Graphene gets bent Free

In both continuous and discrete media, the interplay between defects and the 2D sheets’ bending and stretching energies is illustrated by the deflection of a flat surface into the third dimension by a disclination, such as the one shown in figure 1a. Any smoothly spatially varying out-of-plane displacement $f(r)$ costs a bending energy $(∇2f)2$⁠. If the sheet’s shape has Gaussian curvature $K(r)$⁠, defined as the determinant of the matrix $∂i∂jf(r)$⁠, it also must have strain in the tangent surface—imagine trying to tightly wrap an orange in plastic wrap. The resulting strain field produces a phonon-mediated interaction between the Gaussian curvatures in widely separated regions² and incurs an additional stretching energy penalty.

If the medium is nearly inextensible—that is, if it bends easily but doesn’t stretch—it tends toward shapes that minimize in-plane stress. But that’s only possible if $K(r)$⁠, can configure itself to locally compensate for the disclinations. For example, when a wedge is removed from a material, the leftover sheet forms a conical shape with the Gaussian curvature confined to the vertex, as shown in figure 1a. The shape eliminates the macroscopic stretching energy at the expense of a significant bending energy, which increases logarithmically with the size of the system.³ Reconnecting the edges of the wedge spontaneously forces the deflection of the 2D surface into 3D, and in this instance doesn’t require any stretching at all. Similar ideas appear in theories of stress focusing at vertices and ridges of crumpled elastically stiff sheets.⁴ 

Since a lattice dislocation can be viewed as a bound pair of disclinations, one might expect a related elastic compensation condition to hold. But the obvious mathematical approach of treating the dislocation-induced displacement as a superposition of Gaussian curvatures fails because the stability conditions are nonlinear. A continuous medium can’t screen a dislocation with a Gaussian curvature field focused to a point dipole.³ By abandoning the continuous picture in favor of a new set of rules adapted specifically for a discrete lattice, theorists can define a fully discretized variant that defines a family of shapes caused by dislocations. Randy Kamien at the University of Pennsylvania and his colleagues were the first to derive the variant for topological defects embedded in a 2D honeycomb lattice.¹ In that application, they treated the medium as a network of perfectly incompressible hexagonal units that are linked along easily bendable shared edges.

Kirigami on that honeycomb network takes a form like that shown in figure 1b. First, rows of hexagons are removed from the structure, then the material gap is eliminated by folding the sheet along the honeycomb edges (dashed lines). Closing the surface in that way has an appealing arts-and-crafts aesthetic, which can be appreciated by experimenting with a cut-and-fold paper model: One website (www.futurity.org/kirigami-hexagons-819442/) has a video showing step-by-step directions to build a paper model of the 3D finished product. The reconstituted lattice is a sharply folded structure with right angles and a ridge-and-plateau surface.

The surface is locally flat with the plateau’s height set by the width of the cuts at the ends of the dislocation—that is, the lines from the center of a yellow hexagon to the center of a green hexagon in figure 1b. All of the material’s residual Gaussian curvature lies at its sharp corner vertices, similar to the curvature confinement at the cone’s vertex for the case of a disclination. The model’s target shape nulls the stretching energy, and if the edges hinge perfectly and freely, it has no bending energy either. Composition rules for even more complex 3D surfaces show that all shapes can be reconstructed from just two possible kirigami incisions: cuts that pass through the hexagons’ nodes or those that pass through the hexagons’ centers. Similar in spirit to the continuum solution for an isolated disclination, the common directive for all the constructions is to permit extreme bending in order to avoid a more costly stretching energy.

Lattice kirigami replaces an intractable continuum-shape optimization problem with a discretized problem by prescribing folding rules that are local and essentially geometrical in character. Any mechanical structure with a set of similar composition rules could support similar sharply faceted kirigami surfaces. Those rules can be designed into macroscopic architectures by linking elementary rigid units with folding rules that guide the deflection of a flat surface into a 3D shape. Early researchers speculated that the same strategy could even control the shape and structure of the many robust microscopic honeycomb lattices found in atomically thin 2D materials, such as graphene and transition metal dichalcogenides.

Applications of lattice kirigami to nanomaterials challenge its two central tenets. First, in nanomaterials the bending modulus is nonzero, so the materials can’t form sharply folded edges. Second, the system is compressible and thus can store energy in shear and compressive strains. In practice, the result is an inevitable tradeoff between bending and stretching deformations, and the shape is determined by the solution to a subtle optimization problem in a vastly larger parameter space.

