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Electronic Structure of Boron Flat Holeless Sheet

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Electronic Structure of Boron Flat Holeless Sheet Article Electronic Structure of Boron Flat Holeless Sheet Levan Chkhartishvili 1,2,3,*, Ivane Murusidze 4 and Rick Becker 3 1 Engineering Physics Department, Georgian Technical University, 77 Kostava Ave., Tbilisi 0175, Georgia 2 Boron and Composite Materials Laboratory, Ferdinand Tavadze Institute of Metallurgy and Materials Science, 10 Mindeli Str., Tbilisi 0186, Georgia 3 Boron Metamaterials, Cluster Sciences Research Institute, 39 Topsfield Rd., Ipswich, MA 01938, USA; [email protected] 4 Institute of Applied Physics, Ilia State University, 3/5 Cholokashvili Ave., Tbilisi 0162, Georgia; [email protected] * Correspondence: [email protected] or [email protected] Received: 30 January 2019; Accepted: 26 February 2019; Published: 3 March 2019 Abstract: The electronic band structure, namely energy band surfaces and densities-of-states (DoS), of a hypothetical flat and ideally perfect, i.e., without any type of holes, boron sheet with a triangular network is calculated within a quasi-classical approach. It is shown to have metallic properties as is expected for most of the possible structural modifications of boron sheets. The Fermi curve of the boron flat sheet is found to be consisted of 6 parts of 3 closed curves, which can be approximated by ellipses representing the quadric energy-dispersion of the conduction electrons. The effective mass of electrons at the Fermi level in a boron flat sheet is found to be too small compared with the free electron mass m0 and to be highly anisotropic. Its values distinctly differ in directions G–K and G–M: mG–K/m0 ≈ 0.480 and mG–M/m0 ≈ 0.052, respectively. The low effective mass of conduction electrons, ms/m0 ≈ 0.094, indicates their high mobility and, hence, high conductivity of the boron sheet. The effects of buckling/puckering and the presence of hexagonal or other type of holes expected in real boron sheets can be considered as perturbations of the obtained electronic structure and theoretically taken into account as effects of higher order. Keywords: boron sheet; electronic structure; density-of-states (DoS); Fermi curve; effective mass 1. Introduction Borophene—a one-atom-thick sheet of boron—is one of the most promising nanomaterials [1]. Most all-boron quasi-planar clusters and nanosurfaces, including nanosheets, constructed of triangular atomic rings, which Ihsan Boustani et al. predicted theoretically in 1994 and the following years (quasi-planar clusters [2], nanotubes [2–5], nanotube-to-sheet transitions [6], and sheets [5,7]), are now experimentally confirmed (quasi-planar clusters [8–10], nanotubes [11–13], nanotube-to-sheet transitions [14,15], and sheets [16,17]). 1.1. Why should Boron Sheets be Formed? There are different reasons in favor of the formation of stable 2-D all-boron structures. They can be divided into several groups. Let’s consider them separately. 1.1.1. 3-D All-Boron Structures Boron, the fifth element of the Periodic Table, is located at the intersection of semiconductors and metals. Due to a small covalent radius (only 0.84 Å) and number (only 3) of valence electrons, boron does not form simple three-dimensional structures but crystals with icosahedral clusters with many Condens. Matter 2019, 4, 28; doi:10.3390/condmat4010028 www.mdpi.com/journal/condensedmatter Condens. Matter 2019, 4, 28 2 of 22 Condens. Matter 2019, 4, x 2 of 22 manyatoms atoms in the in unit the cell. unit At cell. least At threeleast three all-boron all-boron allotropes allotropes are known— are known—α- andα- βand-rhombohedral β-rhombohedral and Condens. Matter 2019, 4, x 2 of 22 andhigh-pressure high-pressureγ-orthorhombic γ-orthorhombic phases—for phases—for the experimental the experi phasemental diagram phase of borondiagram see Referenceof boron [see18]. Reference [18]. In addition, α- and β-tetragonal and a number of other boron structures, probably In addition,many atomsα- andin theβ -tetragonalunit cell. At andleast athree number all-boron of other allotropes boron structures,are known— probablyα- and β-rhombohedral stabilized by the stabilized by the presence of impurities/defects, were reported. Theoretical studies of the five boron presenceand high-pressure of impurities/defects, γ-orthorhombic werereported. phases—for Theoretical the experi studiesmental of phase the five diagram boron crystalof boron structures see crystal structures (α, dhcp, sc, fcc, and bcc) were carried out using the LAPW (linearized plane wave) (α,Reference dhcp, sc, [18]. fcc, In and addition, bcc) were α- and carried β-tetragonal out using and the a number LAPW of (linearized other boron plane structures, wave) probably method in methodReferencestabilized in [19Reference