Anisotropic straining of graphene using micropatterned SiN membranes

Graphene displays a range of remarkable properties that have catalyzed—since its discovery in 2004¹—an impressive interest in the scientific community.² Its unique electronic behavior stems from the hexagonal honeycomb structure of the carbon lattice, which forces low-energy conducting electrons to assume linear dispersions that mimic massless relativistic fermions.³ In addition, graphene displays an unusual mechanical strength and strains up to 10% can be applied to it without damaging appreciably its structure.⁴ This factor, combined with its intrinsic two-dimensional nature, opens unique perspectives for the investigation of strain engineering^5–7 and for the development of novel device concepts.⁸ In fact, it has been predicted,^7,9 and in part experimentally demonstrated,^10,11 that mechanical deformations in graphene can be used to tailor its electron properties. As a particularly inspiring possibility, it is known that a suitable deformation of the honeycomb lattice can be equivalent to the application of a pseudomagnetic field.^6,12

Achieving a controlled strain profile in graphene poses non-trivial technical challenges and various alternative approaches have been explored during recent years. Hydrostatic configurations were obtained and studied using circular holes and a uniform differential pressure load,^13–15 and the impact of strain was studied by micro-Raman spectroscopy.^16–18 In this device architecture, local strain was also induced or measured taking advantage of scanning probe techniques.^19,20 Differently, uniform biaxial strain configurations could be induced by taking advantage of piezoelectric effects in the substrate²¹ and by using pressure transmitting media.²² Concerning uniaxial strain, various studies have demonstrated the possibility to anisotropically deform graphene deposited on non-flat substrates,²³ on polydimethylsiloxane (PDMS)²⁴ or on similar stretchable substrates.^25,26 An alternative promising approach consists in anchoring graphene layers to micro-electromechanical actuators.²⁷ More elaborated strain profiles, in particular those giving rise to a pseudomagnetic field, have been hard to demonstrate so far. Interesting experimental evidence has been put forward in the context of random nanobubbles¹⁰ and fascinating results have been obtained in deformed artificial honeycomb structures mimicking the behavior of graphene.¹¹ In practice, the achievement of custom strain profiles has generally proved to be rather elusive.

In the present work, we demonstrate that markedly non-isotropic strain profiles can be obtained in free-standing graphene membranes that are clamped on an edge that is not radially symmetric and are subject to a vertical uniform load using a pressure difference between the two opposite faces of the graphene flake. In particular, we show that loaded elliptical membranes display Raman features that demonstrate the presence of an anisotropic component in the induced strain profile, in good agreement with what is expected with the studied geometry. The possibility to achieve custom strain profiles by choosing a suitable graphene clamping geometry is discussed.

Figure 1 shows the device architecture and setup adopted for this work. Free-standing graphene areas of various shapes and dimensions were obtained using micropatterned SiN membranes as the mechanical support for the graphene layer. Starting from a Si wafer doubly coated in “pre-stressed” $300nm$ of Si₃N₄, a combination of dry and wet etching protocols (see the supplementary material for further details) was adopted to obtain suspended $500×500μm2$ Si₃N₄ membranes with through holes of various geometries. In the present study, we investigated a set of elliptical holes with various major (a) and minor (b) axes: $a×b=5μm×10μm$⁠, $10μm×20μm$⁠, and $20μm×40μm$⁠. Circular holes were also investigated, as reported in the literature.^17,28 Large-scale monocrystalline graphene flakes used in the present work were obtained by CVD growth on Cu²⁹ and transferred on the Si₃N₄/Si chips using a standard “bubbling transfer” technique.³⁰ As sketched in Fig. 1(a), a differential pressure ΔP is applied orthogonally to the free-standing graphene region thanks to a 2mm-thick polydimethylsiloxane (PDMS) coupling layer placed on top of a modified microscope slide. As a result, the top side of the free-standing graphene region is subject to ambient pressure (conventionally $P0=1bar$⁠) while the bottom space can be partially or completely evacuated with a scroll pump. Static vacuum tests were performed to verify the stability of ΔP, which was found to decay over a time scale of various hours; this ensures pressure values measured by the gauge are meaningful. Maps were always performed under active pumping conditions, at ΔP=1bar.

