Ab Initio Study of Single-Walled Boron Nanotubes

Document Type

Book Chapter

Publication Date

2016

Department

Department of Physics

Abstract

According to Tang et al. [21], balance interplay in an α-boron sheet from the competition between 2-c and 3-c bonding is necessary to preserve the bonding stability of this boron sheet. In an α-boron sheet, the hexagonal and triangular structural motifs have distinct electronic properties. The sp2-state of hexagonal structural motif is partially unoccupied, therefore is electrondeficient and prone to accepting electrons, whereas the flat triangular structural motif is found to has a surplus of electrons in antibonding states. From a doping perspective, the 3-c flat triangular regions should act as donors, while the 2-c hexagonal regions should act as acceptors. Therefore, the stability of these 2D boron sheets depend strongly on the ratio of hexagons to triangles, which can be described by a hexagon hole density, η [21]. Under this model, η changes from 0 to 1/3 as the boron sheet alters from the triangular sheet T(0) to the ideal hexagonal graphene-like boron sheet H(1/3). Above all, the most stable boron sheet structure based on this structural motif is found to be an α-boron sheet or A(1/9), as shown in Fig. 4.2 [31], which occurs at η = 1/9. Thus this suggested that if the system is able to turn into a mixture of these two phases in the right proportion, it should benefit from the added stability of both subsystems. Following the α-boron sheet, the next stable configuration is the buckled triangular sheet (Fig. 4.2). It was obtained from the geometry relaxation of a flat triangular sheet through a buckling along the perpendicular direction of the sheet. Subsequently, the buckling mixes in-plane and out-of-plane electronic states and can be thought of as a symmetry reducing distortion that enhances binding. Intuitively, the buckling of boron atoms is a response of the sheet to the internal stress imposed by the arrangement of the atoms in a perfect triangular 2D lattice. For the flat triangular 2D lattice, the delocalized 3-c bonds dominant structural motif is found to be less stable. According to a recent density functional theory (DFT) calculation [47], the atomic lattice of this structure is found to be dynamically unstable with imaginary phonon. In particular, all of its transverse out-of-plane and ZA bending modes are not stable with imaginary ωc ranging from ~261 to 193 cm−1. It therefore appears that the 3-c bonds present in the perfect triangular lattice are not strong enough to bind the boron atoms in a 2D lattice.

In terms of electronic bonding characters, for a perfect triangular boron sheet, both the high atomic coordination and the electronic-deficient character of boron yield a nearly homogenous electron density distribution in the 2D lattice. Its Fermi level is high enough to force some of the electrons to occupy antibonding molecular orbitals, which, in turn, induce a destabilizing effect in the 2D lattice making it to be highly chemically reactive [21]. Thus through a buckling on the triangular boron sheet, it can stabilize both lattice and chemical bonding of this structural motif. As proposed by Tang et al. [21], a perfect hexagonal lattice of boron in 2D sheet is not stable due to its partially occupied sp2 state. The instability of this hexagonal boron sheet has been confirmed by a recent DFT phonon study [47]. However, by breaking the D6hsymmetry of the hexagonal lattice to a lower D2h symmetry of the orthorhombic lattice, the reduced symmetry can make the sheet configuration to be relatively stable (~90% α-rhombohedral [α-B12] boron solid cohesive energy) and subsequently yield the distorted hexagonal boron sheet [20, 47]. In terms of local geometry, this boron sheet is comprised of a network of triangle-square-triangle units in the lattice. Similar to the α-boron sheet and the buckled triangular sheet configurations, the chemical bonding in the distorted hexagonal sheet can also be characterized by both 2-c and 3-c bonds, reminiscent of the electron-deficient features of boron atoms in bulk solid. According to Lau et al. [47], the ab initio DFT results suggest that the nature of the chemical bonding, rather than thermal effects, appears to be the prime factor in determining the stability of atomic monolayers of boron. Despite the fact that they (i.e., α-, buckled triangular, and distorted hexagonal boron sheet) are found to be thermodynamically stable, these 2D phases of elemental boron remain thermodynamically less stable than the 2D graphene of carbon. It is important to point out that thermodynamic stability of these boron sheets relative to boron crystalline bulk solid is substantially less stable than graphene relative to carbon bulk solid, that is, graphite. For a monolayer graphene, its cohesive energy is ~98% of graphite, whereas for these monolayer boron sheets, they are found to be merely ~90%–93% as stable as the commonly known boron bulk crystal, α-B12 rhombohedral solid [20, 47].

