# Moiré tuning of spin excitations: Individual Fe atoms on MoS2/Au(111)

Sergey Trishin  Fachbereich Physik, Freie Universität Berlin, 14195 Berlin,
Germany Christian Lotze  Fachbereich Physik, Freie Universität Berlin, 14195
Berlin, Germany Nils Bogdanoff  Fachbereich Physik, Freie Universität Berlin,
14195 Berlin, Germany Felix von Oppen  Dahlem Center for Complex Quantum
Systems and Fachbereich Physik, Freie Universität Berlin, 14195 Berlin,
Germany Katharina J. Franke  Fachbereich Physik, Freie Universität Berlin,
14195 Berlin, Germany

###### Abstract

Magnetic adatoms on surfaces are exchange coupled to their environment,
inducing fast relaxation of excited spin states and decoherence. These
interactions can be suppressed by inserting decoupling layers between the
magnetic adsorbate and the metallic substrate. Using low-temperature scanning
tunneling microscopy and spectroscopy, we investigate the magnetic properties
of single Fe atoms on Au(111) covered by a monolayer of
$\text{MoS}_{\text{2}}$ and explore the influence of the moiré structure on
the spin excitation spectra. Fe adatoms absorbed at minima of the moiré
structure exhibit almost pure inelastic spin excitations. Kondo correlations
develop for adsorption sites off the moiré minimum, culminating in fully
developed Kondo resonances on moiré maxima. The increase in Kondo correlations
is accompanied by a decrease in the single-ion anisotropy, which we
rationalize by poor-person’s scaling. Local variations in the tunneling
spectra indicate pronounced interference between tunneling paths involving the
spin-carrying orbitals and a hybrid Fe–S orbital, which forms despite the
nominally inert nature of the terminating S layer.

Single magnetic atoms on surfaces hold substantial promise for spin state
manipulation and coherent control of quantum spin states [1]. To protect the
spin state against decoherence, exchanges of energy and angular momentum with
the substrate need to be suppressed [2]. While tunneling currents allow for
addressing individual spins, metallic substrates are problematic as the
creation of electron-hole pairs and spin-spin scattering involving the
conduction electrons provide highly efficient pathways for energy and angular
momentum relaxation [3, 4, 2]. These processes can be efficiently suppressed
when using materials with an energy gap around the Fermi energy, either by
passivating a metallic substrate with nitride [5] or oxide [6, 7, 8] layers,
or by using a superconductor [9].

Monolayers of van der Waals materials have recently been proposed as
interlayers for decoupling magnetic adsorbates from metallic substrates.
Lifetimes of excited spin states as well as coherence times could be extended
using hexagonal boron nitride [10, 11] and graphene [12, 13]. Monolayers of
transition metal dichalcogenides are another promising class of van der Waals
interlayers. Each sheet consists of transition metal atoms sandwiched between
chalcogen layers with all covalent bonds saturated within the layer.
Monolayers of $\text{MoS}_{\text{2}}$ have been shown to decouple organic
molecules from metal substrates [14, 15, 16]. Their decoupling efficiency has
been attributed to the electronic bandgap in combination with the layer
thickness and the van der Waals nature. These properties should also be
favorable for decoupling magnetic atoms from metal substrates.
$\text{MoS}_{\text{2}}$ layers on Au(111) substrates offer further tunability
due to the moiré structure resulting from a mismatch between the chalcogen and
Au lattice constants. This imposes long-range modulations of the electronic
structure and thus of the coupling between adsorbate and substrate [17].
Previous work has shown that the moiré structure of hexagonal boron nitride
(hBN) on Rh(111) induces small variations in the magnetic excitation energies
of hydrogenated Co adatoms with adsorption site [11]. A strongly site-specific
Kondo resonance of Co atoms can be induced on Ru(0001) covered by graphene,
which partially detaches due to periodic rippling [18]