Local graphene deformation is investigated by micro-Raman spectroscopy, using an inVia confocal system by Renishaw equipped with a polarized $λ=532nm$ laser source. Raman signal was collected through a 50× objective with N.A. = 0.75 and analyzed by 1800 grooves/mm grating. Light collection was not polarized, except for a minor polarization selectivity intrinsic in our single-grating monochromator. Raman maps were collected using a laser power of $1mW$ to minimize the impact of local heating. Laser polarization was always set along the ellipse minor axis. In Fig. 1(b) we report the measured Raman spectra collected at the center of a $10μm×20μm$ elliptical graphene region, for $ΔP=0$ (blue curve) and $ΔP=0.9bar$ (red curve); as expected, the G and 2D Raman peaks are significantly red shifted by strain in the suspended graphene region. Importantly, the modification of the Raman spectra was always found to be completely reversible upon removal of the pressure load. In addition, as visible from the maps of Fig. 2, no significant Raman shift was observed in the regions where graphene is supported by the SiN, even in the proximity of the hole. This proves that no measurable adjustment or sliding of graphene occurs during our experiments.

The most evident impact of deformation upon $ΔP$ is visible in Fig. 2, where we compare the map of the 2D peak Raman shift $ω2D$ for $ΔP=0$ (Fig. 2(a)) and $ΔP=1bar$ (Fig. 2(b)). For symmetry reason, its center frequency is sensitive to the hydrostatic component of the strain tensor $εij$⁠, which we name $ε¯=(εxx+εyy)/2$⁠. Experimentally, the 2D peak displays a maximal red shift at the center of the suspended region, similar to what was reported for inflated circular graphene membranes.¹⁷ The quantitative evolution of the shift versus $ΔP$ is highlighted in Fig. 2(c) where we report $ω2D$ measured at the maximal shift region at the center of the ellipse. The phononic mode giving rise to the higher order 2D peak is sketched in Fig. 2(d). Raman shifts at various pressure loads are compared with the adimensional parameter $η=(ΔP/P0)2/3$⁠, where $P0=1bar$⁠; all the components of the strain tensor $εij$ are in fact expected to scale linearly with $η$⁠, as highlighted for specific cases in the recent literature.¹⁶ The validity of such a scaling law is also formally demonstrated, for a general clamping geometry, in the supplementary material. The overall dependence of $ω2D$ as a function of $ΔP$ and of the position on the ellipse is found to be largely consistent with reports on the simpler case of radially symmetric graphene clamping and with the most recent estimates of the Grüneisen parameters.^16,25,31 We also note that, consistently with reported data,¹⁶ the value of $ω2D$ for $ΔP≈0$ is found to display a further surprising red shift. This effect was attributed to uncertainties in the exact determination of $ΔP$⁠. We believe that an additional reason for the shift is possibly related to graphene adhesion on the vertical sidewalls of the SiN hole, which is known to occur in this kind of graphene drums¹⁶ and could be relevant in the low-$ΔP$ regime. The verification of this hypothesis will likely require a combined Raman and atomic force microscopy study at low pressure loads, which goes beyond the scope of the present work. Further details regarding the numerical calculation of the $εij$ tensor as a function of $ΔP$ and scaling rules are reported in the supplementary material.

