pubs.acs.org/NanoLett
Biaxial Strain in Graphene Adhered to Shallow Depressions Constanze Metzger, Sebastian Re´mi, Mengkun Liu, Silvia V. Kusminskiy, Antonio H. Castro Neto, Anna K. Swan,*,† and Bennett B. Goldberg*,† Department of Physics, 590 Commonwealth Avenue, and Department of Electrical and Computer Engineering and The Boston University Photonics Center, 8 Saint Mary’s Street, Boston University, Boston, Massachusetts 02215 ABSTRACT Measurements on graphene exfoliated over a substrate prepatterned with shallow depressions demonstrate that graphene does not remain free-standing but instead adheres to the substrate despite the induced biaxial strain. The strain is homogeneous over the depression bottom as determined by Raman measurements. We find higher Raman shifts and Gru¨neisen parameters of the phonons underlying the G and 2D bands under biaxial strain than previously reported. Interference modeling is used to determine the vertical position of the graphene and to calculate the optimum dielectric substrate stack for maximum Raman signal. KEYWORDS Graphene, Raman spectroscopy, strain, biaxial, reflectivity
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Mechanical exfoliation of natural graphite was employed to transfer single-layer graphene sheets on a SiO2/Si wafer. Silicon wafers with an oxide thickness of 300 nm were used to maximize the optical contrast of graphene on the substrate.11,12 Prior to graphene exfoliation, the wafer was patterned with 6 µm squares etched to a depth of 20 nm with reactive ion etching. We were able to deposit graphene flakes as long as 250 µm over the depressions. The graphene samples were mounted on a stack of piezo positioners in a He bath cryostat as shown in Figure 1a. A 632.8 nm HeNe laser was coupled into a single mode optical fiber guiding the excitation to the head of the Raman microscope. Laser light was focused on the sample with an aspherical lens with numerical aperture 0.65, leading to a diffraction limited full width at half-maximum (fwhm) spot size of 760 nm. The total power on the sample was limited to 200 µW, an intensity low enough to avoid heating. Collected light was directed into a fiber and coupled either to a photodiode to measure reflected power or to a spectrometer for spectrally resolved Raman measurements. Figure 2 (center) shows a reflectivity scan of a 250 µm × 20 µm single graphene layer on top of the patterned
aman spectroscopy is an important, nondestructive tool to study the properties of carbon. For graphene, Raman spectroscopy has been successfully employedtoinvestigatephononproperties1 andelectron-phonon coupling,2,3 to identify the number of graphene layers,1 and to provide information about the doping and disorder.3-5 Graphene has been proposed as a new electronic material because of its tunability with the electric field6 and high carrier mobility.7 Yet, because of its single or few atomic thickness, the material properties are subject to local and global deformations8 and carrier charging and pooling.9 Thus it is critically important to develop both an understanding of and techniques to measure local strain and deformation of graphene layers. Since local strain can be employed to confine and collimate electrons in graphene,10 highresolution strain mapping and control is expected to enable a whole class of graphene electronics. Here, we find graphene spontaneously adhered to the bottom of small aspect ratio depressions, resulting in a biaxial strain of 0.066%. We examine the effect of biaxial strain on the Raman frequency at different temperatures. The Gru¨neisen parameters for the Raman G and 2D bands are found to be substantially higher than those previously reported, with negligible temperature dependence. We show that spatially resolved spectral reflectivity combined with micro-Raman spectroscopy that exhibits a large variation in Raman intensity with oxide thickness can be used to optically determine the vertical position of a graphene layer and whether it is suspended or adhered to the bottom of a trench.
* Corresponding authors:
[email protected] and
[email protected]. † These authors contributed equally. Received for review: 05/22/2009 Published on Web: 11/23/2009
© 2010 American Chemical Society
FIGURE 1. (a) Schematics of confocal Raman scanning setup and (b) cross section through the sample. A single layer graphene sheet covers a shallow square depression in the SiO2/Si substrate. 6
DOI: 10.1021/nl901625v | Nano Lett. 2010, 10, 6-10
FIGURE 2. Center: Confocal reflectivity scan of a graphene sample at 632.8 nm. Dashed gray lines indicate where Raman measurements were recorded. Bright regions are 20 nm deep etched depressions and the darkest regions near edges are graphene bilayers. (a) Raman G and 2D spectra on strained graphene in the 20 nm depression. (b) Raman G and 2D on graphene supported on 300 nm SiO2 without strain. Both (a) and (b) were measured at 4.2 K. (c) Line cut of reflectivity at 633 nm along dashed line in center image with theoretical fit. (d) Similar line cut of reflectivity but at 704 nm with theoretical fit. Data at 760 nm are not shown.
