Two-dimensional magnetic boron

We predict a two-dimensional (2D) antiferromagnetic (AFM) boron (designated as $M$-boron) by using ab initio evolutionary methodology. $M$-boron is entirely composed of ${\mathrm{B}}_{20}$ 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 ${\mathrm{B}}_{20}$ cluster. This flat band originates from the unpaired electrons of the capping atoms and is responsible for magnetism. $M$-boron is thermodynamically metastable and is the first magnetic 2D form of elemental boron.

Figure 1

(a) Total energies from the GGA-PBE and PBE0 calculations for several 2D boron structures. The inset shows the structure of 2D-${\mathrm{B}}_{26}$. (b) and (d) projection of $M$-boron structure along the $\left[001\right]$ and $\left[100\right]$ directions. (c) and (e) projection of $\alpha$-boron along the $\left[111\right]$ and $\left[11\overline{2}\right]$ directions.

Figure 2

Electronic properties of $M$-boron. (a) Band structure of NM $M$-boron. The Fermi level is set to zero. Most of the flat band is colored in blue; part of it is colored in red within the band crossing near the $M$ point, (b) density of states (DOS) of NM $M$-boron, (c) band structure of AFM $M$-boron. The highest valence band (flat band) is colored in blue, (d) DOS of AFM $M$-boron. The majority spin is colored in red, and blue for the minority spin.

Figure 3

Band-decomposed spin charge density and electron localization function (ELF) of AFM $M$-boron: (a) and (c) projection of the flat band decomposed majority-spin charge density (colored in red) along the $\left[001\right]$ and the minority-spin charge density (colored in blue) along the $\left[100\right]$ directions; (b) and (d) projection of the ELF in $M$-boron along the $\left[001\right]$ and $\left[100\right]$ directions. The unpaired electrons are labeled as number 1 in (b) and (d).

Figure 4

Chemical bonding analysis for the model fragment of ${\mathrm{B}}_{20}$ clusters in $M$-boron: (a) ${\mathrm{B}}_{140}{\mathrm{H}}_{50}^{34+}$, where three inequivalent boron atomic positions are labeled as ${\mathrm{B}}_1, {\mathrm{B}}_2$, and ${\mathrm{B}}_3$; (b) fifty $2c-2e$ B-H $\sigma$ bonds with occupation numbers (ONs) in the 1.50–1.70 $\left|e\right|$ range, (c) twelve $3c-2e{\mathrm{B}}_2-{\mathrm{B}}_3-{\mathrm{B}}_2\sigma$ bonds with $\text{ONs}=1.82 \left|e\right|$ forming the side frame of each ${\mathrm{B}}_{20}$ cluster, (d) three $7c-2e\sigma$ bonds with $\mathrm{ONs}=1.84–1.93 \left|e\right|$ found on every cap. Intercluster bonding elements: (e) twenty four classical $2c-2e{\mathrm{B}}_2-{\mathrm{B}}_2\sigma$ bonds with $\mathrm{ONs}=1.81 \left|e\right|$ responsible for the direct bonding between ${\mathrm{B}}_{20}$ building blocks, (f) six $3c-2e{\mathrm{B}}_3-{\mathrm{B}}_3-{\mathrm{B}}_3\sigma$ bonds with $\mathrm{ONs}=1.95 \left|e\right|$ found in the middle layer, (g) twelve $8c-2e\sigma$ bonds $\mathrm{ONs}=1.96 \left|e\right|$ connecting all three layers of the $M$-boron. All the bonding elements of each type are superimposed on a single framework.

Figure 5

Stability of $M$-boron. (a) Phonon dispersion curves of AFM $M$-boron. (b) Phonon density of states (PDOS) of $M$-boron. (c) Temperature dependence of the free energy for the FM and AFM states. The inset shows variation of band gap of $M$-boron compared with $\alpha$-boron $\left(111\right)$ surface via the biaxial tensile (negative strain) or compressive strain (positive strain).