# Fluctuated spin–orbital texture of Rashba-split surface states in real and
reciprocal space

Takuto Nakamura<EMAIL_ADDRESS>Graduate School of Frontier
Biosciences, Osaka University, Suita 565-0871, Japan  Department of Physics,
Graduate School of Science, Osaka University, Toyonaka 560-0043, Japan
Yoshiyuki Ohtsubo<EMAIL_ADDRESS>Institute for Advances Synchrotron Light
Source, National Institutes for Quantum Science and Technology, Sendai
980-8579, Japan  Graduate School of Frontier Biosciences, Osaka University,
Suita 565-0871, Japan  Department of Physics, Graduate School of Science,
Osaka University, Toyonaka 560-0043, Japan  Ayumi Harasawa  Institute for
Solid State Physics, The University of Tokyo, Kashiwa 277-8581, Japan
Koichiro Yaji  National Institute for Materials Science (NIMS), Tsukuba,
Ibaraki 305-0003, Japan  Shik Shin  Fumio Komori  Institute for Solid State
Physics, The University of Tokyo, Kashiwa 277-8581, Japan  Shin-ichi Kimura
<EMAIL_ADDRESS>Graduate School of Frontier Biosciences, Osaka
University, Suita 565-0871, Japan  Department of Physics, Graduate School of
Science, Osaka University, Toyonaka 560-0043, Japan  Institute for Molecular
Science, Okazaki 444-8585, Japan

###### Abstract

Spin-orbit interaction (SOI) in low-dimensional systems, namely Rashba systems
and the edge states of topological materials, is 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 wavenumber. DFT calculations also reveal
that the spin directions of two inequivalent Bi chains on the surface change
from nearly parallel (canted-parallel) to anti-parallel in real space in the
corresponding wavevector 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.

The symmetry breaking of a system often causes fertile unconventional physical
phenomena, such as nematic superconductivity appearing with rotational
asymmetry [1] and orbital-angular-momentum (OAM) polarization of graphene
Dirac-cone states caused by the asymmetry between sublattices of honeycomb
lattice (pseudospin) [2]. In particular, spin–orbit interaction (SOI) in two-
dimensional (2D) surface systems, namely, Rashba effect [3] and edge states of
topological materials [4], is a promising source to host, create, or convert
spin-polarized electrons without magnetic materials and fields [5, 6].

A schematic of SOI-caused helical spin polarization in a 2D system is depicted
in Fig. 1(a), where surface electrons are polarized toward the orientation
perpendicular to both the wavevector and the surface normal. In this model,
the potential gradient at the surface is assumed to be parallel to the surface
normal. An electron moving in such a potential gradient experiences an
effective magnetic field $\vec{B}_{eff}\propto\vec{E}\times\vec{k}$ resulting
in Zeeman-like splitting and spin polarizations parallel and anti-parallel to
$\vec{B}_{eff}$ [3]. This can be regarded as the outcome of the symmetry
breaking in the 2D system along the out-of-plane orientation. The in-plane
asymmetry also contributes to the surface spin polarization, although it is
not included in the simple Rashba model considered above. For example, out-of-
plane spin polarization in triangular surface lattices, i.e., valley
polarization, is caused by the lack of in-plane two-fold rotation symmetry [7,
8, 9].

In the simple Rashba model, one spin-split branch of the surface band always
holds one spin orientation: clockwise (CW) or counter CW (CCW) in the
reciprocal space. This assumption was consistent with various SARPES data
reported by early 2010s except for few cases with valley polarization [7, 10]
and is widely used in the interpretation of surface-spin-mediated energy
conversion experiments [6]. However, recent studies, by using the combination
of the spin- and angle-resolved photoelectron spectroscopy (SARPES) and
density-functional theory (DFT) calculation, revealed the contribution of
complex combinations of spin and orbital terms in the wavefunctions to 2D
spin-polarized surface states, even in a single surface band [11, 12, 13].
These studies were mainly performed by using high-symmetry surface systems,
such as $C_{3v}$ symmetry with three-fold rotation and three mirror planes,
implying that more variants of spin and orbital textures would appear in
lower-symmetry systems. Furthermore, the minimal features of Rashba-type SOI
in the surface electronic structure should be distinguished from coincidental
coexisting features realized by surface symmetry operations, which do not
always exist in general surface systems.

