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Devices such as those shown in Fig. 1a, c, d are made from polycrystalline monolayer graphene that is grown on copper by chemical vapour deposition14, and then transferred to fused silica wafers that are covered with an aluminium release layer. We use optical lithography to pattern both the graphene and the 50-nm-thick gold pads that are deposited on top of the graphene to act as handles. Finally, we release the graphene from the surface by etching away the aluminium in mild acid. The devices remain in aqueous solution with added salts or surfactants as desired. An inverted white-light microscope with a video camera is used to image the sheets, and micromanipulators are used to probe them.

Figure 1: Patterning and manipulating graphene.
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a, Transmission white-light image showing completed devices: a spiral spring, a kirigami pyramid, and a variety of cantilevers. b, Manipulating a large sheet of graphene with a micromanipulator. The sheet folds and crumples like soft paper, and returns to its original shape. c, d, Manipulating devices with gold pads. The devices can be lifted entirely off the surface; see Supplementary Video 1. Scale bars are 10 µm. All images and videos have undergone linear contrast adjustments.

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We move the graphene along the surface or peel it up entirely by pushing a sharp probe tip into the gold pads or against the graphene itself (Fig. 1b–d and Supplementary Video 1). The graphene’s elastic behaviour is reminiscent of that of thin paper: it folds and crumples out of plane, but does not notably stretch in plane (Fig. 1b). The process is almost entirely reversible in the presence of surfactants, even after considerable crumpling of the graphene.

The mechanical properties relevant for kirigami are captured by the Föppl–von Kármán number7,8 for a square sheet of side length L and thickness t: γ = Y2DL2/κ ≈ (L/t)2, that is, the ratio between the two-dimensional Young’s modulus Y2D and the out-of-plane bending stiffness κ, multiplied by the length squared. To determine γ, we measure κ by using the photon pressure from an infrared laser to apply a known force to a pad attached to a graphene cantilever and measuring the resulting displacement (Fig. 2a). We also measure thermal fluctuations of cantilevers to determine their spring constants (Fig. 2b and Extended Data Fig. 4), which, according to the equipartition theorem, are , where T is temperature, kB is Boltzmann’s constant, and is the time-averaged square of the cantilever thermal fluctuation amplitude. Although the presence of water (the aqueous solution in which the device is immersed) slows down the fluctuations, it does not change the spring constant15,16. Cantilevers with lengths of 8–80 μm and widths of 2–15 μm have spring constants of 10−5–10−8 N m−1. These are astonishingly soft springs, as many as eight orders of magnitude softer than a typical atomic force microscope cantilever. The bending stiffness κ is inferred from the measured spring constant using k = 3κW/L3, where W and L are the width and length of the cantilevers, respectively. The values obtained from these thermal measurements and the laser measurements are shown in Fig. 2c, and are seen to be orders of magnitude higher than κ0 = 1.2 eV, which is the value that is predicted from the microscopic bending stiffness of graphene (known from simulations17 and measurements of the phonon modes in graphite18).

Figure 2: Measuring the bending stiffness of monolayer graphene.
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a, Applying controlled forces to a gold pad using an infrared laser. The grey triangle represents the probe tip that holds the device up off the surface; the red triangle represents the focused laser beam. The cantilever displacement gives the spring constant. b, Tracking the motion of a rotated device under thermal fluctuations provides an independent measurement of the spring constant (Extended Data Fig. 3). c, Stacked histogram of bending stiffness, and interference micrographs of devices whose aluminium release layer has been etched away, showing the structure of static ripples (inset). The spring constant relates to the bending stiffness as k = 3κW/L3. The red arrow points to the microscopic bending stiffness, κ0 = 1.2 eV. Scale bars are 10 µm. Interference images were averaged over 180 frames at 90 frames per second.

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Both thermal fluctuations and static ripples are predicted to notably stiffen ultrathin crystalline membranes9,10,11,12,13,19,20 by effectively thickening the membrane, similar to how a crumpled sheet of paper is more rigid than a flat one. For static ripples, the effective bending stiffness is predicted to be9 , where is the space-averaged square of the effective amplitude of the static ripples and Y2D = 340 N m−1 is the two-dimensional Young’s modulus3. For an initially flat membrane with thermal fluctuations, the stiffness is predicted to be κeff ≈ κ0(W/lc) η, where is the Ginzburg length19, and η is a scaling exponent.

