Electronic structure and bonding of $\beta$-rhombohedral boron using cluster fragment approach

Theoretical studies using density functional theory are carried out to understand the electronic structure and bonding and electronic properties of elemental $\beta$-rhombohedral boron. The calculated band structure of ideal $\beta$-rhombohedral boron $\left({\mathrm{B}}_{105}\right)$ shows valence electron deficiency and depicts metallic behavior. This is in contrast to the experimental result that it is a semiconductor. To understand this ambiguity we discuss the electronic structure and bonding of this allotrope with cluster fragment approach using our recently proposed mno rule. This helps us to comprehend in greater detail the structure of ${\mathrm{B}}_{105}$ and materials which are closely related to $\beta$-rhombohedral boron. The molecular structures ${\mathrm{B}}_{12}{\mathrm{H}}_{12}^{-2}$, ${\mathrm{B}}_{28}{\mathrm{H}}_{21}^{+1}$, ${\mathrm{BeB}}_{27}{\mathrm{H}}_{21}$, ${\mathrm{LiB}}_{27}{\mathrm{H}}_{21}^{-1}$, ${\mathrm{CB}}_{27}{\mathrm{H}}_{21}^{+2}$, ${\mathrm{B}}_{57}{\mathrm{H}}_{36}^{+3}$, ${\mathrm{Be}}_3{\mathrm{B}}_{54}{\mathrm{H}}_{36}$, and ${\mathrm{Li}}_2{\mathrm{CB}}_{54}{\mathrm{H}}_{36}$, and corresponding solids ${\mathrm{Li}}_8{\mathrm{Be}}_3{\mathrm{B}}_{102}$ and ${\mathrm{Li}}_{10}{\mathrm{CB}}_{102}$ are arrived at using these ideas and studied using first principles density functional theory calculations.

Figure 1

(Color online) Calculated band structure of $\alpha$-rhombohedral boron and $\beta$-rhombohedral boron along their high symmetry points of Brillouin zone. Fermi energy level is indicated by a dotted line.

Figure 2

${\mathrm{B}}_{28}$ unit is connected to another ${\mathrm{B}}_{28}$ unit through a single boron atom bridge $\left(\mathrm{B}28\text{−}\mathrm{B}\text{−}\mathrm{B}28\right)$. The bonds that are directed out correspond to the $\mathrm{B}–\mathrm{H}$ bonds of the associated ${\mathrm{B}}_{57}{\mathrm{H}}_{36}$ molecule obtained from this fragment.

Figure 3

A and E holes of the optimized rhombohedral ${\mathrm{Li}}_8{\mathrm{Be}}_3{\mathrm{B}}_{102}$ boron showing the $\mathrm{Li}$ positions.

Figure 4

(Color online) Calculated band structure of ${\mathrm{Li}}_8{\mathrm{Be}}_3{\mathrm{B}}_{102}$ and ${\mathrm{Li}}_{10}{\mathrm{CB}}_{102}$ along their high symmetry points of Brillouin zone. Fermi energy level is indicated by a dotted line.

Figure 5

(Color online) Calculated PDOS for ${\mathrm{B}}_{105}$, ${\mathrm{Li}}_8{\mathrm{Be}}_3{\mathrm{B}}_{102}$, and ${\mathrm{Li}}_{10}{\mathrm{CB}}_{102}$. (I) The PDOS of three boron atoms located at the centers of ${\mathrm{B}}_{57}$ (1) $\mathrm{B}\left[15\right]$ and of two ${\mathrm{B}}_{28}$ (2) $\mathrm{B}\left[14\right]$ units in the ${\mathrm{B}}_{105}$. (II) The PDOS of three $\mathrm{Be}$ atoms are located at $\mathrm{B}\left[14\right]$ and $\mathrm{B}\left[15\right]$ positions and $\mathrm{Li}$ atoms at A and E holes in ${\mathrm{Li}}_8{\mathrm{Be}}_3{\mathrm{B}}_{102}$. (III) The PDOS of one C and two $\mathrm{Li}$ at $\mathrm{B}\left[15\right]$ and $\mathrm{B}\left[14\right]$ positions in ${\mathrm{Li}}_{10}{\mathrm{CB}}_{102}$. (IV) The PDOS of 8 $\mathrm{Li}$ at A and E holes in ${\mathrm{Li}}_{10}{\mathrm{CB}}_{102}$.