Abstract
Suspended graphene samples are observed to be gently rippled rather than being flat. In Friedrich et al. (Z Angew Math Phys 69:70, 2018), we have checked that this nonplanarity can be rigorously described within the classical molecular-mechanical frame of configurational-energy minimization. There, we have identified all ground-state configurations with graphene topology with respect to classes of next-to-nearest neighbor interaction energies and classified their fine nonflat geometries. In this second paper on graphene nonflatness, we refine the analysis further and prove the emergence of wave patterning. Moving within the frame of Friedrich et al. (2018), rippling formation in graphene is reduced to a two-dimensional problem for one-dimensional chains. Specifically, we show that almost minimizers of the configurational energy develop waves with specific wavelength, independently of the size of the sample. This corresponds remarkably to experiments and simulations.
Highlights
Carbon forms a variety of different allotropic nanostructures. Among these a prominent role is played by graphene, a pure-carbon structure consisting of a one-atom thick layer of atoms arranged in a hexagonal lattice
Despite the progressive growth of experimental, computational, and theoretical understanding of graphene, the accurate description of its fine geometry remains to date still elusive
Beside the academic interest, the fine geometry of graphene sheets is of a great applicative importance, for it is considered to be the relevant scattering mechanism limiting electronic mobility [13,26]
Summary
Carbon forms a variety of different allotropic nanostructures. Among these a prominent role is played by graphene, a pure-carbon structure consisting of a one-atom thick layer of atoms arranged in a hexagonal lattice. We move within the frame of Molecular Mechanics, which consists in describing the carbon atoms as classical particles and in investigating minimality with respect to a corresponding configurational energy. We analyze the reduced energy (3) taking into account nearest- and next-to-nearest-neighbor interactions and favoring nonaligned consecutive bonds This leads to a large variety of energy minimizers with many different geometries, see Fig. 3. We identify the optimal wavelength λlmax depending on the number of bonds l (later referred to as discrete-wave period). We show that a chain with almost minimal energy essentially consists exclusively of single-period chains with a specific discrete-wave period, which only depends on the choice of the boundary conditions. The arguments rely on a fine interplay of the longer-range contributions and the wavelength λlmax for different discrete-wave periods l
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