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We report electronic structure calculations for BaTi$_2$Sb$_2$O and discussthe results in relation to the observed superconductivity of this material whenhole doped with Na. The Fermi surface shows several sheets. These include anested nearly 2D cylinder. There are also two sheets, which are threedimensional, in spite of the layered crystal structure. A magnetic instabilityassociated with Fermi surface nesting is found. A sign-changing s-wave state,different from the one in the Fe-based superconductors, is predicted within ascenario of spin-fluctuation mediated superconductivity.

Electronic structure, disconnected Fermi surfaces and antiferromagnetism in the layered pnictide superconductor Na$_x$Ba$_{1-x}$Ti$_2$Sb$_2$O