In this work, we studied the spin and orbital texture in the surface
electronic states of Bi/InAs(110)-(2$\times$1) by SARPES and a numerical
simulation based on DFT calculations. The (2$\times$1) superlattice of Bi on
InAs(110) consists of two non-equivalent and tilted Bi chains, as depicted in
Fig. 1(b) [14]. This surface structure holds a small number of surface
symmetry operations, only one mirror plane, and no rotation, and thus, is one
of the ideal test case to study the minimal feature of surface SOI. Actually,
the surface state exhibited an unconventional spin fluctuation for the first
time. The sign on the surface spin polarization modulated independently in the
two nonequivalent surface atomic chains and even in a single band, its sign
inverted depending on the wavenumber. These results point out an
oversimplification of the nature of spin in surface 2D systems, and suggest
more freedom than expected for spin manipulation of photoelectrons.

## Results

### Photoelectron spectra of spin-polarized surface electronic states

Figures 2(a) and 2(b) show conventional (spin-integrated) ARPES band
dispersions along $\bar{\Gamma}$ – $\bar{\rm X}$ using s\- and p-polarized
photons, respectively. The experimental geometry and definitions of the
incident-photon polarizations of the (S)ARPES experiment are shown in Fig.
2(e). As shown in a previous work [15], surface bands (S and S’) were observed
below the Fermi level ($E_{\rm F}$) with paired parabolic dispersion around
the surface Brillouin zone center ($k_{y//[\bar{1}10]}$ = 0 Å-1). The
photoelectron intensity of S with s-polarization was diminished around
$k_{y//[\bar{1}10]}$ = 0 Å-1, suggesting an influence of the photoexcitation
selection rule [16]. Assuming the nearly-free-electron final state,
photoelectrons from the initial states with odd and even parities with respect
to the measurement plane (see Fig. 2(e)) should be excited by s\- and
p-polarized photons, respectively. This suggests that the inner part of the
parabolic S band is mainly composed of wavefunctions with even symmetry,
whereas the outer part is from multiple wavefunctions with both even and odd
symmetries.

Figures 2(c1–c3) and 2(d1–d3) show polarization-dependent SARPES map obtained
by using s\- and p-polarized photons, respectively. These are two-dimensional
color plots of photoelectron intensity and magnitude of spin polarization (see
Figs. 2(f1–f3) for their definitions). The corresponding spin-resolved energy
distribution curves are also shown in supplementary Fig. S2 [17]. The band
dispersion obtained by SARPES intensities shown in Figs. 2(c1–d3) agrees well
with the spin-integrated ones in Figs. 2(a, b). Figures 2(c1, d1) show the
spin polarization along the $S_{x}$ direction. This orientation is in-plane
and perpendicular to the wavevector, which is expected from the simple 2D
model as depicted in Fig. 1(a). However, in contrast to the simple model, Fig.
2(d1) shows that the $S_{x}$ spin polarization direction varies in a single
parabolic band (S) with the same sign of $k_{y//[\bar{1}10]}$. Such a peculiar
spin modulation is not observed in Fig. 2(c1), indicating the effect of
incident photon polarizations (s and p). Thus far, it has been reported in
some topological materials that to switch the photon polarization could change
the spin polarization of photoelectrons [11, 13]. However, the spin modulation
in a single band with the same polarization, which is observed herein with p
polarization, has not been reported yet, to the best of our knowledge. In
contrast to S, another spin-polarized band S’ shows uni-directional
polarization in the entire $k_{y//[\bar{1}10]}$ range. Therefore, we focus on
the spin polarization of S in the following part, in order to investigate this
peculiar behavior.

