Evaluating arbitrary strain configurations and doping in graphene with Raman spectroscopy
Niclas S. Mueller, Sebastian Heeg, Miriam Peña Alvarez, Patryk Kusch, Sören Wasserroth, Nick Clark, Fred Schedin, John Parthenios, Konstantinos Papagelis, Costas Galiotis, Martin Kalbá?, Aravind Vijayaraghavan, Uwe Huebner, Roman Gorbachev, Otakar Frank, Stephanie Reich
EEvaluating arbitrary strain configurations and doping in graphene with Ramanspectroscopy
Niclas S. Mueller, ∗ Sebastian Heeg,
2, 3
Miriam Pe˜na Alvarez, Patryk Kusch, S¨oren Wasserroth, NickClark, Fred Schedin, John Parthenios, Konstantinos Papagelis,
6, 7
Costas Galiotis,
6, 8
Martin Kalb´aˇc, Aravind Vijayaraghavan,
2, 9
Uwe Huebner, Roman Gorbachev, Otakar Frank, and Stephanie Reich † Freie Universit¨at Berlin, Department of Physics, Arnimallee 14, D-14195 Berlin, Germany School of Materials, The University of Manchester, Manchester M13 9PL, UK Photonics Laboratory, ETH Z¨urich, 8093 Z¨urich, Switzerland J. Heyrovsk´y Institute of Physical Chemistry, Academy of Sciences of the Czech Republic,Dolejˇskova 3, CZ-18223 Prague 8, Czech Republic Centre for Mesoscience and Nanotechnology, The University of Manchester, Manchester M13 9PL, UK Institute of Chemical Engineering Sciences, Foundation of Research and Technology - Hellas, Patras 26504, Greece Department of Physics, University of Patras, Patras 26504, Greece Department of Chemical Engineering, University of Patras, Patras 26504, Greece National Graphene Center, The University of Manchester, Manchester M13 9PL, UK Leibnitz Institute of Photonic Technology, 07745 Jena, Germany School of Physics & Astronomy, The University of Manchester, Manchester M13 9PL, UK (Dated: September 20, 2018)Raman spectroscopy is a powerful tool for characterizing the local properties of graphene. Here,we introduce a method for evaluating unknown strain configurations and simultaneous doping. Itrelies on separating the effects of hydrostatic strain (peak shift) and shear strain (peak splitting)on the Raman spectrum of graphene. The peak shifts from hydrostatic strain and doping areseparated with a correlation analysis of the 2D and G frequencies. This enables us to obtain thelocal hydrostatic strain, shear strain and doping without any assumption on the strain configurationprior to the analysis. We demonstrate our approach for two model cases: Graphene under uniaxialstress on a PMMA substrate and graphene suspended on nanostructures that induce an unknownstrain configuration. We measured ω /ω G = 2 . ± .
05 for pure hydrostatic strain. Ramanscattering with circular corotating polarization is ideal for analyzing strain and doping, especiallyfor weak strain when the peak splitting by shear strain cannot be resolved.
I. INTRODUCTION
Being the first two-dimensional material discovered,graphene has attracted a lot of attention for its extraordi-nary properties . Ultra-high carrier mobility, large ther-mal and electrical conductivity, impermeability to anygases and extreme mechanical robustness are combined inone material . Two-dimensional materials like grapheneare strongly affected by their environment and distor-tions introduced during their processing, which affectstheir performance in devices. Locally varying strain isinduced in the graphene lattice upon deposition on asubstrate , thermal annealing and stacking with other2D materials . The substrate and adsorbed or interca-lated substances typically induce a Fermi level shift anddope graphene . Strain and doping both affect theelectronic and chemical properties of graphene.Raman spectroscopy is a powerful and non-destructive tool for characterizing the local propertiesof graphene . Spectral position, width and intensityof the Raman modes give information about layernumber, defects, doping and strain in the graphenelattice . However, these characteristics usually varysimultaneously, which hinders a straightforward analysisof the Raman spectra . Doping and strain both leadto a shift of the Raman modes and strain additionallyto a peak splitting . It was suggested to separate the contributions of strain and doping with a correlationanalysis of the G and 2D frequencies . This ap-proach has been extensively used for accessing the localproperties of graphene on various substrates .A sketch of the correlation analysis is shown in Fig.1a. The frequencies expected for purely strained andpurely doped graphene are plotted as lines resembling acoordinate system in the ω - ω G plot. Extracting strainand doping for a potential frequency pair { ω G , ω } exp isdone by projecting the point onto the strain and dopingaxes. The slope ω /ω G for pure doping is < . This is mostly attributed to anonadiabatic Kohn anomaly at the Γ point in the phonondispersion relation . Strain leads to a slope of ∼ . As one is typically unaware of theunderlying strain configuration, separating the frequencyshifts from strain and doping becomes impossible (seee.g. the ω , ω G pair in