tively. We can rule out different charge densities as the origin of the Raman frequency shifts. While a charge of ∼2 × 1012 cm-2 could account for the 5 cm-1 shift of the G band, such a carrier density difference would only lead to a small shift of 2 cm-1 in the 2D band3,4 whereas we observe ∼13 cm-1. Also, since no charge-induced line broadening for the 2D line is observed experimentally,3,4 differential charging cannot explain the measured 3.5 cm-1 additional broadening in the 2D line width. On the other hand, strain explains the frequency shift. Several recent experimental papers report Raman shifts of graphene due to uniaxial strain induced by bending a flexible substrate13-15 although the results differ by more than a factor of 2. Such a discrepancy is reminiscent of strain experiments in nanotubes, where problems with substrate adhesion and slippage obscured accurate values until spatiallyresolvedmeasurementsonsingletubeswereperformed.16,17 As we show below, our measurements indicate up to 20% higher G band Raman shift with strain than reported by Mohiuddin et al.13 and more than a factor of 2 greater than Huang et al.14 We employed atomic force microscopy (AFM) measurements to accurately determine the amount of biaxial strain in our sample. The graphene sheet was shown to adhere to the bottom of the depression everywhere except a small strip 100 nm ( 10 nm wide along the base edge. In a square depression of 6 µm side length this translates to 0.066% ( 0.006% of biaxial strain. The resulting biaxial Raman shift parameters are 77 cm-1/1% ( 7 cm-1/1% for the G and 203 cm-1/1% ( 20 cm-1/1% for the 2D band, higher than those extrapolated from uniaxial strain data in ref 13. The Gru¨neisen parameter defines the shift of Raman frequency with strain and is given by the relation13,18
FIGURE 3. Raman scans at 4.2 K along y (dashed gray line in Figure 2c). (a) Raman shift of G band, the strain in graphene in the depression leads to a frequency shift of 6 cm-1. (b) fwhm of G band. Inset explains the overshoot behavior while scanning over the edge of the depression. The two black lines depict the Raman lines in strained and relaxed graphene. The red line is the sum of both; it is asymmetric and has a higher fwhm. (c) Raman shift of 2D with a strain-induced shift of 13 cm-1. (d) fwhm of 2D band.
substrate. The graphene covers several 20 nm deep etched depressions (white regions in 2) and near the edges the sheet is folded over, forming a bilayer (darkest regions). The frequencies ωG ) 1594 cm-1, ω2D ) 2647 cm-1 and widths ∆ωG ) 7.4 cm-1, ∆ω2D ) 26.4 cm-1 of the Raman lines (Figure 2b) of the single layer atop the 300 nm SiO2 indicate that our sample had a charge carrier density high enough to block the G phonon electron-hole decay mechanism.3,4 We found substantial differences in the Raman frequency, line width, and intensity of both the Raman G and 2D lines in the depressions versus atop the 300 nm thick SiO2 (compare parts a and b of Figure 2). Line scans along y through depressions were performed, and the frequency and width dependence are displayed in Figure 3. The Raman G band downshifts from ωG ) 1594 cm-1 by 5.1 cm-1 and the 2D from ω2D ) 2647 cm-1 by 13.4 cm-1 in the depression, and the lines broaden from ∆ωG ) 7.4 cm-1 by 1 cm-1 and from ∆ω2D ) 26.4 cm-1 by 3.5 cm-1 for G and 2D, respec© 2010 American Chemical Society
γ)-
7
1 δω ω0 δεh
(1)
DOI: 10.1021/nl901625v | Nano Lett. 2010, 10, 6-10
where ω0 is the Raman frequency without strain and εh ) εl + εt is the hydrostatic strain given by the sum of its longitudinal and transversal components, in our case εt ) εl ) 0.066%. For the G band phonon, we find a Gru¨neisen parameter of 2.4 ( 0.2, exceeding previous measurements13 by 20%. The Gru¨neisen parameter for the 2D line yields 3.8 ( 0.3, slightly higher than prior measurements. There are several possible reasons for the variations of the observed values of Raman shift versus strain between different groups. Uniaxial strain measurement does not actually yield the Gru¨neisen parameter directly but requires knowledge of the material’s Poisson ratio in order to calculate the longitudinal and transverse strain components. Further, it is hard to know precisely the amount of strain transferred from the substrate to the graphene layer. While earlier studies may have suffered from these issues, here we use AFM to measure the region of adherence to directly find the biaxial strain. Another possible reason for the higher values of the Gru¨neisen parameter we observed could be a temperature dependence of γ. To examine this, we repeated the differential measurement of the Gru¨neisen parameters at 300 and 77 K and compared with the values obtained at 4.2 K. While we observe the expected substantial temperaturedependent softening in the Raman frequency with increasing temperature,19 we observe no significant change in the G and 2D Gru¨neisen parameters. Issues related to different thermal expansion coefficients of graphene and SiO2 are reduced in our experiment since we are using a differential measurement between adjacent, nominally unstrained, and strained graphene regions. We expect the observed lack of temperature dependence of the Gru¨neisen