by]. Thethe presence current [19]. stateTheof impu current ofrities/defects, research state on of the wereresearch phase reported. diagramon theTheoretical phase of boron diagram studies from of a of theoreticalthe boron five boron from point a theoreticalof viewcrystal is structuresgiven point inof Referenceview(α, dhcp, is given sc, [20 fcc,]. in Itand Reference should bcc) were be [20]. noted carried It should that, out inusing be the noted the last LAPW decade,that, in (linearized the several last decade, newplane structures wave) several newof boronmethod structures allotropes in Reference of boron were [19].allotropes discovered The current were and discovered somestate of have research and been some disproved.on thehave phase been Currently, diagram disproved. evenof boronCurrently, the numberfrom evena of theallotropes theoreticalnumber of of boronpoint allotropes of is view uncertain. isof given boron The in reasonisReference uncertain. for [20]. this The It is should that reason there be fornoted are this many that, is in suchthat the structures,therelast decade, are many all several of themsuch structures,complex,new structures and all someof of them boron of them complex, allotropes are minimally andwere somediscovered different of them and from some are others. haveminimally Abeen pseudo-cubic disproved. different Currently, tetragonalfrom others. even boron A pseudo-cubicrecentlythe number discovered tetragonalof allotropes under boron high-pressure of boron recently is uncertain. and discov high-temperatureered The underreason forhigh-pressure conditions this is that may thereand also high-temperatureare be many another such form conditionsof boronstructures, allotropes; may all also of however,bethem another complex, its form structure, ofand boron some studied allotropes; of inthem Reference however,are minimally [21 its] using structure, different a DFT studied (densityfrom inothers. functionalReference A [21]theory)pseudo-cubic using calculation, a DFT tetragonal(density is abnormal functional boron compared recently theory) todiscov calculation, otherered allotropes under is abnormal ofhigh-pressure boron compared in many and to ways.high-temperature other allotropes of boronconditions in many may ways. also be another form of boron allotropes; however, its structure, studied in Reference The almost regular icosahedron B12 with B-atoms at the vertices (Figure1) serves as the main [21] using a DFT (density functional theory) calculation, is abnormal compared to other allotropes of structuralThe almost motif notregular only oficosahedron boron allotropes B12 with but B-atoms also of all at known the vertices boron-rich (Figure compounds. 1) serves Inas thethe boronmain boron in many ways. structuralicosahedron, motif each not atom only is surroundedof boron allotropes by 5 neighboring but also atomsof all and,known as usual,boron-rich with onecompounds. more atom In from the The almost regular icosahedron B12 with B-atoms at the vertices (Figure 1) serves as the main boronthe rest icosahedron, of the crystal. each For atom this reason,is surrounded the average by 5 coordination neighboring numberatoms and, of a as boron-rich usual, with lattice one ranges more atomstructural from the motif rest notof theonly crystal. of boron For allotropes this reason but, thealso average of all known coordination boron-rich number compounds. of a boron-rich In the fromboron 5 to icosahedron, 5+1=6. each atom is surrounded by 5 neighboring atoms and, as usual, with one more lattice ranges from 5 to 5+1=6. 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. Figure 1. A regular icosahedron B12 withwith B-atoms B-atoms at at the the vertices. Figure 1. A regular icosahedron B12 with B-atoms at the vertices. However, an isolated regular boron icosahedron is an electron-deficientelectron-deficient structure—the total numberHowever, ofof valencevalence an electrons electronsisolated ofregular of 12 12 boron boronboron atoms icosahedronatoms is not is sufficientnot is sufficientan electron-deficient to fill to all fill the all covalent thestructure—the covalent bonding bonding orbitalstotal orbitalscorrespondingnumber corresponding of valence to such aelectrons cage-molecule.to such of 12a boroncage-molecule. Thus, atoms if it were is not aThus, stable sufficient if structure, it towere fill then alla thestable intra-icosahedral covalent structure, bonding bondsthen wouldintra-icosahedralorbitals be only corresponding partially bonds covalent would to such be but only alsoa cage-molecule. topartially some extent covalent Thus, metallic. but if alsoit As were forto boronsome a stable extent icosahedra structure, metallic. constituting thenAs for boronrealintra-icosahedral crystals, icosahedra it was clearlybondsconstituting would demonstrated,
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