Fig. 1a). A Raman scatteringbased approach for strain-doping analysis irrespective ofthe strain configuration is highly desirable. a r X i v : . [ c ond - m a t . m t r l - s c i ] A p r h y d r o s t a t i c s t r a i n n d o p i n g p d o p i n g t en s il e s t r a i n no strainno doping c o m p r . s t r a i n a b FIG. 1. Correlation analysis of 2D and G frequencies to sep-arate frequency shifts from strain and doping. (a) Referencevalues for strain shift rates deviate from each other and de-pend on the underlying strain configuration. The frequencyshifts from strain and doping cannot be separated for a po-tential data point { ω G , ω } exp . (b) When calculating themean frequencies ω G and ω , the frequency shifts expectedfor pure hydrostatic strain can be used as reference values.The separation of G frequency shifts from doping ∆ ω dG andstrain ∆ ω hG is exemplarily shown for the case of p doping. Here, we show how to extract strain and doping fromthe Raman spectra for any (unknown) strain configu-ration and simultaneous doping. The peak splittingfrom shear strain is removed from the spectra by cal-culating the mean frequencies of the G − and G + aswell as the 2D − and 2D + modes. The frequency shiftsfrom hydrostatic strain and doping are separated witha correlation analysis of the G and 2D mean frequen-cies. The strain and doping levels are calculated fromthe G frequency shifts compared to pristine graphene.We demonstrate two experimental approaches of sep-arating the peak shifts induced by hydrostatic strainand doping and the splitting induced by shear strain.Recording Raman spectra with circularly polarized lightis ideal for extracting the hydrostatic strain componentat small strain where no peak splitting is visible. Thisapproach is demonstrated for graphene under uniaxialstress. We then apply our strain analysis to a com-plex case of graphene covering nanostructures for whichthe strain configuration is totally unknown. Grapheneis suspended on lithographically fabricated gold nan-odiscs forming dimers, which induce strong local andnon-uniform strain in the graphene bridging the gap be-tween the two nanoparticles. Surface-enhanced Ramanscattering probes the local strain and doping levels onthe nanoscale. The two examples serve as model casesfor unraveling the strain and doping levels of grapheneon arbitrary substrates. II. METHODOLOGY
Let us consider a general, unknown strain tensor ε ( r ) = (cid:18) ε xx ( r ) ε xy ( r ) ε yx ( r ) ε yy ( r ) (cid:19) , (1)with in-plane components ε ij . By solving the secularequation, we calculate the phonon shift rates as ∆ ω ± pn = − ω γ pn ( ε xx + ε yy ) ± ω β pn q ( ε xx − ε yy ) + 4 ε xy , (2)with the phonon frequency in the absence of strain ω , the Grueneisen parameter γ pn and the shear-strainphonon deformation potential β pn . Any strain config-uration can be decomposed into a hydrostatic compo-nent ε h = ε xx + ε yy and a shear component ε s = q ( ε xx − ε yy ) + 4 ε xy (assuming ε xy = ε yx ) . Hydro-static strain corresponds to an isotropic increase or de-crease in the size of the graphene lattice; it leads to afrequency shift ∆ ω hpn = − ω γ pn ε h determined by theGrueneisen parameter γ pn34 . Note that from its defini-tion ε h = ε xx + ε yy the hydrostatic strain is twice aslarge as the corresponding biaxial strain ε b = ε xx = ε yy .Shear strain corresponds to an anisotropic distortion ofthe graphene lattice leaving the area of the unit cell un-changed. It leads to a peak splitting ∆ ω spn = ω β pn ε s that depends on the shear deformation potential β pn ,while the mean position of the two peak components re-mains constant . Equation (2) is strictly valid forfirst-order Raman processes such as the G mode. Forhigher-order processes like the 2D mode, the electronicstructure and its dependence on strain have to be con-sidered as well. The general concept of peak splitting byshear strain and peak shift by hydrostatic strain, how-ever, also applies to the 2D mode, as was shown exper-imentally and theoretically . In the following,the peak components are labeled G − and G + for the Gmode and 2D − and 2D + for the 2D mode.The Raman response of graphene subjected to specificstrain configurations, i.e. uniaxial and biaxial strain, hasbeen extensively studied . For biaxial strainthe diagonal tensor components fulfill ε xx = ε yy ; for uni-axial strain they are connected by the Poisson ratio ν as ε xx = − νε yy (stress applied along x axis) . Theoff-diagonal components vanish in both cases. The fre-quency shifts for these very specific strain configurationswere previously used as reference values for the corre-lation analysis of G and 2D frequencies . Thisimplicitly assumed that either biaxial or uniaxial strainis present in the system under study. As the strain con-figuration is generally unknown and varies accross thesample, we suggest a different approach. We propose amethodology for the evaluation of arbitrary strain