# Anomalous excitations of atomically crafted quantum magnets

Sascha Brinker1,∗ 1Peter Grünberg Institut and Institute for Advanced
Simulation, Forschungszentrum Jülich & JARA, Jülich D-52425, Germany
2Max Planck Institute of Microstructure Physics, Halle 06120, Germany
3Faculty of Physics, University of Duisburg-Essen and CENIDE, 47053 Duisburg,
Germany  Felix Küster2 1Peter Grünberg Institut and Institute for Advanced
Simulation, Forschungszentrum Jülich & JARA, Jülich D-52425, Germany
2Max Planck Institute of Microstructure Physics, Halle 06120, Germany
3Faculty of Physics, University of Duisburg-Essen and CENIDE, 47053 Duisburg,
Germany  Stuart S. P. Parkin2 1Peter Grünberg Institut and Institute for
Advanced Simulation, Forschungszentrum Jülich & JARA, Jülich D-52425, Germany
2Max Planck Institute of Microstructure Physics, Halle 06120, Germany
3Faculty of Physics, University of Duisburg-Essen and CENIDE, 47053 Duisburg,
Germany  Paolo Sessi2 1Peter Grünberg Institut and Institute for Advanced
Simulation, Forschungszentrum Jülich & JARA, Jülich D-52425, Germany
2Max Planck Institute of Microstructure Physics, Halle 06120, Germany
3Faculty of Physics, University of Duisburg-Essen and CENIDE, 47053 Duisburg,
Germany  Samir Lounis1,3,∗ 1Peter Grünberg Institut and Institute for Advanced
Simulation, Forschungszentrum Jülich & JARA, Jülich D-52425, Germany
2Max Planck Institute of Microstructure Physics, Halle 06120, Germany
3Faculty of Physics, University of Duisburg-Essen and CENIDE, 47053 Duisburg,
Germany

###### Abstract

High energy resolution spectroscopic studies of quantum magnets proved
extremely valuable in accessing magnetodynamics quantities, such as energy
barriers, magnetic interactions and lifetime of excited states. Here, we
explore the evolution of a new flavor of low-energy spin-excitations for
quantum spins coupled to an electron bath. In sharp contrast to the usual
tunneling signature of two steps symmetrically centered around the Fermi
level, we find a single step in the conductance. Combining time-dependent and
many-body perturbation theories, magnetic field-dependent tunneling spectra
are explained as the result of an interplay between weak magnetic anisotropy
energy, magnetic interactions and Stoner-like electron-hole excitations that
are strongly dependent on the magnetic states of the nanostructures. The
results are rationalized in terms of a non-collinear magnetic ground state and
the dominance of ferro- and antiferromagnetic interactions. The atomically
crafted nanomagnets offer an appealing model for the exploration of
electrically pumped spin systems.

###  0.1 Teaser:

Novel paradigmatic signature of spin-excitations offers a new exploration
platform for electrically pumped spin systems.

## Introduction

Atomic-scale magnetodynamics is at the cornerstone of spin-based nanoscale
devices for future information technology. It defines how and on which
timescales magnetic states can be controllably manipulated. The interaction of
local spins with the local environment plays a crucial role in determining
their properties. Orbital hybridization effects, charge transfer, and the
presence of nearby impurities have all been reported to strongly impact the
magnetic ground state, ultimately determining a range of magnetodynamic
quantities including magnetic anisotropy, spin lifetime, and spin relaxation
mechanisms. (?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?). The development of
experimental techniques capable of directly capturing these properties in
reduced dimensions allows one to clearly visualize the breakdown of classical
or semiclassical descriptions of magnetic phenomena at subnanometer scales,
revealing the emergence of exquisite quantum mechanical effects. (?, ?, ?, ?,
?, ?) These achievements enabled the understanding of the ultimate limits of
classical computational schemes while simultaneously setting the ground for
testing ideas and concepts which might find direct application in innovative
spin-based quantum computational schemes. (?, ?, ?, ?, ?).

In this context, scanning tunneling spectroscopy techniques (STS) occupy a
primary role. They were particularly instrumental in detecting not only
several ground-state phenomena but also a wealth of fundamental excitations
such as those with magnetic (?, ?, ?, ?, ?, ?, ?, ?, ?, ?), vibrational (?),
polaronic (?), excitonic (?) or Kondo characteristics (?, ?, ?, ?, ?, ?). The
analysis of the distinct spectral shapes identified in tunneling transport
sheds light on their origin and ideally a link to a specific excitation
process. For instance, measurements in an applied magnetic field have proven
extremely useful in disentangling magnetic and non-magnetic excitations.
However, dissimilar and in some cases competing processes can be
simultaneously at play, which can make the analysis complicated. This requires
a careful theoretical-experimental investigation to unveil the fundamental
mechanisms at play.

