# Emergence of zero-field non-synthetic single and catenated antiferromagnetic
skyrmions in thin films

Amal Aldarawsheh  Peter Grünberg Institute and Institute for Advanced
Simulation, Forschungszentrum Jülich and JARA, D-52425 Jülich, Germany  Imara
Lima Fernandes  Peter Grünberg Institute and Institute for Advanced
Simulation, Forschungszentrum Jülich and JARA, D-52425 Jülich, Germany  Sascha
Brinker  Peter Grünberg Institute and Institute for Advanced Simulation,
Forschungszentrum Jülich and JARA, D-52425 Jülich, Germany  Moritz Sallermann
Peter Grünberg Institute and Institute for Advanced Simulation,
Forschungszentrum Jülich and JARA, D-52425 Jülich, Germany  RWTH Aachen
University, 52056 Aachen, Germany  Science Institute and Faculty of Physical
Sciences, University of Iceland, VR-III, 107 Reykjavík, Iceland  Muayad Abusaa
Department of Physics, Arab American University, Jenin, Palestine  Stefan
Blügel  Peter Grünberg Institute and Institute for Advanced Simulation,
Forschungszentrum Jülich and JARA, D-52425 Jülich, Germany  Samir Lounis
Peter Grünberg Institute and Institute for Advanced Simulation,
Forschungszentrum Jülich and JARA, D-52425 Jülich, Germany

###### Abstract

Antiferromagnetic (AFM) skyrmions are envisioned as ideal localized
topological magnetic bits in future information technologies. In contrast to
ferromagnetic (FM) skyrmions, they are immune to the skyrmion Hall effect [1,
2], might offer potential terahertz dynamics [3, 4] while being insensitive to
external magnetic fields and dipolar interactions. Although observed in
synthetic AFM structures [5] and as complex meronic textures in intrinsic AFM
bulk materials [6, 7], their realization in non-synthetic AFM films, of
crucial importance in racetrack concepts, has been elusive. Here, we unveil
their presence in a row-wise AFM Cr film deposited on PdFe bilayer grown on
fcc Ir(111) surface. Using first principles, we demonstrate the emergence of
single and strikingly interpenetrating catenated AFM skyrmions, which can co-
exist with the rich inhomogeneous exchange field, including that of FM
skyrmions, hosted by PdFe. Besides the identification of an ideal platform of
materials for intrinsic AFM skyrmions, we anticipate the uncovered knotted
solitons to be promising building blocks in AFM spintronics.

## Introduction

Magnetic skyrmions are particle-like topologically protected twisted magnetic
textures [8, 9, 10] with exquisite and exciting properties [11, 12]. They
often result from the competition between the Heisenberg exchange and the
relativistic Dzyaloshinskii-Moriya interaction (DMI) [13, 14], which is
present in materials, that lack inversion symmetry and have a finite spin
orbit coupling. Since their discovery in multiple systems, ranging from bulk,
thin films, surfaces to multilayers [15, 16, 17, 18, 19, 20, 21, 22, 23],
skyrmions are envisioned as promising candidates for bits, potentially usable
in the transmission and storage of information in the next generation of
spintronic devices [24, 25, 26, 27, 28, 29]. However, requirements for future
(nano-)technologies are not only limited to the generation of information bits
but are also highly stringent from the point of view of simultaneous
efficiency in reading, control, and power consumption [28, 30].
Miniaturization of ferromagnetic (FM) skyrmions suffers from the presence of
dipolar interactions [31], while their stabilization generally requires an
external magnetic field. Another drawback is the skyrmion Hall effect [11],
caused by the Magnus force that deflects FM skyrmions when driven with a
current, which hinders the control of their motion. Additionally, FM skyrmions
exhibit a rather complex dynamical behavior as function of applied currents
[32, 33, 34, 35, 36, 37, 38], under the presence of defects.

