  By the first-principles electronic structure calculations, we have studied
electronic structures of FeSe monolayer and bilayer thin films on SrTiO$_3$
(001) with SrO-termination or TiO$_2$ termination. We find that both FeSe
monolayer and bilayer on either termination behave like a slightly doped
semiconductor and a collinear antiferromagnetic order on Fe ions. There is no
substantial charge transfer between the FeSe layers and the substrate. FeSe is
adhered to the SrTiO$_3$ surface by a dipole-dipole interaction. The Fermi
surface is mainly the contribution of Fe-3d orbitals. A valence band
contributed mainly by the O-$2p$ orbitals in the TiO$_2$ layer is located
slightly below the Fermi level, which can become conducting upon a small doping
of holes.