The Kondo effect versus spin-excitation paradigm is a particularly intriguing
case. (?, ?, ?, ?) Both phenomena share a similar magnetic origin. However,
they are profoundly different in nature, ultimately leading to opposing
technological implications. On the one hand, spin-excitations emerge from the
promotion of a magnetic moment from its ground state to various excited
states. Since spin-excitations play a crucial role in how magnetic bits are
written, understanding these processes is key in information technology. In
tunneling experiments, inelastic spin-excitations are expected to appear in
the differential conductance as symmetric steps centred around the Fermi
energy, which shift in energy in response to an applied magnetic field. (?)
Their spectral shape encodes information directly related to the reading and
manipulation of spins, such as the lifetime of the excited states and the
related energy barriers (e.g. the magnetic anisotropy energy). (?, ?, ?, ?) On
the other hand, the Kondo effect emerges from the entanglement of the spin of
an atom with that of the surrounding bath of electrons (?, ?), which leads to
many-body screening of the spin below the energy scale defined by the Kondo
temperature. Consequently, although of great fundamental interest in the
context of many-body physics, the Kondo effect is of limited technological
importance. Spectroscopically, the Kondo effect is revealed in complicated
spectral shapes that are generally analyzed as Fano resonances. (?, ?, ?)
Consequently, a clear identification of the underlying mechanisms is often
complex since many fitting parameters need to be used, especially when multi-
orbitals are involved. (?, ?) There are several examples, where the same
spectral shape is found whether one invokes a Kondo effect or simply considers
the molecular/atomic orbitals or a combination of both, which sometimes can be
arbitrary (?, ?, ?, ?). In this context, recent ab-initio simulations
challenged the Kondo versus conventional spin-excitation picture. By including
spin-orbit coupling, the well-known zero-bias anomalies experimentally
observed for Co adatoms on Cu, Ag and Au surfaces (?) were systematically
explained as gapped unconventional spin excitations, calling for new
experiments to verify their origin.

In this contribution, we report on the emergence of anomalous spectroscopic
signatures centered around the Fermi energy for Cr single atoms coupled to
metallic niobium, their magnetic field dependence, and their evolution once
adatoms are coupled through the creation of atomically crafted
antiferromagnetic and ferromagnetic nanostructures. Although their spectral
shape might be reminiscent of Kondo-resonances, our first-principles
simulations identify them as new paradigmatic spectroscopic manifestations of
spin excitations. While the observation of spin-excitations of
antiferromagnetic multi-spin nanostructures has so far been explored to a
great extent on thin insulating substrates (see e.g. Refs. ?, ?, ?), here we
demonstrate on a metallic substrate the design of antiferromagnetic or
ferromagnetic magnets composed of a few exchange-coupled Cr atoms that enables
the tailoring of the shape and linewidth of the corresponding spectroscopic
features, which are intimately linked to the adatoms’ weak magnetic anisotropy
energy and the strong dependence of the excitation lifetime on the underlying
magnetic texture.

## Results

###  0.2 Adatoms.

Figure 1:  Experimental and theoretical inelastic STS fingerprint of a Cr
adatom on the Nb(110) surface with varying external magnetic fields. a
Experimental topography image of the adatom. b / c Experimental
$\mathrm{d}I/\mathrm{d}U$ and $\mathrm{d}^{2}I/\mathrm{d}U^{2}$ spectra,
$I=$3\text{\,}\mathrm{nA}$$ and $T=$600\text{\,}\mathrm{mK}$$. d Density of
spin-flip excitations obtained from TD-DFT. e Theoretical inelastic STS
spectrum corresponding to the renormalized local density of states from MBPT
calculated at a height of $4.7\text{\,}\mathrm{\SIUnitSymbolAngstrom}$ above
the adatom. f Similar to e with the spin-resolved theoretical spectra. The
different field-dependent spectra are shifted to improve visibility.

As a prototypical platform, we focus on Cr atoms coupled to the (110) surface
of Nb. The Nb crystal was prepared according to the procedures described in
Ref. ?. Single Cr adatoms were deposited directly onto a cold substrate, as
described in the Methods section. All atoms are found to be adsorbed in hollow
stacking sites, which are energetically most favourable (?). Figure 1a shows a
region of the Nb surface where blue dots correspond to Cr adatoms. After
deposition, atomic manipulation has been used to create a nanostructure with
an isolated Cr atom, which is well-separated (distance
$>$6\text{\,}\mathrm{nm}$$) from all surrounding atoms as shown in Fig. 1a.
This allows us to analyze the effect of local magnetic moments coupled to an
electron bath, while excluding the influence of nearby adatoms which can
couple to each other by indirect interactions mediated by the conduction
electrons, i.e. the RKKY coupling, ultimately influencing the magnetic adatom
ground state (?).

