  We explore theoretically the density of states (LDOS) probed by an STM tip of
2D systems hosting an adatom and a subsurface impurity,both capacitively
coupled to AFM tips and traversed by antiparallel magnetic fields. Two kinds of
setups are analyzed, a monolayer of graphene and a two-dimensional electron gas
(2DEG). The AFM tips set the impurity levels at the Fermi energy, where two
contrasting behaviors emerge: the Fano factor for the graphene diverges, while
in the 2DEG it approaches zero. As result, the spin-degeneracy of the LDOS is
lifted exclusively in the graphene system, in particular for the asymmetric
regime of Fano interference. The aftermath of this limit is a counterintuitive
phenomenon, which consists of a dominant Fano factor due to the subsurface
impurity even with a stronger STM-adatom coupling. Thus we find a full
polarized conductance, achievable just by displacing vertically the position of
the STM tip. To the best knowledge, our work is the first to propose the Fano
effect as the mechanism to filter spins in graphene. This feature arises from
the massless Dirac electrons within the band structure and allows us to employ
the graphene host as a relativistic Fano spin-filter.
