# Light trapping using dimer of spherical nanoparticles based on titanium
nitride for plasmonic solar cells

Nowshin Akhtary  1 Ahmed Zubair1  1 Department of Electrical and Electronic
Engineering, Bangladesh University of Engineering and Technology, Dhaka 1205,
Bangladesh
<EMAIL_ADDRESS>
††journal: opticajournal††articletype: Research Article

Light-trapping mechanisms with plasmonics are an excellent way to increase the
efficiency of photovoltaics. Plasmonic dimer-shaped nanoparticles are
effective in light absorption and scatterings, and there is hardly any
research on dimer TiN nanoparticle-based PV. This paper demonstrated that
titanium nitride could be a suitable substitute for other plasmonic materials
in the visible and near-infrared spectrum. We designed a TiN-based spherical
dimer plasmonic nanoparticle for photovoltaic applications. We conducted
comparison analyses with the metals Ag, Au, and Al to ascertain the
performance of TiN as a plasmonic material. Silicon had an average absorption
power of $\sim$19$\%$, and after incorporating TiN nanoparticles, the average
absorbed power increased significantly to $\sim$75$\%$ over the whole spectral
range. TiN dimer nanoparticle had the highest absorption cross-section,
$Q_{ab}$ value $\sim$6.2 W/m2 greater than Ag, Au, and Al had a fraction of
light scattered into the substrate value greater than Au, Al and comparable to
Ag. TiN dimer exhibited better absorption enhancement, $g$ for the whole
spectral range than Ag, Au, and Al dimers for a radius of 15 nm with a peak
value greater than 1. The maximum optical absorption efficiency of the
plasmonic TiN nanostructures was $\sim$ 35.46$\%$.

##  1 Introduction

Coal, natural gas, oil, biomass, and nuclear energy are non-renewable energy
sources that are becoming scarcer and more depleted every day. Therefore, the
abundance, affordability, and low environmental impact of renewable energy
sources are all very advantageous. The efficiency of photovoltaic (PV) cells
powered by renewable energy sources has been a great focus of research[1, 2].
Light absorption and the frequency of electron-hole pair formation have a
significant role in how well PV cells perform. The most common PV cells in
CMOS technology are based on a silicon absorber layer with the limitation of
high production costs and thinner absorption volume. There are many ways of
increasing efficiency, such as surface texturing[3, 4, 5], metal nanograting
[6, 7], tandem structure [8], optical absorption enhancement by increasing the
effective optical path length or trapping light in the cell by introducing
light scatters in the solar cell [9, 10, 11, 12, 13, 14, 15]. A suitable
choice of materials for active layers can ensure better photon absorption,
generating electron-hole pairs. However, only efficient absorption cannot
generate efficient electron-hole pairs and, consequently, photo-voltage. The
recombination process creates a loss of charge carriers; therefore, an
optically thick semiconductor is unsuitable for better charge carrier
separation. Additionally, more materials are needed for thicker
semiconductors, which is cost-ineffective and wasteful. Thus, a thinner
semiconductor layer is preferred.

Introducing metallic nanoparticles (NPs) has created an alternative approach
to improving absorption efficiency. Surface plasmon resonance (SPR) can
significantly enhance EM waves by placing plasmonic structures in the active
layer. This phenomenon ensures enhanced light absorption providing strong
scattering between the intense plasmon field and the active layer [10, 16].
Localized surface plasmon resonances (LSPRs) generate light scattering by NPs.
The LSPR occurs when the frequency of the optical photon coincides with the
natural frequency of the collective vibration of conduction electrons in NPs,
leading to strong near-field electromagnetic enhancement, acute spectral
absorption, and scattering peaks [17]. The enhancement of optical absorption
of NPs is the foremost attribute of LSPR [18]. Light absorption and photo-
current have been improved by using the LSPR phenomena [19, 20]. Much recent
research has centered on the trade-off between optimal thickness and maximum
field enhancement. The most significant improvement variables ordinarily
happen when the junction between the absorber and NPs is illuminated by
polarized light. Complex structures can achieve a sensitivity that leads to
near-infrared (NIR) sensing and plasmon hybridization. The NPs structure can
get over this restriction since it extends the inside field to the outside
environment, which results in a considerable boost in detection sensitivity.

Plasmonic materials can support electrons or plasmons across a broad spectrum
from infrared to ultraviolet solar radiation. Until recently, researchers were
confined to noble metals like Ag and Au as plasmonic material. Ag and Au are
frequently used plasmonic metals and optical metamaterials because of their
strong DC conductivity or low resistive losses. While an electron in a metal’s
valence band absorbs a photon to jump to the Fermi surface or while an
electron close to the Fermi surface absorbs a photon to fill the ensuing
unoccupied conduction band, there is confinement for plasmonic metals, causing
an excessive loss in conventional plasmonic materials. Ordinary metals have
several drawbacks, including the size of the genuine portion of the
permittivity, the ineffectiveness of tuning or balancing the optical
properties of metals, and their high cost.

Due to the high optical loss of metals, alternative metals with the least
ohmic loss may be preferred for plasmonic devices. To reduce the interband
transition loss, many reports frequently utilized alternative plasmonic NP
[21]. Conventional plasmonic materials have many shortcomings, leading
researchers to seek better alternatives. The alternative plasmonics has the
real permittivity of the same order. Hence, geometric fractions lights can be
promptly tuned to coordinate the plan prerequisites. Conventional plasmonic
metals confront debasement when exposed to air/oxygen or moisture, causing
further problems in device fabrication and integration. These criteria
directly affect optical properties and increase optical loss, resulting in
more significant values of the dielectric function’s imaginary part and
rendering it incompatible with conventional silicon fabrication methods. Metal
nitrides are a better alternative to overcome the shortcomings. Among them,
titanium nitride (TiN) is a non-stoichiometric, interstitial compound with a
high concentration of free carriers. It is refractory and steady, and its
optical properties can be tuned by changing its geometric structure [22].
Moreover, it is consistent with silicon CMOS technology [23, 24] and offers
manufacturing and integration advantages that can help overcome the
challenges. There are several reports on monomer spherical and hemispherical
TiN NPs [25, 26]. However, no dimer spherical TiN NP-based plasmonic solar
cells have been reported.

This paper employed the finite-difference time-domain (FDTD) method to
systematically investigate the scattering cross-section and absorption
enhancement by spherical dimer TiN NPs for photovoltaic application.In order
to ascertain the total scattering cross-section, the percentage of light
scattered into the substrate, the absorption cross-section, and the spatial
mapping of the electric field in this plasmonic nanosystem, we first built and
optimized the dimer of the spherical NPs. We further investigated the effect
of polarization sensitivity on the source. We gained insight into how the
shape of the NPs enhanced the functionality of solar cells when the NPs were
embedded into them. We investigated plasmonic core-shell configuration and
analyzed the effect of dielectric coatings on NPs. Our work provided insights
into using TiN in photovoltaic cells.

