Predicting three-dimensional icosahedron-based boron ${\mathrm{B}}_{60}$

The icosahedron-based bulk boron structures have extremely chemical and structural complexity, and are usually semiconductors at ambient conditions. Here we predict bulk boron phases with a 60-atom orthorhombic unit cell from an ab initio evolutionary structure search, termed as ${\mathrm{B}}_{60}$. The metastable structures can be either a conductor or a semimetal depending on their interstitial atomic positions. In particular, an orthorhombic structure with $Pnma$ symmetry ($Pnma-{\mathrm{B}}_{60}$), consisting of ${\mathrm{B}}_{12}$ icosahedra and twisted interstitial two-atom wide boron ribbons, is identified to be a topological node-line semimetal with potential superior electronic transport. The band structure and simulated electron-diffraction pattern of $Pnma-{\mathrm{B}}_{60}$ are in satisfactory agreement with the experimental data, suggesting that it may exist in the form of nanomaterials.

Figure 1

(a, b) Structural projections of $I{2}_1{2}_1{2}_1-{\mathrm{B}}_{60}$ along [010] and [100] directions. (c, d) Structural projections of $Pnma-{\mathrm{B}}_{60}$ along [010] and [100] directions. For clarity, the $2\times 2\times 2$ supercell is used here and the interstitial boron atoms are plotted by ball and stick form.

Figure 2

Electronic structures of $I{2}_1{2}_1{2}_1-{\mathrm{B}}_{60}$ and $Pnma-{\mathrm{B}}_{60}$. (a, b) Band structure and DOS for $I{2}_1{2}_1{2}_1-{\mathrm{B}}_{60}$. (c, d) Band structure and DOS for $Pnma-{\mathrm{B}}_{60}$. ${\mathrm{A}}_1$ and ${\mathrm{B}}_2$ indicate the irreducible representation of the two crossing bands in the vicinity of the $P$ point.

Figure 3

(a) The calculated Fermi surface of $Pnma-{\mathrm{B}}_{60}$ with corresponding Brillouin zone (BZ) and the high-symmetry points are labeled accordingly. (b) The distribution of nodal lines in the extended BZ. The nodal lines ${\mathrm{L}}_1, {\mathrm{L}}_2$, and ${\mathrm{L}}_3$ are colored in red, green, and blue, respectively. The black dashed lines represent three distinctive closed loops along which the Berry phases are calculated. (c, d) Band structures around the high-symmetry points $S$ and $R$ near the Fermi level. The crossing points are illustrated by white dashed lines.

Figure 4

(a) The band structure of $Pnma-{\mathrm{B}}_{60}$ under a uniaxial tensile strain (1%) along [001] direction. The upper inset shows the enlarged band structure of strained $Pnma-{\mathrm{B}}_{60}$ along the X-S-R-U paths and the lower inset reveals the redistribution of its nodal lines. (b, c) The (100) and (001) surface states of $Pnma-{\mathrm{B}}_{60}$, where the color bar represents the weight of local density of states.

Figure 5

(a, b) Phonon spectra and phonon density of states for $Pnma-{\mathrm{B}}_{60}$ at ambient condition. (c) The simulated electron diffraction (ED) pattern of $Pnma-{\mathrm{B}}_{60}$ along [100] direction. (d) The experimental ED pattern of certain boron nanobelts from Ref. [11].