Here, we exploit a monolayer of $\text{MoS}_{\text{2}}$ on Au(111) as a
decoupling layer for iron (Fe) atoms. Using scanning tunneling microscopy and
spectroscopy at a temperature of $1.1\text{\,}\mathrm{K}$, we identify the
adsorption site of the Fe adatoms and detect their magnetic fingerprint in
tunneling spectra. We show that the data are compatible with a spin state of
$S=1$ and sizable axial anisotropy. The moiré structure of the
$\text{MoS}_{\text{2}}$ overlayer on Au(111) imposes strong variations of the
exchange coupling between Fe adatoms and substrate. When probing directly
above the Fe atoms, we observe almost pure inelastic spin excitations in the
minima of the moiré structure. Away from the minima, an increase of the moiré-
modulated density of states enhances the exchange coupling, culminating in a
fully developed Kondo resonance for Fe atoms located at moiré maxima. Beyond
these dramatic variations with adsorption site, there are pronounced local
variations in the immediate vicinity of the Fe atom, which we trace to
interference of tunneling paths proceeding via different orbitals.

Figure 1a shows an STM image of a monolayer island of $\text{MoS}_{\text{2}}$
on Au(111) with a small coverage of Fe adatoms. The hexagonal corrugation
within the $\text{MoS}_{\text{2}}$ island with a periodicity of $\sim$ 3.3 nm
reflects a moiré structure reflecting a lattice mismatch between the Au and S
layers [19, 20, 17]. An atomically resolved image of a clean
$\text{MoS}_{\text{2}}$ area (Fig. 1b) shows the terminating S layer and
resolves a minute amount of intrinsic point defects (see upper and lower edge
of the image). We focus on adatoms far from the defect sites to avoid any
influence on the magnetic properties.

Figure 1: (a) Large-scale STM image of a monolayer island of
$\text{MoS}_{\text{2}}$ on Au(111) after deposition of a small number of iron
atoms at low temperature. The atoms are statistically distributed over the
bare Au(111) surface and the $\text{MoS}_{\text{2}}$ islands. Image recorded
at $V=$50\text{\,}\mathrm{m}\mathrm{V}$$, $I=$100\text{\,}\mathrm{pA}$$. (b)
Atomic-resolution topography of the S lattice on a clean
$\text{MoS}_{\text{2}}$ island. Atomic defects are seen in the upper and lower
left corners of the image. Image recorded at
$V=$5\text{\,}\mathrm{m}\mathrm{V}$$, $I=$42\text{\,}\mathrm{nA}$$. (c)
Closeup of individual Fe atoms on a $\text{MoS}_{\text{2}}$ island. At low
bias voltages, most atoms appear triangular. Image recorded at
$V=$50\text{\,}\mathrm{m}\mathrm{V}$$, $I=$100\text{\,}\mathrm{pA}$$.

A close-up view of several Fe adatoms on a $\text{MoS}_{\text{2}}$ island is
shown in Fig. 1c. When scanned at low bias voltage, most Fe adatoms appear
with a triangular shape, while a few atoms are imaged as a round protrusion
(see SM). All triangles point in the same direction relative to the underlying
S lattice. The symmetric shape suggests that these Fe atoms are located in
hollow sites of the S layer. There are two distinct hollow adsorption sites on
the $\text{MoS}_{\text{2}}$ islands, with and without a Mo atom underneath.
The uniform orientation of the triangular shapes indicates that only one of
these is occupied. This inference is corroborated by superimposing a hexagonal
lattice representing the S sites on the Fe-covered $\text{MoS}_{\text{2}}$
layer. The positions and orientations of the triangular-shaped Fe atoms are
all consistent with equivalent hollow sites.