In addition to $S_{x}$, spin polarizations of the other spin directions
$S_{y}$ and $S_{z}$ were observed in Figs. 2(c2–c3) and 2(d2–d3). $S_{z}$ is
almost zero with s-polarized photons but has a finite value with p-polarized
photons. Although the $S_{z}$ polarization normal to the surface is not
expected from the simplest model, as shown in Fig. 1(a), this polarization is
allowed with an in-plane potential gradient, as reported in some 2D systems
without two-fold rotation [7, 8, 9]. However, the peculiar spin reversal as
that of $S_{x}$ in the single band (Fig. 2(d3)) has not been reported so far.
Moreover, in this measurement, an evident spin polarization parallel to the
wavevector ($S_{y}$) is also observed with both photon polarizations, as shown
in Figs. 2(c2, d2). The spin polarization along this orientation is not
expected from the Rashba model because the effective magnetic field originates
from the cross product $\vec{E}\times\vec{k}$, which is always orthogonal to
$\vec{k}$. The Bi/InAs(110)-(2$\times$1) surface does not contain any magnetic
element, and hence, the observed complex spin polarization cannot be explained
solely from the initial states. Thus, it is suggested that the the photo-
excitation process is playing some role for the observed complex spin texture.

### Spin polarizations of the ground states obtained from theoretical
calculations

To reveal the initial-state spin texture, we calculated the surface band
structures by DFT, as shown in Fig. 3. The sizes of the circles in Figs. 3(a)
and 3(b) indicate the contribution of surface Bi atoms. Color scales (defined
in Fig. 3(e)) show the degree of the spin polarizations along $S_{x}$ and
$S_{z}$. The shapes of the surface bands (S and S’) are qualitatively
consistent with the results observed by ARPES and SARPES. All the spin-
polarized structures are reversed with the sign of the wavenumber, indicating
that the electronic structure obeys time-reversal symmetry. It should be noted
that $S_{y}$ spin polarization was not obtained in the DFT calculation, as
expected from the time-reversal symmetry of the system confirmed above.

To obtain further insight into the surface spin texture, we decomposed the
contributions to the surface bands from two non-equivalent surface Bi chains:
Bi1-2 and Bi3-4 (see Fig. 1(b) for the definition). The radii of the circles
in Figs. 3(c1–d2) show that both Bi chains contribute to surface band S with
nearly the same size. The in-plane spin polarization $S_{x}$ for chains Bi1-2
and Bi3-4 are shown in Figs. 3(c1) and 3(c2), respectively. Although they
point [00$\bar{1}$] in most of the region of positive $k_{y}$, such spin
polarization decreases near $\bar{\Gamma}$ and even changes the direction
(pointing [001]) for Bi3-4. On the contrary, the out-of-plane spin
polarizations $S_{z}$ of Bi chains are opposite to each other, as shown in
Figs. 3(d1–d2). This indicates that each Bi chain has independent spin
polarization, even at the same (E, $\vec{k}$) points. In contrast to $S_{x}$,
the $S_{z}$ spin polarization is large around $\bar{\Gamma}$ and decreases in
$k_{y}>$ 0.2 Å-1. This behavior of $S_{x}$ and $S_{z}$ indicates the rotation
of the spin polarization of S depending on $k_{y}$.

To visualize the spin texture in real space, the spin rotation angle depending
on the wavenumber was calculated for the S band. The results are shown in Fig.
3(f). The spin rotation angle is defined as $S_{z}$ = +1 to be the origin (0∘)
and $S_{x}$ = +1 to be 90∘. Interestingly, the spin rotation angle of the S
band varies continuously in the xz-plane depending on the wavenumber, and the
sign of the rotation is opposite between Bi1-2 and Bi3-4 in the region from
the $\bar{\Gamma}$ point ($k_{y//[\bar{1}10]}$ = 0.0 Å-1) to near the top of
the parabola ($k_{y//[\bar{1}10]}\sim$ 0.2 Å-1). This indicates that the sign
reversal of the $S_{x}$ polarization in Bi3-4 is the outcome of such a
continuous spin rotation in the xz-plane. Figure 3(h) depicts the spin
orientation of each Bi chain in real space at some specific wavenumbers
indicated in Fig. 3(f). Near $\bar{\Gamma}$ (area (I)), the spin orientation
in each chain is completely different. In area (II), slightly closer to the
apex of the parabolic band, the polarization direction is anti-parallel,
showing anti-ferromagnetic spin order. Near the apex of the parabola (area
III), the spins of the two Bi chains become nearly parallel with a small
$S_{z}$ component. In this $k_{y}$ region, the surface spin texture is similar
to the spin structure expected for a normal Rashba-type spin texture, namely
the “ferromagnetic” case, as compared to region (II). Wavenumber-dependent
spin rotation in the Rashba system has already been reported [7], but such
fertile spin orders, from the anti-ferromagnetic to nearly ferromagnetic cases
between two non-equivalent atomic chains, are not expected from ordinary
Rashba systems.

