Correlation between Magnetocrystalline Anisotropy Energy and Enhanced Upper Critical Fields in η-Carbide-Type Oxide Superconductors

We investigate the relationship between the magnetocrystalline anisotropy energy $E^*$ and superconducting properties, specifically the upper critical field $\mu_0 H_{c2}$, Pauli paramagnetic limit $\mu_0 H_P$, coherence length $\xi$, and penetration depth $\lambda$, in η- carbide-type oxide superconductors. Using reported data for Zr ₄ Pd ₂ O, Zr ₄ Rh ₂ O, and Ti ₄ Ir ₂ O, we calculate $E^*$ along principal crystallographic directions and analyze its correlation with the enhancement $\Delta H = \mu_0 H_{c2} - \mu_0 H_P$. Our results suggest a strong proportionality between $E^*$ and $\Delta H$, indicative of the role of spin-orbit coupling driven by anisotropy energy in violating the Pauli limit. Furthermore, we discuss how $E^*$ influences superconducting coherence length and penetration depth, highlighting its importance in designing high-field oxide superconductors.