Figure 2: Differential conductance spectra taken at the center above
individual Fe adatoms (blue lines) in different local environments as
indicated by the STM topographies in the insets (scale bars are
$1\text{\,}\mathrm{nm}$, the crosses mark the location of the spectra). Fits
are shown as black dashed lines. The spectrum in (a) can be reproduced by
symmetric Fermi-Dirac functions with a width of $0.45\pm 0.07$ meV. The
spectra in (b), (c) and (d) exhibit additional cusps on top of the inelastic
steps. These spectra are fitted within a perturbative approach in the exchange
interaction $\frac{1}{2}J\widehat{\sigma}\cdot\widehat{S}$, with $J$ the
exchange coupling strength and $\widehat{\sigma}$ the vector of Pauli matrices
[21]. Additional potential scattering $U$ leads to bias-asymmetric lineshapes.
The fit uses the code described in [21]. Fit parameters: $J\rho=-0.11$,
$U=-0.32$, $D=$2.5\text{\,}\mathrm{mV}$$ (b), $J\rho=-0.16$, $U=-0.71$,
$D=$2.3\text{\,}\mathrm{mV}$$ (c) and $J\rho=-0.33$, $U=-1$,
$D=$1.73\text{\,}\mathrm{mV}$$ (d). Spectrum (e) exhibits a Frota lineshape
with a dip at the Fermi level. We fitted the spectrum using a Frota lineshape
with an additional Lorentz peak. Fit parameters: Frota phase $\phi=3.87$,
Frota width $\Gamma=$2.21\text{\,}\mathrm{m}\mathrm{V}$$, Frota center
$x_{0}=$1.3\text{\,}\mathrm{m}\mathrm{V}$$, Lorentz width
$\Gamma_{L}=$0.79\text{\,}\mathrm{m}\mathrm{V}$$ and Lorentz position
$x_{L}=$0.11\text{\,}\mathrm{m}\mathrm{V}$$ In (f), the spin excitation gap
has closed, resulting in a fully developed Frota lineshape. Fit parameters:
Frota phase $\phi=4.12$, Frota width
$\Gamma=$3.49\text{\,}\mathrm{m}\mathrm{V}$$, Frota center
$x_{0}=$1.6\text{\,}\mathrm{m}\mathrm{V}$$. The spectra were recorded at a
setpoint of $V=$10\text{\,}\mathrm{m}\mathrm{V}$$,
$I=$1\text{\,}\mathrm{nA}$$.

To investigate the magnetic properties of the individual triangular Fe atoms,
we recorded differential conductance (d$I$/d$V$) spectra of $\sim$ 40 atoms.
Fig. 2 displays a characteristic set of spectra. We find dramatic variations
of the spectral lineshapes depending on the Fe adatom’s position with respect
to the moiré pattern. The Fe atom in Fig. 2a is located in a minimum of the
moiré structure. Its d$I$/d$V$ signal exhibits stepwise increases of intensity
at $\pm 2.7\,\mathrm{mV}$. The Fe atom in Fig. 2b is shifted slightly off the
minimum and shows a small overshoot of the differential conductance at
positive bias voltages above the inelastic excitation threshold. This
overshoot becomes more pronounced for Fe atom located further from a moiré
minimum (Fig. 2c,d). Concomitantly, the step-like feature shifts to lower
energies. Adatoms located close to a moiré maximum exhibit lineshapes which
resemble a Frota peak with an additional dip at the Fermi level (Fig. 2e).
Directly at the moiré maximum, the inelastic gap disappears and gives way to a
fully developed asymmetric Frota peak (Fig. 2f).

To understand these spectroscopic characteristics of individual Fe atoms, we
first discuss their electronic configuration. In the gas phase, Fe atoms host
six electrons in the five-fold degenerate $d$ shell leading to a spin state of
$S=2$. Consistent with the triangular shape of the electronic structure
reflected in the STM images (see further discussion below), the hollow
adsorption site imposes a trigonal pyramidal crystal field which lifts the
degeneracy of the $d$ levels. Simple crystal field considerations suggest a
splitting into two sets of doubly degenerate states, one of mostly
$d_{\mathrm{xz}}$ and $d_{\mathrm{yz}}$ character and another of
$d_{\mathrm{x^{2}-y^{2}}}$ and $d_{\mathrm{xy}}$ character, as well as a high-
lying nondegenerate state deriving from the $d_{\mathrm{z^{2}}}$ orbital.
Depending on the ratio between the crystal field splitting and Hund’s energy,
filling these levels with six $d$ electrons either leads to a spin state of
$S=2$ with a singly-occupied $d_{\mathrm{z^{2}}}$ orbital, or to $S=1$ with an
empty $d_{\mathrm{z^{2}}}$ orbital.

