  We study gate field effects on the Mn$_{12}$O$_{12}$(COOH)$_{16}$(H$_2$O)$_4$
| graphene | GaAs heterostructure via first-principles calculations. We find
that under moderate doping levels electrons can be added to but not taken from
the single-molecule magnet Mn$_{12}$O$_{12}$(COOH)$_{16}$(H$_2$O)$_4$
(Mn$_{12}$). The magnetic anisotropy energy (MAE) of Mn$_{12}$ decreases as the
electron doping level increases, due to electron transfer from graphene to
Mn$_{12}$ and change in the band alignment between Mn$_{12}$ and graphene. At
an electron doping level of $-5.00 \times 10^{13}\, \textrm{cm}^{-2}$, the MAE
decreases by about 18% compared with zero doping. The band alignment between
graphene and GaAs is more sensitive to electron doping than to hole doping
since the valence band of GaAs is close to the Fermi level. The GaAs substrate
induces a small bandgap in the supported graphene under the zero gate field and
a nearly strain-free configuration. Finally, we propose a vertical tunnel
junction for probing the gate dependence of MAE via electron transport
measurements.
