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Journal Club for February 2017: Nanoscale buckling in 2D materials
Graphene as a typical two dimensional (2D) crystal membrane attracts tremendous interest. Geometrical distortion such as nanoscale buckling morphology is widely observed in these 2D materials and is crucial to modulating the electronic properties . The calculated bending stiffness of graphene is found to be the same order as that of the lipid bilayer , therefore buckling instability may easily occur in graphene leading to non-planar configurations. Different from the conventional buckling transition, the buckling instabilities in 2D materials show different physical origins, and its buckling pattern could be down to nanoscale. Proper mechanics models of nanoscale buckles for their formation and controllability need to be developed for various engineering applications. A few examples of nanoscale buckling in 2D materials are introduced to hopefully offer a few commonplace remarks.
Buckling of a free-standing 2D material
The results in Figure 1 show that due to the super flexibility of the free-standing 2D material the nanoscale buckling instability can be easily triggered by various energy dissipations, such as thermal fluctuation [3,4], relaxations of edge stress [5,6], coherent interface stress  and stress field created by topological defects [8-10]. The induced out-of-plane distortion can demonstrate remarkable changes compared to the flat counterparts, such as reductions of dislocation formation energy , and Griffith strength .
Figure 1 Nanoscale buckling structures in a free-standing layer caused by (a) thermal fluctuation , (b) edge stress , (c) coherent interface relaxation ,(d) topological defects 
Buckling of a monolayer on substrates
When graphene-like 2D materials are bonded to substrates, the nanoscale rippling may still be significant although the contribution by thermal fluctuation is neglectable. Figure 2(a)-(b) demonstrates that if the misfit in lattice parameters and/or thermal expansion coefficient between the 2D material and the substrate induces global compression, the monolayer tends to develop wrinkles or fold localization [12-14]. Interestingly, Figure 2(b) also shows that if the monolayer is under global tension due to the mismatch, emergence of interfacial misfit dislocations described by Frenkel–Kontorova model  effectively accommodates the mismatch and localized buckles can arise near the dislocation core where the monolayer is under localized compression . This is the case when Frenkel and Kontorova meet Von Karman, the in-plane deformation due to interfacial misfit dislocations are strongly coupled to out-of-plane deformation. The analytical treatment for the localized buckle is still a challenge. In addition, the results obtained by first principle investigation in Figure 3 show that the local chemical bonding environment may have significant influence on the nanoscale buckling [17,18].
Figure 2 Sketch of nanoscale buckling structures in a monolayer on substrates (a) ,(b)wrinkling to folding transition under global compression [12,13], (b) formation of ordered rippling pattern under localized compression near the dislocation core 
Figure 3 Formation of nanoscale rippled graphene phase on substrates by first principle investigation, (a)bonding energy between the graphene edge and the Ni substrate, (b) total energy change , (c),(d) the bonding structures in the rippled graphene on Ir substrate 
Buckling of twisted bilayer graphene
In bilayer graphene, twist modifies the electronic properties. The twist stacking usually leads to a characteristic interfacial dislocation pattern, known as a Moiré pattern. Emergence of significant buckling in the bilayer graphene governs further relaxation of the Moiré pattern [19-24]. The results in Figure 4 show that relaxed Moiré patterns with out-of-plane displacement and twisted dislocation structures can form [21,22].
Figure 4 Nanoscale buckling arose in twisted bilayer graphene, (a) simulated result by a generalized Peierls−Nabarro model , (b) simulated result by the molecular dynamics code LAMMPS 
The interfacial dislocation with out-of-plane deformation is a so called strain soliton . The results in Figure 5 show that manipulating and visualizing the motion of nanoscale buckled structures have attracted increasing interest since they have substantial effects on the electronic and mechanical properties of such 2D materials [25,26].
Figure 5 Motion and visualization of nanoscale buckled structures (a) movement of the nanoscale wrinkle by thermophoresis  (b) visualization of domain-wall soliton in exfoliated bilayer graphene 
In most 2D materials, out-of-plane distortion as an efficient way to relax compressive stress. It usually competes with other stress relieving modes. The different coupling mechanisms may exhibit different physical origins of the observed nanoscale bucklings. Further theoretical investigations into this fascinating system are to be exploited.
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