rather than the Drude weight.

The effect of uniaxial strain on the crystal lattice and the electronic band structure of graphene was extensively studied theoretically [4, 14, 15, 25–27]. The nearest neighbor hopping parameter t for the π-bands depends strongly on the bond distance a: $t/{{t}_{0}}\approx \exp \left[-3.37\left(a/{{a}_{0}}-1\right)\right]$, where ${{t}_{0}}\approx$ 3 eV and ${{a}_{0}}\approx 0.14$ nm are the equilibrium values. Therefore, the major consequence of the lattice deformation is to modify the three effective hopping parameters and subsequently to shift and uniaxially deform each of the six Dirac cones located in the K-points of the Brillouin zone. This gives rise to elliptical Fermi pockets elongated along the strain axis, regardless of the direction of the strain with respect to the crystal axes [25, 26] (figure 9). Thus, a uniaxial deformation is theoretically expected to anisotropically renormalize the Fermi velocity, producing two different values parallel (${{v}_{\parallel}}$) and perpendicular (${{v}_{\bot}}$) to the strain, with ${{v}_{\parallel}}<{{v}_{\bot}}$. This also modifies the density of states at the Fermi level:

$$\begin{eqnarray}&&g\left({{E}_{\text{F}}}\right)=\frac{2{{E}_{\text{F}}}}{\pi {{\hbar}^{2}}{{v}_{\parallel}}{{v}_{\bot}}}\end{eqnarray} \tag{ 4 }$$

and makes the Drude weight anisotropic [13]:

$$\begin{eqnarray}&&{{D}_{\parallel,\bot}}=\frac{\pi {{e}^{2}}g\left({{E}_{\text{F}}}\right)}{2}v_{\parallel,\bot}^{2}.\end{eqnarray} \tag{ 5 }$$

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Under small deformations, as in our experiment, we can apply a linear expansion for the strain-dependent Fermi energy:

$$\begin{eqnarray}&&\frac{{{E}_{\text{F}}}}{{{E}_{\text{F},0}}}\approx 1+\kappa \epsilon,\end{eqnarray} \tag{ 6 }$$

and the (anisotropic) Fermi velocity:

$$\begin{eqnarray}&&\frac{{{v}_{\parallel,\bot}}}{{{v}_{0}}}\approx 1+{{\nu}_{\parallel,\bot}}\epsilon,\end{eqnarray} \tag{ 7 }$$

where the index 0 refers to zero strain. As we show in the supplementary information, based on the existing ab initio calculations [15], the Fermi-velocity expansion coefficients are equal to ${{\nu}_{\parallel}}\approx -1.8$ and ${{\nu}_{\bot}}\approx +0.5$ and are independent of the strain orientation with respect to the crystal axis. This allows us to ignore the random orientation of the crystalline grains present in CVD graphene.

Using equations (4)–(7), we can obtain a linear expansion for the density of states:

$$\begin{eqnarray}&&\frac{g\left({{E}_{\text{F}}}\right)}{{{g}_{0}}\left({{E}_{\text{F},0}}\right)}\approx 1+\left(\kappa -{{\nu}_{\parallel}}-{{\nu}_{\bot}}\right)\epsilon \end{eqnarray} \tag{ 8 }$$

and the (anisotropic) Drude weight:

$$\begin{eqnarray}&&\frac{{{D}_{\parallel,\bot}}}{{{D}_{0}}}\approx 1+\left(\kappa \pm {{\nu}_{\parallel}}\mp {{\nu}_{\bot}}\right)\epsilon.\end{eqnarray} \tag{ 9 }$$

One can see that ${{D}_{\parallel}}$ decreases, while ${{D}_{\bot}}$ increases with strain, which generates a Drude-weight dichroism $\left({{D}_{\parallel}}-{{D}_{\bot}}\right)/{{D}_{0}}\approx 2\left({{\nu}_{\parallel}}-{{\nu}_{\bot}}\right)\epsilon \approx -4.6\epsilon$ determined solely by the anisotropy of the Fermi velocity and insensitive to the Fermi level. Measuring the anisotropy of the Drude weight would therefore directly probe the anisotropy of the Fermi velocity. Unfortunately, the photoelastic effects in the substrate did not allow us to measure the intrinsic dichroism of graphene with a sufficient precision, as discussed before. On the other hand, the average Drude weight $D=\left({{D}_{\parallel}}+{{D}_{\bot}}\right)/2\approx {{D}_{0}}\left(1+\kappa \epsilon \right)$ is measured very accurately, since the substrate depolarization does not add to the total optical absorption. This value depends only on the shift of the Fermi energy and is not affected by the anisotropy. Therefore, our observation of a constant (within error bars) Drude weight (figure 8(a)) indicates that the Fermi level is essentially strain-independent: $|\kappa |<0.1$.

