Mechanical Characteristics of Diamond-Like Moiré Films

The mechanical characteristics of diamond-like films, such as Dn21.8, Dn27.8, and Dn29.4 moiré diamanes, formed by the hydrogenation of graphene layers twisted at an angle of about 30°, and of conventional diamane (Dn) based on bilayer graphene with the AB packing are simulated using the molecular mechanics approach. The elastic moduli of these materials are calculated. It is shown that the elastic constants for moiré diamanes differ noticeably from similar constants of Dn diamane, and their fracture occurs at higher strains than at those for the latter. The responses to forces applied in the plane of the films turn out to be anisotropic for the Dn21.8 and Dn27.8 structures and almost isotropic for Dn29.4. It is shown that the breakthrough for the Dn29.4 membrane under the action of a tip occurs at a larger force than that for the membrane made of the most energetically stable Dn27.8 diamane.


INTRODUCTION
The properties of twisted bilayer graphene whose layers are rotated relative to each other by an angle Θ have recently been studied in numerous works [1,2]. Moreover, graphene sheets can be imposed on each other at a given angle with an accuracy of about 0.1° [ 3]. This possibility has opened up a whole new research field aimed at finding out and studying the unique characteristics of twisted layers having various atomic compositions, which are coupled by molecular bonds. This research field is now referred to as twistronics [4].
On the other hand, advances in the development of methods for obtaining several covalently bound untwisted graphene layers by functionalizing their surface with light atoms or molecules stimulated the studies of the characteristics and possible applications of such diamond-like films [5][6][7], which were called diamanes in the pioneering work [8], where the structures and main characteristics of these new two-dimensional materials were described in ab initio calculations. Such diamanes have recently been obtained by hydrogenation [9] and fluorination of bilayer graphene [10]. The possible synthesis mechanism of diamanes is described in detail in [11].
The possibility of creating various commensurate moiré superlattices using two monolayers formed by twisting them at certain angles [12] led to the idea of obtaining a two-dimensional moiré material, in which some regions of the bilayer are covalently bound (as a result of the transformation of their carbon atoms from the to hybridization) [2]. Such structures exhibit mechanical characteristics different from those of diamane (Dn) formed using untwisted graphene bilayers. However, the mechanical characteristics of the recently predicted Θ-Dn diamanes formed under the functionalization of the surface of graphene bilayers twisted by angles close to 30° have not been studied [12,13]. The structures described in these works should have electronic properties different from those of ordinary diamanes, such as a wide band gap (E g > 3 eV) and a resonant electron spectrum. Such properties could be quite topical in nanoelectronics and optoelectronics. We are interested in determining their elastic and strength properties, which are important for creating new elements of opto-and electromechanical nanodevices, and in comparing them with the mechanical properties of twisted bilayer graphenes with diamane inclusions [14].

CALCULATION TECHNIQUE
All molecular mechanics and molecular dynamics simulations presented in this paper were performed using the GULP software package [15]. The interatomic interaction was specified in the form of the Brenner and Lennard-Jones potentials. The structures were optimized by the conjugate gradient method.