parameter, since the volume of graphite decreases with T up to room temperature,20 effectively canceling the effect of the Raman phonon softening with increasing T. The Gru¨neisen parameter for the 2D peak is of special interest. As was pointed out in ref 13, uniaxial strain moves the Dirac points, so that the zone edge 2D Raman line measured with the same excitation frequency before and after strain will probe different zone-edge phonons.13,21 Because of this, the measurement will include effects of both the 2D phonon dispersion near the K-point,22 and frequency shifts due to strain. Mohiuddin et al.13 calculated the 2D Gru¨neisen parameters from first principles and found a significantly lower calculated γ2D ) 2.7 than the γ2D ) 3.55 derived from uniaxial measurements. They attributed the discrepancy to the effect of phonon dispersion and noted that a biaxial strain measurement would avoid relative changes of the Dirac cones and therefore should give only the Raman shift due purely to strain. Our results from biaxial strain, γ2D ) 3.8 ( 0.3, are higher but not significantly different from the experimental value derived from the uniaxial strain γ2D ) 3.55. Provided that the strain values are correct, the result implies that the contribution to the downshift from Dirac cone displacement during uniaxial strain is small. It is interesting to note that the high-resolution © 2010 American Chemical Society
Raman spectra show that the strain is nearly homogeneous over the bottom of the hole, with a small shift of about 0.85 cm-1 for the G line indicating a maximum differential strain of less than ∼0.01%. Strain induced by adherence to the bottom of the depression explains the observed frequency shift of the Raman lines, but subtle increase of line widths are still not clearly understood. Increase in line width due to uniaxial strain and development of two nondegenerate G bands13,14 can be eliminated since the strain is both very low and biaxial. The gross behavior of the overshoot of the G and 2D Raman line widths at the depression edge shown in parts b and d of Figure 3, respectively, is simply a result of convoluting shifted peaks from the unstrained and strained regions as the diffraction limited focal spot is scanned across an edge. The two signal components add, yielding a non-Lorentzian shape of greater width. If we fit with two Raman lines and assume a sharp edge, we measure a focal spot of 760 nm fwhm, the same as determined from our optical system. Fitting the data with two Lorentzians for the strained and unstrained contributions, information about the width of the region in which the graphene does not adhere to the substrate can be retrieved. We find that this region must be smaller than 150 nm, consistent with the AFM data of 100 ( 10 nm. The intensity of the Raman signal of the strained graphene is ∼65% lower than the unstrained graphene (see Figure 2). We find that this decrease in signal can be completely understood from interference effects and is not related to strain. The Raman signal strengths differ significantly because they are dependent on the substrate dielectric stack due to interference of the excitation field as well as the Raman emitted light. We first model the reflected intensity of the excitation light shown in Figure 2c, calculated using a transfer matrix method assuming incident parallel light. We used the index for bulk graphite ngr ) 2.6 1.3i,12,23 and nSiO2(633 nm) ) 1.46 was measured with an ellipsometer. The calculations were compared to the confocal reflectivity scans performed at three different laser wavelengths of 632.8, 704 (corresponding to the G band energy), and 760 nm (2D). As shown in parts c and d of Figure 2, the measured absolute reflectivity and contrast change between bare SiO2 on Si, single and bilayer graphene on the 300 nm SiO2 as well as in etched holes at all wavelengths can be reproduced accurately by our simulation. The model agrees well with the data for all three wavelengths used, 633 nm (Figure 2c), 704 nm (Figure 2d), and 760 nm (not shown). We also use this optical model to determine the Raman signal from the graphene surface excited by the excitation field which is composed of the incoming Einc and reflected Er fields at the surface, Iexc ∝ |Einc + Er|2. Next, the combination of the direct and reflected intensity of a Raman photon source at 704 nm (G band) and 760 nm (2D band) at the position of the graphene was approximated by assuming the electromagnetic field split between vertically upward and 8
DOI: 10.1021/nl901625v | Nano Lett. 2010, 10, 6-10
