Broad boron sheets and boron nanotubes: An ab initio study of structural, electronic, and mechanical properties

Based on a numerical ab initio study, we discuss a structure model for a broad boron sheet, which is the analog of a single graphite sheet, and the precursor of boron nanotubes. The sheet has linear chains of $sp$ hybridized $\sigma$ bonds lying only along its armchair direction, a high stiffness, and anisotropic bonds properties. The puckering of the sheet is explained as a mechanism to stabilize the $sp$ $\sigma$ bonds. The anisotropic bond properties of the boron sheet lead to a two-dimensional reference lattice structure, which is rectangular rather than triangular. As a consequence the chiral angles of related boron nanotubes range from 0° to 90°. Given the electronic properties of the boron sheets, we demonstrate that all of the related boron nanotubes are metallic, irrespective of their radius and chiral angle, and we also postulate the existence of helical currents in ideal chiral nanotubes. Furthermore, we show that the strain energy of boron nanotubes will depend on their radii, as well as on their chiral angles. This is a rather unique property among nanotubular systems, and it could be the basis of a different type of structure control within nanotechnology.

Figure 1

(Color online) Top view of a quasiplanar boron sheet. In a planar projection the atoms form an almost perfect triangular lattice. The basic structural unit is a hexagonal pyramidal ${\mathrm{B}}_7$ cluster, as suggested by the Aufbau principle (Ref. 3) (see text).

Figure 2

(Color online) Different structure models for broad boron sheets. Each supercell (thin lines) contains 16 atoms. (a) A simple flat sheet is metastable. (b) A simple up and down puckering seems to be the most stable modulation. Structures (c) and (d) are unstable. Models (b), (c), and (d) are periodic repetitions of structural motives taken from ${\mathrm{B}}_{22}$, ${\mathrm{B}}_{46}$, and ${\mathrm{B}}_{32}$ clusters, described in Ref. 4.

Figure 3

(Color online) The orthorhombic unit cell of model (b) with two basis atoms (see Table ). In a $xy$-projection atom 1 is located at the corners of a rectangular unit cell, while atom 2 is located at the center of the unit cell. Along the $z$ direction the boron atoms will generate a simple up and down puckering, with puckering heights around $\Delta z=0.82\mathrm{\AA }$.

Figure 4

(Color online) Orange (gray): charge density contours of the boron sheet [model (b)] at $0.9e∕{\mathrm{\AA }}^3$. One observes parallel linear chains of $sp$ hybridized $\sigma$ bonds lying along the armchair direction.

Figure 5

The band structure of the model BS. The fatness of the bands indicates their $sp$ character, and it shows that the $\sigma$ bonds in Fig. 4 must be of $sp$ type. The Fermi energy $E_F$ lies at $E=0$, $G$ is the $\Gamma$ point.

Figure 6

(Color online) The two-dimensional Fermi surface of the boron sheet. It consists of two contours in red (black) and yellow (gray), which correspond to the two bands crossing the Fermi energy in Fig. 5.

Figure 7

(Color online) The cross sections of different isomers of a free standing (9,0) zigzag boron nanotube. The big spheres stand for the upper atoms and the small ones for the lower atoms (with respect to the direction of the tube axis). The $\alpha$ and $\gamma$ isomers are the free standing counterparts of the (9,0)C and (9,0)B tubes in Ref. 17, respectively.

Figure 8

(Color online) Cross-sectional view of various isomers of free standing (10,0) and (12,0) zigzag boron nanotubes. Again the big spheres mark the upper atoms and the small ones mark the lower atoms. The $(10,0)\alpha$ and $(10,0)\beta$ isomers are free standing counterparts of the (10,0)B and (10,0)C structures in Ref. 17, respectively.

Figure 9

(Color online) Zigzag boron nanotubes and the presence of straight $\sigma$ bonds along their axial direction, which are indicated by orange (gray) charge density contours at $0.9e∕{\mathrm{\AA }}^3$. Due to a lack of stiff $\sigma$ bonds along the circumferential direction, this type of nanotube might not be stable.

