Prediction of Two-Dimensional Boron Sheets by Particle Swarm

Article pubs.acs.org/JPCC

Prediction of Two-Dimensional Boron Sheets by Particle Swarm Optimization Algorithm Xiao Yu, Lanlan Li, Xue-Wen Xu, and Cheng-Chun Tang* School of Materials Science and Engineering, Hebei University of Technology, Tianjin 300130, China ABSTRACT: We searched for two-dimensional metastable boron sheets by using the particle swarm optimization algorithm combined with first-principles methods. We present several possible boron sheets composed of triangular and hexagonal motifs. The calculated total energy shows that these boron sheets are less stable than α-sheet and two recently predicted 1/8 and 2/15 B-layers. The energy difference between the new proposed struc-1/8 boron sheet and the known α-sheet, 1/8 and 2/15 B-layers, is less than 0.01 eV/atom. The calculated total density states of these boron sheets show that they are metallic. The multicenter chemical bonds of the relatively stable boron sheets are analyzed using the electron localization function. Because two-dimensional sheets are the building blocks of corresponding nanotubes and fullerenes, the proposed metastable boron sheets, not only with the most stable boron sheets, are expected to advance further investigations and understandings on boron nanotubes and fullerenes.

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INTRODUCTION Boron, the fifth element in the periodic table of elements, has received considerable attentions since the discovery of carbon nanostructures.1 It is an electron-deficient atom with three valence electrons in four available orbitals, making it distinct from that of carbon. Bulk boron has several crystal structures: rhombohedral α-phase and β-phase, tetragonal T-phase, and recently synthesized γ-phase,2 whereas the well-known phases of bulk carbon are diamond and graphite. Nevertheless, the boron nanostructures are expected to have some similar structures to that of its neighbor elemental carbon. For example, the structure of recently synthesized B19− cluster is related to those of annulene (C10H10) and circulene (C24H12).3 Because two-dimensional (2D) boron sheets are the building blocks of boron nanotubes and fullerenes, the investigation of new possibilities for boron sheets is important to understanding the structures and properties of boron nanostructures. Early theoretical studies on 2D boron sheets focused on triangular lattices.4−6 These buckled triangular sheets have lower total energy than that of flat triangular sheets. After a particularly stable and highly symmetric B80 fullerene was reported,7 a new boron sheet (α-sheet) that is more stable than the previous triangular lattice was theoretically predicted.8,9 The unit cell of α-sheet comprises triangular and hexagonal motifs, which can also be viewed as hole-doped triangular lattices. The stability of α-sheet is explained by a balance between two-center and three-center bonds, which is used to explain the extreme stability of boron B80 fullerene.7 A detailed chemical bonding analysis of the three types of fragments (B7+7, B22+16, and B30+16) of α-sheet confirms the stability of α-sheet.10 In addition to α-sheet, other 2D boron sheets have been reported. The investigations on the thermodynamic stability of some 2D boron sheets indicate that chemical bonding is the key factor in determining the stability of the boron sheets.11 Tang and Ismail-Beigi proposed several boron sheets based on triangular and hexagonal motifs.12 A new planar sheet (snub© 2012 American Chemical Society

sheet) that is composed of triangular and hexagonal motifs was later predicted.13 All boron atoms in the snub-sheet have five nearest neighbors, and its total energy is 0.02 eV/atom less stable than that of α-sheet. Some boron sheets that comprise close-packed triangular and hexagon boron lattices have recently been shown to have lower total energy than that of α-sheet.14 However, the corresponding works are restricted to the boron sheets with close-packed triangular and hexagonal lattices.14 Despite the favorable features of these presented 2D boron sheets, many metastable boron sheets remain unknown. The recently developed particle swarm optimization (PSO) algorithm for crystal searching regions facilitates the exploration of metastable boron sheets without structural restrictions.15,16 In this article, we investigate 2D boron sheets by using the PSO algorithm combined with first-principles methods. Several metastable boron sheets whose energies are close to the energy of the most stable boron sheet are presented. Because the stability of sheets differs from that of nanotubes and fullerenes, our results are expected to advance further explorations into boron fullerenes and nanotubes, as well as elucidate corresponding stability.

