  We investigate a magnetic proximity effect on Dirac surface states of
topological insulator (TI) induced by a Bi$_2$Se$_3$/EuS interface, using
density-functional theory (DFT) and a low-energy effective model, motivated by
a recent experimental realization of the interface. We consider a thin
ferromagnetic insulator EuS film stacked on top of Bi$_2$Se$_3$(111) slabs of
three or five quintuple layers (QLs) with the magnetization of EuS normal to
the interface ($z$ axis), which breaks both time-reversal and inversion
symmetry. It is found that a charge transfer and surface relaxation makes the
Dirac cones electron-doped. For both 3 and 5 QLs, the top-surface Dirac cone
has an energy gap of 9 meV, while the bottom surface Dirac cone remains
gapless. This feature is due to the short-ranged induced magnetic moment of the
EuS film. For the 5 QLs, an additional Dirac cone with an energy gap of 2 meV
is formed right below the bottom-surface Dirac point, while for 3 QLs, there is
no additional Dirac cone. We also examine the spin-orbital texture of the Dirac
surface states with broken inversion symmetry, using DFT and the effective
model. We find that the $p_z$ orbital is coupled to the $z$ component of the
spin moment in the opposite sign to the $p_x$ and $p_y$ orbitals. The $p_z$ and
radial $p$ orbitals are coupled to the in-plane spin texture in the opposite
handedness to the tangential $p$ orbital. The result obtained from the
effective model agrees with our DFT calculations. The calculated spin-orbital
texture may be tested from spin-polarized angle-resolved photoemission
spectroscopy.
