Topological massive Dirac fermions inβ-tungsten

Ultrathin films of $\ensuremath{\beta}$-tungsten $(\ensuremath{\beta}\text{\ensuremath{-}}\mathrm{W})$ is attracting extensive attentions because of its promising application in spintronics due to giant spin Hall effect and its fundamental interest in superconductivity. By means of first-principles calculations, we have elucidated its nontrivial topological nature with multiple Dirac nodal lines in the $\ensuremath{\beta}\text{\ensuremath{-}}\mathrm{W}$ metal when spin-orbit coupling (SOC) is ignored. The analysis of electronic structures reveal that its topology is associated with the band inversion between the ${d}_xz+{d}_yz$ and ${d}_{{z}²}$ orbitals at the $M$ point in its Brillouin zone. By switching on the SOC, these Dirac nodal lines become gapped, as accompanied with the occurrence of 24 massive Dirac fermions at the energy of 87 meV below the Fermi level. Different from the exact gapless Dirac fermions with the linear band crossings, we have identified that these massive Dirac fermions all open a tiny gap of 7 meV. In addition, we have observed the topologically protected nontrivial surface arc states connecting these massive Dirac fermions. The existence of these massive Dirac fermions in the $\ensuremath{\beta}\text{\ensuremath{-}}\mathrm{W}$ phase would be important to understand its giant spin Hall effect according to a recent study [Nature 555, 638 (2018)] where massive Dirac fermions were identified to generate Berry-curvature-induced Hall conductivity.