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Strain driven onset of non-trivial topological insulating states in Zintl Sr$_2X$ compounds ($X$=Pb, Sn)

We explore the topological behavior of the binary Zintl phase of the alkaline earth metals based compounds Sr$_2$Pb and Sr$_2$Sn using both standard and hybrid density functional theory. It is found that Sr$_2$Pb lies on the verge of a topological instability which can be suitably tuned through the application of a small uniaxial expansion strain ($>$ 3%). The resulting non-trivial topologically insulating state display well-defined metallic states in the Sr$_{2}$Pb(010) surface, whose evolution is studied as a function of the film thickness.