##  2 Methodology

Figure 1: (a) Illustration of spherical dimer TiN nanoparticle on a 30 nm thin
Si3N4 underlayer on Si substrate acting as the absorber layer of the
photovoltaic cell. (b) Perspective view and (c) cross-sectional view of a
single dimer of spherical nanoparticles. Figure 2: (a) $f_{sub}$, (b) $Q_{T}$,
(c) $Q_{sc}$, and (d) $Q_{ab}$ as a function of wavelength for 100 nm radius
dimer NPs consisting of Ag, Au, Al, TiN, and TiN with Si3N4coating on a 30 nm
thick Si3N4 underlayer on Si substrate. Figure 3: Color map showing the xy
distribution of |E| for (a) Ag, (b) Au, (c) Al (d) TiN (e) (d) TiN with Si3N4
coating dimer NP placed on top of a 30 thin Si3N4 on a Si substrate. (f)
Schematic of the LSPR plasmon modes for NP dimers. The coupling of the dipoles
in the two spheres of the dimer, created by the plasmon, can occur in the
direction of the dimer axis or perpendicular to it. Figure 4: The E-field
spectra at (a) yz, and (b) xy plane for spherical dimer NPs comprised of
different materials placed on top of a 30 thin Si3N4 on a Si substrate. Figure
5: Color map showing the yz distribution of |E| for (a) Ag, (b) Au, (c) Al (d)
TiN (e) TiN with Si3N4 coating dimer NP placed on top of a 30 thin Si3N4 on a
Si substrate. (f) Absorption spectra of TiN dimer NP incorporated Si absorber
layer and Solar irradiance spectra. Figure 6: Percentage of power absorbed for
(a) Ag, (b) Au, (c) Al, (d) TiN NP incorporated Si absorber layer for solar
cells. Figure 7: Absorption enhancement, $g$ for the dimer of plasmonic
spherical NPs for r = (a) 15 nm, (b) 25 nm, and (c) 30 nm. (d) The $g$ for TiN
NPs with different radii. Figure 8: (a) $f_{sub}$, (b) $Q_{T}$, (c) $Q_{sc}$,
and (d) $Q_{ab}$ as a function of wavelength for TiN dimer shaped NP with
varying the polarization angle of the source from $\theta$=15∘ to 90∘. Figure
9: Color map showing the distribution of E-field in xy plane at Z=0 for TiN
dimer NP placed on top of a 30 nm thin Si3N4 on a Si Substrate with varying
the polarization angle of the source from $\theta$ = 15∘ to 90∘. Figure 10:
Color map showing the distribution of |E| in the yz plane at x=0 for TiN dimer
NP placed on top of a 30 nm thin Si3N4 on a Si Substrate with varying the
polarization angle of the source from $\theta$ = 15∘ to 90∘.

###  2.1 Structural Design

We developed an alternative plasmonic material, TiN-based spherical dimer NP
on a semi-infinite crystalline silicon substrate as can be seen in Fig. 1 (see
Fig.S1 of Supplement 1). In the visible and longer wavelengths, TiN displays
localized surface plasmon phenomena and metallic characteristics. [27, 28].
The plasmonic particles were separated from the semi-infinite silicon
absorption layer by a thin Si3N4 layer as surface passivation. We compared
cross-sections of NPs based on conventional noble plasmonic metal with TiN
alternative plasmonic NPs. The size of the particles was varied, and their
properties were analyzed. t1 and t2, respectively, presented the thin film and
substrate thickness. The source’s polarization angle represented by $\theta$,
r represented the radius of the sphere, and d represented the distance between
the nanospheres of a dimer.

###  2.2 Simulation Methods

We applied the FDTD method, where Maxwell’s equations were solved numerically,
to study the mentioned nanosystems. The simulation dimensions of the FDTD were
1.2 $\mu$m $\times$ 1.2 $\mu$m $\times$ 1.25 $\mu$m. A mesh size of 0.4 nm was
applied around the NPs. The source was adjusted for polarization perpendicular
to the surface normal of the particles from the air side. The particles were
incident to the total-field scattered-field (TFSF) plane wave along the
negative z-axis. The incident source was a uniform wave with a prominent
wavelength range of 550–1100 nm, which comprised the solar spectrum’s highest
feasible irradiance (AM 1.5). A plane wave with a TFSF was utilized to
separate the incident field from the scattered field to examine the optical
characteristics of NPs. The scattering characteristics were investigated using
an external monitor. The spatial electric field mapping was performed by
adjusting a frequency-domain power monitor. We used light scatterers to
increase the light trapping efficiency, improving the absorber layer’s
absorption. We estimated the scattering and absorption cross-sections, and
optimal values were obtained for better PV application by adjusting various
factors. The electric and magnetic fields around the particle were calculated
by converting the time domain into the frequency domain using a Fourier
transform. The radial Poynting vector, $S(\omega)$, was calculated from the
electric field, $E(\omega$), and magnetic field $H(\omega)$ as a function of
angular frequency, $\omega$. In the scattered field region, the total of power
${P_{s}}$ was determined along the +x, +y, +z, –x, –y, and –z directions. The
ratio of the power in the scattered field region inside the substrate of the
absorber layer to the power in the scattered field region in the air and the
absorber layer is known as the percentage of light scattered into the
substrate, $f_{sub}$. The total scattering cross-section, $Q_{T}(\omega)$ is
defined as the sum of the power per unit area scattered in all directions
divided by the power per unit area of the incident beam.

$Q_{T}(\omega)=\frac{P_{s}(\omega)}{I(\omega)}.$ (1)

Here, $I(\omega)$ is incident power intensity as a function of $\omega$.
Absorption cross-section is a measure of the probability of an absorption
process. The total absorbed power divided by the power per unit area of the
incident light was defined as absorption cross-section, $Q_{ab}$. It can be
calculated from

$dN/dz=-NnQ_{ab}$ (2)

Here, dN/dz is the number of photons absorbed between the points z and z+dz, N
is the number of photons penetrating to depth z, and n is the number of
absorbing molecules per unit volume. The monitors outside the TFSF source
determined the scattering cross-section. [13, 29].

We methodically considered the impact of NPs’ $Q_{T}$, the light scattered
into the substrate $Q_{sc}$, $f_{sub}$, and $Q_{ab}$ by investigating their
structural characteristics. We compared the effects of alternative plasmonic
NPs to plasmonic metals and comprehensively considered the capacity to enhance
absorption within an absorber layer by adding NPs. Moreover, to demonstrate
the effectiveness of the dimer NPs in PV cells, we calculated the proposed
structure’s absorption enhancement and light absorption efficiency.

##  3 Results and discussion

###  3.1 Effect of different material-based plasmonic spherical dimer
nanoparticle

We simulated spherical dimer NPs for different materials and tracked their
scattering and absorbance behavior. Here, t1, t2, and r were regarded as 30
nm, 250 nm, and 100 nm, respectively. We evaluated the performance of NPs made
of different materials, including Au, Ag, Al, and TiN. Moreover, we explored a
core-shell configuration consisting of TiN NP with Si3N4 coating to maximize
the performance. The foremost critical factor representing the path length
enhancement of a scattering light-trapping structure is $f_{sub}$ [30]. As can
be seen in Fig. 2 (a), the overall $f_{sub}$ exhibited Ag> Au> TiN> Tin with
Si3N4 coating> Al in this order. For 800 nm to 1100 nm wavelength,
the$f_{sub}$ of TiN NP was greater than those for Ag, Au, and Al-based NPs.
When we varied the dimer materials, the peak value of $Q_{T}$ was 19 W/m2 for
Au NP, and the values for Ag and Al were comparable to Au. TiN NP had
comparable values of $Q_{T}$ and $Q_{sc}$ from 650 to 1000 nm wavelength.
After adding Si3N4 coating on TiN NP the $Q_{T}$ and $Q_{sc}$ performed better
for 850 to 1000 nm as can be seen Fig. 2(b)-(d). For scattering applications,
Si3N4 coating can be used for TiN NP. $Q_{ab}$ increased for TiN and TiN with
Si3N4 coating NPs. The $Q_{ab}$ was negligible for Ag, Au, and Al NPs.

When a plane wave collides with an object or scatterer, its energy is diverted
in all directions. It is crucial to analyze the optical properties of NPs,
including scattering cross-section and electric field distribution. By
changing the spherical dimer NPs’ material, shown in Fig. 3 electric field
maps in the xy plane were detected. LSPR modes can be produced by these
structures. The plasmonic NP dimers are the equivalent of two atoms sharing
electrons by bonding molecular orbitals. A dimer’s excited dipoles on its two
spheres may couple in the direction of the dimer axis, which is analogous to
$\sigma$-type orbital for atoms or perpendicular to it, which is analogous to
$\pi$-type orbital for atoms. Four additional plasmon modes emerge for dimers,
which are homonuclear diatomic molecules equal to molecular hydrogen(H2),
nitrogen (N2), oxygen (O2), or a halogen (X2)[31]. As shown in Fig. 3(f), when
the charge of the dimers oscillates in the same direction, the charge
accumulates, and electric field enhancement is observed. This phenomenon
occurs both in bonding mode and anti-bonding mode, and they are in-phase
antibonding with the highest energy and in-phase bonding with dipolar plasmon
mode with the lowest energy (highest wavelength) (see Fig. 3(f)).
Additionally, when the charge oscillates in different directions, there is no
field enhancement, which is out-of-phase bonding and antibonding modes [32].
As can be seen in Figs. 3(d)-(e), the scattering spectra TiN and TiN with a
dielectric Si3N$4$ coating exhibited an unprecedented homogeneity for the two
spheres. The in-phase bonding plasmon mode was observed for x-polarized light
The induced dipole moments resulted in two bright modes. Therefore, the
accumulation of high free charge distribution around the surface and center of
the dimer resulted in the enhancement of the electric field.

