Ripples in Graphene: A Variational Approach
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Communications in
Mathematical Physics
Ripples in Graphene: A Variational Approach Manuel Friedrich1 , Ulisse Stefanelli2,3,4 1 Applied Mathematics, Universität Münster, Einsteinstr. 62, 48149 Münster, Germany
E-mail: [email protected] URL: https://www.uni-muenster.de/AMM/Friedrich/index.shtml
2 Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria 3 Vienna Research Platform on Accelerating Photoreaction Discovery, University of Vienna, Währinger Str.
17, 1090 Vienna, Austria
4 Istituto di Matematica Applicata e Tecnologie Informatiche “E. Magenes” - CNR, v. Ferrata 1, 27100 Pavia,
Italy E-mail: [email protected] URL: http://www.mat.univie.ac.at/~stefanelli Received: 15 February 2018 / Accepted: 7 February 2020 Published online: 6 October 2020 – © The Author(s) 2020
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. Contents 1. 2.
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Introduction . . . . . . . . . . . . . . . . . . . . . . . . The Model and Main Results . . . . . . . . . . . . . . . 2.1 Admissible configuration and configurational energy 2.2 Characterization of minimal energy . . . . . . . . . . 2.3 Characterization of almost minimizers . . . . . . . . 2.4 An illustration on a simpler model . . . . . . . . . . The Cell Problem . . . . . . . . . . . . . . . . . . . . . The Single-Period Problem . . . . . . . . . . . . . . . . 4.1 Geometry and length of energy minimizers . . . . . . 4.2 Small perturbations of energy minimizers . . . . . . The Multiple-Period Problem . . . . . . . . . . . . . . . 5.1 The multiple-period problem for the reduced energy . 5.2 The multiple-period problem for the general energy .
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Proof of the Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Upper bound for the minimal en
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