Different shapes of spin textures as a journey through the Brillouin zone

Abstract

Crystallographic space group symmetry (CPGS) such as polar and nonpolar crystal classes has long been known to classify compounds that have spin--orbit--induced spin splitting such as Rashba and Dresselhaus compounds, respectively. However, the enabling symmetries underlying the spin texture prototypes ---the relationship between the expectation value of spin operator 𝑆 !" in Bloch state 𝑢(𝑛, 𝑘) and the wavevector 𝑘 - are instead specific rotations and reflection symmetries in the wavevector point group symmetry rather than the presence or absence of polar field in the global crystallographic space group. Relativistic band structures calculations demonstrate how compounds judged by their macroscopic CPGS to be typical Dresselhaus (GaAs) or Rashba (GeTe) or Wyle (Tellurium) compounds, manifest both Rashba and Dresselhaus spin textures, respectively at different wavevector chiralities and polarities.

Long-range current-induced spin accumulation in chiral crystals

“…Nevertheless, the linear-in-k spin arrangement at the high-symmetry points can be predicted with the help of the group theory and further verified by the DFT calculations. Following the study by Mera Acosta et al, 26 we note that the crystallographic point group D 6 describing TaSi 2 may yield the purely radial Weyl spin texture at the k-vectors possessing the point group symmetry D 3 or D 6 . Using the Bilbao Crystallographic Server, 47,48 we determined little point groups of all the high-symmetry points (Γ, A, K, H and M ) and we found that most of them can be the 'source' or 'sink' of the ra-dial spin texture.…”

“…25 In order to establish a link between CISS and the unconventional spin accumulation, quantitative studies of different chiral crystals are required, including those with complex spin textures, where different types of SOC might be simultaneously present in the momentum space. 26 Before starting the analysis of specific materials, let us briefly overview the computational approach allowing the calculation of the spin accumulation for arbitrary SOC. The equilibrium electron distribution in a crystal can be described by the Fermi distribution function f 0 k .…”

High throughput inverse design and Bayesian optimization of functionalities: spin splitting in two-dimensional compounds

“…These SS prototypes are also characterized by different spin textures, i.e., the SP pattern in the reciprocal space (Figure 1). 29 We propose and implement a two-step automatic design of material properties: i) target property prediction based on the…”

“…In addition to the common DPs previously defined, the Rashba SS in 2D compounds also requires a non-zero electric dipole (allowed by at least one polar atomic site and local dipoles that add up to non-zero, as described by 35). The SS type depends on the wave vector point group 29 and hence, the DPs also include the k-point preserving the time-reversal symmetry, as well as a linear-in-k shift. The unique DPs for the Rashba SS (i.e., non-zero electric dipoles, time-reversal symmetry, and linear-in-k SS) are illustrated in the purple quadrant of Figure 2.…”

“…Note that a given compound can present more than one SS prototype at different k-points, as discussed by Acosta at al. 29 , and thus the SS characterization should identify all SS types in a given compound. As both the Rashba and Dresselhaus SS prototypes share the common characteristic of having linear-in-k Hamiltonians, it will be useful to label them as a Linear SS (LSS) when implementing the SS identification algorithm, as described below, while not losing the symmetry criteria that differentiate them.…”