We look for static ripples in graphene cantilevers using interference microscopy21 (inset of Fig. 2c). The black bands in such images are regions of constant elevation, with the spacing between black and white bands corresponding to changes in z of λ/4 ≈ 100 nm (where λ is the wavelength, corrected for the refractive index of water). With a typical value from these measurements of about (100 nm)2, we obtain an effective bending stiffness of .

Static ripples are present only after releasing graphene from the surface (Extended Data Fig. 2), and likely to be sample specific and influenced by growth, fabrication details, and so on. Developing growth and fabrication protocols that can change the amplitude of the static ripples or eliminate them altogether is of great interest. Other groups have observed ripples in suspended (strained) graphene membranes5,22, although they occur at a much smaller scale and their origin remains a subject of debate. Moreover, the thermal theory outlined above predicts a bending stiffness at room temperature due to thermal fluctuations of for an initially flat membrane. These contradictory findings call for future experiments to firmly establish the relative contribution to bending stiffness of thermal fluctuations and static ripples23. But irrespective of cause, the high bending stiffness notably changes the effective γ value, γeff = YeffL2/κeff. With the predicted renormalization9,11 of Yeff, we find that γeff is of the order of 105–107 for a sheet of graphene 10 μm × 10 μm in size, close to that of a standard sheet of paper.

The mechanical similarity between graphene and paper makes it easy to translate ideas and intuition directly from paper models to graphene devices. For example, the highly stretchable graphene transistors in Fig. 3b, c and Supplementary Video 2 are based on a simple kirigami pattern of alternating, offset cuts and are created using photolithography. Here, the elasticity of the kirigami spring is determined by the pattern of cuts and the bending stiffness (rather than the Young’s modulus) of the graphene. As the reconstruction of the three-dimensional shape of a stretched and lifted device in Fig. 3e shows, the graphene strips pop up and bend out of plane as the spring is stretched.

Figure 3: Stretchable graphene transistors.
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a, b, Paper and graphene in-plane kirigami springs, respectively. c, Graphene spring stretched by about 70%. d, Electrical properties in approximately 10 mM KCl. Conductance G is plotted against liquid-gate voltage VLG at source–drain bias VSD = 100 mV before stretching (blue) and when stretched by 240% (orange). The top (orange-boxed) inset is split because the stretched device was larger than the visible area. e, Three-dimensional reconstruction from a z-scan focal series of a graphene spring. The right side remains stuck to the surface and the left side is lifted. Insets show views of sections of the graphene (right) and paper models (left). Top images show side views; bottom images show top views. The thin grey lines are the bounding box from the three-dimensional reconstruction. The aspect ratio of the side-view paper model was compressed 1.8×. Scale bars are 10 µm. See Supplementary Video 2.

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We measure the electrical response of these stretchable transistors by gating them with an approximately 10 mM KCl solution24 (see Methods for details). Figure 3d plots the liquid-gate dependence of the conductance at a source–drain bias of 100 mV for a device in its initial unstretched state (blue) and when stretched by 240% (orange). The normalized change in conductance with gate voltage per graphene square is 0.7 mS V−1 and the resistance per graphene square at the Dirac point is 12 kΩ, comparable to what has been reported for electrolyte-gated graphene transistors24. Because the graphene lattice itself is not much strained when the kirigami spring is extended, we do not expect or observe a notable change in the conductance curves between the unstretched and stretched states, which is highly desirable for stretchable electronics25. Furthermore, stretching and unstretching a similar device more than 1,000 times did not substantially change its electrical properties.

Figure 4a and Supplementary Video 3 show graphene cut so that it forms an out-of-plane pyramidal spring, along with a paper model. The spring’s force–distance curve in Fig. 4b, measured with the photon pressure from an infrared laser focused on the central pad, gives a spring constant of k = 2 × 10−6 N m−1. This value compares well with the spring constant estimate of (5 × 10−7)–(5 × 10−6) N m−1, obtained from our κ measurements (Fig. 2) and the geometry of the device.

Figure 4: Remote actuation.
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a, Paper model (top) and as-fabricated graphene kirigami pyramid (bottom). We actuate this out-of-plane spring using an infrared laser. b, Schematic and force–distance curve for a pyramid such as the one shown in a. A linear fit at low forces yields k = 2 × 10−6 N m−1. c, A rotating static magnetic field twists and untwists a long strip of graphene. The gold pad is replaced by iron. d, A monolayer graphene hinge actuated by a graphene arm. Stills show the hinge closing. Supplementary Video 3 was taken after opening and closing this hinge 1,000 times; it survived 10,000 cycles. Scale bars are 10 µm.

