First-principles calculations of a robust two-dimensional boron honeycomb sandwiching a triangular molybdenum layer

A graphenelike two-dimensional boron honeycomb is inherently prohibited due to its empty π valence band. Based on chemical intuition and first-principles calculations, we design a two-dimensional crystal ${\mathrm{MoB}}_4$ with two graphenelike boron honeycombs sandwiching a triangular molybdenum layer. It has the attractive electronic structure of double Dirac cones near Fermi level with high Fermi velocity, which are contributed by the coupling of Mo $d$ orbitals and B $p_z$ orbitals. Such a metal stabilized boron honeycomb system could even have both superconductivity and ferromagnetism through appropriate selection of the metal layer, such as manganese. The unique electronic properties of these two-dimensional systems inspire broad interest in nanoelectronics.

Figure 1

The sketch to stabilize B honeycomb. (a) Electronic band dispersion for graphene and B honeycomb. The insets are their atomic primitive cells. The dashed green ellipses mark their Dirac points. The Fermi level is set to the energy zero point. (b) Phonon frequency dispersion for graphene and B honeycomb. (c) graphene and its valence band occupation. (d) The sketch of stabilization for B honeycomb by metal. Gray, orange-yellow, and cyan balls represent C, B, and metal atoms, respectively.

Figure 2

The structure and its stability for ${\mathrm{MoB}}_4$. (a) Left is the top view of ${\mathrm{MoB}}_4$, and right is corresponding STM image with a bias of −1.0 eV. (b) Side view of the structure. To show the structure clearly, the B-Mo bonds are not shown. (c) Phonon dispersion and PDOS of ${\mathrm{MoB}}_4$. (d) Bond length evolution at 1000 K. Color coding of atoms is the same as in Fig. 1.

Figure 3

Snapshot of a (4 × 4) supercell of ${\mathrm{MoB}}_4$ after 10 ps in HTMD kept at 1000 K. (a) Top view. (b) Side view. Color coding of atoms is the same as in Fig. 1.

Figure 4

Bonding mechanism of ${\mathrm{MoB}}_4$ reflected from ELF. (a)–(c) Cutting slabs along the B-B bond, Mo-Mo bond, and B-Mo bond plane, respectively. (d)–(f) Their corresponding ELF. For clarity, a (2 × 2) supercell is used here. Color coding of atoms is the same as in Fig. 1.

Figure 5

Electronic structure of ${\mathrm{MoB}}_4$. (a) Band structure of ${\mathrm{MoB}}_4$. The dashed pink ellipses indicate two Dirac cones, marked by I and II. (b) and (c) Projected EDOS for B and Mo atoms. (d) First Brillouin zone and high-symmetry points along band structure path for ${\mathrm{MoB}}_4$. (e) 3D rendering of Dirac cones.

Figure 6

Charge density plots of states at Dirac points of ${\mathrm{MoB}}_4$ with the isosurface value 0.01 e/$a_0{}^3$. The orbital distribution for Dirac points (a) І and (b) ІІ in Fig. 5 is shown. Here, up and down are for the two crossing states in one cone.

Figure 7

Spin orbit coupling opens the gap for the Dirac points in ${\mathrm{MoB}}_4$ (38 meV for cone I and 55 meV for cone II). Fermi level is set at the energy zero point.

Figure 8

Ferromagnetism and superconductivity of ${\mathrm{MnB}}_4$. (a) Spin-polarized EDOS for ${\mathrm{MnB}}_4$. The blue line and the red line indicate spin-up and spin-down states, respectively, and the green line represents their difference (magnetic EDOS). The inset shows the corresponding structure obtained by replacing Mo with Mn. (b) PDOS and EPC Eliashberg function $\left[{\alpha }^{2}F\left(\omega \right)\right]$ for ${\mathrm{MnB}}_4$.