### Comparison between the observed photoelectron spectra and the calculated
ground states

Here, we compare the calculated spin texture of the initial surface states
with the experimentally observed spin-polarized photoelectron spectra. For
this purpose, we decomposed the calculated Bi orbital wavefunctions to those
odd ($p_{x}$) and even ($p_{y}$ and $p_{z}$) with respect to the measurement
plane, as shown in Figs. 4(a1–b2). The odd and even initial states should
correspond to the photoelectrons excited by s\- and p-polarized photons shown
in Fig. 2, respectively, based on the selection rule of the electric dipole
photo-excitation. The radii of the circles in Figs. 4(a1–b2) are proportional
to the surface Bi wavefunctions with odd or even parities. Odd (even) Bi
wavefunctions shown in Figs. 4(a1, b1) (4(a2, b2)) have a large contribution
around (away from) $\bar{\Gamma}$ for surface band S. This trend is consistent
with the small photoelectron intensity around $\bar{\Gamma}$ by s-polarized
photons (Fig. 2(a)). As for the spin polarization, the in-plane orientation
($S_{x}$) of S could reproduce the observed results. Although the spins of the
odd wavefunctions (Fig. 4(a1)) are intense and monotonic, those of the even
wavefunctions (Fig. 4(a2)) show the sign inversion from 0 Å-1 ($\bar{\Gamma}$)
to 0.2 Å-1 (near the top of the parabolic band) in surface band S. In contrast
to the in-plane case, the situation of the out-of-plane spins ($S_{z}$, shown
in Figs. 4(d1, d2)) is complicated. Although nearly negligible values for the
odd wavefunctions are consistent with the observed results (Figs. 2(c3)), the
observed inversion (Figs. 2(d3)) is not reproduced for the even wavefunctions.
Together with the exactly zero polarization parallel to $k_{y}$ ($S_{y}$) in
the calculation, it is obvious that the initial state calculation is
insufficient for understanding the observed spin polarization of
photoelectrons.

The photoelectron interference effect in the spin-polarized states has been
already discussed for topological materials Bi and Bi2Se3 [18, 11, 12, 13]. If
two or more wavefunctions with different spin polarizations are photo-excited
simultaneously, the interference between them can result in the unique spin
polarization of final-state photoelectrons. As shown in Fig. 3(f),
Bi/InAs(110)-(2$\times$1) has two non-equivalent Bi chains with different spin
orientations at each $k_{y}$. We adopted these two Bi chains as the different
initial states and calculated their interference numerically during the spin-
dependent photo-excitation process in a similar manner to earlier works [18,
13] (see supplementary Note 1 for details). Figures 5(a1) and 5(a2) are the
spin polarization of the initial states of the S bands obtained from our DFT
calculation with the odd and even symmetries, respectively. Figure 5(b1)
(5(b2)) depicts the simulated photoelectron spin polarizations corresponding
to the initial states shown in Fig. 5(a1) (5(a2)). Figures 5(d1) and 5(d2) are
the corresponding spin polarizations of the observed photoelectrons. For the
odd states (Figs. 5(a1) and 5(d1)), monotonic polarizations with sizable
$S_{x}$ and small $S_{z}$ of the calculation agree well with the experimental
results. A decrease in $S_{x}$ in the vicinity of $\bar{\Gamma}$ that appears
in the experimental result does not appear in the calculation. The possible
reason is that the experimental value is not so reliable owing to the
vanishing photoelectron intensities. For the even states (Figs. 5(a2) and
5(d2)), the calculated directions of $S_{x}$ and $S_{z}$ are consistent with
the corresponding experimental ones. The inversion of $S_{z}$ is absent in the
initial states (Fig. 5(a2)) but is propagated from $S_{x}$ by the
photoelectron interference. It should be noted that the spin polarization
parallel to k ($S_{y}$) is also obtained in the calculation, in contrast to
its absence in the initial states. Although this calculation is insufficient
for reproducing some quantitative values, such as the spin polarization ratio
$S_{x}$/$S_{z}$ for the even states, it is sufficient for capturing the
qualitative behavior of this complicated spin texture.

