# Stability and dynamics of magnetic skyrmions in FM/AFM heterostructures

Rajgowrav Cheenikundil  School of Science and Technology, Örebro University,
SE-701 82, Örebro, Sweden  Zhiwei Lu  Department of Applied Physics, School of
Engineering Sciences, KTH Royal Institute of Technology, SE-10691, Stockholm,
Sweden  Manuel Pereiro  Department of Physics and Astronomy, Uppsala
University, SE-75120, Uppsala  Anna Delin  Department of Applied Physics,
School of Engineering Sciences, KTH Royal Institute of Technology, SE-10691,
Stockholm, Sweden  SeRC (Swedish e-Science Research Center), KTH Royal
Institute of Technology Wallenberg Initiative Materials Science for
Sustainability (WISE), KTH Royal Institute of Technology Danny Thonig  School
of Science and Technology, Örebro University, SE-701 82, Örebro, Sweden
Department of Physics and Astronomy, Uppsala University, SE-75120, Uppsala

###### Abstract

Magnetic skyrmions have garnered attention for their potential roles in
spintronic applications, such as information carriers in computation, data
storage, and nano-oscillators due to their small size, topological stability,
and the requirement of small electric currents to manipulate them. Two key
challenges in harnessing skyrmions are the stabilization requirement through a
strong out-of-plane field, and the skyrmion Hall effect (SkHE). Here, we
present a systematic model study of skyrmions in FM/AFM multi-layer structures
by employing both atomistic Monte Carlo and atomistic spin dynamics
simulations. We demonstrate that skyrmions stabilized by exchange bias have
superior stability than field-stabilized skyrmions due to the formation of a
magnetic imprint within the AFM layer. Additionally, stacking two skyrmion
hosting FM layers between two antiferromagnetic (AFM) layers suppresses the
SkHE and enables the transport of AFM-coupled skyrmions with high velocity in
the order of a few Km/s. This proposed multi-layer configuration could serve
as a pathway to overcome existing limitations in the development of skyrmion-
based devices, and the insights obtained through this study contribute
significantly to the broader understanding of topological spin textures in
magnetic materials.

††preprint: APS/123-QED

##  I  Introduction

Magnonics has emerged as a central area for the development of the next
generation of low-power, high-speed computational devices. This progress is
notably illustrated by the use of magnetic principles in magnetic RAMs
(MRAMs), already in commercial use [1], showcasing the effectiveness of
magnetic technologies in enhancing speed and reducing power consumption. In
addition to this, the rapid advancement in artificial intelligence has
intensified the search for low-powered artificial neural networks.
Spintronics-based neural networks have already been theoretically proposed [2,
3], making the research into spintronics devices more relevant and timely [4,
5], especially considering the growing demand for energy-efficient AI
computations. The exploration of spintronics applications in neural networks
could lead to significant breakthroughs in the field of AI, offering a
promising avenue for the development of next-generation computing
technologies.

Magnetic skyrmion-based devices are proposed to be at the forefront of this
technological shift [6, 7]. Skyrmions have attracted significant attention as
potential information carriers in spintronics systems owing to their compact
size [8], ability to be closely packed [9, 10], and exceptional stability.
Their interaction with spin currents via spin transfer torque [9] offers an
opportunity for precise control, a critical factor for practical applications
in advanced computational systems. Moreover, skyrmions require a significantly
lower critical electric current density for movement, usually five orders of
magnitude smaller than the requirement for magnetic domain walls [11, 12, 13].
This reduced need for driving electric current is beneficial, as it minimizes
energy loss due to dissipation and lowers the power requirements for these
devices to two orders of magnitude less than what is needed for conventional
CMOS devices. Skyrmions can also be controlled by other driving forces such as
surface acoustic waves (SAW) [14]. Furthermore, skyrmions could be easily
detected utilizing magneto-transport effects such as tunnel magneto-resistance
and the non-collinear magneto-resistance arising due to band structure changes
[15, 16]. They can also be identified with purely electrical techniques,
making use of the topological Hall effect [17]. The ease of transport and
measurement of skyrmions makes them ideal candidates for logic and computation
applications. Designs and proposals for skyrmion-based devices are already
emerging, including racetrack devices [18, 19, 20, 21, 22], logic gates [23],
and synapses [24].

Our research explores skyrmion-based race-track type devices, in which the
skyrmions are moved in a controlled way with spin currents. The information
bits are carried by a sequence of skyrmions moving along the track. There
exist two significant challenges in the design of such a device. The first
challenge is the skyrmion Hall effect (SkHE) [25, 26], where skyrmions when
propelled by spin currents, exhibit a transverse motion instead of moving
parallel to the current flow. This transverse deviation results in the
annihilation of the skyrmions at the edge of the track and hence it poses a
significant difficulty. This effect arises as a counter-effect to the
topological Hall effect experienced by the charge-carrying electrons, which in
turn originates from the topology of the skyrmion. Various strategies have
been explored, to address the skyrmion Hall effect, including geometric [27,
28] and acoustic[29] control, and the use of skyrmionium structures [30]. A
promising approach is the coupling of two skyrmions with opposite Hall
effects, which can lead to the cancellation of the SkHE [31, 32, 33, 34, 35],
paving the way for more controlled skyrmion-based devices. The second
challenge in the design of skyrmion racetrack devices is the requirement of a
strong out-of-plane magnetic field to keep the skyrmions stable, making the
design of such a device prohibitively complex. Various techniques have been
employed to eliminate the requirement of an external magnetic field to keep
the skyrmions stable. One of the promising approaches is to couple the
skyrmion hosting ferromagnetic layer to an exchange-biased material, where the
exchange bias will substitute the external magnetic field [36, 37]. Such
multilayer setups were already experimentally realized and skyrmions were
stabilized at zero fields in an ultra-thin FM/AFM heterostructure [38]. This
achievement of experimentally realizing these setups marks a significant
advancement. However, to broaden these results to more universal applications
and to gain a deeper understanding of the intricate physical mechanisms
involved, further research is necessary. Particularly, numerical simulation
studies are essential to explore the interplay of various material properties
on skyrmion stability and the strength of exchange bias in such systems. In
this study, we focus on the stabilization of magnetic skyrmions by exchange-
bias in ferromagnetic/antiferromagnetic (FM/AFM) multi-layer structures by
employing atomistic Monte-Carlo and atomistic spin-dynamics simulations [39].
Our objectives include developing a model FM/AFM multilayer structure to study
the dependence of exchange bias effect on various material and geometrical
parameters. Additionally, we aim to investigate the stability of skyrmions
within these multilayers under ambient conditions and explore the potential of
using exchange bias to mitigate the skyrmion Hall effect, ultimately aiming to
develop efficient skyrmion transport devices.

