  We investigate the spin-valley polarization in MoTe$_2$ monolayer on (111)
and (001) surfaces of ferromagnetic semiconductor EuO based on first-principles
calculations. We consider surface reconstructions for EuO(111). We find that
there is no direct chemical bonding between the reconstructed EuO(111) and the
MoTe$_2$ overlayer, in contrast to the case of the ideal EuO(111). However,
there is a strong hybridization between the states of MoTe$_2$ and the
substrate states, which has a substantial impact on the valleys. The valley
polarization due to the magnetic proximity effect is dependent on the detail of
the interface structure, which is in the range of a few meV to about 40 meV.
These values are at least one order of magnitude smaller than that induced by
the ideal EuO(111). When the MoTe$_2$ monolayer is interfaced with EuO(001),
the valley polarization is about 3.2 meV, insensitive to the interface
structure. By a low-energy effective Hamiltonian model, the effective Zeeman
field induced by EuO(001) is about 27 T, comparable to that for WSe$_2$/EuS
obtained by experiment.