Figure 1b shows a sequence of spectra, measured in a small energy range
centered around the Fermi level, where the tip was positioned directly on top
of the Cr adatom. All spectroscopic data have been acquired in the presence of
a magnetic field applied perpendicular to the sample surface. Starting from a
value of $1\text{\,}\mathrm{T}$, which is necessary to drive the otherwise
superconducting Nb substrate ($T_{\mathrm{c}}=$9.3\text{\,}\mathrm{K}$$ and
$B_{\mathrm{c}}=$0.8\text{\,}\mathrm{T}$$) into the normal metallic regime,
the strength of the magnetic field is progressively increased up to
$6\text{\,}\mathrm{T}$. This procedure allows us to map the evolution of the
spectroscopic data and to clearly assess the magnetic origin of the
excitations. The $\mathrm{d}I/\mathrm{d}U$ spectra are characterized by an
asymmetric step-like feature around the Fermi level which shows a significant
magnetic field-dependence. The step starts at negative bias voltage and
reaches its maximum at positive bias. By increasing the magnetic field, the
step broadens. Moreover, an additional feature emerges at negative voltage
starting from $3\text{\,}\mathrm{T}$. This effect is more clearly seen by
looking at the derivative ($\mathrm{d}^{2}I/\mathrm{d}U^{2}$) reported in Fig.
1c, where the magnetic field-dependent broadening is clearly visible. At the
maximum magnetic field available in our set-up, the broadening appears to
split into two distinct features symmetrically centered around the Fermi level
(additional experimental plots and fits are presented in Supplementary Notes
2–4). As described in the following, spin-excitations can account for both the
spectral shape as well as its magnetic field dependence.

To explore the origin of the anomalous spectral feature, we use a combination
of relativistic time-dependent density functional theory (TD-DFT) and many-
body perturbation theory (MBPT) implemented in the framework of the Korringa-
Kohn-Rostoker Green function method (?, ?, ?) (see Methods section). The ab-
initio nature of the theoretical approach allows us to address the
investigated spectroscopic phenomena without the use of adjusting parameters.
Thus, we do not perform or rely on a fitting procedure of the experimental
data. This framework is ideal to treat the ground state and spin-excited state
of adatoms on surfaces as shown in various studies (?, ?, ?) as well as their
theoretical inelastic tunneling spectra (?, ?, ?). The Cr adatom carries a
spin moment of $3.53\text{\,}\mathrm{\mu_{\text{B}}}$ characterized by a weak
magnetic anisotropy energy, $0.20\text{\,}\mathrm{meV}$, slightly favoring an
in-plane orientation of the magnetic moment (the local density of states is
presented in Supplementary Note 1). However, in the presence of a magnetic
field perpendicular to the substrate, the spin moment points out-of-plane.
Owing to the weak magnetic anisotropy of the Cr spin moment, it can be excited
by the tunneling electrons at very low energies. Our theory allows us to
effectively capture these processes by quantifying the excitation probability
in terms of the density of spin-flip excitations, which can be extracted from
the imaginary part of the transverse spin-spin susceptibility,
$-1/\pi\,\mathrm{Im}\,\chi_{+-}(\omega)$, shown in Fig. 1d. The density of
spin-excitations encodes two main features: (i) the energy of the spin
excitation $\omega_{0}$ directly related to the magnetic anisotropy energy and
applied magnetic field, and (ii) their lifetime, which is inversely
proportional to the broadening (full width half maximum). The origin of this
broadening are Stoner-like electron-hole excitations provided by the highly
itinerant electrons of the metallic substrate. Such single particle
excitations are emitted while strongly contributing to the decay of spin-
excitations. (?, ?).

In contrast to a scenario where the adatoms are deposited on a insulator, the
magnetic moments on a metal experience additional effects with respect to
usual interactions forming a Heisenberg Hamiltonian such as the magnetic
anisotropy, Zeeman energy and magnetic exchange interactions. The reason is
the nature of electrons forming moments, which are delocalized and extend
beyond the atomic site, and thus do not respond uniformly to external
perturbation. Therefore, while the magnetic moment undergoes a spin-flip
process, which is by nature a collective excitation process involving many
electrons, single particle excitations, wherein a single electron may be
excited from a majority-spin state below the Fermi level to a minority-spin
state above, kick in and impact the overall dynamical behavior of the magnetic
moment. The inhomogeneous reaction of the electrons broaden then the ideal
response of the spin-moment, leading to an effective decay, or damping, of the
spin-excitation. We note that such electron-hole excitations are associated
with spin-currents discussed in the context of spin-pumping experiments (?, ?,
?). The larger the number of electron-hole excitations, the stronger the
damping and the shorter is the lifetime of the spin-excited states. As can be
recognized from Fig. 1d, our model effectively accounts for all experimental
trends, both in the energy shift as well as in the broadening (Fig. 1b-c).

