Robust Zeeman-type band splitting in sliding ferroelectrics

This dataset contains all source files, including Q.E. input files, of eight bilayer ferroelectric structures, 4 compounds with two antiparallel charge polarization each, of MX<sub>2</sub> (M = Mo, W; X = S, Se) from which all figures inside the paper can be reproduced.
Transition metal dichalcogenides (TMDs) exhibit giant spin-orbit coupling (SOC), and intriguing spin-valley effects, which can be harnessed through proximity in van der Waals (vdW) heterostructures. Remarkably, in hexagonal monolayers, the Zeeman-type band splitting of valence bands, which originate from the prismatic crystal field, can reach values of several hundreds of meV, offering significant potential for both fundamental and applied research. While this effect is suppressed in the commonly studied hexagonal (H)-stacked bilayers due to the presence of inversion symmetry, the recent discovery of sliding ferroelectricity in rhombohedral (R-)stacked MX<sub>2</sub> bilayers (M=Mo, W; X=S, Se) suggest that the Zeeman effect could be present in these non-centrosymmetric configurations, making it even more intriguing to investigate how the spin-resolved bands would evolve during the phase transition. Here, we perform density functional theory calculations complemented by symmetry analysis to unveil the evolution of ferroelectricity during sliding and the behavior of Zeeman splitting along the transition path. While the evolution of the out-of-plane component of the electric polarization vector resembles the conventional ferroelectric transition, switching between positive and negative values, we observe significant in-plane components parallel to the sliding direction, reaching their maximum at the intermediate state. Moreover, we demonstrate that the R-stacked bilayers exhibit substantial Zeeman-type band splitting, akin to monolayers, which persists throughout the transition path, being allowed by the lack of inversion symmetry. Further analysis of different stacking configurations generated by sliding along various directions confirms that the Zeeman effect in MX<sub>2</sub>, primarily arising from the polarity of prismatic ligand coordination of the metal atom, is remarkably robust and completely governs the spin polarization of bands, independently of the sliding direction. This resilience promises to maintain robust spin transport in vdW heterostructures based on MX<sub>2</sub> bilayers, opening new opportunities for ferroelectric spintronics.