Relationship Between Directional Anisotropy Energy and Enhanced Superconducting Properties of η-Carbide-Type Oxide Superconductor Zr₄Pd₂O

We investigate the critical role of directional anisotropy energy E^* in enhancing the superconducting upper critical field u_0 H_{c2} of the η-carbide-type oxide superconductor Zr₄Pd₂O. Using established anisotropy constants and crystallographic direction cosines, E^* is calculated for principal directions. We propose a mathematical model linking E^*to the spin-orbit coupling strength and consequent suppression of the Pauli paramagnetic pair-breaking effect. This relationship explains the experimentally observed violation of the Pauli-Clogston limit by Zr₄Pd₂O. A comparative analysis of model fits to experimental data highlights the superior accuracy of the proposed relation. Our findings elucidate the interplay between crystal anisotropy, spin-orbit coupling, and superconductivity, providing insights into the design of high-field superconductors.