The quasistatic dipole approximation was used to compute the electric field
($E$) enhancement at yz and xy plane around the surface of Ag, Au, Al, and TiN
dimers presented in Fig 4. Due to a lower real permittivity than Ag and Au,
the magnitude of the magnetic field enhancement in TiN nanospheres was
slightly smaller than those of Ag and Au. E-field intensity on the yz plane at
x = 0, which is the center of the dimer, can be seen in Fig. 5 for different
material-based dimer nanosphere. For Ag and Au and TiN with Si3N$4$ coating,
charge distribution was high at the center of the dimers as presented in Figs.
5(a), (b) and (e). For Al and TiN, charge distribution at the center was less
as compared to Ag and Au, as can be seen in Figs. 5(c) and (d). Here, the
charge oscillates in the same direction, so the charge accumulated and filed
enhancement occurred. Resulting LSPR modes were utilized in numerous detection
applications [27]. The LSPR mode of TiN with Si3N$4$ coating in the core-shell
configuration was blued-shifted, and the peak value of the electric field
increased compared to bare TiN dimer.

To determine the effectiveness of the NP in the photovoltaic cells, we
calculated the absorbed power of each layer of nanostructure consisting of
dimer spherical NPs on a 30 nm thin Si3N4 underlayer on Si substrate. NPs were
comprised of Ag, Au, Al, and TiN. The divergence of the Poynting vector was
used to compute the absorption per unit volume as given by,

$p_{abs}=-\frac{1}{2}\,real\,(\vec{\nabla}\,.\,\vec{S}).$ (3)

However, divergence calculations are frequently quite susceptible to numerical
errors. Consequently, the simplest method for calculating absorbed power is,

$p_{abs}=-\frac{1}{2}\,real\,(i\omega\,\vec{E}.\vec{D}^{*}).$ (4)

It can be modified as

$p_{abs}=-\frac{1}{2}\,\omega\,|E|^{2}\,imag(\epsilon).$ (5)

Here, $D$ is the electric displacement field, and $\epsilon$ is the
permittivity. Standard 1.5 ATM solar spectrum and absorption of TiN NP-based
solar cell are presented in Fig.5(f). Solar light absorption is very efficient
for wavelengths from 300 nm to 500 nm. For longer wavelengths than 500 nm, the
absorption decreased gradually. Silicon had an average absorption power of
$\sim$51$\%$ over the 400-500 nm range and $\sim$19$\%$ for the whole spectral
range as presented in Fig. 6. The Si3N$4$ layer enhanced the absorption for
the 400-500 nm spectral range. After the incorporation of TiN NPs, the
absorbed power increased significantly to $\sim$75$\%$ over the whole spectral
range as presented in Fig. 6(d). In comparison to Ag, Au, and Al NPs, TiN NP
integration had a substantially higher absorption power.

###  3.2 Absorption enhancement by TiN-based dimer NP

TThe absorption enhancement defines the increased absorption by the addition
of NPs in the solar absorber layer. The absorption enhancement spectrum, $g$
was presented by,

$g(\lambda)=\frac{EQE_{np}(\lambda)}{EQE_{bs}(\lambda)}.$ (6)

Here, $EQE_{np}$ is the device’s external quantum efficiency when plasmonic
nanoparticles were incorporated on top of a substrate and $EQE_{bs}$ is the
external quantum efficiency of a bare substrate. In this section t1, and t2
were taken to be 30 nm and 250 nm, respectively, in this section. We simulated
the spectra of $g$ for Ag, Au, Al, and TiN plasmonic dimers on a silicon
substrate for r = 15 nm, 25 nm, and 30 nm presented in Figs. 7(a)-(c). TiN
exhibited better absorption than Ag, Au, and Al dimers for r = 15 nm. For r =
25 nm the $g$ of TiN was $\sim$1 for the whole spectral range which was better
than Au and Al. For r = 25 nm, the $g$ of TiN was between 0.9 to 1. For r = 30
nm the $g$ of TiN was better than Al and comparable to Ag, Au for 400 to 800
nm and the $g$ was greater than Ag, Au and Al for the range 800 nm to 1100 nm.
For TiN, Al, Au, and Ag, the average enhancement, G, was discovered to equal
0.997, 0.991, 0.995, and 0.995, respectively.

The light absorption efficiency (LAE) was calculated by,

$\mbox{LAE}=\frac{\int_{400}^{1100}I(\lambda)A(\lambda)d\lambda}{\int_{400}^{1100}I(\lambda)d\lambda}.$
(7)

Here, $I(\lambda)$ is the incident light intensity. And, absorbance,
$A(\lambda)$ was calculated by,

$A(\lambda)=1-R(\lambda)-T(\lambda).$ (8)

Here, $T(\lambda)$ and $R(\lambda)$ are the transmittance and reflectance of
the structure. For the TiN plasmonic nanosphere on a kesterite substrate, we
determined LAE. The values of LAE for r = 15 nm were found to be 35.46$\%$ and
33.78$\%$ for TiN and Ag respectively.

We calculated $g$ for different radii of TiN plasmonic dimer on a silicon
substrate as can be seen from Fig. 7(d). The $g$ decreased with the increase
of r for wavelength which agrees well with the previous study [15].This
happened as a result of plasmonic NP’s strong forward scattering and weak
absorption of light at various radii. While backward scattering prevented
absorption, forward scattering promoted it. Spherical dimer plasmonic NPs with
larger radii often have larger cross-sectional areas of scattering, this
increase of metal NP and higher arrange mode excitation can control light
scattering which improves or diminish the light absorbing productivity into
the substrate [33].

###  3.3 Impact of light polarization

As the polarization of light influences light scattering or absorption, we
varied the polarization angle of the light source to compute the optical
characteristics of the NPs [15]. For the structure design with optimized
scattering and absorption, we changed the source’s polarization angle for
dimer spherical TiN NPs and found the optimal polarization angle, $\theta$.
Here, t1 and r were regarded as being 30 nm and 100 nm, respectively. The
$theta$ had been varied from 0∘ to 90∘. For the wavelength ranging from 550 to
760 nm, the value of $f_{sub}$ decreased with the increase of $\theta$ which
is apparent from Fig. 8(a). With a reduction in $\theta$, $f_{sub}$ remarkably
dropped for the wavelength range of 760 nm to 1100 nm. As the $theta$ was
raised, it was evident from Fig. 8(b) that $Q_{T}$ increased significantly.
The peak wavelength of $Q_{T}$ had a red-shifted spectrum. As can be seen from
Fig. 8(c), $Q_{sc}$ increased as the $\theta$ was increased and from Fig.
8(d), $Q_{ab}$ increased as the angle decreased from 90∘ to 15∘.

Figs. 9(a)–(f) represented E-field intensity for dimer NP with the variation
of $\theta$. The polarization angle defines the direction of the electric
field and magnetic field. When the light was polarized in the x-direction
(i.e., $\theta$=0), field enhancement was observed along the x-axis. As the
angle changed, the induced dipole changed with $\theta$ and consequently, in-
phase bonding and antibonding plasmon modes occurred accordingly [32, 34].
When $\theta$=15 ∘ and the charge of the dimers oscillates in the same
direction the charge accumulated, and electric field enhancement was observed
at $\theta$=15 ∘. This phenomenon occurred both in bonding mode and anti-
bonding mode when they are in-phase plasmon mode. The direction of charge
oscillation changed along the $\theta$ producing an induced dipole moment
[15]. Fig 9(a)-(c) illustrated the strong electric field enhancement that was
observed in the x-direction when charge oscillation was increasingly aligned
to the x-direction. As the polarized angle rose from 45∘ to 90 ∘ strong
electric field enhancement was observed in the y-direction as charge
oscillation alignment changed to y direction and the in-phase antibonding mode
was observed as illustrated in Fig9(d)-(f).