con-figurations and simultaneous doping in graphene:1. The correlation analysis will be based on thephonon frequency shift induced by the hydrostaticstrain component. To obtain it, we eliminate thepeak splitting from shear strain ∆ ω s by calculatingthe mean frequencies ω G = ω G − + ω G + , ω = ω − + ω + . (3)This leads to a data point in the correlation plotthat is only affected by hydrostatic strain and dop-ing (see Fig. 1b).2. The peak shifts from hydrostatic strain ∆ ω hpn anddoping ∆ ω dpn are separated with a correlation anal-ysis of ω G and ω , following Ref. 19. The slopeexpected for hydrostatic strain is used as a refer-ence (illustrated in Fig. 1b).3. Hydrostatic strain ε h is calculated from the G modeshift ∆ ω hG and doping from ∆ ω dG .4. Shear strain ε s is calculated from the G splitting∆ ω sG .The approach requires no assumption on the strain con-figuration as an input parameter. However, the type ofdoping needs to be known for choosing the correct refer-ence values. It can be obtained from reference measure-ments with other techniques; for many graphene-materialcombinations it may be found in literature. III. ANALYSIS WITH CIRCULAR LIGHTPOLARIZATION
In the following, we present two examples for how toapply our methodology. These can be viewed as modelcases for the strain analysis of graphene on arbitrary sub-strates. As a first example, we show that circular lightpolarization is ideal for measuring the mean frequencies ω G and ω . We use a Raman setup where linear andcircular light polarization can be independently chosenfor incoming and scattered light (Fig. 2a, see Methodsfor details). Tensile uniaxial strain is induced in exfo-liated monolayer graphene flakes by the deflection of aPMMA beam (Fig. 2a) . We observe a shift ofthe phonon frequencies to lower wavenumber because ofhydrostatic strain and a peak splitting because of shearstrain (Fig. 2b). For linearly polarized incoming and scat-tered light, the + / − components differ strongly in inten-sity for different polarization configurations (Fig. 2b, ll , ↔↔ and l↔ ). From the intensity ratio of G − and G + peak, we find that the uniaxial strain direction is pri-marily along the zigzag direction with a misorientationof 7.5% [see Methods, Eq. (4)] .The G − and G + peak are of equal intensity for circu-lar corotating polarization as expected from the selectionrules (Fig. 2b, (cid:8)(cid:8) , see Methods for discussion). A simi-lar behavior was observed for the 2D mode. Both G − andG + modes vanished for circular contrarotating polariza-tion (Fig. 2b, (cid:8)(cid:9) ). For this polarization, the 2D mode I n t en s i t y ( a r b . un i t s ) Raman shift (cm -1 ) G 2D laser spectrom.λ/4 or λ/2 BSM NF λ/4 A λ/2graphenePMMA beam beam deflection ab G- G+ 2D+2D-
FIG. 2. Polarized Raman spectroscopy of uniaxially-strainedgraphene. (a) Backscattering Raman setup with control oflight polarization ( λ/ λ/ λ/ λ/ − and 2D − peaksare colored blue, G + and 2D + peaks are colored red. Polar-izations of the incoming- and outgoing light are indicated byarrows ( l - linear along strain, ↔ - linear perpendicular tostrain, (cid:8)(cid:8) - circular corotating, (cid:8)(cid:9) - circular contrarotat-ing). split into components of unequal intensity; the 2D − com-ponent was consistently more intense than the 2D + com-ponent. A similar behavior was visible for strain alongthe armchair direction in the graphene lattice (Supple-mentary Fig. S1).Recording Raman spectra with circular corotatinglight is ideal for measuring the mean frequencies ω G and ω because both peak components are equally intense.For linear light polarization, one of the peak compo-nents can vanish and the mean frequencies cannot beobtained. We demonstrate our approach of using cir-cular light polarization for incrementally increasing uni-axial strain in Fig. 3a. The expected uniaxial strain ε u = ε h / (1 − ν ) = ε s / (1 + ν ) is calculated from the beamdeflection (see plot labels in Fig. 3a) . Both G and 2Dmodes were fit by two Lorentzian peaks of same spectralwidth. We thereby obtained the frequencies of all com-ponents, i.e. ω G − , ω G + , ω − and ω + , which were usedto calculate ω G and ω . For low strain levels, when nopeak splitting was visible, we obtained the mean frequen-cies from single Lorentzian peak fits. This is only possiblebecause the strain-split G and 2D components have equalintensity, i.e. because we use circular-corotating light po-larization.In Fig. 3b, we plot ω and ω G for different strain lev-els in the correlation plot of 2D and G frequency (sam-ple U1). The data points follow a linear trend. Thisshows that we systematically varied the strain at con-stant doping. The same linear trend was observed for twoother graphene flakes, where strain was induced alongthe armchair direction (samples U2 and U3 in Fig. 3b;for spectra see Supplementary Fig. S2). From a linearfit, we obtain the slope for hydrostatic or biaxial strain ω h2D /ω hG = 2 . ± .
