Fluctuating spin-orbital texture of Rashba-split surface states in real and reciprocal space

Spin-orbit interaction in low-dimensional systems, namely, Rashba systems and the edge states of topological materials, has been extensively studied in this decade as a promising source to realize various fascinating spintronic phenomena, such as the source of the spin current and spin-mediated energy conversion. Here, we show the odd fluctuation in the spin-orbital texture in a surface Rashba system on Bi/InAs(110)-($2\times 1$) by spin- and angle-resolved photoelectron spectroscopy and a numerical simulation based on a density-functional theory (DFT) calculation. The surface state shows a paired parabolic dispersion with the spin degeneracy lifted by the Rashba effect. Although its spin polarization should be fixed in a particular direction based on the Rashba model, the observed spin polarization varies greatly and even reverses its sign depending on the wave number. DFT calculations also reveal that the spin directions of two inequivalent Bi chains on the surface change from nearly parallel (canted parallel) to antiparallel in real space in the corresponding wave vector region. These results point out an oversimplification of the nature of spin in Rashba and Dirac systems and provide more freedom than expected for spin manipulation of photoelectrons.

Figure 1

Schematic of conventional Rashba-type spin-split bands and surface atomic structure of Bi/InAs(110)-($2\times 1$). (a) Ideal two-dimensional spin-split electronic bands due to Rashba-type SOI. The spin polarization direction is perpendicular to the wave vector $\vec{k}$, exhibiting a helical spin polarization in reciprocal space. (b) Surface atomic structure of Bi/InAs(110)-($2\times 1$). Tilted atomic chains of Bi on the InAs(110) substrate form the ($2\times 1$) surface superlattice, which consists of four nonequivalent Bi atoms, Bi1, Bi2, Bi3, and Bi4, as indicated. The thin solid rectangle is the ($2\times 1$) surface unit cell, and the dashed line is the mirror plane, which is the unique symmetry operation in this surface atomic structure.

Figure 2

Spin-polarized surface band structures measured by SARPES. (a) and (b) ARPES and (c) and (d) SARPES 2D plots measured with $s$-polarized (top) and $p$-polarized (bottom) photons along $\overline{\mathrm{\Gamma }}\text{−}\overline{X}$ (parallel to the Bi chains). (a) and (b) Spin-integrated ARPES intensity maps. The photon incident angle ${\theta }_i$ is ${50}^{\circ }$. SARPES maps polarized to (c1) and (d1) $S_x$, (c2) and (d2) $S_y$, and (c3) and (d3) $S_z$. The spin orientations are defined in (f). ${\theta }_i$ is ${38}^{\circ }$. (e) Experimental geometry of the SARPES measurements and definitions of the coordinates. The ($2\times 1$) surface Brillouin zone and the common plane of the photon incidence and photoelectron detection are superposed simultaneously. (f) Definitions of the spin directions of the photoelectrons. The colors of the SARPES intensity plots are determined based on the photoelectron intensity and spin polarization, as indicated in these 2D color maps.

Figure 3

Calculated spin-polarized band structure and spin texture in real space. (a)–(d2) Spin-polarized band dispersions obtained by DFT calculations. The radii of the circles are proportional to the sum of the contributions from all surface Bi atoms. Spin orientations are defined as (a), (c1), and (c2) in-plane ($S_x$) and (b), (d1), and (d2) out-of-plane ($S_z$) ones. (c1)–(d2) Surface bands with the spin orientations of each nonequivalent surface Bi chain. (e) Color scales represent spin polarizations: The blue-red (green-purple) color palette corresponds to the spins along the in-plane orientation $S_x$ (out-of-plane orientation $S_z$). (f) Wave number dependence of the spin orientations in each Bi chain. (g) Definition of the spin rotation angle used in (f). (h) Typical cases of the spin orientations depending on $k_{y//\left[\overline{1}10\right]}$ in each Bi chain. Roman numbers correspond to the $k_{y//\left[\overline{1}10\right]}$ position shown by arrows in (f).

Figure 4

Spin polarizations of photoelectrons decomposed based on the symmetries. (a1)–(b2) Calculated band dispersions shown in a manner similar to Fig. 3, but the radii of the circles are obtained by taking the Bi $6p$ ($p_x$, $p_y$, and $p_z$) atomic orbitals, which have (a1) and (b1) odd and (a2) and (b2) even symmetries with respect to the photon-incident plane in the SARPES experimental geometry. The colors of the circles are from the corresponding Bi $6p$ orbitals with $S_x$ and $S_z$ polarizations. (c) Schematics of the Bi $6p$ orbitals superposed on the coordinate axes.

Figure 5

Spin polarizations of the photoelectrons from the computational and experimental analysis. (a1) and (a2) Three-dimensional spin polarizations of the initial state obtained from the DFT calculation with (a1) $p_x$ (odd) and (a2) $p_y+p_z$ (even) Bi orbitals of the $S$ band. (b1) and (b2) Photoelectron spin polarizations with (b1) $p_x$ and (b2) $p_y+p_z$ Bi orbitals calculated numerically by considering the photoelectron interference of two nonequivalent Bi chains. The phase difference of the wave function between the Bi1-2 and Bi3-4 chains was set to $2\pi /3$. See the Supplemental Material for details of the calculation [15]. (c) ARPES band mapping with the $s$-polarized photons. Square rectangles represent the regions where the spin polarizations in (d1) and (d2) are obtained. (d1) and (d2) Experimentally observed spin polarizations measured with (d1) $s$- and (d2) $p$-polarized photons.