Due to magneto-crystalline anisotropy, the spin ${\bf\widehat{S}}$ prefers to
align along a distinct direction even at zero magnetic field. This is
described by the spin Hamiltonian
$\widehat{H}=D\widehat{S}_{z}^{2}+E(\widehat{S}^{2}_{x}-\widehat{S}_{y}^{2})$,
where $D$ is the axial and $E$ the transverse magnetic anisotropy [5].
Tunneling via a surface-adsorbed Fe atom may thus not only proceed
elastically, when leaving the spin projection $m$ unchanged ($\Delta m=0$),
but also inelastically when changing the spin projection ($\Delta m=\pm 1$).
The opening of inelastic tunneling channels results in a stepwise increase of
the junction conductance.

The d$I$/d$V$ spectrum in Fig. 2a can indeed be well reproduced by a broadened
Fermi-Dirac distribution function at $\pm$2.7\text{\,}\mathrm{mV}$$. Thus, the
spectra probe magnetic atoms subject to a local crystal field characteristic
of the adsorption site. The measured width $w=$0.45\pm
0.07\text{\,}\mathrm{meV}$$ of the step is larger than the experimental
broadening at $T=$1.1\text{\,}\mathrm{K}$$, suggesting a lower bound of $\sim$
$1\text{\,}\mathrm{ps}$ for the spin lifetime. Significantly broader steps are
observed for Fe atoms in direct contact with a metal substrate [22, 3, 4, 23].
This confirms that a single layer of $\text{MoS}_{\text{2}}$ indeed acts as an
efficient decoupling layer for the Fe atoms.

At the same time, the decoupling efficiency depends sensitively on the
position of the Fe atom relative to the moiré lattice. As we observe
lineshapes ranging from inelastic steps to fully developed Kondo peaks, there
must be variations in the magnitude of the coupling to the substrate. At and
close to the minimum of the moiré structure (Fig. 2a,b,c), the inelastic steps
can be understood by considering the exchange coupling to the substrate
perturbatively. To describe the conductance overshoot, one needs to go beyond
lowest-order perturbation theory and include processes in second-order Born
approximation [21]. The asymmetry in the bias polarity can be reproduced when
including potential scattering at the impurity in addition to the exchange
coupling. Figure 2a,b,c include corresponding fits [21]. These indicate that
both the exchange coupling $J$ and the potential scattering $U$ increase as
the adatom location moves off the moiré minimum [21, 11]. In addition, there
is a substantial reduction in the longitudinal anisotropy $D$.

For Fe atoms located further from the moiré minima, the pronounced changes in
the lineshape can no longer be captured within this perturbative framework.
The transition to a fully developed Kondo lineshape requires a strong-coupling
approach and can be captured within a poor-person’s scaling approach [24]. In
the present case, one should account for the longitudinal anisotropy as well
as the higher impurity spin. Allowing for anisotropic exchange couplings
$J_{\perp}$ and $J_{z}$ and focusing on the simplest spin state $S=1$
consistent with our experimental data, the scaling equations take the form
[25]

$\displaystyle\frac{dJ_{z}}{d\ell}$ $\displaystyle=$
$\displaystyle\frac{1}{1-\Delta}J_{\perp}^{2}$ (1)
$\displaystyle\frac{dJ_{\perp}}{d\ell}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(1+\frac{1}{1+\Delta})J_{\perp}J_{z}$ (2)
$\displaystyle\frac{d\Delta}{d\ell}$ $\displaystyle=$
$\displaystyle\Delta-(J_{z}^{2}-J_{\perp}^{2})\ln 2.$ (3)

Here, the exchange couplings are made dimensionless by means of the density of
states $\rho$ of the conduction electrons, $\rho J\to J$, the scaling variable
$\ell$ parametrizes the bandwidth $E_{c}$, $\ell=\mathrm{ln}(E_{c0}/E_{c})$,
with initial bandwidth $E_{c0}$, and $\Delta=D/E_{c}$ measures the
longitudinal anisotropy in units of the bandwidth.

