Spin splitting of energy bands can be induced by relativistic spin-orbit interactions in materials without inversion symmetry. Whereas polar space-group symmetries permit Rashba (R-1) spin splitting with helical spin textures in momentum space, which could be reversed upon switching a ferroelectric polarization via applied electric fields, the ordinary Dresselhaus effect $\left(\mathrm{D}\text{−}{1}_{\mathrm{A}}\right)$ is active in materials exhibiting nonpolar noncentrosymmetric crystal classes with atoms occupying exclusively nonpolar lattice sites. Consequently, the spin-momentum locking induced by $\mathrm{D}\text{−}{1}_{\mathrm{A}}$ is not electric field switchable. Here we find a type of ferri-chiral materials with an alternative type of Dresselhaus symmetry, referred to as $\mathrm{D}\text{−}{1}_{\mathrm{B}}$, exhibiting crystal class constraints similar to $\mathrm{D}\text{−}{1}_{\mathrm{A}}$ (all dipoles add up to zero), but unlike $\mathrm{D}\text{−}{1}_{\mathrm{A}}$, at least one polar site is occupied. The spin splitting is associated with the crystalline chirality, which in principle could be reversed upon a change in chirality. Focusing on alkali metal chalcogenides, we identify ${\mathrm{NaCu}}_5{\mathrm{S}}_3$ in the nonenantiomorphic ferri-chiral structure, which exhibits ${\mathrm{CuS}}_3$ chiral units differing in the magnitude of their Cu displacements. We then synthesize ${\mathrm{NaCu}}_5{\mathrm{S}}_3$ (space group $P{6}_{3}22$) and confirm its ferri-chiral structure with powder x-ray diffraction. Our electronic structure calculations demonstrate it exhibits $\mathrm{D}\text{−}{1}_{\mathrm{B}}$ spin splitting as well as a ferri-chiral phase transition, revealing spin splitting interdependent on chirality. Our electronic structure calculations show that a few percent biaxial tensile strain can reduce (or nearly quench) the switching barrier separating the monodomain ferri-chiral $P{6}_{3}22$ states. We compute the circular dichroism absorption spectrum of the two ferri-chiral orientations and discuss what type of external stimuli might switch the chirality so as to reverse the (nonhelical) Dresselhaus $\mathrm{D}\text{−}{1}_{\mathrm{B}}$ spin texture. Our study suggests the design of ferri-chiral crystals as potential spintronic and optoelectronic materials.

• Received 14 June 2020
• Revised 26 October 2020
• Accepted 28 October 2020

DOI:https://doi.org/10.1103/PhysRevB.102.235127