PARAMETERS AND ELASTIC CHARACTERISTICS OF THE STRUCTURES
UNDER STUDY We study four structures of bilayer graphene with adsorbed hydrogen atoms: Dn21.8, Dn27.8, and Dn29.4 diamanes and for comparison the ordinary Dn diamane based on bilayer graphene with the AB (Bernal) type stacking [8].
For the structures under study, we calculated the formation energy E F by the formula where E is the total energy of the system; E C are E H are the energies of carbon or hydrogen atom in the graphene sheet, respectively; N is the total number of atoms; and N C are N H are the numbers of carbon and hydrogen atoms in each cell used in our computations, respectively.
Using the well-known formulas describing the elastic behavior of the structures at small strains, we calculated the effective Young moduli Y and Poisson ratios μ: Here, F is the force acting on the structure along each side of the chosen rectangular computation cell; L and S = D(h + d) are the length and cross-sectional area of this cell, respectively, where D and h are the width and thickness of the cell, respectively (Fig. 1), 3.35 Å is the usually accepted interlayer distance in graphite; and ΔL and ΔD are the changes in the size of the structure along and across the direction of the applied force, respectively.
For this, we chose (as usual in similar calculations [14]) extended unit cells of diamanes. The computation cells are shown in Fig. 1. The results of calculations of structures under study and the corresponding parameters are presented in Table 1.
The Dn diamane turns out to be the most energetically favorable one; this agrees with the results of [12]. Since the Young modulus for different directions of the structures varies within ±2%, we show in Table 1 only the effective Young modulus for displacements along the X direction. For the ordinary diamane, Dn, it is comparable with the Y characteristic of bulk diamond [16], whereas for moiré diamanes, it is smaller. This occurs because the unit area of the Dn diamane contains more interlayer C-C' bonds than that in moiré diamanes, where some of the bonds in the unit cell that are not perpendicular to the film between the C and C' atoms of neighboring layers and C-C and C'-C' "cross" bonds are stressed [12,13]. This means that, at small strains, when Hooke's law is still valid, a smaller force F can be applied in the plane of the film because initially somewhat corrugated surfaces of the moiré diamane are straightened owing to the presence of such C-C' bonds. Moreover, in the Dn27.8 diamane, the number of cross H-C-C-H complexes per unit area is larger, and the modulus Y is smaller than that in the Dn21.8 structure. In the highly symmetric  structure of Dn diamane, the С-Н, C'-H', and С-С' bonds (between layers), as well as С-С and C'-C' bonds, remain uniformly distributed in the film plane almost in the same way as in diamond under the action of forces applied along similar planes. The Poisson ratio for moiré diamane with C-C' bonds, which, in contrast to diamond and Dn, are not directed along the normal, is comparable to the μ value for diamond, since the changes ∆L and ∆D are almost the same. However, the μ value for the ordinary diamane, where C atoms in the upper and lower layers are bound to H atoms more weakly than to C atoms "confining" the structure in adjacent layers of the diamond, is almost a factor of 2.5 smaller than that for diamond (see Table 1). This, as shown below, leads to a larger deflection of a Dn diamane disk than for a Dn-Θ moiré disk of the same diameter with the same force applied by the tip along the normal to the disk. Next, we simulated changes in the energy characteristics of the structures under study under tension along the X and Y axes up to their rupture. To see these changes at sizes exceeding the computation cell size, we chose (as in the previously studied cases [15,18]) rectangular (a × b) cells: 4.0 × 5.2 nm (16 × 12 computation cell) for Dn, 4.6 × 5.3 nm (3 × 6 computation cell) for Dn27.8, and 4.9 × 5.4 nm (2 × 3 computation cell) for Dn29.4. Such cell sizes were chosen to eliminate the contribution of boundary effects, which usually arise in the calculations involving the film rupture at small computation cell sizes. Under tension, we used the method described in [19].
Changes in strains under the action of the force applied in different directions both in moiré diamanes and in the Dn structure [18] are different, i.e., strongly anisotropic. This is clearly seen in the analysis of changes in the angles and bond lengths in Dn, Dn21.8, Dn27.8, and Dn29.4 diamanes under tension.
Initially, all angles in the Dn structure are equal to 109°85′, and the lengths of all C-C bonds are equal to 1.54 Å, which corresponds to the tetrahedral arrangement of carbon atoms. At the tensile strain ( , ), four angles in the structure of ordinary diamane formed upon the complete hydro- genation of bilayer AB graphene decrease significantly to 107°40′, making the structure noticeably asymmetric. Moiré diamanes initially have lower symmetry of the atomic structures with different C-C-C angles and C-C bonds and with somewhat bulging domains in the computation cell (see Fig. 1). Under tension, these domains are flattened relative to their original shape. In ordinary Dn diamane, all C-C bonds become elongated under tension, whereas in the Dn27.8 and Dn29.4 structures at ε = 0.15, not all bonds are significantly elongated (see Fig. 2).
In the case of stress applied to diamane Dn27.8 (within the computation cell) along the X and Y directions, rupture occurs at the strain ε X = 0.34 and at ε Y = 0.3, respectively-see Fig. 3 (the first steps of rupture are shown in the inset near the bottom of the figure).
Ruptures of stressed Dn21.8 diamane begin at strains ε X = 0.31 and at ε Y = 0.33 and are pronounced at strains ε X = 0.35 and at ε Y = 0.4. A clear rupture boundary and nearly complete fracture of the sheet are observed at ε X = 0.36 and at ε Y = 0.37, when only several C-C bonds remain between two parts of the sheet.
The tension of Dn21.8 and Dn27.8 diamanes first leads to the formation of topological pentagon/heptagon carbon defects and then, when individual bonds are broken, results in the formation of "nanoholes" between them (see, e.g., fragments shown in Fig. 4). At ε > 0.3, tension leads to the formation of more and more polygons near the first defects, which line up in a long chain, and then, at ε > 0.35, separated regions go apart from each other, being connected by chains of atoms between them.
For comparison, we show in Fig. 5 the corresponding plots for ordinary Dn diamane. For it, the critical stresses corresponding to the rupture (ε X = 0.21 and at ε Y = 0.23) are in good agreement with the results of [18].
The Young modulus and tensile strength for Dn diamane also depend on the direction of the applied force in the film plane: for different directions, the Young modulus and stress-strain curves deviate from each other by about 10%, as shown in [18]. In our case, for the Dn structure (see Fig. 1a) under tension along the X axis close to the "armchair" direction, the rupture begins earlier than for the force applied along the "zigzag" Y direction. A feature of the Dn29.4 diamane is that the starting points of its lattice transformations (at ε X , ε Y ≈ 0.24) and the critical strains along the X and Y directions corresponding to its fracture (ε X , ε Y ≈ 0.4) are nearly the same according to our calculations illustrated in Figs. 3-5. The reason is that, in the Dn29.4 structure, where the twisting angle of the sheets relative to each other is about 30°, the upper layer contains armchair chains of C atoms having nearly the same direction as   the zigzag chains in the lower layer (see Figs. 6b and 6c). Therefore, the tension along the X axis of the computation cell leads to its distortion similar to that occurring under the action of the force acting along the Y axis, since their action on the Dn29.4 film is effectively "averaged" by the presence of both types of interacting unidirectional chains of C atoms in the structure.
Thus, the calculation shows that, under tension in the plane of moiré Dn21.8, Dn27.8, and Dn29.4 diamanes, their ruptures occur at higher strains than those in the Dn structure. This is directly related to the previously described explanation of the difference in their mechanical characteristics.

