Directional Anisotropy Energy Calculation and Its Role in Enhanced Upper Critical Field of ηCarbide-Type Oxide Superconductor Zr₄Pd₂O

We present a quantitative analysis of the directional anisotropy energy $E^*$ for the η-carbide-type oxide superconductor Zr₄Pd₂O. Using anisotropy constants derived from elemental contributions ₄ ₂ and spin-orbit coupling considerations, we calculate $E^*$ along principal crystallographic directions $[100]$, $[110]$, and $[111]$. Our calculations reveal the highest anisotropy energy along the $[111]$ direction, which correlates with enhanced vortex pinning and strong spin-orbit coupling effects that contribute to the observed violation of the Pauli paramagnetic limit in this material. These results provide insight into the anisotropic nature of superconductivity in Zr₄Pd₂O and highlight the importance of crystallographic directionality in optimizing high-field superconducting properties.