Unlike the carbon, the stability of graphene sheets are mostly sp2 bond driven, and only weakly bind by the interlayer long-range van der Waals force in graphite for carbon bulk solid; however, this unique feature is not found for the boron atoms. Thus to fill up the missing energy gap between these 2D boron sheets and boron bulk solid, the strong interlayer interaction between the boron sheets is thus expected [31]. Similarly, the strong bonding among wall-to-wall interactions for the multiwalled BNTs [18, 31, 33] and the strong intertubular bonding interactions within the crystalline bundles of SWBNTs [28] are therefore predicted. As anticipated from Fig. 4.2, the number of possible atomic configurations for an SWBNT can be enormous with respect to different structural motifs of boron sheets. However, to keep our discussion as vivid as possible, we shall only focus on our discussion on the energetically most stable α-boron sheet-derived SWBNT for the sake of simplicity in the description. Specifically, we will focus on their energetic stability, electronic properties, and elastic mechanical properties as predicted by DFT from literatures. For the SWBNTs that are derived from the buckled triangular boron sheet and the distorted hexagonal boron sheet-derived BNTs, the details can be found from the relevant papers [18, 19, 23, 28, 33].

4.3 Basic Properties of an SWBNT Derived from 4.3.1 Basic Geometry Construction

With a single-layer 2D α-boron sheet as a precursor, the SWBNTs rolled into a 1D hollow cylindrical shape with axial symmetry and in general exhibited a spiral conformation that called chirality. According to the diameter of an SWBNT, it defines the curvature of a nanotube, and chirality of the nanotube is usually defined by the chiral vector along the rolled direction (Fig. 4.4) analogous to an SWCNT. For an SWBNT that is derived from an α-boron sheet, as suggested by Yang et al. [29], two kinds of vectors can be used to determine the BNTs: the primary vectors (a) of the hexagonal lattice used for CNTs and the primary vectors of boron sheet lattice (b), as shown in Fig. 4.4. For the hexagonal lattice, a1 = ax, a2 = a(1/2x + √3/2y), a = √3 lB-B. For the α-boron sheet lattice, b1 = b(3/2x + √3/2y), b2 = b(3/2x – √3/2y), b = 3 lB-B.

To satisfy the definition of the chiral vector, Ch = pb1 + qb2 = na1 + ma2, the chiral vector Ch (or equivalent to (p, q)) of the boron sheet lattice corresponds to the vector (n, m) of the hexagonal lattice with p = (n + 2m)/3 and q = (n − m)/3. Thus, the (p, 0) BNT corresponds to the (n, n) with (n = p) CNTs and the (p, p) BNT corresponds to the (n, 0) with (n = 3p) CNTs. The lattice constants are enlarged since the unit cell of the (p, 0) BNTs contains three unit cells of the (n, n) with (n = p) CNTs. To name an SWBNT, we follow the established standard for an SWCNT [46]; the (p, 0) BNT is thus defined as an armchair SWBNT (i.e., α-SWBNT-arm), whereas for the (p, p) BNTs, it is thus defined as a zigzag BNT (i.e., α-SWBNT-zz). For the (p, q) BNTs, it is thus defined as a chiral SWBNT (i.e., α-SWBNT-chi). Coth zigzag and armchair SWBNTs are thus classified as achiral, whose image has an identical structure to the original one [46].

Publication Title

Handbook of Boron Nanostructures

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