When the magnetic moment of the atom is excited, the triggered spin-
excitations affect the electron density resulting in the spectroscopic
signatures visible in the theoretically calculated tunneling spectra reported
in Fig. 1e, which are obtained from a many-body perturbation theory (see
Methods). In good agreement with the experimental measurements, we recover a
step-like signature of the spin excitation with a clear field-dependent
broadening which progressively evolves towards a double peak structure at high
magnetic field. The origin of this behavior is effectively captured in Fig.
1f, which illustrates the magnetic field evolution of the spin-resolved
tunneling spectra for majority (continuous line) and minority (dashed line)
spins. A single peak feature at negative energies and a step-like function
with onset at positive energies are visible for majority and minority spin
channels, respectively. The energy-separation between the majority and
minority spectral signatures becomes larger by progressively increasing the
strength of the magnetic field. We note that besides Cr adatom, Mn also
displays a similar but weaker step-like zero-bias anomaly, which is nicely
reproduced by our ab-intio simulations as shown in Supplementary Note 5.

To grasp analytically the connection between the various magnetodynamical
parameters shaping the unveiled spectroscopic anomalies, it is useful to map
the TD-DFT results to the equation of motion of a spin-moment, the Landau-
Lifshitz-Gilbert equation (?, ?, ?),
$\frac{d\mathbf{M}}{dt}=\gamma\mathbf{M}\times\frac{d{H}}{d\mathbf{M}}+\frac{\alpha}{M}\mathbf{M}\times\frac{d\mathbf{M}}{dt}$
subjected to a Heisenberg hamiltonian $H$ (more details are given in the
Method section and Supplementary Notes 6 and 8). The field dependence of the
excitation energy obeys $\omega_{0}=\gamma^{\prime}(\mathcal{K}+BM)/M$, while
the broadening (inverse lifetime) of the excitation is given by
$\alpha\gamma^{\prime}(\mathcal{K}+BM)/M$, where $\mathcal{K}$ is the magnetic
anisotropy energy which represents the costs in energy in rotating the
magnetic moments from $z$ towards the $x$ and $y$ directions (corresponding to
the $[1\overline{1}0]$ and $[001]$ direction, respectively). $M$ is the
magnetic moment, $B$ is an out-of-plane magnetic field, $\gamma^{\prime}$ is
the renormalized gyromagnetic ratio, and $\alpha$ is the so-called Gilbert
damping parameter, which quantifies the probability of creating electron-hole
excitations responsible for the decay of the spin-excitations. This
theoretical description allows us to effectively capture the following trends:
(i) the excitation energy becomes higher by increasing the strength of the
magnetic field acting on the magnetic moment, (ii) this is accompanied by the
damping that becomes progressively stronger by enhancing the amplitude of the
electron-hole excitations. Overall, this approach allows the visualization of
the existence of competing trends in the manifestation of spin-excitations.
While an enhanced magnetic anisotropy facilitates their detection, it also
simultaneously enables the creation of more electron-hole pairs, increasing
the broadening, ultimately washing out their spectroscopic signature. Overall,
this could explain why identifying the spin-excitation origin of similar
spectra observed for Co on Cu(001) and Ag(001) surfaces (?, ?, ?),
characterized by a larger magnetic anisotropy energy than for Cr on Nb(110),
is challenging since this requires extremely large magnetic fields and
therefore has escaped experimental observation to date.

###  0.3 Dimers.

Figure 2:  Theoretical and experimental inelastic STS fingerprint of the Cr dimers along the [001], [1$\mathbf{\overline{1}}$0], and [1$\mathbf{\overline{1}}$1] directions on the Nb(110) surface. a Illustration of the dimers (STM topography) and its grounds state magnetic structures. The magnetic structures are obtained from self-consistent non-collinear DFT calculations. b Experimental inelastic STS spectra of the dimers, $I=$3\text{\,}\mathrm{nA}$$ and $T=$600\text{\,}\mathrm{mK}$$ (coloured lines). The measurements on the single Cr adatom are shown as reference (thin grey lines, see also Fig. 1). The different spectra are vertically shifted to improve visibility. c Magnetic susceptibility obtained from TD-DFT. The spectra of the $[001]$ and $[1\overline{1}1]$ dimer are scaled by a factor of $50$ and $2$, respectively.  | Cr1 | Cr${}_{2}[1\overline{1}0]$ | Cr${}_{2}[001]$ | Cr${}_{2}[1\overline{1}1]$ | Cr${}_{3}[1\overline{1}0]$ | Cr${}_{3}[001]$ | Cr${}_{3}[1\overline{1}1]$
---|---|---|---|---|---|---|---
$M$ [$\text{\,}\mathrm{\mu_{\text{B}}}$] | 3.53 | 3.52 | 3.51 | 3.53 | 3.52 | 3.51 | 3.54
$J_{1}$ [$\text{\,}\mathrm{meV}$] | – | 1.24 | 7.94 | -8.95 | 1.31 | 7.97 | -7.35
$J_{2}$ [$\text{\,}\mathrm{meV}$] | – | – | – | – | -0.20 | -0.31 | -2.38
$\mathcal{K}$ [$\text{\,}\mathrm{meV}$] | 0.20 | 0.20 | 0.20 | 0.20 | 0.21 | 0.17 | 0.20
Table 1:  Ground state properties of the considered Cr-based nanostructures
deposited on the Nb(110) surface. Shown are the magnetic moments per atom $M$,
which are averaged in case of the trimers, the next-nearest neighbor (NNN)
isotropic exchange interaction $J_{1}$, the next NNN isotropic exchange
interaction $J_{2}$ along a given direction and the magnetic anisotropy energy
$K$, per atom. Positive (negative) $J$ corresponds to an antiferromagnetic
(FM) coupling, while a positive magnetic anisotropy energy indicates a
preference for an in-plane orientation of the spin moments.

