Multiple robust Dirac states in hexagonal lattice induced by p−d electron-counting rule and bilayer stacking

In this paper, we predict two-dimensional transition metal boride $\mathrm{Ti}{\mathrm{B}}_{2}{\mathrm{C}}_{2}$ and $\mathrm{Hf}{\mathrm{B}}_{2}{\mathrm{C}}_{2}$ nanosheets, which exhibit favorable mechanical and thermal properties. Young's moduli of $\mathrm{Ti}{\mathrm{B}}_{2}{\mathrm{C}}_{2}$ and $\mathrm{Hf}{\mathrm{B}}_{2}{\mathrm{C}}_{2}$ are 137.8 and 128.9 N/m, respectively. Both nanosheets could sustain up to 500 K based on our ab initio molecular dynamics results. In addition, both nanosheets are semimetallic, exhibiting 12 Dirac cones in the first Brillouin zone, six pairs of which form compensated electron-hole pockets. Under different types of external strains, both $\mathrm{Ti}{\mathrm{B}}_{2}{\mathrm{C}}_{2}$ and $\mathrm{Hf}{\mathrm{B}}_{2}{\mathrm{C}}_{2}$ are robust, and the Dirac cones could be tuned to be anisotropic. We propose that the multiple Dirac cones are induced by a combination of the $p\ensuremath{-}d$ electron-counting rule and bilayer stacking; to verify our opinion, we also predicted and proved that $T{\mathrm{B}}_{2}{\mathrm{Si}}_{2}$ ($T$ = Ti, Zr, Hf) are mechanical stable two-dimensional nanosheets with multiple Dirac cones based on phonon and electronic band structure calculations.