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The family of two-dimensional transition metal carbides, so called MXenes,has recently found new members with ordered double transition metalsM$'_2$M$"$C$_2$, where M$'$ and M$"$ stand for transition metals. Here, using aset of first-principles calculations, we demonstrate that some of the newlyadded members, oxide M$'_2$M$"$C$_2$ (M$'$= Mo, W; M$"$= Ti, Zr, Hf) MXenes,are topological insulators. The nontrivial topological states of the predictedMXenes are revealed by the $Z_2$ index, which is evaluated from the parities ofthe 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 thegeneralized gradient approximation and 0.119 -- 0.409 eV within the hybridfunctional. The band gaps are induced by the spin-orbit coupling within thedegenerate 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 phonondispersion calculations find that the predicted topological insulators arestructurally stable.  The predicted W-based MXenes with large band gaps might be suitablecandidates for many topological applications at room temperature. In addition,we study the electronic structures of thicker ordered double transition metalsM$'_2$M$"_2$C$_3$O$_2$ (M$'$= Mo, W; M$"$= Ti, Zr, Hf) and find that they arenontrivial topological semimetals.

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