Coexistence and tunability of orbital and spin Hall effects in RuO$_2$

Altermagnetic materials, especially RuO$_2$, have recently attracted considerable attention for their unique magnetic properties and energy-efficient spintronic applications. However, recent experimental studies have reported highly conflicting signatures regarding altermagnetic spin splitting and charge--spin interconversion (CSI) in RuO$_2$. While some experiments link efficient CSI to non-relativistic altermagnetic spin-splitting effects, others observe large CSI signals in non-spin-splitting RuO$_2$, which are instead explained by relativistic inverse spin Hall effects. In this work, based on first-principles calculations, we reveal that these controversial experimental results originate from a phase-dependent coexistence and relative dominance of the orbital Hall effect (OHE) and spin Hall effect (SHE) in RuO$_2$. We systematically investigate the OHE and SHE in both altermagnetic and nonmagnetic phases of RuO$_2$. Our results show that the altermagnetic state hosts a giant OHE that exceeds the SHE by two orders of magnitude and carries an opposite sign. This dominant OHE can generate experimentally observed "SHE-like" voltages through orbital-to-spin conversion, explaining previously reported altermagnetic CSI signals. In contrast, OHE of nonmagnetic RuO$_2$ is suppressed and a large relativistic SHE emerges, in agreement with recent angle-resolved photoemission and spin-pumping experiments. Finally, we demonstrate that the coexistence of OHE and SHE is tunable via chemical doping, enabling on-demand modulation of CSI in in RuO$_2$. Our work provides a new physical mechanism for understanding CSI in RuO$_2$ and highlights the central role of orbital transport.