#  Direct evidence of a large spin-orbit coupling in the FeSe superconducting
monolayer on STO

Khalil Zakeri<EMAIL_ADDRESS>Heisenberg Spin-dynamics Group,
Physikalisches Institut, Karlsruhe Institute of Technology, Wolfgang-Gaede-
Str. 1, D-76131 Karlsruhe, Germany Dominik Rau Heisenberg Spin-dynamics Group,
Physikalisches Institut, Karlsruhe Institute of Technology, Wolfgang-Gaede-
Str. 1, D-76131 Karlsruhe, Germany Jasmin Jandke Physikalisches Institut,
Karlsruhe Institute of Technology, Wolfgang-Gaede-Str. 1, D-76131 Karlsruhe,
Germany Fang Yang Physikalisches Institut, Karlsruhe Institute of Technology,
Wolfgang-Gaede-Str. 1, D-76131 Karlsruhe, Germany Wulf Wulfhekel
Physikalisches Institut, Karlsruhe Institute of Technology, Wolfgang-Gaede-
Str. 1, D-76131 Karlsruhe, Germany Institute for Quantum Materials and
Technologies, Karlsruhe Institute of Technology, D-76344, Eggenstein-
Leopoldshafen, Germany Christophe Berthod Department of Quantum Matter
Physics, University of Geneva, 1211 Geneva, Switzerland

#  Direct probing of a large spin-orbit coupling in the FeSe superconducting
monolayer on STO: Evidence for nontrivial topological states

Khalil Zakeri<EMAIL_ADDRESS>Heisenberg Spin-dynamics Group,
Physikalisches Institut, Karlsruhe Institute of Technology, Wolfgang-Gaede-
Str. 1, D-76131 Karlsruhe, Germany Dominik Rau Heisenberg Spin-dynamics Group,
Physikalisches Institut, Karlsruhe Institute of Technology, Wolfgang-Gaede-
Str. 1, D-76131 Karlsruhe, Germany Jasmin Jandke Physikalisches Institut,
Karlsruhe Institute of Technology, Wolfgang-Gaede-Str. 1, D-76131 Karlsruhe,
Germany Fang Yang Physikalisches Institut, Karlsruhe Institute of Technology,
Wolfgang-Gaede-Str. 1, D-76131 Karlsruhe, Germany Wulf Wulfhekel
Physikalisches Institut, Karlsruhe Institute of Technology, Wolfgang-Gaede-
Str. 1, D-76131 Karlsruhe, Germany Institute for Quantum Materials and
Technologies, Karlsruhe Institute of Technology, D-76344, Eggenstein-
Leopoldshafen, Germany Christophe Berthod Department of Quantum Matter
Physics, University of Geneva, 1211 Geneva, Switzerland

###### Abstract

Abstract
—————————————————————————————————————————————————–

In condensed-matter physics spin-orbit coupling (SOC) is a fundamental
physical interaction, which describes how the electrons’ spin couples to their
orbital motion. It is the source of a vast variety of fascinating phenomena in
solids such as topological phases of matter, quantum spin Hall states, and
many other exotic quantum states. Although in most theoretical descriptions of
the phenomenon of high-temperature superconductivity SOC has been neglected,
including this interaction can, in principle, revise the microscopic picture
of superconductivity in these compounds. Not only the interaction leading to
Cooper pairing but also the symmetry of the order parameter and the
topological character of the involved states can be determined by SOC. Here by
preforming energy-, momentum-, and spin-resolved spectroscopy experiments with
an unprecedented resolution we demonstrate that while probing the dynamic
charge response of the FeSe monolayer on strontium titanate, a prototype two
dimensional high-temperature superconductor using slow electrons, the
scattering cross-section shows a considerable spin asymmetry. We unravel the
origin of the observed spin asymmetry by developing a model in which SOC is
taken into consideration. Our analysis indicates that SOC in this two
dimensional superconductor is rather strong. We anticipate that such a strong
SOC can have several serious consequences on the electronic structures and can
lead to the formation of topological states. Moreover, a sizable SOC can
compete with other pairing scenarios and is crucial for the mechanism of high-
temperature superconductivity.
——————————————————————————————————————————————————–

##  I Introduction

The fundamental interaction describing the microscopic coupling mechanism
between the spin and orbital degrees of freedom of electrons in solids is the
so-called spin-orbit coupling (SOC) [1, 2]. This interaction, which is a
relativistic effect, is an essential ingredient for describing many emergent
phenomena observed in condensed-matter systems. For instance, a large SOC in
combination with other symmetry aspects can lead to the appearance of
topological phases in solids. Examples of this kind are the topological
insulators, where a large SOC leads to the formation of the topologically
protected surface states and spin momentum locking [3, 4, 5, 6, 7]. Likewise
in magnetically ordered solids SOC in the absence of inversion symmetry can
result in the formation of topologically protected spin textures in the form
of chiral domain walls, skyrmions, antiskyrmions, hopfions, etc. [8].

In order to figure out whether or not a material exhibits topological
electronic states and to which topological classes these states belong, one
requires to quantify the strength of SOC. Assuming that the symmetry
considerations are fulfilled, the presence of a sufficiently large SOC would,
in principle, give rise to the formation of nontrivial topological states in
the system. Although the phenomenon of high temperature superconductivity is,
by itself, a fascinating phenomenon, combined with topological aspects of
matter it would lead to an even more exotic state of matter e.g., topological
superconductivity and the formation of the Majorana states [9, 10, 11]. These
states which obey non-Abelian statistics can be used to realize topological
quantum computers [12]. In most of the proposals for realizing these
interesting concepts it is suggested to attach a low-dimensional
superconductor to a topological material or semiconductor heterostructures
with a large SOC [10]. However, under some circumstances if SOC in a low-
dimensional superconductor is sufficiently large, one expects to observe
topological states in a single material [13]. An ideal candidate for such an
observation would be a single layer of FeSe grown on SrTiO3(001), an ideal
two-dimensional high temperature superconductor (HTSC) [14, 15, 16, 17, 18,
19, 20]. In the case of ultrathin films the inversion symmetry in the
direction perpendicular to the surface is broken. A large SOC together with
the broken inversion symmetry can provide the necessary fundamental basis
required for the observation of topological states in the system [21, 22].
Hence a direct probing of SOC in this class of materials is essential in
connection with the possibility of the formation of topological states.
Unfortunately, the strength of SOC in such two-dimensional superconductors is
hitherto fully unknown.

Irrespective of the importance of SOC for the topological superconductivity,
the presence of this interaction is of prime importance to understand the
underlying physics of HTSC in general [23, 21]. In most theoretical approaches
describing the microscopic mechanism of superconductivity and Cooper pairing
of electrons SOC is assumed to be very small and, therefore, has been
neglected. There are only a few theoretical models which include SOC in bulk
HTSCs, showing that the presence of this interaction is essential in the
determination of the symmetry of the order parameter as well as the electronic
states involved in superconductivity [21, 23]. Generally the impact of SOC
becomes increasingly important when reducing the systems’ dimensionality. This
is due to the emergence of new symmetry aspects in low-dimensional solids.
Surprisingly, so far no direct signature of SOC and its impact in ultrathin
(two-dimensional) HTSCs have been reported experimentally. A direct measure of
SOC would, therefore, be extremely valuable in the context of microscopic
physical mechanism behind high temperature superconductivity in low-
dimensional HTSCs.