For a single sheet of graphene, that tradeoff appears to resolve in favor of the same “bend but do not stretch” algorithm as a sheet of paper. In fact, graphene and paper are remarkably similar elastically. One method to quantify the stretchability of 2D material is the relative in-plane and out-of-plane stiffness, or the dimensionless Föppl–von Kármán number $νK$⁠. For a square sheet of width $L$ and thickness $t$⁠, $νK=YL2/κ≈10(L/t)2$ in terms of the 2D Young’s modulus $Y$ and the effective bending rigidity $K$⁠.^5,6 Larger $νK$ values indicate that a material is less willing to stretch. For a graphene sheet with width $L=0.1$ µm, the estimate is that $νK≈107,$ even larger than $νK$ for an essentially inextensible sheet of paper.

That back-of-the-envelope estimate finds some experimental support. For example, a graphene sheet can be patterned into the same flat 2D spring as paper, see figure 2a. Paul McEuen and his group at Cornell University lithographically cut graphene with alternating grooves on a few-micron scale.⁵ They found that springs with that design are ultrastretchable with a compliance determined by the bending rigidity instead of the Young’s modulus. Bending rather than stretching occurs because the 2D spring buckles into the third dimension in a manner strikingly similar to the shape response from pulling on a macroscopic paper spring with the same design.

That graphene–paper correspondence is only partially reassuring. Crucially, the nanosprings are ultrastretchable only because they contain open perforations that allow the system to bend and not stretch. As a result, the springs lack the reconnected surfaces that contain topological defects, which are the raison d’être of lattice kirigami. A more relevant point of comparison is the fully bonded defect structure of the scars that form on dislocations produced in graphene grown by chemical vapor deposition.⁷ There a very different picture emerges. Even in the extremely stiff limit with $νK>105$⁠, the scars form smoothly elevated humpbacked surfaces, as shown in figure 2b, that connect dislocations. Similar to the nanosprings, those scarred graphene structures are laterally very stiff. Their 3D surfaces, however, are not faceted but smooth, the Gaussian curvature is not focused but distributed, and the nearly unstretchable sheet is stretched.

Theoretical investigations lead to similar conclusions. Several years ago we (Grosso and Mele) carried out large-scale atomistic simulations to study the structures of dislocated graphene sheets that start in faceted lattice-kirigami structures and are then relaxed using accelerated molecular mechanics, a computer code that efficiently solves the equations of motion for many atoms simultaneously.^8,9 In all cases the structures revert to shapes that are warped and softly undulating, as shown in figure 3. Ridge-and-plateau kirigami is replaced by rolling landscapes evocative of the English countryside outside Loughborough University, where the work was carried out. Other researchers obtained similar results for the effect of disclinations in 2D disks of a phosphorus allotrope known as phosphorene.¹⁰ The defected disks relax into domed 3D shapes that also redistribute their Gaussian curvature into smooth surfaces.

The unexpected smooth shapes observed in experiments and calculated in simulations call for a reconsideration of the composition rules for nanoscale lattice kirigami. The different nanoscale behavior is not because prototypical 2D materials such as graphene, phosphorene, and transition metal dichalcogenides are just more compliant to in-plane stresses. On the length scales of those observations, the materials’ $νKs$ are as large as or larger than those of a paper sheet, an assessment that is robust to theory refinements to account for changes in the elastic moduli from quenched disorder or finite temperature.⁶ For instance, the graphene $νK$ cited above includes a factor of 10³ enhancement of its bending moduli as the temperature is increased.

Closer inspection of the graphene landscapes identifies an even deeper conceptual puzzle. Instead of the expected low-strain surfaces separated by stress-focused corners and creases, the smooth surfaces in stiff materials are extremely stress-defocused with low bending energies. That phenomenon is quantified by decomposing the far-field deflection patterns into a sum of separable terms: $f(r,φ)=∑mfm(r)eimφ$⁠. Those partial wave solutions are individually well described by solutions to the biharmonic equation $∇4f=0$⁠, which describes shapes that minimize bending energy. The amplitudes $fm(r)$ are strongly suppressed for larger angular momentum $m$⁠, and for low $m$ they are well fit by combinations of two growing far-field radial solutions.