05. The value is in excellent agree-ment with theoretical calculations of Mohr et al. andmeasured values on graphene blisters reported by Mettenet al. and Lee et al. . It also agrees with the exper-imentally determined slope of 2 . ± .
37 measured byZabel et al. due to its large margin of error.The peak position for pristine graphene (yellow star)lies outside the measured 2D versus G frequency line inFig. 3b. This is due to p-doping of the graphene flakesby the PMMA substrate . ω /ω G for p doping inthe absence of strain is shown as a red line in Fig. 3b.We used the expected G frequency shift under p-typedoping to estimate the doping of the graphene flakes as ∼ × cm − (Ref. 15, see also Ref. 20).From the frequency shift and splitting of the G line wecalculate the hydrostatic and shear strain giving rise tothe spectra in Fig. 3a. We use a Grueneisen parameterof γ G = 1 . and shear deformation potential of β G = 0 . . The shear strain increases linearly withhydrostatic strain, which is expected for uniaxial stress(Fig. 3b inset). Hydrostatic strain and shear strain un-der uniaxial tension are connected by the Poisson ratio ν as ε s = [(1 + ν ) / (1 − ν )] ε h . The Poisson ratio ν = 0 . . The expected slopeagrees well with the data points extracted for sampleU1 and U2 (see dots and squares in the inset of Fig.3b). For sample U3, the shear strain at a given hydro-static strain is larger than expected. In this experimentthe PMMA beam was deflected several times before themeasurement was taken, which led to a non-ideal loadresulting in a different strain configuration. Extractinghydrostatic and shear strain components is useful for de-termining the strain configuration and testing the stress-strain transfer from a substrate to graphene. For samples ab I n t en s i t y ( a r b . un i t s ) Raman shift (cm -1 ) G 2D undoped & unstrained sample U1sample U2sample U3 p d o p i n g FIG. 3. Extracting the hydrostatic strain, shear strainand doping from the Raman spectra of uniaxially-strainedgraphene using circular corotating polarization (laser wave-length 532 nm). (a) G and 2D mode of graphene under uni-axial strain which increases incrementally from top to bottom(sample U1). Fits with one or two Lorentzian peaks are su-perimposed on the data. G − and 2D − peaks are colored blue,G + and 2D + peaks are colored red. Plots are labeled by theexpected uniaxial strain ε u calculated from the beam deflec-tion. (b) Correlation plot of 2D and G mean frequencies forthree samples (U1-U3). A linear fit is plotted as a solid line.The red line indicates the expected spectral positions if onlyp-type doping is present (slope of 0.7, see Ref. 19 and datain Refs. 15,28). The yellow star indicates spectral positionin the absence of doping and strain (extracted from Ref. 15,514 nm laser excitation, assuming a 2D mode dispersion of100 cm − /eV). The inset shows shear strain as a function ofhydrostatic strain. The expected slope for a Poisson ratio ν = 0 .
33 is shown as a black line.
U1 and U2 we observed good agreement of the uniaxialstrain calculated from the beam deflection and the uniax-ial strain calculated from the measured hydrostatic strain(Supplementary Fig. S3).