## Discussion

The consistency of the numerical calculation and SARPES results supports the
peculiar spin orders between neighboring Bi chains, fluctuating depending on
the wavenumber, as shown in Fig. 3. It should be noted that a new mechanism is
not required to explain such a spin texture in the surface ground states. As
already observed in Weyl semimetals [19, 20], spin polarization derived from
SOI can point arbitrary orientations as far as the surface symmetry operations
permit. The reason why the current result seems unnatural could be because
most of earlier works on the surface Rashba states were performed in simple
surfaces. Therein, the number of non-equivalent atoms was smaller, and the
number of symmetry operations was larger than in the present
Bi/InAs(110)-($2\times 1$) case with only one mirror plane. The current
results also exhibit that the excited electrons could have various, almost
arbitrary spin polarizations depending on the photo-excitation conditions. Not
only the fine tuning of photon polarization [13] but also the wavenumber
selection with the electronic states derived from many non-equivalent
wavefunctions can provide highly spin polarized electrons with the
polarization orientations on demand.

Surface band S of Bi/InAs(110)-(2$\times$1) appears to obey surface Rashba
effect, a paired parabolic dispersion with helical spin polarization [15].
However, the current result revealed that the actual spin and orbital
constitution is considerably more complex than the simple Rashba model, as
shown in Fig. 1. This indicates that such a model is too simple to represent
the realistic system. Actually, other typical Rashba and topological systems,
such as Bi2Se3 [18, 12], Bi(111) [13], and Bi/Ag(111) [21], have multiple spin
and orbital components. Because SOI itself is an interaction to mix the off-
diagonal orbital characters with different spins [22, 23], the complex
spin–orbital entanglement would be the general characteristics of the low-
dimensional electronic states influenced by SOI. Some theoretical studies
claim that such a mixture is essential for Rashba-type SOI [24, 25].

## Methods

### Sample preparation

The surface of InAs(110) substrates was cleaned by repeated cycles of Ar ion
sputtering (1 keV) and annealing (700 K). After the cleaning, a sharp
(1$\times$1) electron diffraction pattern was confirmed by using the low-
energy electron diffraction (LEED). Next, a few monolayers of Bi were
deposited on InAs(110)-(1$\times$1) from a Knudsen cell at room temperature.
Subsequent annealing at 600 K for 20 min resulted in a Bi/InAs(110) surface
exhibiting a sharp (2$\times$1) surface reconstruction pattern. The LEED
diffraction patterns are shown in supplementary Fig. S1 [17].

### Laser-ARPES and SARPES experimental setup

ARPES and SARPES measurements were performed at the Institute for Solid State
Physics, the University of Tokyo with a linearly polarized laser source
($h\nu$ = 6.994 eV) [26]. The photoelectrons were detected along the
$\bar{\Gamma}$ – $\bar{\rm X}$ high-symmetry line in the (2$\times$1) surface
Brillouin zone, as shown in Fig. 2(e). In this experimental geometry, the
planes of photon incidence and photoelectron detection were common. The energy
resolution and the position of $E_{\rm F}$ were calibrated by the Fermi edge
of a Cu block attached to the samples. The energy resolution for the ARPES
(SARPES) measurements was set to $\sim$9 meV ($\sim$20 meV). The effective
Sherman function of a very-low-energy-electron-diffraction (VLEED)-type spin
detector was set to 0.27. The sample temperature was kept at 45 K during the
ARPES and SARPES experiments.

