Preparing local strain patterns in graphene by atomic force microscope based indentation

TL;DR

This paper introduces an atomic force microscope-based method to create controlled strain patterns in substrate-supported graphene, enabling new ways to tailor its electronic properties without lithography.

Contribution

The authors develop a novel AFM-based technique for patterning strain in graphene, providing a versatile tool for mechanical property engineering.

Findings

Strain patterns can be precisely induced in graphene using AFM indentation.

The method allows for controlled and localized deformation of graphene.

Potential applications in designing graphene-based electronic devices.

Abstract

Patterning graphene into various mesoscopic devices such as nanoribbons, quantum dots, etc. by lithographic techniques has enabled the guiding and manipulation of graphene's Dirac-type charge carriers. Graphene, with well-defined strain patterns, holds promise of similarly rich physics while avoiding the problems created by the hard to control edge configuration of lithographically prepared devices. To engineer the properties of graphene via mechanical deformation, versatile new techniques are needed to pattern strain profiles in a controlled manner. Here we present a process by which strain can be created in substrate supported graphene layers. Our atomic force microscope-based technique opens up new possibilities in tailoring the properties of graphene using mechanical strain.

Figures

Full text

Preparing local strain patterns in graphene by atomic force microscope based indentation

Péter Nemes-Incze

Centre for Energy Research, Institute of Technical Physics and Materials Science, Nanotechnology Department, 2D NanoFab ERC Research Group, Budapest 1525, POB 49, Hungary

corresponding author email: nemes.incze.peter@energia.mta.hu

Gergő Kukucska

Department of Biological Physics, Eötvös University (ELTE), Pázmány Péter sétány 1/A, 1117 Budapest, Hungary

János Koltai

Department of Biological Physics, Eötvös University (ELTE), Pázmány Péter sétány 1/A, 1117 Budapest, Hungary

Jenő Kürti

Department of Biological Physics, Eötvös University (ELTE), Pázmány Péter sétány 1/A, 1117 Budapest, Hungary

Chanyong Hwang

Korea Research Institute of Standards and Science, Center for Nanometrology, Daejeon 305-340, Republic of Korea

Levente Tapasztó

Centre for Energy Research, Institute of Technical Physics and Materials Science, Nanotechnology Department, 2D NanoFab ERC Research Group, Budapest 1525, POB 49, Hungary

László P. Biró

Centre for Energy Research, Institute of Technical Physics and Materials Science, Nanotechnology Department, Budapest 1525, POB 49, Hungary

**Patterning graphene into various mesoscopic devices such as nanoribbons, quantum dots, etc. by lithographic techniques has enabled the guiding and manipulation of graphene’s Dirac-type charge carriers. Graphene, with well-defined strain patterns, holds promise of similarly rich physics while avoiding the problems created by the hard to control edge configuration of lithographically prepared devices. To engineer the properties of graphene via mechanical deformation, versatile new techniques are needed to pattern strain profiles in a controlled manner. Here we present a process by which strain can be created in substrate supported graphene layers. Our atomic force microscope-based technique opens up new possibilities in tailoring the properties of graphene using mechanical strain. **

Graphene, being a two-dimensional crystal, has an exposed surface which makes it easy to manipulate its atomic and electronic structure [1, 2]. Until recently, focus has been on patterning graphene into nanostructures, mostly with the aim of tailoring its charge transport properties [3, 4, 5]. However, in such nanostructures electronic states of the rough edges obscure quantum confinement effects [6]. One promising alternative to lithographically cutting graphene, is applying mechanical strain to it. Changing the charge transport properties of a material by straining its crystal lattice is not a new idea, it has been realized in the silicon industry with success [7]. Examining the case of graphene, one finds that in order to achieve significant changes in the band structure, large homogeneous strains in the range of 15-20% need to be applied [8], making this approach impractical because the maximum failure strain is also of this order [9]. Furthermore, such a large strain is hard to implement into a working electronic device [10]. However, periodic strain patterns show promise of creating new functionality in graphene devices, through electron wave guiding, pseudo-magnetic fields, valley polarization etc. [11, 12, 13, 14, 15, 16, 17, 18, 19]. Since the graphene lattice remains intact in these cases, new physics can be explored without the plague of edge disorder observed in lithographically processed nanostructures. Strain engineering of this kind has been attempted by harnessing moiré patterns [20, 21, 22], using the different thermal expansion of graphene and its support [20, 21, 22, 23], substrate induced rippling [24, 25] and by placing graphene on a pre-patterned substrate [26, 27, 28, 29, 30, 31, 32]. The drawback of these approaches is that the amount of strain induced, as well as its crystallographic orientation is not easily controlled. Additionally, there is a need for a transfer step to the pre-patterned substrate which may induce defects in the graphene layer. Moiré patterns that form between the graphene and substrate are limited in their applicability, because the pattern is inherently determined by the alignment and lattice parameters of the two materials. Furthermore, with the exception of hexagonal boron nitride [21, 20], such moiré patterns are constrained to metallic substrates, making charge transport measurements problematic. Until now, no truly versatile method of introducing strain into graphene has been demonstrated. Here we show that strain patterns can be prepared in a graphene flake on SiO2, through an AFM indentation approach that combines the ability to write tailor made strain profiles, with the possibility to control the crystallographic orientation of the strain.

