# Electronic Structure of Boron Flat Holeless Sheet

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Why should Boron Sheets be Formed?

#### 1.1.1. 3-D All-Boron Structures

_{12}with B-atoms at the vertices (Figure 1) serves as the main structural motif not only of boron allotropes but also of all known boron-rich compounds. In the boron icosahedron, each atom is surrounded by 5 neighboring atoms and, as usual, with one more atom from the rest of the crystal. For this reason, the average coordination number of a boron-rich lattice ranges from 5 to 5+1=6.

_{3}-type boron monolayer.

#### 1.1.2. Boron Quasi-Planar Clusters

_{36}of hexagonal shape with a central hexagonal hole [9], which is viewed as a potential basis for an extended 2-D boron sheet, and boron fullerene B

_{40}[10], which can be imagined as the fragment of a boron sheet wrapped into the sphere, were discovered experimentally. Photoelectron spectroscopy in combination with ab initio calculations have been carried out to probe the structure and chemical bonding of the B

_{27}

^{−}cluster [56]. A comparison between the experimental spectrum and the theoretical results reveals a 2-D global minimum with a triangular lattice containing a tetragonal defect and two low-lying 2-D isomers, each with a hexagonal vacancy.

#### 1.1.3. Liquid Boron Structure

_{12}icosahedra, a main structural motif of boron crystals and boron-rich solid compounds, are destroyed upon melting. Although atoms form an open packing, they maintain the 6-coordination.

#### 1.1.4. Growing of Boron Sheets

_{12}and χ

_{3}, both exhibiting a triangular lattice but with different arrangements of periodic holes, were observed by scanning tunneling microscopy. DFT simulations indicate that both sheets are planar without obvious vertical undulations.

_{1}/a boron was revealed to possess lower molar energy, indicating the more stable 2-D boron.

#### 1.2. Applications

^{10}B nuclei, solid-state boron allotropies, as well as boron-rich compounds and composites, are good candidates to be used as neutron-protectors. Boron sheets will be especially useful as an absorbing component in composite neutron shields [75]. Materials with the high bulk concentration of B-atoms usually are nonmetals and, therefore, not suitable for electromagnetic shielding purposes. However, frequently, the simultaneous protection against both the neutron irradiation and electromagnetic waves is needed, in particular, because neutron absorption by

^{10}B nuclei is accompanied by a gamma-radiation. For this reason, in the boron-containing nanocomposites designed for neutron-protection, it is necessary to introduce some foreign components with metallic conductivity. Utilizing of the metallic boron sheet as a component may resolve this problem [76].

#### 1.3. Available Electron Structure Calculations

_{n}for $n$ up to 46, considered to be fragments of bare boron quasi-planar surfaces, have to possess a singly occupied bonding orbital [29]. Assuming that conduction band of the infinite surface is generated from the HOMO (highest-occupied-molecular-orbital) of a finite fragment, it means the partial filling of the conduction band, i.e., the metallic mechanism of conductance.

^{5}m/s, close to that of graphene. However, in H-borophene [53] constructed by tiling 7-membered rings side by side, a Dirac point appeared at about 0.33 eV below the Fermi level.

_{20}clusters in a hexagonal arrangement. Most strikingly, the highest valence band of M-boron is isolated, strongly localized, and quite flat, which induces spin polarization on either cap of the B

_{20}cluster. This flat band originates from the unpaired electrons of the capping atoms and is responsible for magnetism. M-boron is thermodynamically metastable.

_{1}/a was revealed to possess double anisotropic Dirac cones. It is the most stable 2-D boron with particular Dirac cones. The puckered structure of P2

_{1}/a boron results in the peculiar Dirac cones.

## 2. Theoretical Approach

- The construction of matrix elements for secular equation, which, within the initial quasi-classical approximation, reduces to a geometric task of determining the volume of the intersection of three spheres [89], and
- The solving of the secular equation, which determines the crystalline electronic energy spectrum [90].

_{2}. Analyses of the isolated layer instead of multilayered structure also seems to be quite sufficient for the initial approximation because, in such structures, only intra-layer conductivity is metallic, while interlayer conductivity is nonmetallic due to larger interlayer bond lengths if compared with those in layers.