Antiferromagnetic (AFM) skyrmions are expected to resolve several of the
previous issues and offer various advantages. Indeed, AFM materials being at
the heart of the rapidly evolving field of AFM spintronics [39] are much more
ubiquitous than ferromagnets. Their compensated spin structure inherently
forbids dipolar interactions, which should allow the stabilization of rather
small skyrmions while enhancing their robustness against magnetic
perturbations. AFM skyrmions were predicted early on using continuum models
[40], followed with multiple phenomenology-based studies on a plethora of
properties and applications, see e.g. Refs. [41, 42, 2, 43, 44, 45, 46, 47,
48, 49, 50, 51, 52]. The predicted disappearance of the Magnus force, which
triggers the skyrmion Hall effect, would then enable a better control of the
skyrmion’s motion [2, 1], which has been partially illustrated experimentally
in a ferrimagnet [53].

Intrinsic AFM meronic spin-textures (complexes made of half-skyrmions) were
recently detected in bulk phases [6, 54, 7] while synthetic AFM skyrmions were
found within multilayers [5]. However, the observation of intrinsic AFM
skyrmions has so far been elusive, in particular at surfaces and interfaces,
where they are highly desirable for racetrack concepts. A synthetic AFM
skyrmion consists of two FM skyrmions realized in two different magnetic
layers, which are antiferromagnetically coupled through a non-magnetic spacer
layer. In contrast to that an intrinsic AFM skyrmion is a unique magnetic
entity since it is entirely located in a single layer. Here we predict from
first-principles (see Method section) intrinsic AFM skyrmions in a monolayer
of Cr deposited on a surface known to host ferromagnetic skyrmions: A PdFe
bilayer grown on Ir(111) fcc surface as illustrated in Fig. 1a. The AFM nature
of Cr coupled antiferromagnetically to PdFe remarkably offers the right
conditions for the emergence of a rich set of complex AFM textures. The ground
state is collinear row-wise AFM (RW-AFM) within the Cr layer (see inset of
Fig. 1c), a configuration hosted by a triangular lattice so far observed
experimentally only in Mn/Re(0001) [55, 56]. The difference to the latter,
however, is that although being collinear, the Cr layer interfaces with a
magnetic surface, the highly non-collinear PdFe bilayer.

A plethora of localized chiral AFM-skyrmionic spin textures (Fig. 1c) and
metastable AFM domain walls (see Supplementary Figure 1) emerge in the Cr
overlayer. Besides isolated topological AFM solitons, we identify strikingly
unusual interpenetrating AFM skyrmions, which are reminiscent of crossing
rings (see schematic Fig. 1b), the building blocks of knot theory where
topological concepts such as Brunnian links are a major concept [57]. The
latter has far reaching consequences in various fields of research, not only
in mathematics or physics but extends to chemistry and biology. For instance,
the exciting and intriguing process of catenation is paramount in carbon-,
molecular-, protein- or DNA-based assemblies [58, 59, 60]. We discuss the
mechanisms enforcing the stability of the unveiled catenated topological
objects, their response to magnetic fields and the subtle dependence on the
underlying magnetic textures hosted in PdFe bilayer. Our findings are of prime
importance in promoting AFM localized entities as information carriers in
future AFM spintronic devices.

Figure 1: Catenated AFM skyrmions in CrPdFe trilayer on Ir(111). a Schematic
representation of the investigated trilayer deposited on Ir(111) following fcc
stacking. b The catenation of skyrmions is reminiscent of interpenetrating
rings, which realize topologically protected phases. c The ground state being
RW-AFM (see inset) can host AFM skyrmions that can be isolated or catenated to
form multimers of skyrmions, here we show examples ranging from dimers to
pentamers. The AFM skyrmions can be decomposed into FM skyrmions living in
sublattices illustrated in d. In case of the single AFM skyrmion, two of the
sublattices, L1 and L2, are occupied by the FM skyrmions shown in e. L3 and L4
host quasi-collinear AFM spins in contrast to the FM skyrmions emerging in the
case of the double AFM skyrmion presented in f. Note that the separation of
sublattices L1, L2, L3 and L4 shown in e and f is only done for illustration.

