Competition between local magnetic disorder and altermagnetism in doped FeSb$_2$

Recent experimental reports suggest that the narrow-gap nonmagnetic semiconductor FeSb$_2$ can be transformed into an altermagnetic metal through Co doping (Co$_{0.15}$Fe$_{0.85}$Sb$_2$), or into a magnetically disordered or short-range-ordered state through Cr doping (Cr$_{0.15}$Fe$_{0.85}$Sb$_2$). Here we explore the energy landscape and magnetic states of these doped systems from first principles, relying on Hubbard-augmented density-functional theory (DFT+U) combined with the Romeo ground-state search algorithm. Within the established virtual-crystal approximation (VCA), we show that \texttt{Romeo} finds several non-trivial magnetic states, which inform targeted explicit simulations of doping in supercells. We rely on these findings to discuss strengths and limitations of the VCA-Romeo approach versus the explicit-doping supercell approach, and how they can be used in synergy. Overall, our simulations suggest that the ground state of the Cr-doped system is a Locally Disordered Spin-Compensated (LDSC) configuration, formally compatible with N\'eel's L-type fully compensated ferrimagnetism, whereas the ground state of the Co-doped system is found to be altermagnetic (AFMo). This work shows how approximate and explicit simulations of magnetic alloys can be mutually informative, and establishes a protocol for studying candidate metallic altermagnets.