IV. COMPLEX NANOSCALE STRAINCONFIGURATIONS
So far, we presented one application of our methodol-ogy for strain evaluation. Using circular light polariza-tion turned out to be useful for unraveling the local strainand doping levels, especially for weak strain when nopeak splitting is detectable. As a second example, we an-alyze complex and unknown strain configurations on thenanoscale. Exfoliated flakes of graphene were suspendedon multiple lithographically-fabricated pairs of gold nan-odiscs with diameters of ∼
100 nm, heights of 40 − −
30 nm (Fig. 4a) . After transferon top of the nanodimers, the substrate adhesion pulledthe graphene into the gap between the two gold nanopar-ticles. This induced strong local strain with completelyunknown configurations (see AFM topography, Fig. 4b).The Raman spectrum from the graphene bridging thedimer gap is strongly enhanced by surface-enhanced Ra-man scattering . The plasmon of the gold nanodimerinduces strong electromagnetic near fields that increasethe local Raman cross section by three to four ordersof magnitude . The near field enhancement providesnanoscale spatial resolution; it also fixes the polarizationof the field to be parallel to the nanodimer axis.Figure 4c shows two representative Raman spectrawith different levels of intrinsic strain. They belong tothe same graphene flake but were recorded on differ-ent nanodimers. Both G and 2D modes are split intothree components of different intensity. The two peakswith the lowest Raman shift arise from the strainedgraphene in the interparticle gap . The third peak isonly slightly shifted from the peak position expected forpristine graphene (grey dashed lines in Fig. 4c). It arisesfrom the unstrained graphene around the nanodimer thatcontributes to all Raman spectra without plasmonic en-hancement. In the following, we focus on the two peaksthat are shifted to lower wavenumbers and thereby an-alyze the local strain in the dimer cavity. We identifythese 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 anddoping locally induced in different positions of a grapheneflake by various nanodimers. The dimers were individu-ally addressed with a confocal Raman microscope. Werecorded the Raman spectra of 60 graphene-covered goldnanodimers, all of which showed surface-enhanced Ra-man scattering. We calculated the mean frequencies ω G and ω from fits similar to Fig. 4c for each nanodimer.The contributions of doping and hydrostatic strain wereseparated with a correlation plot of the mean 2D and G frequencies shown in Fig. 4d. The peak positions arestrongly shifted compared to the reference measurements1 µ m away from each nanodimer (grey triangles) and ofpristine graphene (yellow star). To determine the localhydrostatic strain, we needed to analyse the contribu-tion of local charge doping first. We assumed an n-typedoping in the graphene bridging the dimer cavity, as pre-viously observed for graphene on gold nanoparticles .The ω /ω G positions for n-type doping are plotted asa green line in Fig. 4c . The local n-type doping wascalculated from the expected G frequency shifts . Itreached levels of up to 10 cm − , see Fig. 4e. This ismore than one order of magnitude higher than the max-imum doping levels of 6 × cm − for a comparableplasmonic system reported in Ref. 57. This is expectedbecause we probe the local doping in graphene in thevicinity of the gold nanoparticles, whereas Fang. et al. probed the charge doping of the entire graphene sheetby transport measurements. Our approach complementsthe macroscopic measurements with an opportunity toprobe local doping levels in an area of ∼ ×
20 nm .Interestingly, the local charge doping increased statisti-cally with the hydrostatic strain (Fig. 4e, calculated usinga Grueneisen parameter of γ G = 1 . ). We ex-pect that with increasing local strain, graphene is pulleddeeper into the dimer cavity. This leads to a stronger in-teraction between graphene and the gold nanostructureand explains the increase in doping with strain.The nanodimers induced strong local strain in thegraphene lattice with a hydrostatic component rangingfrom 1% to 2.6% (Fig. 4e). The shear component wascalculated from the G splitting using a shear deforma-tion potential of β G = 0 . . With a magnitude from0.75% to 1.8% it was weaker than the hydrostatic com-ponent (Fig. 4e). This differs from uniaxial strain, wherethe shear component was twice as large as the hydro-static component (Fig. 3b inset) and biaxial strain wherethe shear component vanishes. Shear strain and hydro-static strain are largely uncorrelated for the nanodimers(Fig. 4e), from which we conclude that the nanodimersinduce different strain configurations in the suspendedgraphene. Our methodology for strain evaluation is evenapplicable for this complicated case because it does notrequire any assumption of the strain configuration as aninput parameter.Finally, we demonstrate our analysis of local strain anddoping for several different graphene flakes covering ar-rays of gold nanodimers. Representative Raman spectraof three graphene flakes (samples D1-D3) are shown inFig. 