Here by performing high resolution spectroscopy of spin-polarized slow
electrons on epitaxial FeSe monolayers grown on Nb-doped strontium titanate
SrTiO3(001), a prototypical two-dimensional superconductor, we demonstrate
that the frequency and momentum dependent scattering cross-section depends
strongly on the spin of the incoming electron. A careful analysis of the
spectra reveals that the observed effect is due to the presence of a
considerably large SOC in this system. Such a large SOC together with other
symmetry aspects provides the required ingredients for the formation of
topologically nontrivial states and would shed light on the mysterious origin
of superconductivity in this system.

##  II Results

The epitaxial FeSe monolayer was grown by molecular beam epitaxy on Nb-doped
SrTiO3(001) (hereafter STO). The dynamic charge response of the system was
probed by means of spin-polarized high-resolution electron energy-loss
spectroscopy (SPHREELS) (see Sec. V.1 of Materials and Methods for details on
the substrate preparation, film growth and SPHREELS experiments). The
scattering geometry is sketched in Fig. 1a. The scattering plane was chosen to
be parallel to the [100]-direction of STO(001), as indicated in Fig. 1b and c.
This would allow probing the dynamic response of the system along the high
symmetry $\bar{\Gamma}$–$\bar{\rm X}$ direction of the surface Brillouin zone
(SBZ). In order to be sensitive to the spin-dependent effects associated with
the broken inversion symmetry in the direction perpendicular to the surface,
we used a longitudinally spin-polarized electron beam with the spin
orientation being parallel and antiparallel to the scattering’s plane normal
vector $\bm{n}$ (see Supplementary Note 1 for an explanation). These incoming
spin states are denoted by $|+\rangle$ and $|-\rangle$, respectively. Figure
1d shows the spin-resolved spectra recorded in the superconducting state of
the sample and using an incident electron beam energy $E_{i}=4.07$ eV. The
spectra were recorded at the $\bar{\Gamma}$–point of SBZ. Beside the so-called
zero loss peak at the energy-loss $\hbar\omega=0$, one observes several
features as a result of the excitation of several collective modes. The peaks
with lower intensity at $\hbar\omega=11.8$, 20.5, 24.8 and 36.7 meV represent
the various phonon modes of the FeSe film itself [24, 25, 26]. More obviously
the so-called Fuchs-Kliewer (FK) phonon modes of the underlying STO substrate
can also be recognized at the loss energies $\hbar\omega=59.3$ and 94.5 meV
[27, 28].

Figure 1: Evidence of a large SOC in FeSe ML on STO. a The scattering geometry
used for probing the dynamic charge response. The electrons are represented by
the blue and red balls. Their spin in the laboratory frame is shown by the red
and blue arrows. The incident energy and wavevector are denoted by $E_{i}$ and
$\bm{k}_{i}$, respectively. The energy and the wavevector after the scattering
event are given by $E_{f}$ and $\bm{k}_{s}$, respectively. The laboratory
frame is depicted by black arrows with x, y, and z labels. The incident and
outgoing angles are called $\theta_{i}$ and $\theta_{s}$. The total scattering
angle is $\theta_{0}$ and was set to 80∘. b The top view of STO(001) and the
FeSe(001) film. The spin polarization of the beam is either parallel or
antiparallel to the y-axis that is the [010]-direction of the STO(001)
surface. These spin states are called $|+\rangle$ and $|-\rangle$,
respectively. c Atomically resolved scanning tunneling microscopy topography
image of the FeSe surface, showing the atomic resolution of the topmost Se
atoms, indicated by the gray balls in b. The field of view is $7\times 4.5$
nm2. The constant current topography image was recorded at $T=0.9$ K and using
a tunneling current of 180 pA and a bias voltage of 1.0 V. The corresponding
reciprocal lattice is shown in the right side. The $\bar{\Gamma}$–point
represents the SBZ center and the $\bar{\rm X}$ and $\bar{\rm M}$ points
represent the edges of SBZ. d Blue upward and red downward triangles represent
the experimental spectra recorded for the spin of the incoming beam being
parallel and antiparllel to the y-axis, respectively. The open circles denote
the spin asymmetry
$A=\left(I_{|+\rangle}-I_{|-\rangle}\right)/\left(I_{|+\rangle}+I_{|-\rangle}\right)$.
The spectra were recorded at the specular geometry i.e., the wavevector of
$q=0$, at the $\bar{\Gamma}$–point. The error bars represent the statistical
uncertainties.

Generally the spectral function $\mathcal{S}(q,\omega)$ measured by SPHREELS
directly reflects the dynamic response of the collective charge excitations in
the system. This quantity is proportional to the imaginary part of the dynamic
charge susceptibility $\mathfrak{Im}\chi(q,\omega)$ [29, 30, 31]. The
electrons are scattered by the total charge distribution of the sample and,
hence, the scattering intensity carries information regarding collective
excitations of the lattice i.e, phonons, collective electronic excitations
i.e., plasmons and any type of excitation representing a hybrid mode of these
two [31]. The most interesting observation here is that
$\mathcal{S}(q,\omega)$ depends strongly on the spin. The spin asymmetry
defined as
$A=\left(I_{|+\rangle}-I_{|-\rangle}\right)/\left(I_{|+\rangle}+I_{|-\rangle}\right)$
is shown in the lower part of Fig. 1d. Here $I_{|+\rangle}$ and
$I_{|-\rangle}$ denote the intensity of the scattered electrons when the
incoming electron’s spin is parallel and antiparallel to $\bm{n}$,
respectively.

In order to shed light on the origin of the observed spin asymmetry, its
dependency on the physical variables e.g., temperature, incident energy, and
wavevector transfer $q$ was measured and the results are summarized in Fig. 2.
Data presented in Fig. 2a clearly demonstrate that the spin asymmetry does not
depend on temperature. In both superconducting and normal states one observes
a value as large as 11%. This fact indicates that the spin asymmetry is not
related to the superconducting (or magnetic) phase transition and is due to
the intrinsic SOC of the system. Next we check the dependence of the spin
asymmetry on the incident beam energy $E_{i}$. Generally for very low incident
energies ($E_{i}<3$ eV) the intensity may be influenced by the space charge
effects. On the other hand for incident energies higher than 12 eV the
intensity is determined by the multiple scattering and electron diffraction
processes (the so-called low-energy electron diffraction or LEED states).
Hence, the relevant energy window would be between 3 and 11 eV. Such data are
presented in Fig. 2b. For this set of measurements first the incident and
scattered beam angles were fixed to $\theta_{i}=\theta_{s}=40^{\circ}$ (see
Fig. 1a). The incident beam energy $E_{i}$ was precisely defined and the
electrons with the final energy $E_{f}=E_{i}\pm\delta E$ were collected. Here
$\delta E$ represents the energy width of the elastic scattering (the hatched
area in Fig. 2a). In order to make sure that all the elastically scattered
electrons are collected, we recorded the intensity for $\delta E=8$ meV (this
value is two times the energy resolution). The spin asymmetry shows a strong
dependence on the incident beam energy and exhibits a maximum near 4 eV. As
the next physical variable we check the dependence of the spin asymmetry on
$q$. Spectra recorded for different values of $q$ near the zone center (in the
vicinity of the $\bar{\Gamma}$–point) indicate that the spin asymmetry does
not depend on $q$. This is demonstrated in Figs. 2c and d, where the spin
asymmetry recorded for different values of $q$ is presented. The data shown in
Fig. 2c were recorded with an incident beam energy of $E_{i}=6.0$ eV and those
in Fig. 2d were recorded with $E_{i}=7.25$ eV. A careful inspection of the
data shown in Figs. 2c and d indicates that although the spin asymmetry
depends strongly on $E_{i}$, it does not depend on $q$. The strong
$E_{i}$-dependence of spin asymmetry and its $q$-independence is an
unambiguous evidence that the observed spin asymmetry is originating from a
substantially large SOC at the surface (see the discussion below).

