Topological insulators in ordered double transition metals M$'_2$M$"$C$_2$ (M$'$= Mo, W; M$"$= Ti, Zr, Hf) MXenes

The family of two-dimensional transition metal carbides, so called MXenes, has recently found new members with ordered double transition metals M$'_2$M$"$C$_2$, where M$'$ and M$"$ stand for transition metals. Here, using a set of first-principles calculations, we demonstrate that some of the newly added members, oxide M$'_2$M$"$C$_2$ (M$'$= Mo, W; M$"$= 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$'_2$M$"$C$_2$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 $d_{x^2-y^2}$ and $d_{xy}$ characters of M$'$ and M$"$, while the band inversion occurs at the $\Gamma$ point among the degenerate $d_{x^2-y^2}$/$d_{xy}$ orbitals and a non-degenerate $d_{3z^2-r^2}$ 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$'_2$M$"_2$C$_3$O$_2$ (M$'$= Mo, W; M$"$= Ti, Zr, Hf) and find that they are nontrivial topological semimetals.