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.

中文翻译：

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石墨烯中的波纹：一种变体方法

观察到悬浮的石墨烯样品有轻微的波纹而不是平坦的。在弗里德里希等人。(Z Angew Math Phys 69:70, 2018)，我们已经检查了这种非平面性可以在构型能量最小化的经典分子力学框架内严格描述。在那里，我们已经确定了所有具有石墨烯拓扑结构的基态配置，涉及到紧邻相互作用能的类别，并对其精细的非平坦几何形状进行了分类。在关于石墨烯非平坦性的第二篇论文中，我们进一步完善了分析并证明了波形图案的出现。在弗里德里希等人的框架内移动。(2018)，石墨烯中的波纹形成被简化为一维链的二维问题。具体来说，我们表明，构型能量的几乎最小值会产生具有特定波长的波，而与样品的大小无关。这非常符合实验和模拟。