  The layered material Mn$_3$Si$_2$Te$_6$, with alternating stacking honeycomb
and triangular layers, is attracting considerable attention due to its rich
physical properties. Here, using density functional theory and classical Monte
Carlo (MC) methods, we systematically study this system. Near the Fermi level,
the states are mainly contributed by Te $5p$ orbitals hybridized with Mn $3d$
orbitals, resembling a charge transfer system. Furthermore, the spin
orientations of the ferrimagnetic (FiM) ground state display different
conductive behaviors when along the $ab$ plane or out-of-plane directions:
insulating vs. metallic states. The energy difference between the FiM [110]
insulating and FiM [001] metallic phases is very small($ \sim 0.71$ meV/Mn).
Changing the angle $\theta$ of spin orientation from in-plane to out-of-plane
directions, the band gaps of this system are gradually reduced, leading to an
insulator-metal transition, resulting in an enhanced electrical conductivity,
related to the colossal angular magnetoresistance (MR) effect. In addition, we
also constructed the magnetic phase diagram using the classical $XY$ spin model
studied with the MC method. Three magnetic phases were obtained including
antiferromagnetic order, noncollinear spin patterns, and FiM order. Moreover,
we also investigated the Se- and Ge- doping into the Mn$_3$Si$_2$Te$_6$ system:
the FiM state has the lowest energy among the magnetic candidates for both Se-
or Ge- doped cases. The magnetic anisotropy energy (MAE) decreases in the
Se-doped case because the Mn orbital moment is reduced as the doping $x$
increases. Due to the small spin-orbital coupling effect of Se, the
insulator-metal transition caused by the spin orientation disappears in the
Se-doped case, resulting in an insulating phase in the FiM [001] phase. This
causes a reduced colossal angular MR.