## Results

###  0.1 AFM skyrmions in CrPdFe/Ir(111) surface.

PdFe deposited on Ir(111) surface hosts a homo-chiral spin spiral as a ground
state [21, 61] emerging from the interplay of the Heisenberg exchange
interactions and DMI. The latter is induced by the heavy Ir substrate, which
has a strong spin-orbit coupling. Upon application of a magnetic field, sub
10-nm FM skyrmions are formed [21, 61, 62, 63, 25, 64, 65]. After deposition
of the Cr overlayer, the magnetic interactions characterizing Fe are strongly
modified (see comparison plotted in Supplementary Figure 2) due to changes
induced in the electronic structure (Supplementary Figure 3). The Heisenberg
exchange interaction among Fe nearest neighbors (n.n.) reduces by 5.5 meV (a
decrease of 33%). This enhances the non-collinear magnetic behavior of Fe,
which leads to FM skyrmions even without the application of a magnetic field
(see Supplementary Figure 4). The n.n. Cr atoms couple strongly
antiferromagnetically (-51.93 meV), which along with the antiferromagnetic
coupling of the second n.n (-6.69 meV) favors the Néel state. The subtle
competition with the ferromagnetic exchange interactions of the third n.n.
(5.32 meV) stabilizes the RW-AFM state independently from the AFM interaction
with the Fe substrate (the detailed magnetic interactions are shown in
Supplementary Figure 2). As illustrated in Fig. 1c, the RW-AFM configuration
is characterized by parallel magnetic moments along a close-packed atomic row,
with antiparallel alignment between adjacent rows. Due to the hexagonal
symmetry of the atomic lattice, the AFM rows can be rotated in three
symmetrically equivalent directions. We note that the moments point out-of-
plane due to the magnetic anisotropy energy (0.5 meV per magnetic atom).

The n.n. Cr DMI (1.13 meV) is of the same chiral nature and order of magnitude
as that of Fe atoms (1.56 meV), which gives rise to the chiral non-collinear
behavior illustrated in Fig. 1c. We note that the solitons are only observed
if Cr magnetic interactions beyond the n.n. are incorporated, which signals
the significance of the long-range coupling in stabilizing the observed
textures. Since the Heisenberg exchange interaction among the Cr atoms is much
larger than that of Fe, the AFM solitons are bigger, about a factor of three
larger than the FM skyrmions found in Fe.

Figure 2: .Energetics of two catenated AFM skyrmions. a Two overlapping AFM
skyrmions decoupled from the PdFe bilayer with black and blue line
representing two examples of paths along which the lower skyrmion is rigid-
shifted with respect to the upper one, which is pinned. b Two-dimensional map
of the total energy difference with respect to the magnetic state shown in a
as a function of the distance between the skyrmion centers. c Energy profile
along the blue line shown in a. A double minimum is found once the skyrmions
swap their positions and become truly degenerate once the rigidity of the spin
state is removed (see the red circle). d The Heisenberg exchange is the most
prominent contribution to the skyrmion stabilization, as shown along the path
hosting a single minimum. The total skyrmion-skyrmion repulsive homo-
interaction is dominated by the attractive hetero-interaction, red curves in e
and f respectively. The DMI contribution, shown in insets, is smaller and
sublattice independent. It favors the overlap of AFM skyrmions.

While the RW-AFM state is defined by two sublattices, the different AFM
skyrmions, isolated or catenated, can be decomposed into interpenetrating FM
skyrmions living in 4 sublattices illustrated in Fig. 1d and denoted as L1,
L2, L3 and L4. In the RW-AFM phase, L1 and L4 are equivalent and likewise for
L2 and L3. It is evident that the moments in L1 and L4 are antiparallel to the
ones in L2 and L3. Taking a closer look at the the isolated AFM magnetic
texture, one can dismantle it into two FM skyrmions with opposite topological
charges anchored in the distinct antiparallel FM sublattices L1 and L2, while
L3 and L4 carry rather collinear magnetization (Fig. 1e). In the case of the
overlapped AFM skyrmions, however, no sublattice remains in the collinear
state. As an example, the dimer consists of two couples of
antiferromagnetically aligned skyrmions, each being embedded in one of the
four sublattices (Fig. 1f). The total topological charge naturally cancels
out, indicating the absence of the skyrmion Hall effect in the Cr layer.