Figure 2: SOC as the origin of the observed spin asymmetry. a Spin asymmetry
measured below ($T=15$ K) and above ($T=300$ K) the superconducting transition
temperature. The data are recorded at an incident beam energy of 4.07 eV and
at the specular geometry ($q=0$). The error bars represent the statistical
uncertainties. b Spin asymmetry as a function of the incident beam energy. The
error bars represent the systematic uncertainties. c,d The pattern of the spin
asymmetry over the energy loss for different values of wavevector along the
$\bar{\Gamma}$–$\bar{\rm{X}}$ direction. The graphs represent the spin
asymmetry of the spectral function $\mathcal{S}(q,\omega)$. The data shown in
c are recorded at an incident beam energy of 6.0 eV and those shown in d are
recorded at 7.25 eV.

##  III Discussion

It is well-known that when a beam of spin-polarized slow electrons is
scattered from a free atom with a large atomic number, and consequently a
large SOC, the scattering cross-section can be spin dependent [32, 33]. The
effect is understood based on the fact that due to the relativistic effects
the electrons with different spins feel different scattering potentials, while
scattered off the atom. The apparent different scattering potentials are the
direct consequence of the relativistic motion of electrons in the vicinity of
the atom [33]. The same phenomenon has also been observed when such a beam is
scattered from a surface i.e., an array of atoms ordered in a two-dimensional
fashion [33, 34, 35, 36]. The effect is attributed to the intrinsic SOC of the
involved atomic orbitals. As a consequence of the broken translation symmetry
at the surface, the largest effect is observed when the spin of the incoming
electron beam is parallel and antiparallel to the scattering’s plane normal
vector $\bm{n}$ and hence the spin asymmetry is maximum in this case.

In a simplified version the spin dependent elastic scattering cross-section
$dS/d\Omega$ may be written as

$\dfrac{dS}{d\Omega}=\dfrac{dS_{0}}{d\Omega}\left\\{1+\sum_{\gamma}\delta_{l_{\gamma}}\left[\bm{\xi}\cdot\left(\bm{\hat{{k}}}_{i}\times\bm{\hat{{k}}}_{s}\right)\right]\right\\},$
(1)

where $dS_{0}/d\Omega$ is the scattering cross-section without considering
SOC, $\sum_{\gamma}\delta_{l_{\gamma}}$ represents the sum over all the
possible scattering phase shifts $\delta_{l_{\gamma}}$ between partial waves
with total angular momenta between $l-1/2$ and $l+1/2$ ($l$ is the orbital
quantum number of the involved atomic orbitals), $\bm{\xi}$ represents the
spin of the incoming beam, $\bm{\hat{{k}}}_{i}$ and $\bm{\hat{{k}}}_{s}$
denote the unit vectors of the momentum of the incoming and scattered beam,
respectively [33, 35, 37]. Note that Eq. (1) is valid for the energies above
the so-called centrifugal barrier, which is on the order of 1–3 eV.

Equation (1) indicates that the value of the spin asymmetry depends only on
$\sum_{\gamma}\delta_{l_{\gamma}}$, which, in turn, depends on the incident
energy $E_{i}$. The spin asymmetry does not depend on $q$, since different
values of $q$ are achieved by changing the scattering geometry, keeping the
total scattering angle $\theta_{0}=\theta_{i}+\theta_{s}=80^{\circ}$ fixed.
For a longitudinally spin-polarized beam, since the asymmetry is given by the
vector product $\bm{\hat{{k}}}_{i}\times\bm{\hat{{k}}}_{s}$, it does not
change as long as the angle between $\bm{\hat{{k}}}_{i}$ and
$\bm{\hat{{k}}}_{s}$ is unchanged. More importantly the maximum asymmetry is
expected when the second term in Eq. (1) is maximum. Such a condition is
realized for a longitudinally spin-polarized beam in which the direction of
spin polarization vector is parallel or antiparallel to the cross product
$\bm{\hat{{k}}}_{i}\times\bm{\hat{{k}}}_{s}$ or the scattering’s plane normal
vector $\bm{n}$ (see Supplementary Note 1 for details).

Figure 3: Dependence of the spin asymmetry on the energy of the scattered
beam. Spin asymmetry $A$ versus scattered beam energy $E_{f}$, when the
incident beam energy $E_{i}$ is kept constant. For each set of data the energy
of the incident beam $E_{i}$ is also shown using the top axis; $E_{i}=4$ (teal
color), 5 (blue color), 6.2 (red color), 7.3 (green color), 8 (brown color)
and 9 (orange color) eV. The filled circles represent the elastic scattering
i.e., cases in which $E_{i}=E_{f}$. The gray thick curve is a guide to the eye
and describes the $E_{i}$-dependence of $A$ shown in Fig. 2b. The data
indicate that the $E_{f}$-dependence of $A$ is the same as its
$E_{i}$-dependence. The error bars represent the systematic uncertainties.

When electrons are scattered from a surface the scattering process near the
specular angle is governed by the dipolar scattering mechanism [38]. In the
so-called dipolar lobe, the electrons interact with the charge density
fluctuations of the sample. The interaction is of the long range dipolar
(Coulomb) nature. Hence, not only the charge density fluctuations near the
surface region but also those located far below the surface can be observed in
the spectra, depending on the kinetic energy of the incoming beam [39, 40,
41]. Since we are interested in the properties of the FeSe/STO interface, we
use electrons with kinetic energies as low as 4 up to 10 eV. This choice of
incident energy is also important to, on the one hand keep the incident
electron energy above the centrifugal barrier and, on the other hand, avoid
multiple scattering processes, whose presence would add to the complexity of
the problem when one aims to understand the spin dependent scattering cross-
section.

Beside the elastic part of the spectrum, an even more interesting part is the
energy-loss region, where the collective excitations of the system show up.
We, therefore, carefully analyzed the spin asymmetry as a function of
electrons’ energy in the final state after scattering $E_{f}$, while keeping
the energy of the incident electrons constant. The data are presented in Fig.
3. For this set of measurements we first optimized the incoming beam at a
given energy $E_{i}$ and probed the spin asymmetry of the elastic scattering
at $E_{f}=E_{i}\pm\delta E$. The results are shown by the filled circles in
Fig. 3. Then for each value of $E_{i}$ we probed the spin asymmetry as a
function of $E_{f}$ over an energy range for which we could obtain reasonable
count rates. The experiment was performed for various values between 4 and 9
eV. The results clearly demonstrate that the spin asymmetry versus scattered
beam energy follows the same trend as that of the elastic scattering for
different energies. For example for the electrons with the incident energy of
$E_{i}=4$ eV the asymmetry decreases when decreasing $E_{f}$ from 4 to 3.5 eV.
However, for the higher incident energies the spin asymmetry gradually
increases when moving towards lower values of $E_{f}$. This means that the
observed spin asymmetry in the energy loss region has the same origin as that
of the elastic scattering. In simple words electrons contributing to the
surface loss processes are then affected by the SOC potential before they are
finally scattered out. In the limit of small energy losses
($\hbar\omega=E_{i}-E_{f}\ll E_{i}$) the spin asymmetry is almost entirely
determined by the incident energy $E_{i}$.