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Remote actuation of the kirigami devices is possible using magnetic fields or linked graphene elements. Figure 4c illustrates magnetic actuation, with magnetic forces and torques acting on an attached iron pad allowing for parallel (many hinges being controlled simultaneously) and complex manipulations to be made. The opening and closing of the graphene hinge (which is 1 μm long and 10 μm wide) in Fig. 4d uses a longer graphene strip (out of focus in the image) that extends in a loop over the hinge to a gold pad, so that moving this probe along the surface opens and closes the hinge (Supplementary Video 3). Although the hinge is only one atom thick, it survived more than 10,000 open-and-close cycles (at which point the gold pads started to warp). This remarkable resilience and the scope for scaling down to tens of nanometres make monolayer graphene hinges ideally suited for use in microscale moving parts.

We envisage that graphene kirigami will have many useful applications. For example, springs like those in Figs 3 and 4 are easily designed, with spring constants that range from 1 N m−1 to 10−9 N m−1 (which covers the full range from atomic force microscopes to optical traps), for use as force measurement devices with a simple visual readout and femtonewton force resolution. The addition of elements such as bimorphs26 or chemical tags to graphene kirigami devices27 will create environment-responsive metamaterials. Kirigami techniques can also easily be applied to other two-dimensional materials that have different optical, electronic, and mechanical properties, which creates opportunities for further development of self-actuated two-dimensional functional devices that respond to light or magnetic fields, changes in temperature, or chemical signals. Such atomically thin membrane devices may be used for sensing, manipulation, complex origami, and nanoscale robotics.

Methods

Graphene growth

We grow graphene on copper following a standard chemical vapour deposition process14. The copper foil is purchased from Alpha Aesar, stock number 13382. The copper is annealed for 36 min at 980 °C with a H2 flow of 60 standard cubic centimetres per minute (s.c.c.m.). Graphene is grown at 980 °C for 20 min with a H2 flow of 60 s.c.c.m. and a CH4 flow of 36 s.c.c.m. The foil is then cooled in a matching environment as quickly as possible.

Graphene characterization

Typical Raman spectra, scanning electron microscope images, and bright-field transmission electron microscope (TEM) images all confirm that the growths yielded mostly single-layer graphene with small bilayer regions (Extended Data Fig. 1). Dark-field TEM on a variety of growths reveal that typical grain sizes are of the order of hundreds of nanometres to micrometres.

Fabrication of cantilevers and kirigami devices

Fabrication follows standard graphene processing methods, with the addition of an aluminium release layer. We evaporate 40 nm of aluminium on 170-µm double-side-polished fused silica wafers from Mark Optics. We dice the wafers into 2 cm × 2 cm chips, and transfer graphene to the chips using 2% poly(methyl methacrylate) (PMMA). We then etch the copper in ferric chloride (Transene, CE-200) for one hour and rinse with five consecutive deionized water baths. We transfer the graphene onto the aluminium-coated chip, and soak overnight in acetone to remove the PMMA. Next, we use photolithography to pattern the pads and evaporate 50 nm of gold. We pattern the graphene strips and etch away the unwanted graphene with a 25-s oxygen plasma. Finally, we soak the chip in a mild (10:1) deionized water/HCl solution until the aluminium release layer has completely disappeared. The chip is transferred directly to a deionized water bath, which is kept refrigerated between uses to discourage bacterial growth.

Atomic force microscope characterization

Atomic force microscope measurements on aluminium-free chips that are run in parallel with measured devices usually give step heights of 1–3 nm above that of pristine exfoliated graphene (Extended Data Fig. 2). Although it is impossible to completely avoid polymer residues from standard transfer and fabrication processing, a 2-nm layer of the stiffest PMMA (Young’s modulus Y = 3.3 GPa, Poisson ratio σ = 0.4) should add only about 20 eV to the stiffness (since κ = Yt3/[12(1−σ)]), which is negligible compared to the measured values.

Influence of surfactants

The addition of a surfactant reduces the graphene’s adhesion to the surface and prevents the graphene from permanently sticking to itself. We performed bending stiffness measurements with and without surfactant, and found that the presence of surfactant does not measurably affect the bending stiffness. A surfactant was used in all kirigami experiments. We used sodium dodecylbenzenesulfonate (SDBS) from Sigma–Aldrich (product number 289957), dissolved in deionized water to a concentration of approximately 3 mM. During the measurement process some water evaporates and is replaced with deionized water, so that the concentration of SDBS remains approximately constant over time.