To further explore the consistency of the experimental data with the spin-
excitation paradigm, we use atomic manipulation techniques to artificially
create magnetic dimers oriented along different crystallographic directions on
the Nb(110) surface. We consider three next-nearest neighbor configurations:
$[1\overline{1}0]$, $[1\overline{1}1]$, and $[001]$, illustrated in Fig. 2a.
The highly anisotropic (110) surface enables us to create and scrutinize
distinct direction-dependent magnetic ground states utilizing the coupling
between the adatoms mediated by the conduction electrons. This effect is
expected to be directly imprinted in the spin-excitation spectra which should
show a clear direction-dependence. STS measurements confirm the existence of a
strong direction-dependence behavior, as clearly visible in Fig. 2b. A single
step-like spectroscopic signature centered around the Fermi level is visible
for all the three distinct directions. However, a strong direction-dependent
deviation with respect to the signal obtained for the isolated single Cr atom
is evident, both in the intensity and in the broadening. The
$[1\overline{1}0]$ dimer is characterized by the largest separation and
therefore the weakest coupling between the adatoms, resulting in only very
minor differences with respect to the isolated atom case. In sharp contrast,
both the $[1\overline{1}1]$ and the $[001]$ dimers show a significant
broadening of the step-like feature. The spectrum corresponding to the $[001]$
dimer hosts a deep-like feature offset from zero-bias, surrounded by two
maxima that are not symmetrically located with respect to the Fermi energy. In
general, these observations are in line with the theoretically calculated
strength of the magnetic exchange interactions along different
crystallographic directions summarized in Table 1.

In the following, we present a theoretical analysis which allows us to
effectively understand the origin of the distinct behaviour observed for
dimers aligned along $[1\overline{1}1]$ and $[001]$ direction, where the
magnetic coupling is the strongest. Since the adatoms are not nearest
neighbors, the single ion anisotropy of each of the adatoms remains constant
(see Table 1). However, the nature of the coupling is opposite along the two
different directions, resulting in a strong antiferromagnetic
($7.94\text{\,}\mathrm{meV}$) and ferromagnetic interaction
($-8.95\text{\,}\mathrm{meV}$) for $[001]$ along the $[1\overline{1}1]$
directions, respectively. These predictions are in line with the experimental
observations reported in Supplementary Note 11, where the magnetic coupling
between next-nearest neighbours adatoms has been scrutinized by using a
functionalized superconducting tip (a similar approach was introduced in Ref.
?). As demonstrated in the adatoms subsection, the analysis of the spin-
excitations spectra can be carried out on the basis of the density of spin-
flip excitations, which provide the intrinsic inelastic spectra. These are
obtained from the first-principles ground states illustrated in Fig. 2a
resulting from the interplay of the single ion anisotropy, magnetic exchange
interaction and magnetic field. The density of spin-excitations are shown and
compared to that of the isolated Cr adatom in Fig. 2c for each of the atoms
forming the dimers.

Since the $[1\overline{1}1]$ dimer is ferromagnetic (see Fig. 2a and
Supplementary Note 7), the density of spin-excitation hosts a double peak
structure, leading to a redistribution of the spectral weight observed for the
isolated adatom. They represent the acoustical and optical excitation modes,
where in a classical picture the spin moments precess either clockwise or
counter-clockwise, respectively. A simple description on the basis of a
quantum spin model is given in Supplementary Note 9. The position of the
lowest mode, also observed experimentally, is dictated by the magnitude of the
magnetic anisotropy energy ($\gamma^{\prime}\mathcal{K}/M$) and shows a weaker
intensity than that of the excitation mode of the isolated adatom with a
broadening given by $\gamma^{\prime}\alpha\mathcal{K}/M$ (See Supplementary
Note 8). The optical mode however occurs at energies not resolvable within the
experiment since it is dictated by the magnitude of the exchange interaction
$\gamma^{\prime}(\mathcal{K}-J)/M$ leading to a much broader feature
($\gamma^{\prime}\alpha(\mathcal{K}-J)/M$) than the acoustical one.

