INTRODUCTION

Carbon has numerous allotropic modifications [1–3]; their two-dimensional compounds are of great interest [4, 5], in particular, the most well-known and studied two-dimensional allotrope called graphene [6], which is an isolated single sheet of carbon atoms all being in the sp2 hybridized state, which makes it possible to stabilize a common π system and to ensure the unique electronic properties of graphene. The creation of a two-dimensional allotrope consisting of carbon atoms with various hybridization types is fundamentally relevant and will allow the expansion of applications of low-dimensional carbon forms for electronic devices with tunable properties. The most widely discussed and disputable is the T12 carbon phase [7]; its representation in the form of a monolayer allowed the authors of [4] to propose penta-graphene (PG), which is a hypothetical single-layer carbon allotrope consisting of five-membered rings. Its atomic structure and electronic properties were studied in detail using the density functional theory [8, 9]. In addition, molecular dynamics studies of its thermal properties show that the thermal conductivity at room temperature is about 167 W/mK, which is much lower than that for graphene [10–12], and it is thermally stable up to a temperature of 1000 K [4, 13], which is lower compared to graphene [14]. In addition, PG has a negative Poisson ratio (–0.068); superhigh strength, which can be higher than the strength of graphene [4]; and a band gap width of 2.2 to 3.3 eV, depending on the method of calculation [15]. Thus, because of a low thermal conductivity, the existence of the band gap [4, 9], a negative Poisson’s ratio [4], and a high strength [4], penta-graphene is attractive for various potential applications, e.g., as gas sensors [16, 17]. The authors of [15] demonstrate that multilayer penta-graphene compounds will be promising for applications in electronic and opto-electronic nanodevices because of the possibility of controlling the width of the band gap. The heat transfer properties of the bilayer and multilayer penta-graphene were also considered [18]. Furthermore, it was shown that PG, in contrast to graphene, is insensitive to a change in the number of layers [19]. Previous studies demonstrate numerous fields of the potential application of penta-graphene but its energy stability is lower than that of graphene [4]. This indicates that this carbon form is a metastable phase. One of the possible ways to stabilize this structure is the reduction of out-of-plane vibrational degrees of freedom through the formation of a strong bond with a substrate or an increase in the dimension of the structure through the formation of multilayer structures consisting of PG [20]. At the same time, the authors of [4] predicted the possibility of obtaining penta-graphene from T12 carbon by means of mechanical separation. It is remarkable that covalently bonded PG compounds were considered in [15], whereas only van der Waals (vdW) bonded structures were considered in [19]. Therefore, the possible existence of stable structures with different types of bonding between penta-graphene sheets can be discussed.

The results obtained in [21–24] for the stability of penta-graphene indicate that penta-graphene is not mechanically stable, undergoing bending and twisting deformations of the crystal structure in the periodic and limited representations. However, the authors of [25] showed that bilayer penta-graphene nanoclusters with the АВ stacking order keep a perfect two-dimensional structure. In this case, the bending and tensile forces induced by perpendicular \(s{{p}^{2}}\) dimers of different PG sheets compensate each other and a nanocluster keeps the perfect two-dimensional planar structure. The authors of [26] also showed that the AA-T12 phase has a very high Vickers hardness number (\( \approx \)62 GPa), which is close to the hardness of diamond, and discussed the chemically induced phase transition between this phase and penta-graphene.

In this work, we study the formation and stability of bilayer penta-graphene structures, the thermal and dynamic stability of two-dimensional penta-graphene compounds with various stacking orders, the dependence of the electronic and optical properties on the type of bonding between layers, and the barrier for the transition between covalent and vdW compounds.

CALCULATION METHODS

The study was performed using the density functional theory [27, 28] implemented in VASP [29–31]. The exchange correlation functional was calculated in the generalized gradient approximation with the Perdew–Burke–Ernzerhof parameterization [32]. The projector augmented-wave method [33] was used in the Monkhorst–Pack scheme [34] on a 20 × 20 × 1 k‑point grid. The relaxation of the atomic structure was performed until the maximum interatomic force became smaller than 0.05 eV/Å and the energy spread became smaller than 10−5 eV. A vacuum region no shorter than 15 Å was chosen in the nonperiodic direction. Phonon spectra were calculated with the PH-ONOPY package [35]. To study the stability of bilayer PG structures disregarding the effect of periodic boundary conditions, we chose a cluster with a dimension of 3 × 3 unit cells (19 × 19 Å). The ab initio molecular dynamics simulation at a temperature of 1200 K for 3 ps was carried out to study the stability of the nanocluster. When calculating the optical properties, we used the random phase approximation [28] to evaluate the imaginary part of the dielectric function \({{\varepsilon }_{{{\text{Im}}}}}\) and the Kramers–Kronig relations [36] to determine the real part of the dielectric function \({{\varepsilon }_{{{\text{Re}}}}}\).