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CALCULATION METHODS Structural search simulations for boron sheets are performed using the CALYPSO code with the local PSO minimization schemes within the unit cell of 16 atoms.16 Local structural relaxations are performed using density functional theory within the Perdew−Burke−Ernzerhof (PBE) parametrization of generalized gradient approximation (GGA), as implemented in the Vienna ab initio Simulation Package code.17 The allelectron projector augmented wave method is adopted with Received: June 6, 2012 Revised: August 18, 2012 Published: August 30, 2012 20075

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2s22p1 treated as valence electrons. A vacuum slab of 15 Å is selected. A plane-wave basis set with an energy cutoff of 400 eV and k points of 0.02 1/Å are used. The structures studied here are relaxed at zero pressure until the total energy difference is less than 1 × 10−5 eV/atom, and the stress is less than 0.02 GPa. The band structures and density of states of boron sheets are calculated using the CASTEP code in GGA with ultrasoft pseudopotential.18

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RESULTS AND DISCUSSION Figure 1 shows the calculation results for the boron sheets with relatively lower total energy. In our simulations, several

Figure 1. Energy difference vs hexagon hole density η for boron sheets. The vertical dashed red lines show cases with several fixed η. This figure can also be interpreted as depicting energy difference vs the binary composition map of flat triangular boron sheets and hexagonal boron sheets. The boron sheets represented by the blue boxes are obtained by the PSO searching in this work, whereas those represented by the red boxes are obtained from Yakobson and coworkers. All these structures are relaxed using first-principles methods. A solid blue box indicates that a sheet was predicted by previous studies.

Figure 2. Structures of relatively stable boron sheets with fixed hexagon hole density η: struc-η (η = 1/4, 1/5, 1/6, 1/7, 1/8, 1/9, 1/ 10, and 1/12). The black solid lines show the unit cell, while the blue lines show the primitive cell.

structures that were proposed earlier are reproduced: α-, β-, γ-, and snub- boron sheets.7,8,10,13−16 Four recently predicted boron sheets by Yakobson and co-workers (named as 1/8, 1/ 10, 2/15, and 4/27 B-layers in this article), which are not obtained in the PSO searching, are also calculated within firstprinciples methods and shown in Figure 1 (marked as red boxes) for comparison.14 Most of the proposed boron sheets comprise triangular and hexagonal motifs. Our calculation results show that the most energetically favorable structure of the α-sheet in the previous calculations8,9 is less stable than that of the recently predicted 1/8 and 2/15 B-layers.14 The relative energy difference is defined as Ed = Esheet − Eα‑sheet, where Esheet is the total energy per atom of the calculated boron sheet. These sheets can be obtained by the patterned holes from a flat triangular sheet. We select hexagon hole density, defined as8 η=

considered as the mixtures of flat triangular and hexagonal boron sheets, Figure 1 can be converted into the binary composition map of AxBy, where A is defined as the flat triangular boron sheet, B is defined as the flat hexagonal boron sheet, and x and y are the numbers of the flat triangular and hexagonal boron sheets in the unit cell, respectively. Therefore, some relatively stable compositions of boron sheets can be obtained directly from Figure 1. From the perspective of a binary composition map, the boxes crossed by the solid lines shown in Figure 1 should be the relatively stable composition. Thus, 1/8, 1/10, and 2/15 B-layers and struc-η with η equal to 1/12, 1/9, 1/5, and 1/4 should be the relatively stable boron sheets. On the basis of the evenly or linearly distributed hexagons and hexagon hole density of boron sheets, Tang and IsmailBeigi showed that the flat boron sheets are located at the region 1/9 < η < 1/5.12 The flat boron sheets should be converted into buckled structures with η < 1/9, and complex unstable boron sheets should be obtained with η > 1/5.12 Our results are in accordance with those of Tang and Ismail-Beigi’s12 because our proposed boron sheets (Figure 1) are located at the region 1/9 < η < 1/5. Figure 3 shows the boron sheets with a total energy less than that of struc-1/10 (horizontal dashed line in Figure 1).