Yet the question whether or not the dynamic charge response depends on the
spin of the incident electron remains unanswered. In order to answer this
question and shed light onto the origin of the observed spin asymmetry we
developed a model to simulate the SPHREEL spectra. The simulation is based on
the scattering theory of spin-polarized slow electrons from a surface with a
nonnegligible SOC. The theory is an extension of the original theory of Evan
and Mills [42, 43, 38], which describes the scattering cross-section of an
unpolarized electron beam. In our modeling of the scattering event and in the
calculation of the scattering cross-section we further account for SOC and
spin dependent electron reflection (in addition to the Hartree or Coulomb
scattering, see Sec. V.2 of Materials and Methods for details). Our theory
indicates that for small values of momentum transfers the observed spin
dependent asymmetry can be explained based on the spin dependence of electron
reflectivity. Hence, the dynamic charge response itself is rather spin
independent. The observed spin asymmetry is almost entirely due to the spin
dependent electron reflection from the surface, which in turn is a result of
SOC. In order to verify this hypothesis we simulated the SPHREEL spectra using
the values of spin dependent reflection coefficient measured for the elastic
scattering (data shown in Fig. 2b). The results of simulation for two
different values of $E_{i}$ are shown in Fig. 4 together with the experimental
spectra. Our theory is able to perfectly reproduce the experimental spectra.
In the simulation we considered a system composed of an atomic layer of FeSe
(Se-Fe-Se trilayer structure) on $17$ unit cells of charge free insulating
STO(001) on top of a semi-infinite Nb-STO(001). In this model the Fe plane in
FeSe ML is placed at $d_{\rm{FeSe}}=0.43$ nm above the insulating STO(001)
surface [44]. Only in this way both the peak position and amplitude of the
excitations associated with the FK modes agree with those measured
experimentally. The discovery of a charge depletion layer below the FeSe layer
has been discussed in details in Ref. [45]. Both simulations and experiments
reveal that the higher harmonics of the principal FK modes are strongly
suppressed. This is mainly due to the presence of the free carriers in FeSe ML
as well as in the interior part of the substrate, below the depletion region.
As discussed in Ref. [45] the presence of a charge depletion layer, which is
due to a considerably large charge transfer from the Nb-doped STO to the FeSe
film, has several serious consequences on the system. One of the consequences
is that it generates a rather large electric field and band bending near and
below the interface. As a rule of thumb one may simply divide the value of the
total band bending, probed by the experiment (2.1 V), by the depletion layer
thickness (6.5 nm). This results in an electric field on the order of 0.3
GV/m. Such a large electric field greatly influences the electronic states in
FeSe ML and boosts SOC at and near the surface region. The effect is very
similar to the Rashba and Dresselhaus effects observed for semiconductor
quantum-well states and two-dimensional electron gasses formed at the
semiconductor surfaces and interfaces [2].

Experiments performed on bulk FeSe using neutron scattering [46] and on other
Fe-based superconductors using angle-resolved photoemission spectroscopy
(ARPES) [23] have shown that the Fe atoms possess a nonnegligible SOC. Since
in the present case the FeSe monolayer is subject to a rather strong electric
field as a result of the charge transfer and the dielectric depletion layer,
we conclude that the observed large SOC in this system is, therefore, an
additive effect. It includes the intrinsic SOC of FeSe ML as well as the
electric field induced SOC, as discussed above. In order to provide an
estimation of the strength of SOC one may compare the results to that of the
W(110) surface. In the case of W(110) the largest spin asymmetry was observed
for $E_{i}=4.7$ eV and was about 64% with an electron beam with a polarization
of about 72% (this would mean a corrected value of about $A=62\%/72\%\approx
86\%$). For the case of Cu(001) the spin asymmetry is negligible (it is below
0.5%) over the same range of incident energy. Comparing these results to the
value of about 8% for the same beam (or 11% after correction), one may
conclude that SOC in FeSe ML on STO is by a factor of about 8 smaller than
that in W(110). The value is, however, sufficiently large to cause topological
states [47, 22, 48].

Figure 4: Spin dependence of dynamic charge response. a and b The measured and
simulated SPHREEL spectra at an incident electron energy of $E_{i}=4.07$ eV. c
and d The measured and simulated SPHREEL spectra at an incident electron
energy of $E_{i}=6.0$ eV. Blue upward and red downward triangles represent the
spectra recorded or simulated for the incoming spin states of $|+\rangle$ and
$|-\rangle$ (spin polarization parallel and antiparallel to the y-axis in Fig.
1a), respectively. The open circles denote the spin asymmetry
$A=\left(I_{|+\rangle}-I_{|-\rangle}\right)/\left(I_{|+\rangle}+I_{|-\rangle}\right)$.
All the experimental spectra are recorded at $T=15$ K in the superconducting
state of the sample and at the specular geometry i.e., the wavevector of
$q=0$, at the $\bar{\Gamma}$–point. The error bars in spin asymmetry represent
the statistical uncertainties.

The conditions under which FeSe ML on STO can exhibit topological characters
have theoretically been examined in Refs. [47, 48]. It has been discussed that
the appearance of a topological phase in this system is associated with (i) a
trivial bandgap in the electronic band structure near the
$\bar{\rm{M}}$-point, (ii) the parity-broken coupling at the interface, and
(iii) a sufficiently large SOC [47]. The presence of the bandgap near the
$\bar{\rm{M}}$–point has already been well established by several ARPES
experiments [20, 49, 16, 50, 51, 52, 53]. The parity-broken coupling at the
interface does exist in the system and, in addition, can be tuned by the
interface engineering. Our results indicate that SOC in this system is large
and hence one would, in principle, be able to realize the nontrivial
topological states in the system. This would open an avenue for investigation
of topological superconductivity and Majorana states in a model system with a
rather high $T_{c}$. Evidence of topological states in FeSe ML has been
reported by scanning tunneling spectroscopy [48]. However, the attribution of
the observed peaks in the tunneling spectra to the topological states has
entirely been based on the first-principles calculations, in which SOC was
taken into account. An important piece of this puzzle i.e., a direct
experimental probe of SOC has been missing. This is now provided by our
experimental results.

In a similar system, namely bulk FeSexTe1-x, the presence of topological
surface superconductivity has been proposed to originate from a band inversion
of the $p_{z}$ and $d_{xz}$ bulk bands. The resulting topological surface
states have, therefore, been described to be Z2 invariant (and hence be
protected by time-reversal symmetry) [54, 55, 56]. ARPES experiments have
revealed that in addition to the superconducting gap in the vicinity of the
Fermi level the surface states become gapped several mili-electron-volts below
the Fermi level [57, 58, 59]. This gap opening, which emerges together with
the superconducting phase has been attributed to the appearance of a weak
ferromagnetism at the surface. It has been discussed that the appearance of
ferromagnetism breaks the time reversal symmetry, which is in contradiction to
the picture of topological superconductivity induced by the band inversion.
Recently, it has been discussed that the emergence of topological
superconductivity on surfaces can be the consequence of the interplay between
the $s_{\pm}$-wave symmetry of the superconducting order parameter, a Rashba
type SOC and the emergent surface magnetism [60]. This mechanism explains
several experimentally observed fingerprints of topological superconductivity
at the surface of bulk FeSeTe superconductor [54, 61, 56, 58, 62, 63] and thin
films [57, 64]. The proposed mechanism can also be important to either
understand or to realize topological edge superconductivity in the FeSe ML on
STO, by introducing magnetic impurities. However, such a realization would
require a more detailed study of the involved states by means of ARPES or
similar techniques [49].