Our study reveals that in contrast to catenated magnetic textures, single AFM
skyrmions are significantly sensitive to the magnetic environment hosted by
the underlying PdFe bilayer. For brevity, we focus in the next sections on the
single and two-overlapping AFM skyrmions and address the mechanisms dictating
their stability.

###  0.2 Stabilization mechanism of the overlapping AFM skyrmions.

The formation of overlapped solitons is an unusual phenomenon since FM
skyrmions repel each other. It results from competing interactions among the
skyrmions living in the different sublattices, which finds origin in the
natural AFM coupling between the n.n. magnetic moments. Depending on the
hosting sublattice, the four skyrmions shown in Fig. 1f experience attraction
or repulsion. We can define the skyrmion-skyrmion homo-interaction if the
skyrmions are living in a similar magnetic background such as sublattices (L1,
L4) and (L2, L3) and the hetero-interaction if the background is different as
it is for (L1, L2), (L2, L4), (L3, L4) and (L1, L3). The homo-interaction
induces the known repulsion experienced by the usual FM skyrmion and is
expected to be out-numbered by the hetero-interaction, which leads to
attraction. Both are obviously dictated by the AFM coupling between the
magnetic moments and a compromise is found with an equilibrium distance of
2.58 nm between the two AFM skyrmions, which can then overlap.

To substantiate the proposed mechanism, we quantify the skyrmion-skyrmion
interaction. We simplify the analysis by neglecting the Cr-Fe magnetic
interactions, which puts aside the impact of the rich non-collinear magnetic
behavior hosted by the PdFe bilayer. In this case, single AFM skyrmions
disappear and only the overlapping ones are observed. We take the skyrmion
dimer illustrated in Fig. 2a and proceed to a rigid shift of the lower AFM
skyrmion while pinning the upper one at the equilibrium position. We extract
the skyrmion-skyrmion interaction map as a function of distance, as shown in
Fig. 2b, which clearly demonstrates that as soon as the AFM skyrmions are
pulled away from each other, the energy of the system increases. Note that
within this procedure, the contributions from the sublattice interactions (L1,
L2) and (L3, L4) do not contribute to the plots since they are assigned to
each of the AFM skyrmions moved apart from each other. Two minima are
identified along a single direction as favored by the symmetry reduction due
to the AFM arrangement of the magnetic moments in which the skyrmions are
created. Indeed, one notices in Fig. 1d that due to the sublattice
decomposition symmetry operations are reduced to C2, i.e. rotation by
$180^{\circ}$, while mirror symmetries, for example, originally present in the
fcc(111) lattice are broken. Figs. 2c-d depict the skyrmion-skyrmion
interaction, which hosts either one or two minima, as a function of distance
along two directions indicated by the dashed lines, blue and black, in Fig.
2a. The two minima found along the blue line should be degenerate and
correspond to the swapping of the two AFM skyrmions. The breaking of
degeneracy is an artifact of the rigid shift assumed in the simulations, which
can be corrected by allowing the moments to relax (see red circle in Fig. 2c).
The maximum of repulsion is realized when the two AFM skyrmions perfectly
overlap (see inset).

The interaction profile shown in Fig. 2d is decomposed into two contributions:
the skyrmion-skyrmion homo- and hetero-interactions, which we plot in Fig. 2e
and f respectively. The data clearly reveals the strong repulsive nature of
the homo-interaction mediated by the Heisenberg exchange, which competes with
the attractive hetero-interaction driven by both the Heisenberg exchange
coupling and DMI. The latter skyrmion-skyrmion interaction is strong enough to
impose the unusual compromise of having strongly overlapping solitons.