^{2}2s

^{2}2p configuration is considered, not the 1s

^{2}2s2p

^{2}configuration, from which the ground state and first excited states of some boron-like ions arise [99].

## 3. Results and Discussion

_{2}[100], which is believed to be a structural analog of the hypothetical multilayered boron sheet, where metal Me atoms in a metal diboride MeB

_{2}structure are replaced by B-atoms themselves.

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

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**Figure 3.**The borophene structure on a silver substrate: the top and side views of the monolayer structure (unit cell indicated by the box) [16].

**Figure 7.**The transform from the $({\alpha}_{1},{\alpha}_{2})$ domain to the $({k}_{x},{k}_{y})$ domain of reciprocal space.

**Figure 8.**The hexagonal (first Brillouin zone) and rhombic unit cells of a reciprocal lattice of a flat boron sheet.

**Figure 12.**The section of the conduction band surface along the main diagonal of a rhombic unit cell (direction Γ–K) of reciprocal space (in atomic units).

**Figure 13.**The section of the conduction band surface along a small diagonal of a rhombic unit cell (direction Γ–M) of reciprocal space (in atomic units).

**Figure 14.**Curves of the intersection of the band surface with the Fermi plane in neighboring rhombic unit cells of reciprocal space.

**Figure 16.**The density-of-electron-states renormalized to the Fermi level in a valence band and the lower and upper conduction bands of a boron flat sheet in two different scales: general view (

**a**) and in Fermi level vicinity (

**b**).

**Table 1.**The inner and outer classical turning point radii ${{r}^{\prime}}_{i}$ and ${{r}^{\u2033}}_{i}$ of electrons in boron atom.

Orbital | State | $-{\mathit{E}}_{\mathit{i}}$ (a.u.) | ${{\mathit{r}}^{\prime}}_{\mathit{i}}$ (a.u.) | ${{\mathit{r}}^{\u2033}}_{\mathit{i}}$ (a.u.) |
---|---|---|---|---|

1 | 1s | 7.695335 | 0 | 0.509802 |

2 | 2s | 0.494706 | 0 | 4.021346 |

3 | 2p | 0.309856 | 0.744122 | 4.337060 |

4 | 2p | 0.214595 | 0.894159 | 5.211538 |

**Table 2.**The radii ${r}_{\lambda}$ of radial layers of quasi-classical averaging of potential in boron atoms and the averaged values of potential ${\phi}_{\lambda}$.

$\mathit{\lambda}$ | ${\mathit{r}}_{\mathit{\lambda}}$ (a.u.) | ${\mathit{\phi}}_{\mathit{\lambda}}$ (a.u.) |
---|---|---|

0 | 0 | – |

1 | 0.027585 | 210.5468 |

2 | 0.509802 | 8.882329 |

3 | 0.744122 | 3.65292 |

4 | 4.021346 | 0.206072 |

5 | 4.33706 | 0.000614 |

Band | $\mathsf{\Delta}{\mathit{E}}_{\mathit{i}},$ eV | $\mathsf{\Delta}{\mathit{E}}_{\mathit{i}\mathit{j}},$ eV |
---|---|---|

1 | 0 | |

2 | 17.36 | 239 |

3 | 5.92 | −3.23 |

4 | 9.57 | −0.49 |

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**MDPI and ACS Style**

Chkhartishvili, L.; Murusidze, I.; Becker, R. Electronic Structure of Boron Flat Holeless Sheet. *Condens. Matter* **2019**, *4*, 28.
https://doi.org/10.3390/condmat4010028

**AMA Style**

Chkhartishvili L, Murusidze I, Becker R. Electronic Structure of Boron Flat Holeless Sheet. *Condensed Matter*. 2019; 4(1):28.
https://doi.org/10.3390/condmat4010028

**Chicago/Turabian Style**

Chkhartishvili, Levan, Ivane Murusidze, and Rick Becker. 2019. "Electronic Structure of Boron Flat Holeless Sheet" *Condensed Matter* 4, no. 1: 28.
https://doi.org/10.3390/condmat4010028