While hydrostatic deformations explain well the coarse evolution of the Raman spectra, the strain profiles in our elliptically clamped graphene membranes are expected to display a marked deviation from a uniform strain configuration and a larger strain can be expected along the shorter axis of the ellipse. More in general, the anisotropic component of the strain field can be expressed through the invariant

corresponding to the difference between two eigenvalues of the strain tensor $ε=ε¯±Δε/2$⁠. It is well known³¹ that strain anisotropy, when sufficiently large, can be detected in Raman spectroscopy as a splitting of the degenerate phononic modes $G±$⁠. In Fig. 3 we report a detailed study of the Raman spectrum of the G peak region as a function of $ΔP$⁠. Data reported in Fig. 3 refer to the largest explored Si₃N₄ elliptical $20×40μm2$ hole; larger suspended areas in fact correspond—for a given value of ΔP—to a larger anisotropic strain $Δε$⁠.

A first rough analysis was performed by fitting the Raman data with a single Lorentzian peak. As visible in Fig. 3(a), the resulting map of the Raman shift $ωG$ at $ΔP=1bar$ is found to be consistent with the $ω2D$ map reported in Fig. 2(a). For comparison, we report in Fig. 3(b) the map of $ε¯$ calculated for the same pressure load. A top hydrostatic strain of 0.68% is expected, in agreement with the observed red shift of the G peak and known values of the corresponding Grüneisen parameter (see the supplementary material for further details). As argued in the following, on the other hand, hydrostatic strain is not sufficient to satisfactorily describe the evolution of the G peak as a function of the pressure load. Indeed, a non-trivial broadening is visible in Fig. 3(c), where we report the FWHM $ΓG$ resulting from the same fitting procedure. The observed broadening displays a peculiar “saddle point” spatial evolution, with large $ΓG$ values close to the central part of the ellipse, while a significantly smaller effect is observed at the top and bottom apexes. Remarkably, a very similar pattern is visible in Fig. 3(d), displaying the calculated $Δε$ for the at $ΔP=1bar$⁠. This suggests that, beyond mechanisms highlighted in recent works,¹⁶ broadening in our experiment is also in part connected to strain anisotropy.

A more detailed investigation of the broadening mechanism was performed through the analysis of the G peak measured at the center of the elliptical hole, which represents a good trade-off between the expected value of $Δε$ and the minimization of the impact of the borders of the Si₃N₄ hole. In this position, numerical estimates indicate that a top anisotropic strain component $Δε$ = 0.64% can be expected. The Raman spectrum in the G peak region for $ΔP=1bar$ is reported in Fig. 3(e); the peak displays a lineshape which is clearly consistent with the superposition of two nearby Lorentzian peaks, which we interpret as corresponding to the G₊ and G₋ modes in uniaxially strained graphene (see Fig. 3(f)). A similar analysis (see the supplementary material for further information concerning the fitting procedure) was performed for various values of $ΔP$ and the resulting peak positions are reported in the inset to Fig. 3(e). Two divergent peaks are obtained with a top splitting of about 10 cm⁻¹, which is in very good agreement³¹ with what is expected for an anisotropy $Δε=0.64%$⁠. A weighted linear regression of the two peak positions $ωG±$ yields two remarkably linear trends in $η$ that converge almost exactly for $ΔP=0$⁠, as expected. It is important to note that, since light collection in our experiment is not polarized, both G₊ and G_$−$ are expected to be visible. In fact, only a minor amplitude difference is observed, as a likely consequence of the residual polarization sensitivity of our single-grating monochromator. Finally, we note that no evidence of a clear splitting of the 2D peak could be detected in the current experiments, differently from what was reported in other works.³² Since the 2D splitting strongly depends on the strain direction, this behavior could be linked to the orientation of the studied flakes with respect to the ellipse axes.