###  0.3 Impact of magnetic field.

Figure 3: Impact of magnetic field of AFM skyrmion radius. Radius of the
sublattice FM skyrmions for two-catenated AFM skyrmions a decoupled from and b
coupled to the Fe magnetization. c A single case is shown for the isolated AFM
skyrmion since it disappears without the inhomogeneous magnetic field emerging
from the substrate. Examples of snapshots of the AFM skyrmions are illustrated
as insets of the different figures. In Fe, the amount of FM skyrmions and
antiskyrmions increases once applying a magnetic field, which erases the
ground state spin-spiral. The coupling to the Fe magnetization affects the
evolution of the AFM skyrmions as function of the magnetic field dramatically.
d depicts the dependence of the AFM skyrmion under a magnetic field of 70
Tesla on the surrounding magnetic environment, by sequentially deleting one FM
skyrmion or antiskyrmion in the Fe layer and relaxing the spin structure. At
some point, removing any of the single FM skyrmions in the lower left of d
annihilates the AFM skyrmion.

Prior to discussing stability aspects pertaining to the single AFM skyrmion in
detail, we apply a magnetic field perpendicular to the substrate and disclose
pivotal ingredients for the formation of the isolated solitons. In general,
the reaction of FM and AFM skyrmions to an external magnetic field is expected
to be deeply different. When applied along the direction of the background
magnetization, FM skyrmions reduce in size while recent predictions expect a
size expansion of AFM skyrmions [45, 46, 41], thereby enhancing their
stability.

To inspect the response of AFM skyrmions to a magnetic field perpendicular to
the substrate, we first remove, as done in the previous section, the Cr-Fe
interaction since it gives rise to a non-homogeneous and strong effective
exchange field. In this particular case, we can only explore the case of AFM
skyrmion dimers. As illustrated in Fig. 3a, the size of each of the sublattice
skyrmions, which together form the AFM skyrmion dimer, increases with an
increasing magnetic field. The type of the hosting sublattice, with the
magnetization being parallel or antiparallel to the applied field, seems
important in shaping the skyrmions dimension. Strikingly, and in strong
contrast to what is known for FM skyrmions, the AFM skyrmions, single and
multimers, were found to be stable up to extremely large magnetic fields.
Although the assumed fields are unrealistic in the lab, they can be emulated
by the exchange field induced by the underlying magnetic substrate. Indeed,
the magnetic interaction between Cr and its nearest neighboring Fe atoms,
carrying each a spin moment of 2.51 $\mu_{B}$, reaches 3.05 meV, which
translates to an effective field of about 21 T. At this value, the average
skyrmion radius is about 1.6 nm, which is 30% smaller than the one found once
the Cr-Fe magnetic coupling is enabled (see Fig. 3b). We note that the we
defined the skyrmion radius by the position where the spin moments lie in-
plane. The significant size difference is induced by the strong inhomogenous
exchange field emanating from the Fe sub-layer, which can host spirals,
skyrmions and antiskyrmions.

If the Cr-Fe interaction is included, the size dependence changes completely.
Instead of the rather monotonic increase with the field, the size of the
skyrmion is barely affected until reaching about 50 T, which is accompanied by
substantial miniaturization of the AFM skyrmions. Here, a phase transition
occurs in Fe, which initially hosts spin spirals that turn into FM skyrmions
(see Supplementary Figure 5). After being squeezed down to an average radius
of 1.48 nm at 140 T, the size expansion observed without the Cr-Fe interaction
is recovered because the substrate magnetization is fully homogeneous and
parallel to the Zeeman field. Likewise, single AFM skyrmions, found only once
the coupling to the substrate is enabled, react in a similar fashion to the
field as depicted in Fig. 3c. The substantial difference, however, is that
fields larger than 80 T destroy the AFM skyrmions due to the annihilation of
the Fe FM skyrmions. This highlights an enhanced sensitivity to the underlying
magnetic environment and clearly demonstrates the robustness enabled by
skyrmion catenation.