E-field intensity on the yz plane represented the center of the dimers, can be
seen from Figs. 10(a)-(f) with the variation of $\theta$. When $\theta$
increased from 15∘ to 45∘, the effective induced dipole moment decreased,
which originated the in-phase antibonding mode, due to the E-field
polarization alignment changed toward y direction presented in Figs.
10(a)-(c). When $\theta$ increased from 60∘ to 90∘, the charge was distributed
along the y-direction, which was the polarization angle of the E-field.
Consequently, the out-of-phase antibonding mode was apparent when $\theta$ was
near 90∘, as can be seen in Figs. 10(d)-(f).

###  3.4 Impact of the distance between the spheres of dimer

Figure 11: (a) $f_{sub}$, (b) $Q_{T}$, (c) $Q_{sc}$, and (d) $Q_{ab}$ as a
function of wavelength for TiN dimer spherical NP with distance, d = 0, 20,
70, and 100 nm placed on top of a 30 nm Si3N4 on a Si substrate. Figure 12:
(a) $f_{sub}$, (b) $Q_{T}$, (c) $Q_{sc}$, and (d) $Q_{ab}$ as a function of
wavelength for TiN dimer spherical NP for 30∘ the polarization angle of the
source with distance, d = 0, 20, 70, and 100 nm placed on top of a 30 nm Si3N4
on a Si substrate.

We considered the impact of changing the distance between the spheres of TiN
dimer NPs for the light source polarization angle at 0∘ and 30∘. We simulated
dimer spherical NPs with various distances between the spheres for $\theta$=0∘
as can be seen from Fig. 11. We varied the distance, d from 0 nm to 50 nm to
determine the structure for the optimized scattering cross-section. Here, t1
and t2 were considered 30 nm and 250 nm, respectively. The $f_{sub}$ decreased
with the increase of the d from d = 0 nm to 20 nm for 550 nm to 850 nm. When d
= 50 nm the distance between the dimers increased, and they started to behave
like a single sphere, As a result, $f_{sub}$ increased. When d = 0 nm, the
$Q_{T}$ spectrum was the lowest and increased as d increased. The $Q_{sc}$
decreased as d increased from 0 to 20 nm for the 550 nm to 820 nm range. For
longer wavelengths than 820 nm, $Q_{sc}$ increased with the increase of d.
$Q_{ab}$ was highest when d = 10 nm. For the wavelength 550 to 750 nm, the
$Q_{ab}$ decreased with the increase of d. For the dimers with d = 50 nm, as
the distance increased, it started to behave like two independent monomer
nanospheres.

We simulated a spherical dimer NP for 30∘ the polarization angle of the source
varying the distance between the spheres as can be seen in Fig. 12. Here, t1
and t2 were considered 30 nm and 250 nm, respectively. The $f_{sub}$ and
$Q_{sc}$ were comparatively higher for the whole spectral range for d = 0 nm
and lowest for d=20 nm. When d was smaller than 50 nm the $f_{sub}$ and
$Q_{sc}$ increased with the decrease of d, and $Q_{T}$ increased with the
increase of d. The $Q_{T}$ was highest for d = 10 nm and performed better for
the wavelength range 740 to 880 nm. Dimers performed better when there was no
distance between the spheres. For the polarization angle from 0∘ to 30∘, the
scattering spectra red-shifted.

###  3.5 Dependency on the radius of the dimer spherical NP

We simulated spherical dimer NPs with various radii and observed their optical
characteristics, as seen from Fig. 13. To find the best structure for
scattering cross-section, we varied the r from 50 nm to 120 nm. When the
radius was 50 nm, $Q_{T}$ was highest and the peak was almost 50 W/m2.

The most important element to consider when modeling a light-trapping
structure’s increased path length is $f_{sub}$ [30]. As the radius increased,
the value of $f_{sub}$ decreased significantly [15]. For the whole spectral
range, $f_{sub}$ did not vary appreciably at 50 nm and 70 nm radius. This made
it possible to efficiently couple the part of the scattered light with a high
in-plane wave vector that is transient in the air but engendered in silicon.
As can be seen in Figs. 13(b)-(c), $Q_{T}$ and the $Q_{sc}$ decreased as the r
increased from 50 nm to 120 nm. $Q_{ab}$ exhibited the highest value for the
whole spectral range for r = 100 nm and the peak was at 5.5 W/m2 as can be
seen from Figs. 13(d). Therefore 100 nm radius is optimum for dimers
applications.

Figure 13: (a) $f_{sub}$, (b) $Q_{T}$, (c) $Q_{sc}$, and (d) $Q_{ab}$ as a
function of wavelength for dimer spherical TiN NPs with radius, r = 50, 70,
100, and 120 nm placed on top of a 30 nm Si3N4 on a Si substrate.

##  4 Conclusions

Dimer of spherical TiN NPs appeared as an efficient alternative plasmonic
material for plasmonic and metamaterial applications.The results of our
investigation into the effect of TiN dimer spherical NPs on the enhancement of
the thin solar cell’s light absorption were promising. After the incorporation
of TiN NPs on a silicon substrate, the average absorbed power increased
significantly from $\sim$19$\%$ to $\sim$75$\%$ over the whole spectral range.
TiN exhibited better absorption enhancement, $g$ and percentage absorbed power
to Ag, Au, and Al dimers for r = 15. The average enhancement, G for TiN, Au,
and Ag were found to be 0.9972, 0.9953. and 0.9954, respectively, for r = 15
nm. TiN dimer NP had the highest $Q_{ab}$ value of $\sim$6.2 W/m2 which were
greater than Ag, Au, and Al. By changing the size of TiN dimer NPs, the
absorption enhancement peak may be tailored to the required solar spectrum.
TiN dimer NPs demonstrated to be beneficial when inserted in tandem solar
cells because of their cost-effectiveness along with their abundance and ease
to manufacture.

Funding A.Z acknowledges the Basic Research Grant (Sonstha/R-60/Ref-4747)
provided by the Bangladesh University of Engineering and Technology.

Acknowledgments N.A. and A.Z. acknowledge the technical support of the
Department of Electrical and Electronic Engineering at Bangladesh University
of Engineering and Technology (BUET), Dhaka, Bangladesh, for the completion of
the work.

Disclosures The authors declare no conflict of interest.

Data availability Data underlying the results presented in this paper are not
publicly available at this time but may be obtained from the authors upon
reasonable request.

Supplemental document See Supplement 1 for supporting content.

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# Current-driven dynamics of antiferromagnetic skyrmions: from skyrmion Hall
effects to hybrid inter-skyrmion scattering

Amal Aldarawsheh1,2∗ Moritz Sallermann1,3,4 Muayad Abusaa5 and Samir
Lounis1,2∗

Peter Grünberg Institute and Institute for Advanced Simulation,
Forschungszentrum Jülich and JARA, D-52425 Jülich, Germany

Faculty of Physics, University of Duisburg-Essen and CENIDE, 47053 Duisburg,
Germany

RWTH Aachen University, 52056 Aachen, Germany

Science Institute and Faculty of Physical Sciences, University of Iceland, VR-
III, 107 Reykjavík, Iceland

Department of Physics, Arab American University, Jenin, Palestine

*<EMAIL_ADDRESS><EMAIL_ADDRESS>
## Abstract

Antiferromagnetic (AFM) skyrmions have emerged as a highly promising avenue in
the realm of spintronics, particularly for the development of advanced
racetrack memory devices. A distinguishing feature of AFM skyrmions is their
zero topological charge and hence anticipated zero skyrmion Hall effect
(SkHE). Here, we unveil that the latter is surprisingly finite under the
influence of spin-transfer torque, depending on the direction of the injected
current impinging on intrinsic AFM skyrmions emerging in CrPdFe trilayer on
Ir(111) surface. Hinging on first-principles combined with atomistic spin
dynamics simulations, we identify the origin of the SkHE and uncover that FM
skyrmions in the underlying Fe layer act as effective traps for AFM skyrmions,
confining them and reducing their velocity. These findings hold significant
promise for spintronic applications, the design of multi-purpose skyrmion-
tracks, advancing our understanding of AFM-FM skyrmion interactions and hybrid
soliton dynamics in heterostructures.