This paper is organized as follows. In Section II, the theory and techniques
used for the numerical simulations are discussed in detail. In Section III,
the results are presented in deductive order. Finally, in the last section,
Section IV, we provide a summary and an outlook.

##  II  Methodology

###  II.1 Theory and modeling

The simulations are performed using the atomistic simulation tool UppASD [39,
40]. Utilizing an atomistic model allows an accurate simulation of the
magnetic system when compared to continuum models, as this approach enables
detailed calculation of the intricate interactions among atoms and considers
factors like interracial effects, antiferromagnetic coupling between
neighboring atoms, and localized properties, such as anisotropies.

In this study, the properties of the system are modeled by the Hamiltonian
given by

$\displaystyle\mathscr{H}=$ $\displaystyle-\frac{1}{2}\sum_{i\neq
j}J_{ij}\mathbf{S}_{i}\cdot\mathbf{S}_{j}-\frac{1}{2}\sum_{i\neq
j}\mathbf{D}_{ij}\cdot(\mathbf{S}_{i}\times\mathbf{S}_{j})$ (1)
$\displaystyle-\sum_{i}K_{i}(\mathbf{S}_{i}\cdot\mathbf{e}^{k}_{i})^{2}-\mu_{0}\sum_{i}\mathbf{H}_{\text{ext}}\cdot\mathbf{S}_{i}$

where $\mathbf{S}_{i}$ is the spin vector, ${J_{ij}}$ is the isotropic part of
the exchange coupling between the $i^{th}$ and $j^{th}$ atomic spin, $J>0$
will result in ferromagnetic coupling and $J<0$ will give antiferromagnetic
coupling. The breaking of the inversion symmetry at the FM/AFM interface could
also lead to the rise of anisotropic terms of the symmetric exchange. We do
not consider the contribution of these interactions in our Heisenberg term.
$\mathbf{D}_{ij}$ is the Dzyaloshinski Moriya interaction (DMI) vector between
the $i^{th}$ and $j^{th}$ spins. In our model, we assumed $\mathbf{D}_{ij}$ to
have a direction parallel to the distance vector
$\mathbf{r}_{ij}=\mathbf{r}_{j}-\mathbf{r}_{i}$ between the position
$\mathbf{r}_{i}$ and $\mathbf{r}_{j}$ of the $i^{th}$ and $j^{th}$atom,
respectively, which is a scenario commonly observed in bulk materials with
non-centrosymmetric crystal structure [6, 41] when there is an n-fold symmetry
along the bond direction [42].

$K_{i}$ is the uniaxial anisotropy constant and $\mathbf{e}^{k}_{i}$ is the
direction of easy axis for the $i^{th}$ atom. The constant $\mu_{0}$ is the
permeability of free space. The external magnetic field is given by
${\mathbf{H}}_{\text{ext}}$. The temperature-induced effects of the system can
be simulated using Monte Carlo simulations. Here, the parametrized energy
given by Eq.(1) was then used to obtain the thermal equilibrium state as a
function of temperature and external magnetic field. [40] . The dynamics of
the spins are governed by the Landau-Lifshitz-Gilbert equation (LLG), given by
[43],

$\displaystyle\frac{\partial\mathbf{S}_{i}}{\partial t}=$
$\displaystyle-\gamma\mathbf{S}_{i}\times\mathbf{H}_{i}^{\text{eff}}+\frac{\alpha}{M_{s}}\mathbf{S}_{i}\times\frac{\partial\mathbf{S}_{i}}{\partial
t}$ (2) $\displaystyle-
u_{i}\mathbf{S}_{i}\times\mathbf{S}_{i}\times\frac{\partial\mathbf{S}_{i}}{\partial
x}-\beta u_{i}\mathbf{S}_{i}\times\frac{\partial\mathbf{S}_{i}}{\partial x}$

In this equation, the first term represents the precession of the atomistic
magnetic moments, the second is the damping term, followed by the spin
transfer torque (STT) term introduced by Sloncewski [44, 45]. The STT term
describes the magnetization dynamics under the influence of a spin-polarized
current along the $x$-direction. The constant $\alpha$ is the Gilbert damping
parameter, $\gamma$ is the gyromagnetic ratio, and $\beta$ is the adiabaticity
constant. The term $\mathbf{H}_{i}^{\text{eff}}$ is the effective field, which
is defined as the partial derivative of the Hamiltonian (1) with respect to
the magnetic moment at each site $\bm{S}{{}_{i}}$, ie.,

$\mathbf{H}^{\text{eff}}_{i}=-\frac{\partial\mathscr{H}}{\partial\mathbf{S}_{i}}$
(3)

The term $u_{i}$ is the STT coefficient, given by

$u_{i}=j_{e}Pg\mu_{B}/\left(2eM_{s}\right)$ (4)

where $j_{e}$ is the current density along the $x$-direction, $P$ is the spin
polarization, and $M_{s}$ is the saturation magnetization. For the
simulations, we are using a simple cubic lattice with localised magnetic
moments at the atoms. A periodic boundary condition is applied along the
$xy$-plane, mimicking the physics of an infinite plane to avoid any boundary
effects, especially during the current induced dynamics of skyrmions.  The
Monte-Carlo simulation typically has steps in the order of $10^{6}$, depending
on the complexity of the configuration. These steps are divided into different
phases with decreasing temperature, chosen to establish that the final state
obtained represents the global minimum and the noise in the average of the
observable is minimized. This is confirmed by monitoring the variation in the
average energy of the system and ensuring that there is no significant change
after each step.

