Optically Forged Diffraction-Unlimited Ripples in Graphene

In nanofabrication, just as in any other craft, the scale of spatial details is limited by the dimensions of the tool at hand. For example, the smallest details of direct laser writing with far-field light are set by the diffraction limit, which is approximately half of the used wavelength. In this work, we overcome this universal assertion by optically forging graphene ripples that show features with dimensions unlimited by diffraction. Thin sheet elasticity simulations suggest that the scaled-down ripples originate from the interplay between substrate adhesion, in-plane strain, and circular symmetry. The optical forging technique thus offers an accurate way to modify and shape 2D materials and facilitates the creation of controllable nanostructures for plasmonics, resonators, and nano-optics.

* S Supporting Information ABSTRACT: In nanofabrication, just as in any other craft, the scale of spatial details is limited by the dimensions of the tool at hand. For example, the smallest details of direct laser writing with far-field light are set by the diffraction limit, which is approximately half of the used wavelength. In this work, we overcome this universal assertion by optically forging graphene ripples that show features with dimensions unlimited by diffraction. Thin sheet elasticity simulations suggest that the scaled-down ripples originate from the interplay between substrate adhesion, in-plane strain, and circular symmetry. The optical forging technique thus offers an accurate way to modify and shape 2D materials and facilitates the creation of controllable nanostructures for plasmonics, resonators, and nanooptics.
O ne of the central aims in nanoscience is to be able to modify nanostructures at will. Modifications are necessary because it is rarely the pristine materials but the modified and engineered materials that establish functionalities for practical applications. 1,2 Modifications are particularly necessary for 2D materials. 3,4 Graphene, for instance, gains specific functionalities once modified into ribbons, 5,6 introduced with pores or adsorbants, 7−10 or curved into 3D shapes. 11−13 However, all modification techniques have their limitations. Direct mechanical manipulation is either slow and accurate, 14 or fast, coarse, and nonreproducible. 15,16 Thermal annealing, 17,18 electron irradiation, 19,20 chemical treatment, 21, 22 and Joule heating 23 may be scalable but spatially imprecise due to their random character. It is particularly challenging to modify 2D materials into customized ripples and other 3D shapes. Such modifications frequently require dedicated experimental apparatuses 24 or specially prepared substrates. 25 The difficulty for 3D modification lies partly in substrate adhesion. Although often of weak van der Waals type, adhesion effectively prevents controlled detachment of 2D membranes from the substrate.
Limitations exist also in optical patterning. Although optical techniques may be scalable and easy to apply, the spatial details are determined by the size of the focused laser beam. Creating patterns with details finer than beam size is just as difficult as scribbling equations on a piece of paper with a spray can. Still, optical techniques have plenty of potential for exploration because irradiation provides various mechanisms to modify 2D materials, depending on laser energy and ambient atmosphere. 26,27 One particularly promising, still mostly untapped technique is the so-called optical forging, which alone enables controlled and on-the-fly 3D shaping of graphene. 28 Given the ubiquity of various limitations, there is urgency to improve techniques to modify and engineer 2D materials scalably, accurately, and preferably in situ, without customized preparations.
In this work, we demonstrate optical forging of graphene into circular ripples with features much smaller than the size of the laser beam. By using thin sheet elasticity simulations, the rippling is shown to arise from the interplay between substrate adhesion, in-plane stress due to optical forging, and the underlying circular symmetry. Being based on direct irradiation of graphene without specially prepared experimental settings, optical forging provides a practical technique and thereby substantially broadens our abilities to modify and enhance the functionalities of graphene and maybe even other 2D materials.
To prepare the sample, we grew single-layer graphene by chemical vapor deposition (CVD) on a Cu substrate 29 and transferred it to thermally grown SiO 2 . For fabrication details and graphene characterization, see the Supporting Information (SI).
Selected points in the sample were then irradiated by a 515 nm femtosecond laser focused with an objective lens (N.A. of 0.8) to a single Gaussian spot. To prevent photoinduced oxidation during the irradiation, the sample was installed inside a closed chamber purged with N 2 . 27 The laser produced 250 fs pulses at 5−25 pJ/pulse energy and 600 kHz repetition rate for a tunable irradiation time τ. This process is called optical forging and results in blistering of the graphene membrane ( Figure 1a and Movie 1 in the SI). Blistering occurs due to local expansion of the membrane, caused by laser-induced defects and the related compressive in-plane stress. 28 The local expansion field ε(r) therefore depends on the time-integrated laser intensity profile I(r), which enables accurate control over the expansion and blister height via the irradiation time τ. Consequently, we irradiated the sample at separate spots for irradiation times ranging from τ = 0.1 to 3600 s. Finally, the blistered sample was characterized by Raman spectroscopy and measured by an atomic force microscope (AFM; see the SI).
The systematic increase in irradiation time produced a nontrivial but beautiful and reproducible pattern of blisters ( Figure 1b and Figure S4). In particular, blisters had profiles more complex than the usual domes. 30 At short irradiation times (τ < 0.4 s) the graphene remained flat on the substrate (Figure 1c). At intermediate irradiation times (0.4 ≤ τ ≤ 2 s), the graphene developed blisters with one circular ripple (Figure 1d). At long irradiation times (τ ≥ 5 s) the graphene developed concentric ripples in progressively increasing numbers and a gradually developing central dome ( Figure 1e and Movie 2 in the SI). Parts of the area between the blisters were detached from the substrate because the laser irradiated also during the movement from one spot to another. Note that the radial features in the ripples have dimensions down to 100 nm, nearly 10 times smaller than the laser spot and the ripple diameters themselves. Optical forging can thus reach 3D shaping of graphene that beats the diffraction limit. This is our main result.
To quantify the expansion of the graphene membrane, we used AFM height profiles to measure the increase in the surface area of the blisters. Within the projected areas of ∼1 μm 2 , the corrugated membrane areas increase nearly monotonously upon increasing irradiation time, reaching 10 −2 μm 2 (∼1%) area increase at τ = 1 h ( Figure 2a). Initially, the area increases linearly in irradiation time, at rate 22 nm 2 /s ( Figure 2b). This area increase was used to determine radiusdependent linear expansion, ε(r). By assuming here a onephoton process and a Gaussian laser intensity profile I(r), we obtain