##  1 Introduction

Topologically protected magnetic textures, such as magnetic skyrmions [1, 2,
3, 4, 5, 6, 7, 8], have emerged as a subject of immense interest due to their
potential to become non-volatile information carriers in future spintronic
devices [9, 10, 11, 12]. The discovery of magnetic skyrmions in numerous
materials hosting bulk or interfacial Dzyaloshinskii-Moriya interaction (DMI)
has spurred extensive research to explore their unique properties and
applications [3, 4, 5, 6]. Particularly, the concept of skyrmion-based
racetrack memory [2, 3, 13, 14] has garnered considerable attention, where
skyrmions are dynamically driven by numerous means. These skyrmions possess
several advantages, including nanoscale size, topological stability, ability
to overcome pinning sites [15, 16, 17, 18, 19, 20], and low depinning current
compared to domain walls [21, 22, 23, 24].

Various methods have been proposed to drive magnetic skyrmions, encompassing
electric currents [2, 13, 14, 25], spin waves [26], magnetic field gradients
[27], temperature gradients [28], and voltage-controlled magnetic anisotropy
[29, 30, 31, 32]. However, one significant challenge that arises during their
manipulation via electrical means or magnetic field gradient [33] is the
skyrmion Hall effect (SkHE), wherein skyrmion trajectories deviate from the
driving current direction due to the Magnus force [34, 35, 36] proportional to
the topological charge [3]. This undesired effect hampers the precise control
and movement of skyrmions in spintronic devices.

In contrast, antiferromagnetic (AFM) skyrmions are expected to be transparent
to the SkHE since the building-blocks skyrmions carry opposite topological
charge, which enforce the motion along the direction imposed by the applied
current, as predicted theoretically [37, 38, 39] and observed for significant
distances experimentally [40]. Moreover, AFM skyrmions are in general
insensitive to external magnetic fields [41, 42, 43, 44, 45, 46], which
overall promotes their discovery and control for promising implementation in
devices. Following extensive phenomenology-based predictions [37, 39, 43, 47,
48, 49, 50], FM skyrmions coupled antiferromagnetically through a spacer, so-
called synthetic AFM skyrmions, were realized in multilayers [40, 51, 52, 53,
54] while more complex AFM topological objects were identified in bulk
materials [55, 56, 57]. Ab-initio simulations predicted the emergence of
intrinsic AFM skyrmions in Cr films and frustrated multimeronic states in Mn
films interfaced with Ir(111) surface [45, 58]. Intrinsic means that the AFM
solitons are hosted within the same AFM material.

Here, we explore the dynamical response of intrinsic AFM skyrmions to an
applied current. We consider the scenario of a magnetic tunnel junction (MTJ),
where a magnetic electrode injects a perpendicular-to-plane spin-polarized
current (SP-CPP) with in-plane polarization on CrPdFe thin films deposited on
Ir(111) surface predicted from first-principles to host intrinsic AFM
skyrmions [45](Fig. 1). Counter-intuitively, we demonstrate that these
skyrmions exhibit a significant SkHE, which is strongly anisotropic, i.e. that
is dependent on the polarization direction of the applied spin-current. We
identify the origin of the SkHE and its vanishing conditions, which permits
the design of track with or without the SkHE. We unveil complex interactions
when interfacing intrinsic AFM skyrmions hosted in Cr with the spin-textures,
including individual FM skyrmions, found in Fe. This unique hybrid scenario
enables the exploration of AFM-FM inter-skyrmion dynamics. The mutual inter-
skyrmion interactions design a non-trivial two-dimensional energetical map,
with pinning and repulsive centers, which impact on both the trajectory and
velocity of AFM skyrmions and provide pinning and deflection processes. These
findings pave the way for further exploration and control of skyrmion-based
devices and applications in antiferromagnetic storage systems.

Figure 1: Current-driven dynamics of AFM skyrmions. a, b Schematic
representations of CPP induced motion of an elliptical AFM skyrmion showing
the SkHE a or not b depending on the alignment of the polarization of the
applied current $\mathbf{S}_{P}$ with respect to the skyrmion. c, d Top view
of a and b, respectively. e Schematic representation of the material hosting
intrinsic AFM skyrmions at the triangular lattice of a Cr layer grown on PdFe
film deposited on an fcc(111) surface of Ir. The ground state in the Cr layer
is the row-wise AFM configuration illustrated in the top view of the surface
as red and blue spheres for different orientation of the spins shown in inset
of f. Snapshots of single f and double g overlapping AFM skyrmions emerging in
the Cr film with the spins distribution among four sublattices L1-L4.

##  2 Results

###  2.1 Trajectories of AFM skyrmions driven by perpendicular-to-plane
currents.

We investigate current-induced dynamics of single and interchained AFM
skyrmions (Fig. 1 f-g), emerging at the Cr layer when deposited on
PdFe/Ir(111) with fcc stacking (Fig. 1 e). PdFe/Ir(111) is a well established
thin film material that hosts spin spirals, which can be deformed into FM
skyrmions upon application of a magnetic field [15, 59, 60, 61, 62, 18, 63,
64, 65]. Once covered with a Cr layer, a row-wise AFM (RW-AFM) state emerges,
which hosts individual or catenated AFM skyrmions [45] even without applying
an external magnetic field. RW-AFM states in a triangular lattice are not
common in nature [66, 67], and arise due to the interplay of the neighboring
Heisenberg exchange interactions ($J$) [68, 46]. The presence of both the DMI
and an out-of-plane magnetic anisotropy (K) stabilizes the highly sought
intrinsic AFM skyrmions. With the magnetic interactions among Cr atoms
calculated from ab-initio (see Method section) and depicted in Supplementary
Figure 1, the spins in the Cr layer are distributed among four sublattices
(L1-L4). Ferromagnetically aligned spins on one sublattice host a single FM
skyrmion, which is coupled to FM skyrmions emerging in other sublattices to
form AFM skyrmions. Single and interchained AFM skyrmions can form by
populating in different fashion the distinct sublattices (Fig.1 f-g). Here we
employ atomistic spin dynamics simulations, applying Landau-Lifshitz-Gilbert
(LLG) equation augmented with spin transfer torque (STT) terms [69, 70, 38,
37, 71], as described in the Methods section, to explore the current-driven
motion of the AFM skyrmions, by injecting perpendicular to plane currents with
in-plane polarization (Fig. 1 a-d). In this scenario, one expects a straight
motion of an AFM skyrmion along the direction perpendicular to the
polarization $\mathbf{S}_{p}$ of the applied spin current $\mathbf{j}_{s}$
[38, 37], as illustrated in Fig.1 b-d.

Figure 2: Transitional motion of elliptical AFM skyrmion driven by SP-CPP
currents. a-d Snapshots of the AFM skyrmion at Cr layer in four different
cases: a Cr-free case, where the magnetic interactions with Fe layer are not
included (AFM skyrmion radius is 2 nm); b Fe interactions are included with a
finite magnetic field saturating Fe into a FM state (AFM skyrmion radius is
2.1 nm) while c a weaker magnetic field leads to a skyrmion lattice (SkX)
which slightly enlarges the AFM skyrmion (radius of 2.5 nm); d in the absence
of a magnetic field spirals emerge at Fe layer (AFM skyrmion reaches a radius
of 3.2 nm). e Impact of the current parameter ratio $\eta/\alpha$ on the
skyrmion Hall angle, and f on velocity. g The trajectories of the AFM skyrmion
for cases in a-d with $\eta/\alpha$ =0.01 meV.

We initiate our study by investigating the case of single AFM skyrmions (Fig.
1 a-d) and consider two possibilities: either (i) by neglecting the Cr-Fe
magnetic exchange interactions, which corresponds to a free standing Cr film
(Fig. 2 a), or (ii) not by scrutinizing various magnetic states in PdFe, which
can be tuned by applying a magnetic field (Fig. 2 b-d). For the latter, we
consider the case of a saturated FM state in the Fe film (Fig. 2 b), which is
obtained upon application of a large magnetic field, while a moderate field
can transition the spin-spiraling state shown in Fig. 2 d to a skyrmion
lattice (SkX) illustrated in Fig. 2 c [15, 61]. The effective impact of the
spin current can be monitored via the current parameter $\eta$, which is
directly proportional to ${j}_{s}$, divided by the Gilbert damping $\alpha$
(see Method section).

