Symmetry Guide to Ferroaxial Transitions

The 212 species of the structural phase transitions with a macroscopic symmetry breaking are inspected with respect to the occurrence of the ferroaxial order parameter, the electric toroidal moment. In total, 124 ferroaxial species are found, some of them being also fully ferroelectric (62) or fully ferroelastic ones (61). This ensures a possibility of electrical or mechanical switching of ferroaxial domains. Moreover, there are 12 ferroaxial species that are neither ferroelectric nor ferroelastic. For each species, we have also explicitly worked out a canonical form for a set of representative equilibrium property tensors of polar and axial nature in both high-symmetry and low-symmetry phases. This information was gathered into the set of 212 mutually different symbolic matrices, expressing graphically the presence of nonzero independent tensorial components and the symmetry-imposed links between them, for both phases simultaneously. Symmetry analysis reveals the ferroaxiality in several currently debated materials, such as ${\mathrm{VO}}_2$, ${\mathrm{LuFe}}_2{\mathrm{O}}_4$, and ${\mathrm{URu}}_2{\mathrm{Si}}_2$.

Figure 1

Triclinic (left panel), monoclinic (middle panel), and orthorhombic (right panel) species. Each vertex pair connected by a top-down path starting in a bold-frame rectangle corresponds to one species. Note that a species is identified by the top and bottom groups so that several paths may give the same species. Numbers in ovals give the order of the point group.

Figure 2

Subordination of tetragonal and cubic species. Each top-down connected path starting in one of the bold-frame vertices corresponds to one of the tetragonal (left panel) or cubic (right panel) species. Note that several paths may give the same species. The insets clarify the meaning of the $m_{|}$, $m_{-}$, $m_{\backslash }$, and $m_{+}$ symbols introduced in the text. Full, dashed, and dotted lines indicate group-subgroup relations of index 2, 3, and 4, respectively. Numbers in ovals give the order of the point group.

Figure 3

List of 212 nonmagnetic species $G>F$. “No.,” $G$, and $F$ stand for the serial number of species and the symmetry class of the high-symmetry and low-symmetry phases. A total number of the orientational domain states appears in column $n$. Full, semiopen, and open circles in column $a$ stand for full ferroelectric, partial ferroelectric, and nonferroelectric transitions. Corresponding symbols in column $b$ and $c$ refer to ferroelasticity and ferroaxiality.

Figure 4

Form of the canonical property tensors. (a) Layout of the extended Nye’s matrix. Original entries of the Nye’s matrix [24], labeled by Jahn’s symbols [25] $V,[V^2]$, $V[V^2]$, and $[V^2{]}^2$, stand for polar tensors with the internal symmetry of the pyroelectricity, permittivity, piezoelectricity, and elastic tensors, respectively. Added entries, labeled by Jahn’s symbols $ϵ$, $ϵV$, $ϵ[V^2]$, and $ϵV[V^2]$, stand for axial tensors with the symmetry of pseudoscalar, pyroaxiality, optical activity (symmetric gyration), and electrogyration tensor, respectively. Tensors marked with a $\star$ are contracted as in Refs. [24, 26]. (b) Example of dichromatic matrix for species No. 151. Red (black) symbols refer to the components nonzero in the low-symmetry phase (in both phases). Lines are connecting equal components.