Once the ground states are obtained using MC simulation, the dynamics are
studied with atomistic spin dynamics calculations. The spin dynamics
simulations are conducted at $0\,\mathrm{K}$ temperature with a time-step
$dt=10^{-16}$ seconds. To study the current-induced dynamics of the skyrmions,
a spin-polarized current is applied along the +$x$-direction and the evolution
of the magnetic moment at each point is recorded at appropriate time
intervals. Simulations are repeated by varying the values of current strength
$j_{e}$ from $1.75\times 10^{11}$ $A/m^{2}$ to $30\times 10^{11}$ $A/m^{2}$
and adiabaticity term $\beta$ from 0.01 to 0.5. A value of 0.1 is typically
used for the damping term $\alpha$. The number of simulation steps is
appropriately chosen based on the velocity of the skyrmion’s motion. This
choice ensures that we can record a trajectory of sufficient length to
calculate the average velocity accurately. From this, the skyrmion Hall angle
is calculated as, $\theta_{SHE}=\tan^{-1}\left(\frac{V_{y}}{V_{x}}\right)$,
where $V_{y}$ and $V_{x}$ are the vertical and horizontal components of the
skyrmion velocity.

###  II.2 Exchange bias: model setup

In the first stage, we investigated the exchange bias effect in AFM/FM
multilayer structures by simulating a hysteresis loop. Our study included a
comprehensive analysis of the dependence of the exchange bias field on various
simulation parameters, such as the strength of the interlayer coupling
($J_{il}$) and the number of ferromagnetic layers. An illustrative diagram of
an example setup which we used for our simulation is given in Fig. 1. This
model system is comprised of three ferromagnetic (FM) layers (labeled as layer
1, 2, 3 respectively), stacked over seven uncompensated antiferromagnetic
(AFM) layers. In the AFM layers, the magnetization was pinned in the out-of-
plane direction (along the Z direction), with a strong uniaxial anisotropy.
This pinning is crucial for ensuring the stability of the magnetic
orientations within these layers while the external field is varied. On the
other hand, the FM layers’ magnetization was not fixed and it was allowed to
couple with the adjacent AFM layers through interlayer exchange coupling. This
coupling plays a significant role in the exchange bias phenomenon observed at
FM/AFM interfaces. The AFM/FM layers are stacked along the Z-axis direction.
Atoms in the AFM layers are coupled to their nearest neighbors with in-plane
FM coupling ($J_{\text{AFM}}$) and with out-of-plane AFM coupling ($J_{AFM-
il}$). The $4^{th}$ AFM layer is coupled to the $3^{rd}$ FM layer with an
exchange strength given by $J_{il}$. The atoms in the FM layers are coupled to
all neighbors with strength given by $J_{\mathrm{FM}}$. The hysteresis loop is
simulated by varying the external magnetic field from 1 T to -15 T and back to
1 T in steps of 0.5 T. . The first simulation occurs at a field step of 1 T,
where the MC simulation begins from an initial state, in which the
magnetization in the ferromagnetic (FM) layers is aligned along the +Z
direction, while in the antiferromagnetic (AFM) layers, it alternates between
-Z and +Z directions, as illustrated in Fig. 1. This simulation comprises four
stages, with a total 250,000 steps, during which the temperature is decreased
from 100 K to 10 K, and then to 1 K. This will yield the relaxed ground state
at this specific field. This final state obtained here serves as the initial
state for the subsequent field stages of the hysteresis loop.

Table 1: Material parameters used for the simulation of the model exchange bias setup Parameter |  Symbol |  Value
---|---|---
Saturation Magnetization |  $M_{s}$ |  $2.23\mu_{B}$
Exchange coupling in FM Layer |  $J_{\mathrm{FM}}$ |  1.33 mRy
In-plane AFM coupling |  $J_{\text{AFM}}$ |  $J_{\mathrm{FM}}$
Out of plane AFM coupling (along Z) |  $J_{AFM-il}$ |  $-J_{\mathrm{FM}}$
Interlayer exchange coupling between FM/AFM Layers |  $J_{il}$ |  $0.025\,J_{\mathrm{FM}}$ to $0.4\,J_{\mathrm{FM}}$
Out of plane anisotropy in AFM Layers |  $K_{AFM}$ |  $-3\,J_{\mathrm{FM}}$
Number of FM layers |  $N_{FM}$ |  1 to 10

We used ideal material parameters which are given in table 1.  In our exchange
bias simulation, we only considered the nearest-neighbor exchange
interactions. This is based on typical behavior observed in the materials of
interest, where nearest-neighbor coupling tends to be the dominant interaction
and higher-order couplings, such as next-nearest neighbors, typically exhibit
a magnitude approximately eight times smaller [46, 47, 48, 49, 50]. Since all
the interactions are given in terms of $J_{\mathrm{FM}}$, the model can be
scaled up or down according to suit various material combinations.

Figure 1: The model exchange bias setup, with different exchange interactions
marked with arrows. Only three out of the seven AFM layers are shown. Figure
2:  Exchange bias effect in the model system. (a) Simulated hysteresis loop of
the model system with three FM layers and $J_{il}/J_{\mathrm{FM}}=0.07$. (b)
Comparison of the exchange-bias strength with the inter-layer coupling
strength $J_{il}$ (red) and the number of FM layers ($N_{FM}$) (blue).

###  II.3 Skyrmions in FM Layers: formation field

To study the formation and stability of skyrmions, a ferromagnetic monolayer,
with an in-plane DMI is studied. We did a range of large-scale simulations,
systematically varying the strength of the ferromagnetic exchange strength
from $0.1J_{\mathrm{FM}}$ to $J_{\mathrm{FM}}$, the DMI strength $D$ from
$0.7J_{\mathrm{FM}}$ to $J_{\mathrm{FM}}$, and the temperature from
$0\,\mathrm{K}$ to $300\,\mathrm{K}$. The direction of the $\mathbf{D}$ was
aligned parallel to the bond direction. These simulations identified the
parameter space in which skyrmions could remain stable.

