Robust Zeeman-type band splitting in sliding ferroelectrics

Homayoun Jafari, Evgenii Barts, Przemysław Przybysz, Karma Tenzin, Paweł J. Kowalczyk, Paweł Dabrowski, and Jagoda Sławińska

Phys. Rev. Materials 8, 024005 – Published 28 February 2024

Abstract

Transition metal dichalcogenides are known for their intriguing spin-valley effects, which can be harnessed through proximity in van der Waals heterostructures. Their hexagonal monolayers exhibit significant Zeeman band splitting of valence bands, reaching up to several hundred meV and giving rise to spin textures that yield long spin lifetimes. However, this effect is suppressed in the hexagonal bilayers due to inversion symmetry. The recent discovery of sliding ferroelectricity in ${M X}_{2}$ bilayers ( $M = Mo$ , W; $X = S$ , Se) opens a pathway for the electrical control of the spin properties in these materials. While substantial Zeeman splitting is found in the rhombohedral structure, its behavior during the ferroelectric transition remains unclear. In this paper, we explore different stacking configurations of ${M X}_{2}$ bilayers by sliding and demonstrate, using density functional theory calculations and symmetry analysis, that the Zeeman effect completely dominates the spin polarization of bands throughout the structural transition. This promises to maintain robust spin transport in polar ${M X}_{2}$ bilayers, opening potential avenues for ferroelectric spintronics.

Received 10 November 2023
Accepted 29 January 2024

DOI:https://doi.org/10.1103/PhysRevMaterials.8.024005

Physics Subject Headings (PhySH)

Electric polarization Spin relaxation Spintronics

2-dimensional systems Ferroelectrics

Density functional theory

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Homayoun Jafari ^1,*, Evgenii Barts ^1,*, Przemysław Przybysz ^1,2,*, Karma Tenzin ^1,3, Paweł J. Kowalczyk ², Paweł Dabrowski², and Jagoda Sławińska ^1,†

¹Zernike Institute for Advanced Materials, University of Groningen, Nijenborgh 4, 9747 AG Groningen, The Netherlands
²Department of Solid State Physics, Faculty of Physics and Applied Informatics, University of Łódź, Pomorska 149/153, 90-236 Łódź, Poland
³Department of Physical Science, Sherubtse College, Royal University of Bhutan, 42007 Kanglung, Trashigang, Bhutan

^*These authors contributed equally to this work.
^†Corresponding author: jagoda.slawinska@rug.nl

Article Text (Subscription Required)

Click to Expand

Supplemental Material (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand

Issue

Vol. 8, Iss. 2 — February 2024

Reuse & Permissions

Access Options

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
(a) Crystal structure of an R-stacked ${M X}_{2}$ bilayer with polarization upwards ( $P_{↑}$ ) for M stacking and downwards ( $P_{↓}$ ) for N stacking. Top and side views are shown in the top and bottom panels, respectively. Black dashed lines denote the hexagonal unit cell. The red arrows show the three equivalent sliding directions. The interlayer distance $d$ is indicated by the black arrow. (b) Out-of-plane electric polarization $P_{z}$ versus lateral sliding between the M and N configurations calculated for all the considered compounds. The sliding path is along the [120] crystallographic direction ( $y$ axis). ( $b^{'}$ ) The same for the in-plane component $P_{y}$ ; $P_{x} = 0$ from symmetry. (c) Schematic view of different stacking configurations. The bottom layer is represented by the semitransparent honeycomb, and the different positions of the top layer are marked by single hexagons.
Reuse & Permissions
Figure 2
(a)–(d) Band structures of M and N stacking configurations calculated for different ${M X}_{2}$ . The inset shows the Brillouin zone (BZ) and the high-symmetry lines. The arrows indicate the band parameters: indirect energy gaps $E_{g}$ and layer splitting at the $Γ$ point $Δ_{layer}^{Γ}$ . (e) A zoom of the electronic structure around the $K$ point marked by the green rectangle in (c). The colors denote out-of-plane spin polarization originating from the Zeeman interaction, and the solid and dashed lines represent the top and bottom layers, respectively. The spin splitting ( $Δ_{spin}^{K}$ ) and layer splitting ( $Δ_{layer}^{K}$ ) parameters are indicated by the arrows. (f) A zoom of the electronic structure around the $Γ$ point marked by the orange rectangle in (c). $E_{K Γ}$ denotes the offset between the valence band edge at the $K$ and $Γ$ points. (g) A zoom of the electronic structure around the $Γ$ point marked by the yellow rectangle in (d). The colored lines represent the spin texture perpendicular to the momentum, indicating a purely Rashba spin texture with the Rashba coefficient defined as $α_{R} = 2 E_{R} / k_{R}$ . The numerical values of all the parameters are listed in Table 1.
Reuse & Permissions
Figure 3
(a) and (b) Spin-resolved band structure of N-stacked ${WSe}_{2}$ . (a) shows the $S_{x}$ component of the spin texture; $S_{y}$ is omitted, as it emerges only along the $Γ - K$ line close to the BZ center. (b) shows the out-of-plane component $S_{z}$ . (c) $S_{z}$ component calculated for the topmost valence band in the entire BZ. (d) Illustration of trigonal prismatic ligand coordination in hexagonal ${M X}_{2}$ ; top and side views are displayed in the top and bottom panels. The dashed lines represent the connection between the ligands and metal atoms located in different layers. (e) Orbital-resolved band structure of ${WSe}_{2}$ along $K − Γ − K^{'}$ . For the sake of completeness, the layer-resolved bands are additionally shown in Fig. S1 in the Supplemental Material [30].
Reuse & Permissions
Figure 4
(a) Energy of the topmost valence band calculated over the entire BZ for the intermediate state (A) of ${WSe}_{2}$ . The bright spots at the corners of the BZ represent the valence band maximum. The labels indicate the high-symmetry points. The energy splitting and spin polarization of this band are shown in (b) and (c). The dashed line represents the $[\bar{1} 10]$ crystallographic direction defined in the reciprocal lattice. In (c), only the $S_{z}$ component of the spin texture is shown, as $S_{x}$ and $S_{y}$ are nearly negligible and are present only in the close vicinity of $Γ$ . Analogous data are reported for the rest of considered materials in Fig. S2 in the Supplemental Material [30]. ( $a^{'}$ )–( $c^{'}$ ) The same as (a)–(c), but calculated for the stacking configuration between the N and A points.
Reuse & Permissions

Physical Review Materials