### Calculation of photoelectron spin polarization

The initial states and their spin polarization of the Bi/InAs(110) surface
were obtained by using a DFT calculation based on the “ augmented plane wave
$+$ local orbitals ” method implemented in the WIEN2k code with SOI considered
[27]. We adopted the modified Becke–Johnson potential combined with a local
density approximation to construct the exchange and correlation potentials
[28, 29]. The surface atomic structure was modeled by 20 layers of InAs with a
surface covered with the (2$\times$1) zig-zag Bi chains. The surface atomic
structure was energetically optimized down to the third In and As layers, and
the rest of the substrate atom positions were fixed to those in the bulk InAs
single crystal. The eventual surface atomic structure model was almost
identical to the model obtained experimentally [14].

The spin polarization of the photoelectrons was numerically calculated
considering the interference in the final-state spinors. For this calculation,
we assumed the photoexcitation process from the non-equivalent initial-state
wavefunctions, the two different Bi chains, obtained by the DFT calculation
above. The final state was set as a spin-integrated free electron. The spinor
interference was calculated in a similar manner to refs. [11, 13] (see
supplementary note 1 [17] for details).

## Acknowledgements

The SARPES measurements were jointly conducted with ISSP, the University of
Tokyo. This work was also supported by JSPS KAKENHI (Grants Nos. JP20K03859,
JP19H01830, JP18K03484, and JP20H04453).

## Author contributions

T.N. and Y.O. conducted the SARPES experiments with assistance from A.H.,
K.Y., S.S., and F.K. Y.O. performed the DFT calculation, and its numerical
analysis to simulate photoelectron spin polarizations was performed by T.N.
and Y.O. T.N., Y.O., and S.-i.K. wrote the text and were responsible for the
overall direction of the research project. All authors contributed to the
scientific planning and discussions.

## Competing financial interests

The authors declare no competing financial interests.

## Data availability

The dataset generated during and/or analyzed during the current study are
available from the corresponding author on reasonable request.