As the adsorption site moves towards the moiré maximum, the associated change
in the density of states increases the bare exchange couplings and hence the
Kondo temperature. A fully developed Kondo peak is expected once the Kondo
temperature exceeds the longitudinal anisotropy $D$. The scaling equations
suggest that this effect is reinforced by the renormalization of $D$. For
easy-axis anisotropy ($D>0$ and $J_{z}>J_{\perp}$) as well as easy-plane
anisotropy ($D<0$ and $J_{z}<J_{\perp}$), the anisotropy $D$ is reduced in
magnitude during renormalization [see the second term on the right hand side
of Eq. (3)]. This further favors the crossover to fully developed Kondo
correlations and is consistent with the observation that the threshold voltage
of spin excitations decreases as the adsorption site approaches the moiré
maximum. We note that a renormalization of the anisotropy has also been
observed in model calculations using the one-crossing approximation [26].

Figure 3: (a) STM topography image of an Fe atom located on a moiré minimum.
(b) Differential conductance spectra at low energies on the center (indicated
by blue cross in (a)) and vertex (indicated with a purple cross in (a)) of the
triangular shape of the Fe adatoms. Fits as described in [21] are shown as
black dashed lines. Fit parameters: $J\rho=-0.13$, $U=-0.14$,
$D=$2.75\text{\,}\mathrm{mV}$$ (blue) and $J\rho=-0.13$, $U=-0.78$,
$D=$2.74\text{\,}\mathrm{mV}$$ (red). (c) Differential conductance spectra
over a larger energy range on the center and the vertex of the triangular
shape of the Fe adatoms. (d) d$I$/d$V$ spectra (orange) recorded across the
Fe–S complex along the orange line in (e). Fits shown as black dashed lines;
$J\rho$ was kept constant in all fits, while $U$ and the transmittivity of the
junctions were adjusted. Values of $U$ are shown in SM [27]. Spectra are
offset for clarity. (e-g) STM topographies (blue-yellow, background) with
superimposed differential conductance signal at the indicated bias voltage
(black-white dots, scale given below panels) extracted from a densely spaced
grid of spectra across the Fe atom at the indicated energies. The spectra in
(b), (d), (e), and (f) were recorded at a setpoint of
$V=$10\text{\,}\mathrm{m}\mathrm{V}$$, $I=$1\text{\,}\mathrm{nA}$$ with an
additional retraction of the tip by $20\text{\,}\mathrm{p}\mathrm{m}$; the
spectra in (c) and (g) at $V=$10\text{\,}\mathrm{m}\mathrm{V}$$,
$I=$20\text{\,}\mathrm{pA}$$ with an additional retraction of the tip by
$20\text{\,}\mathrm{p}\mathrm{m}$.

Beyond the dramatic variations in lineshape as a function of the adsorption
site relative to the moiré lattice, recorded at the centers of the Fe atoms,
we observe remarkable spatial variations also in the immediate vicinity of the
adatoms. We illustrate these local variations for Fe atoms located at the
minimum of the moiré structure (Fig. S4). (Corresponding data for an Fe atom
close to the maximum are shown in the SM [27].) At a moiré minimum, spectra
recorded directly above the center of the Fe atom show pure inelastic steps.
Significant additional structure is observed off center near one of the
vertices of the triangle (purple cross in Fig. S4a). Here, the spectrum
exhibits substantial bias asymmetry, a conductance overshoot just above the
inelastic threshold at positive bias, and an additional dip just below
threshold at negative bias (Fig. S4b). We can fit this spectrum within the
perturbative approach [21] using the same exchange coupling as for the
spectrum taken above the center, but a larger potential scattering amplitude
$U$. More generally, all spectra recorded in the immediate vicinity of the Fe
atom can be described by adjusting $U$ (as well as the overall tunneling
strength) and keeping $J\rho$ constant (see Fig. S4d for a set of fits for
spectra taken along a high-symmetry axis of the Fe–S complex).

