Mathematical Relation Between Directional Anisotropy Energy and Superconducting Properties of η-Carbide-Type Oxide Superconductor Zr₄Pd₂O

The η-carbide-type oxide superconductor Zr₄Pd₂O exhibits unconventional superconducting behavior characterized by an enhanced upper critical field $\mu_0 H_{c2}$ that exceeds the Pauli paramagnetic limit. In this paper, we establish a quantitative relation between the directional anisotropy energy $E^*$, arising from spin-orbit coupling (SOC) and crystal anisotropy, and the superconducting properties of Zr₄Pd₂O. We demonstrate how $E^*$ influences SOC strength, suppresses Pauli paramagnetic pair-breaking, and leads to enhancement of $\mu_0 H_{c2}$. The derived expression provides a mechanistic framework to understand the robustness of superconductivity in Zr₄Pd₂O under high magnetic fields.