The different shapes of spin textures as a journey through Brillouin zone chiral and polar symmetries

TL;DR

This paper classifies spin texture patterns in crystals based on symmetry properties of the wavevector point group, revealing unexpected spin-momentum locking effects and guiding the selection of materials with desired spin properties.

Contribution

It provides a comprehensive classification of spin textures based on WPGS symmetry, including new insights into spin-momentum locking in polar WPGS without electric dipoles.

Findings

Classified spin textures according to WPGS symmetry operations.

Discovered spin-momentum locking in polar WPGS without electric dipoles.

Identified 37 materials with diverse spin textures near band edges.

Abstract

Crystallographic space group symmetry (CPGS) such as polar and nonpolar crystal classes have long been known to classify compounds that have spin-orbit-induced spin splitting. While taking a journey through the Brillouin Zone (BZ) from one k-point to another for a fixed CPGS, it is expected that the wavevector point group symmetry (WPGS) can change, and consequently a qualitative change in the texture of the spin polarization (the expectation value of spin operator $\vec{S}^{nk_{0}}$ in Bloch state $u(n,k)$ and the wavevector $k_0$). However, the nature of the spin texture (ST) change is generally unsuspected. In this work, we determine a full classification of the linear-in-$k$ spin texture patterns based on the polarity and chirality reflected in the WPGS at $k_0$. The spin-polarization vector $\vec{S}^{nk_{0}}$ controlling the ST is bound to be parallel to the rotation axis and perpendicular to the mirror planes and hence, symmetry operation types in WPGSs impose symmetry restriction to the ST. For instance, the ST is always parallel to the wavevector $k$ in non-polar chiral WPGSs since they contain only rotational symmetries. Some consequences of the ST classification based on the symmetry operations in the WPGS include the observation of ST patterns that are unexpected according to the symmetry of the crystal. For example, it is usually established that spin-momentum locking effect requires the crystal inversion symmetry breaking by an asymmetric electric potential. However, we find that polar WPGS can have this effect even in compounds without electric dipoles or external electric fields. We use the determined relation between WPGS and ST as a design principle to select compounds with multiple ST near band edges at different $k$-valleys. Based on high-throughput calculations for 1481 compounds, we find 37 previously fabricated materials with different ST near band edges.