Topological insulators in the ordered double transition metalsM2′M″C2MXenes (M′

The family of two-dimensional transition metal carbides, so called MXenes, has recently found new members with ordered double transition metals ${M}_{2}^{\ensuremath{'}}{M}^{\ensuremath{''}}{\mathrm{C}}_{2}$, where ${M}^{\ensuremath{'}}$ and ${M}^{\ensuremath{''}}$ stand for transition metals. Here, using a set of first-principles calculations, we demonstrate that some of the newly added members, oxide ${M}_{2}^{\ensuremath{'}}{M}^{\ensuremath{''}}{\mathrm{C}}_{2}$ (${M}^{\ensuremath{'}}=\mathrm{Mo}$, W; ${M}^{\ensuremath{''}}=\mathrm{Ti}$, Zr, Hf) MXenes, are topological insulators. The nontrivial topological states of the predicted MXenes are revealed by the ${Z}_{2}$ index, which is evaluated from the parities of the occupied bands below the Fermi energy at time reversal invariant momenta, and also by the presence of the edge states. The predicted ${M}_{2}^{\ensuremath{'}}{M}^{\ensuremath{''}}{\mathrm{C}}_{2}{\mathrm{O}}_{2}$ MXenes show nontrivial gaps in the range of 0.041--0.285 eV within the generalized gradient approximation and 0.119--0.409 eV within the hybrid functional. The band gaps are induced by the spin-orbit coupling within the degenerate states with ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ and ${d}_{xy}$ characters of ${M}^{\ensuremath{'}}$ and ${M}^{\ensuremath{''}}$, while the band inversion occurs at the $\mathrm{\ensuremath{\Gamma}}$ point among the degenerate ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}/{d}_{xy}$ orbitals and a nondegenerate ${d}_{3{z}^{2}\ensuremath{-}{r}^{2}}$ orbital, which is driven by the hybridization of the neighboring orbitals. The phonon dispersion calculations find that the predicted topological insulators are structurally stable. The predicted W-based MXenes with large band gaps might be suitable candidates for many topological applications at room temperature. In addition, we study the electronic structures of thicker ordered double transition metals ${M}_{2}^{\ensuremath{'}}{M}_{2}^{\ensuremath{''}}{\mathrm{C}}_{3}{\mathrm{O}}_{2}$ (${M}^{\ensuremath{'}}=\mathrm{Mo}$, W; ${M}^{\ensuremath{''}}=\mathrm{Ti}$, Zr, Hf) and find that they are nontrivial topological semimetals. Among the predicted topological insulators and topological semimetals, ${\mathrm{Mo}}_{2}{\mathrm{TiC}}_{2}$ and ${\mathrm{Mo}}_{2}{\mathrm{Ti}}_{2}{\mathrm{C}}_{3}$ functionalized with a mixture of F, O, and OH have already been synthesized, and therefore some of the topological materials proposed here can be experimentally accessed.