Figure 10

(Color online) Three basic structure elements. The cross sections of the most stable zigzag boron nanotubes in Figs. 7, 8 may be composed using these elements, only.

Figure 11

(Color online) Top and side view of various free standing armchair boron nanotubes and the presence of $\sigma$ bonds, which are indicated by orange (gray) charge density contours at $0.95e∕{\mathrm{\AA }}^3$. All armchair nanotubes have bent $\sigma$ bonds along the circumferential direction, which basically generate the strain energies of the tubes. The black bar on the right indicates the height of a supercell in axial direction that was used for our simulations; for aesthetical reasons we actually displayed three identical units cells.

Figure 12

(Color online) The strain energy of $\alpha$ isomers as a function of the mean radius $\overline{R}$ [Eq. (4)]; for armchair boron nanotubes we used ${\overline{R}}_{\sigma }$ [Eq. (7)]. In orange (gray) we show the universal strain energy curve for carbon nanotubes (◻); the energy obviously depends on their radii, but not on their chiral angles. For armchair boron nanotubes (◇) we find a similar curve, but those boron tubes have more strain energy. For zigzag boron nanotubes (▵) we cannot really plot a strain energy curve, as different nanotubes of different radii are almost isoenergetic. Ideal zigzag boron nanotubes (엯) have less strain energy than their armchair counterparts, but they are metastable.

Figure 13

(Color online) The triangular (t), the rectangular (r), and the honeycomb-derived (h) primitive cells that are used to characterize boron nanotubes. They contain one, two, and three atoms, respectively. Only the rectangular cell may properly describe the puckering of the boron sheet (indicated by black and gray atoms in the background).

Figure 14

(Color online) The geometrical construction of an ideal boron nanotube from a boron sheet: the red (gray) area is cut and rolled up such that ${\mathit{W}}^{\mathrm{r}}$ will become the circumference of the nanotube. $\mathit{O}$ is the origin, ${\mathit{W}}^{\mathrm{r}}$ is the wrapping vector, $\mathit{T}$ is the translational vector, $\theta$ is the chiral angle measured with respect to the zigzag direction, ${\mathit{a}}_{1,2}$ are the primitive vectors of the underlying rectangular lattice, and $A$ and $B$ are the lattice constants (see text). The puckering of the boron sheet is indicated by black and gray atoms in the background. The zigzag and the armchair directions are perpendicular to each other. This figure corresponds to ${\mathit{W}}^{\mathrm{r}}=(5,3)$ and $A∕B=\sqrt{3}$, which implies $\mathit{T}=(-1,5)$.

Figure 15

(Color online) Two different ways of “building up” a nanotube: the tubular unit cell in light gray (see also Fig. 14) is repeated along the nanotube’s axis, which lies parallel to $\mathit{T}$. The helical unit cell in red (dark gray) is translated along spirals (represented by the dotted lines) on the surface of the nanotube; it is defined by the helical vector $\mathit{H}$ and vector $\mathit{K}$. It holds ${\mathit{W}}^{\mathrm{r}}=\mathit{K}+\mathit{L}$. Here $\mathit{H}=(0,1)$, $\mathit{K}=(2,0)$, and $\mathit{L}=(0,9)$, and therefore ${\mathit{W}}^{\mathrm{r}}=(2,9)$. The length of $\mathit{T}=(-3,2)$ was artificially reduced by choosing $A∕B=\sqrt{3}$.

Figure 16

(Color online) Properties of a flat boron sheet: (a) The two-dimensional band structure. (b) Black lines indicate the triangular unit cells, black spheres are boron atoms, and the orange (gray) contours show the electron localization function (ELF) at contours of 0.7. We observe a simple network of two- and three-center bonds.

Figure 17

(Color online) A kinked boron sheet based on a supercell (thin lines) that contains 16 atoms (see text). Apart from the kink its surface is slightly puckered.