No. of hexagon holes No. of atoms in the original triangular sheet

Because the flat boron sheets that are composed of only triangular lattice or hexagonal lattices are unstable, the corresponding two limiting cases of η = 0 and η = 1/3 are not shown in Figure 1. We label the boron sheet with the lowest energy difference in our calculations at fixed η as struc-η. The corresponding crystal structures of struc-η with η equal to 1/4, 1/5, 1/6, 1/7, 1/8, 1/9, 1/10, and 1/12 are shown in Figure 2. Given that the predicted boron sheets can also be 20076

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that all the boron sheets investigated in the current work are metallic. Figure 5 illustrates the chemical bonding analyses through the electron localization function (ELF) for the representative 2D boron sheets of 1/8 and 2/15 B-layers and struc-1/7, 1/8, and 1/9. Because the σ-bonds in plane are more localized than π-bonds, we select an isosurface of 0.78. The chemical bonding of the α-sheet was analyzed by the adaptive natural density partitioning (AdNDP) method through various boron fragments.10 The unit cell of the α-sheet is composed of threecenter-two-electron (3c-2e) σ-bonds, 4c-2e σ-bonds, and delocalized π-bonds. None of the 2c-2e σ-bonds are observed in the α-sheet when examined using AdNDP methods.10 As shown in Figure 5c, three 4c-2e σ-bonds and six 3c-2e σ-bonds are present in the unit cell of the α-sheet. The other electrons occupy the delocalized π-bonds in the α-sheet. The ELF results for the α-sheet agree with those of the AdNDP methods. Given the high symmetry, all the 4c-2e σ-bonds and 3c-2e σ-bonds in the α-sheet are the same. In comparison with the α-sheet, which have uniform 3c-2e σ-bonds, struc-1/8 has two types of 3c-2e σ-bonds (Figure 5b). Two 3c-2e σ-bonds are similar to those of the α-sheet, which are symmetrical to the direction of the corresponding B−B bond, whereas the other four 3c-2e σbonds are asymmetrical. The 4c-2e σ-bonds are also not as perfectly symmetrical as those in the α-sheet. In comparison to the chemical bonds of α-sheet, perfectly symmetrical 2c-2e σbonds are presented in struc-1/7, 1/8, and 2/15 B-layers (Figure 5a,d,e), connecting two hexagons in the unit cell. Figure 5d indicates that one 2c-2e σ-bond, nine 4c-2e σ-bonds, and 16 3c-2e σ-bonds are present in the unit cell of the 1/8 Blayer. The 3c-2e σ-bonds that are around 2c-2e σ-bonds are more asymmetrical than that of the α-sheet, whereas the other chemical bonds are similar to that of the α-sheet. As the delocalized π-bonds have been proposed in the α-sheet, it can be expected that the delocalized π-bonds also exist in these boron sheets. Previous investigations have shown that the most stable αsheet explains the extreme stability of B80 fullerenes.7,8 However, recent studies have shown that the stability of planar boron sheets cannot be responsible for the stability of boron clusters.23 Several new boron fullerenes with a total energy lower than that of B80 fullerenes have been theoretically predicted.24 This phenomenon is also observed in the carbon system. Although graphene is the most stable carbon sheet, the C60 fullerene is composed of 12 pentagons and 20 hexagons and not purely composed of hexagons. The theoretical calculations have shown that the graphene, which is composed of only hexagons, has lower total energy than carbon sheets composed of pentagons and hexagons. Therefore, the new proposed metastable boron sheets facilitate investigations on boron fullerene and nanotubes, even although the proposed 2D boron sheets are less stable than α-sheet8,9 and recently predicted close-packed triangular and hexagon boron sheets.14 B100 fullerenes with considerably lower total energy than B80 have been presented in recent investigations.23 The proposed precursor of B100 fullerenes is struc-h (γ-sheet). As shown in Figure 1 and Table 1, several boron sheets are located between struc-h and α-sheet. These boron sheets are potential precursors for other stable boron clusters.

Figure 3. Structures of other boron sheets. The black solid lines show the unit cell, and the blue solid lines show the primitive cell. The structures are arranged according to increasing formation energy.

Previously proposed boron sheets with higher total energy are also included in Figure 3 for comparison. All the abovementioned boron sheets are summarized in Table 1. The table also shows several undiscovered boron sheets that correspond to that shown in Figure 1. Compared with the total energy of the α-sheet, the energy difference of the proposed boron sheets is less than 0.1 eV/atom. The hexagon hole density η of these metastable boron sheets are located mainly with η between 1/9 and 1/7. As shown in Figures 2 and 3, the unit cells of the predicted boron sheets with relatively lower total energy are usually composed of either 1, 2, or 3 hexagon holes. Furthermore, none of the boron pentagonal, heptagonal, and octagonal motifs are found in the boron sheets shown in Figure 1, making these boron sheets distinct from 2D carbon sheets. The permutations and combinations of pentagonal, hexagonal, heptagonal, and octagonal motifs in carbon sheets have been observed in both experimental and theoretical works.19−22 A nine-membered ring is present in the unit cell of struc-1/4 (Figure 2a), but the unit cell of struc-1/4 can be obtained by removing three boron atoms in the flat triangular sheet. Figure 4 shows the electronic density of states of the predicted boron sheets. On the basis of the total density of states, we conclude