###  0.4 Stabilization mechanism for single AFM skyrmions.

Figure 4: Interaction map of the single AFM skyrmion with the magnetic
substrate. In the first row of figures, sublattice decomposition of Cr
skyrmion including the underlying Fe skyrmions shown in four columns a, d, g
and j, corresponding respectively to L1, L2, L3 and L4. The AFM skyrmion is
made of two FM skyrmions hosted by sublattices L1 and L2. In Fe, FM skyrmions
and antiskyrmions can be found in all four lattices. The second row (b, e, h
and k) illustrates the sublattice dependent two dimensional Heisenberg
exchange energy map corresponding to the areas plotted in the first row,
followed by the third row (c, f, i and l) corresponding to DMI. Note that the
energy difference $\Delta E$ is defined with respect to the RW-AFM background.

We learned that single AFM skyrmions can be deleted after application of an
external magnetic field or by switching off the exchange coupling to the
magnetic substrate. Both effects find their origin in the magnetization
behavior of the PdFe bilayer. To explore the underlying correlation, we
consider as an example the magnetic configuration obtained with a field of 70
T and delete one after the other the skyrmions and antiskyrmions found in Fe,
then check whether the AFM skyrmion in Cr survives (see example in Fig. 3d).
We notice that the AFM skyrmion disappears by deleting the FM solitons located
directly underneath or even a bit away. Supplementary Figure 6 shows that when
shifted across the lattice, the AFM skyrmion disappears if fixed above a
magnetically collinear Fe area.

We proceed in Fig. 4 to an analysis of the Fe-Cr interaction pertaining to the
lower-right snapshot presented in Fig. 3d by separating the Heisenberg
exchange contribution from that of DMI and plotting the corresponding heat
maps of the site-dependent of these two contributions for each sublattice.
Here, we consider as reference energy that of the RW-AFM collinear state
surrounding the non-collinear states in Fig. 3d. The building-blocks of the
AFM skyrmion are shown in Figs. 4a and 4d, where one can recognize the
underlying Fe FM skyrmions in the background. The latter are more
distinguishable in the sublattices free from the AFM skyrmion as illustrated
in Figs. 4g and j. The order of magnitude of the interactions clearly
indicates that the DMI plays a minor role and that one can basically neglect
the interactions arising in the skyrmion-free sublattices, namely L3 and L4.
It is the Heisenberg exchange interaction emerging in the sublattices L1 and
L2 that dictates the overall stability of the AFM skyrmion.

In L2, the core of the magnetization of the Cr FM skyrmion points along the
same direction as that of the underlying Fe atoms, which obviously is
disfavored by the AFM coupling between Cr and Fe (-3.05 meV for nearest
neighbors). This induces the red exchange area surrounding the core of the AFM
skyrmion (black circle in Fig. 4e), which is nevertheless sputtered with blue
spots induced by the magnetization of the core of the Fe FM skyrmions pointing
in the direction opposite to that of the Cr moments in L2. The latter is a
mechanism reducing the instability of the Cr skyrmion. Overall, the total
energy cost in having the Cr skyrmion in L2 reaches +693.7 meV and is
compensated by the exchange energy of -712.4 meV generated by the one living
in sublattice L1. Here, the scenario is completely reversed since the core of
the Cr skyrmion has its magnetization pointing in the opposite direction than
that of the neighboring Fe atoms and therefore the large negative blue area
with the surrounding area being sputtered by the Fe skyrmions, similar to the
observation made in L2 (see Fig. 4d). Overall, the Cr AFM skyrmion arranges
its building blocks such that the energy is lowered by the skyrmion anchored
in sublattice L1. Here, the details of the non-collinear magnetic textures
hosted by Fe play a primary role in offering the right balance to enable
stabilization. This explains the sensitivity of the single AFM skyrmion to the
number and location of the underlying FM Fe skyrmions. Removing non-
collinearity in Fe makes both building blocks of the AFM skyrmion equivalent
without any gain in energy from the Cr-Fe interaction, which facilitates the
annihilation of the Cr skyrmion.