Extraction of from thermal motion

To extract from the thermal motion of the gold pad on the free end of the graphene cantilever, we recorded the motion at 90 frames per second for about 20 min to ensure that the entire phase space of the cantilever motion was sampled. The first 20 s from the trace of the free gold pad on a 40 µm × 10 µm cantilever are shown in Extended Data Fig. 3a. We tracked the motion of the pad centroid frame by frame using image analysis to extract the x position of the pad over time; the x direction is perpendicular to the profile of the free gold pad (see inset of Extended Data Fig. 3a). To extract from this thermal motion, we calculated the power spectral density (PSD), which is the Fourier transform of the autocorrelation of the data, shown in Extended Data Fig. 3b. In all devices we observed low-frequency 1/f noise from the long-timescale motion of the supporting probe (shown in red). This low-frequency noise was excluded from further analysis. We fit the data plotted in blue, which resulted from the thermal motion of the free gold pad, with the theoretical one-sided PSD for Brownian thermal motion15,16 (dashed line): Sxx (f) = S0/[1 + (f/fc)2], where S0 is the low-frequency value of the Brownian motion PSD, and fc is the corner frequency. The integral of this fitted function yields15,16 : . Using , where = (130 nm)2 for the device shown in Extended Data Fig. 3, we find that the spring constant for this 40 µm × 10 µm cantilever is k = 2.4 × 10−7 N m−1 and that the bending stiffness κ = 3 keV.

Interferometric measurements

The wavelength of the laser that was used for the interferometric measurements is 436 nm; in water this means that the separation of the black and white bands is 436/4/1.33 = 82 nm. We used a 10-nm full-width-half-maximum bandpass filter with a centre wavelength of 430 nm on the 436-nm line of a mercury arc lamp. The reflectivity of the glass–water–graphene–water cavity creates a situation where the reflectivity changes from 0.0026 to 0.0067 between dark and light bands (based on thin-film equations). A single sheet of graphene in water has a reflectivity of 0.0002, so our geometry greatly enhances the visibility of graphene.

Electrical measurements of stretchable transistors

Electrical measurements were conducted in an approximately 10 mM KCl solution, with a few drops of about 3 mM SDBS solution added. Since water evaporates during the measurement process, we periodically add deionized water to keep the concentration of SDBS approximately constant. The solution was gated by a gold wire, and the gate–drain current was minimized by contacting the drain electrode with a parylene-C-coated tungsten probe. An Ithaco 1211 current preamplifier was used to measure the current, and the system had a negligible gate–drain leakage current of about 10 nA. The device geometry in Fig. 3d is equivalent to approximately 40 squares in series.

Laser force actuation and calibration

Force–displacement curves for cantilevers and a pyramid were measured using the radiation pressure of a 1,064-nm laser. The spring constant was determined by finding the slope of a linear fit to the data. The power of the laser was adjusted using an acousto-optic modulator. The force delivered to a gold pad was calibrated experimentally using the known weight of the gold. For a given power, the laser was focused near the centre of the gold pad, and the change in displacement was measured using a piezo attached to the objective of the optical microscope.

Three-dimensional reconstruction of kirigami devices

We reconstructed the three-dimensional shape of graphene kirigami devices from a z-scanned focal series (Supplementary Video 2). We first acquire a series of images, varying the position of the lens to scan through a z depth of 100 µm in 10-nm steps. The images are background-subtracted to remove fixed-pattern features on the lens and camera. We then find the focal plane of each x–y pixel by using a refined minimum-intensity algorithm to look for the z plane with the highest contrast. Because the graphene is much thinner than the depth of focus, we assume it behaves like a point object in the z direction. In this model, we use Gaussian fits to find the z centre of the dip in intensity for each x–y pixel. For graphene near the gold pads, we restrict the fit positions to exclude shadowing effects from the pads as they go out of focus. The resulting matrix of z positions is cropped to the size of our object, converted to a three-dimensional matrix of intensities, and smoothed with a three-dimensional Gaussian blur of about 400 nm to reduce noise. Finally, we use TomViz (a development of Paraview that is optimized for tomography visualizations) to render the three-dimensional object. The colour map is based on the intensity of the original video.