5a. While the intensity ratio of the 2D − and 2D + components is similar in all spectra, the G − /G + -intensityratio varies from one flake to the other. This is explainedby the different lattice orientation of the graphene flakeswith respect to the direction of the local shear-straincomponent (see Methods section) . Based on sym-metry arguments as discussed in detail in Ref. 51 we as-sume that the local shear strain is directed along and per-pendicular to the dimer axis. We determined the angle x : . μ m y : . μ m Au AuSiO /Sigraphene a SERS cd dimer 1 dimer 2 G 2D
Raman shift (cm -1 ) Raman shift (cm -1 ) R a m an i n t en s i t y ( a r b . un i t s ) dimer 1dimer 2 S h e a r s t r a i n ( % ) b e mean frequencies on dimersreference frequenciespristine grapheneonly hydrostatic strainonly n-type dopingonly p-type doping FIG. 4. Raman characterization of graphene on gold nanodimers. (a) Geometry of plasmonic nanostructure and substrate. Ar-rows indicate surface-enhanced Raman scattering (SERS) using graphene as Raman scatterer. (b) Representative 3-dimensionalAFM profile of the graphene-covered nanodimer. (c) Exemplary Raman spectra of graphene covering two Au nanodimers withstrong (top) and weak (bottom) strain. G − and 2D − peaks are colored blue, G + and 2D + peaks are colored red. The spectralpositions for unstrained graphene are indicated by grey dashed lines (sample D1, laser wavelength 638 nm, linear light pol.along dimer axis). (d) Correlation plot of 2D and G mean frequencies (circles). Spectral positions are indicated by trianglesfor reference measurements on SiO /Si. The yellow star shows the frequencies for unstrained and undoped graphene (deducedfrom experimental data in Ref. 18, 633 nm laser excitation, assuming a 2D mode dispersion of 100 cm − /eV). Expected peakpositions are shown by the blue line for hydrostatic strain (no doping), the red line for p-type doping (Ref. 19, data fromRefs. 15,28; no strain) and the green line for n-type doping (no strain). (e) Extracted magnitude of doping, shear strain andhydrostatic strain induced in graphene by the nanodimers. A linear fit is superimposed on the data in the upper plot. between the zigzag direction of graphene and the dimeraxis for 86 nanodimers covered by five different grapheneflakes (Supplementary Fig. S4a). The estimated anglesrange from (9 ± ◦ for sample D1 to (19 ± ◦ for D3.We also repeated the correlation analysis of the 2D andG mean frequencies for all graphene flakes (Supplemen-tary Fig. S4b). The peak positions nicely follow the sametrend as for sample D1 in Fig. 4d. We found statisticallythe same strain and doping levels in all graphene flakes. V. CONCLUSION
In conclusion, we proposed a method for analyzing ar-bitrary strain configurations and simultaneous doping ingraphene using Raman spectroscopy. First, the shift dueto pure hydrostatic strain is determined through explicitor implicit (circular light) averaging of the G and 2Dmode components. Second, the peak shifts induced byhydrostatic strain and doping are separated by a corre-lation analysis of the G and 2D mean frequencies. Thisoffers the possibility to calculate the local shear strain,hydrostatic strain and doping without any assumption onthe underlying strain configuration. We demonstrated
G 2D sample D1sample D2sample D3
Raman shift (cm -1 ) Raman shift (cm -1 ) N o r m a li z ed R a m an i n t en s i t y ( a r b . un i t s )
9° ± 2°16° ± 1°19° ± 1°
FIG. 5. Strain evaluation for several graphene flakes coveringgold nanodimers (samples D1-D3, laser wavelength 638 nm,linear light pol. along dimer axes). Exemplary Raman spec-tra are shown for three samples. Fits with three Lorentzianpeaks are superimposed on the experimental data and offsetfor clarity. G − and 2D − components are colored blue andG + and 2D + components red. The angles between the zigzagdirection in the graphene lattice and the dimer axes were cal-culated from I G − /I G + and are given next to the spectra. this strain analysis for graphene on two different sub-strates. First, graphene was subjected to uniaxial stressby deflection of a PMMA beam. We demonstrate thatcircular corotating light polarization is ideal for obtain-ing the mean frequencies of the G and 2D mode understrain. This approach is even applicable for low strainlevels when no peak splitting is visible. As a second ex-ample, graphene was suspended on pairs of gold nanopar-ticles, which induced strong local strain of completelyunknown configuration. Surface-enhanced Raman scat-tering allowed extracting the local strain and doping lev-els with nanoscale resolution. The two examples serveas model cases for the strain analysis of arbitrary strainconfigurations in graphene. Our methodology also carriesover to other 2D materials, such as MoS . VI. METHODSA. Sample fabrication and characterization