## Discussion

Following a two-pronged approach based on first-principles simulations
combined with atomistic spin dynamics, we identify a thin film that can host
intrinsic, i.e. non-synthetic, AFM skyrmions at zero magnetic field. A Cr
monolayer deposited on a substrate known to host FM skyrmions, PdFe/Ir(111),
offers the right AFM interface combination enabling the emergence of a rich
set of AFM topological solitons. Owing to the AFM nature of Cr, its ground
state is RW-AFM as induced by magnetic interactions beyond nearest neighbors.
We strikingly discovered catenated AFM skyrmions, which can be cut and
decoupled into isolated solitons via the inhomogenous exchange field emanating
from PdFe bilayer. The intra-overlayer skyrmion-skyrmion interaction favors
the important overlap of the AFM skyrmions, and makes them robust against the
rich magnetic nature of the PdFe substrate. In contrast, the single AFM
skyrmion annihilates if positioned on a homogeneously magnetized substrate and
it is only through the presence of various spin-textures such as spin spirals
and multiples of skyrmions or antiskyrmions that one of the building-blocks of
the AFM skyrmion can lower the energy enough to enable stability.

Since the experimental observation of intrinsic AFM skyrmions has so far been
elusive at interfaces, our predictions open the door for their realization in
well defined materials and offer the opportunity to explore them in thin film
geometries. The robustness of the catenated skyrmions qualify them as ideal
particles for racetrack memory devices to be driven with currents while
avoiding the skyrmion Hall effect. The ability to control the substrate
magnetization makes the system, we studied, a rich playground to design and
tune the highly sensitive single AFM skyrmion living in the overlayer. We
envision patterning the ferromagnetic surface with regions hosting different
magnetic textures, being trivial, such as the ferromagnetic regions, or
topological, to define areas where the AFM skyrmion can be confined or driven
within specific paths. We envisage a rich and complex response to in-plane
currents, which could move the underlying FM solitons along specific
directions subjected to the Magnus effect. The latter would affect the
response of the overlaying isolated AFM skyrmions in a non-trivial fashion.

We anticipate that the proposed material and the FM-substrate on which AFM
films can be deposited offer an ideal platform to explore the physics of AFM
skyrmions. Noticing already that different catenated AFM skyrmions can co-
exist establishes a novel set of multi-soliton objects worth exploring in
future studies. Besides their fundamental importance enhanced by the potential
parallel with topological concepts known in knot theory, such unusual AFM
localized entities might become exciting and useful constituents of future
nanotechnology devices resting on non-collinear spin-textures and the emerging
field of antiferromagnetic spintronics.

###  0.5 First-principles calculations.

The relaxation parameters were obtained using the Quantum-Espresso
computational package[66]. The projector augmented wave pseudo potentials from
the PS Library [67] and a $28\times 28\times 1$ k-point grid were used for the
calculations. The Cr, Pd, Fe and Ir interface layer were fcc-stacked along the
[111] direction and relaxed by 4%, 5.8%, 8.1% and -1% with respect to the Ir
ideal interlayer distance, respectively. Positive (negative) numbers refer to
atomic relaxations towards (outward of) the Ir surface.

The electronic structure and magnetic properties were simulated using all
electron full potential scalar relativistic Koringa -Kohn-Rostoker (KKR) Green
function method [68, 69] in the local spin density approximation. The slab
contains 30 layers (3 vacuum + 1 Cr + 1 Pd + 1 Fe + 20 Ir + 4 vacuum). The
momentum expansion of the Green function was truncated at
$\ell_{\text{max}}=3$. The self-consistent calculations were performed with a
k-mesh of $30\times 30$ points and the energy contour contained 23 complex
energy points in the upper complex plane with 9 Matsubara poles. The
Heisenberg exchange interactions and DM vectors were extracted using the
infinitesimal rotation method [70, 71] with a k-mesh of a $200\times 200$.