In contrast to the ferromagnetic dimer, the antiferromagnetic one exhibits a
single broad mode settled by the magnetic exchange interaction
($\gamma^{\prime}\sqrt{4\mathcal{K}(\mathcal{K}+J)-\alpha^{2}\left(J-2\mathcal{K}\right)^{2}}/2M$)
since the excitation process promotes the dimer from a state where the total
spin is 0 to 1. The broadening of the state is given by
$\gamma^{\prime}\alpha(2\mathcal{K}+J)/2M$, which explains the significantly
large damping observed in both the experimental and theoretical spectra shown
for the $[001]$ dimers in Fig. 2b-c.

Overall, our theoretical analysis highlights that antiferromagnetic dimers are
subjected to stronger electron-hole excitations, resulting in spin-excitations
characterized by significantly larger energy broadening, i.e. shorter
lifetimes, compared to the ferromagnetic case. This behavior manifests itself
in a larger energy broadening along the $[001]$ direction as compared to the
$[1\overline{1}1]$ direction, in agreement with our experimental observations.

Similarly to the adatom, we expect the spectra pertaining to the spin-
excitation to result from a complex combination of excitation features hosted
by majority and minority spin electrons at either sides of the bias voltage.
This might explain the peculiar spectrum detected experimentally on the [001]
dimer (see Fig. 2b), which hosts a deep-like feature offset from zero-bias
that is induced by the coupling of the two adatoms (see Supplementary Note
10). A similar kink is detected on the theoretical data of the adatom after
application of a magnetic field (Figs. 1e-f).

###  0.4 From trimers to long chains.

Figure 3:  Cr trimers along the [$\mathbf{001}$], [$\mathbf{1\overline{1}0}$],
and [$\mathbf{1\overline{1}1}$] directions on the Nb(110) surface. a
Illustration of the trimers (STM topography) and its grounds state magnetic
structures at $5\text{\,}\mathrm{T}$. b Experimental inelastic STS spectra of
the trimers at $1\text{\,}\mathrm{T}$ (coloured lines). The measurements on
the single Cr adatom are shown as reference (thin grey lines, see also Fig.
1). c Magnetic susceptibility obtained from TD-DFT at $5\text{\,}\mathrm{T}$.
The spectra of atom B in the $[1\overline{1}0]$ direction and in the $[001]$
direction are scaled by a factor $10$ and $30$, respectively. All trimer
spectra are vertically shifted to improve visibility. The insets show the
magnetic structures obtained from self-consistent non-collinear DFT
calculations.

As identified for the dimers, the distinct spin-excitations spectra of the AFM
and FM dimer is triggered by the alignment of their spin-moments. To confirm
this aspect, we artificially create a trimer along all distinct directions.
This breaks the perfect antiferromagnetic balance while it should not
significantly affect the excitation behavior of the FM configurations. STM
images for all configurations are shown in Fig. 3a. Due to the $C_{2}$
symmetry, the two end atoms (A and A’) can be distinguished from the central
one (B) for all trimers. The inelastic STS spectra obtained at
$1\text{\,}\mathrm{T}$ and $600\text{\,}\mathrm{mK}$ for each Cr atom are
shown in Fig. 3b. Similar to the dimers, a clear correlation between the
strength of the magnetic coupling (see Table 1) and the deviation of the
inelastic STS signal from the single adatom case is visible. Moreover, we also
recover similar trends: the $[1\overline{1}0]$ trimer exhibits the weakest
magnetic coupling and therefore only a weak deviation from the single adatom
spectra is observed, while the observed broadening increases for the stronger
coupled trimers along the $[1\overline{1}1]$ and $[001]$ directions.
Noteworthy, we find two distinct trends depending on the nature of the
magnetic coupling: The ferromagnetic trimer along the $[1\overline{1}1]$
direction shows only a very weak difference between the end atoms (A) and the
central atom (B) in contrast to the antiferromagnetic trimers, with the
$[001]$ case hosting the largest discrepancy.

To explain the observed spectra theoretically, we exploit the density of spin
excitations of the trimers shown in Fig. 3c. For all trimers, the end atoms
host a peak close to the excitation energy of the isolated adatom. The
ferromagnetic $[1\overline{1}0]$ trimer shows peaks at higher energies
depending on the nature of the considered atom. Due to the increased
coordination number of the central atom B, its high-energy peak is higher in
energy than the one of atom A. For the antiferromagnetically coupled trimers
along the $[001]$ and $[1\overline{1}0]$ directions, two fundamentally
different spectra are observed depending on whether atom A or B is excited.
The low energy excitations contrast those observed for the dimers, which
showed only a single broad feature. The fundamental difference between the
antiferromagnetic dimers and trimers is that the net magnetic moment cancels
out in the former while it is finite for the latter. This triggers a mixture
of modes similar to those of the ferromagnetic and antiferromagnetic dimers.
Thus, unlike the dimers, the trimers can respond to an external magnetic field
while preserving their collinear antiferromagnetic configuration. The lowest
mode corresponds to a collinear in-phase motion of all three atoms at an
energy $\omega\approx 3\gamma\mathcal{K}/M$ (see Supplementary Note 8), and
the density of spin excitations in Fig. 3c shows that this mode is possible to
be excited via the end atoms A and A’. In contrast, the central atom B only
excites the modes at higher energy and with significantly increased
broadening, i.e. translating to a larger coupling to electron-hole
excitations, which explains to the observed inelastic STS signatures in Fig.
3b. By progressively increasing the size of the wires, we recover a
spectroscopic behavior similar to that of the trimers (see Supplementary Note
12). We note that a similar sequence of modes was observed in Ref. ?, which
can equivalently be explained in terms of the nodal structure of the
collective spin-excitations hosted by the nanomagnet.