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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 wavevector $\vec{k}$, 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$1) surface superlattice, which consists of four
non-equivalent Bi atoms, Bi1, Bi2, Bi3, and Bi4, as indicated. The thin solid
rectangle is the (2$\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, b) ARPES and (c–d) SARPES 2D plots measured with (a, c1–c3) s\-
and (b, d1–d3) p-polarized photons along $\bar{\Gamma}$–$\bar{\rm X}$
(parallel to the Bi chains). (a,b) Spin-integrated ARPES intensity maps. The
photon incident angle ($\theta_{i}$) was $50^{\circ}$. (c1–d3) SARPES maps
polarized to $S_{x}$ (c1, d1), $S_{y}$ (c2, d2), and $S_{z}$ (c3, d3). The
spin orientations are defined in (f). $\theta_{i}$ was $38^{\circ}$. (e)
Experimental geometry of the SARPES measurements and definitions of the
coordinates. The (2$\times$1) surface Brillouin zone and the common plane of
the photon incidence and photoelectron detection are superposed
simultaneously. (f) Definitions of the spin directions of the photoelectrons.
The colors of the SARPES intensity plots are determined based on the
photoelectron intensity and spin polarization, as indicated in these 2D color
maps.  Figure 3:  Calculated spin-polarized band structure and spin-texture in
real space. (a–d2) Spin-polarized band dispersions obtained by DFT
calculations. The radii of the circles are proportional to the sum of the
contributions from all surface Bi atoms. Spin orientations are defined as in-
plane ($S_{x}$, (a, c1–c2)) and out-of-plane ($S_{z}$, (b, d1–d2)) ones.
(c1–c2, d1–d2) Surface bands with the spin orientations of each non-equivalent
surface Bi chain. (e) Color scales represent spin polarizations: Blue–red
(green–purple) color palette corresponds to the spins along the in-plane
($S_{x}$) (out-of-plane ($S_{z}$)) orientations. (f) Wavenumber 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
$k_{y//[\bar{1}10]}$ in each Bi chain. Roman numbers correspond to the
$k_{y//[\bar{1}10]}$ 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 similar manner to Fig. 3 (a), but the
radii of the circles are obtained by taking the Bi 6$p$ ($p_{x}$, $p_{y}$, and
$p_{z}$) atomic orbitals, which have the odd (a1, b1) and even (a2, b2)
symmetries with respect to the photon-incident plane in the SARPES
experimental geometry. The colors of the circles are from the corresponding Bi
6$p$ orbitals with $S_{x}$ and $S_{z}$ polarizations. (c) Schematics of the Bi
6$p$ orbitals superposed on the coordinate axes.  Figure 5:  Spin
polarizations of the photoelectrons from the computational and experimental
analysis. (a1, 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, 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
non-equivalent Bi chains. The phase difference of the wave function between
the Bi1-2 and Bi3-4 chains was set to 2$\pi$/3. See SM for details of the
calculation [17]. (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, d2) Experimentally observed spin polarizations measured
with (d1) s\- and (d2) p-polarized photons.  Supplementary Figure S1:  Low-
energy electron diffraction patterns of (a) a clean InAs(110) substrate and
(a) the Bi/InAs(110)-(2$\times$1) surface. Both images were taken at room
temperature. The red and blue dashed lines indicate (1$\times$1) and
(2$\times$1) reciprocal lattices, respectively.  Supplementary Figure S2:
SARPES energy distribution curves (EDCs) with s\- (a1, b1, c1) and p\- (a2,
b2, c2) polarized photons along $\bar{\Gamma}$–$\bar{\rm X}$. The intensities
are normalized by the maximum values of the spin integrated spectra at each
$k_{y//[\bar{1}10]}$ in the measured energy range (binding energy = 0 $\sim$
0.8 eV).

## Supplementary Note 1. Numerical calculation of spin-polarized
photoelectrons

We describe the initial-state wavefunction as follows:

$\ket{i}=\phi_{1}\ket{\theta_{1}}+\phi_{2}\ket{\theta_{2}},$ (1)

where

$\ket{\theta_{n}}=\left(\begin{array}[]{c}\rm{cos}\frac{\theta_{n}}{2}\\\
\rm{sin}\frac{\theta_{n}}{2}\par\end{array}\right),$ (2)

represents the initial-state spin polarization of the nth Bi chain (n = 1, 2)
pointing the angle $\theta_{n}$ in the xz plane, and $\phi_{n}$ are the atomic
wavefunctions from each Bi atom. Note that $\ket{0}(\ket{\pi})$ is the
eigenspinor of the Pauli matrix $\sigma_{z}$ with an eigenvalue of 1 (-1).
Assuming the simple electric dipole transition to the nearly free-electron
final state, the spinor field of the photoelectron $\chi$ is described by

$\ket{\chi_{\alpha}}=A_{1\alpha}\ket{\theta_{1}}+A_{2\alpha}\ket{\theta_{2}},$
(3)

where $\alpha$ = odd or even and $A_{n\alpha}$ is the complex matrix element
from the atomic wavefunctions from the nth Bi chain with $\alpha$ symmetry
combined with the free-electron final state.