The thermal stability of the atomic structure of penta-graphene was also simulated with the L-AMMPS molecular dynamics simulator [37]. The interatomic interaction was described in the ReaxFF model [38], which was successively applied to simulate carbon nanostructures [39]. The average dimensions of the supercell in molecular dynamics simulations were 72 × 72 × 60 Å. The calculations were performed with periodic boundary conditions along the X and Y axes in the plane of the considered structures; the size of the cell along the Z axis perpendicular to the plane of the structures was equal to 60 Å, which is much larger than the cutoff radius of the potential (12 Å). The same method was used to study the thermal stability disregarding periodic boundary conditions. The simulation was carried out for a circular nanocluster with a diameter of 50 Å. Before heating, the structure was optimized using the conjugate gr-adient method until the forces became no more than 10−5 kcal/(mol Å). The Verlet algorithm with a time step of t = 0.15 fs was used to numerically solve the equations of motion. Using the Nosé–Hoover thermostat (NVT ensemble), the structures were heated from 500 to 6000 K for 0.5 ns, i.e., at a heating rate of 1.1 × 1013 K/s. The chosen time intervals correspond to the used potential and were used by other authors, e.g., in [40].

RESULTS AND DISCUSSION

Structural Characteristics

Vertical PG compounds can be represented in the form of various stacking orders of layers by analogy with graphite. The АА stacking order is formed by two identical PG sheets one over the other, the АВ stacking order is composed of penta-graphene sheets shifted with respect each other by the half-period of the lattice a/2, the AA' stacking order is obtained by the inversion of PG sheets, and the AB' stacking order is formed by inverted PG sheets shifted by a/2. The calculated equilibrium distances between the PG sheets with the AA, AB, AA'_2, and AB'_2 stacking orders are 2.7, 3.6, 3.8, and 3.6 Å, respectively, and correspond to the vdW bonding. Equilibrium structures for the AA'_1 and AB'_1 stacking orders with the interlayer distances of 1.6 and 1.3 Å, respectively, and covalent bonding correspond to the AA-T12 and T12 phases, respectively. The AA-T12 and T12 phases are denoted below as AA'_1 and AB'_1, respectively.

The calculated binding energies of the AA, AB, AA'_2, and AB'_2 structures (see Figs. 1b and 1d) are in the range from 5 to 20 meV/atom, corresponding to vdW materials [14, 41]. At the same time, the binding energies of the AA'_1 and AB'_1 structures (see Fig. 1c) are 250 and 300 meV/atom, respectively, which corresponds to the covalent (nvdW) interaction. Thus, covalently bonding PG compound is formed in the AA'_1 and AB'_1 structures, whereas the AA, AB, AA'_2, and AB'_2 structures have vdW bonding.

Fig. 1.
figure 1

(Color online) (a) Top views of all considered carbon structures with different stacking orders; (b–d) side views of (b) AA and AB stacking orders, (c) AA'_1 and AB'_1 stacking orders, and (d) AA'_2 and AB'_2 stacking orders.

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Two energy minima for different interlayer distances were found only with the AA' and AB' stacking orders, which indicates the existence of two states with vdW and covalent bonding types with the possible barrier for the transition between these states. Since periodic boundary conditions were used and the lattice parameters for the covalent and vdW structures are noticeably different, barriers for the transition were calculated by calculating the energies of the structures with increasing and decreasing interlayer distance in the covalent and vdW structures, respectively, without structural optimization. The intersection of the resulting dependences of the energy on the interlayer distance (see Figs. 2a and 2d) corresponds to the barrier for the transition. Blue lines are the dependences of the energy of the structure in the energetically favorable covalently bonded state on the distance between two penta-graphene sheets with its increase. Brown lines correspond to approaching the PG sheets, beginning with the energy favorable vdW state. Thus, the barrier for the transition from the vdW to covalent structure for the AA' stacking order is 0.11 eV/atom, whereas the barrier for the transition from the cov-alently bonded to vdW bonded structure is 0.38 eV/atom (see Fig. 2a). The determined barrier for the transition from the vdW state corresponds to the energy of the transformation of ethylene to cyclobutane with the rupture of double bonds and the transition of carbon atoms from the sp2 to sp3 hybridized state in the planar representation of the cyclobutane molecule [42]. The respective barriers for the AB' stacking order are 0.03 and 0.11 eV/atom for the transition from the vdW to covalent structure and back, respectively (see Fig. 2d). Thus, for the transition between these states, it is necessary to overcome the energy barrier without additional conditions as, e.g., in the case of diamanes, where the functionalization of the surface with chlorine and bromine or hydrogen and fluorine is required [43, 44] in order to reduce the barrier for the transition to the covalent state and to stabilize the resulting film. Nevertheless, the barrier for such a transition from multilayer graphene to diamane 0.4 eV/atom is higher than that for penta-graphene [45]. Moreover, the functionalization of the surface also requires additional energy for the rupture of the π bond of carbon.