For the measurements under uniaxial strain, monolayergraphene flakes were obtained by micromechanical exfoli-ation and deposited on a flexible polymethylmethacrylat(PMMA) beam. Prior to graphene transfer, the PMMAsubstrate was spin coated with SU8 photoresist (SU82000.5, MicroChem). The samples were soft-cured at 80 ◦ C for 30 min and exposed to UV radiation (366 nm, 30sec). Graphene was transferred onto the samples with thescotch-tape method. Finally, the samples were covered with PMMA (1% in anisole) to improve the strain trans-fer efficiency. Uniaxial stress was induced by bending thePMMA beam with a jig (see e.g. Refs. 44,45). We assumethat the doping does not change when inducing strain ingraphene by deflection of the PMMA substrate, as weobserved no systematic linewidth change of the G peakcomponents with increasing strain. To avoid heating andaccompanying structural change of the graphene-PMMAinterface, we used laser powers below 150 µ W for Ramancharacterization (532 nm laser excitation).The graphene-covered gold nanodimers were fabricatedusing the same procedure as described in Ref. 51. Inshort, gold nanodisk dimers were produced with electron-beam lithography, followed by metallization (5 nm Cror Ti and 40 to 80 nm Au) and lift-off in an ultrasonicbath. Micromechanically cleaved large flakes of single-layer graphene were then transferred on top of the plas-monic nanostructures using a dry transfer method. Thespacing between the nanodimers beneath the grapheneflakes was much larger than the laser-spot size. This en-abled us to record Raman spectra of graphene interactingwith an individual nanodimer.Raman spectra were recorded with a Horiba XploRAsingle-grating confocal Raman spectrometer, equippedwith a 1200 groves per mm grating, which leads to aspectral resolution of 2-3 cm − . The spectrometer wascalibrated with the Raman response of diamond. 532 nmlaser excitation was used for measurements on grapheneunder uniaxial stress (2 ×
120 sec acquisition time, 130 µ W laser power, spot size of 340 nm). Reference spectraof the PMMA substrate were recorded with the same pa-rameters next to the graphene flakes and subtracted fromthe Raman spectra of graphene. The graphene-coveredgold nanodimers were characterized with 638 nm laserexcitation (2 ×
120 sec acquisition time, 280 µ W laserpower, spot size of 605 nm); at this wavelength largeplasmonic enhancement is expected . The laser was fo-cused on each plasmonic nanodimer using a piezo stageand steps of 100 nm in x -, y - and z direction until a maxi-mum intensity of the 2D mode was achieved. Linear lightpolarization was chosen along the dimer axis. Referencespectra were recorded 1 µ m away from each nanodimer. B. Polarization dependent measurements
The Raman intensities of the G components can becalculated with the macroscopic theory of Raman scat-tering as I G ± ∝ | e i R G ± e s | . e i is the polarization ofthe incoming light and e s of the Raman-scattered light. R G − is the Raman tensor for the G − peak and R G + forthe G + peak . For uniaxial stress applied at an angle ϑ with respect to the crystallographic zigzag direction ingraphene, the intensities are calculated with the Ramantensors given in Ref. 17 as I G + ∝ d cos ( ϕ + ψ + 3 ϑ ) ,I G − ∝ d sin ( ϕ + ψ + 3 ϑ ) , (4)where d is the Raman tensor component of the G modefor unstrained graphene. ϕ is the angle of the incominglight and ψ the angle of the Raman scattered light to thestrain axis. From Eq. (4) it is apparent that I G − and I G + can be very different; for ϕ + ψ + 3 ϑ = n · π/ n integer) one of the components vanishes. The situation isdifferent for circular light polarization with polarizationvectors e (cid:8) = 1 √ (cid:18) − i (cid:19) , e (cid:9) = 1 √ (cid:18) (cid:19) . (5)For circular corotating polarization (i.e. (cid:8)(cid:8) or (cid:9)(cid:9) ascombinations for e i and e s ) we obtain I G − ∝ d and I G + ∝ d ; the G mode splits into components of equalintensity. When using circular contrarotating polariza-tion (i.e. (cid:8)(cid:9) or (cid:9)(cid:8) ), both G mode components vanish.The polarization dependence of the 2D mode is morecomplex because it is a second-order Raman process. Theintensities of the 2D − and 2D + components are deter-mined by double-resonance processes and the simple the-oretical treatment that was used for the G mode cannotbe applied . If incoming- and outgoing light are lin-early polarized along the strain axis ( ll ), I − > I + .The opposite case I − < I + is observed for light po-larizations perpendicular to the strain axis ( ↔↔ ). Ingeneral there is a complicated functional dependence of I − and I + on the polarization directions of incom-ing and outgoing light . A reliable fit of the 2D peakcomponents is only possible for a large peak splitting in-duced by shear strain. For circular light polarization,the overall peak intensity is determined by the selectionrules of the Raman process. The 2D mode has symme-try A g ⊕ E g . Under circular corotating polarizationprocesses with A g symmetry vanish whereas for circularcontrarotating polarization processes with E g symmetryare zero . This explains the non-vanishing 2D mode forboth, circular corotating and contrarotating light polar- ization.For Raman measurements with circular light polariza-tion, we used an optical setup as schematically depictedin Fig. 2a. The laser beam with initially linear lightpolarization was directed through a quarter-wave plate( λ/
4) to induce circular light polarization. The lightwas focused onto the sample with an optical microscopewhich 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 linearlight polarization. With an analyzer, we selected eitherthe circular-corotating or the circular-contrarotatingpolarization in the Raman experiment. In front of thespectrometer entrance we placed a half-wave plate ( λ/ λ/ .Combinations of linear light polarization were realizedby using λ/ λ/ ACKNOWLEDGMENTS
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2, 3
Miriam Pe˜na Alvarez, Patryk Kusch, S¨oren Wasserroth, Nick Clark, Fred Schedin, John Parthenios, KonstantinosPapagelis,
6, 7
Costas Galiotis,
6, 8
Martin Kalb´aˇc, Aravind Vijayaraghavan,
2, 9
Uwe Huebner, Roman Gorbachev, Otakar Frank, and Stephanie Reich † Freie Universit¨at Berlin, Department of Physics,Arnimallee 14, D-14195 Berlin, Germany School of Materials, The University of Manchester, Manchester M13 9PL, UK Photonics Laboratory, ETH Z¨urich, 8093 Z¨urich, Switzerland J. Heyrovsk´y Institute of Physical Chemistry,Academy of Sciences of the Czech Republic,Dolejˇskova 3, CZ-18223 Prague 8, Czech Republic Centre for Mesoscience and Nanotechnology,The University of Manchester, Manchester M13 9PL, UK Institute of Chemical Engineering Sciences,Foundation of Research and Technology - Hellas, Patras 26504, Greece Department of Physics, University of Patras, Patras 26504, Greece Department of Chemical Engineering,University of Patras, Patras 26504, Greece National Graphene Center, The University of Manchester, Manchester M13 9PL, UK Leibnitz Institute of Photonic Technology, 07745 Jena, Germany School of Physics & Astronomy, The University of Manchester, Manchester M13 9PL, UK (Dated: September 20, 2018) a r X i v : . [ c ond - m a t . m t r l - s c i ] A p r -1 ) I n t en s i t y ( a r b . un i t s ) FIG. S1. Polarization dependence of the Raman spectrum of graphene with induced uniaxialstress along the armchair direction (sample U2, laser wavelength 532 nm). Fits with one or twoLorentzian peaks are superimposed on the experimental data. Reliable fitting of the 2D mode iscomplicated by the weak splitting. Polarizations of incoming- and outgoing light are indicated byarrows ( l - linear along strain, ↔ - linear perpendicular to strain, (cid:8)(cid:8) - circular corotating, (cid:8)(cid:9) -circular contrarotating). n t en s i t y ( a r b . un i t s ) I n t en s i t y ( a r b . un i t s ) a b G 2D G 2D
Raman shift (cm -1 ) Raman shift (cm -1 ) FIG. S2. Raman spectra of uniaxially-strained graphene using circular corotating light polarization,for (a) sample U2 and (b) sample U3. The strain was incrementally increased from top to bottom.Fits with two Lorentzian peaks are superimposed on the data. The G − and 2D − peaks are coloredblue and the G + and 2D + peaks are colored red. A single Lorentzian peak fit was used if no peaksplitting was visible. The fits were used to obtain the mean frequencies ω G and ω in Fig. 3b. IG. S3. Measured uniaxial strain plotted vs expected uniaxial strain. The measured uniaxialstrain ε u was obtained from the measured hydrostatic strain ε h as ε u = ε h / (1 − ν ) with a Poissonratio of ν = 0 .