As aforementioned, AFM skyrmions exposed to spin-polarized currents via STT,
typically do not experience a SkHE [38, 37, 72, 39, 40] while their velocity
is expected to be proportional to $\eta/\alpha$ in the case of CPP injection
[72]. Surprisingly, our AFM skyrmions exhibit an unexpected dynamical
behavior, deviating from conventional expectations. Independently from the Cr-
Fe interaction and the nature of the magnetic state pertaining to the PdFe
film, the Hall angle is found overall to be around -5∘ (negative sign means
the deviation is clockwise), which remains consistent across various
$\eta/\alpha$ values (Fig. 2 e). Deviations occur, however, for weak driving
forces (small $\eta/\alpha$) when Cr is placed atop Fe spin spirals. Indeed,
the AFM skyrmions display then an irregular ’Brownian-like’ motion as depicted
in Fig. 2 g, due to uncontrolled scattering at various spin-textures emerging
in Fe. In this particular case, the extraction of the Hall angle is not
trivial, since the skyrmion trajectories are not straight. Impressively, the
Fe spirals can strongly deflect the AFM skyrmions, which can lead to effective
Hall angles that are larger than 10∘, as calculated up to average distances of
about 90 nm. Overall, the skyrmion Hall angles are found to be the smallest
(largest) atop the SkX Fe (Fe spirals).

The velocity of the skyrmions is linear with $\eta/\alpha$, with the largest
speed found when Fe host a SkX state (Fig. 2f). Intriguingly, the Cr-Fe
interaction in general favors large skyrmion velocities, which can be traced
back to the size of the skyrmions. Indeed, the Cr-Fe interaction enlarges the
diameter of the AFM skyrmion, which is known to increase its velocity [48,
73], as unveiled in the upcoming analysis. Examples of skyrmion trajectories
are shown in supplementary Movies 1-3.

Fig. 2 e-f is just the tip of the iceberg. By scrutinizing the skyrmion
dynamics as function of the direction of the applied current, we unveil a rich
anisotropic responses: both the skyrmion Hall angles and velocities are
modified and we identify directions along which the SkHE cancels out. Before
discussing the anisotropic current-driven dynamical response, we briefly
address the origin of this behavior, which is induced by the ellipticity of
the AFM skyrmions emerging in Cr/PdFe/Ir(111) surface. By carefully
scrutinizing the AFM skyrmions, one can identify an elliptical shape. For
instance, the single AFM skyrmion shown in Fig. 3c has a major and minor axis
of 2.2, and 1.8 nm, respectively. Upon formation of a double AFM skyrmion, the
shape of the skyrmions remains elliptical. The size of the skyrmions forming
the solitonic dimer, experiences a significant increase, enlarging both the
major and minor axes of the skyrmion building-blocks to 3 nm and 2.4 nm,
respectively. The origin of the observed ellipticity can be traced back to the
anisotropic magnetic environment experienced by the spins residing in the AFM
skyrmion due to the interplay between the neighboring exchange interactions
and the triangular lattice, which can be decomposed into four sublattices
carrying distinct magnetic states (a more detailed analysis is provided in
Supplementary Note 1 and Supplementary Figure 2). Phenomenologically, one can
demonstrate that by tuning the underlying interactions, the skyrmions can be
reshaped into an isotropic form [46]. We note that this is clearly a material
dependent property.

###  2.2 Directionality of the current-driven elliptical AFM skyrmions.

Figure 3: Influence of the direction of current polarization on the skyrmion
Hall angle and velocity of elliptical AFM skyrmions. a The skyrmion Hall angle
and b velocity of AFM skyrmions as a function of the angle $\theta_{j_{s}}$
between the current polarization direction and the major ellipse axis, for
single (brown) and double(blue) AFM skyrmions, with $\eta/\alpha=$ 0.05 meV. c
Schematic representation of the AFM skyrmion showing an elliptical shape, with
a long (short) axis defining the $x$-axis ($y$-) axis. The angles associated
with the spin-polarization of the current $\mathbf{S}_{\mathrm{P}}$ and the
SkHE are displayed. The colored regions within the ellipse correspond to those
shown in a and b.

To explore the anisotropic current-driven response of the AFM skyrmions, we
focus here on the case of the free-standing Cr layer, i.e. with the Cr-Fe
interaction switched-off, which is also representative of the behavior found
when the interaction is switched on while the Fe film hosts either the
skyrmion lattice or the saturated FM state. As an example, we inject a current
with $\eta/\alpha=0.05$ meV, but varying systematically the angle
$\theta_{j_{s}}$ between the in-plane current and the major axis of the
ellipse, which is represented by the orange dashed line in Fig. 3c.

The skyrmion Hall angle as function of $\theta_{j_{s}}$ is illustrated in Fig.
3a, which clearly shows an oscillating behavior for both the single (brown)
and double (blue) AFM skyrmions, with color coded regions corresponding to
colored areas depicted in Fig. 3c. From Fig. 3a, one can notice that
$\theta_{\text{Hall}}$ is suppressed when the current is polarized along the
two ellipse axes, and reaches its maximum value of about 6∘ when
$\theta_{j_{s}}$ = 42∘. Notably, it is not only the Hall angle that changes
with $\theta_{j_{s}}$; the velocity of AFM skyrmions also varies as shown in
Fig. 3b, exhibiting the maximum (minimum) velocity when the skyrmions move
along the ellipse major (minor) axis. Interestingly as depicted in Fig. 3a,
double and single AFM skyrmions show the same Hall angle when subjected to the
same polarized currents, however the double AFM skyrmion moves faster than the
single (Fig. 3b).

###  2.3 Thiele equation for elliptical AFM skyrmions.

To comprehend these intriguing findings, we analyze the Thiele equation
governing AFM skyrmions driven by CPP [74, 38, 37, 36] (see more details in
Supplementary Note 2):

$-\alpha\mathcal{D}\cdot\textbf{v}+\mathcal{B}\cdot\textbf{S}_{p}=0,$ (1)

where v denotes the skyrmion velocity while
$\textbf{S}_{p}=(\cos\theta_{j_{s}},\sin\theta_{j_{s}})$ stands for the unit
vector defining the spin polarization direction of the injected current. Note
that the $x$ and $y$ axes are defined respectively by the semi-major and semi-
minor axes of the elliptical skyrmion (see Fig. 3c). $\mathcal{B}$ is the
driving force related tensor, and $\mathcal{D}$ is the dissipative tensor for
the whole AFM skyrmion. If the skyrmions were isotropic in shape, the velocity
would be perpendicular to the polarization of the current and there would not
skyrmion Hall effect.

For each building-block FM skyrmion $i$ forming the AFM skyrmion, the
associated components of the dissipative tensor [75], assuming skyrmions of
identical size and shape, are given by:
$(\mathcal{D}^{i}_{xx},\mathcal{D}^{i}_{yy})=\frac{\pi^{2}}{8}\left(\frac{b}{a},\frac{a}{b}\right)$,
where $a$, and $b$ are the semi-major and semi-minor ellipse axes, while
$\mathcal{D}^{i}_{yx}=\mathcal{D}^{i}_{xy}=0$. The components of the driving
force tensor are
$(\mathcal{B}^{i}_{xy},\mathcal{B}^{i}_{yx})=\frac{\gamma\pi}{8\mu_{s}}\eta(-b,a)$
and $\mathcal{B}^{i}_{xx}=\mathcal{B}^{i}_{yy}=0$. The skyrmion velocity is
then given by:

v $\displaystyle=$
$\displaystyle\frac{1}{\alpha}\left(\frac{\mathcal{B}_{xy}}{\mathcal{D}_{xx}}\sin\theta_{j_{s}},\frac{\mathcal{B}_{yx}}{\mathcal{D}_{yy}}\cos\theta_{j_{s}}\right)=\frac{\gamma\eta
a}{\pi\mu_{s}\alpha}\left(-\sin\theta_{j_{s}},\frac{b}{a}\cos\theta_{j_{s}}\right),$
(2) $\displaystyle|\textbf{v}|$ $\displaystyle=$
$\displaystyle\frac{\gamma\eta
a}{\pi\mu_{s}\alpha}\sqrt{\sin^{2}{\theta_{j_{s}}}+\frac{b^{2}_{sk}}{a^{2}_{sk}}\cos^{2}{\theta_{j_{s}}}},$
(3)

where one immediately notices that if the skyrmions were circular isotropic,
the polarization of the spin-current is perpendicular to the velocity since
$\textbf{v}\cdot\textbf{S}_{p}=0$. The propagation direction associated to the
isotropic case case defines the reference angle from which the skyrmion Hall
angle is measured
$\theta_{\text{ref}}=\arctan{\frac{v_{y}}{v_{x}}}=\theta_{j_{s}}+\frac{\pi}{2}$.
Therefore, the Hall angle is evaluated from