###  II.4 Stability of skyrmions: GNEB method

To assess the thermal stability of skyrmions, we explore the energy and
transition mechanisms involved in skyrmion annihilation using the geodesic
nudged-elastic band method (GNEB), [51]. Magnetic skyrmions are separated from
a topologically trivial colinear state by a finite energy barrier. To get an
accurate description of their stability, and the mechanism involving their
magnetic transitions the minimum energy path (MEP) involved and the
corresponding activation energy should be studied. The activation energy is
defined as the difference between the energy of the initial state and that of
the highest saddle point (SP) along the MEP. The geodesic-nudged-elastic-band
(GNEB) method is a technique that is adept at tracing the minimum energy paths
over the complex, curved energy landscapes of such magnetic transitions. This
involves the identification of the local minima at the initial
($\mathcal{G}^{int}$) and final ($\mathcal{G}^{fin}$) states and then
connecting these two states by an initial path. This path is discretized to
several points called the images. These images give the magnetic state of the
system at these intermediate points and they are an approximation to a
continuous path. The number of images in the energy path is kept at an optimum
value so as to have a good resolution while still minimizing computational
cost. The initial path between $\mathcal{G}^{int}$ and $\mathcal{G}^{fin}$ is
brought to the nearest minimum energy path, by means of an iterative algorithm
that tries to minimize the force in the tangent space, which is calculated
similarly to Eq. 3. Once the MEP is identified, the energy barrier or the
activation energy can be calculated as

$E_{ac}=E(\mathcal{G}^{SP})-E(\mathcal{G}^{int})$ (5)

Where $\mathcal{G}^{SP}$ is the saddle point. This $E_{ac}$ is a measure of
the thermal stability of the skyrmion. It relates to the lifetime of the
skyrmion in the form of an Arrhenius rule [52, 53, 54]. A detailed description
of the theory and the algorithm of this technique can be found in Ref. [51].

##  III Results and discussion

###  III.1 Exchange bias

The exchange bias effect observed in FM/AFM multilayer structures is
characterized by an asymmetric magnetization reversal and a consequent shift
in the coercive field. This is caused by the presence of a unidirectional
anisotropy at the FM/AFM interface due to interfacial coupling, which acts as
an additional field term during magnetization reversal [55, 56, 57]. The shift
in the coercive field gives the strength of the exchange bias field
$H_{\text{Exb}}$. The hysteresis loop obtained for the model setup with three
FM layers and an FM-AFM interlayer coupling strength ($J_{il}$) of 0.075
$J_{\mathrm{FM}}$ is shown in Figure 2. The loop is shifted towards the
negative X-axis, signifying the presence of an exchange bias field of 7.7 T.
This simulation is repeated, systematically varying the interlayer exchange
coupling $J_{il}$ from 0.01 to 0.4 $J_{\mathrm{FM}}$ and by increasing the
number of FM layers from 1 to 10. The results are shown in Fig. 2. We observed
that the exchange bias field strength was directly proportional to the
interlayer coupling strength (Fig. 2). This result underscores the fundamental
role of inter-layer coupling in generating the exchange bias effect and agrees
with the existing theoretical models [58, 59] and previously published
simulation results, e.g., in Ref. [60, 61, 55]. In our model, the AFM spins at
the interface are pinned along the Z direction with strong uniaxial anisotropy
while the free FM spins are coupled to these pinned spins by the interlayer
coupling $J_{il}$. Thus the exchange field experienced at the interface is
directly proportional to $J_{il}$. Conversely, we observed an inverse
relationship between the exchange bias field and the number of ferromagnetic
(FM) layers, as shown in figure. 2 (b). This phenomenon can be attributed to
the specific characteristics of our model, where only first-neighbor coupling
occurs between atoms at the FM/AFM interface. Consequently, the exchange bias
is predominantly an interfacial effect, confined primarily to the first FM
layer adjacent to it. The influence of the exchange bias extends to the above-
lying FM layers via the $J_{ij}$ coupling. As the number of layers increases,
the intensity of the exchange bias effect diminishes in the layers farther
from the interface, explaining the observed inverse relationship [62]. Based
on these results, parameter combinations that result in desired values of
exchange bias field strength could be identified. These findings are
particularly relevant for the next stage, where we stabilize skyrmions with
exchange bias.

###  III.2 Skyrmion in Ferromagnetic layers

In the study of skyrmions within FM/AFM multi-layers, we introduce an in-plane
Dzyaloshinskii-Moriya interaction (DMI) in the ferromagnetic layer. This
incorporation of DMI in the FM layer may influence the exchange bias field.
Consequently, the field impacting a FM layer with DMI might vary from that
affecting a FM layer without it, assuming all other conditions are identical.
The discrepancy in the exchange bias field is challenging to measure
accurately due to the difficulty in modeling a collinear state at zero fields
caused by the presence of DMI. This complication hinders identifying the
hystereses loop and, consequently, the coercive field. Although there have
been theoretical explorations of the exchange bias field strength at FM/AFM
interfaces, [63], our simulations suggest that the variation caused by DMI in
our setup is likely minor. Therefore, for simplicity, we overlook any
potential differences in the exchange bias field due to the effect of DMI in
the FM layer.

To study the formation of skyrmions, with the Monte Carlo simulation approach,
we initiate the magnetization process from a state where each site’s
magnetization is randomly oriented, simulating the physical case of thermal
annealing. From this initial state the magnetization is relaxed to a zero-
field state. In the absence of an external field, the magnetization naturally
settles into a labyrinthine domain state, as shown in Fig. 5. This state is
distinguished by its intricate, tangled pattern. The width of these domains
typically aligns with the size of skyrmions found in the material [64]. To
this state, an out-of-plane field is applied. When we gradually increase the
out-of-plane magnetic field strength, these domain structures begin to evolve.
Domains aligned with the external field expand, while those in opposition
contract. Increasing the field further leads to the formation of skyrmions
within the layer, as depicted in Fig. 5. Continuing to raise the field
strength causes the skyrmions to reduce in size until, at a certain high field
strength, they are annihilated, resulting in a saturated or collinear magnetic
state. This field strength at which a skyrmion is formed: the formation field,
which is crucial for the creation of a skyrmion, results from the interplay of
the exchange coupling strength $J$ and the Dzyaloshinskii-Moriya interaction
strength $D$. Consequently, the formation field is influenced by the $D/J$
ratio inherent to the material. To investigate this, we conducted a series of
simulations varying the $D$ and $J$ values systematically. In each simulation,
we identified the formation field as the strength of the field at which the
skyrmions are formed when transitioning from a labyrinth state. The results
are illustrated in Fig. 3. It is observed that the formation field is directly
proportional to the in-plane exchange coupling $J$ and the $D/J$ ratio. Based
on the values of $J$ and $D$, the formation field varied considerably from
$0.7\,\mathrm{T}$ to $85\,\mathrm{T}$. The material parameters associated with
different exchange bias field strengths were already identified based on the
studies conducted in section III.1. Thus, combining these two results,
exchange bias-stabilized skyrmions are simulated by stacking the FM layers
with in-plane DMI on top of a pinned AFM layer. The simulated skyrmion and an
illustration of the setup are given in Fig. 4. The material parameters used in
this setup are given in Table 2. These values are carefully chosen by
comparing the previous results so that the exchange bias field arising from
the FM/AFM interface exactly matches the formation field of the skyrmions in
the FM layer.

