  We theoretically investigate the spin resolved Friedel oscillation (FO) and
quasiparticle interference (QPI) in graphene induced by an Anderson impurity.
Once the impurity becomes magnetic, the resulted FO becomes spin dependent,
which gives rise to a local magnetic moment oscillation with an envelop
decaying as $r^{-2}$ in real space in the doping cases. Meanwhile, at half
filling, the charge density and local magnetic moment will not oscillate but
decay as $r^{-3}$. Such spin resolved FO has both sublattice and spin
asymmetry. Interestingly, the local magnetic moment decay at half filling only
occurs at one sublattice of graphene, which is quite like the phenomenon
observed in a recent STM experiment [H. Gonzalez-Herrero et al., Science 352,
437 (2016)]. We further give an analytic formula about such spin dependent FO
based on the stationary phase approximation. Finally, we study the interference
of quasiparticles around the magnetic impurity by calculating the
spin-dependent Fourier-transformed local density of states (FT-LDOS). Our work
gives a comprehensive understanding about the local magnetic moment oscillation
around an Anderson impurity on graphene.
