Cluster multipole orderings composed of atomic high-rank multipole moments are theoretically investigated with a 5$d$-electron compound Ca$_5$Ir$_3$O$_{12}$ in mind. Ca$_5$Ir$_3$O$_{12}$ exhibits two hidden orders: One is an intermediate-temperature phase with time-reversal symmetry and the other is a low-temperature phase without time-reversal symmetry. By performing the symmetry and augmented multipole analyses for a $d$-orbital model under the hexagonal point group $D_{\rm 3h}$, we find that the 120$^{\circ}$-type ordering of the electric quadrupole corresponds to cluster electric toroidal dipole ordering with the electric ferroaxial moment, which can become the microscopic origin of the intermediate-temperature phase in Ca$_5$Ir$_3$O$_{12}$. Furthermore, based on ${}^{193}$Ir synchrotron-radiation-based M\"{o}ssbauer spectroscopy, we propose that the low-temperature phase in Ca$_5$Ir$_3$O$_{12}$ is regarded as a coexisting state with cluster electric toroidal dipole and cluster magnetic toroidal quadrupole, the latter of which is formed by the 120$^{\circ}$-type ordering of the magnetic octupole and accompanies a small uniform magnetization as a secondary effect. Our results provide a clue to two hidden phases in Ca$_5$Ir$_3$O$_{12}$. 

/pdf/cluster-toroidal-multipoles-formed-by-electric-quadrupole-30wamoxf.pdf

Cluster Toroidal Multipoles Formed by Electric-Quadrupole and Magnetic-Octupole Trimers: A Possible Scenario for Hidden Orders in Ca5Ir3O12

The so-called ferroaxial transition characterized by a rotational structural distortion that breaks a mirror symmetry has gained growing interest in terms of a new class of ferroic state in crystalline materials. RbFe(MoO4)2, which belongs to glaserite-type compounds, X(□;1)Y(□;2)[M(TO4)2], is one of the most representative materials showing a ferroaxial transition, i.e., ferroaxial materials. Considering a variety of glaserite-type compounds, we expect that they provide a good arena for ferroaxial materials. In this work, we explored new ferroaxial materials by formula-based screening using a regular expression search and the symmetry detection algorithm. As a result, we found that a glaserite-type compound, K2Zr(PO4)2, is one of the promising candidates for ferroaxial materials. Experimentally, we demonstrate that K2Zr(PO4)2 shows a ferroaxial transition at about 700 K, which is well explained by ab initio phonon calculations. The ferroaxial nature of K2Zr(PO4)2 is further confirmed by the observation of its domain structures using a linear electrogyration effect, i.e., optical rotation in proportion to an applied electric field. Our work provides an effective approach to exploring ferroaxial materials. 

Ferroaxial Transitions in Glaserite-type Compounds: Database Screening, Phonon Calculations, and Experimental Verification

Ferroaxial order, characterized by a rotational arrangement of electric dipoles, attracts increasing attention in terms of a new family of ferroic orders. However, there has been no chemical guideline for exploring crystalline materials showing ferroaxial order, namely ferroaxial materials. Here, we present a chemical guideline grounded in staggered polyhedral connectivity, which we propose as a structural prerequisite for ferroaxial order, and the second-order Jahn-Teller (SOJT) theory extended from molecular orbitals to electronic band structures. Na-superionic conductors (NASICON) including NaM2(PO4)3 (M = early-transition or post-transition metal) are identified as potential ferroaxial materials because of their staggered structures composed of MO6 octahedra and PO4 tetrahedra. However, ferroaxial phase transitions hardly occur in some of the NASICON systems, which offers a platform to uncover a hidden factor playing an important role in driving this system into ferroaxial states. Our first-principles calculations demonstrate that a ferroaxial phase transition in NASICON systems occurs only when SOJT interaction is symmetrically allowed, that is, energy-lowering chemical bonds are formed as a consequence of the distortion. Our proposals would be not limited to NASICON systems but applicable to a variety of compounds and provide new insight into the exploration of displacive-type ferroaxial materials.

Chemical Aspect of Displacive-Type Ferroaxial Phase Transition from Perspective of Second-Order Jahn-Teller Effect: NASICON Systems as an Example.

An electric toroidal dipole (ETD) moment is one of the fundamental dipole moments as well as electric and magnetic ones. Although it directly couples to neither an electric nor magnetic field due to its spatial inversion and time-reversal parities, its ordered state leads to unconventional transverse responses of the conjugate physical quantities. We here theoretically investigate nonlinear transverse magnetic susceptibility under the ETD ordering. By performing a self-consistent mean-field calculation for a five $d$-orbital model under a tetragonal crystalline electric field and using the nonlinear Kubo formula, we show that a third-order transverse magnetic susceptibility corresponding to a uniform magnetization perpendicular to the external magnetic field becomes nonzero once the ETD moment is ordered under tetragonal crystalline electric field. Moreover, we find that spin-orbital entanglement and a low-lying first excited crystal-field level are important for realizing large transverse responses.

Nonlinear Transverse Magnetic Susceptibility under Electric Toroidal Dipole Ordering

Cluster multipole orderings composed of atomic high-rank multipole moments are theoretically investigated with a 5$d$-electron compound Ca$_5$Ir$_3$O$_{12}$ in mind. Ca$_5$Ir$_3$O$_{12}$ exhibits two hidden orders: One is an intermediate-temperature phase with time-reversal symmetry and the other is a low-temperature phase without time-reversal symmetry. By performing the symmetry and augmented multipole analyses for a $d$-orbital model under the hexagonal point group $D_{\rm 3h}$, we find that the 120$^{\circ}$-type ordering of the electric quadrupole corresponds to cluster electric toroidal dipole ordering with the electric ferroaxial moment, which can become the microscopic origin of the intermediate-temperature phase in Ca$_5$Ir$_3$O$_{12}$. Furthermore, based on ${}^{193}$Ir synchrotron-radiation-based M\"{o}ssbauer spectroscopy, we propose that the low-temperature phase in Ca$_5$Ir$_3$O$_{12}$ is regarded as a coexisting state with cluster electric toroidal dipole and cluster magnetic toroidal quadrupole, the latter of which is formed by the 120$^{\circ}$-type ordering of the magnetic octupole and accompanies a small uniform magnetization as a secondary effect. Our results provide a clue to two hidden phases in Ca$_5$Ir$_3$O$_{12}$.

Cluster Toroidal Multipoles Formed by Electric-Quadrupole and Magnetic-Octupole Trimers: A Possible Scenario for Hidden Orders in Ca5Ir3O12

Abstract The spontaneous symmetry breakdown of matter is one of the most important concepts in materials physics and leads to a phase transition into an ordered phase and domain formation in its consequence. The so-called ‘ferroaxial order’ characterized by a rotational structural distortion with an axial vector symmetry has gained growing interest as a new class of ordered state. However, the observation of ferroaxial domain states, that is, clockwise and counterclockwise rotational states, is not straightforward and has been little investigated. Here, we propose that the circular intensity difference in second harmonic generation (CID-SHG) offers an experimental technique to investigate ferroaxial order and its domain states through the transition process of higher-order multipoles such as magnetic-dipole and electric-quadrupole. By using CID-SHG microscopy, we successfully visualize three-dimensional images of ferroaxial domain structures in NiTiO 3 . Our results indicate that CID-SHG is a sensitive probe of ferroaxial order and opens possibilities for the use of ferroaxial materials in nonlinear optical manipulations. 

https://www.nature.com/articles/s41535-022-00515-w.pdf

Dan Kitahara

Papers

Three-dimensional imaging of ferroaxial domains using circularly polarized second harmonic generation microscopy