We study the properties of a family of anti-pervoskite materials, which are topological crystalline insulators with an insulating bulk but a conducting surface. Using ab-initio DFT calculations, we investigate the bulk and surface topology and show that these materials exhibit type-I as well as type-II Dirac surface states protected by reflection symmetry. While type-I Dirac states give rise to closed circular Fermi surfaces, type-II Dirac surface states are characterized by open electron and hole pockets that touch each other. We find that the type-II Dirac states exhibit characteristic van-Hove singularities in their dispersion, which can serve as an experimental fingerprint. In addition, we study the response of the surface states to magnetic fields.
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