Fig. 2.
figure 2

(Color online) (a, c) Analysis of the barrier for the transition between covalent and van der Waals bonding in the (a) AA' and (c) AB' structures. The energy versus the interlayer distance in the case of (blue lines) an increase in the interlayer distance from the equilibrium position with covalent bonding and (brown lines) approaching the penta-graphene sheets from the van der Waals state. The dashed line marks the barrier for the transition between the structures. Arrows indicate the barrier for the transition from the corresponding position. The inset of panel (a) shows the width of the band gap versus the distance between the nearest carbon atoms in different penta-graphene sheets. (b) Phonon spectra of periodic covalently (nvdW) bonded penta-graphene structures. (d) Time dependences of the relative energy of the cluster obtained by (top) the ab initio molecular dynamics simulation at 1200 K and (bottom) the ReaxFF method at 600 K.

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The dynamic stability of the considered periodic structures AA' and AB' was estimated by calculating phonon spectra (see Fig. 2b). Imaginary modes in the phonon modes are absent in all considered structures both covalently and vdW bonded, which indicates their possible dynamic stability. Furthermore, the molecular dynamics simulation shows that the amorphization temperatures for the covalently bonded structures with the AA'_1 and AB'_1 stacking orders are 1350 and 1300 K, respectively, which are higher than those for diamanes, which keep their structural stability to a temperature of 1200 K [44].

Moreover, in much the same way as in [24], to study covalently bonded nanoclusters of bilayer compounds with the АА' and AB' stacking orders, we performed the ab initio molecular dynamics simulation of the behavior of a cluster with a dimension of 3 × 3 unit cells at a temperature of 1200 K. The ReaxFF method was also used to examine the thermal stability of larger nanoclusters at a temperature of 600 K (see Fig. 2d). In both studies, the nanoclusters keep a perfect planar structure during the entire simulation time. Both layers in the considered configurations form a united structure with two perpendicular pairs of sp2 hybridized carbon atoms. The resulting bonding force induced by perpendicular sp2 hybridization provides a strong opposite response to tension because of a significant thickness of the bilayer penta-graphene compound. As a result, the bending and tensile forces compensate each other and the structure keeps the perfect planar shape. Thus, the covalently bonded l-ayers with the АА' and AB' stacking orders satisfy the  topology conservation theorem proposed by P.V. Avramov [46]. The time dependence of the total energy of penta-graphene nanoclusters with a dimension of 3 × 3 unit cells for 3 ps is shown in Fig. 2d.

Thus, the energy minimum, the absence of imaginary modes in phonon spectra, and the stability of two-dimensional penta-graphene nanoclusters at 600 and 1200 K indicate the stability of the considered covalently bonded carbon thin films.

To estimate the mechanical properties of the considered compounds, we calculated the elastic constants C11 and C12, the Poisson ratio

$$\nu = {{C}_{{12}}}{\text{/}}{{C}_{{11}}},$$

and the Young modulus [47]

$$E = \frac{{C_{{11}}^{2} - C_{{12}}^{2}}}{{{{C}_{{11}}}}} .$$

The results are summarized in Table 1 in comparison with the data for diamane, graphene, and the h-P‑C18 phase. It is seen that Young moduli of covalent PG compounds are noticeably higher than those for other presented two-dimensional carbon allotropes. Furthermore, the AA'_1 structure, as well as multilayer penta-graphene, has a negative Poisson ratio.

Table 1. Mechanical properties of covalently bonded penta-graphene compounds compared to diamane, graphene, and hP-C18
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Electronic Properties

We calculated the electron band structures of the proposed compounds with the AA' and AB' stacking orders. The width of the band gap Eg in the AA'_2 and AB'_2 vdW structures is 2.0 eV, which is smaller than a value of 2.2 eV in multilayer PG [15]. However, it was previously shown that the width of the band gap decreases to 1.58 eV with an increase in the number of layers with the АА stacking order of penta-graphene with the vdW interaction [9]. The analysis of the evolution of the band gap during the transition from vdW to covalent bonding under the variation of the distance between the nearest carbon atoms of neighboring penta-graphene sheets (see the inset of Fig. 2а) showed that the band gap decreases with a decrease in the distance dC–C between the nearest carbon atoms to ~2.1 Å and further increases with a decrease in the interlayer distance to the equilibrium value corresponding to the covalent bonding of carbon atoms from neighboring layers. This is in agreement with the evolution of the electronic structure in layered materials such as InSe [51], but in the considered materials, this can be due to a change in the hybridization of carbon atoms. The widths of the band gap in the covalently bonded AA'_1 and AB'_1 structures are 1.3 and 1.5 eV, respectively, in agreement with previous results [15].