We map out the asymmetry underlying the variations of $U$ by plotting the
d$I$/d$V$ signal at the energies of the dip and the overshoot (Fig. S4e,f).
Both maps reveal that these features are most pronounced at the vertices of
the topographic triangular shape (see Fig. S4a). The variation in $U$ can be
understood as an interference effect originating from parallel tunneling paths
[28, 29, 30], similar to tunneling through metal-organic complexes [31, 32].
To gain insight into the tunneling paths, we record d$I$/d$V$ spectra for a
larger bias-voltage range (Fig. S4c). Spectra taken at the center of the Fe
atoms reveal a wide slope across the Fermi level, but are otherwise
featureless, lacking distinct resonances. In contrast, we observe a resonance
at $\sim$150\text{\,}\mathrm{m}\mathrm{V}$$ at the triangle’s vertices. This
resonance has its largest intensity at the vertices, as seen in the spatial
map in Fig. S4g, suggesting the formation of a hybrid Fe–S state with weight
concentrated on the S atoms. The hybridization is sufficiently strong to give
the Fe atoms their triangular shape in the STM images, despite the saturated
bonds within the $\text{MoS}_{\text{2}}$ layer.

The nearly symmetric lineshape at the center of the Fe–S complex indicates a
dominant tunneling path with negligible potential scattering, which is due to
cotunneling via the spin-carrying orbitals. The more pronounced bias asymmetry
at the vertices originates from enhanced potential scattering $U$, which we
associate with cotunneling via another orbital. Most likely, this orbital can
be identified with the resonance at
$\sim$150\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}$$ as the spatial variations
of its map are correlated with the spatial variations of $U$. The total
tunneling amplitude emerges from interference between these cotunneling paths.
As a function of tip location, their relative contributions vary in accordance
with the orbital wave functions, explaining the spatial variations in the
potential scattering strength $U$ [32].

In conclusion, monolayers of $\text{MoS}_{\text{2}}$ on Au(111) act as
decoupling layers for paramagnetic Fe atoms on metallic substrates. The
decoupling efficiency is widely tunable due to the moiré modulation of the
density of states. As a function of adatom location relative to the moiré
lattice, we observe lineshapes ranging all the way from pure inelastic
excitations to fully developed Kondo resonances. Concomitant with increasing
Kondo correlations, the single-ion anisotropy is reduced. Moiré structures can
thus be employed for tuning the magnetic coupling strength, highlighting the
importance of local properties of decoupling layers. Moreover, atomic-scale
variations of the tunneling lineshapes originate from inelastic excitations
via different orbitals, indicating the formation of Fe–S hybrid states despite
the van der Waals character of $\text{MoS}_{\text{2}}$.

##  I Methods

We have performed STM experiments on Fe adatoms deposited on single-layer
molybdenum disulfide islands grown on a Au(111) single crystal. The
experiments were carried out at a base temperature of
$1.1\text{\,}\mathrm{K}$. The Au single-crystal surface was cleaned by
repeated sputter-anneal cycles until a clean, atomically flat surface was
obtained. The $\text{MoS}_{\text{2}}$ islands were grown by depositing Mo
atoms on the Au(111) surface and subsequent annealing to 800 K in H2S gas at a
pressure of p=$10^{-5}$ mbar [19, 17]. The as-prepared layers were decorated
with Fe atoms by evaporation onto the sample held at
$\sim$10\text{\,}\mathrm{K}$$. Differential conductance spectra were recorded
in constant-height mode, using a lock-in amplifier. We used a modulation
frequency of $f=$911\text{\,}\mathrm{Hz}$$. Modulation voltage and set point
conditions are given in the figure captions. The grid of spectra were analyzed
using the freeware SpectraFox [33].

###### Acknowledgements.

We thank David Jacob for discussions and acknowledge financial support by
Deutsche Forschungsgemeinschaft through TRR 227, project B05.

