Dirac Fermions in Borophene

Honeycomb structures of group IV elements can host massless Dirac fermions with nontrivial Berry phases. Their potential for electronic applications has attracted great interest and spurred a broad search for new Dirac materials especially in monolayer structures. We present a detailed investigation of the ${\beta }_{12}$ sheet, which is a borophene structure that can form spontaneously on a Ag(111) surface. Our tight-binding analysis revealed that the lattice of the ${\beta }_{12}$ sheet could be decomposed into two triangular sublattices in a way similar to that for a honeycomb lattice, thereby hosting Dirac cones. Furthermore, each Dirac cone could be split by introducing periodic perturbations representing overlayer-substrate interactions. These unusual electronic structures were confirmed by angle-resolved photoemission spectroscopy and validated by first-principles calculations. Our results suggest monolayer boron as a new platform for realizing novel high-speed low-dissipation devices.

Figure 1

Schematic drawing of the Dirac cones and lattices. (a) Honeycomb structure. (b) ${\beta }_{12}$ sheet. (c) The ${\beta }_{12}$ sheet with a $3\times 1$ perturbation. The blue and green balls indicate the boron atoms with different on-site energies in our TB analysis. The top and bottom panels are the band structures and atomic structures, respectively. The basic vectors of the primitive unit cell are indicated by the blue arrows.

Figure 2

TB model and first-principles calculations of the ${\beta }_{12}$ sheet. (a) and (b) The wave function of each boron atom, as indicated by the amplitude near the red balls; the blue balls represent boron atoms with a vanishing amplitude. The boron atoms at site $c$ always have a vanishing amplitude. The red balls are equivalent to the sublattices of the honeycomb lattice in (c). (d) Schematic drawing of the band folding process. The green hexagon and black rectangles indicate the BZ of the equivalent honeycomb lattice and the ${\beta }_{12}$ sheet, respectively. The blue dots indicate the original Dirac cones (DC) from the honeycomb lattice; the red dots indicate the folded Dirac cones. (e) and (f) Band structures of free-standing ${\beta }_{12}$ sheet from the TB model and first-principles calculations, respectively. $E_D$ in the figure corresponds to the Dirac point, which is approximately 2 eV above the Fermi level from our first-principles calculations. The black arrows indicate the Dirac cones. (g) TB band structures of the ${\beta }_{12}$ sheet under a modulated potential. The Dirac cone is split in the $\mathrm{\Gamma }\text{−}Y$ direction. (h) Schematic drawing of the folding and splitting (blue arrow) of the Dirac cones.

Figure 3

Band structures of the ${\beta }_{12}$ sheet on Ag(111). (a) The Fermi surface of the ${\beta }_{12}$ sheet on Ag(111). The black, green, and blue rectangles indicate the BZ of three equivalent domains; the grey hexagon indicates the BZ of Ag(111). The black and red arrows indicate the bands of the boron layer. The surface state (SS) and bulk $sp$ band of Ag(111) are also observed because the coverage of boron is less than 1 ML. The pink lines indicate cuts 1–3 where the ARPES intensity plots in (c)–(f) were measured. (b) CECs derived from the second-derivative energy distribution curves measured in the black dotted rectangle in (a). $E_F$ in the figure corresponds to the Fermi level. All the data in (a) and (b) were measured with $p$ polarized light. (c) ARPES intensity plot measured along cut 1 with $s$ polarized light. (d)–(f) ARPES intensity plots measured with $p$ polarized light along cut 1 to cut 3, respectively. The yellow dashed lines indicate the Dirac cones (DC). All the ARPES data in (a)–(f) were measured with a photon energy of 80 eV. (g) Schematic drawing of the Dirac cones according to our experimental results. (h) Relaxed structure model of the ${\beta }_{12}$ sheet on Ag(111) from our first-principles calculations. The orange and blue balls indicate the B and Ag atoms, respectively. (i) and (j) Calculated PDOS of B atoms and Ag atoms, respectively. (k) Calculated band structure along cut 1.