Properties of electronic axiality in bulk crystals

Abstract Axial materials, with an axial vector as their order parameter, represent a novel class of ferroic materials with the potential for unique functionalities due to their intrinsic asymmetry. We recently proposed the order parameter in axial materials, namely axiality, based on relativistic quantum theory. The axiality is defined by the spatial integral of the axial vector over the unit cell, which depends on the choice of the unit cell and its origin. In this study, we examine these dependencies on axiality for the axial crystal K 2 Zr(PO 4 ) 2 . We investigate two unit cell choices: the Wigner–Seitz cell and the simple unit cell spanned by the primitive lattice vectors a 1 , a 2 , a 3 . We find that in the simple unit cell, the x - and y -components of axiality are finite and comparable in magnitude to the z -component. In contrast, in the Wigner–Seitz cell, the x - and y -componentsof axiality become zero due to the three-fold rotational symmetry. While the x - and y -components of axiality exhibit distinct behaviors in these cells, we demonstrate that the z -component of axiality remains identical in both cells.