The spin expectation values of photoelectrons $P_{\beta}$ ($\beta=x,y,z$) are
obtained with Pauli matrices $\sigma_{\beta}$ as follows:

$P_{\beta}=\frac{\bra{\chi_{\alpha}}\sigma_{\beta}\ket{\chi_{\alpha}}}{\braket{\chi_{\alpha}}{\chi_{\alpha}}},$
(4)

$\displaystyle\braket{\chi_{\alpha}}{\chi_{\alpha}}$ $\displaystyle=$
$\displaystyle|A_{1\alpha}|^{2}+|A_{2\alpha}|^{2}+2Re(A_{1\alpha}^{*}A_{2\alpha})\rm{cos}\frac{\theta_{1}-\theta_{2}}{2},$
(5) $\displaystyle P_{x}\braket{\chi_{\alpha}}{\chi_{\alpha}}$
$\displaystyle=$
$\displaystyle|A_{1\alpha}|^{2}\rm{sin}\theta_{1}+|A_{2\alpha}|^{2}\rm{sin}\theta_{2}+2Re(A_{1\alpha}^{*}A_{2\alpha})\rm{sin}\frac{\theta_{1}+\theta_{2}}{2},$
(6) $\displaystyle P_{y}\braket{\chi_{\alpha}}{\chi_{\alpha}}$
$\displaystyle=$ $\displaystyle
2Im(A_{1\alpha}A_{2\alpha}^{*})\rm{sin}\frac{\theta_{1}-\theta_{2}}{2},$ (7)
$\displaystyle P_{z}\braket{\chi_{\alpha}}{\chi_{\alpha}}$ $\displaystyle=$
$\displaystyle|A_{1\alpha}|^{2}\rm{cos}\theta_{1}+|A_{2\alpha}|^{2}\rm{cos}\theta_{2}+2Re(A_{1\alpha}^{*}A_{2\alpha})\rm{cos}\frac{\theta_{1}+\theta_{2}}{2}.$
(8)

The first and second terms of $P_{x}$ and $P_{z}$ are from each initial state,
and the third term originates from the interference between each Bi chain,
namely, spin–orbital entanglement. The same term also appears in $P_{y}$ where
no initial state polarization exists. Such photoelectron spin polarization
from the interference during photoexcitation is similar to what has been
reported for atomic wavefunctions in different layers [S1] and mixtures of
even and odd orbitals [S2].

By setting the phase difference between $A_{1}$ and $A_{2}$ to be
$\frac{2\pi}{3}$, we calculated the expected spin polarizations based on the
ground states with the spin polarization orientations $\ket{\theta_{n}}$ and
amplitudes of the atomic wavefunctions obtained by a DFT calculation. As
discussed in the main text, the obtained values shown in Figs. 5(d1) and (d2)
in the main text agree well with the SARPES observation.

## Supplementary Note 2. Low-energy electron diffraction patterns of InAs(110)
and Bi/InAs(110)-(2$\times$1).

Fig. S1(a) shows the low-energy electron diffraction (LEED) pattern of a
InAs(110) substrate after Ar sputtering and annealing cycles. Detailed
conditions are described in the method section of the main text. The Sharp and
low-background pattern was observed, indicating a successfull surface
cleaning. Fig. S1(b) shows the LEED pattern after Bi deposition and subsequent
annealing as explained in the method section. As shown by the blue dotted
line, fractional order spots appeared, indicating the formation of the
Bi/InAs(110)-(2$\times$1) surface.

## Supplementary Note 3. Spin-resolved energy distribution curves taken along
$\bar{\Gamma}$ – $\bar{\rm X}$.

Figures S2(a1, b1, c1) and S2(a1, b1, c1) show the polarization-dependent
SARPES energy distribution curves (EDCs) by using s\- and p-polarized photons,
respectively. The definitions of spin polarization ($S_{x}$, $S_{y}$, and
$S_{z}$) are the same as those in Fig. 2(f) in the main text. The intensities
of SARPES EDCs are normalized by the maximum values of the spin integrated
spectra (sum of the corresponding EDC pair for each $k_{y//[\bar{1}10]}$) in
the measured energy range (binding energy = 0 $\sim$ 0.8 eV).

### References of Supplementary Note

  * S1.

Zhu, Z. -H. et al. Photoelectron Spin-Polarization Control in the Topological
Insulator Bi2Se3. Phys. Rev. Lett. 112, 076802 (2014).

  * S2.

Yaji, K. et al. Spin-dependent quantum interference in photoemission process
from spin-orbit coupled states, Nature Commun. 8 14588 (2017).