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CONCLUSIONS We investigated the structures of 2D boron sheets by using the PSO algorithm combined with first-principles methods. Several 20077

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Table 1. Calculated Equilibrium Structural Parameters, the Hexagon Hole Density η, and Energy Differences (Ed) of Predicted Boron Sheetsa structure

a (Å)

b (Å)

γ (deg)

η

Ed (eV/atom)

1/8 B-layer 2/15 B-layer struc-1/9 struc-1/8 struc-1/7 struc-a struc-b struc-c 4/27 B-layer struc-d struc-e struc-f struc-g struc-1/6 1/10 B-layer struc-1/10 struc-1/5 struc-h struc-1/12 struc-1/4

11.70 14.63 5.06 6.75 11.82 6.77 7.35 11.75 15.18 7.79 4.47 7.70 5.09 10.14 14.62 8.46 8.41 2.93 5.85 5.88

5.06 5.06 5.06 5.85 2.93 5.86 5.85 6.74 8.79 4.44 4.47 7.39 4.43 2.92 5.06 5.84 2.91 5.07 5.85 5.88

90.0 90.0 120.0 90.0 90.0 90.0 66.8 90.0 90.0 90.0 120.0 90.0 79.1 90.0 90.0 90.0 90.0 90.0 120.0 120.0

1/8 2/15 1/9 1/8 1/7 1/8 1/8 1/8 1/10 1/7 1/7 3/23 1/9 1/6 1/10 1/10 1/5 1/6 1/12 1/4

−0.004 −0.001 0.000 0.003 0.009 0.017 0.017 0.020 0.022 0.023 0.024 0.025 0.027 0.028 0.029 0.036 0.041 0.053 0.066 0.092

b b α-sheetc,d β-sheetc,e

b snub-sheetf,g

b h γ-sheetf,i,j f

a The four boron sheets (1/8, 1/10, 2/15, and 4/27 B-layers) proposed by Yakobson and co-workers are also calculated within first-principles methods and listed for comparison. The structures are sorted with the increasing energy differences. bReference 14. cReference 8. dReference 9. e Reference 11. fReference 12. gReference 13. hReference 25. iReference 23. jReference 26.

Figure 4. Calculated total density states of boron sheets. The Fermi energy is indicated by the dashed vertical lines.

undiscovered metastable boron sheets have an energy difference less than 0.1 eV/atom compared with the α-sheet and recently predicted boron sheets composed with close-packed triangular and hexagonal lattices. The energy difference of new proposed struc-1/8 sheet is 0.01 eV/atom less stable than the know α-sheet and recently predicted 1/8 and 2/15 B-layers. The unit cells of these boron sheets are composed of hexagonal

Figure 5. Isosurfaces of the electron localization function with a value of 0.78 for struc-η (η = 1/7, 1/8, and 1/9) and 1/8 and 2/15 B-layers. The black solid lines show the unit cell.

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(25) De, S.; Willand, A.; Amsler, M.; Pochet, P.; Genovese, L.; Goedecker, S. Phys. Rev. Lett. 2011, 106, 225502. (26) Tang, H.; Ismail-Beigi, S. Phys. Rev. B 2009, 80, 134113. (27) Wu, X. J.; Dai, J.; Zhao, Y.; Zhuo, Z. W.; Yang, J. L.; Zeng, X. C. ACS Nano 2012, 6, 7443−7453.

and triangular motifs. Pentagonal, heptagonal, and octagonal motifs were not observed in these boron sheets. The binary composition map of the boron sheets show that the struc-η of most of the relative stable boron sheets is located at η between 1/9 and 1/7. The calculated total density of states demonstrates that all the predicted boron sheets are metallic. The multicenter chemical bonds in the relatively stable boron sheets are analyzed using the ELF. Our work should advance investigations on boron fullerenes and nanotubes.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest. After revision request of this paper, a PSO searching on 2D boron sheets was reported.27

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ACKNOWLEDGMENTS This work was supported by the Natural Science Foundation of Hebei Province of China (Grant No. B2012202044).

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REFERENCES

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