$\displaystyle\theta_{\text{Hall}}$ $\displaystyle=$
$\displaystyle\arctan{\left[\frac{b}{a}\tan\left(\theta_{j_{s}}+\frac{\pi}{2}\right)\right]}-\left(\theta_{j_{s}}+\frac{\pi}{2}\right)\,.$
(4)

With these findings at hand, we can explain the behavior of the AFM skyrmions.
If the current is polarized along the major or minor axes of the spins, i.e.
$\theta_{j_{s}}$ is a multiple of $\frac{\pi}{2}$, the Hall angle cancels out,
which define the extrema of the velocity given by $\frac{\gamma\eta
b}{\pi\mu_{s}\alpha}$ for $\theta_{j_{s}}=0,\pi$ (minimum) and
$\frac{\gamma\eta a}{\pi\mu_{s}\alpha}$ for
$\theta_{j_{s}}=\frac{\pi}{2},\frac{3\pi}{2}$ (maximum). The ratio
$\frac{b}{a}$, which is about 0.8 for both the single and double AFM
skyrmions, shapes the magnitude of the Hall angle as well as the range of
oscillations in the velocity. This means with elliptical AFM skyrmions, we
have two more degrees of freedom to manipulate the CPP induced motion of the
AFM skyrmions, where the skyrmion exhibits its maximum velocity when injecting
currents polarized along its minor axis, resulting in a Hall free motion along
the major axis. This analysis goes along with our findings depicted in Fig. 3
b, where the double AFM skyrmion with dimensions of $(a,b)=(3,2.4)$ nm moves
with maximum velocity of 355 while the single AFM skyrmion with smaller size
(dimensions of $(2.2,1.8)$ nm), and hence slower motion according to eq. 2,
where its maximum velocity reaches 260 m/s. The maximum Hall angle is expected
for $\theta_{j_{s}}+\pi/2=\arccos{\sqrt{\frac{b}{a+b}}}$, which leads to
$\theta_{j_{s}}=42^{\circ}$ and $\theta_{\text{Hall}}=6.2^{\circ}$ in
agreement with the numerical findings of the previous section. Notably, the
impact of ellipticity on the motion of FM skyrmions subjected to spin currents
was discussed in Refs. [76, 75, 77].

###  2.4 Current-driven dynamics of AFM skyrmions interacting with FM
skyrmions.

Figure 4: Position dependent deflection of AFM skyrmions due to the AFM-FM
skyrmionic interaction.  a, b Schematic representation of the forces acting on
the AFM skyrmion: The driving force due to the applied spin current and the
AFM-FM skyrmionic interacting induced by the FM skyrmion in the Fe layer
through the Pd-spacer. In a two trajectories with different starting points
are illustrated. c Schematic representation of the effect of the Fe FM
skyrmion on the velocity of the AFM skyrmion with the blue and orange lines
corresponding to the paths shown in a. d The trajectory of the AFM skyrmion
shown in a when current-driven toward a pinned FM skyrmion considering
different initial positions (motion from left to right). Deflection in the
motion direction occurs depending on the relative position between the AFM and
FM skyrmions.

When the Fe substrate hosts spirals with an inhomogeneous distribution of FM
skyrmions, we unveiled in Fig. 2g that the intrinsic AFM skyrmions driven in
the Cr overlayer exhibit typical dynamics pertaining to interactions with
defects. In this section, we explore the synthetic configuration of an AFM
skyrmion interacting with a FM one through a Pd film. This scenario is
trivially realized in Cr/PdFe/Ir(111) surface by applying an out-of-plane
magnetic field, which reduces the size of the FM skyrmions in Fe and
transforms the lattice configuration into individual topological objects. By
applying spin-polarized current, as done previously, we drive an AFM skyrmion
living in the Cr film towards a pinned FM skyrmion hosted by the Fe layer. We
consider two cases, either the planned skyrmion trajectory passes trough the
FM skyrmion, or it is shifted (see Fig. 4 a-b).

We notice that the AFM skyrmion gets pinned at the FM skyrmion for a weak
applied current, see example $\eta/\alpha=$ 0.001 meV in Supplementary movie
4, which clearly indicates the attractive nature of the FM-AFM skyrmion
interaction. A stronger current, e.g. $\eta/\alpha$ = 0.017 meV, enables the
AFM skyrmion to escape the trapping FM soliton following a roller-coaster-like
motion, with a velocity increase from the initial 75 m/s to reach 130 m/s once
getting close to the FM skyrmion, which leads to a ”speeding up zone” as shown
in Fig. 4c. Once the AFM skyrmion overtakes the FM skyrmion, the velocity
reduces by about 88$\%$ down to around 15 m/s due to the FM-AFM skyrmion
interaction that opposes the driving force and leads to a ”slowing-down zone”
. As the AFM skyrmion moves away from the FM skyrmion, only the driving force
dictates its motion, resulting in a ”constant motion regime,” depicted in Fig.
4 c and Supplementary Figure 3, where the velocity stabilizes at approximately
75 m/s.

If the AFM skyrmion is off-centered with respect to the FM one, their mutual
attractive interaction is capable of deviating the underlying trajectory to
bring the AFM skyrmion to the vicinity of the FM one as depicted schematically
in Fig. 4b and demonstrated systematically for different paths illustrated in
Fig. 4d and Supplementary Figures 3-5. Due to the AFM-FM skyrmion attraction
the AFM skyrmion deflects at the vicinity of the FM one and then continues its
motion along a straight line with a velocity of 75 m/s. Intriguingly, a second
deflection manifests when the AFM skyrmion starts an approach from positions
(11) and (12) after passing the FM skyrmion, which signals a non-trivial
energy profile of the hybrid AFM-FM skyrmionic interaction.

###  2.5 AFM-FM skyrmion interaction profile.

Figure 5: AFM-FM skyrmion interaction profile. a Two-dimensional heat map of
the total energy difference resulting when rigid-shifting the AFM skyrmion all
over the lattice with the presence of Fe-FM skyrmion at the center of the Fe
layer. The energy difference is taken with respect to the case when the two
skyrmions are not interacting. This energy difference is further decomposed
into the interaction with the AFM skyrmions building-blocks: the red-cored FM
skyrmion residing at sublattice L2 b, which is of attractive nature, and the
interaction with the blue-cored skyrmion residing at sublattice L1 c, which is
of repulsive nature. d Energy profile along the purple line indicated in a,
decomposed in e into the L2 blue and L1 red contributions. f The z component
of the spin in the two sublattices L1 (blue) and L2 (red) along the purple
line when the AFM skyrmion positioned at the second minimum shown in a. Inset
in d is a schematic representation of the nature of the interaction between
the building blocks of the AFM skyrmion and the FM Fe-skyrmion.

To elucidate the underlying reason for the unanticipated second deflection
mentioned in the previous section, we analyse the AFM-FM skyrmion interaction
profile. Since the AFM skyrmion in Cr film is made of two FM skyrmions
oppositely oriented with respect to each other, we expect two competing
interactions with the FM skyrmion in Fe layer, as shown in the inset of Fig. 5
d. The FM skyrmions having their cores pointing in the same direction and
residing in the same ferromagnetic background would repel each other, while
those having an opposite magnetic alignment would attract each other. This is
clearly illustrated in Fig. 5 b, and c, respectively, which shows the energy
contribution of both types of coupling to the energy profile, that is obtained
by rigidly shifting the AFM skyrmion all over the Cr lattice atop the FM
skyrmion pinned in center of the Fe film. The total interaction heat map
depicted in Fig. 5 a exhibits a minimum when the AFM skyrmion overlaps with
the FM skyrmion, signifying their mutual attraction. A second minimum appears
when the AFM skyrmion is positioned in the lower right part of the FM
skyrmion, which explains the aforementioned second deflection experienced by
the AFM skyrmion as noted in the previous section.