Regarding the origin of superconductivity in FeSe ML on STO the main attention
is paid to the so-called replica bands observed in ARPES experiments. They
have been considered as an indication of a phonon-mediated superconductivity
in this system [20, 49, 16, 50, 51, 52, 53]. This suggestion is merely based
on probing the quasiparticle band dispersions. However, no solid evidence,
indicating a strong electron-phonon coupling, has been reported by probing the
phononic excitations. On the other hand it has also been discussed that the
pairing mechanism can be of unconventional nature and be mediated by spin
fluctuations [65, 66]. The signature of such a coupling has experimentally
been observed by means of tunneling spectroscopy experiments [28, 67].
Moreover, it has been suggested that a cooperative effect of several bosonic
excitations may be the main reason for such a high transition temperature of
this system [68, 69, 52]. Beside the mechanism discussed above, it has
theoretically been shown that considering both, the effect of nematic
fluctuations and SOC in the absence of inversion symmetry, a $s-$wave state is
favored [21]. More importantly, if SOC and the broken inversion symmetry are
strong enough so that they overcome the mismatch between the electron pockets
near the $\bar{\rm{M}}$-point, the gap function measured in the experiment can
be well reproduced [21]. Our result, indicating a large SOC in FeSe/STO, can
now provide a clue to understand the gap function and the nature of gap
opening in great details. Together with the observation of spin fluctuations,
we have reported earlier [28], we conclude that the superconductivity in the
FeSe monolayer on STO is of $s_{\pm}$ character. Our results provide critical
insights into the symmetry of the paring and the origin of superconductivity
in this system.

##  IV Conclusions

While probing the dynamic charge response of the FeSe superconducting ML on
STO by means of SPHREELS, we observed that the scattering cross-section is
strongly spin dependent. The observed spin asymmetry is attributed to a large
SOC at and near the surface region. This large SOC, which very likely
originates from the $d$ orbitals of the Fe atoms [21] and the presence of a
dielectric depletion layer below the FeSe ML, has several consequences on the
properties of the system. One of the important consequences is the formation
of topological states. The depletion layer is a result of charge transfer from
the Nb-STO into the FeSe layer, which leads to a band bending and the
generation of an electric field at the interface.

Since the charge transfer, the formation of a depletion layer at the interface
and the associated electric field is a general phenomenon, we anticipate that
the observed effect is also general and shall be observed for other Fe-based
superconducting monolayers brought in contact with STO or other dielectric
oxides. This would allow an interfacial engineering of the superconducting
states by growing ML iron chalcogenides or other HTSC on dielectric oxides.
Realization of topological states in superconducting MLs would provide a
platform for investigation and realization of Majorana states in,
structurally, very simple systems.

In addition to the facts mentioned above, our results suggest that the
superconductivity in FeSe ML is very likely of $s_{\pm}$ nature and,
therefore, shed light on the long-standing question regarding the nature of
superconducting order parameter and the pairing symmetry in this system.

##  V Methods

###  V.1 Experiments

####  V.1.1 Sample preparation and characterization

All the sample growth and characterizations were performed under ultrahigh
vacuum conditions. FeSe ML was grown by means of molecular beam epitaxy
technique on the Nb-doped STO(001) substrates in a separate chamber [28, 70].
The Nb-doping level was 0.6%. Prior to the film growth, the substrate was
annealed at temperatures up to about 1000 ∘C and was then etched by a selenium
flux for 20 minutes. The sample was kept at the elevated temperature for 30
minutes and then was gradually cooled down to 480 ∘C. Fe and Se were co-
deposited with a growth rate of 0.059 ML/min at 480 ∘C at a flux ratio of
Fe:Se $\equiv$ 1:10. The sample was post annealed at 500 ∘C right after the
deposition for several hours to ensure a good morphological quality. This also
ensures desorption of the residual Se atoms on the surface. The film growth
was monitored by reflection high energy electron diffraction.

For the further measurements, the sample was transferred using an ultrahigh
vacuum suitcase to the scanning tunneling microscopy and spin-polarized high-
resolution electron energy-loss spectroscopy chambers.

The morphological and electronic properties were investigated by means of
scanning tunneling microscopy [28]. Our samples show a superconducting gap of
about $\Delta=11\pm 3$ meV at a temperature of about 0.9 K.

####  V.1.2 Spin-polarized high-resolution electron energy-loss spectroscopy

The dynamic response of the samples was investigated by means of spin-
polarized high-resolution electron energy-loss spectroscopy [71].

A spin-polarized monochromatic electron beam with an energy resolution between
4 and 11 meV was used [72, 71]. The spin-polarized electron beam is generated
by photoemission from a strained GaAsP photocathode. In order to observe
maximum spin asymmetry according to Eq. (1), a longitudinally spin-polarized
beam was used, meaning that the polarization vector of the incoming electron
beam was either parallel or antiparallel to the scattering’s plane normal
vector $\bm{\hat{n}}$. For the parallel case we call these incoming spin
states $|+\rangle$ and for the antiparallel case we call them $|-\rangle$. The
degree of the beam polarization was estimated by performing spin-polarized
elastic reflectivity from a W(110) single crystal, resulting in a value of
about $72\pm 5$%. The total scattering angle i.e., the angle between the
incident and the scattered beam was kept constant ($\theta_{0}=80^{\circ}$).
The scattering intensity was recorded simultaneously for the two possible spin
polarizations of the incoming electron beam $|+\rangle$ and $|-\rangle$. This
means that first the scattering geometry was adjusted and then the intensity
of the scattered electrons was measured after energy analysis. When recording
the intensity of the scattered beam two values for the intensity were
recorded; one for electrons with $|+\rangle$ spin state and the other one for
electrons with $|-\rangle$ spin state. Changing the incoming spin state from
$|+\rangle$ to $|-\rangle$ was realized by reversing the helicity of the laser
beam used for the excitation and emission of the spin-polarized electrons from
the photocathode. The scattered electron beam was energy analyzed without any
further spin analysis.

In order to collect the spectra in off-specular geometry at a certain
wavevector transfer $q$ the scattering geometry was adjusted to realize the
required wavevector transfer. The in-plane wavevector transfer is given by
$q=k_{i}\sin\theta_{i}-k_{s}\sin(\theta_{0}-\theta_{i})$, where $k_{i}$
($k_{s}$) is the magnitude of the wavevector of the incident (scattered)
electrons, and $\theta_{i}$ ($\theta_{0}$) is the angle between the incident
beam and sample normal (the scattered beam). Different wavevectors transfers
were achieved by changing the scattering angles, i.e., by rotating the sample
about its main axis. In the experiments $\theta_{0}$ was kept at $80^{\circ}$.
The spectra were recorded along the $\bar{\Gamma}$–$\bar{\rm{X}}$ direction of
the surface Brillouin zone. The wavevector resolution of the experiment is
given by $\Delta
q=\sqrt{2mE_{i}}/\hbar[\cos\theta_{i}+\cos(\theta_{0}-\theta_{i})]\Delta\theta_{i}$.
$E_{i}$ denotes the energy of the incident beam and $\Delta\theta_{i}$ depends
of the spectrometer design (in our case $\Delta\theta_{i}=2^{\circ}$). The
resolution of the spectrometer in momentum space is about 0.03 Å-1.