surface interactions at the graphene SiO2 interface that makes graphene prefer to adhere to substrates even at the cost of getting strained. To explore the tendency of graphene sticking to the substrate, we have estimated the free energy of a graphene sheet placed across a trench,26 using the Deryagin approximation.27,28 This is a mean field approximation in which the influence of the substrate is modeled as a harmonic potential characterized by a strength v which encapsulates the details of the substrate-membrane interaction. The equilibrium profile of the membrane over the trench is obtained by minimizing the free energy and solving the resulting Euler-Lagrange equations constrained by the geometry of the trench: (κ∇4 - γ∇2 + v) δh(F) ) vzs(F) where κ is the bending rigidity, γ the tension, and h(F), zs(F) are the heights of the membrane and substrate respectively with respect to a plane parametrized by a planar vector F in the Monge representation. This calculation assumes that the graphene membrane is unstrained before it adheres to the substrate. Initial strain in the sheet can safely be neglected, because the energy cost of straining graphene is very low compared with its bending energy. The solution to this equation can be expressed in terms of two relevant length scales ξκ ) (κ/v)1/4 ≈ 2 nm and ξ ) (γ/v)1/2 ≈ 1.3 nm, where the numerical values are taken from experimental data on the corrugation of graphene sheets.29,30 We find that the lowest free energy state for a graphene sheet placed across a 6 µm wide trench of 20 nm depth is reached when the sheet sticks to the bottom of the trench; similar results are obtained for a cylindrical pit in accord with the experimental setup. In summary, high-resolution Raman studies on locally biaxially strained graphene sheets together with detailed optical analysis of the reflectivity and Raman intensity show that exfoliated graphene adheres to the bottom of shallow depressions under a modest strain (∼0.07%) instead of remaining suspended, a finding confirmed by AFM measurements. Our results on biaxial strain yield a G band Gru¨neisen parameter that is substantially higher than that extrapolated from uniaxial data by Mohiuddin et al., while the 2D Gru¨neisen parameter is higher, but not significantly different than earlier measurements.13 No temperature dependence of the Gru¨neisen parameters between 4 and 300 K is discernible in our experiments. Finally, our modeling can be used to determine the optimum substrate to the maximize Raman signal. For example, a 100 nm SiO2 layer would lead to an increase of Raman signal of about a factor of 2 for the G and a factor of 5 for the 2D band at 632.8 nm excitation light compared to commonly used 300 nm thick SiO2 layer.
FIGURE 4. Interference enhanced G and 2D band Raman signal calculated at λ ) 632.8 nm excitation light for a single layer graphene sheet. (a) G band Raman signal and (b) 2D band Raman signal as a function of oxide thickness atop silicon with varying thickness. (c) G and 2D band Raman signal strength from (a) and (b) split up in contributions from excitation light |Einc + Er|2 and outgoing Raman light for single-layer graphene. A multiplication of the excitation light and the outgoing Raman signal leads to the G and 2D signal strength shown in (a) and (b). (d) For comparison, G Raman signal from a single layer sheet suspended over a hole with varying depth in a fixed 300 nm SiO2 layer.
downward propagating fields with a ratio of ERaman,up/ ERaman,down ) 1/nSiO2, taking account of the emitting behavior of a dipole at a dielectric interface.24 We then calculated |ERaman,up + rERaman,down|2 with r the reflection coefficient of the dielectric stack. The total Raman signal is then proportional to the product |Einc + Er|2 × |ERaman,up + rERaman,down|2. Figure 4a shows the calculated Raman signal strength of the G band excited with a HeNe laser for a graphene layer versus oxide thickness and versus depression depth into a 300 nm SiO2 layer atop silicon in (b). Adhesion of the graphene sheet to the bottom of the depression yields the relative Raman intensity indicated by the dashed lines in Figure 4a, which accurately match the 65% loss in Raman signal from outside to within the depression we observe in the data (see parts a and b of Figure 2. Had the graphene been suspended, Figure 4d shows that we would observe a signal loss of only 10%. It is important to note that it is critical to combine the Raman intensity information together with reflectivity to accurately determine optically if a graphene sheet is free-standing or adhering to the bottom. Since the reflectivity contrast differs only slightly between suspension and adherence, it is impossible to unambiguously distinguish between the two situations without the use of the Raman signal strength information. In an experiment with graphene exfoliated over deep holes in SiO2, Lee et al.25 found that the sheets adhere up to 10 nm to the vertical wall of micrometer-diameter holes. While the graphene sheets still remain suspended in their studies, their length elongation corresponds to an isotropic strain of 1-2%, substantially higher than the strain exhibited in our sample. This indicates strong © 2010 American Chemical Society
Acknowledgment. The work presented here was supported by a donation from Schlumberger-Doll Research. A.H.C.N. acknowledges the partial support of the U.S. Department of Energy under Grant DE-FG02-08ER46512 and A.K.S. acknowledges support from the National Science 9
DOI: 10.1021/nl901625v | Nano Lett. 2010, 10, 6-10
Foundation Grant DMR-0706574. We thank David H. Newby for performing the AFM measurements.
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© 2010 American Chemical Society
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DOI: 10.1021/nl901625v | Nano Lett. 2010, 10, 6-10