The surprising second minimum finds its origin in the intrinsic asymmetric
shape of the AFM skyrmion with respect to the skyrmion core residing in one of
the sublattices, as illustrated in Supplementary Figure 6. When interfaced
with the Fe FM spins, the Cr spins residing in the background of sublattice L1
tilt away from their initial direction, while those residing in sublattice L2
get more collinear and antiferromagnetically aligned with respect to the Fe-
magnetization, see the red and blue plots respectively in Fig. 5 f obtained
along the purple line in Fig. 5 a when positioning the Cr skyrmion at the
second minimum. At the vicinity of the Fe FM-skyrmion, the interaction picture
gets reversed, which leads to the sublattice-dependent interaction profile
shown in Fig. 5 e. In particular, the asymmetric profile of the AFM skyrmions
together with the magnetic interaction across the sublattices enable an energy
gain in an area of sublattice L1, where the Cr spins benefit from the AFM
coupling with the core of the Fe skyrmion, see the kink in the spin profile
highlighted in the inset of Fig. 5 f. Unveiling the interaction profile
between AFM and FM skyrmions holds significant importance, as it offers an
opportunity for manipulating and regulating the trajectories and dynamics of
AFM skyrmions by strategically positioning pinned FM skyrmions at the Fe
layer.

##  3 Discussion

In this study, we uncovered the intricate dynamics of intrinsic AFM skyrmions
subjected to perpendicular to plane spin polarised currents, with a particular
attention to the impact of FM skyrmions emerging in a hybrid heterostructure
(Cr/PdFe/Ir(111) surface) made of an AFM layer (Cr) separated from a FM layer
(Fe) by a Pd spacer layer. In contrast to expectations, even in AFM skyrmions
we demonstrate the emergence of the SkHE stemming from the elliptical shape of
the topological states. Both the SkHE and skyrmion velocity are anisotropic
and follow well defined dependencies with respect to the polarization
direction of the applied currents. The ability to manipulate the polarization
direction of impinging spin currents provides a clear avenue for designing
tracks where the skyrmion Hall effect (SkHE) either diminishes or persists.
The presence of non-trivial magnetic states in the FM film can impact the
dynamics of the AFM skyrmions by tuning both their velocity and trajectory.
For instance, FM skyrmions act as pinning centers , which depending on the
applied current can deflect AFM skyrmions. The seeding of FM skyrmions
modifies non-trivially the emergent hybrid AFM-FM skyrmionic interaction
profile, which can host several minima, offering the potential of customizing
pathways for the motion of AFM skyrmions (see examples illustrated in
Supplementary Figure 7).

In summary, our study advances the understanding of AFM skyrmion dynamics and
their interplay with FM skyrmions. These insights hold great promise for the
development of innovative spintronic devices that harness the unique
properties of AFM and FM spin textures. As the field of AFM spintronics
continues to evolve, this research contributes to the foundation for efficient
information processing and storage schemes, potentially revolutionizing the
realm of next-generation spintronic technologies.

To simulate the trilayer of CrPdFe deposited on Ir(111) with fcc stacking, we
employ DFT. To relax the magnetic layers we use the Quantum-Espresso
computational package[78]. The projector augmented wave pseudo potentials from
the PS Library [79] and a $28\times 28\times 1$ k-point grid were used for the
calculations. Then the electronic structure and magnetic properties were
simulated using all electron full potential relativistic Koringa -Kohn-
Rostoker (KKR) Green function method [80, 81, 82] in the local spin density
approximation. The Heisenberg exchange interactions and DM vectors were
extracted using the infinitesimal rotation method [83, 84] with a k-mesh of a
$200\times 200$. More details on the simulation procedure can be found in Ref.
[45].

###  3.1 Spin atomistic dynamics:

To explore the intricate dynamics of the AFM skyrmions, we conducted spin
atomistic simulations. 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:

$\mathcal{H}=\mathcal{H}_{\text{Exc }}+\mathcal{H}_{\text{
DMI}}+\mathcal{H}_{\text{Ani}}+\mathcal{H}_{\text{Zeem}},$ (5)

with:

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

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 Heisenberg
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 $K$. The latter favors the
out-of-plane orientation of the magnetization. In this study, in order to
promote the formation of a single AFM skyrmion, independent of the Fe magnetic
state’s influence; we reduced the Cr atom magnetic anisotropy from 0.5 to 0.4
meV per magnetic atom. $h_{i}=\mu_{i}B$ describes the Zeeman coupling to the
atomic spin moment $\mu$ at site $i$ assuming an out-of-plane field.

###  3.2 Landau–Lifshitz–Gilbert (LLG) equation for CPP.

We apply the spin-polarized CPP injection to induce transitional motion of AFM
skyrmions. In the CPP case, the current is perpendicular to the film plane,
but polarized in in-plane direction. The dynamics of the magnetization
$\mathbf{\textbf{S}}_{i}$ at the lattice site $i$ is then governed by the
extended LLG equation taking into account the STT term [69, 70, 38, 37, 71],

$\displaystyle\frac{\mathrm{d}\mathbf{\textbf{S}}_{i}}{\mathrm{~{}d}t}=$
$\displaystyle-\frac{\gamma}{(1+\alpha^{2})\mu_{i}}\mathbf{\textbf{S}}_{i}\times\mathbf{\textbf{B}}_{\text{eff
}}^{i}-\frac{\gamma\alpha}{(1+\alpha^{2})\mu_{i}}\mathbf{\textbf{S}}_{i}\times\left(\mathbf{\textbf{S}}_{i}\times\mathbf{\textbf{B}}_{\text{eff
}}^{i}\right)$ (6)
$\displaystyle+\frac{\gamma\alpha\eta}{(1+\alpha^{2})\mu_{B}}\mathbf{\textbf{S}}_{i}\times\mathbf{\textbf{S}}_{p}-\frac{\gamma\eta}{(1+\alpha^{2})\mu_{B}}\mathbf{\textbf{S}}_{i}\times\left(\mathbf{\textbf{S}}_{i}\times\textbf{S}_{p}\right),$

where $\gamma$ is the gyromagnetic ratio, $\alpha$ is the Gilbert damping
controlling the dissipation of angular momentum and energy from the magnetic
subsystem, $\mathbf{\textbf{B}}_{\text{eff }}^{i}$ is the effective field
related to the energy gradient
$(-\frac{\partial\mathcal{H}}{\partial\mathbf{\textbf{S}}_{i}})$, and
$\textbf{S}_{p}$ is the current polarisation direction with the current
amplitude monitored by the parameter
$\eta=\frac{j_{s}Pg\mu_{B}}{2edM_{s}\gamma}$, with the absolute value of the
current density $j_{s}$ , polarisation $P$, saturation magnetisation $M_{s}$,
the Landé $g$-factor $g$, magnitude of electron charge $e$ and Bohr magneton
$\mu_{B}$, d is the film thickness, and $\gamma$ is the gyromagnetic ratio.
In our study, we utilize the LLG equation as implemented in the Spirit code
[85]. We assumed periodic boundary conditions to model the extended two-
dimensional system with cells containing $200^{2}$.

###  3.3 Data availability

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

###  3.4 Code availability

We used SPIRIT code which can be found at https://github.com/spirit-
code/spirit. Additionally, the KKR code, a sophisticated ab-initio DFT tool,
can be found at https://github.com/JuDFTteam/JuKKR.

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). The authors gratefully acknowledge
the computing time granted through JARA on the supercomputer JURECA at
Forschungszentrum Jülich. M.S. acknowledges the fund from the European
Research Council (ERC) under the European Union’s Horizon 2020 research and
innovation program (Grant No. 856538, project “3D MAGiC”.

S.L. initiated, designed and supervised the project. A.A. performed the
simulations and analysed the results with the supervision of S.L.. 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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