Table 2: Material parameters used for the simulation of skyrmions Parameter |  Symbol |  Value
---|---|---
Exchange coupling in FM layer |  $J_{\text{FM}}$ |  $1.3\,\text{mRy}$
In-plane DMI strength |  $D$ |  $0.1J_{\mathrm{FM}}$
In-plane AFM coupling |  $J_{\text{AFM}}$ |  $J_{\text{FM}}$
Out-of-plane AFM coupling |  $J_{\text{AFM-il}}$ |  $-J_{\text{FM}}$
Interlayer exchange coupling between FM/AFM Layers |  $J_{\text{il}}(\text{N1})$ |  $0.011\,J_{\text{FM}}$
Figure 3: Skyrmion formation field, at various values of exchange strength $J$
and $D$. The formation field is directly proportional to $J$ and $D/J$ ratio.
Figure 4:  (a) Zero-field skyrmion stabilized by the exchange bias field in
the FM/AFM multilayer setup. The colors describes the out of plane (Z)
component of the Magnetization as indicated in the legend. Red color indicates
a large spin up component perpendicular to the surface, blue color indicates a
large spin down component perpendicular to the surface and a white color
indicates that the spin is parallel to the surface. The bottom purple layer
represents the AFM layer (b) Zoomed view showing the top FM and bottom AFM
layer (c) Inter-atomic exchange ($J$) and DMI coupling ($D$) in this setup.
Figure 5: The stability of skyrmions with respect to temperature and the
interlayer coupling $J_{il}$ in (a) an FM monolayer under exchange bias (Fig.
4), and (b) an AFM-coupled double layer under exchange bias (Fig. 8(d)). The
yellow region is the parameter space in which the labyrinth domain states are
observed, and the red part is the area in which the system exists in a
colinear state. Stable skyrmions are observed in the green area.

A key metric for assessing the stability of skyrmion states within this setup
is the range of field strengths within which the skyrmion maintains its
stability. To determine this, one begins by establishing a stable skyrmion as
the initial state. Subsequently, the exchange bias field is methodically
incremented and decremented by varying the interlayer coupling strength
$J_{il}$. This process identifies the upper and lower limits of the magnetic
field strengths at which the skyrmion ceases to be stable. The span of the
field strengths between these two limits offers a quantifiable indication of
the skyrmion’s stability against field fluctuations, in the given material. To
study the effect of temperature on this, the process is repeated by varying
the temperature. From this, the temperature-field space inside which the
skyrmion is stable, can be identified. The results of these calculations are
given in Fig. 5(a). Due to the FM layers being stacked atop a pinned AFM layer
and the exchange bias acting as the field source, the field strength is
quantified in terms of the interlayer exchange coupling $J_{il}$. In lower
field regions, the labyrinth phase is predominant, whereas at higher field
strengths, the system transitions into a collinear state. Notably, there
exists a well-defined range of field strengths, sandwiched between these two
phases, where the skyrmions exhibit stability. It was observed that as the
temperature increases, this stability field range for the skyrmions
progressively narrows. Beyond a critical temperature threshold of 160K, it
ceases to be a field range where the skyrmion can maintain stability. In
addition to the single FM monolayer, we investigated a setup involving two
AFM-coupled FM layers (Fig. 5 (b)), sandwiched between two pinned AFM layers.
This configuration is significant later in our discussion and more details
about this setup are given in section III.5. We observed qualitatively similar
results as the monolayer, however, the double layer configuration demonstrated
enhanced temperature stability, persisting up to $230\,\mathrm{K}$, compared
to $160\,\mathrm{K}$ the monolayer.

###  III.3 Stability: annihilation and activation energy

To better understand the thermal stability of the skyrmions in these
structures, their annihilation mechanism in an FM monolayer and an FM/AFM
multilayer were studied in detail with GNEB, as explained in the methodology
section II.4. For this, an initial path discretized into $20$ images was
created, where the an isolated skyrmion is chosen as the initial state and a
collinear FM state is chosen as the final state. The GNEB calculation gives
the minimum energy path (MEP) with respect to the reaction coordinate ( see
Fig. 6) and the corresponding magnetization dynamics. Reaction coordinate is a
dimensionless number representing the progress of the magnetic transition and
it is defined as a variation along the path connecting the initial and final
states. It is normalized so that it is equal to zero at the initial state and
one at the final state. The energy of the system at each point along the path
is plotted on the Y-axis, taking the energy of the skyrmion state as zero. In
both these configurations, the annihilation of the skyrmion follows a similar
path. Initially, there is a progressive shrinking of the skyrmion’s core, the
energy of the skyrmion is gradually increased during this phase. The energy
reaches its peak value when the skyrmion is in the smallest size. This is the
saddle point (SP). The SP is followed by an abrupt reversal of the outer
region leading to the destruction of the skyrmion leaving behind a non-
colinear spin structure at the core. These spins undergo a smooth rotation,
while the energy of the system is gradually decreased, eventually leading to a
collinear state. This observed collapse mechanism is consistent with previous
simulation studies [65] and  recent experimental observations [66]. In the
energy landscape, the skyrmion state represents a metastable local minima
state and the global energy minimum is given by a colinear magnetic state.
The activation energy ($E_{ac}$) for the transition from the skyrmion to the
collinear state is given by the height of the saddle point. This $E_{ac}$ is a
direct measure of the thermal stability and the mean lifetime of a skyrmion.

When comparing the activation energies required for the annihilation process
in two distinct setups, we notice a significant 40% increase when the skyrmion
is stabilized by an exchange bias field. This points to a notably higher
stability of the skyrmion in the exchange bias setup compared to a monolayer
configuration. The reason for this increased stability is linked to the
formation of a skyrmion “imprint” [67, 68, 69, 70] in the neighboring layer, a
process aided by exchange interactions. Due to the strong exchange coupling at
the interface, the magnetizations in the antiferromagnetic (AFM) and
ferromagnetic (FM) layers tend to align with one another. As a result, any
magnetization texture that forms in the FM layer is mirrored in the AFM layer,
leading to the creation of an “imprint” of the skyrmion in the lower layer,
hence increasing the stability of the skyrmion. This is further verified by
comparing the activation energy of the skyrmion to a case where there is a
high uniaxial anisotropy in the AFM layers, restricting the formation of these
imprints. In this scenario, the activation energy was the same as that of the
case stabilized by an external field. The details of the theory and the
mechanism that governs this interaction have been discussed in [71].