###  0.6 Hamiltonian Model and atomistic spin dynamics.

In our study, we consider a two dimensional Heisenberg model on a triangular
lattice, equipped with Heisenberg exchange coupling, DMI, the magnetic
anisotropy energy, and Zeeman term. All parameters were obtained from ab-
initio. The energy functional reads as follows:

$H=H_{\text{Exc}}+H_{\text{ DMI}}+H_{\text{Ani}}+H_{\text{Zeem}},$ (1)

with:

$H_{\text{Exc}}=-\sum\limits_{<i,j>}J^{\text{Cr-
Cr}}_{ij}\;\textbf{S}_{i}\cdot\textbf{S}_{j}-\sum\limits_{<i,j>}J^{\text{Fe-
Cr}}_{ij}\;\textbf{S}_{i}\cdot\textbf{S}_{j}-\sum\limits_{<i,j>}J^{\text{Fe-
Fe}}_{ij}\;\textbf{S}_{i}\cdot\textbf{S}_{j},$
$H_{\text{DMI}}=\sum\limits_{<i,j>}\textbf{D}^{\text{Cr-
Cr}}_{ij}\cdot[\textbf{S}_{i}\times\textbf{S}_{j}]+\sum\limits_{<i,j>}\textbf{D}^{\text{Fe-
Cr}}_{ij}\cdot[\textbf{S}_{i}\times\textbf{S}_{j}]+\sum\limits_{<i,j>}\textbf{D}^{\text{Fe-
Fe}}_{ij}\cdot[\textbf{S}_{i}\times\textbf{S}_{j}],$
$H_{\text{Ani}}=-K^{\text{Cr}}\sum\limits_{i}(S_{i}^{z})^{2}-K^{\text{Fe}}\sum\limits_{i}(S_{i}^{z})^{2},$
$H_{\text{Zeem}}=-\sum\limits_{i}h_{i}S_{i}^{z},$

where $i$ and $j$ are site indices carrying each magnetic moments. S is a unit
vector of the magnetic moment. $J^{\text{X-Y}}_{ij}$ is the Heisenberge
exchange coupling strength, being $<$ 0 for AFM interaction, between an X atom
on site $i$ and a Y atom on site $j$. A similar notation is adopted for the
DMI vector D and the magnetic anisotropy energy $K$ (0.5 meV per magnetic
atom). The latter favors the out-of-plane orientation of the magnetization,
and $h_{i}=\mu_{i}B$ describes the Zeeman coupling to the atomic spin moment
$\mu$ at site $i$ assuming an out-of-plane field.

To explore the magnetic properties and emerging complex states we utilize the
Landau-Lifshitz-equation (LLG) as implemented in the Spirit code [72]. We
assumed periodic boundary conditions to model the extended two-dimensional
system with cells containing $100^{2}$, $200^{2}$, $300^{2}$, $400^{2}$ sites.

###  0.7 Data availability

The data that support the findings of this study is available from the
corresponding author upon reasonable request.

We thank Markus Hoffmann for fruitful discussions. This work was supported by
the Federal Ministry of Education and Research of Germany in the framework of
the Palestinian-German Science Bridge (BMBF grant number 01DH16027) and the
Deutsche Forschungsgemeinschaft (DFG) through SPP 2137 “Skyrmionics” (Projects
LO 1659/8-1, BL 444/16-2). The authors gratefully acknowledge the computing
time granted through JARA on the supercomputer JURECA at Forschungszentrum
Jülich.

S.L. initiated, designed and supervised the project. A.A. performed the
simulations with support and supervision from I.L.F., S.Br. and M.S. All
authors discussed the results. A.A. and S.L. wrote the manuscript to which all
co-authors contributed.

The authors declare no competing interests.

Correspondence and requests for materials should be addressed to A.A. (email:
<EMAIL_ADDRESS>or to S.L. (email: s.lounis@fz-juelich.de).

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