Investigation of the intrinsic hidden spin texture and spin-state segregation in centrosymmetric monolayer dichalcogenide: effectiveness of the electric-field approach

TL;DR

This work addressses hidden spin polarization in centrosymmetric two-dimensional dichalcogenides, using PtTe$_2$ monolayer as a prototype. It introduces a perpendicular electric-field approach, $E_{app}$, to lift degeneracy and directly reveal intrinsic hidden spin textures and the spatial distribution of spin-separated states from density functional theory. The study shows that the upper valence bands split into spin-polarized branches $ ext{α}_{ ext{up/dw}}$ and $ ext{β}_{ ext{up/dw}}$, with layer-resolved probability and magnetization densities that manifest spin-layer locking and symmetry-governed textures across the Brillouin zone, including near $ ext{Γ}$ and at $ ext{K}$/$ ext{K}'$. It also elucidates the competition between bonding, Rashba, and non-magnetic Zeeman effects in determining the degree of spin-state segregation and demonstrates that the method yields symmetry-based predictions for spin textures, offering a practical first-principles toolkit for studying hidden spin polarization in centrosymmetric materials with potential spintronic applications.

Abstract

The emergence of hidden spin polarization in centrosymmetric nonmagnetic crystals due to local symmetry breaking has created new opportunities for potential spintronic applications and for enhancing our understanding of mechanisms to electrically manipulate spin-related phenomena. In this work, we investigate within density functional theory the properties of the hidden spin texture and spin-layer segregation in a prototype centrosymmetric dichalcogenide-monolayer material using an electric-field-based method. This method is shown to yield a precise and robust alternative to traditional layer-projected spin-polarization techniques for obtaining the intrinsic hidden spin textures in such materials. Moreover, it gives access at the same time to the spatial distribution within the monolayer of the individual spin-segregated states responsible for the hidden spin textures, not provided by other techniques. With this approach we determine and study the hidden spin textures of the upper valence bands of the PtTe2 monolayer together with the spatial behavior of the probability densities and spin polarization densities of the corresponding maximally segregated spin states. This combined study enabled by the electric-field method yields new insights into the mechanisms controlling the spin-layer segregation and resulting hidden spin texture in such systems. We also discuss the symmetry rules governing the shape in the Brillouin zone of the hidden spin texture, which can be straightforwardly predicted within the present framework.

Figures (9)

• Figure 1: (a) Top and (b) side view of the $1T$-$PtTe_2$ monolayer structure, with lattice vectors a and b indicated in the figure. (c) The first Brillouin zone of the monolayer $PtTe_2$ , with indicated in red the $k$-points considered in this study to investigate the spatial behavior of the segregated spin states. These points include the four high symmetry points marked with red circles: $\Gamma$, K, M and K', and the three $k$-points, $k_1$, $k_2$, and $k_3$, marked with red crosses.
• Figure 2: Planar averaged plots of the total electrostatic potential along the $\hat{z}$ direction, including the local component of the ionic pseudopotential ($V_{ions}$), Hartree potential ($V_{H}$), and sawtooth potential ($V_{saw}$) for an applied electric field, $E_{app}=$ 0.05 V/Å. The gray hatched region indicates the part subjected to the electric field. The orange and cyan circles show the positions of the Te and Pt atoms, respectively, along the $\hat{z}$ direction. The red arrow represents the slope corresponding to the electric field.
• Figure 3: The band structure of $PtTe_2$ monolayer with SOC under an electric field of 0.2 V/Å. The first and second upper valence bands, indicated by a solid red and a dashed blue line, are labeled as $\alpha_{up}$ and $\alpha_{dw}$, respectively. The third and fourth upper valence bands, shown by a solid purple and green dashed line, are denoted as $\beta_{up}$ and $\beta_{dw}$, respectively. The inset shows a magnified view of the bands within a smaller energy range, outlined by the solid black box, highlighting the splitting of the $\alpha_{up}$ and $\alpha_{dw}$ bands.
• Figure 4: Planar average of the probability density (in ${Bohr}^{-3}$) for the electronic states of the $\alpha$ (panels (a)-(b)) and $\beta$ (panels (c)-(d)) valence bands of the $PtTe_2$ monolayer at the point $k_{1}~(0.167,0,0)2\pi/a$ in the presence of different electric fields, (a) and (c) $E_{app}=$ 0.05 V/Å, (b) and (d) $E_{app}=$ 0.1 V/Å. The orange and cyan circles show the positions of the Te and Pt atoms, respectively, along the z direction.
• Figure 5: Spin texture of the $\alpha$ and $\beta$ valence bands of the $PtTe_2$ monolayer in the presence of a small electric field ($E_{app}=$ 0.05 V/Å). The in-plane spin components $(S_{x}, S_{y})$ are shown by black arrows at the $k$-grid points in the 2D BZ, while the out-of-plane spin texture component is represented by the map of $S_{z}$ isovalues. The length of the arrows represents the magnitude of the in-plane spin components, with the scale provided in the bottom-right corner of the plot.