Topological phases in layered pyrochlore oxide thin films along the [111] direction

We theoretically study a multi-band Hubbard model of pyrochlore oxides of the form A$_2$B$_2$O$_7$, where B is a heavy transition metal ion with strong spin-orbit coupling, in a thin film geometry orientated along the [111] direction. Along this direction, the pyrochlore lattice consists of alternating kagome and triangular lattice planes of B ions. We consider a single kagome layer, a bilayer, and the two different trilayers. As a function of the strength of the spin-orbit coupling, the direct and indirect $d$-orbital hopping, and the band filling, we identify a number of scenarios where a non-interacting time-reversal invariant Z$_2$ topological phase is expected and we suggest some candidate materials. We study the interactions in the half-filled $d$-shell within Hatree-Fock theory and identify parameter regimes where a zero magnetic field Chern insulator with Chern number $\pm1$ can be found. The most promising geometries for topological phases appear to be the bilayer which supports both a Z$_2$ topological insulator and a Chern insulator, and the triangular-kagome-triangular trilayer which supports a relatively robust Chern insulator phase.