Figure 6: Minimum energy path illustrating the mechanism and energy barrier
involved in the annihilation of the skyrmion. Each image depicts the
magnetization state at a specific point along the reaction path. The blue
curve represents the minimum energy path (MEP) for the skyrmion stabilized by
an external field. The top green curve illustrates the MEP for the skyrmion in
the absence of anisotropy in the AFM layers. The increase in $E_{ac}$ is
clearly visible. The red line corresponds to the MEP for a skyrmion stabilized
by an exchange bias in the presence of high anisotropy in the
antiferromagnetic (AFM) layers.

###  III.4 Dynamics

Having established a deep understanding of the stability of the skyrmions in
different configurations and material parameters, we can proceed to look at
their current-induced dynamics. In the first part, we investigated the
dynamics of isolated skyrmions stabilized by exchange bias in monolayers,
focusing on skyrmions of up and down polarity, but with the same chirality. We
use the following values for the simulation parameters: exchange coupling
within the FM layer $J_{\mathrm{FM}}$, DMI strength within the FM layer
$D=0.1J_{\mathrm{FM}}$, inter-layer exchange coupling between the FM and AFM
layers, $J^{1}_{il}=0.01J_{\mathrm{FM}}$. These values are chosen based on the
previous results so that we can have skyrmions with high stability, small
size, and realistic values of the exchange bias field. A smaller skyrmion will
let us use a smaller simulation box, thus reducing the computational cost.
Simulations were repeated at different values of current strength $j_{e}$,
$\alpha$, and $\beta$. From these series of calculations, the velocity and
$\theta_{SHE}$ are calculated, which are given in Fig. 7(a).

In this scenario, a skyrmion with a polarity of -1 demonstrates negative Hall
angles when $\beta$ is less than $\alpha$, set at 0.1. Consequently, it drifts
towards the lower section of the racetrack due to the Hall effect. When
$\beta$ equals $\alpha$, the skyrmion shows no Hall effect, maintaining a
linear trajectory devoid of any vertical displacement. As $\beta$ surpasses
$\alpha$, the skyrmion exhibits positive Hall angles. The magnitude of this
Hall angle increases linearly with $\beta$, escalating by a factor of 40 as
$\beta$ progresses from 0.1 to 1, while the skyrmion’s velocity experiences
only a slight rise from 280 to 300 m/s (see Fig. 7(b)). Our investigation
affirms the theoretical projections [72, 73, 25, 74], that skyrmions of
opposing polarities present Hall angles that are equal in magnitude yet
opposite in sign. Thus replacing a skyrmion with opposite polarity but the
same chirality results in Hall angles at different $\beta$ values that are
equal in size but opposite in direction.

Initially, we hypothesized that pairing two such skyrmions with opposite
polarities [31], each exhibiting equal and opposite Hall angles individually,
and coupling them through an antiferromagnetic (AFM) interaction would lead to
their movement without any Hall angles. To test this, we created a bilayer
arrangement of skyrmions, as depicted in Fig. 8. This system, referred to as
setup 1, consists of four layers. The first ferromagnetic layer (layer 1) is
magnetized in the -Z direction and is coupled to the second layer (layer 2)
through a first neighbor antiferromagnetic coupling ($J^{1}_{sk}$). Layer 2 is
further coupled to a pinned antiferromagnetic layer (layer 3) via nearest
neighbor ferromagnetic coupling ($J^{1}_{il}$). In contrast to the previous
monolayer setup, this configuration additionally incorporates a next-nearest-
neighbor ($N2$) AFM coupling ($J^{2}_{il}$) between layer 1 and layer 3. For
the simulation, we used the following values: $J^{1}_{sk}=J_{\mathrm{FM}}$,
$J^{1}_{il}=0.01J_{\mathrm{FM}}$, $J^{2}_{il}=-0.005J_{\mathrm{FM}}$. In this
setup, layers 1 and 2 each host a skyrmion. These skyrmions, with opposite
polarities, are antiferromagnetically coupled to each other, effectively
functioning as a unified entity. This unique arrangement of coupled skyrmions
with contrasting polarities and their stabilization through exchange
interactions creates a novel platform. The figure depicted in Fig. 7
illustrates the velocity and Hall angle of the coupled skyrmion at various
values of $\beta$, with $\alpha$ fixed constant at 0.1. Here, the velocity of
the skyrmion shows a linear increase with $\beta$. Within the range
$\beta<0.1$, the coupled skyrmion moves at a velocity slower than that of the
isolated monolayer skyrmions. However, as $\beta$ surpasses 0.1, the coupled
skyrmions demonstrate velocities significantly faster than those of the
isolated skyrmions, reaching speeds of up to 2.5 Km/s when $\beta=1$.
Nevertheless, we observed an unusual behavior regarding the skyrmion Hall
angle. In the range $\beta>0.4$, the skyrmion Hall angle, $\theta_{SHE}$ of
the coupled skyrmions drops below the individual $\theta_{SHE}$ of the
monolayer skyrmions. The coupled skyrmions move with very low Hall angles
($\theta_{SHE}<10\degree$), despite the isolated skyrmions showing very high
Hall angles ($>20\degree$) in this region. However, when $\beta<0.3$ the
coupled skyrmion exhibits a Hall angle significantly greater than the sum of
the individual skyrmions’ angles, tending towards $90\degree$. This unexpected
result can be attributed to the asymmetry in the exchange fields experienced
by FM layers 1 and 2. Layer 1 is coupled to the AFM layer via second-neighbor
coupling ($N2$), whereas layer 2 experiences first-nearest-neighbor ($N1$)
coupling. This disparity leads to unequal exchange fields for the top and
bottom skyrmions, resulting in their size deformation and, consequently, a
heightened Hall effect in the coupled state.

To address the observed anomaly, we propose the creation of an artificial
setup (Fig. 8, Setup 2) that ensures uniformity in the exchange field
experienced by both layers. In this revised configuration, both the top and
bottom ferromagnetic (FM) layers are coupled to the underlying
antiferromagnetic (AFM) layers using second-nearest neighbor ($N2$) coupling,
and there is no coupling between layers 2 and 3. Such a design makes sure that
the exchange field across both layers is equal, thereby eliminating the
asymmetry observed in the original setup. In this artificial setup, the
uniform coupling results in the coupled skyrmion moving with a zero net Hall
effect across the entire range of ($\beta$). This outcome is significant as it
demonstrates the ability to control and negate the skyrmion Hall effect
through careful design of the magnetic layer coupling. By ensuring that both
layers are subjected to the same magnetic influences, the coupled skyrmion
behaves consistently, regardless of the value of $\beta$.