We note that this observation is a rather non-trivial one. Indeed, as the density of states varies with strain [7, 14, 25, 28], one has to assume a strain-induced charge transfer between graphene and the environment (in particular, the substrate) in order to maintain ${{E}_{\text{F}}}$ constant. It is therefore likely that the PET substrate acts as a charge reservoir maintaining the chemical potential in graphene at a fixed level. Interestingly, our KPFM experiments show a rather small spatial variation of the chemical potential (about 3.4 meV), which may be also due to the stabilizing effect of the substrate on the chemical potential.

Next we discuss the surprisingly strong strain-induced increase of the scattering rate shown in figure 8(b). The data are well described by a linear dependence (dashed lines):

$$\begin{eqnarray}&&\frac{\hbar /\tau}{{{\left(\hbar /\tau \right)}_{0}}}\approx 1+c\epsilon,\end{eqnarray} \tag{ 10 }$$

where the values $c=12.5\pm 1$ and $14.2\pm 1$ are found for the increasing and decreasing strain respectively. By taking the average value, we obtain the linear expansion coefficient $c=13.4\pm 0.7$. Speaking differently, the scattering rate increases by about 13.5 % per 1% of the applied strain, which indicates that the Drude scattering is highly sensitive to the mechanical deformation.

In a recent theoretical report [28], the effect of strain-dependent density of states on the scattering rate was calculated in a connection to transport measurements, and a significant increase of $\hbar /\tau$ with increasing strain was found. While the predicted effect was characterized as giant, the results of [28] are in fact smaller than what we observe here.

As it is widely discussed in the literature [28–38], the charge scattering in graphene is affected by short-range point defects (pd), long-range charged impurities (ci), acoustic phonons (ap), surface-phonons (sp) in the substrate and the graphene grain boundaries. According to the Matthiessen's rule, the total scattering rate is a sum of these respective contributions: $\hbar /\tau ={\sum}_{i}\hbar /{{\tau}_{i}}$. In principle, each of them may contribute to the total strain dependence. We can therefore attribute to each scattering channel (i) its respective strain-induced increase rate c_i:

$$\begin{eqnarray}&&{{c}_{i}}=\frac{\partial \ln \left(\hbar /{{\tau}_{i}}\right)}{\partial \epsilon}.\end{eqnarray} \tag{ 11 }$$

Even though we cannot distinguish the different scattering-rate contributions experimentally, it is clear that $c\leqslant \max \left({{c}_{i}}\right)$, i.e. there should be at least one scattering channel, for which ${{c}_{i}}\gtrsim 13.4$. In the following discussion, we therefore consider each mechanism separately in order to identify candidate channels responsible for the large effect observed here. As we analyze the polarization-averaged spectra, and in order to use the existing theoretical literature, where only the isotropic case was considered, we neglect the strain-induce anisotropy of the Fermi surface and assume that the density of states changes with strain according to equation (8), where we furthermore set $\kappa =0$, in agreement with our experiment.

Point defects

Point defects are assumed to be uncharged short-range scatterers with a potential $V\left(\vec{r}\right)={{V}_{\text{pd}}}\delta \left(\vec{r}\right)$ and the concentration ${{n}_{\text{pd}}}$ located directly in the graphene plane. They contribute to the scattering rate as follows [31, 32]:

$$\begin{eqnarray}&&\frac{\hbar}{{{\tau}_{\text{pd}}}}=\frac{\pi}{4}{{n}_{\text{pd}}}g\left({{E}_{\text{F}}}\right)V_{\text{pd}}^{2}.\end{eqnarray} \tag{ 12 }$$

As the density of defects per unit cell remains the same, the spatial density changes inversely with the area of the unit cell, leading to ${{n}_{\text{pd}}}/{{n}_{\text{pd},0}}\approx 1-\epsilon$, where we ignore a small Poisson ratio (0.16 in graphite [14, 39] but possibly even smaller in graphene on a substrate). Assuming that ${{V}_{\text{pd}}}$ does not change, we obtain:

$$\begin{eqnarray}&&{{c}_{\text{pd}}}\approx -1-2\nu \approx 0.3,\end{eqnarray} \tag{ 13 }$$

where $\nu =\left({{\nu}_{\parallel}}+{{\nu}_{\bot}}\right)/2\approx -0.65$ is the average Fermi-velocity expansion coefficient. Such a small theoretical value of ${{c}_{\text{pd}}}$ allows us to safely exclude point defects as a candidate mechanism.