Results and Discussion

Our sample system consists of exfoliated graphene flakes on a silicon support, with a 300 nm SiO2 capping layer. Being one of the strongest materials known [9], it is natural to assume that in an AFM indentation experiment graphene does not get damaged in the initial phase of the indentation, while the SiO2 substrate can undergo plastic deformation. Stopping the indentation before rupture of the graphene occurs, can leave the graphene membrane pinned to the deformed substrate. During indentation, the tip is lowered towards the sample surface until a pre-set cantilever deflection (see Methods). In the next step, the tip is either retracted or moved along a line on the sample surface (Figure 1a). The procedure can be repeated with changing the tip location, resulting in an indentation dot or line pattern (Figure 2a,b). No significant damage to the graphene has been observed either through AFM or Raman measurements up to a final indentation depth of 1.5 nm. With deeper indentation, the rupture of the graphene layer becomes increasingly likely (see Supplementary Figure S1). Imaging of the resulting patterns is done using the same tip in tapping mode, unless otherwise noted. Importantly, the crystallographic directions of the graphene can be revealed before patterning, by imaging the surface using a softer cantilever (typically 0.1 N/m force constant) in contact mode. In this case the frictional forces experienced by the tip are modulated by the atomic lattice (see inset in Figure 1b).

To determine the magnitude of the strain, we have measured Raman spectroscopy maps of the indentation patterns, with the help of a confocal Raman microscope, using a 532 nm or 633 nm excitation laser. If graphene is subjected to tensile strain, both the G and 2D peak positions shift down in wave number, by a factor determined by the respective Grüneisen parameter [34, 33]. These parameters are in the range of $\partial\omega_{2D}/\partial\epsilon\approx-83cm^{-1}/\%$, $\partial\omega_{G^{+}}/\partial\epsilon\approx-36cm^{-1}/\%$ and $\partial\omega_{G^{-}}/\partial\epsilon\approx-18cm^{-1}/\%$ for uniaxally applied strain, as measured by Mohiuddin et al. [34]. The G peak has two Grüneisen parameters, because if the strain has a uniaxial character it will split into two subpeaks called G+ and G-. From these parameters it is clear that the 2D peak shows much more shift as a function of strain than the G peak, making possible the detection of strains in the range of 0.01% [34]. Because of this property we choose to plot the 2D peak wave number in our Raman maps, to make the strain variations induced by the AFM tip clearer. In Figure 1c a plot of the 2D peak position can be seen across a sample area containing $2\mathrm{\mu m\times 2.5\mu m}$ arrays of line patterns similar to the one in Figure 1b, each array being composed of 50 indent lines of $2\mathrm{\mu m}$ length (AFM images of the patterns: supplementary Figure S4). As the indentation depth is varied from array to array, from 0.15 nm to 0.5 nm, the downshift in G and 2D peak position becomes stronger, meaning increased strain (see Figure 1f,g). Of course it has to be noted that the strain distribution within the indentation lines will be far from constant [15] and Raman spectroscopy only probes the average of the strain in the graphene inside the laser spot of roughly 500 nm diameter. Within these limitations we will quantify the average strain in these structures. In Figure 1d we show a correlation plot of the G and 2D peak positions, measured with 532 nm excitation. If the G and 2D peak shifts are due to strain effects, their shift is only determined by their respective Grüneisen parameters ($\Delta\omega_{G}$, $\Delta\omega_{2D}$) [34]. Thus, the measurement points in the correlation plot will lie along a line, the slope of which is determined by the ratio of $\Delta\omega_{2D}/\Delta\omega_{G}$ (blue line). This ratio lies within a range of 2.2 to 2.8, depending on the anisotropy in the strain distribution and the crystallographic direction of the strain in the pure uniaxial case [33]. In addition to strain, the change in the graphene chemical potential can also shift the peak positions. If doping effects are significant, the data points will show a deviation from the blue slope. If purely doping is the source of the peak shifts the G peak is more strongly affected than the 2D peak and the slope of the line corresponding to it is 0.75, as shown by the red line in Figure 1d. Following the evolution of the Raman peak positions with increasing indentation depth, the data points move along the blue line. The largest 2D peak shift of 8 cm*-1*, with respect to the unperturbed graphene is observed for the 0.5 nm deep indentation marks, corresponding to an average strain of 0.1%, using the Grüneisen parameter shown above. Although splitting of the G peak can be expected, we do not observe this due to the small overall Raman shift. Examples of raw Raman spectra, used to create the map and correlation plot in Figure 1c,d can be seen in Figure 1e-g. Notice the absence of any disorder induced peak around 1350 cm*-1*, indicating that the number of lattice defects introduced during indentation is negligible.