The Journal of Physical Chemistry Letters
Letter where r = 0 at the center of the spot and fwhm = 800 nm is the full width at half-maximum of the laser beam. Because the laser focal spot was difficult to maintain, fwhm had to be treated as a parameter and adjusted to give the best overall fit to the observed lateral dimensions in the experiment. The maximal expansion ε 0 increases at the rate 1.5 × 10 −3 %/s at short irradiation times and saturates at almost 1% at long irradiation times ( Figure 2). The initial linear rate and the saturation are in good agreement with previous experiments. 28 The diffraction-unlimited rippling suggests a mechanism that involves competition between surface adhesion and expansioninduced stress. To investigate the mechanism in detail, we simulate blister growth by classical thin sheet elasticity model. 31 Such models have proven successful in the modeling of deformed graphene membranes. 16,32−36 The energy in the model contains in-plane strain energy, out-of-plane bending energy, and surface adhesion. The laser-induced isotropic expansion is introduced via the diagonal of the in-plane strain tensor as e αβ (r) = e αβ 0 (r) − δ αβ ε(r), where ε(r) is the expansion field and e αβ 0 (r) is the strain tensor of the unexpanded, pristine graphene. 28 The adhesion is modeled by the generic 12−6 Lennard-Jones potential. 37 This model was discretized, implemented in two computer codes (with and without circular symmetry), and used to optimize blister geometries for given ε 0 and adhesion energy ε adh . 38 For details, see the SI. Before analyzing the model in full, it is instructive first to ignore adhesion and calculate a few analytical results. Because of the smallness of the graphene bending modulus, on micrometer-length scales the mechanical behavior is domi-nated by in-plane strain energy. 39 The strain energy is minimized when e αβ ≈ 0 or e αβ 0 ≈ δ αβ ε(r). To a first approximation, eq 1 then implies an area increase of ΔA = [π/ (2 log 2)] × fwhm 2 ε 0 . (This relation was previously used to transform ΔA into ε 0 .) With the displacement vector a⃗ (r) = a r (r)r̂+ a z (r)z, the diagonal components of the strain tensor become e r a r a r a r r e r a r r a r r r where r refers to radial and t refers to tangential in-plane component, and prime stands for a derivative with respect to r. Because the in-plane strain energy minimizes at e αβ ≈ 0, we obtain a r (r) ≈ rε(r) and a r r r ( ) 16 log 2 ( ) ( /fwhm) That is, when the membrane adapts to isotropic expansion under radial symmetry, energy gets minimized by adjusting the slope into a fixed absolue value. When the slope is negative for all r, integration yields the profile a r r ( ) fwhm ( )/log 2 z ε = × . This profile corresponds to a blister with one central dome and a maximum height of h fwhm /log 2 max 0 ε = × . The numerically optimized blister profile follows this analytical estimate accurately (Figure 3a).

