Spatially dependent f-π exchange interaction within a single-molecule magnet TbPc$_2$

Electrically probing the spin state of localized f electrons in a rare-earth-based single-molecule magnet, along with understanding its intramolecular magnetic coupling, is of crucial importance for applications in quantum information and advanced spintronics, yet it remains experimentally challenging. Herein, within a single-molecule magnet terbium(III) bis(phthalocyaninato) (TbPc2) double-decker molecule adsorbed on a bilayer graphene epitaxially grown on a SiC(0001) substrate, we experimentally demonstrate a spatially dependent exchange interaction between the magnetic moment of the localized Tb 4f electron and the unpaired spin of the Pc π-radical. The magnetic state of TbPc2, associated with the f-π interaction, is evidently detected through the spectroscopic Kondo resonance and a zero-field Kondo splitting, which can be reversibly switched in a charge/discharge process triggered by the tip-molecule distance. Furthermore, we theoretically describe how the Kondo resonance evolves at the molecular scale, which is mediated by the f-π exchange interaction with its strength varying spatially in a radial decay fashion. Our spatially resolved Kondo characteristics offer a quantitative understanding of the many-body spin correlation, which is coupled with the charge states in a nonuniform and spatially extended system.