Charged impurities

The scattering potential of charged impurities is given by: $V(q)=\left(2\pi {{e}^{2}}/\tilde{\kappa}q\right){{\text{e}}^{-qz}}$, where $\tilde{\kappa}=\left({{\epsilon}_{\text{PET}}}+1\right)/2\approx 1.8$ is the average dielectric constant of the media surrounding the graphene and z is the distance between the layer containing these impurities (typically the substrate) and the graphene sheet [31, 32]. The contribution of charged impurities to scattering, which takes into account a dynamical self-screening by the graphene electrons, can be expressed as follows [29, 31, 32] (see the supplementary information):

$$\begin{eqnarray}&&\frac{\hbar}{{{\tau}_{\text{ci}}}}=\frac{{{n}_{\text{ci}}}}{g\left({{E}_{\text{F}}}\right)}{\int}_{0}^{\pi}\frac{{{\text{e}}^{-4{{k}_{\text{F}}}z\sin \frac{\theta}{2}}}{{\sin}^{2}}\theta \text{d}\theta}{{{\left[1-\left(\frac{\pi}{4}-\frac{1}{2{{\alpha}_{\text{g}}}}\right)\sin \frac{\theta}{2}\right]}^{2}}}.\end{eqnarray} \tag{ 14 }$$

Here ${{n}_{\text{ci}}}$ is the impurity concentration, ${{k}_{\text{F}}}={{E}_{\text{F}}}/\left(\hbar v\right)$ is the Fermi momentum and ${{\alpha}_{\text{g}}}={{e}^{2}}/\left(\hbar v\tilde{\kappa}\right)\approx 1.2$ is the effective graphene fine-structure constant.

One can see that multiple parameters affect $\hbar /{{\tau}_{\text{ci}}}$, contributing rather differently to its strain dependence. First, the density of states $g\left({{E}_{\text{F}}}\right)$, which is now in the denominator, tends to decrease the scattering rate, in contrast to the case of point defects. Second, applying strain increases, peculiarly, the graphene fine-structure constant via a reduced Fermi velocity: ${{\alpha}_{\text{g}}}/{{\alpha}_{\text{g},0}}\approx 1-\nu \epsilon$. From equation (14) it follows that increasing ${{\alpha}_{\text{g}}}$ leads to the growth of the scattering rate. Third, the scattering rate is affected by the graphene-impurity distance and by the Fermi momentum via their dimensionless product ${{k}_{\text{F}}}z$. As the Fermi energy is fixed, the Fermi momentum increases with strain: ${{k}_{\text{F}}}/{{k}_{\text{F},0}}\approx 1-\nu \epsilon$, which reduces scattering. However, if z decreases with strain, this may potentially increase of $\hbar /{{\tau}_{\text{ci}}}$. Since we apply strain by bending the PET substrate, it is possible that this brings the graphene closer to the substrate (where we assume the charged scatterers are located). If $z\sim 1-2$ nm then ${{k}_{\text{F}}}z\sim$ 1, which a regime where the scattering is highly sensitive to the graphene-substrate separation [31].

By combining these competing factors, we obtain:

$$\begin{eqnarray}&&{{c}_{\text{ci}}}\approx -1+2\nu -\frac{\partial \ln I}{\partial \ln {{\alpha}_{\text{g}}}}\nu +\frac{\partial \ln I}{\partial \ln \left({{k}_{\text{F}}}z\right)}\left(-\nu +\zeta \right),\end{eqnarray} \tag{ 15 }$$

where I is the integral in the equation (14) and $\zeta =\partial \ln z/\partial \epsilon$. Specifically, assuming that ${{k}_{\text{F}}}z=1$, we numerically obtain ${{c}_{\text{ci}}}\approx -3.5-2.1\zeta$. One can see that the cumulative contribution of other factors than the distance z would result in the opposite strain dependence as compared to the experiment. Thus only the change of z may account for the effect that we observe. Specifically, if $\zeta <-8$, i.e. z decreases by more than 8% at 1% of applied strain, then ${{c}_{\text{ci}}}>13.4$, which would quantitatively explain our experimental observation. Although we cannot measure z and ζ directly, these estimates do not seem to be unreasonable, given that CVD graphene is bound very weakly to the substrate. Consequently, we speculate that the increase of scattering by charged impurities due to the reduction of the effective graphene-PET distance might be an explanation of our experimental observation.