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Letter However, positive and negative slopes in eq 3 are equally acceptable. Because the energy cost of bending is small, it is cheap to create a kink that reverses the sign of a z ′(r) abruptly. This kink appears topologically as a perfectly round ripple (Figure 3b). Multiple kinks at different radii produce concentric ripples of varying heights and diameters ( Figure  3c). Compared with the scale of in-plane strain energy, blisters of different ripple counts are nearly isoenergetic. When the number of ripples increases, the slopes progressively deviate from eq 3. Otherwise, the analytical description of the blister profiles without adhesion is apparent.
The role of adhesion, then, is to pull the membrane down, toward the substrate. Understanding the behavior of adhesionfree membranes is helpful, but when elastic and adhesive energies compete, we have to rely on numerical simulations. We took a closer look at the blister with τ = 1 s, which is near the onset of blistering (Figure 1d). This 4 nm high blister has a 0.97 μm ripple diameter and ε 0 = 0.017% expansion, as given by the AFM profile. We simulated this blister using the experimental ε 0 and adhesion in the range ε adh = 0...1 eV/nm 2 .
When ε adh < 1 μeV/nm 2 , the ripple is broad and the middle of the blister is mostly detached from the substrate, disagreeing with the experiment (Figure 4a); adhesion remains a minor perturbation to the zero-adhesion profile (Figure 3b). When ε adh > 10 μeV/nm 2 , in turn, the ripple becomes too narrow and shallow, also disagreeing with the experiment; when ε adh ≳ 100 μeV/nm 2 , the membrane ultimately snaps flat on the substrate. However, around ε adh ≈ 3 μeV/nm 2 , adhesion pulls the membrane down so that both the ripple width and height agree with the experiment. Using the adhesion ε adh = 3 μeV/nm 2 , one-ripple blistering occurs at ε 0 ≈ 0.005%, and the blister height increases linearly when ε 0 further increases (Figure 4b). This simulated trend agrees with the experimental trend in one-ripple blisters (τ ≲ 2 s). These agreements suggest that the adhesion between laser-modified graphene and SiO 2 is observable but substantially smaller than typically observed for pristine van der Waals solids and clean interfaces. 40 For completeness, we optimized all 18 blisters by using ε adh = 3 μeV/nm 2 and by adopting the observed set of ripples as initial guesses. After optimization, the resulting pattern of blisters turned out similar to the experimental ones (Figure 4f). At small ε 0 , stable blisters have only one ripple (Figure 4c), but at larger ε 0 , stable blisters have multiple ripples (Figure 4d). Simulations capture the main features of the experimental blisters, even if they deviate with respect to certain details, presumably due to the asymmetric expansion field and small variations in the initial conditions of the graphene membrane, generated during the sample fabrication.
Yet a question remains: Why do blisters initially appear with one circular ripple? This question can be addressed by considering eq 3. The preferred slope has a maximum at r fwhm/2 ln 2 0 = = 0.48 μm. In other words, around radius r 0 , the energy to keep the membrane flat is the largest. When the in-plane stress in a flat membrane increases upon increasing ε 0 , it becomes energetically favorable to release the stress by creating the kink right at r 0 and making a circular ripple with diameter 2r 0 = 0.96 μm. This result agrees well with the observations. Upon continuous irradiation, after the initial ripple has appeared, the ripple height increases until it becomes energetically favorable to create more ripples. This implies a process-dependent rippling of ever-increasing complexity.
This scenario for rippling was confirmed by performing global optimizations for blisters with ε adh = 0.1−100 μeV/nm 2 , ε 0 = 0.001...1%, and various types of initial guesses. First, at sufficiently small ε 0 , the membrane remains flat without blistering. A critical limit for blistering is around ε 0 c ≈ 0.02 × (ε adh /eV nm −2 ) 1/2 . Second, when ε 0 increases just above the critical limit, the first blisters always have one ripple with diameter D 0 ≈ 1 μm, independent of ε adh . This result is in agreement with the experiments and with the maximum-slope argument given above (D 0 ≈ 2r 0 ). Third, at intermediate values of ε 0 , blisters show a complex pattern of ripples of varying heights and diameters. Fourth, at the limit of large ε 0 , the in-plane strain energy dominates, and the minimum energy blisters always have one central dome (Figure 3a).
Compared with the typical magnitude of adhesion (1 to 2 eV/nm 2 ) between clean interfaces of van der Waals solids, 40−43 the adhesion in the model (∼1 μeV/nm 2 ) is small. The smallness, however, is apparent even in a back-of-the-envelope calculation. Namely, upon blistering, the gain in elastic energy density is k s ε 0 2 /(1 − ν), and the cost of adhesion energy density is ε adh . At the onset of blistering, the two energies are equal, ε adh ≈ k s ε 0 2 /(1 − ν). Because the blisters appear at ε 0 ≈ 0.02%, the adhesion has to be around 1−10 μeV/nm 2 . The small adhesion may be due to water or functional groups, 44 topographic corrections, 45

The Journal of Physical Chemistry Letters
Letter investigation of the laser-modified adhesion will be pursued later.
To summarize, by using the optical forging technique, we created diffraction-unlimited circular ripples in graphene on SiO 2 . The rippling could be explained by the presence of circular symmetry amid the competition between substrate adhesion and in-plane compressive stress. In other words, the tiny rippling results spontaneously after creating an inhomogeneous expansion field on a much larger length scale. We can therefore straightforwardly predict that upon shrinking the size of the laser beam, the ripples will get smaller still. Once the mechanism responsible for the expansion of graphene is understood better, the technique could also be applied to other substrates and 2D materials. However, already now the technique and our observations provide many openings for novel research. A straightforward extension will be to control the rippling by engineering beam shapes. The technique produces beautiful circular blisters that probably have welldefined vibrational frequencies and can be used in resonators. 49 Via the formation of circular ripples, the technique also produced controllable curvatures that can be used to launch localized plasmons. 50 Thus, in addition to producing new physics and posing fundamental questions such as that of the laser-modified adhesion, the technique opens new avenues in the research of 2D materials.

Notes
The authors declare no competing financial interest.

■ ACKNOWLEDGMENTS
We acknowledge the Academy of Finland for funding (projects 297115 and 311330).