Electronic structure and defect properties of ${\mathrm{B}}_6\mathrm{O}$ from hybrid functional and many-body perturbation theory calculations: A possible ambipolar transparent conductor

${\mathrm{B}}_6\mathrm{O}$ is a member of icosahedral boron-rich solids known for their physical hardness and stability under irradiation bombardment, but it has also recently emerged as a promising high mobility $p-$type transparent conducting oxide. Using a combination of hybrid functional and many-body perturbation theory calculations, we report on the electronic structure and defect properties of this material. Our calculations identify ${\mathrm{B}}_6\mathrm{O}$ has a direct band gap in excess of 3.0 eV and possesses largely isotropic and low effective masses for both holes and electrons. Of the native defects, we identify no intrinsic origin to the reported $p-$type conductivity and confirm that $p-$type doping is not prevented by intrinsic defects such as oxygen vacancies, which we find act exclusively as neutral defects rather than hole-killing donors. We also investigate a number of common impurities and plausible dopants, finding that isolated acceptor candidates tend to yield deep states within the band gap or act instead as donors, and cannot account for $p-$type conductivity. Our calculations identify the only shallow acceptor candidate to be a complex consisting of interstitial H bonded to C substituting on the O site ${\left(\mathrm{CH}\right)}_{\mathrm{O}}$. We therefore attribute the origins of $p-$type conductivity to these complexes formed during growth or more likely via isolated ${\mathrm{C}}_{\mathrm{O}}$ which later binds with H within the crystal. Lastly, we identify Si as a plausible $n-$type dopant, as it favorably acts as a shallow donor and does not suffer from self-compensation as may the C-related defects. Thus, in addition to the observed $p-$type conductivity, ${\mathrm{B}}_6\mathrm{O}$ exhibits promise of $n-$type dopability if the stoichiometry and both native and extrinsic sources of compensation can be sufficiently controlled.

Figure 1

(a) Structure of ${\mathrm{B}}_6\mathrm{O}$ that shows the ${\mathrm{B}}_{12}$ icosahedral units bridged by B atoms $({\mathrm{B}}_{\mathrm{I}}-$${\mathrm{B}}_{\mathrm{I}}$ linkages) and threefold coordinated O atoms (O-${\mathrm{B}}_{\mathrm{II}}$ linkages). (b) The band structure along symmetry lines as calculated using HSE (red and blue) and $G_0{W}_0:\mathrm{GGA}$ (black). For comparison purposes, the $G_0{W}_0$ band structure was computed using the relaxed atomic structure determined with HSE. Charge density isosurfaces of the valence band maximum located at (c) the $Z$ point and (d) the $\Gamma$ point. All isosurfaces are shown at 20% of their maximum value.

Figure 2

Defect formation energy vs Fermi level for intrinsic defects in ${\mathrm{B}}_6\mathrm{O}$ for (a) boron-rich and (b) oxygen-rich conditions as defined in the text. The Fermi level is referenced to zero at the VBM. Defects on the O site are always more favorable than interstitials and native defects on the B sites.

Figure 3

Defect formation energies vs Fermi level for various impurities in ${\mathrm{B}}_6\mathrm{O}$. H and N-related defects are included in the first two panels for B-rich (a) and O-rich (b) conditions, while C and Si-related defects are shown in the last two for B-rich (c) and O-rich (d) conditions. Complexes with H are also shown, with the ${\left(\mathrm{CH}\right)}_{\mathrm{O}}$ complex on the O site found to be the only shallow acceptor candidate in ${\mathrm{B}}_6\mathrm{O}$.

Figure 4

Structure of the isolated C${}_{\mathrm{O}}^0$ defect in ${\mathrm{B}}_6\mathrm{O}$ shown in (a) and its complex with ${\mathrm{H}}_i$, (CH)${}_{\mathrm{O}}^0$, shown in (b). The charge density isosurfaces associated with the hole state are included to show the significant hole localization on the C${}_{\mathrm{O}}^0$ in (a) and the (CH)${}_{\mathrm{O}}^0$ in (b). The isosurfaces are set at 15% of their maximum value.

Figure 5

The $X-$B-O phase diagrams shown as a function of the oxygen chemical potential $\left({\mu }_{\mathrm{O}}\right)$ versus (a) ${\mu }_{\mathrm{N}}$ for $X=\text{N}$, (b) ${\mu }_{\mathrm{C}}$ for $X=\text{C}$, and (c) ${\mu }_{\mathrm{Si}}$ for $X=\text{Si}$. The highest ${\mu }_{\mathrm{O}}$ that ensures the stability of ${\mathrm{B}}_6\mathrm{O}$ represents the O-rich limit, while the lowest value represents the B-rich limit.

Figure 6

The N-H-B-O phase diagrams shown for (a) B-rich, reducing conditions and (b) O-rich, oxidizing conditions. The chemical potentials ${\mu }_{\mathrm{H}}$ and ${\mu }_{\mathrm{N}}$ used in Figs. 3 and 3 were chosen to correspond to their maximum values that simultaneously ensure the stability of ${\mathrm{B}}_6\mathrm{O}$.

Figure 7

The C-H-B-O phase diagrams shown for (a) B-rich, reducing conditions and (b) O-rich, oxidizing conditions. The chemical potentials ${\mu }_{\mathrm{H}}$ and ${\mu }_{\mathrm{C}}$ used in Figs. 3 and 3 were chosen to correspond to their maximum values that simultaneously ensure the stability of ${\mathrm{B}}_6\mathrm{O}$.