###  0.5 Spin-resolved spectroscopy.

Figure 4:  Spin-polarized STS acquired on a 7 adatoms chain along [001].  a
Spectroscopic data collected by positioning the tip on top of each adatom
along the chain (see numbers). The data reveal the existence of
antiferromagnetic coupling. b Comparison between the STS spectra acquired on
neighboring adatom (3 and 4 in a), highlighting the opposite spin contrast.
Measurements have been acquired at $5\text{\,}\mathrm{T}$ with a Cr-terminated
tip. Stabilization parameters: $V$ = -10 meV, $I$ = 3 nA, and modulation
$V_{rms}$ = 200 $\mu$V.

Finally, we experimentally scrutinize the spin-resolved spectroscopic
signatures for chains oriented along the [001], i.e. when neighboring adatoms
are strongly antiferromagnetically coupled (see also related Supplementary
Notes 11 and 12). This is a particularly interesting scenario, since the
splitting of the zero bias anomaly is the strongest (see section on dimers).
This is expected to result in well-separated spectroscopic signatures for
opposite spin channels, a direct result of the magnetic interaction between
the adatoms which renormalizes the spectrum while simultaneously removing the
spin degeneracy (?). This situation is analogue to the theoretical plots
presented in Fig. 1f for a single adatom. Indeed, the coupling between adatoms
can be considered as an effective magnetic field. Figure 4 a shows the spin
resolved spectroscopic data acquired with a Cr-terminated W tip under a
magnetic field strength of $5\text{\,}\mathrm{T}$ applied perpendicular to the
sample surface. Spectra obtained by positioning the tip on top of each one of
the adatom in the chain are indicated by numbers. An alternating contrast is
clearly visible along the chain, with a slightly different signature for
adatoms localized at the end of the chain (number 1 and 7 in panel a), a
direct consequence of their different local environment compared to adatoms in
the bulk of the chain. The spectral shape is better highlighted in Fig. 4b,
which directly compares the spectra acquired on antiferromagnetically coupled
adatoms i.e. number 3 and 4 in Fig. 4a. The spectra for opposite spin channels
are in very good agreement with the theoretically calculated spectra shown in
Fig. 1f. i.e. a relatively sharp peak center below the Fermi level for the
majority-spin channel, and a broader step-like signature at positive energy
for minority spin electrons.

## Discussion

In conclusion, we systematically unveiled zero-bias anomalies in Cr atom
nanostructures deposited on the metallic Nb(110) surface. Our inelastic STS
measurments reveal an asymmetric single step-like zero bias feature, which is
preserved but can be modulated depending on the strength of the external
magnetic field, the size of the nanostructure, and its spatial orientation on
the substrate. The latter is used to engineer inter-atom magnetic coupling to
be ferromagnetic or antiferromagnetic. We demonstrated the possibility of
tracking these complex spectral line shapes through ab-initio simulations
combining time-dependent density functional and many-body perturbation
theories, evidencing that by constructing the nano-object with the desired
magnetic state, one can control the underlying excitation energies and related
lifetimes by impacting on the number of emitted electron-hole excitations. The
emitted spin-currents accompanying the electron-hole excitations
characterizing each excitation mode might prove useful for spin-pumping
nanoscale devices to explore innovative spintronics concepts based on spin-
transfer/spin-orbit torques and spin-Hall effects, for the manipulation of
quantum spins. Our results shed new light on the interpretation of zero-bias
anomalies and pave a way to engineering low-energy features by manipulating
the spin excitation spectra by utilizing the underlying magnetic structure.

###  0.6 Sample and tip preparation.

To obtain an atomically flat and clean substrate, a commercial Nb single
crystal polished on one side with (110) surface normal was heated up to
$2300\text{\,}\mathrm{K}$ multiple times by short flashes of electron
bombardment and under UHV conditions (?). The procedure is repeated until the
main contaminant oxygen gives way to large areas of clean Nb. In-situ
deposition of Cr atoms is done by keeping the substrate cold below
$15\text{\,}\mathrm{K}$ to prevent migration and to avoid clustering of
adsorbed single atoms. A commercial STM head from Scienta Omicron is employed
to carry out topographic imaging, scanning tunneling spectroscopy measurements
and atomic manipulation of Cr adatoms. Data is acquired in
$10^{-10}$\mathrm{mbar}$$ pressure at a temperare of $600\text{\,}\mathrm{mK}$
and with an applied external magnetic field of $>$$1\text{\,}\mathrm{T}$ using
an etched tungsten tip characterized on Ag(111). Bringing the tip close to the
adatoms by increasing the constant tunneling current set point to about
$70\text{\,}\mathrm{nA}$ at a bias voltage of $-5\text{\,}\mathrm{mV}$ enabled
us to drag the Cr single atoms to the desired lattice positions and to isolate
all the nanostructures under investigation from surrounding Cr atoms by at
least $4\text{\,}\mathrm{nm}$.