Acoustic phonons

Scattering from the acoustic phonons was calculated in [32, 34]:

$$\begin{eqnarray}&&\frac{\hbar}{{{\tau}_{\text{ap}}}}=\frac{\pi}{4}g\left({{E}_{\text{F}}}\right)\left(\frac{{{k}_{B}}T}{\rho v_{\text{ap}}^{2}}\right){{F}^{2}},\end{eqnarray} \tag{ 16 }$$

where ρ is the mass density, ${{v}_{\text{ap}}}$ is the sound velocity (averaged over the longitudinal and transverse branches), F is the deformation potential, and k_B is the Boltzmann constant. When strain is applied, both the mass density and the sound velocity decrease: $\rho /{{\rho}_{0}}\approx 1-\epsilon$ and ${{v}_{\text{ap}}}/{{v}_{\text{ap},0}}\approx 1-h\epsilon$. Based on theoretical results of [27], we can estimate that $h\approx 1.4$. Ignoring a possible strain-induced decrease of the deformation potential, we obtain:

$$\begin{eqnarray}&&{{c}_{\text{ap}}}\approx -2\nu +1+h\approx 3.7.\end{eqnarray} \tag{ 17 }$$

Since ${{c}_{\text{ap}}}$ is well below the experimental value of 13.4, one can also exclude the acoustic phonons as a dominant contribution to the strain-induced increase of the total scattering rate.

Surface phonons

Grain boundaries

Based on the observed growth of the scattering rate we can estimate its effect on other important parameters, such as the mobility and the terahertz/microwave absorption. If the Fermi level is fixed, the anisotropic mobility is determined by the anisotropic Fermi velocity and the scattering rate (which we assume to be isotropic):

$$\begin{eqnarray}&&{{\mu}_{\parallel,\bot}}=\frac{{{e}^{2}}\tau v_{\parallel,\bot}^{2}}{{{E}_{\text{F}}}},\end{eqnarray} \tag{ 18 }$$

from which it follows that

$$\begin{eqnarray}&&\frac{{{\mu}_{\parallel,\bot}}}{{{\mu}_{0}}}\approx 1+\left(2{{\nu}_{\parallel,\bot}}-c\right)\epsilon.\end{eqnarray} \tag{ 19 }$$

Using the experimentally obtained value of c and theoretical values for ${{\nu}_{\parallel,\bot}}$ we find that ${{\mu}_{\parallel}}$ and ${{\mu}_{\bot}}$ should decrease with a rate of 17% and 12.5% per 1% of strain, respectively.

The (anisotropic) optical absorption in the low-THz and the microwave range, where $\omega \tau \ll 1$, is proportional to the zero-frequency limit of the (anisotropic) optical conductivity:

$$\begin{eqnarray}&&{{\sigma}_{\text{dc},\parallel,\bot}}=\frac{{{D}_{\parallel,\bot}}\tau}{\pi},\end{eqnarray} \tag{ 20 }$$

which yields in the linear approximation:

$$\begin{eqnarray}&&\frac{{{\sigma}_{\text{dc},\parallel,\bot}}}{{{\sigma}_{\text{dc},0}}}\approx 1+\left(\pm {{\nu}_{\parallel}}\mp {{\nu}_{\bot}}-c\right)\epsilon.\end{eqnarray} \tag{ 21 }$$

Therefore ${{\sigma}_{\text{dc},\parallel}}$ and ${{\sigma}_{\text{dc},\bot}}$, and the corresponding low-frequency absorption coefficients should decrease with a rate of 15.7% and 11.1% per 1% of strain, respectively.

Although the strain value in our experiment was limited to about 0.8%, graphene can in principle support strains of up to 25% [4, 5] depending on the substrate used. The large numbers discussed above therefore suggest the possibility to controllably and reproducibly vary the mobility and optical absorption of graphene by several tens of percent, even if the linear dependence observed in our present study will eventually saturate.