Stress focusing, as expected in elastically stiff media, would produce sharp edges instead and have only a slow, nearly power-law suppression of the large $m$ amplitudes. The directive from lattice kirigami to “bend but don’t stretch” is thus replaced by the organic principle to not “fold, spindle, or mutilate,” as dictated by the geometrical constraints from the topological defects and external boundaries.

Analyzing the simulations of dislocation-induced 3D shapes reveals three bending rules for kirigami in nanomaterials.⁸ First is the law of amplitudes: As mentioned previously, the far-field patterns are well represented by a rapidly converging expansion in the first few cylindrical harmonics $m$⁠. Second is the law of ratios: Radial solutions to the biharmonic equation contain two terms that grow with distance. For each symmetry-allowed $m$⁠, the terms always have opposite signs such that the ratio of their amplitudes counterbalances to reach zero area-integrated Gaussian curvature. For example, disks that have been kirigamied, with allowed $m=2$ solutions, have a shape similar to a potato chip.

Third is the law of boundaries: Although the law of ratios specifies the amplitude ratio, the overall amplitude of the solution is controlled by a boundary energy that’s proportional to the perimeter’s length. As a result, graphene disks of different radii adopt the same shape so long as the height and lateral position are both scaled to the disk radius. Together the three rules provide a compact, analytic, and reasonably accurate theory of far-field shapes, which can’t have a sharp ridge-and-plateau motif.⁸ 

The bending energy’s ascendancy in the problem reveals an oversight in the extrapolation of lattice kirigami to nanomaterials. In continuum theory, Gaussian curvature is a source of in-plane strain, which is unfavorable in any ultrastiff elastic medium. To avoid a stretching energy penalty, which grows faster than the system size, the area integral of the curvature must vanish. But strain couples remote regions of the Gaussian curvature, and with increasing separation, that coupling grows rather than decays; in reciprocal space it diverges² as $1/q4$ as $q→0$⁠. That strong coupling at large distances means that the curvature doesn’t need to exactly cancel locally. More generally stress-focusing the Gaussian curvature to dislocation/disclination cores is not an inevitable or even a desirable outcome in the shape responses of various nanomaterials. Many 2D materials have taken advantage of that nonlocal physics all along to economize on their bending energies.

Bending rules describe kirigami shapes on finite systems with open boundaries and with the ability to compensate the Gaussian curvature globally. Structures with periodic boundary conditions, which are formed by cutting a periodic pattern of dislocations in an effectively infinite sheet, present an entirely different class of solutions: kirigami without borders. Under periodic boundary conditions with a globally flat surface, the integral of the Gaussian curvature over the simulation area is zero. But the Gaussian curvature field has indirect phonon-mediated interactions, which are present only for the discrete wavelengths that satisfy the periodic boundary conditions, and the $q→0$ divergence of the coupling is thereby eliminated.

Atomistic simulations with periodic boundary conditions predict that a nanosheet won’t optimize to avoid bending and instead will produce elevated shapes with sharp edges, as shown in figure 4a. Such a graphene microstructure represents a completely new allotrope of “planar” carbon: a defect-stabilized mesophase poised between 2D and 3D. A disk with the same dislocation but open boundary conditions, as in figure 4b, has a smooth 3D surface because the ultralong-range strain coupling returns and the system resumes its primary task of minimizing the bending energy.

Kirigami can change the electronic behavior of 2D nanomaterials, and graphene is a prototypical example famous for its interesting low-energy electronic behavior (see the article by Andrey Geim and Allan MacDonald, Physics Today, August 2007, page 35). That behavior derives from the geometry of two point singularities in its band structure, which occur at two time-reversed momenta, or valleys. At those singularities, known as Dirac points, the bands have linear dispersion relations that form two cones, and the electron and hole bands touch at the shared tip of the cones. Kinematics near the Dirac points follow a solid-state version of the Dirac Hamiltonian for a massless particle.

In graphene that masslessness is enforced by the presence of both time reversal symmetry and twofold rotation symmetry. Uniform in-plane strain preserves both symmetries and leaves the Dirac points intact, although their location in momentum space shifts proportionally to the strain. Microscopically the strain is coupled to the Dirac kinetic energy through a gauge potential that has opposite signs in the two valleys.¹¹ For a uniformly strained sheet, that vector potential is a pure gauge, and it has no observable consequence. In that case, a strained graphene sheet is still just a sheet of graphene.

