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Rippling and crumpling in disordered free-standing graphene

机译：无序自立石墨烯的起皱和起皱

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摘要

Graphene is a famous realization of an elastic crystalline two-dimensional (2D) membrane. Thermal fluctuations of a 2D membrane tend to destroy the long-range order in the system. Such fluctuations are stabilized by strong anharmonicity effects, which preserve thermodynamic stability. The anharmonic effects demonstrate critical behavior on scales larger than the Ginzburg scale. In particular, a clean suspended flake of graphene shows a power-law increase of the bending rigidity with the system size, x ∝ L~η, due to anharmonic interaction between the in-plane and out-of-plane (flexural) phonon modes. We demonstrate that random fluctuations of the membrane curvature caused by static disorder may change dramatically the scaling of the bending rigidity and lead to a nonmonotonous dependence of x on L. We derive coupled renormalization-group equations describing the combined flow of x and effective disorder strength b, find a critical curve b(x) separating flat and crumpled phases, and explore the behavior of disorder in the flat phase. Deep in the flat phase, disorder decays in a power-law way at scales larger than the Ginzburg length, which therefore sets a characteristic size for the ripples-static out-of-plane deformations observed experimentally in suspended graphene. We find that in the limit L → ∞ ripples are characterized by an anomalous exponent 2η in contrast to dynamical fluctuations governed by η. For sufficiently strong disorder, there exists an intermediate range of spatial scales where ripples decay much slower, with exponent η/4. In the near-critical regime, disorder first increases with L, then reaches a maximum and starts to decrease. In this case, the membrane shows fractal properties implying a multiple folding starting from a certain length scale L_1 and finally flattens at a much larger scale L_2 (which diverges at criticality). We conclude the paper by a comparison of our results with available experimental data on graphene ripples.
机译：石墨烯是弹性晶体二维（2D）膜的著名实现。 2D膜的热波动往往会破坏系统中的远距离顺序。强烈的非谐效应可稳定此类波动，从而保持热力学稳定性。非谐效应在大于金兹堡音阶的音阶上表现出临界行为。特别地，由于平面内和平面外（弯曲）声子模式之间的非谐相互作用，干净的石墨烯悬浮片显示出随系统尺寸x ∝ L〜η的弯曲刚度的幂律增加。 。我们证明了由静态紊乱引起的膜曲率的随机波动可能会极大地改变弯曲刚度的比例，并导致x对L的非单调依赖。 b，找到一条临界曲线b（x），它将平坦相和皱折相分开，并探索平坦相中的无序行为。在平坦阶段的深处，无序以幂律的方式在大于Ginzburg长度的尺度上衰减，因此为在悬浮石墨烯中实验观察到的波纹静态平面外变形设定了特征尺寸。我们发现，在极限情况下，L→∞波纹的特征是反常指数2η，而动态波动则受η支配。对于足够强的无序度，存在一个空间尺度的中间范围，在该范围内，波纹的衰减要慢得多，其指数为η/ 4。在近临界状态下，混乱首先随着L增加，然后达到最大值并开始减少。在这种情况下，膜表现出分形特性，暗示着从一定的长度尺度L_1开始多次折叠，最终以更大的尺度L_2（在临界点发散）变平。通过将我们的结果与石墨烯波纹的可用实验数据进行比较，可以得出本文的结论。

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  • 来源
    《Physical review》 |2015年第15期|155428.1-155428.22|共22页
  • 作者

    I. V. Gornyi; V. Yu. Kachorovskii; A. D. Mirlin;

  • 作者单位

    Institut fuer Nanotechnologie, Karlsruhe Institute of Technology, 76021 Karlsruhe, Germany,A.F. Ioffe Physico-Technical Institute, 194021 St. Petersburg, Russia,Institut fuer Theorie der kondensierten Materie, Karlsruhe Institute of Technology, 76128 Karlsruhe, Germany,L.D. Landau Institute for Theoretical Physics, Kosygina street 2, 119334 Moscow, Russia;

    Institut fuer Nanotechnologie, Karlsruhe Institute of Technology, 76021 Karlsruhe, Germany,A.F. Ioffe Physico-Technical Institute, 194021 St. Petersburg, Russia,Institut fuer Theorie der kondensierten Materie, Karlsruhe Institute of Technology, 76128 Karlsruhe, Germany,L.D. Landau Institute for Theoretical Physics, Kosygina street 2, 119334 Moscow, Russia;

    Institut fuer Nanotechnologie, Karlsruhe Institute of Technology, 76021 Karlsruhe, Germany,Institut fuer Theorie der kondensierten Materie, Karlsruhe Institute of Technology, 76128 Karlsruhe, Germany,L.D. Landau Institute for Theoretical Physics, Kosygina street 2, 119334 Moscow, Russia,Petersburg Nuclear Physics Institute, 188300, St. Petersburg, Russia;

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  • 正文语种 eng
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  • 关键词

    ballistic transport; nanocrystalline materials;

    机译：弹道运输;纳米晶材料;

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