Effects of altermagnetic order, strain and doping in RuO$_2$

RuO$_2$, one of the most widely studied transition metal oxides, was recently predicted to host a novel form of collinear magnetic order referred to as altermagnetism. In this study we combine experiment (reflectance, transmittance, ellipsometry and Raman measurements) and first-principles calculations to elucidate the potential role of altermagnetic order, strain and doping on the optical and vibrational properties of RuO$_2$ grown on TiO$_2$ (001), (101) and (110) substrates. Our combined experimental and theoretical results point toward a nonmagnetic ground state as the most consistent description of the optical and vibrational properties of bulk RuO$_2$. RuO$_2$ strained to TiO$_2$ (001) remains nonmagnetic in our calculations. Straining to TiO$_2$ (110) stabilizes ferromagnetic and altermagnetic states that are nearly degenerate and both lower in energy than the nonmagnetic state. The relative energetic ordering of these states is highly sensitive to the level of strain and the choice of exchange-correlation approximation.