###  0.7 First-principles calculations.

The methodology followed in our work is based on a first-principles approach
as implemented in the framework of the scalar-relativistic full-electron
Korringa-Kohn-Rostoker (KKR) Green function augmented self-consistently with
spin-orbit interaction (?, ?). The spin-excitations are described in a
formalism based on time-dependent density functional theory (TD-DFT) (?, ?, ?,
?) including spin-orbit interaction. Many-body effects triggered by the
presence of spin-excitations are approached via many-body perturbation theory
(?, ?, ?) extended to account for relativistic effects (?). The utilized
Multiple-scattering theory enables an embedding scheme, which is versatile for
the treatment of nanostructures in real space. The local spin density
approximation (LSDA) is employed for the evaluation of the exchange-
correlation potential (?) while the full charge density is computed within the
atomic-sphere approximation (ASA). We assume an angular momentum cutoff at
$\ell_{\text{max}}=3$ for the orbital expansion of the Green function and when
extracting the local density of states a k-mesh of $450\times 450$ is
considered. The Cr atoms sit on the hollow stacking site relaxed towards the
surface by $20\%$ of the interlayer distance of the underlying Nb(110)
surface, as found in previous studies (?).

The single-particle Green functions corresponding to the ground-state are
employed for the construction of the tensor of dynamical magnetic
susceptibilities, $\underline{\chi}(\omega)$, within time-dependent density
functional theory (TD-DFT) (?, ?, ?) including spin-orbit interaction. In
practice, we solve the Dyson-like equation relating the full susceptibility to
the Kohn-Sham susceptibility, $\underline{\chi}_{\text{KS}}(\omega)$ as

$\underline{\chi}(\omega)=\underline{\chi}_{\text{KS}}(\omega)+\underline{\chi}_{\text{KS}}(\omega)\,\underline{\mathcal{K}}\,\underline{\chi}(\omega)\quad,$
(1)

where $\underline{\chi}_{\text{KS}}(\omega)$ describes uncorrelated electron-
hole excitations, while $\underline{\mathcal{K}}$ represents the exchange-
correlation kernel, taken in adiabatic LSDA (such that this quantity is local
in space and frequency-independent (?)). The energy gap in the spin excitation
spectrum is accurately evaluated using a magnetization sum rule (?, ?, ?).
The magnetic exchange interactions in the dimers and trimers were obtained
using the magnetic force theorem in the frozen-potential approximation and the
infinitesimal rotation method (?). The on-site magnetic anisotropy of the Cr
atoms in all systems is obtained from the method of constraining fields (?).
More details are given in Supplementary Notes 6 and 7.

Considering the reasonable theoretical recovery of the experimentally observed
low-energy anomaly, we discuss here the impact of the assumptions made within
the simulations. It is clear that the assumed LSDA is performing well in
describing the magnitude of the magnetic anisotropy energy, which dictates the
position of the spin-excitation, and of the magnetic exchange interactions
since the spin-polarized STS experiments are in nice agreement with the ab-
initio simulations. However, it is very possible that a slight shift in the
magnetic anisotropy energy and of the electronic states at the Fermi energy,
which defines the lifetime/broadening of the zero-bias anomalies, by changing
the exchange and correlation potential could improve the agreement between
theory and experiment.

S.B. and S.L. thank M. dos Santos Dias, A. M. Montero, Filipe S. M. Guimarães
for fruitful discussions. Funding: This work is supported by the European
Research Council (ERC) under the European Union’s Horizon 2020 research and
innovation programme (ERC-consolidator grant 681405 — DYNASORE). We
acknowledge the computing time granted by the JARA-HPC Vergabegremium and VSR
commission on the supercomputer JURECA at Forschungszentrum Jülich. Authors
contributions: P.S. and S.L. initiated, designed and supervised the project.
F.K. performed all the STM measurements supervised by S.S.P and P.S.. S.B.
performed the ab-initio simulations supervised by S.L.. S.B. and S.L. wrote
the manuscript to which all authors contributed via discussions and
corrections. Competing Interests: The authors declare that they have no
competing interests. Data and materials availability: All data needed to
evaluate the conclusions in the paper are present in the paper.

This PDF file includes:

Sections S1 to S12
Figs. S1 to S10
Tables S1 to S2

## References and notes

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