###  V.2 Theory

####  V.2.1 Theory of spin dependent scattering cross-section

We define $|m\rangle$, $|n\rangle$ as many-body states of the sample with
energies $E_{m}$, $E_{n}$ and $|i\rangle$, $|f\rangle$ as initial and final
states of the electron with energies $E_{i}$, $E_{f}$. The general definition
of the differential scattering cross-section is given by [73]

$\frac{d^{2}S}{d\Omega
d\hbar\omega}=\left(\frac{2\pi}{\hbar}\right)^{4}m_{e}^{2}\frac{k_{s}}{k_{i}}\sum_{mn}\frac{e^{-E_{m}/k_{\mathrm{B}}T}}{Z}|\langle
n,f|\hat{T}(E_{m}+E_{i})|m,i\rangle|^{2}\delta(E_{m}+E_{i}-E_{n}-E_{f}),$ (2)

where $m_{e}$ is the electron mass, $k_{i}=|\bm{k}_{i}|$ and
$k_{s}=|\bm{k}_{s}|$ are the norms of the three-dimensional wavevectors of the
incident and scattered electron, respectively, $k_{\mathrm{B}}$ is the
Boltzmann constant, $Z=\sum_{m}e^{-E_{m}/k_{\mathrm{B}}T}$ is the partition
function, and $\hbar\omega=E_{i}-E_{f}$ is the energy-loss of electrons during
the scattering process. $\hat{T}(E)$ is the many-body t-matrix given by
$\hat{T}(E)=\hat{V}+\hat{V}(E+i0-\hat{H})^{-1}\hat{T}(E)$ with $\hat{H}$ the
Hamiltonian of the sample and $\hat{V}$ the interaction energy of the incident
electron with the sample. We use the first-order (Born) approximation,
$\hat{T}(E)\approx\hat{V}$.

The interaction energy of the incident electron with the sample is written as
$\hat{V}=\hat{V}_{\mathrm{H}}+\hat{V}_{\mathrm{SOC}}$, ignoring the exchange-
correlation term. $\hat{V}_{\mathrm{H}}$ is the Hartree energy, i.e., the
electrostatic interaction with the charge density in the sample. We define the
total charge density operator $\hat{\rho}(\bm{R})$ given in the units of
Coulomb per unit volume, where $\bm{R}$ is a three-dimensional position
vector. For the given geometry it can be decomposed into the in- and out-of-
plane components $\bm{R}=(\bm{r},z)$. Here $z$ represents the coordinate
normal to the surface and the sample is placed in the $x$–$y$ plane at $z<0$
(the surface is located at $z=0$). Note that $\hat{\rho}(\bm{R})$ includes
both negative as well as positive charges in the sample. This operator acts on
the many-body states with matrix elements $\langle
n|\hat{\rho}(\bm{R})|m\rangle$. The Hartree energy is

$\hat{V}_{\mathrm{H}}(\bm{R})=e\int
d\bm{R}^{\prime}\,\frac{\hat{\rho}(\bm{R}^{\prime})}{4\pi\epsilon_{0}|\bm{R}-\bm{R}^{\prime}|}\sigma_{0},$
(3)

where $\sigma_{0}$ is the identity matrix in spin space, since the Hartree
potential conserves the spin of the electron during the scattering process.
$\hat{V}_{\mathrm{SOC}}$ is the spin-orbit interaction given by

$\hat{V}_{\mathrm{SOC}}(\bm{R})=\frac{e\hbar}{4m_{e}^{2}c^{2}}\bm{\sigma}\cdot\left[\hat{\bm{E}}(\bm{R})\times\bm{p}\right].$
(4)

Here $\bm{\sigma}$ is the vector of Pauli matrices,
$\hat{\bm{E}}(\bm{R})=-\bm{\nabla}\hat{V}_{\mathrm{H}}(\bm{R})/e$ is the
electric field due to the charge distribution $\hat{\rho}(\bm{R})$, and
$\bm{p}=-i\hbar\bm{\nabla}$ is the momentum operator.

In the next step we introduce the wavefunctions of the incident and scattered
electrons, taking into account spin-dependent reflection coefficients

$\displaystyle\psi_{i}(\bm{R})$
$\displaystyle=N_{i}e^{i\bm{k}_{i}\cdot\bm{r}}\left(\bm{i}e^{ik_{i}^{z}z}+\mathcal{R}\bm{i}e^{-ik_{i}^{z}z}\right)\theta(z)$
(5a) $\displaystyle\psi_{s}(\bm{R})$
$\displaystyle=N_{s}e^{i\bm{k}_{s}\cdot\bm{r}}\left(\bm{s}e^{ik_{s}^{z}z}+\mathcal{R}\bm{s}e^{-ik_{s}^{z}z}\right)\theta(z).$
(5b)

$N_{i}$ and $N_{s}$ are normalization factors, $\theta(z)$ denotes the
Heaviside step function, $\bm{i}$ and $\bm{s}$ are two-component vectors
representing the initial and final states in some spin-1/2 basis (for instance
the $\\{|+\rangle,|-\rangle\\}$ basis), and $\mathcal{R}$ is the reflection
matrix expressed in the same basis. For definiteness, we note that a matrix
$\mathcal{R}$ expressed in the usual basis
$\\{|\uparrow\rangle,|\downarrow\rangle\\}$ with the spin quantization axis
along $z$ can be rotated to a basis with quantization axis along a direction
defined by the polar and azimuthal angles $(\vartheta,\varphi)$ by means of
the unitary transformation $\mathcal{U}^{\dagger}\mathcal{R}\mathcal{U}$ with
$\mathcal{U}=\begin{pmatrix}\cos\frac{\vartheta}{2}&-\sin\frac{\vartheta}{2}\\\
\sin\frac{\vartheta}{2}e^{i\varphi}&\cos\frac{\vartheta}{2}e^{i\varphi}\end{pmatrix}$.
The $\\{|+\rangle,|-\rangle\\}$ basis corresponds to
$(\vartheta,\varphi)=(\pi/2,\pi/2)$ (see the Cartesian coordinates in Fig.
1a). Inserting Eqs. (5) and (3) in Eq. (2) we find that the scattering cross-
section including only the Hartree term can be written as

$\frac{d^{2}S_{\mathrm{H}}}{d\Omega
d\hbar\omega}=\left(\frac{2\pi}{\hbar}\right)^{4}m_{e}^{2}\frac{k_{s}}{k_{i}}\left(\frac{e}{2\epsilon_{0}q}\right)^{2}(N_{s}N_{i})^{2}\left|\frac{\bm{s}^{\dagger}\\!\cdot\bm{i}}{q+iq_{z}^{-}}+\frac{\bm{s}^{\dagger}\\!\cdot(\mathcal{R}\bm{i})}{q+iq_{z}^{+}}+\frac{(\mathcal{R}\bm{s})^{\dagger}\\!\cdot\bm{i}}{q-iq_{z}^{+}}+\frac{(\mathcal{R}\bm{s})^{\dagger}\\!\cdot(\mathcal{R}\bm{i})}{q-iq_{z}^{-}}\right|^{2}\\\
\times\int_{-\infty}^{0}dzdz^{\prime}\,\mathcal{S}(\bm{q},z,z^{\prime},\omega)e^{-q|z+z^{\prime}|},$
(6)

where $q_{z}^{\pm}=k_{s}^{z}\pm k_{i}^{z}$,
$\mathcal{S}(\bm{q},z,z^{\prime},\omega)=\frac{1}{Z}\sum_{mn}e^{-E_{m}/k_{\mathrm{B}}T}\langle
m|\hat{\rho}(-\bm{q},z)|n\rangle\langle
n|\hat{\rho}(\bm{q},z^{\prime})|m\rangle\delta(\hbar\omega+E_{m}-E_{n})$, and
$\bm{q}$ is a two-dimensional vector with $|\bm{q}|=q$.

