Crystalline electric field and magnetic anisotropy in Dy-based icosahedral quasicrystal and approximant

The lack of the theory crystalline electric field (CEF) in rare-earth based quasicrystal (QC) and approximant crystal (AC) has prevented us from understanding electronic states. Recent success formulation CEF on basis point charge model made it possible to analyze microscopically. Here, by applying this QC Au-SM-Dy (SM=Si, Ge, Al, Ga) AC, we theoretically CEF. In Dy$^{3+}$ ion with $4f^9$ configuration, Hamiltonian is diagonalized set for total angular momentum $J=15/2$. ratio valences screened ligand ions $\alpha=Z_{\rm SM}/Z_{\rm Au}$ plays an important role characterizing ground state. For $0\le\alpha<0.30$, magnetic easy axis state shown be perpendicular mirror plane. On other hand, $\alpha>0.30$, lying plane as $\alpha$ increases, rotates clockwise direction at Dy site tends approach pseudo 5 fold axis. Possible relevance these results experiments discussed.