Comparative Analysis of η-Carbide Superconductors: Structural, Electronic, and Superconducting Properties of Zr₄Rh₂O, Zr₄Pd₂O, and Ti₄Ir₂O

This study presents a comprehensive investigation of three η-carbide-type superconductors (**Zr₄Rh₂O, Zr₄Pd₂O, and Ti₄Ir₂O) through combined theoretical modeling and experimental data analysis. We introduce a novel unified parameter (*E*) that quantifies superconducting potential based on elemental composition and spin-orbit coupling (SOC) effects. Our key findings reveal:  Zr₄Rh₂O exhibits conventional behavior, adhering to the Pauli paramagnetic limit (Hc₂/Hp = 0.81)  Zr₄Pd₂O (Hc₂/Hp = 1.27) and Ti₄Ir₂O (Hc₂/Hp = 1.63) demonstrate significant Pauli limit violation  -Ti₄Ir₂O achieves the highest upper critical field (μ₀Hc₂ = 16.1 T) due to strong SOC from iridium  - The parameter E shows strong correlation with Hc₂ enhancement (R² = 0.91), serving as an effective predictive tool  These results provide crucial insights into the relationship between chemical composition, spin-orbit interactions, and superconducting performance in η-carbide materials. The findings highlight **Ti₄Ir₂O** as particularly promising for high-field applications while establishing parameter **E** as a valuable metric for materials design.