Raman spectroscopy also gives us the means to demonstrate that not only can we tune the magnitude of the strain in the patterns, but also to influence the direction and symmetry of the strain. In graphene, the crystal momentum of the scattered electron is selected by the polarization of the excitation laser [35]. This means that changing the polarization of the laser we can probe the strain in the graphene in various directions. Keeping the laser light in the same spot on a line pattern (40 nm line spacing) and a hexagonal dot pattern (40 nm nearest neighbor dot distance), we have measured the dependence of the 2D peak shift with rotating the polarizer of the incident laser beam. In Figure 2e,f we show a polar plot of the resulting peak shift (red dots), compared with the same measurement performed on the unperturbed graphene next to the patterns (blue dots). The data points on the unperturbed graphene form a circle, meaning the average strain distribution within the laser spot is isotropic, as would be expected for graphene on SiO2. On the other hand, the measurement on the line pattern shows a 2D peak position that is up to 1 cm*-1* smaller if the polarization vector of the laser is perpendicular to the indent lines (at $\theta=15^{\circ}$). Thus, the strain has a uniaxial character, being larger in the direction perpendicular to the indent lines [36]. In the case of the dot patterns, the 2D peak shift has a slight hexagonal character, which is aligned with the dot pattern (inset in Figure 2d). In this case the peak is shifted to higher values by up to 2 cm*-1*, if the polarization is perpendicular to the close packed direction of the indentation dots. Therefore, selecting the crystallographic orientation of the pattern, the direction of the strain with respect to the graphene lattice can be set.

The remarkable observation that graphene stays in the strained configuration after the AFM tip is retracted, leads us to explore the energetics of adhesion. The pinning of graphene onto a corrugated substrate can be achieved if the adhesion energy due to van der Waals forces ($E_{vdW}$) is larger than the elastic energy ($E_{el}$) induced in the graphene. To be able to compare the two quantities in the present experiment it is necessary to know the exact geometry of the graphene in the pinned configuration. AFM probes with a nominal tip radius of curvature of 2 nm have been used to image indentation patterns (see Figure 3a). Gaussians of the form: $h_{0}(1-exp(-r^{2}/2\sigma^{2}))$, with a variance $\sigma$ in the 7 nm range and depths ($h_{0}$) from 0.7 to 1 nm, fit the AFM height data very well (Fig. 3c). In estimating $E_{el}$ for the present graphene geometry, the bending energy can be safely disregarded, so that the elastic energy is assumed to be dominated by the in-plane stretching of the graphene membrane. In this regime we can apply the calculations of Kusminskiy et al. [37] for graphene adhered to a Gaussian depression, where the ratio of the Gauss depth to the variance determines the onset of depinning from the substrate. For a conservative assumption of graphene-SiO2 adhesion energy [38, 39] of 2 meV/Å2 the $h_{0}$/$\sigma<0.28$ ratio is needed for stable pinning of the graphene to the substrate. In the case of the dot patterns prepared here, this ratio is up to 0.14. From a mechanical stability point of view, this means that the graphene in the dot patterns is still well within the pinned configuration. Estimating the strain from the geometry, one obtains for this dot pattern 0.15%. As the strain in the deformation also scales with $h_{0}$/$\sigma$, an increase in the possible strain by a factor of 2 could be achieved if AFM tips with smaller tip radius are used for patterning. The above calculation assumes that the graphene is adhered by van der Waals forces to the whole surface of the Gaussian shaped hole [37]. This is a reasonable assumption, since the graphene is pushed into close contact with the support during indentation. Therefore, it is expected that the adhesion is improved with respect to exfoliated graphene on SiO2, where the graphene layer is partially suspended [40, 41].

The effect of strain on the orbital motion of electrons in graphene can be described using a vector potential, corresponding to a time reversal symmetric pseudo-magnetic field [2]. This vector potential is of the form: $\vec{A}=\frac{\beta\hbar}{2ae}(u_{xx}-u_{yy},-2u_{xy})$, where $\beta\approx 2$, $a$ is the lattice constant, $e$ is the elementary charge and $u_{ij}$ is the strain tensor [27, 2]. The resulting pseudo-magnetic field is given by $B_{\mathrm{ps}}=(\nabla\times\vec{A})_{z}$, it’s effect on graphene’s electronic states having been measured previously by scanning tunneling microscopy [42, 41, 22].

