First-principles study of the stability and electronic properties of sheets and nanotubes of elemental boron

The structural and electronic properties of sheets and nanotubes of boron are investigated using density functional theory. The calculations predict the stability of a novel reconstructed {1221} sheet over the idealized triangular {1212} sheet. Nanotubes formed by wrapping the half-metallic {1221} sheet show a curvature-induced transition in their electronic properties. Analysis of the charge density reveals a mixed metallicand covalent-type of bonding in the reconstructed {1221} sheet and the corresponding nanotubes, in contrast to metallic-type bonding in the idealized {1212} sheet and its analogous nanotubes. 2005 Elsevier B.V. All rights reserved. Boron holds a unique place among the elements of the periodic table by having the most varied polymorphism, which includes quasicrystal [1] and novel nanostructures [2–6], in addition to the complex icosahedral networks observed in conventional boron-rich solids [7]. Based on a generalization of the Euler–Poincaré formula for a cylinder [8], it was suggested that boron nanotubes (BNTs) could be constructed by the appropriate wrapping of an idealized triangular boron sheet, referred to as the {1212} sheet. It is noteworthy to point out that the formation of the triangular boron sheet has not yet been verified by experiments. Analogous to the case of carbon nanotubes (CNTs), the wrapping in BNTs was described by a chiral vector R = na + mb, denoted as (n,m), where n and m are integers, and the BNTs were suggested to form zigzag , armchair and chiral structures depending upon the values of n and m [9]. A tubular structure consisting of the hexagonal pyramidal boron units was postulated [10], and non-self-consistent, non-orthogonal tight-binding (TB) calculations [11] were carried out on the nanotubular 0009-2614/$ see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2005.10.104 * Corresponding author. E-mail address: [email protected] (R. Pandey). bundle structure which predicted the structure to be metallic. A new form of radially constricted bundles (ropes) have also been proposed [12], and pristine single wall 3 nm BNTs have been synthesized [4]. Most recently, a planar to tubular structural transition in the B20 cluster has been confirmed by photoelectron spectroscopy, suggesting that B20 may be considered as the basic building block for a boron tubular structure of diameter 0.52 nm [13]. Interestingly, the subtle questions about the stability and electronic properties of either the idealized triangular boron sheet or the pristine, single-walled BNT have so far not been addressed. In this Letter, we propose to address these important questions using a state-of-the-art theoretical method based on periodic, gradient-corrected density functional theory. The calculated results show that the reconstructed {1221} sheet is more stable than the idealized {1212} sheet. Surprisingly, in contrast to CNTs, the nanotubes formed by wrapping boron sheets are predicted to be energetically more favorable than their planar counterparts. The BNTs obtained by wrapping the {1221} sheet are found to be metallic showing a curvature-induced transition in their electronic properties. No curvature induced changes in electronic properties are found for the 550 K.C. Lau et al. / Chemical Physics Letters 418 (2006) 549–554 case of the {1212} sheet. Furthermore, an analysis of the charge density reveals a prominent feature consisting of a mixed metallicand covalent-type of bonding in the {1221} sheet and the corresponding nanotubes. First-principles calculations were performed using the spin-polarized gradient-corrected density functional theory (DFT) with the Perdew–Wang (PW91) exchange and correlation functionals [14]. A plane wave basis set was used, and the valence–core interaction was described by the ultra-soft pseudopotential as implemented in the VIENNA AB INITIO SIMULATION PACKAGE (VASP) [15,16]. An energy cutoff of 208 eV in the plane wave expansion and of 443 eV for the augmented charge was used. A supercell was constructed in which a BNT was placed in a rectangular grid. The wall to wall distance between the nanotubes was 15 Å ensuring negligible interaction between the tube and its images, in contrast to the very small ( 1.8 Å) inter-tubular spacing considered in previous studies [11]. The infinite open-end BNT is then built by stacking up the supercell in the z-direction, and the repeated basic unit is arranged at the centre of the xyplane. For the single-layer boron sheet, we constructed a supercell by placing a part of the infinite boron sheet inside Fig. 1. (Top) Idealized {1212} sheet, and the reconstructed {1221} sheet. (C Projected density of states of the {1212} and {1221} sheets. The Fermi level a rectangular grid with a surface-to-surface separation of 10 Å. Calculations were deemed converged when changes in total energy were less than 10 5 eV and those in the inter-atomic forces were less than 0.01 eV/Å. Boron sheet: we begin by considering four possible configurations for the boron sheet. As shown in Fig. 1, the idealized {1212} configuration consists of a triangular three-atom unit uniformly repeated in the y-direction in which each atom has the same number of nearest-neighbors. In the {1212} configuration, the {1212} monolayer is buckled with a small atomic displacement of 0.2 Å in the alternate chains. On the other hand, the pair-buckled {1212} configuration consists of a sheet which is buckled in the alternate pair chains. Lastly, we consider a reconstructed {1221} configuration with inversion symmetry in the unit cell. It can be considered as a triangularsquare-triangular unit network repeated in the y-direction. The DFT calculations show the reconstructed {1221} configuration to be the most stable configuration: it is stable by 0.23 eV/atom relative to the idealized {1212} configuration. Both the {1212} and {1212} configurations tend to converge to the idealized {1212} configuration when relaxed during the geometry optimization. The results are, enter) Total charge density of the {1212} and {1221} sheets. (Bottom) is taken to be zero. K.C. Lau et al. / Chemical Physics Letters 418 (2006) 549–554 551 therefore, not in agreement with the previously reported cluster calculations, based on an ab initio plane wave approach, which found the buckled configuration to be stable with respect to the unbuckled configuration by 0.03 eV/atom [11]. The bonding in the idealized {1212} sheet is found to be dominated by out-of-plane p-type interactions resulting in delocalized charge density. We note that the idealized sheet consists of six coordinated boron atoms. However, this is not the case for the reconstructed {1221} sheet which shows anisotropic chemical bonding. Here, the coordination index of the boron atoms is five in the triangular-square-triangular network, and in-plane r-type interactions contribute significantly to the bonding. This results in a contraction of about 9% in RB–B (Table 1). Analysis of the charge density, shown in Fig. 1, reveals that charge transfer between the delocalized multi-center triangular networks is saturated by the formation of directional covalent bonds that interconnect these triangular units. Fig. 1 also shows the projected density of states of both sheets. Note that the occupied states in the idealized {1212} sheet are dominated by contributions from p orbitals, whereas contributions from s orbitals become significant for the reconstructed {1221} sheet, resulting in a mixture of strong covalent and weak metallic-type bonding in {1221}. In this context, it is worth pointing out that the 2D sheets share some similarities in electronic properties with their fragments, namely, the planar boron clusters [6,17]. The dominant feature of out-of-plane p-type bonding interactions resulting in delocalized charge density in the idealized {1212} sheet can be attributed to the delocalized nature of the p-electrons in the planar boron clusters, which renders aromaticity and antiaromaticity to the corresponding clusters in analogy to bonding in planar hydrocarbons [6,17]. The uniform delocalized charge density found in the {1212} sheet yields isotropic features of metallic-like character in the band dispersion along the KX and KY directions. Whereas the anisotropic nature of the chemical bonding in the {1221} sheet yields different dispersion along KX and KY, thereby leading to direction-dependent electronic properties (Fig. 2). The directional r-type