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## Supplementary Material

##  S1 Identification of Fe adatoms on defects

Figure S1: a) Close-up view of Fe atoms adsorbed on a $\text{MoS}_{\text{2}}$
island. The moiré structure is seen in the background. The Fe atoms appear
either triangular or circular shaped. Image recorded at
$V=$50\text{\,}\mathrm{m}\mathrm{V}$$, $I=$100\text{\,}\mathrm{pA}$$. b)
Differential conductance spectra recorded on a triangular (blue) and circular
(orange) shaped Fe atom. The spectrum on $\text{MoS}_{\text{2}}$ (gray) is
shown for reference, revealing the well-known band gap with conduction band
onset at 0.5 V and valence band onset at -1.4 V [1, 2, 3]. Spectra taken after
opening the current feedback at a setpoint of $V=$1.5\text{\,}\mathrm{V}$$,
$I=$300\text{\,}\mathrm{pA}$$, lock-in modulation
$V_{\mathrm{rms}}=$5\text{\,}\mathrm{mV}$$.

As described in the main manuscript, the majority of Fe atoms are imaged as
triangular shaped protrusions at low bias voltage. In contrast, a minority of
protrusions is of circular shape (Fig. S1a). A clear distinction between the
triangular and circular shaped Fe atoms can also be found in their
differential conductance spectra in a larger energy range (Fig. S1b). The
triangular atoms exhibit states above 0.5 V, but no states within the occupied
region. This is in contrast to the findings on the round-shaped Fe atoms,
which show substantial conductance in a broad energy range below -0.2 V. We
also observe a narrow peak at zero bias, which may be associated to a Kondo
resonance.

To unravel the origin of the different types of Fe atoms, we remove all Fe
atoms by scanning at small tip-sample distance ($I=$25\text{\,}\mathrm{nA}$$,
$V=$50\text{\,}\mathrm{mV}$$). The subsequently recorded STM image reveals
several local defects (Fig. S2). Comparison to the previous STM image with the
Fe atoms, we conclude that the round shaped Fe atoms had been located on
defects. While we have shown an example of one Fe atom on a defect site in
Fig. S1b, we remark that Fe atoms on different defects showed a large variety
of d$I$/d$V$ features which we disregard in this paper.

Figure S2: a) STM topography image of an Fe-decorated $\text{MoS}_{\text{2}}$
island. The majority of the Fe atoms appears triangular shaped, while some
atoms take on different shapes (marked by orange circles). b) STM topography
image of the same area as in (a) taken after removing all Fe atoms. This has
been achieved by scanning at small bias voltage and large currents
($V=$4.7\text{\,}\mathrm{m}\mathrm{V}$$, $I=$25\text{\,}\mathrm{nA}$$). The
locations of the atoms which had been imaged with non-triangular shape are
marked again by orange circles. Topographies were recorded at a setpoint of
$V=$50\text{\,}\mathrm{m}\mathrm{V}$$, $I=$100\text{\,}\mathrm{pA}$$.

##  S2 Determination of the potential scattering $U$ at an Fe atom close to
the moiré minimum

In the main manuscript we discussed the differences between spectra recorded
on and in the immediate vicinity of the Fe atoms. When the Fe atoms sit on or
close to the minima of the moiré structure, the d$I$/d$V$ lineshapes can be
fitted by calculating the conductance including inelastic electron scattering
up to second order in Born approximation and potential scattering [4]. These
fits allow for an evaluation of the variations in exchange and potential
scattering strength. Here, we present a set of spectra taken along one of the
high-symmetry axes of the Fe–S complex (spectra in Fig. S3a along dashed line
in S3b). From the corresponding fits, we deduce that the exchange scattering
$J\rho$ does not vary, while the potential scattering $U$ varies across the
Fe–S complex (S3c). We observe the largest values of $U$ at the triangle’s
vertex and a gradual decrease towards the center.

Figure S3: a) Stacked plot of d$I$/d$V$ spectra taken along the orange dashed
line in (b), spectra are offset for clarity. Every second spectrum was also
plotted in Fig. 3d of the main manuscript. b) Differential conductance signal
extracted from a densely-spaced grid of spectra across an Fe atom on the moiré
minimum at $V=$-2.5\text{\,}\mathrm{m}\mathrm{V}$$ (same as in Fig. 3e in the
main text). c) Extracted values for the potential scattering parameter $U$
obtained by fits according to Ref. [4] with $J\rho=-0.132$ and $D=2.75$ meV
kept constant (fits shown as black dashed lines in (a)). The spectra in (a)
and (b) were recorded at a setpoint of $V=$10\text{\,}\mathrm{m}\mathrm{V}$$,
$I=$1\text{\,}\mathrm{nA}$$, with a lock-in modulation of
$V_{\mathrm{rms}}=$50\text{\,}\mathrm{\SIUnitSymbolMicro V}$$.

