Gateways to Orbital and Spin Hall Effects in Rh-Doped Altermagnetic RuO$_2$

Abstract: Ruthenium dioxide (RuO$_2$) has recently emerged as a prototypical material for exploring the fundamental properties of altermagnets. In this work, we investigate the impact of Rhodium (Rh) doping on the electronic and transport characteristics of altermagnetic RuO$_2$ using first-principles calculations. We show that Rh substitution at Ru sites modifies the spin-splitting of electronic bands across momentum space and reshapes the spin-resolved Fermi surface topology. These changes are found to significantly influence both the spin Hall and orbital Hall effects. In particular, we demonstrate that the orbital Berry curvature is strongly modulated by the doping concentration, opening new avenues for tuning orbital transport responses in multi-orbital systems without relying on strong spin-orbit coupling. Our results suggest that Rh-doped RuO$_2$ provides a versatile platform for engineering spin and orbital Hall effects in altermagnetic materials, and contributes to the growing efforts in designing next-generation orbitronic and spintronic devices.