  Proximity coupling of ferromagnetic insulator EuS to the topological
insulator Bi$_2$Se$_3$ has been proposed to break time-reversal symmetry near
the surface of Bi$_2$Se$_3$, introducing an energy gap or a tilt in the surface
Dirac cone. As an inverse proximity effect, strong spin-orbit coupling
available in the topological surface states can enhance the Curie temperature
of ferromagnetism in EuS largely beyond its bulk value, and also generate a
magnetic anisotropy. This can result in a canting of the magnetic moment of Eu
ions in a plane perpendicular to the interface. Here, we investigate
theoretically electronic transport properties arising from the Bi$_2$Se$_3$/EuS
interfaces in the planar Hall geometry. Our analysis, based on a realistic
model Hamiltonian and a semi-classical formalism for the Boltzmann transport
equation, reveals distinct intriguing features of anisotropic planar Hall
conductivity, depending on different scenarios for the canting of the Eu
moments: fixed Eu moment canting, and freely-orientable Eu moment in response
to the external in-plane magnetic field. The anisotropy in the planar Hall
conductivity arises from the asymmetric Berry curvature of the gapped
topological surface states. We also explore topological Hall effect of the
Dirac surface states, coupled to a skyrmion crystal which can emerge in the EuS
due to the interplay of ferromagnetic Heisenberg exchange, interfacial
Dzyaloshinskii-Moriya interaction, and perpendicular alignment of the Eu
moment. Our study provides new impetus for probing complex interplay between
magnetic exchange interactions and topological surface states via anisotropic
planar Hall effects.
