Odd-Parity Crystalline Electric Field and Multipoles in Icosahedral Quasicrystal and Approximant: Magnetic Toroidal and Electric Dipoles in Ce-Based Systems

Quasicrystals (QCs) lack periodic arrangement of atoms but possess unique structural order distinct from periodic crystals and random systems. The electronic property is far from complete understanding. Here we have formulated not only even-parity but also odd-parity crystalline electric field (CEF) in the Ce-based icosahedral QC and approximant crystal (AC) on the basis of the point charge model. Under the odd-parity CEF due to local-inversion symmetry breaking at the Ce site, the hybrid wavefunction of 4f and 5d states at Ce is shown to give rise to the odd-parity multipoles such as electric dipole and magnetic toroidal (MT) dipole. We have found that the vortex order in the 1/1 AC where the magnetic moments on each icosahedron swirl is characterized by the cluster-type MT moment. The ferroic alignment of the cluster MT moments in the vortex order gives rise to the electro-magnetic effect. The cluster-MT moment is shown to be resonated with the MT dipole moments in each icosahedron by tuning the 4f energy level and 5d energy level so as to approach. We have revealed a noncollinear magnetic structure is stabilized in the ground state of the icosahedral QC cluster, behind which there exists underlying mechanism to prefer the ferroic interaction for the net MT dipole moments between the icosahedrons. We point out that the 4f and 5d energy levels can be made approach at the valence transition or the valence crossover, which is ubiquitous in the rare-earth based ACs and QCs.