Our work demonstrates that non-trivial strain profiles can be obtained in Si₃N₄ holes with an elliptical shape and that a sizable anisotropic component in the graphene strain can be obtained. Similarly, we expect that more in general clamping geometries can be used to design even more advanced non-uniform and non-isotropic strain profiles, taking advantage of a relatively robust implementation with no free graphene edges. The relevant case of triaxial strain, which however also requires a correct crystalline orientation to give rise to a pseudo magnetic field, is reported in the supplementary material. In view of the possibility to impact the electronic states via the engineering of custom strain profiles, the observed $G±$ splitting phenomenology has also been compared with a first-principle calculation on an atomistic model system mimicking the experimental setup. To this end, we have simulated an unstrained suspended graphene layer with an elliptic-shape depression (see the inset of Fig. 4). To reproduce the experimental configuration, we have fixed the position of the carbon atoms external to the ellipse to a “zero” height, as it happens for graphene on Si₃N₄ substrate, and the effect of the vertical load has been reproduced by fixing the two carbon atoms at the center of the ellipse (blue dots in the inset) at a lower vertical position and leaving all the other carbon atoms in the ellipse free to relax in order to reach the minimum energy structure. The simulation has been performed on a $7.4Å×12.8Å$ ellipse containing a total of 22 carbon atoms. The Raman spectra of the system have been calculated by means of density functional perturbation theory³³ as implemented in QUANTUM-ESPRESSO code,³⁴ with local density approximation and norm-conserving pseudopotential for the carbon atoms.³⁵ We used a plane wave expansion up to $80Ry$ cutoff and $4×4×1$ Monkhorst-Pack mesh³⁶ for the sampling of Brillouin zone. The ellipse depression was created in the central part of a $5×5$ graphene supercell with the minimum energy configuration lattice parameter $a0=12.24Å$⁠.

In the absence of strain, the frequency of the degenerate G phononic mode is $ωG=1607.5$ cm⁻¹. For $ΔP≠0$⁠, the value of the effective strain along the ellipse axes has been estimated from the depth of the two central carbon atoms, $δ$ and the length of the principal axes of the ellipse, a and b; e.g., for the major axis we have $εa∼(2L−a)/a$ where $L∼(a/2)2+δ2$ is the profile length for the depression along the a direction; the same holds for the minor axis. The results of the calculation are shown in Fig. 4; the asymmetry of the strain due to the elliptical geometry of the depression is responsible of an averaged splitting of the G peak mode that is found to be of the same order of magnitude as the one experimentally induced by uniaxial strain on the graphene layer.²⁵ 

In conclusion, we have provided evidence of an incipient splitting of the G mode in free-standing graphene regions clamped to elliptical holes in Si₃N₄ and subject to a uniform differential pressure load. Our results indicate a promising route to induce custom strain profiles, which can be controlled by the applied pressure load and designed according to the chosen geometry of the supporting Si₃N₄ frame. We also highlight that our experiment has been performed using large scale CVD monocrystalline graphene flakes, thus providing a route for scalable strain-engineered graphene devices. Finally, we would like to stress that present results have been obtained using a vacuum chamber to induce a maximal load $ΔP=1bar$⁠. A recent report¹⁶ demonstrates that using pressurized gas as a load it is possible to reach $ΔP=14bar$⁠, potentially leading to significantly increased achievable strain magnitude (about a factor six larger strain can be expected) and/or to less stringent limits on the minimal area of the Si₃N₄ holes.

See the supplementary material for more details about sample fabrication, numerical calculations, demonstration of the scaling law, and Raman measurements. An example of a hole geometry which induces a triaxial strain profile with the related pseudo-magnetic field is also given.

This work was supported by the EC under the Graphene Flagship Program (Contract No. CNECT-ICT-604391) and by the ERC advanced grant SoulMan (No. G.A. 321122). The authors acknowledge the “IT center” of the University of Pisa for the computational support and the allocation of computer resources from the CINECA, ISCRA C Project Nos. HP10C6H6O1 and HP10CAI9PV. F.C., A.P., and S.R. thank S. Gröblacher and R. Norte for their help in the initial development of this experiment. S.R. acknowledges the support of the CNR through the bilateral CNR-RFBR projects. F.C. acknowledges support from Fondazione Silvio Tronchetti Provera.