▎ 摘　　要

We explore theoretically the density of states (LDOS) probed by a scanning tunneling microscope (STM) tip of two-dimensional systems hosting an adatom and a subsurface impurity, both capacitively coupled to atomic force microscope (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. Our work proposes 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.