Figure 7: (a) Variation in the skyrmion Hall angle of the isolated skyrmions
with +1 and -1 polarity in the monolayer and coupled skyrmions in setup 1 and
the conduit with different values of $\beta$ when $\alpha=0.1$ and
$j_{e}=14\times 10^{11}$ A/m2. (b) Horizontal velocity of these skyrmions with
$\beta$. The red line shows the velocity of the coupled skyrmion in the
conduit with different current densities ($j_{e}$) keeping $\alpha=0.1$ and
$\beta=0.5$.  Figure 8: (a) Model system for studying skyrmion dynamics.
Layers 1 and 2 are the skyrmion-hosting FM layers, while layers 3 and 4 are
the pinned AFM layers. Skyrmions of up/down polarity in the two layers are
stabilized by exchange bias from the bottom AFM layer (b) Exchange interaction
in setup 1. The FM layers 1 and 2 are coupled to the AFM layers with a first
nearest neighbor (N1) and second nearest neighbor (N2) coupling. (c) In Setup
2, both FM layers are coupled to the AFM layers with N2 coupling. (d) Exchange
bias stabilized skyrmion conduit. The coupled skyrmion is hosted in the middle
FM layers. Exchange bias arising from the two AFM layers keeps the skyrmion
stable while the AFM coupling between the first and second FM layers cancels
the skyrmion Hall effect. The arrows in (b) and (c) represent the exchange
coupling between the layers. The white arrow represents the nearest neighbor
(N1) AFM coupling, while the yellow arrow shows the nearest neighbor FM
coupling, and the green arrow gives the next-nearest-neighbor (N2) coupling.

###  III.5 Exchange-bias stabilized skyrmion conduit

Based on the insights gained from these studies, we propose the design for a
skyrmion transport device capable of hosting stable skyrmions at zero magnetic
fields and transporting them at high velocities while completely negating the
Hall effect. This device is structured by stacking two skyrmion-hosting
ferromagnetic (FM) layers on top of one another, with these layers being
coupled via nearest-neighbor antiferromagnetic (AFM) interactions. The entire
arrangement is then enclosed between two additional antiferromagnetic layers,
as depicted in Fig. 8. In this configuration, the first FM layer (FM layer 1)
is connected to the second FM layer (FM layer 2) through AFM coupling and to
the bottom AFM layer via FM coupling. The second FM layer (FM layer 2) is also
coupled to the top AFM layer with FM coupling. The exchange bias field arising
from the coupling to the AFM layers keeps the skyrmions stable and as the two
skyrmions within the FM layers are coupled in an antiferromagnetic manner,
their respective Hall effects counteract each other. This reciprocal
cancellation ensures that the skyrmions can be transported at high velocities
without experiencing any Hall effect (Fig. 7). The velocity of the coupled
skyrmion in the conduit at different values of $\beta$ and $j_{e}$, while
fixing the value of $\alpha$ to 0.1 are given in Fig. 7(b).The skyrmion
velocity increased linearly with $j_{e}$ and $\beta$ and we observed high
velocities up to 3 Km/s. In comparing the velocities of coupled skyrmions in
the conduit versus those in setup 1, it’s evident that the velocity of the
skyrmion in the conduit experiences only a marginal decrease, while
successfully circumventing the Hall effect entirely. The high velocity
reported here is at least six times faster than the skyrmion velocities
generally reported before [75, 76, 77]. Thus, such a device, leveraging the
principles of magnetic layer coupling and exchange bias stabilization, offers
a promising avenue for efficient and controlled skyrmion transport in
spintronic applications.

###  III.6 Skyrmions in antiferromagnetic layers

In addition to our primary study on skyrmions in FM layers stabilized by
exchange bias from AFM layers, we extended our research to include skyrmions
in two distinct antiferromagnetic (AFM) structures: A-type AFM structures
stabilized by exchange coupling from a pinned FM layer, and in a G-Type AFM
interface[78].

### Skyrmions in A-type AFM Structures Stabilized by Pinned FM Layers

The simulation setup for this investigation mirrors our primary model system.
It consists of three pinned FM layers which are coupled to two A-type AFM
layers by an antiferromagnetic coupling ($J_{il}<0$). The distinguishing
feature in this setup is the location of the Dzyaloshinskii-Moriya
Interaction, which is present only in the AFM layers. We adopted the same
material parameters as used in our primary study to maintain consistency and
comparability of results. An illustration of the model is given in Fig. 9 (a).
Here, the Dzyaloshinskii Moriya interaction is confined to the
antiferromagnetic (AFM) layers, and the magnetization within the FM layers is
fixed along the Z-axis, with a strong uniaxial anisotropy. This setup leads to
the formation of magnetic skyrmions in the AFM layers as shown in Fig. 9(b).
The exchange coupling from the stationary ferromagnetic layers stabilize these
skyrmions. Given that the exchange interaction and other relevant parameters
closely resemble those in earlier configurations, the skyrmions within the AFM
layers exhibit current-driven dynamics akin to those outlined in section
III-D, in the sense that they move with equal and opposite Hall angles when
isolated and exhibit the anomaly in the low $\beta$ region when coupled
together.

Figure 9: (a) Model setup for the simulation of skyrmions in an AFM layer.
Layers 1 and 2 are the AFM layers and layers 2 and 3 are the pinned FM layers.
The arrows represent the direction of magnetization of the layer. (b) Skyrmion
lattice formed in the AFM layer.

### Skyrmions in G-type AFM Interface

The G-type AFM setup presents a more complex scenario. It can be
conceptualized as two interlaced FM sublattices [78], akin to a chessboard
pattern, with atoms in each lattice coupled through distinct exchange
interactions. The second nearest neighbor atoms, belonging to the same
sublattice are coupled with ferromagnetic exchange coupling ($J^{2}$), while
the next nearest atoms in the opposing lattice are linked via
antiferromagnetic coupling ($J^{1}$). Additionally, this structure
incorporates an in-plane DMI. The magnetizations within these lattices are
pinned in the out-of-plane or Z direction, with the first lattice aligning
along the +Z direction and the second lattice along the -Z direction. The
material parameters used for the simulation of skyrmion in this G-type AFM
structure are given in table 3. These parameters were chosen to optimize the
conditions under which skyrmions could exist stably within the G-type AFM
structure.

