Structural motifs and bonding in two families of boron structures predicted at megabar pressures

The complex crystal chemistry of elemental boron has led to numerous proposed structures with distinctive motifs as well as contradictory findings. Herein, evolutionary structure searches performed at 100 GPa have uncovered a series of metastable phases of boron, and bonding analyses were carried out to elucidate their electronic structure. These polymorphs, dynamically stable at 100 GPa, were grouped into two families. The first was derived from the thermodynamic minimum at these conditions, $\alpha$-Ga, whereas channels comprised the second. Two additional intergrowth structures were uncovered, and it was shown they could be constructed by stacking layers of $\alpha$-Ga-like and channel-like allotropes on top of each other. A detailed bonding analysis revealed networks of four-center $\sigma$-bonding functions linked by two-center B-B bonds in the $\alpha$-Ga based structures, and networks that were largely composed of three-center $\sigma$-bonding functions in the channel-based structures. Seven of these high-pressure phases were found to be metastable at atmospheric conditions, and their Vickers hardnesses were estimated to be $\approx 36$ GPa.

Figure 1

Select experimentally observed allotropes of boron. At ambient pressures the (a) $\alpha$- and (b) $\beta$-rhombohedral forms predominate, with transitions to the (c) $\gamma$-orthorhombic and (d) $\alpha$-Ga phases at 19 and 89 GPa, respectively. The characteristic icosahedra of boron-based structures (green, free standing; blue, face- and vertex-sharing) are evident in the phases that are preferred at lower pressures.

Figure 2

Illustration of the $\alpha$-Ga structure of elemental boron ($\zeta$-B) [48]. It can be viewed as (a) a layered structure consisting of sheared honeycomb nets, with thicker contacts along short B-B distances, and (b) layers of buckled trigonal nets connected by the short B-B contacts highlighted in (a). The light green and dark green boron atoms do not lie in the same plane.

Figure 3

Boron allotropes related to $\alpha$-Ga. The (a) $\alpha$-Ga'-I and (b) $\alpha$-Ga'-II structures are based on stacks of sheared honeycomb nets with different stacking orders and shearing patterns. The (c, d) $\alpha$-Ga'-III phase can be directly derived from $\alpha$-Ga by rotating every buckled layer by ${90}^{\circ }$ from the layer above it.

Figure 4

Channel-based family of boron structures that contain diamondlike ${\mathrm{B}}_4$ motifs made from two edge-sharing triangles, along with more open parallelograms (strips connecting buckled layers are shown in the insets), with different arrangements producing different phases: (a) channel-I, (b) channel-II, and (c) channel-III. The buckled layers comprising these phases are made from two adjacent rows of diamondlike motifs separated by a single row of parallelogram units in (d) channel-I, or alternating strips of diamondlike ${\mathrm{B}}_4$ units and parallelograms as in the (e) channel-II and (f) channel-III structures.

Figure 5

(a) The channel-IV structure is built up of corrugated strips of B triangles (inset) connecting buckled layers of B atoms, leading to open channels in the phase. (b) It can be alternately viewed as another derivative of the $\alpha$-Ga structure built up of sheared hexagonal nets. (c) The buckled layers of this structure reveal a complex pattern of diamondlike ${\mathrm{B}}_4$ and open parallelogram motifs.

Figure 6

(a) The intergrowth-I structure, and one of its (b) buckled layers. (c) The intergrowth-II structure, and (d) one of its buckled layers. Both phases are based on the $\alpha$-Ga and channel structures, with thicker bonds highlighting $\alpha$-Ga-based regions. Gray backgrounds span channel-based regions (shown in the insets), which are larger in the intergrowth-II structure.

Figure 7

(a) Enthalpies of select phases of boron, including those listed in Table 1, with respect to that of $\alpha \text{−}{\mathrm{B}}_{12}$ as a function of pressure. (b) The internal energy of these boron phases relative to $\alpha \text{−}{\mathrm{B}}_{12}$. (c) The $PV$ contribution to the enthalpy of these boron phases relative to that of the $\alpha$-Ga polymorph. Channel-based phases tend to have a lower internal energy, while those based on $\alpha$-Ga have smaller $PV$ contributions to the enthalpy.

Figure 8

Isosurface (isovalue = 0.8) plots of the ELF for the (a) $\alpha$-Ga, (b) channel-I, and (c) channel-II structures at 100 GPa. Plots of the ELF in the (d) (100) plane of $\alpha$-Ga, (e) (010) plane of channel-I, and (f) (010) plane of channel-II are also provided.

Figure 9

Isosurfaces (isovalue=0.08 a.u.) of the bonding functions obtained with the SSAdNDP method for $\alpha$-Ga at 100 GPa. (a) Four 2c-2e $\sigma$-bonds between buckled layers of boron atoms and (b) eight 4c-2e $\sigma$-bonds within buckled layers of boron atoms account for all 24 valence electrons in the structure.

Figure 10

Electron-density difference isosurface plots (the positive isovalue of 0.015 a.u. is shown) for the (a) $\alpha$-Ga, (b) channel-I, and (c) channel-II structures. Bottom, buckled layers; top, chains connecting them.

Figure 11

Isosurfaces (isovalue=0.08 a.u.) of the bonding functions obtained for the channel-I structure of boron at 100 GPa. (a) Two 2c-2e and four 3c-2e $\sigma$-bonds between buckled layers of boron atoms and (b) eight 3c-2e and four 4c-2e $\sigma$-bonds within buckled layers of boron atoms account for all 36 valence electrons in the phase.

Figure 12

Isosurfaces (isovalue=0.08 a.u.) of the bonding functions obtained for the channel-II structure of boron at 100 GPa. (a) Four 3c-2e $\sigma$-bonds between buckled layers of boron atoms and (b) eight 3c-2e $\sigma$-bonds within buckled layers of boron atoms account for all 24 valence electrons in the phase.