In order to calculate the pseudo-magnetic field induced by the indentation, we need to quantify $u_{ij}$. Since, displacements in the $z$ direction (perpendicular to the graphene plane) are much bigger than displacements in-plane, we can safely neglect the in-plane component [43, 41], resulting in a strain tensor: $u_{ij}=\frac{1}{2}\partial_{i}h\partial_{j}h$, where $h$ is the out of plane displacement of the graphene layer. We can measure $h$ by AFM topography maps, as long as the AFM tips used for imaging the indentation patterns are much sharper than the ones used to prepare the patterns (see Methods). As an example, the AFM topography of an indentation hole pattern (see Fig. 3a) has been used to calculate the strain tensor and the resulting pseudo-magnetic field (Fig. 3b) by numerical differentiation of $h$. The resulting pattern of $B_{\mathrm{ps}}$ is largest around the indentation marks (see dashed circle in Fig. 3b) and forms a petal-like structure with alternating positive and negative values of $B_{\mathrm{ps}}\approx$ 4T. This flower-like $B_{\mathrm{ps}}$ pattern is characteristic of circularly symmetric deformations [15, 44, 45, 41] and we can compare this to the $B_{\mathrm{ps}}$ of an ideal Gaussian, because the indentation dots are well fitted by Gaussians (Fig. 3c). Fig. 3d shows the calculated $B_{\mathrm{ps}}$ pattern of a Gaussian having a depth of 1 nm and a variance of 7 nm. The maximum pseudo-magnetic field in this pattern is 5 T, in good agreement with the $B_{\mathrm{ps}}$ map calculated from the AFM topography data. To put the $\sim$4 Tesla pseudo-magnetic field induced by the indentation into perspective, it is instructive to compare it to the pseudo-magnetic field fluctuations resulting from the substrate induced rippling of graphene on SiO2. From magneto-transport measurements, such fluctuations were estimated to be in the 1T range [46]. Therefore, AFM indentation can be used to significantly perturb in a tunable fashion the electronic properties of graphene.

In summary, scanning probe based techniques have demonstrated remarkable versatility in lithographically cutting nanostructures into graphene [47, 4]. Here we have shown that in an analogous fashion, strain can be induced in SiO2 supported graphene by AFM indentation. The crystallographic orientation, magnitude, periodicity of the strain patterns can all be tuned. The versatility of the strain patterning technique is demonstrated in Figure 3e, where we have prepared a strain pattern in graphene showing the initials of our institute. These results open up the way to the exploration of tailor made strain profiles in graphene and enable new device concepts, using strain engineering. For example, creating periodic strain patterns to realize exotic quantum states, such as a valley ordered ground state [48].

Methods

AFM patterning and imaging details. For the indentation experiments, a Bruker Multimode 8 AFM, equipped with a closed loop scanner, is used. For all indentation experiments diamond-like carbon coated silicon AFM probes (Tap300DLC, Budget Sensors) are used, with a nominal force constant of 40 N/m and a tip radius of 15 nm. Imaging of the indentation patterns was carried out, using AFM tip having a 2 nm nominal tip radius (SSS-NCH type, NanoWorld)

For the indentation experiments the NanoMan lithography software of Bruker has been used. Between indentation steps the tip was moved in tapping mode. At the begin of indentation the tip was lowered towards the sample surface with a $z$ velocity of 400 nm/s, until deflection of the cantilever has taken place. In the case of the dot patterns the tip was retracted with the same $z$ velocity and moved to a new position for the next indentation step. In the case of the line patterns after moving towards the sample the tip was dragged across the surface in contact mode without feedback with a velocity of 200 nm/s. Finally the tip was retracted and moved in tapping mode to the new line location. The z displacement of the tip was controlled either by setting a cantilever deflection threshold or by moving the tip towards the sample by 40-100 nm. The final indent depth was used as a control parameter during indentation experiments because of the variability in cantilever spring constant and tip sharpness. The typical cantilever spring constant was 40 N/m, with a tip radius of $\sim$15 nm (Tap300DLC, Budget Sensors). However, due to large variability in these parameters, the z movement was incrementally adjusted. An indentation experiment was always followed by imaging the patterned location for the onset of plastic deformation of the SiO2.

Raman measurements. Raman measurements were carried out using a Witec 300rsa+ confocal Raman spectrometer, using a 532 nm or 633 nm excitation laser.

Author contributions statement

PN-I designed and carried out the experiments. GK has done the calculations. CH, LT, JKo, JKü and LPB reviewed the data. JKü, JKo and LPB supervised the project. PN-I wrote the manuscript. All authors discussed the data and reviewed the manuscript.