Likewise, the spin-orbit cross-section can be expressed as

$\frac{d^{2}S_{\mathrm{SOC}}}{d\Omega
d\hbar\omega}=\left(\frac{2\pi}{\hbar}\right)^{4}m_{e}^{2}\frac{k_{s}}{k_{i}}\left(\frac{e}{2\epsilon_{0}q}\right)^{2}(N_{s}N_{i})^{2}|\alpha|^{2}\left|(\mathcal{G_{+}})^{+-}q\left(k_{i}^{x}-ik_{i}^{y}\right)+(\mathcal{G_{-}})^{+-}\left(q_{x}-iq_{y}\right)ik_{i}^{z}\right.\\\
\left.-(\mathcal{G_{+}})^{-+}q\left(k_{i}^{x}+ik_{i}^{y}\right)-(\mathcal{G_{-}})^{-+}\left(q_{x}+iq_{y}\right)ik_{i}^{z}+\left[(\mathcal{G_{+}})^{++}-(\mathcal{G_{+}})^{--}\right]\left(q_{x}k_{i}^{y}-q_{y}k_{i}^{x}\right)\right|^{2}\\\
\times\int_{-\infty}^{0}dzdz^{\prime}\,\mathcal{S}(\bm{q},z,z^{\prime},\omega)e^{-q|z+z^{\prime}|},$
(7)

where ($\mathcal{G_{\pm}}$) is given by

$(\mathcal{G_{\pm}})^{\sigma\sigma^{\prime}}=\frac{s^{*}_{\sigma}i_{\sigma^{\prime}}}{q+iq_{z}^{-}}\pm\frac{s^{*}_{\sigma}\left(\mathcal{R}_{\sigma^{\prime}\sigma^{\prime}}i_{\sigma^{\prime}}+\mathcal{R}_{\sigma^{\prime}\bar{\sigma}^{\prime}}i_{\bar{\sigma}^{\prime}}\right)}{q+iq_{z}^{+}}+\frac{\left(\mathcal{R}^{*}_{\sigma\sigma}s^{*}_{\sigma}+\mathcal{R}^{*}_{\sigma\bar{\sigma}}s^{*}_{\bar{\sigma}}\right)i_{\sigma^{\prime}}}{q-iq_{z}^{+}}\pm\frac{\left(\mathcal{R}^{*}_{\sigma\sigma}s^{*}_{\sigma}+\mathcal{R}^{*}_{\sigma\bar{\sigma}}s^{*}_{\bar{\sigma}}\right)\left(\mathcal{R}_{\sigma^{\prime}\sigma^{\prime}}i_{\sigma^{\prime}}+\mathcal{R}_{\sigma^{\prime}\bar{\sigma}^{\prime}}i_{\bar{\sigma}^{\prime}}\right)}{q-iq_{z}^{-}}.$
(8)

Here $\alpha=-i\hbar^{2}/(4m_{e}^{2}c^{2})$, $\bar{\sigma}\equiv-\sigma$, and
$\mathcal{R}_{\sigma\sigma^{\prime}}$ represent the reflection coefficients
when an electron with a spin $\sigma$ is impinged onto the sample and an
electron with spin $\sigma^{\prime}$ is detected in the final state after the
scattering event.

Equations (6) and (7) shall provide a description for the scattering
intensities for any possible spin directions of the incoming and scattered
beam, when the Hartree and SOC terms are treated separately. However, the
quantities $\mathcal{R}_{\sigma\sigma^{\prime}}$ are not known in practice. In
order to overcome this problem simplifications are needed. As it is apparent
from Eq. (7) the spin-orbit cross-section is by a factor $\sim|\alpha
qk_{i}|^{2}$ smaller than the Hartree cross-section and hence its contribution
to the intensity may be neglected. The Hartree contribution by itself should,
in principle, conserve the spin during the scattering process. In such a
scenario no spin asymmetry is expected. In order to account for the spin
dependent effects one may assume that the reflection coefficients are spin
dependent even for the case of Hartree scattering. In this case one would
observe a spin asymmetry. This means that the role of SOC is to break the spin
degeneracy and thereby lead to spin-dependent reflection coefficients.

Our analysis showed that, indeed, in this case the asymmetry caused by the
Hartree term introduced in Eq. (6) does not depend, in first approximation, on
$q$ and $\hbar\omega$ and is given by (for an extended discussion see
Supplementary Note 2)

$\frac{I_{|+\rangle}-I_{|-\rangle}}{I_{|+\rangle}+I_{|-\rangle}}\approx\frac{\mathcal{R}_{++}^{2}-\mathcal{R}_{--}^{2}}{\mathcal{R}_{++}^{2}+|\mathcal{R}_{+-}|^{2}+|\mathcal{R}_{-+}|^{2}+\mathcal{R}_{--}^{2}}.$
(9)

We define the quantities
$|\mathcal{R}_{|+\rangle}|^{2}=|\mathcal{R}_{++}|^{2}+|\mathcal{R}_{+-}|^{2}$
and
$|\mathcal{R}_{|-\rangle}|^{2}=|\mathcal{R}_{--}|^{2}+|\mathcal{R}_{-+}|^{2}$,
which represent the intensity of the scattered electrons when the spin of the
incoming beam is of $|+\rangle$ and $|-\rangle$ character, respectively. Such
quantities can be extracted from the spin-dependent elastic reflectivity data.
The asymmetry strongly depends on the incident (scattered) energy as well as
the scattering geometry, since $|\mathcal{R}_{|+\rangle}|^{2}$ and
$|\mathcal{R}_{|-\rangle}|^{2}$ depend on these variables.

In our simulations we use exactly the same formalism introduced by Evan and
Mills [42, 43] and implemented by Lucas and Šunjić [74, 75, 76]). In order to
account for the spin dependent effects we use reflection coefficients measured
by the elastic reflectivity measurements (see below).

####  V.2.2 Simulation

Simulation of the spectra were performed by a numerical scheme based on the
dipolar scattering theory. To calculate $P^{\mathrm{sl}}(\omega)$ we first
calculate the single-loss probability

$P(\omega)=\frac{e^{2}|\mathcal{R}_{|\sigma\rangle}(E_{i})|^{2}}{4\pi\epsilon_{0}\hbar
v_{\bot}}\frac{4}{\pi^{2}}\iint_{\Omega}d^{2}q\,\frac{|\bm{q}|(v_{\bot})^{3}}{\left[(\omega-\bm{q}\cdot\bm{v}_{\parallel})^{2}+(|\bm{q}|v_{\bot})^{2}\right]^{2}}\mathfrak{Im}\left[\dfrac{-1}{g(\bm{q},\omega)+1}\right].$
(10)

Here $\epsilon_{0}$ is the vacuum permittivity, $\hbar$ is the reduced Planck
constant, $e$ is the electron charge, $v_{\bot}$ ($v_{\parallel}$) represents
the perpendicular (parallel) component of the velocity of the incident
electron, $|\mathcal{R}_{|\sigma\rangle}|=|\mathcal{R}_{|+\rangle}|$ or
$|\mathcal{R}_{|-\rangle}|$ denotes reflection coefficient of electrons with
the incoming spin state $|+\rangle$ or $|-\rangle$ and strongly depends on the
incident energy $E_{i}$. The integration range $\Omega$ denotes the range of
momentum covered by the exit and entrance slits of the monochromator and
analyzer. We emphasize that in Eq. (10) $\bm{q}$ represents the two-
dimensional vector of the momentum transfer parallel to the surface.

