  The study of ultracold atomic spin systems with long-range interaction
provides the possibility of searching for magnetic supersolid phases in quantum
many-body scenarios. In this paper, we consider two-species Bose gases with
spin-orbit coupling and nearest-neighbor interaction confined in a
two-dimensional optical lattice. The competition between spin-orbit coupling
and interactions creates rich ground-state diagrams with supersolid phases
exhibiting phase modulations or magnetic orderings. We obtain the phase-twisted
and phase-striped pair checkboard supersolid phases that are generated by the
combination of spin-orbit coupling and intraspecies nearest-neighbor
interaction. The introduction of interspecies nearest-neighbor interaction
enriches the quantum phases of the system. It leads to the appearance of the
phase-twisted and phase-striped lattice supersolid phases. In addition to the
lattice supersolid phase, we find the emergence of nontrivial supersolid phases
that depend on the interspecies on-site interaction strength. The
lattice-insulated supersolid phase with supersolidility in one species but
insulation in the other exists in the miscible domain, while the pair striped
supersolid phase with stripe structures in each species is in the immiscible
domain. Finally, to further characterize each phase, we discuss their
spin-dependent momentum distributions and spin textures. The magnetic textures,
such as antiferromagnetic, spiral and stripe orders, are shown in SS phases.
The results here could help in the observe for these magnetic supersolid phases
in ultracold atomic experiments with nearest-neighbor interaction and
spin-orbit coupling in optical lattice.