The width of the band gap in the bilayer AA' and AB' compounds is much smaller than that in diamanes [5, 44], which can be favorable for optoelectronic applications.

Next, we consider the electron localization function for the AA' structure with covalent (see Fig. 3a) and vdW (see Fig. 3b) bonding types. In the case of the covalent bonding, sp2 hybridized states are transformed to sp3 hybridized states for atoms involved in covalent bonds, which is demonstrated in Fig. 3a. It is noteworthy that prolate electron localization clouds for vdW structures (see Fig. 3b) correspond to the formation of π bonds between p orbitals of carbon atoms in the sp2 hybridized state. The electron localization functions for the AB' structure with the covalent and vdW bonding types are shown in Figs. 3c and 3d, respectively. It is remarkable that atoms of these compounds with different bonding types are both in sp3 and in sp2 hybridized states in contrast to diamanes, where all carbon atoms are sp3 hybridized [52].

Fig. 3.
figure 3

(Color online) Electron localization function for the AA' and AB' structures with (a, c) covalent and (b, d) van der Waals bonding types (0.6 isovalue).

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Optical Characteristics

The optical absorbance in the perpendicular direction for the AA' and AB' structures was calculated by the formula [53]

$$\alpha = \frac{{2\omega }}{c}k,$$

where

$$k = \sqrt {\frac{{\sqrt {\varepsilon _{{{\text{Re}}}}^{2} + \varepsilon _{{{\text{Im}}}}^{2}} - {{\varepsilon }_{{{\text{Re}}}}}}}{2}} $$

is the extinction coefficient obtained from the dielectric function.

The energy dependences of the absorbance obtained for the AA' and AB' structures with the covalent and vdW bonding types are shown in Fig. 4 by blue and brown lines, respectively. The absorption region for the structures with the covalent bonding begins below 2 eV and corresponds to the red visible range. The absorption region in the case of the vdW bonding begins with ~2 eV in agreement with the width of the band gap in these structures (2.1 eV). Thus, the structure with the covalent bonding absorbs in a wide range because the width of its band gap is smaller than that in the structure with the vdW bonding.

Fig. 4.
figure 4

(Color online) Spectrum of the absorption coefficient in the perpendicular direction for the (a) AA' and (b) AB' structures in the (blue line) covalently (nvdW) and (brown line) van der Waals bonded compounds. The circle marks the absorption peak appearing at the covalent bonding of AA' and AB'.

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CONCLUSIONS

To summarize, we have considered thin films of covalently bonded penta-graphene sheets in the AA' and AB' stacking orders and have examined their thermal and dynamic stability. It has been shown that AA' and AB' compounds have two energy minima at van der Waals and covalent bonding types unlike the AA and AB stacking orders. Since imaginary modes are absent, it is reasonable to examine the stability of the considered structures. The stability of the two-dimensional structure of finite carbon nanoclusters with the AA' and AB' stacking orders at temperatures of 600 and 1200 K has been demonstrated; i.e., they satisfy the topology conservation theorem. The detailed analysis of the barrier for the transition between the van der Waals and covalently bonded compounds has shown that the latter compounds can be obtained with lower energy consumptions compared to the formation of thin diamond films, i.e., diamanes. The barriers for the transition from the van der Waals to covalent structures with the AA' and AB' stacking orders are 0.11 and 0.03 eV/atom, respectively, whereas higher barriers of 0.38 and 0.33 eV/atom should be overcome for the transition from the covalent to van der Waals structures with the AA' and AB' stacking orders, respectively. It has been found that the width of the band gap first decreases with a decrease in the distance between layers and then increases sharply because of the formation of covalent \(s{{p}^{3}}\) bonds between neighboring penta-graphene sheets. In addition, the formation of covalently bonded penta-graphene thin films results in the appearance of the absorption peak below 2 eV, which corresponds to the red visible range. The determined physicochemical characteristics indicate that the creation of two-dimensional covalent penta-graphene compounds is promising for applications in elements of optoelectronic devices, which can keep their stability in a high temperature range.