The most important entities in Eq. (10) are
$|\mathcal{R}_{|\sigma\rangle}(E_{i})|$ and the dielectric response function
$g(\bm{q},\omega)$. The former is obtained from the experimental elastic
reflection measured for different spin directions $|+\rangle$ and $|-\rangle$.
For layered systems, such as our case, the latter can be related to the
dielectric function of each individual layers $\varepsilon^{p}(q,\omega)$,
where $p$ is the layer index (for details see Refs. [74, 75, 76, 77]).

Equation (10) describes only the single-loss probability for an electron
having a wavevector $k_{i}$ and spin $|\sigma\rangle$ to be scattered from a
semi-infinite slab system and loose the energy $\hbar\omega$ at $T=0$ K. The
multiple scattering events, the elastic peak, and temperature effects were
included using the approach introduced by Lucas and Šunjić [74, 75, 76, 77].

We first construct a multi-slab system by considering one unit cell of FeSe on
17 unit cells (about 6.5 nm) of insulating STO on a semi infinite Nb-doped
STO(001) (a sketch of the structure is provided in Supplementary Figure 1).
The dielectric function of each individual layer is then written in different
contributions i.e., (i) a frequency independent background dielectric constant
$\varepsilon^{p}_{\infty}$, (ii) a phononic contribution
$\varepsilon^{p}_{\mathrm{phonon}}$, and (iii) an electronic contribution
$\varepsilon^{p}_{\mathrm{plasmon}}$

$\varepsilon^{p}(q,\omega)=\varepsilon^{p}_{\infty}+\varepsilon^{p}_{\mathrm{phonon}}+\varepsilon^{p}_{\mathrm{plasmon}}.$
(11)

We use the literature values of $\varepsilon^{\rm{FeSe}}_{\infty}=15$ and
$\varepsilon^{\rm{STO}}_{\infty}=5.7$ for FeSe and STO, respectively [78, 79,
80, 81].

The phononic contribution to the dielectric function of each layer can be
expressed in terms of different phonon contributions

$\varepsilon^{p}_{\mathrm{phonon}}=\sum_{j=1}^{m}\frac{Q_{j}\omega_{\mathrm{TO},j}^{2}}{\omega_{\mathrm{TO},j}^{2}-\omega^{2}-i\gamma_{\mathrm{TO},j}\omega},$
(12)

where $m$ is the number of all transverse optical (TO) phonon modes with the
oscillator strength $Q_{j}$, which depends on the splitting between TO and
longitudinal optical (LO) modes

$Q_{j}=\frac{\varepsilon_{\infty}}{\omega_{\mathrm{TO},j}^{2}}\frac{\prod\limits_{l}\left(\omega_{\mathrm{LO},l}^{2}-\omega_{\mathrm{TO},j}^{2}\right)}{\prod\limits_{l\neq
j}\left(\omega_{\mathrm{TO},l}^{2}-\omega_{\mathrm{TO},j}^{2}\right)},$ (13)

where $\omega_{\mathrm{TO},j}$ and $\omega_{\mathrm{LO},j}$ denote the
frequency of the $j$-th TO and LO phonon modes, respectively.
$\gamma_{\mathrm{TO},j}$ and $\gamma_{\mathrm{LO},j}$ represent their
corresponding damping.

In addition, we consider a Drude-like term in the dielectric function of each
layer, in order to account for the contribution of the charge carriers

$\varepsilon^{p}_{\mathrm{plasmon}}=-\varepsilon_{\infty}\frac{\omega_{pl}^{2}-i(\gamma_{pl}-\gamma_{0})\omega}{\omega(\omega+i\gamma_{0})},$
(14)

where $\omega_{pl}$ denotes the plasma frequency associated with the charge
carriers and is directly related to the carrier density $n_{c}$ and carriers’
effective mass $m_{\rm eff}$ by
$\omega_{pl}=\sqrt{\frac{n_{c}e^{2}}{\varepsilon_{\infty}\epsilon_{0}m_{\rm
eff}}}$. The quantities $\gamma_{pl}$ and $\gamma_{0}$ are the linewidth
broadening of the plasmon peak and are determined by the plasmon relaxation
time. For $\gamma_{pl}=\gamma_{0}$ in Eq. (14) one arrives at the well-known
Drude term. For the Nb-STO we use $\omega_{pl}=83$ meV, $\gamma_{pl}=75$ meV
and $\gamma_{0}=5$ meV. The values are estimated by extrapolating the values
measured by optical techniques at liquid nitrogen temperature to our
measurement temperature ($T=15$ K) [79, 82]. The extrapolation is based on the
temperature dependence of effective mass as discussed in detail in Ref. [83].
For FeSe ML we use a Drude term with $\omega_{pl}=334$ meV estimated based on
$\omega_{pl}=\sqrt{\frac{n_{c}e^{2}}{\varepsilon_{\infty}\epsilon_{0}m_{\rm
eff}}}$, assuming $m_{\rm eff}\simeq 3m_{e}$ and $n_{c}=0.12$ $e^{-}$/Fe. The
damping parameter was $\gamma_{pl}=\gamma_{0}=270$ meV.

The TO phonon frequencies and their damping as well as
$|\mathcal{R}_{|\sigma\rangle}|^{2}$ serve as the input of the simulations.
The values we used for our simulations are provided in Supplementary Table 1.
The values of $|\mathcal{R}_{|\sigma\rangle}|^{2}$ depend on the incident
energy. We take the values from Fig. 2b based on the expression
$|\mathcal{R}_{|+\rangle}|^{2}=\frac{1+A(E_{i})}{2}$ and
$|\mathcal{R}_{|-\rangle}|^{2}=\frac{1-A(E_{i})}{2}$. For the data shown in
Fig. 4b the values are
$|\mathcal{R}_{|+\rangle}|^{2}(E_{i}=4~{}\rm{eV})=0.555$ and
$|\mathcal{R}_{|-\rangle}|^{2}(E_{i}=4~{}\rm{eV})=0.445$ for those shown in
Fig. 4d they are $|\mathcal{R}_{|+\rangle}|^{2}(E_{i}=6~{}\rm{eV})=0.53$ and
$|\mathcal{R}_{|-\rangle}|^{2}(E_{i}=6~{}\rm{eV})=0.47$.

## Data availability

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

## Code availability

The codes associated with this manuscript are available from the corresponding
author on reasonable request.

## Author contributions

Kh.Z. initiated the idea of the study, supervised the project, conceived and
planned the experiments, analyzed the experimental data, performed the
simulations and wrote the paper. D.R. contributed to carrying out the SPHREELS
experiments. F.Y., J.J. and W.W. prepared the samples and performed the STM
experiments. C.B. carried out the theoretical modeling and derived all the
analytical expressions for the spin-dependent scattering cross-sections.

## Competing interests

The authors declare no competing interests.

## Additional information

Correspondence and requests for materials should be addressed to Kh.Z
(khalil.zakeri@kit.edu).

## Acknowledgements

Kh.Z. acknowledges funding from the Deutsche Forschungsgemeinschaft (DFG)
through the Heisenberg Programme ZA 902/3-1 and ZA 902/6-1 and the DFG Grant
No. ZA 902/5-1. The research of J.J. and W.W. was supported by DFG through
Grant No. Wu 394/12-1. F.Y. acknowledges funding from the Alexander von
Humboldt Foundation. Kh.Z. thanks the Physikalisches Institut for hosting the
group and providing the necessary infrastructure. We thank Janek Wettstein and
Markus Döttling for developing the first version of the simulation code.

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