##  S3 Local variations of lineshapes on Fe atoms close to the moiré maximum

In the main manuscript we investigated the spatial variations of spectra in
close vicinity of an Fe atom adsorbed on a minimum of the moiré structure. The
reason for this choice was that the Fe atom was one of the most weakly coupled
to the substrate such that the excitations could be described within second-
order perturbation theory, and the lineshape could be fitted by the code
developed by Ternes [4]. In Fig. S4 we provide complementary data on an Fe
atom adsorbed at a rim towards a maximum of the moiré structure. As explained
in the main manuscript, the d$I$/d$V$ lineshape of the low-energy spectrum
recorded on the center of the atom is highly asymmetric in bias voltage (blue
spectrum in Fig. S4b). The lineshape at the vertex of the triangular shape is
again different compared to the center (purple spectrum in Fig. S4b). The
spectral variation along one of the (expected) threefold symmetry axes of the
Fe–S complex is plotted in Fig. S4d. The spatial variations are additionally
mapped by plotting the d$I$/d$V$ signal at $-0.9$ mV and $1.6$ mV from a
densely-spaced grid of d$I$/d$V$ spectra (Fig. S4e,f). Similar to the case of
the Fe atoms on the moiré minima, the asymmetry is enhanced at the vertices of
the Fe–S complex. However, the expected threefold symmetry of the Fe–S complex
is broken by the superposition of the moiré modulation, which additionally
contributes to the lineshape variations.

To probe the correlation of the low-energy variations with the higher energy
electronic structure, we plot the d$I$/d$V$ spectra in a larger energy range
in Fig. S4c. On the Fe center, the spectrum consists of a broad slope, while a
spectrum on a vertex shows a faint peak at
$\sim$85\text{\,}\mathrm{m}\mathrm{V}$$. We extract its spatial intensity
distribution from a densely-spaced grid of spectra (Fig. S4g). It reflects the
same symmetry as the low-energy features in Fig. S4e,f, corroborating again
the correlation between the orbital structure and lineshape of inelastic
excitations. Although we cannot fit these spectra and therefore extract values
of $U$, the asymmetry variations are in agreement with a variation of
potential scattering as has been quantified from Fe atoms on the moiré
minimum. We thus suggest that the variations in spectral features can be
explained by the different contributions to the interfering tunneling paths.

Figure S4:  a) STM topography image of an Fe atom close to a maximum of the
moiré structure ($V=$10\text{\,}\mathrm{m}\mathrm{V}$$,
$I=$1\text{\,}\mathrm{nA}$$). b,c) Differential conductance spectra on the
center (blue cross) and vertex (purple cross) of triangular shape of the Fe
adatoms. d) Stacked plot of spectra (offset for clarity) along the orange
dashed line indicated in e). e-g) STM topographies (blue-yellow, background)
with superimposed differential conductance signal at the indicated bias
voltage (black-white dots, scale given below panels) extracted from a densely
spaced grid of spectra across the Fe atom at the indicated energies. The
spectra in (b), (e) and (f) were recorded at a setpoint of
$V=$10\text{\,}\mathrm{m}\mathrm{V}$$, $I=$1\text{\,}\mathrm{nA}$$, the
spectra in (c) and (g) at $V=$10\text{\,}\mathrm{m}\mathrm{V}$$,
$I=$20\text{\,}\mathrm{pA}$$, with an additional retraction of the tip by
$z=$20\text{\,}\mathrm{pm}$$. The lock-in modulation amplitude for (b), (e)
and (f) was $V_{\mathrm{rms}}=$50\text{\,}\mathrm{\SIUnitSymbolMicro V}$$, for
(c) and (g) $V_{\mathrm{rms}}=$1\text{\,}\mathrm{mV}$$.

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