background: Per this abstract

Silicene, germanene and stanene are part of a monoelemental class of two-dimensional (2D) crystals termed 2D-Xenes (X = Si, Ge, Sn and so on) which, together with their ligand-functionalized derivatives referred to as Xanes, are comprised of group IVA atoms arranged in a honeycomb lattice — similar to graphene but with varying degrees of buckling.

question: This comment notes

Graphene is not flat, either. Cannot be, because only 3D crystallites have perfect long-range order. Landau-Peierls instability this is called.

How general is this? Can we expect all stable 2D crystals to be buckled, and any that would be flat and not buckled to "self-destruct" due to instabilities related to long-range order?

  • 1
    $\begingroup$ "X"enes? You need a better word. Also you didn't get the comment right. Graphene isn't "buckled", it's just that it's much like a scroll - without "support" or "gluing" to something it won't stay in plane. $\endgroup$
    – Mithoron
    Nov 7, 2019 at 22:24
  • 1
    $\begingroup$ "bucked" sounds like a rigid feature, wobbly is perhaps a better word. Oriented smectic liquid crystals are another well know example for this $\endgroup$
    – Karl
    Nov 7, 2019 at 23:01
  • $\begingroup$ @Mithoron perhaps you can tell these authors as well that they need a better word? 1, 2, 3, 4 Also, I'm pretty sure I have the word buckled misspelled (a missing "l" now included) but otherwise correct. See Nature paper (1st ref). Buckling refers to the periodic displacement, some unit cell atoms above/below the plane. $\endgroup$
    – uhoh
    Nov 8, 2019 at 1:20
  • $\begingroup$ @Mithoron for graphene buckling see Buckling Analysis of Single-Layer Graphene Sheets Using Molecular Mechanics I've restored the title because I'm confident that it is clear, and correctly uses its terms. $\endgroup$
    – uhoh
    Nov 8, 2019 at 1:23
  • 1
    $\begingroup$ Well, if even in paper title there was explanation what's X then your clarity of title is dubious... $\endgroup$
    – Mithoron
    Nov 8, 2019 at 1:44

1 Answer 1


According to the authors of Ref. 1 this is a general property. They provide an explanation for the stability of such sheets based on formation of particular buckled geometries:

The discovery of a flat two-dimensional crystal known as graphene has contradicted Landau−Peierls−Mermin−Wagner arguments that there is no stable flat form of such crystals. Here, we show that the “flat” shape of graphene arises due to a microscopic buckling at the smallest possible interatomic scale. We show that the graphene, silicene, and other two-dimensional crystals are stable due to transverse short-range displacements of appropriate atoms.

This explanation is corroborated by experimental observations:

On the other hand, the generality of our prediction of the buckled universal shape has been revealed in recent experiments on silicene [...]

The authors note in the introduction that their explanation is only one among three that at that time were being seriously considered. The two other arguments are that (1) the sheets are trapped in a quenched metastable state that is not subject to large thermal perturbations compared to the lattice strength and (2) the sheets are slightly warped in the third dimension, the deformation suppressing thermal excursions.

The reference cited above is from 2012. Low-buckled 2D lattices have been reported more recently, for instance for germanene in Ref. 2.


  1. A. O’Hare, F. V. Kusmartsev, and K. I. Kugel . A Stable “Flat″ Form of Two-Dimensional Crystals: Could Graphene, Silicene, Germanene Be Minigap Semiconductors? Nano Letters 2012 12 (2), 1045-1052. DOI: 10.1021/nl204283q

  2. Jincheng Zhuang, Nan Gao, Zhi Li, Xun Xu, Jiaou Wang, Jijun Zhao, Shi Xue Dou, and Yi Du . Cooperative Electron–Phonon Coupling and Buckled Structure in Germanene on Au(111). ACS Nano 2017, 11 (4) , 3553-3559. DOI: 10.1021/acsnano.7b00687.

  • $\begingroup$ Thanks to Karl for the comment - a good lead. Also, evidently Mithoron's comment is one of the proposed alternative explanations justifying the stability of the nanosheets. $\endgroup$
    – Buck Thorn
    Nov 7, 2019 at 23:28
  • $\begingroup$ Thank you for your answer! It will take some time to dig in to these references. In the mean time, do I understand correctly that what's often termed buckling in Xene parlance (the periodic out-of-plane displacement with each unit cell) does need to be present to preserve stability? $\endgroup$
    – uhoh
    Nov 8, 2019 at 1:29
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    $\begingroup$ I have not followed developments in this field closely enough to know which among the various theories is preferred to explain imperfections (deviations from planarity) or whether there is a universal structure, but there is agreement that they exist and are necessary (expected?). I suspect you could explain the necessity of deviations from planarity based on the presence of ZPE associated with out-of-plane vibrations and entropic considerations. I will have to look further to see what they suggest is limiting behavior is at 0 K. $\endgroup$
    – Buck Thorn
    Nov 8, 2019 at 7:02
  • $\begingroup$ I just wrote this answer at the edge of my comfort zone; how does it look? I can finally accept now that I've learned enough to understand your answer sufficiently :-) $\endgroup$
    – uhoh
    Jun 22, 2021 at 5:28

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