MEMBRANE RUPTURE UNDER THE ACTION OF A PROBE
The behavior of diamane membranes affected by a probe was first reported in 2009 [8] and then recently studied in more detail using membranes made of partially hydrogenated bilayer graphene with a twist angle  of 7.34° [14] and a diamane quasicrystal [20]. Here, we simulated the rupture of circular Dn27.8 and Dn29.4 membranes 7 nm in diameter, in the situation where the tip of the nanoprobe is pressed against their center with fixed edges (according to the method described in [8]). The calculation was carried out until the time when the tip breaks through the membrane (Fig. 7). The dynamics of the deflection of these membranes differs from those described earlier [8,20]: the pressed Dn27.8 membrane is deflected almost uniformly from its center to the edge, while the observed deflection of the Dn29.4 diamane membrane is more localized near the center, apparently owing to the existence of rigid inhomogeneously directed bonds between carbon atoms [12,20].
The Dn27.8 and Dn29.4 diamane membranes are deflected without fracture down to critical depths δ c = 11 Å and 9.4 Å, respectively. In this case, the "critical" force F c = 265 nN applied to the Dn29.4 membrane is 4% higher than that for the Dn27.8 membrane. This suggests a higher rigidity of Dn29.4 diamane modeling the quasicrystal, which is reported to be more rigid [20] as compared to untwisted Dn diamane.
To summarize, we have found that the two-dimensional Dn21.8, Dn27.8, and Dn29.4 moiré structures have elastic characteristics different from those of untwisted Dn diamane. Their rupture occurs at higher longitudinal stresses, and to break through such moiré membranes with a tip, it is necessary to apply a larger force, which is a signature of their higher rigidity. Note that the two-dimensional diamond-like films under study having a moiré atomic superlattice should exhibit characteristic opto-and electromechanical effects under the action of mechanical forces [21]. Since resonant peaks in the electron density of states in bilayer moiré structures can be changed by applied mechanical stresses [22], it would be interesting to use this feature in nonlinear optics to control the generation of higher harmonics and mixing of high-order waves (see, e.g., [23]). Such expected effects require a separate study.
Our studies suggest that moiré diamanes with such diamond-like structure can meet the necessary requirements for creating various layered nanosystems with unique mechanical characteristics. This implies that they are quite promising materials.