Table 3: Material parameters for the simulation of stable skyrmions in a G-type AFM interface III.  Parameter |  Symbol |  Value
---|---|---
Exchange coupling between second nearest neighbors in the FM Layer |  $J_{\mathrm{FM}}^{2}$ |  $J_{\mathrm{FM}}$
Exchange coupling between first nearest neighbors from opposing lattices in the AFM layers |  $J_{\text{AFM}}^{1}$ |  -2 $J_{\mathrm{FM}}$
DMI strength |  $D$ |  0.75 $J_{\text{FM}}$
Uniaxial anisotropy constant |  $K$ |  1.3 $J_{\mathrm{FM}}$

Skyrmion formations within the G-type AFM structure are depicted in Fig. 10.
This AFM lattice is divisible into two opposite sublattices, each being
individually ferromagnetic. The antiferromagnetic coupling between the nearest
neighbors ensures that these sublattices are essentially mirror images of each
other. Here, the uniaxial anisotropy provides the out-of-plane field needed to
stabilize skyrmion structures because using an external field is not feasible.
This is because the sublattices are arranged oppositely, meaning an external
field in any direction would energetically benefit one sublattice while
disadvantaging the other. The magnetization in the layer evolves with the
anisotropy strength analogous to the response found in the FM layer to the
external field. At low anisotropy, the magnetization is in an AFM-labyrinth
state as shown in Fig. 10 (a). When this state is divided into two sublattices
and analyzed separately, it reveals that it is made of two FM labyrinth domain
patterns which we saw before. With an increase in the anisotropy field
strength, these domains are disrupted and bubble-like structures are formed,
these are the AFM-skyrmions. Each bubble is characterized by a ring-shaped
area where the magnetization lies in the plane. Inside and outside this ring,
the magnetization adheres to the typical G-type pattern. When the
magnetization structure of the bubble is divided and analyzed separately for
each sublattice, it reveals the presence of two skyrmions, each with an
opposite polarity in its respective sublattice. The discovery of these
magnetic configurations in G-type antiferromagnetic (AFM) materials is
noteworthy. Numerical simulations have demonstrated that these structures can
move at high speeds in AFM lattices [79]. Additionally, the formation of these
structures involves the coupling of two skyrmions with opposite topological
characteristics. This unique coupling suggests the potential for these
skyrmions to neutralize each other’s Hall effect, which is a significant
aspect of their behavior in magnetic materials.

Figure 10: Magnetization structures formed in the AFM layer. (a) Labyrinth
domain state formed at low anisotropy. This can be split into two sublattices,
as shown in (b) and (c). (d) Skyrmions formed at a G-type AFM interface. (e)
Zoomed view of one of the skyrmionsSkyrmion formed in the (f) sublattice 1 and
(g) sublattice 2.

##  IV Conclusion

In our study, we employed atomistic spin dynamics and Monte Carlo simulations
to investigate how various material properties influence the exchange bias
field at ferromagnetic FM/AFM interfaces. We observed that the exchange bias
field is directly proportional to the interlayer coupling strength,
underscoring its critical role. However, it is important to note that our
model assumes a perfectly uncompensated interface and does not take into
account factors like defects, interlayer diffusion, or inter-layer roughness.
These elements could potentially modify the mechanism of exchange bias and may
reduce the strength of the bias field, as suggested in previous studies [57,
80, 81, 82, 83]. Future research that incorporates these effects could yield
more comprehensive results. Furthermore, we conducted a detailed analysis of
the stability of skyrmions in FM/AFM multilayers. Our findings indicate that
skyrmions formed in the FM monolayer via an exchange bias interface exhibit
greater stability compared to those stabilized by external fields, as
determined through GNEB calculations. This is, due to the formation of
magnetic imprints of the skyrmion in the AFM layer at the interface. However,
when the number of FM layers was increased to three or more, the exchange bias
from the interface was insufficient to maintain the skyrmion stability in the
upper layers. This observation aligns with our previous findings,
demonstrating an inverse relationship between the exchange bias field and the
number of FM layers, and is consistent with the experimental work on exchange
bias stabilized skyrmions by Rana et al. [37]. In their study, skyrmions are
stabilized by exchange bias in a ferromagnetic NiFe layer of approximately 0.3
nm thickness, equivalent to two or fewer atomic layers. Future research
efforts aiming to extend the exchange bias strength across more layers,
thereby stabilizing skyrmion tubes with additional FM layers, could be
particularly intriguing.

Additionally, our studies on the current-induced dynamics of isolated
skyrmions on monolayers and AFM-coupled skyrmions on bilayers have provided
insights into the fundamental mechanisms of skyrmion Hall effects. Based on
these insights, we proposed the design of an exchange-stabilized skyrmion
conduit. This setup is envisioned to host stable skyrmions in the absence of
external fields and facilitate their high-velocity movement, reaching speeds
up to 3 Km/s:  which is several times faster than generally reported
velocities [75, 76, 77, 26, 84]. The practical realization of such a setup
could mark a significant milestone in the field of spintronics. Nevertheless,
further investigations are required, particularly to identify materials with
optimal properties for the device’s construction. Given our existing knowledge
of material parameters like $J$, $D$, etc., density functional theory
calculations could be instrumental in identifying suitable materials or
material combinations for this purpose.

##  V Acknowledgment

Financial support from Carl Tryggers stiftelse Grant No. CTS:21-1400, the
Swedish Research Council (Vetenskapsrådet, VR) Grant No. 2016-05980, Grant No.
2019-05304, and Grant No. 2023-04239, and the Knut and Alice Wallenberg
foundation Grant No. 2018.0060, Grant No. 2021.0246, and Grant No. 2022.0108
is acknowledged. The China Scholarship Council (CSC) is acknowledged by Z.L.
The Wallenberg Initiative Materials Science for Sustainability (WISE) funded
by the Knut and Alice Wallenberg Foundation is also acknowledged. The
computations/data handling were enabled by resources provided by the National
Academic Infrastructure for Supercomputing in Sweden (NAISS), partially funded
by the Swedish Research Council through grant agreement no. 2022-06725.

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