A nonuniform strain field can eliminate the band degeneracy, and that lifted degeneracy restores band mass to the electrons and possibly even renders graphene electrically insulating. The curl of the strain-induced vector potential acts like a pseudomagnetic field that pierces closed electron trajectories on the lattice. The strain produced by Gaussian curvature is nonuniform and does restore band mass, although with a catch.¹² In graphene the curl is carried out on a discrete lattice, which endows the Dirac electrons with a mass proportional to the third spatial derivative of the curvature. As a result, a smoothly varying surface-height profile generates a spatially varying mass. The Dirac theory admits the possibility of a negative mass, and here the mass oscillates in sign as a function of angle proportional to $sin(3φ)$⁠. Because of those oscillations, the effect averages out to zero and consequently eludes most measurements made on large length scales.

By inverting that thinking, one can imagine structures that would eliminate the angular oscillations and support instead a spatially uniform pseudomagnetic field. The recipe for doing so is a theorist’s dream and an experimentalist’s nightmare: The shape’s strain field must also oscillate with the same threefold angular anisotropy.¹¹ In such a system, the lattice curl “eats” its sign changes, and the combination restores a robust nonvanishing contribution in the pseudofield. In practice, the recipe requires alternately pulling and pushing on a graphene sheet in the threefold pattern shown in figure 5a. Electrons propagating on that warped background deflect into orbits as though they were moving in a strong perpendicular magnetic field, although the senses of their orbital circulations are opposite in the two valleys. And since the mechanism for the deflection involves site-to-site hopping and not a conventional magnetic deflection, the magnitude of the equivalent strain-induced pseudomagnetic field can be impressively large, even as high as a few megagauss.

The phenomenon shows up unexpectedly in experiments on graphene sheets grown on nanoscale prismatic platinum islands, shown in figure 5b, which imprint anisotropic strain patterns into the graphene.¹³ Spatially resolved scanning tunneling spectroscopy reveals a reorganization of the continuous electronic spectrum of unstrained graphene into resonances corresponding to the quantized Landau levels in the relativistic Hall effect for a ~300 T effective pseudofield.

Graphene kirigami can refine that proof-of-concept demonstration by producing structures that control the size, shape, and symmetry of the strain fields. For example, a threefold symmetric pattern of short dislocation segments (see figure 5c) automatically buckles the sheet into a domed tricorner surface with the requisite spatial symmetry. The deformation releases strain into the dome such that the magnitude and range of the strain-induced pseudofield can be selected by choosing the lengths and orientations of the dislocation segments. Calculations predict related band-structure engineering in other insulating 2D materials, in which strain coupling produces even larger changes to an existing energy gap.¹⁰ The strain could tune the insulators’ optical responses or confine the motion of free carriers.

Implementing those ideas requires practical methods for cutting and joining defect patterns with specified widths, lengths, and orientations in otherwise ordered nanomaterials. That process is perhaps the biggest distinction between the macroscopic and microscopic forms of kirigami: A literal interpretation of the cut-and-fold instructions is untenable for nanomaterials. Fortunately, some promising alternatives are being developed in other materials contexts. For example, graphene nanoribbons grow from organic precursors with atomically precise control of the ribbon axis.¹⁴ If they’re grown on a 2D structure designed to frustrate perfect alignment, topological defects can form in the lattice during the growth process and serve as a scaffold for building a kirigami network.

A related strategy is to directly grow a material on a target 3D shape that has Gaussian curvature. The technique already works for caps of single-layer transition-metal dichalcogenides grown on spherical surfaces.¹⁵ Alternatively, patterning graphene or other 2D materials with functional units can produce hybrid materials that change their shapes in response to external mechanical or chemical stimuli.¹⁶ The strain distributions can, by design, buckle the structure and modify electronic behavior, or, reciprocally, an electronic or optical trigger can produce a shape change in situ. Versions of the concept have appeared in flexible electronics made from graphene and silicon.^16,17 

Progress in nano-kirigami is now informed by a confluence of ideas from mathematics, physics, chemistry, materials science, and the arts. That eclectic combination seems poised to turn up yet more cutting-edge science.

We thank Randy Kamien and Marc Miskin for discussions on this topic. Eugene Mele’s work is supported by the US Department of Energy under grant no. DE-FG02-84ER45118.

Bastien Grosso is a PhD candidate in the department of materials at ETH Zürich. Eugene Mele is Christopher H. Browne Distinguished Professor of Physics at the University of Pennsylvania.